TSTP Solution File: NUM924+7 by Leo-III---1.7.7
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%------------------------------------------------------------------------------
% File : Leo-III---1.7.7
% Problem : NUM924+7 : TPTP v8.1.2. Released v5.3.0.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n027.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 : Fri May 19 11:42:57 EDT 2023
% Result : Theorem 31.28s 9.31s
% Output : Refutation 32.91s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 1363
% Syntax : Number of formulae : 2620 ( 820 unt; 108 typ; 0 def)
% Number of atoms : 6598 (1744 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 77822 ( 481 ~; 149 |; 644 &;73255 @)
% ( 247 <=>;3046 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 9 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 130 ( 130 >; 0 *; 0 +; 0 <<)
% Number of symbols : 111 ( 108 usr; 15 con; 0-5 aty)
% Number of variables : 5593 ( 0 ^;5541 !; 52 ?;5593 :)
% Comments :
%------------------------------------------------------------------------------
thf(hBOOL_type,type,
hBOOL: $i > $o ).
thf(hAPP_type,type,
hAPP: $i > $i > $i > $i > $i ).
thf(int_type,type,
int: $i ).
thf(bool_type,type,
bool: $i ).
thf(fun_type,type,
fun: $i > $i > $i ).
thf(ord_less_type,type,
ord_less: $i > $i ).
thf(plus_plus_type,type,
plus_plus: $i > $i > $i ).
thf(nat_type,type,
nat: $i ).
thf(power_power_type,type,
power_power: $i > $i > $i ).
thf(s_type,type,
s: $i ).
thf(number_number_of_type,type,
number_number_of: $i > $i > $i ).
thf(bit0_type,type,
bit0: $i > $i ).
thf(bit1_type,type,
bit1: $i > $i ).
thf(pls_type,type,
pls: $i ).
thf(one_one_type,type,
one_one: $i > $i ).
thf(zero_zero_type,type,
zero_zero: $i > $i ).
thf(number_type,type,
number: $i > $o ).
thf(ring_type,type,
ring: $i > $o ).
thf(times_times_type,type,
times_times: $i > $i > $i ).
thf(minus_minus_type,type,
minus_minus: $i > $i > $i ).
thf(power_type,type,
power: $i > $o ).
thf(real_type,type,
real: $i ).
thf(linordered_semidom_type,type,
linordered_semidom: $i > $o ).
thf(cancel146912293up_add_type,type,
cancel146912293up_add: $i > $o ).
thf(group_add_type,type,
group_add: $i > $o ).
thf(ti_type,type,
ti: $i > $i > $i ).
thf(zcong_type,type,
zcong: $i > $i > $i ).
thf(semiring_div_type,type,
semiring_div: $i > $o ).
thf(div_mod_type,type,
div_mod: $i > $i > $i ).
thf(ordere1490568538miring_type,type,
ordere1490568538miring: $i > $o ).
thf(semiri456707255roduct_type,type,
semiri456707255roduct: $i > $o ).
thf(wset_type,type,
wset: $i > $i > $i ).
thf(no_zero_divisors_type,type,
no_zero_divisors: $i > $o ).
thf(zero_neq_one_type,type,
zero_neq_one: $i > $o ).
thf(standardRes_type,type,
standardRes: $i > $i > $i ).
thf(semiring_type,type,
semiring: $i > $o ).
thf(comm_semiring_1_type,type,
comm_semiring_1: $i > $o ).
thf(comm_monoid_add_type,type,
comm_monoid_add: $i > $o ).
thf(ab_semigroup_mult_type,type,
ab_semigroup_mult: $i > $o ).
thf(multInv_type,type,
multInv: $i > $i > $i ).
thf(zero_type,type,
zero: $i > $o ).
thf(ring_n68954251visors_type,type,
ring_n68954251visors: $i > $o ).
thf(linord20386208strict_type,type,
linord20386208strict: $i > $o ).
thf(inv_type,type,
inv: $i > $i > $i ).
thf(linord1278240602ring_1_type,type,
linord1278240602ring_1: $i > $o ).
thf(comm_ring_1_type,type,
comm_ring_1: $i > $o ).
thf(ab_group_add_type,type,
ab_group_add: $i > $o ).
thf(linord626643107strict_type,type,
linord626643107strict: $i > $o ).
thf(ordered_semiring_type,type,
ordered_semiring: $i > $o ).
thf(linordered_idom_type,type,
linordered_idom: $i > $o ).
thf(ordered_ring_type,type,
ordered_ring: $i > $o ).
thf(ring_div_type,type,
ring_div: $i > $o ).
thf(t_type,type,
t: $i ).
thf(number_ring_type,type,
number_ring: $i > $o ).
thf(legendre_type,type,
legendre: $i > $i > $i ).
thf(mult_zero_type,type,
mult_zero: $i > $o ).
thf(m_type,type,
m: $i ).
thf(real_normed_algebra_type,type,
real_normed_algebra: $i > $o ).
thf(ring_char_0_type,type,
ring_char_0: $i > $o ).
thf(cancel_semigroup_add_type,type,
cancel_semigroup_add: $i > $o ).
thf(monoid_add_type,type,
monoid_add: $i > $o ).
thf(min_type,type,
min: $i ).
thf(ab_semigroup_add_type,type,
ab_semigroup_add: $i > $o ).
thf(ordere236663937imp_le_type,type,
ordere236663937imp_le: $i > $o ).
thf(number_semiring_type,type,
number_semiring: $i > $o ).
thf(combc_type,type,
combc: $i > $i > $i > $i > $i > $i ).
thf(dvd_type,type,
dvd: $i > $o ).
thf(linordered_semiring_type,type,
linordered_semiring: $i > $o ).
thf(undefined_type,type,
undefined: $i > $i ).
thf(member_type,type,
member: $i > $i > $i ).
thf(quadRes_type,type,
quadRes: $i > $i ).
thf(ordere223160158up_add_type,type,
ordere223160158up_add: $i > $o ).
thf(linord219039673up_add_type,type,
linord219039673up_add: $i > $o ).
thf(one_type,type,
one: $i > $o ).
thf(linord893533164strict_type,type,
linord893533164strict: $i > $o ).
thf(semiring_1_type,type,
semiring_1: $i > $o ).
thf(linordered_ring_type,type,
linordered_ring: $i > $o ).
thf(comm_semiring_type,type,
comm_semiring: $i > $o ).
thf(linorder_type,type,
linorder: $i > $o ).
thf(idom_type,type,
idom: $i > $o ).
thf(plus_type,type,
plus: $i > $o ).
thf(monoid_mult_type,type,
monoid_mult: $i > $o ).
thf(comm_ring_type,type,
comm_ring: $i > $o ).
thf(ordere779506340up_add_type,type,
ordere779506340up_add: $i > $o ).
thf(linord581940658strict_type,type,
linord581940658strict: $i > $o ).
thf(ordere216010020id_add_type,type,
ordere216010020id_add: $i > $o ).
thf(ordered_ab_group_add_type,type,
ordered_ab_group_add: $i > $o ).
thf(ordere453448008miring_type,type,
ordere453448008miring: $i > $o ).
thf(ring_11004092258visors_type,type,
ring_11004092258visors: $i > $o ).
thf(d22set_type,type,
d22set: $i > $i ).
thf(ring_1_type,type,
ring_1: $i > $o ).
thf(semiring_0_type,type,
semiring_0: $i > $o ).
thf(order_type,type,
order: $i > $o ).
thf(comm_monoid_mult_type,type,
comm_monoid_mult: $i > $o ).
thf(dvd_dvd_type,type,
dvd_dvd: $i > $i ).
thf(combs_type,type,
combs: $i > $i > $i > $i > $i > $i ).
thf(zprime_type,type,
zprime: $i ).
thf(ord_less_eq_type,type,
ord_less_eq: $i > $i ).
thf(combb_type,type,
combb: $i > $i > $i > $i > $i > $i ).
thf(collect_type,type,
collect: $i > $i > $i ).
thf(zfact_type,type,
zfact: $i > $i ).
thf(twoSqu1929807760sum2sq_type,type,
twoSqu1929807760sum2sq: $i > $i ).
thf(product_Pair_type,type,
product_Pair: $i > $i > $i > $i > $i ).
thf(s1_type,type,
s1: $i ).
thf(fconj_type,type,
fconj: $i ).
thf(twoSqu658283162sum2sq_type,type,
twoSqu658283162sum2sq: $i ).
thf(product_prod_type,type,
product_prod: $i > $i > $i ).
thf(sr_type,type,
sr: $i > $i ).
thf(323,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) @ C ) ) @ B )
= ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ C ) ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_911_zpower__zmod) ).
thf(2163,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) @ C ) ) @ B )
= ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ C ) ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).
thf(105,axiom,
! [A: $i] :
( ( ring_div @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ A @ A @ ( div_mod @ A @ D ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ F ) @ E ) )
=> ( ( ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ E ) )
=> ( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ D ) @ B ) ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ F ) @ C ) ) @ E ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_947_mod__diff__cong) ).
thf(1501,plain,
! [A: $i] :
( ( ring_div @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ A @ A @ ( div_mod @ A @ D ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ F ) @ E ) )
=> ( ( ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ E ) )
=> ( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ D ) @ B ) ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ F ) @ C ) ) @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(38,axiom,
! [A: $i] :
( ( ab_group_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( minus_minus @ A @ D ) @ E ) )
=> ( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
<=> ( ( ti @ A @ D )
= ( ti @ A @ E ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_766_diff__eq__diff__eq) ).
thf(1337,plain,
! [A: $i] :
( ( ab_group_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( minus_minus @ A @ D ) @ E ) )
=> ( ( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
=> ( ( ti @ A @ D )
= ( ti @ A @ E ) ) )
& ( ( ( ti @ A @ D )
= ( ti @ A @ E ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(828,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_334_dvd__mult__left) ).
thf(3944,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[828]) ).
thf(810,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( zero_zero @ nat ) ) @ A )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_621_plus__nat_Oadd__0) ).
thf(3883,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( zero_zero @ nat ) ) @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[810]) ).
thf(91,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ B )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_946_mod__mult__self1__is__0) ).
thf(1469,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ B )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(133,axiom,
plus @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Oplus) ).
thf(1559,plain,
plus @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(392,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B )
= ( zero_zero @ nat ) )
=> ( ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ A )
= ( zero_zero @ nat ) )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_539_diffs0__imp__equal) ).
thf(2412,plain,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B )
= ( zero_zero @ nat ) )
=> ( ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ A )
= ( zero_zero @ nat ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).
thf(841,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( zfact @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( one_one @ int ) ) ) @ ( number_number_of @ int @ min ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_755_Wilson__Russ) ).
thf(3979,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( zfact @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( one_one @ int ) ) ) @ ( number_number_of @ int @ min ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[841]) ).
thf(224,axiom,
one @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oone) ).
thf(1840,plain,
one @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).
thf(397,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ A ) )
=> ( ( d22set @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( wset @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_857_d22set__eq__wset) ).
thf(2427,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ A ) )
=> ( ( d22set @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( wset @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[397]) ).
thf(427,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ E ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( plus_plus @ int @ C ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ D ) @ B ) ) @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_357_zcong__zadd) ).
thf(2544,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ E ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( plus_plus @ int @ C ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ D ) @ B ) ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[427]) ).
thf(824,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_939_neg__mod__bound) ).
thf(3927,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[824]) ).
thf(1043,axiom,
! [A: $i,B: $i] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ( ( ti @ ( fun @ B @ B ) @ ( plus_plus @ B @ A ) )
= ( plus_plus @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Oplus__class_Oplus_0_res) ).
thf(4682,plain,
! [A: $i,B: $i] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ( ( ti @ ( fun @ B @ B ) @ ( plus_plus @ B @ A ) )
= ( plus_plus @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1043]) ).
thf(217,axiom,
! [A: $i] :
( ( idom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) )
<=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_405_dvd__mult__cancel__left) ).
thf(1818,plain,
! [A: $i] :
( ( idom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) )
=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) ) ) )
& ( ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).
thf(178,axiom,
! [A: $i] :
( ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= ( zero_zero @ int ) )
<=> ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( one_one @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_971_neq__one__mod__two) ).
thf(1686,plain,
! [A: $i] :
( ( ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= ( zero_zero @ int ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( one_one @ int ) ) )
& ( ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( one_one @ int ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= ( zero_zero @ int ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(333,axiom,
! [A: $i] :
( ( linord1278240602ring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B )
= ( one_one @ A ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ E ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) @ F ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_453_convex__bound__le) ).
thf(2199,plain,
! [A: $i] :
( ( linord1278240602ring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B )
= ( one_one @ A ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ E ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) @ F ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[333]) ).
thf(573,axiom,
semiring @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Osemiring) ).
thf(3021,plain,
semiring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[573]) ).
thf(31,axiom,
linord20386208strict @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Olinordered__semiring__strict) ).
thf(1322,plain,
linord20386208strict @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(1090,axiom,
! [A: $i] :
( ( one @ A )
=> ( ( ti @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Oone__class_Oone_res) ).
thf(4835,plain,
! [A: $i] :
( ( one @ A )
=> ( ( ti @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1090]) ).
thf(533,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) @ C ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_927_mod__add__left__eq) ).
thf(2889,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) @ C ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[533]) ).
thf(879,axiom,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_764_add__right__imp__eq) ).
thf(4101,plain,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[879]) ).
thf(991,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_721_nat__less__add__iff1) ).
thf(4450,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[991]) ).
thf(659,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_562_dvd_Oless__imp__le) ).
thf(3345,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[659]) ).
thf(1216,axiom,
( ( ti @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ fconj )
= fconj ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_fconj_res) ).
thf(5260,plain,
( ( ti @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ fconj )
= fconj ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1216]) ).
thf(757,axiom,
comm_semiring_1 @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Ocomm__semiring__1) ).
thf(3661,plain,
comm_semiring_1 @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[757]) ).
thf(870,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ int @ ( power_power @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) @ B ) ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_504_zdiff__power2) ).
thf(4062,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ int @ ( power_power @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) @ B ) ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[870]) ).
thf(675,axiom,
no_zero_divisors @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Ono__zero__divisors) ).
thf(3399,plain,
no_zero_divisors @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[675]) ).
thf(266,axiom,
! [A: $i] :
( ( mult_zero @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( zero_zero @ A ) ) @ B )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_319_mult__zero__left) ).
thf(1995,plain,
! [A: $i] :
( ( mult_zero @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( zero_zero @ A ) ) @ B )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).
thf(355,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_800_add__nonneg__nonneg) ).
thf(2263,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[355]) ).
thf(760,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( one_one @ int ) ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_84_zle__add1__eq__le) ).
thf(3666,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( one_one @ int ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[760]) ).
thf(190,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ ( one_one @ real ) ) @ ( hAPP @ nat @ real @ ( power_power @ real @ ( number_number_of @ real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_716_two__realpow__ge__one) ).
thf(1729,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ ( one_one @ real ) ) @ ( hAPP @ nat @ real @ ( power_power @ real @ ( number_number_of @ real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(30,axiom,
! [A: $i] :
( ( ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) )
<=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ E ) ) @ C ) ) @ D )
= ( ti @ A @ F ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_479_eq__add__iff1) ).
thf(1315,plain,
! [A: $i] :
( ( ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ E ) ) @ C ) ) @ D )
= ( ti @ A @ F ) ) )
& ( ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ E ) ) @ C ) ) @ D )
= ( ti @ A @ F ) )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(502,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_487_zcong__def) ).
thf(2787,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[502]) ).
thf(252,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) )
<=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_376_zero__le__mult__iff) ).
thf(1950,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) ) ) ) )
& ( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).
thf(127,axiom,
ordere216010020id_add @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Oordered__comm__monoid__add) ).
thf(1549,plain,
ordere216010020id_add @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(360,axiom,
! [A: $i] :
( ( number_semiring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_168_power2__sum) ).
thf(2280,plain,
! [A: $i] :
( ( number_semiring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).
thf(130,axiom,
ordere453448008miring @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Oordered__cancel__semiring) ).
thf(1554,plain,
ordere453448008miring @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(336,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ~ ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ B ) @ ( d22set @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_858_d22set__le__swap) ).
thf(2210,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ~ ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ B ) @ ( d22set @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).
thf(724,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ ( bit1 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_less__eq__int__code_I16_J) ).
thf(3549,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ ( bit1 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ ( bit1 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[724]) ).
thf(443,axiom,
comm_monoid_mult @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ocomm__monoid__mult) ).
thf(2590,plain,
comm_monoid_mult @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[443]) ).
thf(591,axiom,
! [A: $i,B: $i] :
( ( ab_semigroup_add @ B )
=> ( ( plus_plus @ B @ ( ti @ B @ A ) )
= ( plus_plus @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Oplus__class_Oplus_3_arg1) ).
thf(3098,plain,
! [A: $i,B: $i] :
( ( ab_semigroup_add @ B )
=> ( ( plus_plus @ B @ ( ti @ B @ A ) )
= ( plus_plus @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[591]) ).
thf(595,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ ( zero_zero @ int ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_295_bin__less__0__simps_I2_J) ).
thf(3108,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ ( zero_zero @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[595]) ).
thf(1052,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_713_mult__le__cancel1) ).
thf(4711,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1052]) ).
thf(300,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) ).
thf(2099,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).
thf(514,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ A ) @ ( one_one @ int ) ) @ B ) )
=> ( ( ( ti @ int @ A )
= ( one_one @ int ) )
| ( ( ti @ int @ A )
= ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_500_zcong__square__zless) ).
thf(2837,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ A ) @ ( one_one @ int ) ) @ B ) )
=> ( ( ( ti @ int @ A )
= ( one_one @ int ) )
| ( ( ti @ int @ A )
= ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[514]) ).
thf(1001,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_381_mult__strict__left__mono) ).
thf(4479,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1001]) ).
thf(558,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ B @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_831_wset__g__1) ).
thf(2976,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ B @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[558]) ).
thf(965,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ t ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_t__l__p) ).
thf(4376,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ t ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[965]) ).
thf(545,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ B ) ) @ ( one_one @ A ) )
= ( number_number_of @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( bit1 @ pls ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_add__special_I3_J) ).
thf(2940,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ B ) ) @ ( one_one @ A ) )
= ( number_number_of @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( bit1 @ pls ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[545]) ).
thf(150,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) @ C )
= ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_920_mod__mod__cancel) ).
thf(1596,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) @ C )
= ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(67,axiom,
! [A: $i,B: $i] :
( ( power @ B )
=> ( ( times_times @ B @ ( ti @ B @ A ) )
= ( times_times @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_1_arg1) ).
thf(1406,plain,
! [A: $i,B: $i] :
( ( power @ B )
=> ( ( times_times @ B @ ( ti @ B @ A ) )
= ( times_times @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(343,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ C ) ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) @ ( zero_zero @ int ) ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_460_zcong__zprime__prod__zero__contra) ).
thf(2231,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ C ) ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) @ ( zero_zero @ int ) ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[343]) ).
thf(456,axiom,
( ( number_number_of @ nat @ ( bit1 @ pls ) )
= ( one_one @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_nat__numeral__1__eq__1) ).
thf(2622,plain,
( ( number_number_of @ nat @ ( bit1 @ pls ) )
= ( one_one @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[456]) ).
thf(942,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_388_mult__pos__neg2) ).
thf(4311,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[942]) ).
thf(1016,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
<=> ? [C: $i] :
( B
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_660_le__iff__add) ).
thf(4515,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ? [C: $i] :
( B
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) )
& ( ? [C: $i] :
( B
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1016]) ).
thf(1142,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_68_zadd__zmult__distrib) ).
thf(5000,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1142]) ).
thf(213,axiom,
linord581940658strict @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__ring__strict) ).
thf(1808,plain,
linord581940658strict @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(468,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( A
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ C ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_527_divides__div__not) ).
thf(2659,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( A
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ C ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[468]) ).
thf(682,axiom,
! [A: $i,B: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_638_not__add__less2) ).
thf(3417,plain,
! [A: $i,B: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[682]) ).
thf(710,axiom,
ab_semigroup_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oab__semigroup__add) ).
thf(3509,plain,
ab_semigroup_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[710]) ).
thf(156,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_642_add__less__mono1) ).
thf(1618,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(580,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ pls ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ pls ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_rel__simps_I27_J) ).
thf(3040,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ pls ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ pls ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ pls ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ pls ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[580]) ).
thf(970,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ A ) )
=> ( ( ti @ int @ A )
= ( ti @ int @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_zle__antisym) ).
thf(4391,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ A ) )
=> ( ( ti @ int @ A )
= ( ti @ int @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[970]) ).
thf(548,axiom,
! [A: $i] :
( ( ordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_368_mult__left__mono) ).
thf(2947,plain,
! [A: $i] :
( ( ordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[548]) ).
thf(347,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( ( ti @ int @ A )
!= ( inv @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_824_inv__distinct) ).
thf(2248,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( ( ti @ int @ A )
!= ( inv @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[347]) ).
thf(473,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( A = B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_652_less__or__eq__imp__le) ).
thf(2671,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( A = B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[473]) ).
thf(474,axiom,
linorder @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Orderings_Olinorder) ).
thf(2676,plain,
linorder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[474]) ).
thf(818,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( number_number_of @ int @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( number_number_of @ int @ ( bit0 @ A ) ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( number_number_of @ int @ ( bit1 @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_739_number__of1) ).
thf(3910,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( number_number_of @ int @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( number_number_of @ int @ ( bit0 @ A ) ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( number_number_of @ int @ ( bit1 @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[818]) ).
thf(946,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_330_dvd__mult2) ).
thf(4326,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[946]) ).
thf(1218,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_225_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) ).
thf(5265,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1218]) ).
thf(384,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ti @ ( fun @ A @ B ) @ ( combs @ A @ C @ B @ D @ E ) )
= ( combs @ A @ C @ B @ D @ E ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_COMBS_res) ).
thf(2380,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ti @ ( fun @ A @ B ) @ ( combs @ A @ C @ B @ D @ E ) )
= ( combs @ A @ C @ B @ D @ E ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[384]) ).
thf(258,axiom,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_852_d22set__induct__old) ).
thf(1971,plain,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).
thf(207,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ C )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( one_one @ A ) ) ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_254_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) ).
thf(1793,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ C )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( one_one @ A ) ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(1004,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) )
= ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_97_zadd__left__commute) ).
thf(4487,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) )
= ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1004]) ).
thf(1036,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_674_le__diff__conv2) ).
thf(4592,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1036]) ).
thf(801,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( zero_zero @ nat ) ) @ A )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_630_mult__0) ).
thf(3848,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( zero_zero @ nat ) ) @ A )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[801]) ).
thf(719,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( one_one @ nat ) ) )
=> ( A
= ( one_one @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_875_gcd__lcm__complete__lattice__nat_Ole__bot) ).
thf(3530,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( one_one @ nat ) ) )
=> ( A
= ( one_one @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[719]) ).
thf(1249,axiom,
( ( twoSqu1929807760sum2sq @ ( product_Pair @ int @ int @ s @ ( one_one @ int ) ) )
= ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ t ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_751__096sum2sq_A_Is_M_A1_J_A_061_A_I4_A_K_Am_A_L_A1_J_A_K_At_096) ).
thf(5361,plain,
( ( twoSqu1929807760sum2sq @ ( product_Pair @ int @ int @ s @ ( one_one @ int ) ) )
= ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ t ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1249]) ).
thf(27,axiom,
! [A: $i,B: $i] :
( ( multInv @ ( ti @ int @ A ) @ B )
= ( multInv @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Int2_OMultInv_arg1) ).
thf(1310,plain,
! [A: $i,B: $i] :
( ( multInv @ ( ti @ int @ A ) @ B )
= ( multInv @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(857,axiom,
number @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Int_Onumber) ).
thf(4026,plain,
number @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[857]) ).
thf(372,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ C ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_459_zcong__zprime__prod__zero) ).
thf(2316,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ C ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[372]) ).
thf(402,axiom,
! [A: $i] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_320_zero__neq__one) ).
thf(2448,plain,
! [A: $i] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[402]) ).
thf(12,axiom,
semiri456707255roduct @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossp) ).
thf(1275,plain,
semiri456707255roduct @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(1098,axiom,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ C )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_796_add__diff__cancel) ).
thf(4856,plain,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ C )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1098]) ).
thf(766,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ C ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ A ) ) @ B ) )
= ( zero_zero @ nat ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ C ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ A ) ) @ B ) )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ A ) ) ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_231_nat__number__of__mult__left) ).
thf(3693,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ C ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ A ) ) @ B ) )
= ( zero_zero @ nat ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ C ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ A ) ) @ B ) )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ A ) ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[766]) ).
thf(409,axiom,
! [A: $i] :
( ( idom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
<=> ( ( ( ti @ A @ C )
= ( zero_zero @ A ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_406_dvd__mult__cancel__right) ).
thf(2476,plain,
! [A: $i] :
( ( idom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
=> ( ( ( ti @ A @ C )
= ( zero_zero @ A ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ D ) ) ) )
& ( ( ( ( ti @ A @ C )
= ( zero_zero @ A ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[409]) ).
thf(264,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( one_one @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_407_power__increasing) ).
thf(1988,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( one_one @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).
thf(39,axiom,
ab_semigroup_mult @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Oab__semigroup__mult) ).
thf(1342,plain,
ab_semigroup_mult @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(516,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ B ) ) @ E ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ D ) @ B ) ) @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_352_zcong__zmult__prop1) ).
thf(2844,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ D ) @ E ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ B ) ) @ E ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ D ) @ B ) ) @ E ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ D ) @ B ) ) @ E ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ B ) ) @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[516]) ).
thf(45,axiom,
! [A: $i,B: $i] :
( ( ti @ int @ ( inv @ A @ B ) )
= ( inv @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_WilsonRuss_Oinv_res) ).
thf(1354,plain,
! [A: $i,B: $i] :
( ( ti @ int @ ( inv @ A @ B ) )
= ( inv @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(437,axiom,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( zero_zero @ A ) ) @ B )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_886_mult_Ozero__left) ).
thf(2575,plain,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( zero_zero @ A ) ) @ B )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[437]) ).
thf(974,axiom,
! [A: $i,B: $i] :
( ( ti @ ( fun @ int @ bool ) @ ( wset @ A @ B ) )
= ( wset @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_WilsonRuss_Owset_res) ).
thf(4403,plain,
! [A: $i,B: $i] :
( ( ti @ ( fun @ int @ bool ) @ ( wset @ A @ B ) )
= ( wset @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[974]) ).
thf(689,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( combc @ A @ B @ C @ ( ti @ ( fun @ A @ ( fun @ B @ C ) ) @ D ) @ E )
= ( combc @ A @ B @ C @ D @ E ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_COMBC_arg1) ).
thf(3439,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( combc @ A @ B @ C @ ( ti @ ( fun @ A @ ( fun @ B @ C ) ) @ D ) @ E )
= ( combc @ A @ B @ C @ D @ E ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[689]) ).
thf(569,axiom,
! [A: $i] :
( ( ab_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_759_ab__semigroup__add__class_Oadd__ac_I1_J) ).
thf(3004,plain,
! [A: $i] :
( ( ab_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[569]) ).
thf(280,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) )
<=> ( ( A
= ( zero_zero @ nat ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_730_nat__mult__dvd__cancel__disj) ).
thf(2035,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) )
=> ( ( A
= ( zero_zero @ nat ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) ) ) )
& ( ( ( A
= ( zero_zero @ nat ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).
thf(1092,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_103_zero__le__power2) ).
thf(4839,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1092]) ).
thf(850,axiom,
cancel_semigroup_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ocancel__semigroup__add) ).
thf(4011,plain,
cancel_semigroup_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[850]) ).
thf(752,axiom,
( ( ti @ int @ pls )
= pls ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Int_OPls_res) ).
thf(3649,plain,
( ( ti @ int @ pls )
= pls ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[752]) ).
thf(665,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B )
= ( ti @ int @ C ) )
=> ( ( ti @ int @ A )
= ( hAPP @ int @ int @ ( plus_plus @ int @ C ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_475_Int2_Oaux1) ).
thf(3365,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B )
= ( ti @ int @ C ) )
=> ( ( ti @ int @ A )
= ( hAPP @ int @ int @ ( plus_plus @ int @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[665]) ).
thf(540,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ C ) ) ) @ ( zero_zero @ A ) ) )
<=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_sum__squares__le__zero__iff) ).
thf(2918,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ C ) ) ) @ ( zero_zero @ A ) ) )
=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) )
& ( ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ C ) ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[540]) ).
thf(882,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_434_mult__right__le__one__le) ).
thf(4110,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[882]) ).
thf(1019,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ B )
<=> ( hBOOL @ A ) ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( hBOOL @ B ) )
<=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( hBOOL @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_742_imp__le__cong) ).
thf(4528,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( ( hBOOL @ B )
=> ( hBOOL @ A ) )
& ( ( hBOOL @ A )
=> ( hBOOL @ B ) ) ) )
=> ( ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( hBOOL @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( hBOOL @ A ) ) )
& ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( hBOOL @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( hBOOL @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1019]) ).
thf(330,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ C ) )
= ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ B ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_zpower__zadd__distrib) ).
thf(2184,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ C ) )
= ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ B ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).
thf(165,axiom,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) )
<=> ( ( ti @ A @ C )
= ( ti @ A @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_760_add__left__cancel) ).
thf(1640,plain,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) )
=> ( ( ti @ A @ C )
= ( ti @ A @ D ) ) )
& ( ( ( ti @ A @ C )
= ( ti @ A @ D ) )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(439,axiom,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_362_split__mult__pos__le) ).
thf(2580,plain,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[439]) ).
thf(814,axiom,
! [A: $i,B: $i] :
( ( ti @ ( product_prod @ int @ int ) @ ( product_Pair @ int @ int @ A @ B ) )
= ( product_Pair @ int @ int @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Product__Type_OPair_res) ).
thf(3895,plain,
! [A: $i,B: $i] :
( ( ti @ ( product_prod @ int @ int ) @ ( product_Pair @ int @ int @ A @ B ) )
= ( product_Pair @ int @ int @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[814]) ).
thf(772,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( one_one @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_398_zero__less__one) ).
thf(3717,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( one_one @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[772]) ).
thf(382,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( zero_zero @ A ) ) )
<=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_375_mult__le__0__iff) ).
thf(2360,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( zero_zero @ A ) ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) ) ) ) )
& ( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[382]) ).
thf(1068,axiom,
ordere1490568538miring @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Oordered__comm__semiring) ).
thf(4762,plain,
ordere1490568538miring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1068]) ).
thf(86,axiom,
mult_zero @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Omult__zero) ).
thf(1458,plain,
mult_zero @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(667,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ C @ B ) ) )
=> ( ( ( ti @ int @ A )
= ( ti @ int @ C ) )
| ( ( ti @ int @ A )
= ( inv @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_833_wset__mem__imp__or) ).
thf(3370,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ C @ B ) ) )
=> ( ( ( ti @ int @ A )
= ( ti @ int @ C ) )
| ( ( ti @ int @ A )
= ( inv @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[667]) ).
thf(661,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_401_zero__le__power) ).
thf(3350,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[661]) ).
thf(1214,axiom,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) )
=> ( ( ti @ A @ C )
= ( ti @ A @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_762_add__left__imp__eq) ).
thf(5252,plain,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) )
=> ( ( ti @ A @ C )
= ( ti @ A @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1214]) ).
thf(786,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ ( div_mod @ nat @ B ) @ C ) ) )
<=> ( ( ( C
= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ B ) ) )
& ( ( C
!= ( zero_zero @ nat ) )
=> ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ E ) @ C ) )
=> ( ( B
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ D ) ) @ E ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ E ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_993_split__mod) ).
thf(3766,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ ( div_mod @ nat @ B ) @ C ) ) )
=> ( ( ( C
= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ B ) ) )
& ( ( C
!= ( zero_zero @ nat ) )
=> ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ E ) @ C ) )
=> ( ( B
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ D ) ) @ E ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ E ) ) ) ) ) ) )
& ( ( ( ( C
= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ B ) ) )
& ( ( C
!= ( zero_zero @ nat ) )
=> ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ E ) @ C ) )
=> ( ( B
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ D ) ) @ E ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ E ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ ( div_mod @ nat @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[786]) ).
thf(498,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) )
<=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ A ) @ B ) )
| ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_543_less__eq__real__def) ).
thf(2763,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ A ) @ B ) )
| ( A = B ) ) )
& ( ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ A ) @ B ) )
| ( A = B ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[498]) ).
thf(142,axiom,
order @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Orderings_Oorder) ).
thf(1576,plain,
order @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(81,axiom,
! [A: $i,B: $i] :
( ( mult_zero @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( times_times @ B @ A ) )
= ( times_times @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_4_res) ).
thf(1441,plain,
! [A: $i,B: $i] :
( ( mult_zero @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( times_times @ B @ A ) )
= ( times_times @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(712,axiom,
! [A: $i,B: $i] :
( ( number @ B )
=> ( ( number_number_of @ B @ ( ti @ int @ A ) )
= ( number_number_of @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Int_Onumber__class_Onumber__of_arg1) ).
thf(3513,plain,
! [A: $i,B: $i] :
( ( number @ B )
=> ( ( number_number_of @ B @ ( ti @ int @ A ) )
= ( number_number_of @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[712]) ).
thf(154,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( multInv @ B @ A ) ) ) @ ( multInv @ B @ ( multInv @ B @ A ) ) ) @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_848_Int2_Oaux____2) ).
thf(1613,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( multInv @ B @ A ) ) ) @ ( multInv @ B @ ( multInv @ B @ A ) ) ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(531,axiom,
comm_semiring @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Ocomm__semiring) ).
thf(2886,plain,
comm_semiring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[531]) ).
thf(1227,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26_less__eq__number__of__int__code) ).
thf(5301,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1227]) ).
thf(815,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ B ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ C ) @ B ) )
& ! [D: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ D ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ D ) @ B ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ D ) @ B ) ) )
=> ( ( ti @ int @ D )
= ( ti @ int @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_750_zcong__zless__unique) ).
thf(3898,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ B ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ C ) @ B ) )
& ! [D: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ D ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ D ) @ B ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ D ) @ B ) ) )
=> ( ( ti @ int @ D )
= ( ti @ int @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[815]) ).
thf(997,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ ( bit0 @ A ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_294_rel__simps_I8_J) ).
thf(4467,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ ( bit0 @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ ( bit0 @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[997]) ).
thf(1013,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ min ) @ min ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_289_rel__simps_I24_J) ).
thf(4511,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ min ) @ min ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1013]) ).
thf(553,axiom,
! [A: $i] :
( ( cancel146912293up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) )
=> ( ( ti @ A @ C )
= ( ti @ A @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_763_add__imp__eq) ).
thf(2957,plain,
! [A: $i] :
( ( cancel146912293up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) )
=> ( ( ti @ A @ C )
= ( ti @ A @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[553]) ).
thf(1078,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) @ ( zero_zero @ int ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_485_less__bin__lemma) ).
thf(4792,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) @ ( zero_zero @ int ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1078]) ).
thf(222,axiom,
( ( ti @ int @ s )
= s ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_v_s_____res) ).
thf(1835,plain,
( ( ti @ int @ s )
= s ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).
thf(271,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_944_mod__by__1) ).
thf(2010,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).
thf(1081,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ D ) @ E ) ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ D ) @ E ) ) @ F ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ F ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ E ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_266_zdiv__mono2__lemma) ).
thf(4803,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ D ) @ E ) ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ D ) @ E ) ) @ F ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ F ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ E ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1081]) ).
thf(888,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
= ( number_number_of @ A @ ( bit0 @ ( bit1 @ pls ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_one__add__one__is__two) ).
thf(4124,plain,
! [A: $i] :
( ( number_ring @ A )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
= ( number_number_of @ A @ ( bit0 @ ( bit1 @ pls ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[888]) ).
thf(859,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_686_mult__le__mono2) ).
thf(4030,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[859]) ).
thf(891,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_436_power__less__imp__less__base) ).
thf(4137,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[891]) ).
thf(188,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ B ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ C ) ) @ D ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_150_add__number__of__left) ).
thf(1724,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ B ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ C ) ) @ D ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(93,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_907_mod__mult__right__eq) ).
thf(1475,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(326,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ A ) @ ( one_one @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( one_one @ int ) ) @ B ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_501_zcong__square) ).
thf(2173,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ A ) @ ( one_one @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( one_one @ int ) ) @ B ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[326]) ).
thf(49,axiom,
ordered_semiring @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Oordered__semiring) ).
thf(1366,plain,
ordered_semiring @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(119,axiom,
! [A: $i,B: $i] :
( ( power @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( times_times @ B @ A ) )
= ( times_times @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_1_res) ).
thf(1529,plain,
! [A: $i,B: $i] :
( ( power @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( times_times @ B @ A ) )
= ( times_times @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(256,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( quadRes @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ ( number_number_of @ int @ min ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_462__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096) ).
thf(1967,plain,
hBOOL @ ( hAPP @ int @ bool @ ( quadRes @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ ( number_number_of @ int @ min ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).
thf(1220,axiom,
! [A: $i] :
( pls
!= ( bit1 @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_rel__simps_I39_J) ).
thf(5269,plain,
! [A: $i] :
( pls
!= ( bit1 @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1220]) ).
thf(809,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_239_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) ).
thf(3880,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[809]) ).
thf(764,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ti @ ( fun @ A @ B ) @ ( combc @ A @ C @ B @ D @ E ) )
= ( combc @ A @ C @ B @ D @ E ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_COMBC_res) ).
thf(3688,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ti @ ( fun @ A @ B ) @ ( combc @ A @ C @ B @ D @ E ) )
= ( combc @ A @ C @ B @ D @ E ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[764]) ).
thf(524,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_445_power__strict__increasing__iff) ).
thf(2869,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ D ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[524]) ).
thf(466,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ A ) )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_672_le__add__diff__inverse) ).
thf(2654,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ A ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[466]) ).
thf(987,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( B != A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_548_less__not__refl2) ).
thf(4439,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( B != A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[987]) ).
thf(449,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( inv @ B @ A ) ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_820_inv__less__p__minus__1) ).
thf(2603,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( inv @ B @ A ) ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[449]) ).
thf(642,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) @ ( one_one @ int ) ) @ B ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_498_inv__not__p__minus__1__aux) ).
thf(3275,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) @ ( one_one @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) @ ( one_one @ int ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[642]) ).
thf(1225,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B )
= ( one_one @ int ) )
<=> ( ( ( ( ti @ int @ A )
= ( one_one @ int ) )
& ( ( ti @ int @ B )
= ( one_one @ int ) ) )
| ( ( ( ti @ int @ A )
= ( number_number_of @ int @ min ) )
& ( ( ti @ int @ B )
= ( number_number_of @ int @ min ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_304_zmult__eq__1__iff) ).
thf(5284,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B )
= ( one_one @ int ) )
=> ( ( ( ( ti @ int @ A )
= ( one_one @ int ) )
& ( ( ti @ int @ B )
= ( one_one @ int ) ) )
| ( ( ( ti @ int @ A )
= ( number_number_of @ int @ min ) )
& ( ( ti @ int @ B )
= ( number_number_of @ int @ min ) ) ) ) )
& ( ( ( ( ( ti @ int @ A )
= ( one_one @ int ) )
& ( ( ti @ int @ B )
= ( one_one @ int ) ) )
| ( ( ( ti @ int @ A )
= ( number_number_of @ int @ min ) )
& ( ( ti @ int @ B )
= ( number_number_of @ int @ min ) ) ) )
=> ( ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B )
= ( one_one @ int ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1225]) ).
thf(656,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J) ).
thf(3342,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[656]) ).
thf(705,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ D ) ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) @ E ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_340_power__le__dvd) ).
thf(3495,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ D ) ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[705]) ).
thf(733,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) )
<=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_729_nat__mult__eq__cancel1) ).
thf(3574,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( ( ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) )
=> ( A = B ) )
& ( ( A = B )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[733]) ).
thf(794,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B )
= ( zero_zero @ int ) )
<=> ? [C: $i] :
( ( ti @ int @ A )
= ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_935_zmod__eq__0__iff) ).
thf(3821,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B )
= ( zero_zero @ int ) )
=> ? [C: $i] :
( ( ti @ int @ A )
= ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) )
& ( ? [C: $i] :
( ( ti @ int @ A )
= ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B )
= ( zero_zero @ int ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[794]) ).
thf(28,axiom,
zero @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Ozero) ).
thf(1313,plain,
zero @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(1029,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( one_one @ int ) ) @ A ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_int__one__le__iff__zero__less) ).
thf(4568,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( one_one @ int ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1029]) ).
thf(380,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_zadd__left__mono) ).
thf(2356,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[380]) ).
thf(634,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ real @ real @ ( plus_plus @ real @ ( hAPP @ nat @ real @ ( power_power @ real @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ real @ ( power_power @ real @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( zero_zero @ real ) )
<=> ( ( A
= ( zero_zero @ real ) )
& ( B
= ( zero_zero @ real ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_528_realpow__two__sum__zero__iff) ).
thf(3248,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ real @ real @ ( plus_plus @ real @ ( hAPP @ nat @ real @ ( power_power @ real @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ real @ ( power_power @ real @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( zero_zero @ real ) )
=> ( ( A
= ( zero_zero @ real ) )
& ( B
= ( zero_zero @ real ) ) ) )
& ( ( ( A
= ( zero_zero @ real ) )
& ( B
= ( zero_zero @ real ) ) )
=> ( ( hAPP @ real @ real @ ( plus_plus @ real @ ( hAPP @ nat @ real @ ( power_power @ real @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ real @ ( power_power @ real @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( zero_zero @ real ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[634]) ).
thf(68,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( one_one @ A ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_245_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) ).
thf(1409,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( one_one @ A ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(1021,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ B )
| ~ ( hBOOL @ A )
| ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fconj @ B ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fconj_1_1_U) ).
thf(4539,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ B )
| ~ ( hBOOL @ A )
| ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fconj @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1021]) ).
thf(690,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ ( zero_zero @ int ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( zero_zero @ int ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_160_bin__less__0__simps_I3_J) ).
thf(3442,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( zero_zero @ int ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ ( zero_zero @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[690]) ).
thf(827,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ C ) )
<=> ( B
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_738_nat__eq__add__iff2) ).
thf(3938,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ C ) )
=> ( B
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ C ) ) )
& ( ( B
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[827]) ).
thf(577,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ t ) @ ( one_one @ int ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1_calculation_I1_J) ).
thf(3029,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ t ) @ ( one_one @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[577]) ).
thf(957,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ ( bit1 @ A ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_291_rel__simps_I9_J) ).
thf(4348,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ ( bit1 @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ ( bit1 @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[957]) ).
thf(684,axiom,
! [A: $i] :
( ( ( plus @ A )
& ( linorder @ A )
& ( dvd @ A ) )
=> ! [B: $i,C: $i] :
? [D: $i] :
! [E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ C ) ) )
<=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_867_pinf_I10_J) ).
thf(3425,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[684]) ).
thf(1228,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_564_dvd_Oless__imp__neq) ).
thf(5306,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1228]) ).
thf(1102,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ ( bit1 @ A ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_rel__simps_I5_J) ).
thf(4866,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ ( bit1 @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ ( bit1 @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1102]) ).
thf(482,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_zdvd__reduce) ).
thf(2698,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[482]) ).
thf(1131,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ ( number_number_of @ int @ min ) ) @ A ) @ ( hAPP @ nat @ int @ ( power_power @ int @ ( number_number_of @ int @ min ) ) @ B ) ) @ C ) )
=> ( ( hAPP @ nat @ int @ ( power_power @ int @ ( number_number_of @ int @ min ) ) @ A )
= ( hAPP @ nat @ int @ ( power_power @ int @ ( number_number_of @ int @ min ) ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_464_neg__one__power__eq__mod__m) ).
thf(4956,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ ( number_number_of @ int @ min ) ) @ A ) @ ( hAPP @ nat @ int @ ( power_power @ int @ ( number_number_of @ int @ min ) ) @ B ) ) @ C ) )
=> ( ( hAPP @ nat @ int @ ( power_power @ int @ ( number_number_of @ int @ min ) ) @ A )
= ( hAPP @ nat @ int @ ( power_power @ int @ ( number_number_of @ int @ min ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1131]) ).
thf(873,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ C ) )
<=> ( ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ B )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_940_dvd__eq__mod__eq__0) ).
thf(4070,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ C ) )
=> ( ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ B )
= ( zero_zero @ A ) ) )
& ( ( ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ B )
= ( zero_zero @ A ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[873]) ).
thf(471,axiom,
! [A: $i,B: $i] :
( ( product_Pair @ int @ int @ A @ ( ti @ int @ B ) )
= ( product_Pair @ int @ int @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Product__Type_OPair_arg2) ).
thf(2667,plain,
! [A: $i,B: $i] :
( ( product_Pair @ int @ int @ A @ ( ti @ int @ B ) )
= ( product_Pair @ int @ int @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[471]) ).
thf(7,axiom,
! [A: $i,B: $i] :
( ( group_add @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( minus_minus @ B @ A ) )
= ( minus_minus @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Ominus__class_Ominus_0_res) ).
thf(1264,plain,
! [A: $i,B: $i] :
( ( group_add @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( minus_minus @ B @ A ) )
= ( minus_minus @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(951,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_325_dvd__0__left) ).
thf(4334,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[951]) ).
thf(32,axiom,
! [A: $i,B: $i] :
( ( zcong @ A @ ( ti @ int @ B ) )
= ( zcong @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_IntPrimes_Ozcong_arg2) ).
thf(1323,plain,
! [A: $i,B: $i] :
( ( zcong @ A @ ( ti @ int @ B ) )
= ( zcong @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(605,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ D ) )
=> ( ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ D )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_644_less__add__eq__less) ).
thf(3148,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ D ) )
=> ( ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ D )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[605]) ).
thf(1180,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
| ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_574_dvd_Ole__imp__less__or__eq) ).
thf(5130,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
| ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1180]) ).
thf(143,axiom,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_894_mult_Odiff__right) ).
thf(1577,plain,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(643,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_220_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) ).
thf(3280,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[643]) ).
thf(1195,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_563_dvd_Oless__not__sym) ).
thf(5182,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1195]) ).
thf(433,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ ( zero_zero @ int ) ) @ A )
= ( zero_zero @ int ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_918_zmod__zero) ).
thf(2562,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ ( zero_zero @ int ) ) @ A )
= ( zero_zero @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[433]) ).
thf(856,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_673_add__diff__assoc) ).
thf(4023,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[856]) ).
thf(589,axiom,
! [A: $i] :
( ( power @ A )
=> ! [B: $i,C: $i] :
( ( ( C
= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C )
= ( one_one @ A ) ) )
& ( ( C
!= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ ( one_one @ nat ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_718_realpow__num__eq__if) ).
thf(3089,plain,
! [A: $i] :
( ( power @ A )
=> ! [B: $i,C: $i] :
( ( ( C
= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C )
= ( one_one @ A ) ) )
& ( ( C
!= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ ( one_one @ nat ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[589]) ).
thf(220,axiom,
linord20386208strict @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__semiring__strict) ).
thf(1832,plain,
linord20386208strict @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(536,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( number_number_of @ nat @ A ) ) @ ( number_number_of @ nat @ B ) ) )
<=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ B ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_less__nat__number__of) ).
thf(2897,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( number_number_of @ nat @ A ) ) @ ( number_number_of @ nat @ B ) ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ B ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ) )
& ( ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ B ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( number_number_of @ nat @ A ) ) @ ( number_number_of @ nat @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[536]) ).
thf(1070,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_664_add__le__mono1) ).
thf(4766,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1070]) ).
thf(88,axiom,
real_normed_algebra @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___RealVector_Oreal__normed__algebra) ).
thf(1462,plain,
real_normed_algebra @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(1223,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_432_mult__le__cancel__left__pos) ).
thf(5279,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1223]) ).
thf(920,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ C ) @ A ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_zadd__zless__mono) ).
thf(4243,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ C ) @ A ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[920]) ).
thf(483,axiom,
! [A: $i] :
( ( linord893533164strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_380_comm__mult__strict__left__mono) ).
thf(2703,plain,
! [A: $i] :
( ( linord893533164strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[483]) ).
thf(1209,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ ( bit1 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_less__eq__int__code_I14_J) ).
thf(5231,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ ( bit1 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ ( bit1 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1209]) ).
thf(1238,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( ( inv @ B @ A )
!= ( zero_zero @ int ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_825_inv__not__0) ).
thf(5338,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( ( inv @ B @ A )
!= ( zero_zero @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1238]) ).
thf(672,axiom,
! [A: $i] :
( ( bit0 @ A )
!= min ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_286_rel__simps_I45_J) ).
thf(3385,plain,
! [A: $i] :
( ( bit0 @ A )
!= min ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[672]) ).
thf(958,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_400_power__mono) ).
thf(4353,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[958]) ).
thf(181,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_307_dvd__refl) ).
thf(1704,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(893,axiom,
! [A: $i,B: $i] :
( ( ( standardRes @ A @ B )
= ( zero_zero @ int ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_988_StandardRes__eq__zcong) ).
thf(4141,plain,
! [A: $i,B: $i] :
( ( ( ( standardRes @ A @ B )
= ( zero_zero @ int ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ A ) )
=> ( ( standardRes @ A @ B )
= ( zero_zero @ int ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[893]) ).
thf(321,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_391_mult__less__cancel__left__pos) ).
thf(2158,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[321]) ).
thf(798,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
<=> ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_871_gcd__lcm__complete__lattice__nat_Otop__unique) ).
thf(3835,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( A
= ( zero_zero @ nat ) ) )
& ( ( A
= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[798]) ).
thf(563,axiom,
! [A: $i,B: $i] :
( ( order @ B )
=> ( order @ ( fun @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_fun___Orderings_Oorder) ).
thf(2989,plain,
! [A: $i,B: $i] :
( ( order @ B )
=> ( order @ ( fun @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[563]) ).
thf(1169,axiom,
zero_neq_one @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Ozero__neq__one) ).
thf(5091,plain,
zero_neq_one @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1169]) ).
thf(233,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_663_trans__le__add2) ).
thf(1875,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).
thf(923,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_596_le__antisym) ).
thf(4253,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[923]) ).
thf(1037,axiom,
! [A: $i] :
( ( bit1 @ A )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( one_one @ int ) ) @ A ) ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_164_Bit1__def) ).
thf(4595,plain,
! [A: $i] :
( ( bit1 @ A )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( one_one @ int ) ) @ A ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1037]) ).
thf(487,axiom,
ab_semigroup_add @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oab__semigroup__add) ).
thf(2713,plain,
ab_semigroup_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[487]) ).
thf(886,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_601_le__diff__iff) ).
thf(4119,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[886]) ).
thf(325,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( zero_zero @ nat ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ B @ A ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ A ) )
& ! [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ B @ D ) ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_862_ex__least__nat__le) ).
thf(2169,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( zero_zero @ nat ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ B @ A ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ A ) )
& ! [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ B @ D ) ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[325]) ).
thf(112,axiom,
! [A: $i,B: $i] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ( ( plus_plus @ B @ ( ti @ B @ A ) )
= ( plus_plus @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Oplus__class_Oplus_0_arg1) ).
thf(1514,plain,
! [A: $i,B: $i] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ( ( plus_plus @ B @ ( ti @ B @ A ) )
= ( plus_plus @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(906,axiom,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( number_number_of @ A @ B ) ) @ ( one_one @ A ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( bit1 @ pls ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_less__special_I4_J) ).
thf(4194,plain,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( number_number_of @ A @ B ) ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( bit1 @ pls ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( bit1 @ pls ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( number_number_of @ A @ B ) ) @ ( one_one @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[906]) ).
thf(1179,axiom,
! [A: $i,B: $i] :
( ( monoid_add @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( plus_plus @ B @ A ) )
= ( plus_plus @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Oplus__class_Oplus_1_res) ).
thf(5127,plain,
! [A: $i,B: $i] :
( ( monoid_add @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( plus_plus @ B @ A ) )
= ( plus_plus @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1179]) ).
thf(430,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ pls ) @ A )
= ( ti @ int @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_add__Pls) ).
thf(2552,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ pls ) @ A )
= ( ti @ int @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[430]) ).
thf(1006,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hAPP @ A @ B @ ( combb @ C @ B @ A @ D @ E ) @ F )
= ( hAPP @ C @ B @ D @ ( hAPP @ A @ C @ E @ F ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_COMBB_1_1_U) ).
thf(4492,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hAPP @ A @ B @ ( combb @ C @ B @ A @ D @ E ) @ F )
= ( hAPP @ C @ B @ D @ ( hAPP @ A @ C @ E @ F ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1006]) ).
thf(1119,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_957_pos__mod__sign) ).
thf(4928,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1119]) ).
thf(140,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_215_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) ).
thf(1572,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(821,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_328_dvd__triv__left) ).
thf(3919,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[821]) ).
thf(461,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( minus_minus @ int @ ( bit1 @ A ) ) @ ( bit1 @ B ) )
= ( bit0 @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_483_diff__bin__simps_I10_J) ).
thf(2635,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( minus_minus @ int @ ( bit1 @ A ) ) @ ( bit1 @ B ) )
= ( bit0 @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[461]) ).
thf(697,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
!= ( zero_zero @ real ) )
=> ( ( ( hAPP @ real @ real @ ( times_times @ real @ C ) @ A )
= ( hAPP @ real @ real @ ( times_times @ real @ C ) @ B ) )
<=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_635_real__mult__left__cancel) ).
thf(3470,plain,
! [A: $i,B: $i,C: $i] :
( ( C
!= ( zero_zero @ real ) )
=> ( ( ( ( hAPP @ real @ real @ ( times_times @ real @ C ) @ A )
= ( hAPP @ real @ real @ ( times_times @ real @ C ) @ B ) )
=> ( A = B ) )
& ( ( A = B )
=> ( ( hAPP @ real @ real @ ( times_times @ real @ C ) @ A )
= ( hAPP @ real @ real @ ( times_times @ real @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[697]) ).
thf(649,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( div_mod @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ C ) ) @ B )
= ( hAPP @ nat @ nat @ ( div_mod @ nat @ C ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_982_mod__mult__self3) ).
thf(3312,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( div_mod @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ C ) ) @ B )
= ( hAPP @ nat @ nat @ ( div_mod @ nat @ C ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[649]) ).
thf(138,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ C ) ) @ D ) ) @ C )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_902_zmod__simps_I4_J) ).
thf(1568,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ C ) ) @ D ) ) @ C )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(996,axiom,
! [A: $i] :
( ( A
!= ( zero_zero @ nat ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_615_neq0__conv) ).
thf(4460,plain,
! [A: $i] :
( ( ( A
!= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( A
!= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[996]) ).
thf(100,axiom,
linord219039673up_add @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Olinordered__ab__group__add) ).
thf(1494,plain,
linord219039673up_add @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(519,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_731_nat__mult__dvd__cancel1) ).
thf(2852,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[519]) ).
thf(1073,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( ( inv @ B @ A )
!= ( one_one @ int ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_823_inv__not__1) ).
thf(4771,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( ( inv @ B @ A )
!= ( one_one @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1073]) ).
thf(1164,axiom,
plus @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oplus) ).
thf(5077,plain,
plus @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1164]) ).
thf(172,axiom,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ E ) ) @ C ) ) @ D ) ) @ F ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_488_le__add__iff1) ).
thf(1664,plain,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ E ) ) @ C ) ) @ D ) ) @ F ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ E ) ) @ C ) ) @ D ) ) @ F ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(622,axiom,
! [A: $i] :
( ( ti @ ( fun @ int @ bool ) @ ( d22set @ A ) )
= ( d22set @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_IntFact_Od22set_res) ).
thf(3199,plain,
! [A: $i] :
( ( ti @ ( fun @ int @ bool ) @ ( d22set @ A ) )
= ( d22set @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[622]) ).
thf(565,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_421_mult__left__le__imp__le) ).
thf(2996,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[565]) ).
thf(993,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ B ) ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) ) )
=> ( hBOOL @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_558_dvd_Oless__imp__triv) ).
thf(4454,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ B ) ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) ) )
=> ( hBOOL @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[993]) ).
thf(352,axiom,
one @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oone) ).
thf(2259,plain,
one @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[352]) ).
thf(403,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_309_zcong__sym) ).
thf(2451,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ A ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[403]) ).
thf(768,axiom,
group_add @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Ogroup__add) ).
thf(3702,plain,
group_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[768]) ).
thf(1095,axiom,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_547_less__irrefl__nat) ).
thf(4847,plain,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1095]) ).
thf(1188,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_503_power2__diff) ).
thf(5156,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1188]) ).
thf(364,axiom,
linord20386208strict @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Olinordered__semiring__strict) ).
thf(2291,plain,
linord20386208strict @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).
thf(1103,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_671_le__add__diff) ).
thf(4871,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1103]) ).
thf(16,axiom,
no_zero_divisors @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Ono__zero__divisors) ).
thf(1283,plain,
no_zero_divisors @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(802,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ D ) ) ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_969_zmult2__lemma__aux3) ).
thf(3851,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ D ) ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[802]) ).
thf(985,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_zle__refl) ).
thf(4432,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[985]) ).
thf(404,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B )
= ( zero_zero @ nat ) )
<=> ( ( A
= ( zero_zero @ nat ) )
& ( B
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_623_add__is__0) ).
thf(2456,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B )
= ( zero_zero @ nat ) )
=> ( ( A
= ( zero_zero @ nat ) )
& ( B
= ( zero_zero @ nat ) ) ) )
& ( ( ( A
= ( zero_zero @ nat ) )
& ( B
= ( zero_zero @ nat ) ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[404]) ).
thf(269,axiom,
number_semiring @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Int_Onumber__semiring) ).
thf(2007,plain,
number_semiring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).
thf(328,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( number_number_of @ A @ ( bit0 @ B ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ ( number_number_of @ A @ B ) ) ) @ ( number_number_of @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_number__of__Bit0) ).
thf(2178,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( number_number_of @ A @ ( bit0 @ B ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ ( number_number_of @ A @ B ) ) ) @ ( number_number_of @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).
thf(339,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_443_power__strict__increasing) ).
thf(2222,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).
thf(977,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_221_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) ).
thf(4411,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[977]) ).
thf(550,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ B ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_706_real__mult__order) ).
thf(2950,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[550]) ).
thf(909,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( d22set @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_859_d22set__le) ).
thf(4205,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( d22set @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[909]) ).
thf(834,axiom,
number_ring @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Int_Onumber__ring) ).
thf(3966,plain,
number_ring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[834]) ).
thf(277,axiom,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ ( one_one @ nat ) ) ) ) @ B )
= ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_715_realpow__minus__mult) ).
thf(2027,plain,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ ( one_one @ nat ) ) ) ) @ B )
= ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).
thf(988,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ E ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ D ) @ B ) ) @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_356_zcong__zmult) ).
thf(4443,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ E ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ D ) @ B ) ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[988]) ).
thf(84,axiom,
! [A: $i] :
( ( number @ A )
=> ! [B: $i,C: $i] :
( ( ( number_number_of @ A @ B )
= ( ti @ A @ C ) )
<=> ( ( ti @ A @ C )
= ( number_number_of @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_number__of__reorient) ).
thf(1450,plain,
! [A: $i] :
( ( number @ A )
=> ! [B: $i,C: $i] :
( ( ( ( number_number_of @ A @ B )
= ( ti @ A @ C ) )
=> ( ( ti @ A @ C )
= ( number_number_of @ A @ B ) ) )
& ( ( ( ti @ A @ C )
= ( number_number_of @ A @ B ) )
=> ( ( number_number_of @ A @ B )
= ( ti @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(1032,axiom,
! [A: $i] :
( ( min
= ( bit1 @ A ) )
<=> ( min
= ( ti @ int @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_282_rel__simps_I43_J) ).
thf(4578,plain,
! [A: $i] :
( ( ( min
= ( bit1 @ A ) )
=> ( min
= ( ti @ int @ A ) ) )
& ( ( min
= ( ti @ int @ A ) )
=> ( min
= ( bit1 @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1032]) ).
thf(302,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ ( bit0 @ A ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_158_rel__simps_I4_J) ).
thf(2105,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ ( bit0 @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ ( bit0 @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).
thf(854,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hAPP @ A @ B @ ( combc @ A @ C @ B @ D @ E ) @ F )
= ( hAPP @ C @ B @ ( hAPP @ A @ ( fun @ C @ B ) @ D @ F ) @ E ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_COMBC_1_1_U) ).
thf(4018,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hAPP @ A @ B @ ( combc @ A @ C @ B @ D @ E ) @ F )
= ( hAPP @ C @ B @ ( hAPP @ A @ ( fun @ C @ B ) @ D @ F ) @ E ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[854]) ).
thf(297,axiom,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_power3__eq__cube) ).
thf(2090,plain,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).
thf(162,axiom,
! [A: $i] :
( ( number_semiring @ A )
=> ( ( number_number_of @ A @ ( bit1 @ pls ) )
= ( one_one @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_171_semiring__numeral__1__eq__1) ).
thf(1632,plain,
! [A: $i] :
( ( number_semiring @ A )
=> ( ( number_number_of @ A @ ( bit1 @ pls ) )
= ( one_one @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(406,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ C ) )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_734_nat__diff__add__eq1) ).
thf(2469,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ C ) )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[406]) ).
thf(1236,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_519_diff__square) ).
thf(5333,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1236]) ).
thf(1087,axiom,
min != pls,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_285_rel__simps_I40_J) ).
thf(4820,plain,
min != pls,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1087]) ).
thf(717,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_314_order__le__neq__implies__less) ).
thf(3525,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[717]) ).
thf(866,axiom,
ordere223160158up_add @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add) ).
thf(4050,plain,
ordere223160158up_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[866]) ).
thf(253,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_936_Divides_Otransfer__nat__int__function__closures_I2_J) ).
thf(1960,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(1181,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_619_diff__less) ).
thf(5135,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1181]) ).
thf(431,axiom,
order @ bool,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_HOL_Obool___Orderings_Oorder) ).
thf(2555,plain,
order @ bool,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[431]) ).
thf(645,axiom,
! [A: $i] :
( ( number_semiring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( number_number_of @ A @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_semiring__mult__2__right) ).
thf(3290,plain,
! [A: $i] :
( ( number_semiring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( number_number_of @ A @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[645]) ).
thf(884,axiom,
! [A: $i] :
( ( bit0 @ A )
= ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_Bit0__def) ).
thf(4115,plain,
! [A: $i] :
( ( bit0 @ A )
= ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[884]) ).
thf(145,axiom,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_896_mult_Odiff__left) ).
thf(1581,plain,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(500,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ A ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ C ) )
& ( ( hAPP @ nat @ real @ ( power_power @ real @ C ) @ B )
= A )
& ! [D: $i] :
( ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ D ) )
& ( ( hAPP @ nat @ real @ ( power_power @ real @ D ) @ B )
= A ) )
=> ( D = C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_529_realpow__pos__nth__unique) ).
thf(2777,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ A ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ C ) )
& ( ( hAPP @ nat @ real @ ( power_power @ real @ C ) @ B )
= A )
& ! [D: $i] :
( ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ D ) )
& ( ( hAPP @ nat @ real @ ( power_power @ real @ D ) @ B )
= A ) )
=> ( D = C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[500]) ).
thf(607,axiom,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ E ) @ B ) ) @ C ) ) @ F ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_489_le__add__iff2) ).
thf(3152,plain,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ E ) @ B ) ) @ C ) ) @ F ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ E ) @ B ) ) @ C ) ) @ F ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[607]) ).
thf(673,axiom,
! [A: $i,B: $i,C: $i] :
( ( B = C )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_566_dvd_Oord__eq__less__trans) ).
thf(3389,plain,
! [A: $i,B: $i,C: $i] :
( ( B = C )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[673]) ).
thf(843,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ ( standardRes @ C @ A ) ) @ ( standardRes @ C @ B ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_975_StandardRes__prop4) ).
thf(3986,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ ( standardRes @ C @ A ) ) @ ( standardRes @ C @ B ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[843]) ).
thf(618,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ C ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ D ) ) ) @ C ) ) @ ( zero_zero @ int ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_966_zmult2__lemma__aux2) ).
thf(3189,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ C ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ D ) ) ) @ C ) ) @ ( zero_zero @ int ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[618]) ).
thf(1206,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( number_number_of @ A @ ( bit1 @ pls ) ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_mult__numeral__1__right) ).
thf(5220,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( number_number_of @ A @ ( bit1 @ pls ) ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1206]) ).
thf(1256,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_104_power2__less__imp__less) ).
thf(5384,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1256]) ).
thf(83,axiom,
! [A: $i,B: $i] :
( ( dvd @ B )
=> ( ( times_times @ B @ ( ti @ B @ A ) )
= ( times_times @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_2_arg1) ).
thf(1447,plain,
! [A: $i,B: $i] :
( ( dvd @ B )
=> ( ( times_times @ B @ ( ti @ B @ A ) )
= ( times_times @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(763,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ti @ ( fun @ A @ B ) @ ( combb @ C @ B @ A @ D @ E ) )
= ( combb @ C @ B @ A @ D @ E ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_COMBB_res) ).
thf(3685,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ti @ ( fun @ A @ B ) @ ( combb @ C @ B @ A @ D @ E ) )
= ( combb @ C @ B @ A @ D @ E ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[763]) ).
thf(396,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ B )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_931_mod__self) ).
thf(2424,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ B )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[396]) ).
thf(1240,axiom,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_778_add__le__imp__le__left) ).
thf(5343,plain,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1240]) ).
thf(507,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( plus_plus @ int @ D ) @ C ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( plus_plus @ int @ D ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_749_incr__mult__lemma) ).
thf(2809,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( plus_plus @ int @ D ) @ C ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( plus_plus @ int @ D ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[507]) ).
thf(77,axiom,
ordere236663937imp_le @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add__imp__le) ).
thf(1433,plain,
ordere236663937imp_le @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(1182,axiom,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_le__number__of) ).
thf(5137,plain,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1182]) ).
thf(741,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ real @ real @ ( plus_plus @ real @ ( hAPP @ real @ real @ ( times_times @ real @ A ) @ A ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ B ) @ B ) )
= ( zero_zero @ real ) )
<=> ( ( A
= ( zero_zero @ real ) )
& ( B
= ( zero_zero @ real ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_709_real__two__squares__add__zero__iff) ).
thf(3606,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ real @ real @ ( plus_plus @ real @ ( hAPP @ real @ real @ ( times_times @ real @ A ) @ A ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ B ) @ B ) )
= ( zero_zero @ real ) )
=> ( ( A
= ( zero_zero @ real ) )
& ( B
= ( zero_zero @ real ) ) ) )
& ( ( ( A
= ( zero_zero @ real ) )
& ( B
= ( zero_zero @ real ) ) )
=> ( ( hAPP @ real @ real @ ( plus_plus @ real @ ( hAPP @ real @ real @ ( times_times @ real @ A ) @ A ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ B ) @ B ) )
= ( zero_zero @ real ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[741]) ).
thf(1235,axiom,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_805_zero__less__double__add__iff__zero__less__single__add) ).
thf(5328,plain,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1235]) ).
thf(678,axiom,
~ ! [A: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( one_one @ int ) )
!= ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_) ).
thf(3406,plain,
~ ! [A: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( one_one @ int ) )
!= ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[678]) ).
thf(1053,axiom,
! [A: $i,B: $i] :
( ( A != B )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_577_dvd_Oneq__le__trans) ).
thf(4718,plain,
! [A: $i,B: $i] :
( ( A != B )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1053]) ).
thf(1064,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_454_power__Suc__less) ).
thf(4751,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1064]) ).
thf(55,axiom,
number_ring @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Int_Onumber__ring) ).
thf(1377,plain,
number_ring @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(1245,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ t ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ ( zero_zero @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2__096_I4_A_K_Am_A_L_A1_J_A_K_At_A_060_A_I4_A_K_Am_A_L_A1_J_A_K_A0_096) ).
thf(5353,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ t ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ ( zero_zero @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1245]) ).
thf(208,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B ) ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_976_mod__le__divisor) ).
thf(1796,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(422,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_383_mult__neg__neg) ).
thf(2530,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[422]) ).
thf(62,axiom,
( ( ti @ int @ m )
= m ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_v_m_res) ).
thf(1394,plain,
( ( ti @ int @ m )
= m ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(499,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_620_zero__less__diff) ).
thf(2772,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[499]) ).
thf(608,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_680_add__mult__distrib2) ).
thf(3157,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[608]) ).
thf(44,axiom,
! [A: $i,B: $i] :
( ( number @ B )
=> ( ( ti @ B @ ( number_number_of @ B @ A ) )
= ( number_number_of @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Int_Onumber__class_Onumber__of_res) ).
thf(1351,plain,
! [A: $i,B: $i] :
( ( number @ B )
=> ( ( ti @ B @ ( number_number_of @ B @ A ) )
= ( number_number_of @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(1193,axiom,
! [A: $i,B: $i] :
( ( A != B )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_545_nat__neq__iff) ).
thf(5169,plain,
! [A: $i,B: $i] :
( ( ( A != B )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) ) )
=> ( A != B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1193]) ).
thf(42,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_943_mod__mult__self1) ).
thf(1345,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(863,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( one_one @ int ) ) @ A ) ) @ A )
!= ( zero_zero @ int ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_165_odd__nonzero) ).
thf(4038,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( one_one @ int ) ) @ A ) ) @ A )
!= ( zero_zero @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[863]) ).
thf(616,axiom,
! [A: $i] :
( ( ring_11004092258visors @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ C )
!= ( zero_zero @ A ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B )
!= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_324_field__power__not__zero) ).
thf(3184,plain,
! [A: $i] :
( ( ring_11004092258visors @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ C )
!= ( zero_zero @ A ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B )
!= ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[616]) ).
thf(561,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( quadRes @ A ) @ B ) )
<=> ? [C: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_509_QuadRes__def) ).
thf(2982,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( quadRes @ A ) @ B ) )
=> ? [C: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) @ A ) ) )
& ( ? [C: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( quadRes @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[561]) ).
thf(530,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29_zle__trans) ).
thf(2884,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[530]) ).
thf(1003,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_560_dvd_Oless__imp__not__eq) ).
thf(4483,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1003]) ).
thf(711,axiom,
! [A: $i] :
( ( ti @ ( fun @ int @ bool ) @ ( quadRes @ A ) )
= ( quadRes @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Residues_OQuadRes_res) ).
thf(3510,plain,
! [A: $i] :
( ( ti @ ( fun @ int @ bool ) @ ( quadRes @ A ) )
= ( quadRes @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[711]) ).
thf(871,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_646_add__diff__inverse) ).
thf(4065,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[871]) ).
thf(552,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
!= ( zero_zero @ nat ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ C ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_524_divides__exp2) ).
thf(2954,plain,
! [A: $i,B: $i,C: $i] :
( ( C
!= ( zero_zero @ nat ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ C ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[552]) ).
thf(730,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C )
= ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_456_power__eq__imp__eq__base) ).
thf(3567,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C )
= ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[730]) ).
thf(80,axiom,
! [A: $i,B: $i] :
( ( power @ B )
=> ( ( power_power @ B @ ( ti @ B @ A ) )
= ( power_power @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Power_Opower__class_Opower_arg1) ).
thf(1438,plain,
! [A: $i,B: $i] :
( ( power @ B )
=> ( ( power_power @ B @ ( ti @ B @ A ) )
= ( power_power @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(639,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_397_not__one__le__zero) ).
thf(3265,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[639]) ).
thf(1148,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ C ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ D ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ D ) ) ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_967_zmult2__lemma__aux1) ).
thf(5017,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ C ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ D ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ D ) ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1148]) ).
thf(306,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_654_less__diff__iff) ).
thf(2118,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).
thf(1183,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ? [B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_740_zprime__factor__exists) ).
thf(5142,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ? [B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1183]) ).
thf(1031,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_422_mult__right__le__imp__le) ).
thf(4576,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1031]) ).
thf(1113,axiom,
! [A: $i] :
( ( sr @ ( ti @ int @ A ) )
= ( sr @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Residues_OSR_arg1) ).
thf(4910,plain,
! [A: $i] :
( ( sr @ ( ti @ int @ A ) )
= ( sr @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1113]) ).
thf(1118,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( zero_zero @ int ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) @ ( zero_zero @ int ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_961_neg__mod__conj) ).
thf(4925,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( zero_zero @ int ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) @ ( zero_zero @ int ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1118]) ).
thf(579,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_925_zmod__simps_I2_J) ).
thf(3037,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[579]) ).
thf(606,axiom,
linordered_idom @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__idom) ).
thf(3151,plain,
linordered_idom @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[606]) ).
thf(590,axiom,
linordered_ring @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__ring) ).
thf(3097,plain,
linordered_ring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[590]) ).
thf(1097,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( number_number_of @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_number__of__mult) ).
thf(4853,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( number_number_of @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1097]) ).
thf(735,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_607_Nat_Odiff__le__self) ).
thf(3582,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[735]) ).
thf(1146,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_258_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) ).
thf(5012,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1146]) ).
thf(708,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ B ) )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_592_diff__cancel2) ).
thf(3503,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ B ) )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[708]) ).
thf(907,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( number_number_of @ A @ B ) )
= ( number_number_of @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( bit1 @ pls ) ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_add__special_I2_J) ).
thf(4199,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( number_number_of @ A @ B ) )
= ( number_number_of @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( bit1 @ pls ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[907]) ).
thf(912,axiom,
ordered_semiring @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oordered__semiring) ).
thf(4216,plain,
ordered_semiring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[912]) ).
thf(1109,axiom,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_777_add__le__imp__le__right) ).
thf(4893,plain,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1109]) ).
thf(945,axiom,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ B )
= ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_327_power__commutes) ).
thf(4323,plain,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ B )
= ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[945]) ).
thf(1133,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) )
<=> ( ( ( ti @ A @ B )
!= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
!= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8_sum__power2__gt__zero__iff) ).
thf(4962,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) )
=> ( ( ( ti @ A @ B )
!= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
!= ( zero_zero @ A ) ) ) )
& ( ( ( ( ti @ A @ B )
!= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
!= ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1133]) ).
thf(1065,axiom,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) @ ( zero_zero @ A ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_806_double__add__less__zero__iff__single__add__less__zero) ).
thf(4753,plain,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1065]) ).
thf(282,axiom,
idom @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oidom) ).
thf(2047,plain,
idom @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).
thf(651,axiom,
! [A: $i] :
( ( semiri456707255roduct @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( ( ti @ A @ D )
!= ( ti @ A @ E ) )
& ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) )
<=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) )
!= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_240_crossproduct__noteq) ).
thf(3318,plain,
! [A: $i] :
( ( semiri456707255roduct @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( ( ( ti @ A @ D )
!= ( ti @ A @ E ) )
& ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) )
!= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ B ) ) ) )
& ( ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) )
!= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ B ) ) )
=> ( ( ( ti @ A @ D )
!= ( ti @ A @ E ) )
& ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[651]) ).
thf(242,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_880_dvd__pos__nat) ).
thf(1915,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).
thf(298,axiom,
! [A: $i] :
( ( ti @ int @ ( zfact @ A ) )
= ( zfact @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_IntFact_Ozfact_res) ).
thf(2093,plain,
! [A: $i] :
( ( ti @ int @ ( zfact @ A ) )
= ( zfact @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).
thf(203,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_zmult__zless__mono2) ).
thf(1783,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(964,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ C ) )
<=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ B )
= C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_735_nat__eq__add__iff1) ).
thf(4370,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ B )
= C ) )
& ( ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ B )
= C )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[964]) ).
thf(493,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ti @ int @ A )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) @ D ) )
=> ( ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ D ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ D ) @ B ) ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ D ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ ( zero_zero @ int ) ) ) ) ) )
=> ( ( ( ti @ int @ B )
!= ( zero_zero @ int ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B )
= ( ti @ int @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_965_divmod__int__rel__mod__eq) ).
thf(2737,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ti @ int @ A )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) @ D ) )
=> ( ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ D ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ D ) @ B ) ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ D ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ ( zero_zero @ int ) ) ) ) ) )
=> ( ( ( ti @ int @ B )
!= ( zero_zero @ int ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B )
= ( ti @ int @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[493]) ).
thf(537,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ D ) ) @ E ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ E ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_265_unique__quotient__lemma) ).
thf(2905,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ D ) ) @ E ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ E ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[537]) ).
thf(231,axiom,
! [A: $i] :
( ( dvd @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( B
= ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_333_dvdI) ).
thf(1868,plain,
! [A: $i] :
( ( dvd @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( B
= ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).
thf(779,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ( A != B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_579_dvd_Oless__le) ).
thf(3735,plain,
! [A: $i,B: $i] :
( ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ( A != B ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ( A != B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[779]) ).
thf(484,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_685_mult__le__mono1) ).
thf(2705,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[484]) ).
thf(426,axiom,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_889_mult__right_Ozero) ).
thf(2541,plain,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[426]) ).
thf(144,axiom,
comm_monoid_mult @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Ocomm__monoid__mult) ).
thf(1580,plain,
comm_monoid_mult @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(986,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( member @ A @ B ) @ C ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_mem__def) ).
thf(4434,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( member @ A @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ C @ B ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ C @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( member @ A @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[986]) ).
thf(317,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_431_mult__le__cancel__left__neg) ).
thf(2147,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( zero_zero @ A ) ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).
thf(574,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( zero_zero @ int ) )
= ( ti @ int @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_zadd__0__right) ).
thf(3022,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( zero_zero @ int ) )
= ( ti @ int @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[574]) ).
thf(97,axiom,
ordere223160158up_add @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Oordered__cancel__ab__semigroup__add) ).
thf(1487,plain,
ordere223160158up_add @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(633,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( one_one @ int ) ) ) )
<=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
| ( ( ti @ int @ A )
= ( ti @ int @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_166_zless__add1__eq) ).
thf(3239,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
| ( ( ti @ int @ A )
= ( ti @ int @ B ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
| ( ( ti @ int @ A )
= ( ti @ int @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( one_one @ int ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[633]) ).
thf(1009,axiom,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( one_one @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_792_mult__1__left) ).
thf(4499,plain,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( one_one @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1009]) ).
thf(312,axiom,
( ( bit1 @ min )
= min ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_283_Bit1__Min) ).
thf(2132,plain,
( ( bit1 @ min )
= min ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).
thf(134,axiom,
! [A: $i] :
( ( d22set @ ( ti @ int @ A ) )
= ( d22set @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_IntFact_Od22set_arg1) ).
thf(1560,plain,
! [A: $i] :
( ( d22set @ ( ti @ int @ A ) )
= ( d22set @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(11,axiom,
group_add @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Ogroup__add) ).
thf(1274,plain,
group_add @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(1204,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_649_le__eq__less__or__eq) ).
thf(5206,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( A = B ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( A = B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1204]) ).
thf(59,axiom,
! [A: $i] :
( ( ab_semigroup_mult @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_758_ab__semigroup__mult__class_Omult__ac_I1_J) ).
thf(1387,plain,
! [A: $i] :
( ( ab_semigroup_mult @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(239,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( hAPP @ real @ real @ ( times_times @ real @ A ) @ C ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ B ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_707_real__mult__less__iff1) ).
thf(1900,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( hAPP @ real @ real @ ( times_times @ real @ A ) @ C ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ B ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( hAPP @ real @ real @ ( times_times @ real @ A ) @ C ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).
thf(187,axiom,
ordered_ab_group_add @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oordered__ab__group__add) ).
thf(1723,plain,
ordered_ab_group_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(40,axiom,
linord626643107strict @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Olinordered__semiring__1__strict) ).
thf(1343,plain,
linord626643107strict @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(837,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ nat @ int @ ( power_power @ int @ D ) @ A ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ nat @ int @ ( power_power @ int @ D ) @ A ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_zprime__power__zdvd__cancel__left) ).
thf(3972,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ nat @ int @ ( power_power @ int @ D ) @ A ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ nat @ int @ ( power_power @ int @ D ) @ A ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[837]) ).
thf(446,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( zero_zero @ int ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_249_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J) ).
thf(2596,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( zero_zero @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[446]) ).
thf(806,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ C ) ) ) )
<=> ( ( ( ti @ A @ B )
!= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
!= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_sum__squares__gt__zero__iff) ).
thf(3867,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ C ) ) ) )
=> ( ( ( ti @ A @ B )
!= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
!= ( zero_zero @ A ) ) ) )
& ( ( ( ( ti @ A @ B )
!= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
!= ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[806]) ).
thf(1042,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) ) )
<=> ( ( ( ( ti @ int @ C )
= ( zero_zero @ int ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ! [D: $i,E: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ E ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ E ) @ C ) )
& ( ( ti @ int @ B )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ D ) ) @ E ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ E ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ ( zero_zero @ int ) ) )
=> ! [D: $i,E: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ E ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ E ) @ ( zero_zero @ int ) ) )
& ( ( ti @ int @ B )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ D ) ) @ E ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ E ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_970_split__zmod) ).
thf(4609,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) ) )
=> ( ( ( ( ti @ int @ C )
= ( zero_zero @ int ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ! [D: $i,E: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ E ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ E ) @ C ) )
& ( ( ti @ int @ B )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ D ) ) @ E ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ E ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ ( zero_zero @ int ) ) )
=> ! [D: $i,E: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ E ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ E ) @ ( zero_zero @ int ) ) )
& ( ( ti @ int @ B )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ D ) ) @ E ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ E ) ) ) ) ) )
& ( ( ( ( ( ti @ int @ C )
= ( zero_zero @ int ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ! [D: $i,E: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ E ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ E ) @ C ) )
& ( ( ti @ int @ B )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ D ) ) @ E ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ E ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ ( zero_zero @ int ) ) )
=> ! [D: $i,E: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ E ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ E ) @ ( zero_zero @ int ) ) )
& ( ( ti @ int @ B )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ D ) ) @ E ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ E ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1042]) ).
thf(1155,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_355_zcong__scalar2) ).
thf(5044,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1155]) ).
thf(1243,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( multInv @ B @ ( multInv @ B @ A ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( multInv @ B @ A ) ) ) @ ( multInv @ B @ ( multInv @ B @ A ) ) ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_849_Int2_Oaux____1) ).
thf(5349,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( multInv @ B @ ( multInv @ B @ A ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( multInv @ B @ A ) ) ) @ ( multInv @ B @ ( multInv @ B @ A ) ) ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1243]) ).
thf(176,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_928_mod__add__right__eq) ).
thf(1676,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(955,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_802_add__increasing) ).
thf(4344,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[955]) ).
thf(877,axiom,
! [A: $i] :
( ( ( zero_zero @ nat )
= ( number_number_of @ nat @ A ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ pls ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_eq__0__number__of) ).
thf(4091,plain,
! [A: $i] :
( ( ( ( zero_zero @ nat )
= ( number_number_of @ nat @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ pls ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ pls ) )
=> ( ( zero_zero @ nat )
= ( number_number_of @ nat @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[877]) ).
thf(318,axiom,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ min ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_292_rel__simps_I3_J) ).
thf(2150,plain,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ min ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[318]) ).
thf(51,axiom,
linordered_idom @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Olinordered__idom) ).
thf(1370,plain,
linordered_idom @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(279,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( zero_zero @ int ) ) @ A )
= ( ti @ int @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_zadd__0) ).
thf(2032,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( zero_zero @ int ) ) @ A )
= ( ti @ int @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).
thf(1172,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( one_one @ int ) ) @ A )
= ( ti @ int @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_66_zmult__1) ).
thf(5102,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( one_one @ int ) ) @ A )
= ( ti @ int @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1172]) ).
thf(625,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ min ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ min ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_302_rel__simps_I11_J) ).
thf(3207,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ min ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ min ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ min ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ min ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[625]) ).
thf(762,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ti @ A @ ( hAPP @ B @ A @ C @ D ) )
= ( hAPP @ B @ A @ C @ D ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_hAPP_res) ).
thf(3682,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ti @ A @ ( hAPP @ B @ A @ C @ D ) )
= ( hAPP @ B @ A @ C @ D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[762]) ).
thf(855,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ A ) ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_937_zmod__le__nonneg__dividend) ).
thf(4021,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ A ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[855]) ).
thf(655,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( div_mod @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_983_mod__geq) ).
thf(3339,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( div_mod @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[655]) ).
thf(812,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_631_mult__0__right) ).
thf(3889,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[812]) ).
thf(72,axiom,
monoid_add @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Omonoid__add) ).
thf(1423,plain,
monoid_add @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(472,axiom,
zero @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Ozero) ).
thf(2670,plain,
zero @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[472]) ).
thf(807,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_567_dvd__diff__nat) ).
thf(3877,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[807]) ).
thf(85,axiom,
linordered_semiring @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Olinordered__semiring) ).
thf(1457,plain,
linordered_semiring @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(452,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ A ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_310_zcong__refl) ).
thf(2609,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[452]) ).
thf(1063,axiom,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_773_add__le__cancel__left) ).
thf(4746,plain,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1063]) ).
thf(168,axiom,
number_semiring @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Int_Onumber__semiring) ).
thf(1656,plain,
number_semiring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(576,axiom,
semiri456707255roduct @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) ).
thf(3028,plain,
semiri456707255roduct @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[576]) ).
thf(20,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) )
= ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_904_mod__mult__mult1) ).
thf(1291,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) )
= ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(195,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( number_number_of @ nat @ A ) ) @ ( number_number_of @ nat @ B ) ) )
<=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ pls ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_211_le__nat__number__of) ).
thf(1740,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( number_number_of @ nat @ A ) ) @ ( number_number_of @ nat @ B ) ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ pls ) ) ) )
& ( ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ pls ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( number_number_of @ nat @ A ) ) @ ( number_number_of @ nat @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(788,axiom,
! [A: $i] :
( ( order @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( ord_less_eq @ A ) )
= ( ord_less_eq @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Orderings_Oord__class_Oless__eq_res) ).
thf(3799,plain,
! [A: $i] :
( ( order @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( ord_less_eq @ A ) )
= ( ord_less_eq @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[788]) ).
thf(831,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( one_one @ int ) @ ( number_number_of @ int @ min ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_465_one__not__neg__one__mod__m) ).
thf(3958,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( one_one @ int ) @ ( number_number_of @ int @ min ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[831]) ).
thf(401,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( one_one @ int ) )
= ( ti @ int @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_zmult__1__right) ).
thf(2445,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( one_one @ int ) )
= ( ti @ int @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[401]) ).
thf(936,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ int @ ( power_power @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ B ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ A ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_182_zadd__power3) ).
thf(4295,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ int @ ( power_power @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ B ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ A ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[936]) ).
thf(619,axiom,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( one_one @ A ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_794_mult__1__right) ).
thf(3191,plain,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( one_one @ A ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[619]) ).
thf(929,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ s1 ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193__096_091s_A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_J_096) ).
thf(4280,plain,
hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ s1 ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[929]) ).
thf(223,axiom,
! [A: $i] :
( ( ring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( one_one @ A ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( one_one @ A ) ) ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ ( one_one @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_710_real__squared__diff__one__factored) ).
thf(1837,plain,
! [A: $i] :
( ( ring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( one_one @ A ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( one_one @ A ) ) ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ ( one_one @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).
thf(152,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
| ( B
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_344_nat__zero__less__power__iff) ).
thf(1602,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
| ( B
= ( zero_zero @ nat ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
| ( B
= ( zero_zero @ nat ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(1041,axiom,
semiri456707255roduct @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) ).
thf(4608,plain,
semiri456707255roduct @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1041]) ).
thf(244,axiom,
! [A: $i,B: $i] :
( ( A != B )
=> ( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_546_linorder__neqE__nat) ).
thf(1920,plain,
! [A: $i,B: $i] :
( ( A != B )
=> ( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).
thf(1033,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ ( hAPP @ int @ int @ ( div_mod @ int @ C ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_915_zdvd__zmod) ).
thf(4585,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ ( hAPP @ int @ int @ ( div_mod @ int @ C ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1033]) ).
thf(399,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B ) ) )
<=> ( ( B
= ( zero_zero @ nat ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_zero__less__power__nat__eq) ).
thf(2433,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B ) ) )
=> ( ( B
= ( zero_zero @ nat ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ) )
& ( ( ( B
= ( zero_zero @ nat ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[399]) ).
thf(677,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_power2__less__0) ).
thf(3403,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[677]) ).
thf(1055,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_403_zero__less__power) ).
thf(4725,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1055]) ).
thf(257,axiom,
! [A: $i,B: $i] :
( ( ti @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A @ B ) )
= ( member @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_member_res) ).
thf(1968,plain,
! [A: $i,B: $i] :
( ( ti @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A @ B ) )
= ( member @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).
thf(434,axiom,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C )
= ( zero_zero @ A ) )
<=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_786_right__minus__eq) ).
thf(2565,plain,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C )
= ( zero_zero @ A ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) )
& ( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
=> ( ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[434]) ).
thf(469,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_329_dvd__triv__right) ).
thf(2663,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[469]) ).
thf(131,axiom,
! [A: $i] :
( ( minus_minus @ nat @ ( ti @ nat @ A ) )
= ( minus_minus @ nat @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Ominus__class_Ominus_1_arg1) ).
thf(1555,plain,
! [A: $i] :
( ( minus_minus @ nat @ ( ti @ nat @ A ) )
= ( minus_minus @ nat @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(510,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_277_zpower__zdvd__prop2) ).
thf(2826,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[510]) ).
thf(896,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_433_mult__left__le__one__le) ).
thf(4155,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[896]) ).
thf(1127,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_440_power__less__power__Suc) ).
thf(4947,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1127]) ).
thf(707,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( B
!= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_617_gr__implies__not0) ).
thf(3499,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( B
!= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[707]) ).
thf(585,axiom,
! [A: $i] :
( ( A
!= ( one_one @ nat ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( one_one @ nat ) ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( one_one @ nat ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_877_gcd__lcm__complete__lattice__nat_Obot__less) ).
thf(3065,plain,
! [A: $i] :
( ( ( A
!= ( one_one @ nat ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( one_one @ nat ) ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( one_one @ nat ) ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( one_one @ nat ) ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( one_one @ nat ) ) ) )
=> ( A
!= ( one_one @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[585]) ).
thf(781,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ D ) @ E ) ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ D ) @ E ) ) @ F ) ) @ ( zero_zero @ int ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ E ) @ B ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_268_zdiv__mono2__neg__lemma) ).
thf(3750,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ D ) @ E ) ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ D ) @ E ) ) @ F ) ) @ ( zero_zero @ int ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ E ) @ B ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[781]) ).
thf(525,axiom,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_779_add__less__cancel__right) ).
thf(2872,plain,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[525]) ).
thf(968,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( bit0 @ A ) ) @ B )
= ( bit0 @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_mult__Bit0) ).
thf(4385,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( bit0 @ A ) ) @ B )
= ( bit0 @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[968]) ).
thf(904,axiom,
! [A: $i] :
( ( ordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_369_mult__right__mono) ).
thf(4187,plain,
! [A: $i] :
( ( ordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[904]) ).
thf(1239,axiom,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_371_mult__nonpos__nonneg) ).
thf(5341,plain,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1239]) ).
thf(205,axiom,
! [A: $i] :
( ( comm_monoid_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_768_add__0) ).
thf(1787,plain,
! [A: $i] :
( ( comm_monoid_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(1175,axiom,
! [A: $i] :
( ( ordered_ab_group_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_812_less__iff__diff__less__0) ).
thf(5113,plain,
! [A: $i] :
( ( ordered_ab_group_add @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1175]) ).
thf(329,axiom,
! [A: $i] :
( ( ti @ int @ ( twoSqu1929807760sum2sq @ A ) )
= ( twoSqu1929807760sum2sq @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_TwoSquares__Mirabelle__lldtdwpyrp_Osum2sq_res) ).
thf(2181,plain,
! [A: $i] :
( ( ti @ int @ ( twoSqu1929807760sum2sq @ A ) )
= ( twoSqu1929807760sum2sq @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[329]) ).
thf(1124,axiom,
ring_11004092258visors @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oring__1__no__zero__divisors) ).
thf(4942,plain,
ring_11004092258visors @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1124]) ).
thf(371,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ B ) @ A ) )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_725_real__le__antisym) ).
thf(2313,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ B ) @ A ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[371]) ).
thf(785,axiom,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ D ) )
= ( hAPP @ nat @ A @ ( power_power @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_345_power__mult) ).
thf(3763,plain,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ D ) )
= ( hAPP @ nat @ A @ ( power_power @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[785]) ).
thf(411,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ B @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_832_wset__less) ).
thf(2492,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ B @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[411]) ).
thf(880,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_958_pos__mod__conj) ).
thf(4104,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[880]) ).
thf(739,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ C ) )
= ( zero_zero @ A ) )
<=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_sum__squares__eq__zero__iff) ).
thf(3595,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ C ) )
= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) )
& ( ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ C ) )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[739]) ).
thf(858,axiom,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_544_less__not__refl) ).
thf(4027,plain,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[858]) ).
thf(65,axiom,
ring_char_0 @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Int_Oring__char__0) ).
thf(1402,plain,
ring_char_0 @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(1184,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( hAPP @ nat @ nat @ ( div_mod @ nat @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( div_mod @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_977_mod__mult__distrib2) ).
thf(5145,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( hAPP @ nat @ nat @ ( div_mod @ nat @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( div_mod @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1184]) ).
thf(505,axiom,
linordered_semidom @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Olinordered__semidom) ).
thf(2801,plain,
linordered_semidom @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[505]) ).
thf(805,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ pls ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_157_rel__simps_I10_J) ).
thf(3862,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ pls ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ pls ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[805]) ).
thf(54,axiom,
( ( ti @ int @ t )
= t ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_v_t_____res) ).
thf(1375,plain,
( ( ti @ int @ t )
= t ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(670,axiom,
! [A: $i,B: $i,C: $i] :
( ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ C @ D ) ) @ ( hAPP @ nat @ nat @ C @ E ) ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ C @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ C @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_856_mono__nat__linear__lb) ).
thf(3381,plain,
! [A: $i,B: $i,C: $i] :
( ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ C @ D ) ) @ ( hAPP @ nat @ nat @ C @ E ) ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ C @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ C @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[670]) ).
thf(235,axiom,
! [A: $i] :
( ( ( number_ring @ A )
& ( ring_char_0 @ A ) )
=> ! [B: $i,C: $i] :
( ( ( number_number_of @ A @ B )
= ( number_number_of @ A @ C ) )
<=> ( ( ti @ int @ B )
= ( ti @ int @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_eq__number__of) ).
thf(1878,plain,
! [A: $i] :
( ( ( number_ring @ A )
& ( ring_char_0 @ A ) )
=> ! [B: $i,C: $i] :
( ( ( ( number_number_of @ A @ B )
= ( number_number_of @ A @ C ) )
=> ( ( ti @ int @ B )
= ( ti @ int @ C ) ) )
& ( ( ( ti @ int @ B )
= ( ti @ int @ C ) )
=> ( ( number_number_of @ A @ B )
= ( number_number_of @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).
thf(1120,axiom,
! [A: $i] :
( ( ordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_364_mult__mono_H) ).
thf(4930,plain,
! [A: $i] :
( ( ordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1120]) ).
thf(600,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ ( bit1 @ pls ) ) ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_mult__numeral__1) ).
thf(3119,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ ( bit1 @ pls ) ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[600]) ).
thf(1107,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_714_mult__le__cancel2) ).
thf(4883,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ C ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1107]) ).
thf(851,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_430_mult__strict__mono) ).
thf(4012,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[851]) ).
thf(462,axiom,
! [A: $i] :
( ( ( number_number_of @ nat @ A )
= ( zero_zero @ nat ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ pls ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_eq__number__of__0) ).
thf(2638,plain,
! [A: $i] :
( ( ( ( number_number_of @ nat @ A )
= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ pls ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ pls ) )
=> ( ( number_number_of @ nat @ A )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[462]) ).
thf(567,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_561_dvd_Oless__imp__not__less) ).
thf(3001,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[567]) ).
thf(1083,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( minus_minus @ int @ ( bit0 @ A ) ) @ ( bit0 @ B ) )
= ( bit0 @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_472_diff__bin__simps_I7_J) ).
thf(4808,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( minus_minus @ int @ ( bit0 @ A ) ) @ ( bit0 @ B ) )
= ( bit0 @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1083]) ).
thf(47,axiom,
! [A: $i,B: $i] :
( ( ti @ int @ ( multInv @ A @ B ) )
= ( multInv @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Int2_OMultInv_res) ).
thf(1360,plain,
! [A: $i,B: $i] :
( ( ti @ int @ ( multInv @ A @ B ) )
= ( multInv @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(346,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ( ( zero_zero @ A )
= ( number_number_of @ A @ pls ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_146_semiring__norm_I112_J) ).
thf(2245,plain,
! [A: $i] :
( ( number_ring @ A )
=> ( ( zero_zero @ A )
= ( number_number_of @ A @ pls ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).
thf(1198,axiom,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_149_less__number__of) ).
thf(5189,plain,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1198]) ).
thf(369,axiom,
! [A: $i] :
( ( ring_n68954251visors @ A )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C )
= ( zero_zero @ A ) )
<=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_317_mult__eq__0__iff) ).
thf(2300,plain,
! [A: $i] :
( ( ring_n68954251visors @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C )
= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) )
& ( ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
= ( zero_zero @ A ) ) )
=> ( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[369]) ).
thf(413,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ ( bit1 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_rel__simps_I32_J) ).
thf(2498,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ ( bit1 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ ( bit1 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[413]) ).
thf(742,axiom,
! [A: $i] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ! [E: $i,F: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ B ) ) )
<=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ E ) @ ( hAPP @ A @ A @ ( times_times @ A @ F ) @ D ) ) ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_746_inf__period_I4_J) ).
thf(3615,plain,
! [A: $i] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ! [E: $i,F: $i] :
( ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ B ) ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ E ) @ ( hAPP @ A @ A @ ( times_times @ A @ F ) @ D ) ) ) @ B ) ) ) )
& ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ E ) @ ( hAPP @ A @ A @ ( times_times @ A @ F ) @ D ) ) ) @ B ) ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[742]) ).
thf(797,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
=> ( ( C = A )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_571_dvd_Oord__le__eq__trans) ).
thf(3831,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
=> ( ( C = A )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[797]) ).
thf(515,axiom,
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ s ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ s ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ s1 @ s ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_s0p) ).
thf(2840,plain,
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ s ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ s ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ s1 @ s ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[515]) ).
thf(304,axiom,
hBOOL @ ( hAPP @ int @ bool @ zprime @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_p) ).
thf(2115,plain,
hBOOL @ ( hAPP @ int @ bool @ zprime @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).
thf(9,axiom,
! [A: $i,B: $i] :
( ( semiring_div @ B )
=> ( ( div_mod @ B @ ( ti @ B @ A ) )
= ( div_mod @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Divides_Odiv__class_Omod_arg1) ).
thf(1270,plain,
! [A: $i,B: $i] :
( ( semiring_div @ B )
=> ( ( div_mod @ B @ ( ti @ B @ A ) )
= ( div_mod @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(228,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
<=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
| ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_580_dvd_Ole__less) ).
thf(1851,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
| ( A = B ) ) )
& ( ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
| ( A = B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).
thf(200,axiom,
! [A: $i] :
( ( semiri456707255roduct @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ B )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) )
<=> ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_238_add__0__iff) ).
thf(1772,plain,
! [A: $i] :
( ( semiri456707255roduct @ A )
=> ! [B: $i,C: $i] :
( ( ( ( ti @ A @ B )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) )
=> ( ( ti @ A @ C )
= ( zero_zero @ A ) ) )
& ( ( ( ti @ A @ C )
= ( zero_zero @ A ) )
=> ( ( ti @ A @ B )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(601,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ ( bit1 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_rel__simps_I17_J) ).
thf(3122,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ ( bit1 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ ( bit1 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[601]) ).
thf(1211,axiom,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_798_zero__le__double__add__iff__zero__le__single__add) ).
thf(5238,plain,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1211]) ).
thf(1014,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ min ) @ pls ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_298_rel__simps_I23_J) ).
thf(4512,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ min ) @ pls ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1014]) ).
thf(1144,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ min ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ min ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_290_rel__simps_I13_J) ).
thf(5006,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ min ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ min ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ min ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ min ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1144]) ).
thf(1241,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_379_mult__strict__right__mono__neg) ).
thf(5345,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1241]) ).
thf(922,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ min ) @ ( bit1 @ A ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ min ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_296_rel__simps_I26_J) ).
thf(4248,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ min ) @ ( bit1 @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ min ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ min ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ min ) @ ( bit1 @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[922]) ).
thf(1077,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_242_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) ).
thf(4789,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1077]) ).
thf(1023,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ ( bit0 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_rel__simps_I31_J) ).
thf(4544,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ ( bit0 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ ( bit0 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1023]) ).
thf(984,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ C ) )
= ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_474_zdiff__zmult__distrib2) ).
thf(4429,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ C ) )
= ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[984]) ).
thf(479,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( one_one @ int ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_87_le__imp__0__less) ).
thf(2692,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( one_one @ int ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[479]) ).
thf(102,axiom,
one @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Oone) ).
thf(1498,plain,
one @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(148,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ A ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_413_zcong__not) ).
thf(1587,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ A ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(480,axiom,
linordered_semiring @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__semiring) ).
thf(2694,plain,
linordered_semiring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[480]) ).
thf(1018,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( zero_zero @ int ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ A )
= ( ti @ int @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_962_mod__neg__neg__trivial) ).
thf(4525,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( zero_zero @ int ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ A )
= ( ti @ int @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1018]) ).
thf(416,axiom,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ min ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_288_rel__simps_I7_J) ).
thf(2509,plain,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ min ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[416]) ).
thf(174,axiom,
linord1278240602ring_1 @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__semiring__1) ).
thf(1672,plain,
linord1278240602ring_1 @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(124,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ A @ A @ ( div_mod @ A @ D ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ F ) @ E ) )
=> ( ( ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ E ) )
=> ( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ F ) @ C ) ) @ E ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_901_mod__mult__cong) ).
thf(1540,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ A @ A @ ( div_mod @ A @ D ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ F ) @ E ) )
=> ( ( ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ E ) )
=> ( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ F ) @ C ) ) @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(714,axiom,
! [A: $i] :
( ( monoid_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_770_add__0__right) ).
thf(3518,plain,
! [A: $i] :
( ( monoid_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[714]) ).
thf(833,axiom,
! [A: $i] :
~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( one_one @ nat ) ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( one_one @ nat ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_879_gcd__lcm__complete__lattice__nat_Onot__less__bot) ).
thf(3963,plain,
! [A: $i] :
~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( one_one @ nat ) ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( one_one @ nat ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[833]) ).
thf(820,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) ).
thf(3916,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[820]) ).
thf(632,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fconj @ A ) @ B ) )
| ( hBOOL @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fconj_3_1_U) ).
thf(3237,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fconj @ A ) @ B ) )
| ( hBOOL @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[632]) ).
thf(624,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ ( hAPP @ real @ real @ ( plus_plus @ real @ A ) @ B ) ) @ ( hAPP @ real @ real @ ( plus_plus @ real @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_612_real__add__left__mono) ).
thf(3205,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ ( hAPP @ real @ real @ ( plus_plus @ real @ A ) @ B ) ) @ ( hAPP @ real @ real @ ( plus_plus @ real @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[624]) ).
thf(914,axiom,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ C ) @ E ) ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ D ) ) @ E ) ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ C ) @ E ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_884_mult_Oprod__diff__prod) ).
thf(4219,plain,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ C ) @ E ) ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ D ) ) @ E ) ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ C ) @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[914]) ).
thf(584,axiom,
! [A: $i] :
( ( hBOOL @ ( ti @ bool @ A ) )
<=> ( hBOOL @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_hBOOL_arg1) ).
thf(3060,plain,
! [A: $i] :
( ( ( hBOOL @ ( ti @ bool @ A ) )
=> ( hBOOL @ A ) )
& ( ( hBOOL @ A )
=> ( hBOOL @ ( ti @ bool @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[584]) ).
thf(614,axiom,
! [A: $i] :
( ( ( monoid_mult @ A )
& ( number @ A ) )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_167_power2__eq__square__number__of) ).
thf(3176,plain,
! [A: $i] :
( ( ( monoid_mult @ A )
& ( number @ A ) )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[614]) ).
thf(572,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fconj @ A ) @ B ) )
| ( hBOOL @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fconj_2_1_U) ).
thf(3018,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fconj @ A ) @ B ) )
| ( hBOOL @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[572]) ).
thf(1086,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ pls )
= ( ti @ int @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_add__Pls__right) ).
thf(4817,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ pls )
= ( ti @ int @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1086]) ).
thf(926,axiom,
! [A: $i] :
( ( A
!= ( zero_zero @ nat ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( zero_zero @ nat ) ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_872_gcd__lcm__complete__lattice__nat_Oless__top) ).
thf(4263,plain,
! [A: $i] :
( ( ( A
!= ( zero_zero @ nat ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( zero_zero @ nat ) ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( zero_zero @ nat ) ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) )
=> ( A
!= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[926]) ).
thf(526,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_414_zcong__not__zero) ).
thf(2877,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[526]) ).
thf(1192,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ ( number_number_of @ int @ min ) )
= ( zero_zero @ int ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_954_zmod__minus1__right) ).
thf(5166,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ ( number_number_of @ int @ min ) )
= ( zero_zero @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1192]) ).
thf(901,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B )
= ( zero_zero @ nat ) )
<=> ( ( B
!= ( zero_zero @ nat ) )
& ( A
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_523_nat__power__eq__0__iff) ).
thf(4167,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B )
= ( zero_zero @ nat ) )
=> ( ( B
!= ( zero_zero @ nat ) )
& ( A
= ( zero_zero @ nat ) ) ) )
& ( ( ( B
!= ( zero_zero @ nat ) )
& ( A
= ( zero_zero @ nat ) ) )
=> ( ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[901]) ).
thf(776,axiom,
order @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Orderings_Oorder) ).
thf(3728,plain,
order @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[776]) ).
thf(332,axiom,
! [A: $i] :
( ( comm_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_323_comm__semiring__class_Odistrib) ).
thf(2196,plain,
! [A: $i] :
( ( comm_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).
thf(620,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ D ) ) ) @ C ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_968_zmult2__lemma__aux4) ).
thf(3194,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ D ) ) ) @ C ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[620]) ).
thf(761,axiom,
! [A: $i,B: $i] :
( ( A = B )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_581_dvd_Oeq__iff) ).
thf(3671,plain,
! [A: $i,B: $i] :
( ( ( A = B )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[761]) ).
thf(1059,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( ti @ int @ C )
!= ( zero_zero @ int ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ A ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_743_zdvd__mono) ).
thf(4736,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ti @ int @ C )
!= ( zero_zero @ int ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ A ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ B ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ A ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1059]) ).
thf(137,axiom,
semiring_0 @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Osemiring__0) ).
thf(1567,plain,
semiring_0 @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(934,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_427_mult__le__less__imp__less) ).
thf(4291,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[934]) ).
thf(950,axiom,
ring_div @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Divides_Oring__div) ).
thf(4333,plain,
ring_div @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[950]) ).
thf(34,axiom,
! [A: $i,B: $i] :
( ( ti @ A @ ( ti @ A @ B ) )
= ( ti @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_ti_idem) ).
thf(1329,plain,
! [A: $i,B: $i] :
( ( ti @ A @ ( ti @ A @ B ) )
= ( ti @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(249,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( zero_zero @ A ) ) @ B )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_933_mod__0) ).
thf(1942,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( zero_zero @ A ) ) @ B )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(170,axiom,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_533_less__zeroE) ).
thf(1659,plain,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(375,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
<=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_393_mult__less__cancel__right__disj) ).
thf(2323,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ B ) ) ) ) )
& ( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ B ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[375]) ).
thf(615,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_494_zle__diff1__eq) ).
thf(3179,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[615]) ).
thf(980,axiom,
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( one_one @ int ) )
= ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ t ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_t) ).
thf(4421,plain,
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( one_one @ int ) )
= ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ t ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[980]) ).
thf(73,axiom,
! [A: $i,B: $i] :
( ( legendre @ ( ti @ int @ A ) @ B )
= ( legendre @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Residues_OLegendre_arg1) ).
thf(1424,plain,
! [A: $i,B: $i] :
( ( legendre @ ( ti @ int @ A ) @ B )
= ( legendre @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(571,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
<=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
& ( ( ti @ int @ A )
!= ( ti @ int @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_zless__le) ).
thf(3010,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
& ( ( ti @ int @ A )
!= ( ti @ int @ B ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
& ( ( ti @ int @ A )
!= ( ti @ int @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[571]) ).
thf(975,axiom,
! [A: $i] :
~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_874_gcd__lcm__complete__lattice__nat_Onot__top__less) ).
thf(4406,plain,
! [A: $i] :
~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[975]) ).
thf(146,axiom,
linord893533164strict @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__comm__semiring__strict) ).
thf(1584,plain,
linord893533164strict @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(1015,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( multInv @ D @ B ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( multInv @ D @ C ) ) ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_840_MultInv__zcong__prop1) ).
thf(4513,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( multInv @ D @ B ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( multInv @ D @ C ) ) ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1015]) ).
thf(1062,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ E ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ D ) @ B ) ) @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_476_zcong__zdiff) ).
thf(4744,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ E ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ D ) @ B ) ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1062]) ).
thf(454,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) )
| ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ B ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_723_real__le__linear) ).
thf(2613,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) )
| ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[454]) ).
thf(248,axiom,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ ( number_number_of @ A @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ pls ) ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_180_less__special_I2_J) ).
thf(1937,plain,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ ( number_number_of @ A @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ pls ) ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ pls ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ ( number_number_of @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).
thf(215,axiom,
! [A: $i] :
( ( A
!= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_618_gr0I) ).
thf(1810,plain,
! [A: $i] :
( ( A
!= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(1100,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_378_mult__strict__left__mono__neg) ).
thf(4861,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1100]) ).
thf(868,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_700_mult__less__cancel2) ).
thf(4054,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ C ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[868]) ).
thf(726,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_473_zdiff__zmult__distrib) ).
thf(3555,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[726]) ).
thf(1085,axiom,
! [A: $i,B: $i] :
( ( ti @ ( fun @ A @ bool ) @ ( collect @ A @ B ) )
= ( collect @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Set_OCollect_res) ).
thf(4814,plain,
! [A: $i,B: $i] :
( ( ti @ ( fun @ A @ bool ) @ ( collect @ A @ B ) )
= ( collect @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1085]) ).
thf(559,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( minus_minus @ int @ pls ) @ ( bit1 @ A ) )
= ( bit1 @ ( hAPP @ int @ int @ ( minus_minus @ int @ min ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_495_diff__bin__simps_I4_J) ).
thf(2978,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( minus_minus @ int @ pls ) @ ( bit1 @ A ) )
= ( bit1 @ ( hAPP @ int @ int @ ( minus_minus @ int @ min ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[559]) ).
thf(407,axiom,
! [A: $i] :
( ( no_zero_divisors @ A )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C )
= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_315_divisors__zero) ).
thf(2472,plain,
! [A: $i] :
( ( no_zero_divisors @ A )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C )
= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[407]) ).
thf(94,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_932_mod__by__0) ).
thf(1478,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(22,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_212_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) ).
thf(1297,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(123,axiom,
linord581940658strict @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Olinordered__ring__strict) ).
thf(1539,plain,
linord581940658strict @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(259,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ pls ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_rel__simps_I29_J) ).
thf(1974,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ pls ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ pls ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).
thf(405,axiom,
! [A: $i,B: $i] :
( ( bit1 @ A )
!= ( bit0 @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_rel__simps_I50_J) ).
thf(2465,plain,
! [A: $i,B: $i] :
( ( bit1 @ A )
!= ( bit0 @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[405]) ).
thf(210,axiom,
! [A: $i,B: $i] :
( ( ab_semigroup_add @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( plus_plus @ B @ A ) )
= ( plus_plus @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Oplus__class_Oplus_3_res) ).
thf(1799,plain,
! [A: $i,B: $i] :
( ( ab_semigroup_add @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( plus_plus @ B @ A ) )
= ( plus_plus @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).
thf(251,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ D )
= ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) ).
thf(1947,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ D )
= ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).
thf(262,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_657_dvd__diffD1) ).
thf(1984,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).
thf(151,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( combs @ A @ B @ C @ D @ ( ti @ ( fun @ A @ B ) @ E ) )
= ( combs @ A @ B @ C @ D @ E ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_COMBS_arg2) ).
thf(1599,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( combs @ A @ B @ C @ D @ ( ti @ ( fun @ A @ B ) @ E ) )
= ( combs @ A @ B @ C @ D @ E ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(1130,axiom,
! [A: $i] :
( ( sr @ A )
= ( collect @ int @ ( combs @ int @ bool @ bool @ ( combb @ bool @ ( fun @ bool @ bool ) @ int @ fconj @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) ) @ ( combc @ int @ int @ bool @ ( ord_less @ int ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_885_SR__def) ).
thf(4953,plain,
! [A: $i] :
( ( sr @ A )
= ( collect @ int @ ( combs @ int @ bool @ bool @ ( combb @ bool @ ( fun @ bool @ bool ) @ int @ fconj @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) ) @ ( combc @ int @ int @ bool @ ( ord_less @ int ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1130]) ).
thf(876,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( number_number_of @ nat @ A ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_less__0__number__of) ).
thf(4086,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( number_number_of @ nat @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( number_number_of @ nat @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[876]) ).
thf(61,axiom,
semiring @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Osemiring) ).
thf(1393,plain,
semiring @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(769,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_263_q__pos__lemma) ).
thf(3703,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[769]) ).
thf(1161,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
& ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C )
= B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_863_less__imp__add__positive) ).
thf(5063,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
& ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1161]) ).
thf(362,axiom,
( ( number_number_of @ int @ pls )
!= ( number_number_of @ int @ min ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_301_eq__number__of__Pls__Min) ).
thf(2285,plain,
( ( number_number_of @ int @ pls )
!= ( number_number_of @ int @ min ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[362]) ).
thf(1232,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ ( bit0 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_126_less__int__code_I13_J) ).
thf(5317,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ ( bit0 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ ( bit0 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1232]) ).
thf(1221,axiom,
! [A: $i] :
( ( comm_monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( one_one @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_793_mult__1) ).
thf(5273,plain,
! [A: $i] :
( ( comm_monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( one_one @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1221]) ).
thf(6,axiom,
cancel146912293up_add @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Ocancel__ab__semigroup__add) ).
thf(1263,plain,
cancel146912293up_add @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(701,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ A ) ) @ ( one_one @ int ) ) ) )
=> ( ( legendre @ ( number_number_of @ int @ min ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ A ) ) @ ( one_one @ int ) ) )
= ( one_one @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_507_Legendre__1mod4) ).
thf(3486,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ A ) ) @ ( one_one @ int ) ) ) )
=> ( ( legendre @ ( number_number_of @ int @ min ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ A ) ) @ ( one_one @ int ) ) )
= ( one_one @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[701]) ).
thf(309,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ D ) ) @ E ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ E ) @ ( zero_zero @ int ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_267_unique__quotient__lemma__neg) ).
thf(2125,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ D ) ) @ E ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ E ) @ ( zero_zero @ int ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).
thf(755,axiom,
hBOOL @ ( hAPP @ int @ bool @ twoSqu658283162sum2sq @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ t ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_qf1pt) ).
thf(3654,plain,
hBOOL @ ( hAPP @ int @ bool @ twoSqu658283162sum2sq @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ t ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[755]) ).
thf(872,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_435_zero__less__two) ).
thf(4068,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[872]) ).
thf(320,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_639_nat__add__left__cancel__less) ).
thf(2153,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).
thf(793,axiom,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_372_mult__nonneg__nonpos2) ).
thf(3819,plain,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[793]) ).
thf(917,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( one_one @ A ) ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_338_one__dvd) ).
thf(4237,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( one_one @ A ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[917]) ).
thf(628,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ D ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ D ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ E ) ) ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_zdvd__period) ).
thf(3222,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ D ) @ E ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ D ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ D ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ E ) ) ) @ C ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ D ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ E ) ) ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ D ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[628]) ).
thf(1122,axiom,
! [A: $i] :
( ( number_number_of @ int @ A )
= ( ti @ int @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_number__of__is__id) ).
thf(4934,plain,
! [A: $i] :
( ( number_number_of @ int @ A )
= ( ti @ int @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1122]) ).
thf(722,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) ).
thf(3540,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[722]) ).
thf(746,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ B ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_955_zmod__zdvd__zmod) ).
thf(3628,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ B ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[746]) ).
thf(598,axiom,
! [A: $i] :
( ( number_semiring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ B ) )
=> ( ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ C ) ) @ ( number_number_of @ A @ B ) )
= ( number_number_of @ A @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_semiring__mult__number__of) ).
thf(3115,plain,
! [A: $i] :
( ( number_semiring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ B ) )
=> ( ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ C ) ) @ ( number_number_of @ A @ B ) )
= ( number_number_of @ A @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[598]) ).
thf(75,axiom,
! [A: $i,B: $i] :
( ( ti @ int @ ( standardRes @ A @ B ) )
= ( standardRes @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Residues_OStandardRes_res) ).
thf(1429,plain,
! [A: $i,B: $i] :
( ( ti @ int @ ( standardRes @ A @ B ) )
= ( standardRes @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(835,axiom,
! [A: $i] :
( ( dvd @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( dvd_dvd @ A ) )
= ( dvd_dvd @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Rings_Odvd__class_Odvd_res) ).
thf(3967,plain,
! [A: $i] :
( ( dvd @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( dvd_dvd @ A ) )
= ( dvd_dvd @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[835]) ).
thf(890,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_817_add__neg__nonpos) ).
thf(4135,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[890]) ).
thf(895,axiom,
! [A: $i] :
( ( twoSqu1929807760sum2sq @ ( ti @ ( product_prod @ int @ int ) @ A ) )
= ( twoSqu1929807760sum2sq @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_TwoSquares__Mirabelle__lldtdwpyrp_Osum2sq_arg1) ).
thf(4152,plain,
! [A: $i] :
( ( twoSqu1929807760sum2sq @ ( ti @ ( product_prod @ int @ int ) @ A ) )
= ( twoSqu1929807760sum2sq @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[895]) ).
thf(185,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_259_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) ).
thf(1720,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(596,axiom,
! [A: $i] :
( ( power @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_411_power__0) ).
thf(3109,plain,
! [A: $i] :
( ( power @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[596]) ).
thf(260,axiom,
! [A: $i] :
( ( linordered_ring @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_sum__squares__ge__zero) ).
thf(1979,plain,
! [A: $i] :
( ( linordered_ring @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).
thf(586,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ B ) ) )
=> ( ( C = A )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_557_dvd_Oord__less__eq__trans) ).
thf(3074,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ B ) ) )
=> ( ( C = A )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[586]) ).
thf(159,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_653_diff__less__mono) ).
thf(1627,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(383,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B )
= ( one_one @ nat ) )
<=> ( ( A
= ( one_one @ nat ) )
| ( B
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_526_exp__eq__1) ).
thf(2370,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B )
= ( one_one @ nat ) )
=> ( ( A
= ( one_one @ nat ) )
| ( B
= ( zero_zero @ nat ) ) ) )
& ( ( ( A
= ( one_one @ nat ) )
| ( B
= ( zero_zero @ nat ) ) )
=> ( ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B )
= ( one_one @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).
thf(117,axiom,
! [A: $i,B: $i] :
( ( no_zero_divisors @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( times_times @ B @ A ) )
= ( times_times @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_5_res) ).
thf(1525,plain,
! [A: $i,B: $i] :
( ( no_zero_divisors @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( times_times @ B @ A ) )
= ( times_times @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(783,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( B
= ( zero_zero @ nat ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_520_divides__ge) ).
thf(3755,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( B
= ( zero_zero @ nat ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[783]) ).
thf(629,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B )
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_629_diff__is__0__eq_H) ).
thf(3225,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B )
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[629]) ).
thf(428,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ ( hAPP @ real @ real @ ( times_times @ real @ C ) @ A ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ C ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_704_real__mult__le__cancel__iff2) ).
thf(2546,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ ( hAPP @ real @ real @ ( times_times @ real @ C ) @ A ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ ( hAPP @ real @ real @ ( times_times @ real @ C ) @ A ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[428]) ).
thf(201,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( standardRes @ B @ A ) ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_990_StandardRes__prop1) ).
thf(1779,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( standardRes @ B @ A ) ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).
thf(875,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ ( bit1 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_rel__simps_I34_J) ).
thf(4081,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ ( bit1 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ ( bit1 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[875]) ).
thf(737,axiom,
pls != min,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_284_rel__simps_I37_J) ).
thf(3587,plain,
pls != min,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[737]) ).
thf(135,axiom,
! [A: $i,B: $i] :
( ( inv @ ( ti @ int @ A ) @ B )
= ( inv @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_WilsonRuss_Oinv_arg1) ).
thf(1563,plain,
! [A: $i,B: $i] :
( ( inv @ ( ti @ int @ A ) @ B )
= ( inv @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(556,axiom,
! [A: $i] :
( ( linord626643107strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B )
= ( one_one @ A ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ E ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) @ F ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_276_convex__bound__lt) ).
thf(2963,plain,
! [A: $i] :
( ( linord626643107strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B )
= ( one_one @ A ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ E ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) @ F ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[556]) ).
thf(90,axiom,
! [A: $i,B: $i] :
( ( member @ A @ ( ti @ A @ B ) )
= ( member @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_member_arg1) ).
thf(1466,plain,
! [A: $i,B: $i] :
( ( member @ A @ ( ti @ A @ B ) )
= ( member @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(990,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_515_divides__mul__l) ).
thf(4448,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[990]) ).
thf(391,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_855_int__gr__induct) ).
thf(2408,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[391]) ).
thf(862,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_250_zero__le__even__power_H) ).
thf(4036,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[862]) ).
thf(237,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B )
= ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_zmult__commute) ).
thf(1895,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B )
= ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).
thf(463,axiom,
! [A: $i] :
( ( linordered_ring @ A )
=> ! [B: $i,C: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ C ) ) ) @ ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_not__sum__squares__lt__zero) ).
thf(2645,plain,
! [A: $i] :
( ( linordered_ring @ A )
=> ! [B: $i,C: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ C ) ) ) @ ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[463]) ).
thf(421,axiom,
ordere453448008miring @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oordered__cancel__semiring) ).
thf(2529,plain,
ordere453448008miring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[421]) ).
thf(192,axiom,
! [A: $i] :
( ( ( number @ A )
& ( linorder @ A ) )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) )
<=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( number_number_of @ A @ C ) ) @ ( number_number_of @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50_le__number__of__eq__not__less) ).
thf(1732,plain,
! [A: $i] :
( ( ( number @ A )
& ( linorder @ A ) )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( number_number_of @ A @ C ) ) @ ( number_number_of @ A @ B ) ) ) )
& ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( number_number_of @ A @ C ) ) @ ( number_number_of @ A @ B ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(539,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( zero_zero @ A ) )
<=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_sum__power2__eq__zero__iff) ).
thf(2909,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) )
& ( ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[539]) ).
thf(899,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_922_dvd__mod__imp__dvd) ).
thf(4163,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[899]) ).
thf(693,axiom,
( ( ti @ int @ s1 )
= s1 ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_v_s1_____res) ).
thf(3454,plain,
( ( ti @ int @ s1 )
= s1 ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[693]) ).
thf(698,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B )
= A )
<=> ( C
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_676_le__imp__diff__is__add) ).
thf(3476,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B )
= A )
=> ( C
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) )
& ( ( C
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B )
= A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[698]) ).
thf(36,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_903_mod__mult__mult2) ).
thf(1333,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(894,axiom,
! [A: $i] :
( min
!= ( bit0 @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_287_rel__simps_I42_J) ).
thf(4148,plain,
! [A: $i] :
( min
!= ( bit0 @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[894]) ).
thf(495,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_807_add__pos__pos) ).
thf(2749,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[495]) ).
thf(78,axiom,
number_semiring @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Int_Onumber__semiring) ).
thf(1434,plain,
number_semiring @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(789,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( one_one @ nat ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_226_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) ).
thf(3802,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( one_one @ nat ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[789]) ).
thf(1229,axiom,
( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) ) @ ( one_one @ nat ) )
= ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_nat__1__add__1) ).
thf(5310,plain,
( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) ) @ ( one_one @ nat ) )
= ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1229]) ).
thf(1187,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( number_number_of @ A @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_141_mult__2__right) ).
thf(5153,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( number_number_of @ A @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1187]) ).
thf(529,axiom,
plus @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oplus) ).
thf(2883,plain,
plus @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[529]) ).
thf(613,axiom,
! [A: $i] :
( ( number_semiring @ A )
=> ( ( number_number_of @ A @ pls )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_144_semiring__numeral__0__eq__0) ).
thf(3173,plain,
! [A: $i] :
( ( number_semiring @ A )
=> ( ( number_number_of @ A @ pls )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[613]) ).
thf(1000,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_556_dvd_Oless__le__trans) ).
thf(4476,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1000]) ).
thf(288,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ ( zero_zero @ int ) ) @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( multInv @ D @ C ) ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( multInv @ D @ B ) ) ) @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_846_MultInv__zcong__prop2) ).
thf(2057,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ ( zero_zero @ int ) ) @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( multInv @ D @ C ) ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( multInv @ D @ B ) ) ) @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).
thf(648,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) )
<=> ( C = A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_603_eq__diff__iff) ).
thf(3306,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( ( ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) )
=> ( C = A ) )
& ( ( C = A )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[648]) ).
thf(226,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) )
= ( number_number_of @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_151_add__number__of__eq) ).
thf(1846,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) )
= ( number_number_of @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).
thf(847,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ C ) ) @ D ) ) @ C )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_924_zmod__simps_I1_J) ).
thf(4004,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ C ) ) @ D ) ) @ C )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[847]) ).
thf(378,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) @ ( zero_zero @ int ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_960_neg__mod__sign) ).
thf(2338,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) @ ( zero_zero @ int ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[378]) ).
thf(1116,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_zle__linear) ).
thf(4921,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1116]) ).
thf(544,axiom,
! [A: $i] :
( ( bit1 @ A )
!= pls ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_rel__simps_I46_J) ).
thf(2936,plain,
! [A: $i] :
( ( bit1 @ A )
!= pls ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[544]) ).
thf(388,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C ) )
=> ~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_668_add__leE) ).
thf(2391,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C ) )
=> ~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).
thf(432,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_698_nat__0__less__mult__iff) ).
thf(2556,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[432]) ).
thf(349,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ ( number_number_of @ int @ min ) ) @ A )
= ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( one_one @ int ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_972_zmod__minus1) ).
thf(2253,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ ( number_number_of @ int @ min ) ) @ A )
= ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( one_one @ int ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[349]) ).
thf(1248,axiom,
( ( ti @ ( fun @ int @ bool ) @ zprime )
= zprime ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_IntPrimes_Ozprime_res) ).
thf(5359,plain,
( ( ti @ ( fun @ int @ bool ) @ zprime )
= zprime ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1248]) ).
thf(703,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_712_dvd__mult__cancel) ).
thf(3491,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[703]) ).
thf(10,axiom,
ordere1490568538miring @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Oordered__comm__semiring) ).
thf(1273,plain,
ordere1490568538miring @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(246,axiom,
! [A: $i] :
( ( ( semiring_div @ A )
& ( number @ A ) )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) )
<=> ( ( hAPP @ A @ A @ ( div_mod @ A @ ( number_number_of @ A @ C ) ) @ ( number_number_of @ A @ B ) )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_900_dvd__eq__mod__eq__0__number__of) ).
thf(1927,plain,
! [A: $i] :
( ( ( semiring_div @ A )
& ( number @ A ) )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) )
=> ( ( hAPP @ A @ A @ ( div_mod @ A @ ( number_number_of @ A @ C ) ) @ ( number_number_of @ A @ B ) )
= ( zero_zero @ A ) ) )
& ( ( ( hAPP @ A @ A @ ( div_mod @ A @ ( number_number_of @ A @ C ) ) @ ( number_number_of @ A @ B ) )
= ( zero_zero @ A ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).
thf(637,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ A ) )
=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ B ) @ ( d22set @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_861_d22set__mem) ).
thf(3262,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ A ) )
=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ B ) @ ( d22set @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[637]) ).
thf(1079,axiom,
! [A: $i,B: $i] :
( ( cancel_semigroup_add @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( plus_plus @ B @ A ) )
= ( plus_plus @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Oplus__class_Oplus_2_res) ).
thf(4797,plain,
! [A: $i,B: $i] :
( ( cancel_semigroup_add @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( plus_plus @ B @ A ) )
= ( plus_plus @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1079]) ).
thf(1011,axiom,
! [A: $i] :
( ( hAPP @ real @ real @ ( times_times @ real @ ( one_one @ real ) ) @ A )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_595_real__mult__1) ).
thf(4503,plain,
! [A: $i] :
( ( hAPP @ real @ real @ ( times_times @ real @ ( one_one @ real ) ) @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1011]) ).
thf(1171,axiom,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) )
<=> ( ( ti @ A @ B )
= ( ti @ A @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_761_add__right__cancel) ).
thf(5095,plain,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ D ) ) )
& ( ( ( ti @ A @ B )
= ( ti @ A @ D ) )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1171]) ).
thf(853,axiom,
cancel146912293up_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add) ).
thf(4017,plain,
cancel146912293up_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[853]) ).
thf(161,axiom,
cancel_semigroup_add @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Ocancel__semigroup__add) ).
thf(1631,plain,
cancel_semigroup_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(307,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_532_dvd_Oorder__refl) ).
thf(2121,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).
thf(345,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) )
<=> ( ( A = C )
| ( B
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_634_mult__cancel2) ).
thf(2235,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) )
=> ( ( A = C )
| ( B
= ( zero_zero @ nat ) ) ) )
& ( ( ( A = C )
| ( B
= ( zero_zero @ nat ) ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[345]) ).
thf(316,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_229_nat__mult__2) ).
thf(2144,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).
thf(771,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) )
<=> ( ( B = C )
| ( A
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_633_mult__cancel1) ).
thf(3707,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) )
=> ( ( B = C )
| ( A
= ( zero_zero @ nat ) ) ) )
& ( ( ( B = C )
| ( A
= ( zero_zero @ nat ) ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[771]) ).
thf(189,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( one_one @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_499_power__dvd__imp__le) ).
thf(1727,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( one_one @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(1115,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( inv @ B @ A ) ) @ ( one_one @ int ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_826_inv__is__inv) ).
thf(4919,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( inv @ B @ A ) ) @ ( one_one @ int ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1115]) ).
thf(603,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) ) )
<=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( zero_zero @ nat ) ) ) )
& ! [D: $i] :
( ( B
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_697_nat__diff__split) ).
thf(3130,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( zero_zero @ nat ) ) ) )
& ! [D: $i] :
( ( B
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ D ) ) ) ) )
& ( ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( zero_zero @ nat ) ) ) )
& ! [D: $i] :
( ( B
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ D ) ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[603]) ).
thf(475,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_724_real__le__trans) ).
thf(2677,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[475]) ).
thf(1050,axiom,
ring_n68954251visors @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oring__no__zero__divisors) ).
thf(4708,plain,
ring_n68954251visors @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1050]) ).
thf(1044,axiom,
! [A: $i] :
( ( no_zero_divisors @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ C )
!= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ B )
!= ( zero_zero @ A ) )
=> ( ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B )
!= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_316_no__zero__divisors) ).
thf(4685,plain,
! [A: $i] :
( ( no_zero_divisors @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ C )
!= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ B )
!= ( zero_zero @ A ) )
=> ( ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B )
!= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1044]) ).
thf(198,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_611_diff__mult__distrib) ).
thf(1754,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(365,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_power2__le__imp__le) ).
thf(2292,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[365]) ).
thf(635,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( quadRes @ C ) @ B ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( multInv @ C @ A ) ) ) @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_835_MultInvPair__distinct) ).
thf(3257,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( quadRes @ C ) @ B ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( multInv @ C @ A ) ) ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[635]) ).
thf(489,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_cube__square) ).
thf(2717,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[489]) ).
thf(924,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) ) ) @ ( zero_zero @ A ) ) )
=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_233_even__power__le__0__imp__0) ).
thf(4256,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) ) ) @ ( zero_zero @ A ) ) )
=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[924]) ).
thf(196,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_650_less__imp__le__nat) ).
thf(1747,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(24,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( one_one @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_244_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) ).
thf(1301,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( one_one @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(492,axiom,
! [A: $i] :
( ( ti @ int @ ( bit0 @ A ) )
= ( bit0 @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Int_OBit0_res) ).
thf(2734,plain,
! [A: $i] :
( ( ti @ int @ ( bit0 @ A ) )
= ( bit0 @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[492]) ).
thf(535,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B )
= ( zero_zero @ int ) )
=> ? [C: $i] :
( ( ti @ int @ A )
= ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_974_zmod__eq__0D) ).
thf(2894,plain,
! [A: $i,B: $i] :
( ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B )
= ( zero_zero @ int ) )
=> ? [C: $i] :
( ( ti @ int @ A )
= ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[535]) ).
thf(962,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_441_power__gt1__lemma) ).
thf(4365,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[962]) ).
thf(674,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_248_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) ).
thf(3396,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[674]) ).
thf(992,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_p0) ).
thf(4453,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[992]) ).
thf(657,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( number_number_of @ int @ ( bit1 @ ( bit1 @ pls ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_262_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J) ).
thf(3343,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( number_number_of @ int @ ( bit1 @ ( bit1 @ pls ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[657]) ).
thf(353,axiom,
ordere236663937imp_le @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le) ).
thf(2260,plain,
ordere236663937imp_le @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[353]) ).
thf(221,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_754_norR__mem__unique__aux) ).
thf(1833,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).
thf(283,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_256_pos__zmult__pos) ).
thf(2048,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).
thf(467,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_395_pos__add__strict) ).
thf(2657,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[467]) ).
thf(588,axiom,
! [A: $i] :
( ( semiring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) @ E ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) @ C ) ) @ E ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_322_combine__common__factor) ).
thf(3086,plain,
! [A: $i] :
( ( semiring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) @ E ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) @ C ) ) @ E ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[588]) ).
thf(887,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ A ) ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_551_less__imp__diff__less) ).
thf(4122,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ A ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[887]) ).
thf(740,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_428_mult__less__le__imp__less) ).
thf(3604,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[740]) ).
thf(255,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ A ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_420_zprime__zdvd__zmult__better) ).
thf(1965,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ A ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).
thf(1126,axiom,
dvd @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Odvd) ).
thf(4946,plain,
dvd @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1126]) ).
thf(313,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ ( bit0 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_155_rel__simps_I16_J) ).
thf(2134,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ ( bit0 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ ( bit0 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).
thf(1076,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ A ) )
=> ( ( ti @ int @ B )
= ( zero_zero @ int ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_447_Int2_Ozcong__zero) ).
thf(4786,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ A ) )
=> ( ( ti @ int @ B )
= ( zero_zero @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1076]) ).
thf(1125,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_870_gcd__lcm__complete__lattice__nat_Otop__le) ).
thf(4943,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1125]) ).
thf(1141,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_275_power__strict__mono) ).
thf(4998,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1141]) ).
thf(617,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_429_mult__strict__mono_H) ).
thf(3187,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[617]) ).
thf(954,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ( ( number_number_of @ A @ ( bit1 @ pls ) )
= ( one_one @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_numeral__1__eq__1) ).
thf(4341,plain,
! [A: $i] :
( ( number_ring @ A )
=> ( ( number_number_of @ A @ ( bit1 @ pls ) )
= ( one_one @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[954]) ).
thf(491,axiom,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B )
= ( zero_zero @ A ) )
<=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_114_double__eq__0__iff) ).
thf(2727,plain,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( ( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B )
= ( zero_zero @ A ) )
=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) )
& ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[491]) ).
thf(931,axiom,
! [A: $i] :
( ( mult_zero @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_318_mult__zero__right) ).
thf(4283,plain,
! [A: $i] :
( ( mult_zero @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[931]) ).
thf(155,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ( ( one_one @ A )
= ( number_number_of @ A @ ( bit1 @ pls ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_semiring__norm_I110_J) ).
thf(1615,plain,
! [A: $i] :
( ( number_ring @ A )
=> ( ( one_one @ A )
= ( number_number_of @ A @ ( bit1 @ pls ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(874,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
| ( ( ti @ A @ B )
= ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_458_dvd__power) ).
thf(4077,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
| ( ( ti @ A @ B )
= ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[874]) ).
thf(560,axiom,
ring @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oring) ).
thf(2981,plain,
ring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[560]) ).
thf(1162,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ pls ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_293_rel__simps_I6_J) ).
thf(5067,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ pls ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1162]) ).
thf(941,axiom,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_781_add__strict__right__mono) ).
thf(4309,plain,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[941]) ).
thf(157,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ ( bit1 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_74_less__int__code_I14_J) ).
thf(1620,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ ( bit1 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ ( bit1 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(1177,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ C ) ) @ D ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_mult__number__of__left) ).
thf(5121,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ C ) ) @ D ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1177]) ).
thf(587,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ ( bit0 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_less__eq__int__code_I15_J) ).
thf(3081,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ ( bit0 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ ( bit0 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[587]) ).
thf(784,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) )
<=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ ( hAPP @ real @ real @ ( minus_minus @ real @ A ) @ B ) ) @ ( zero_zero @ real ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_541_real__le__eq__diff) ).
thf(3758,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ ( hAPP @ real @ real @ ( minus_minus @ real @ A ) @ B ) ) @ ( zero_zero @ real ) ) ) )
& ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ ( hAPP @ real @ real @ ( minus_minus @ real @ A ) @ B ) ) @ ( zero_zero @ real ) ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[784]) ).
thf(644,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_695_add__gr__0) ).
thf(3283,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[644]) ).
thf(273,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( one_one @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_404_less__add__one) ).
thf(2016,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( one_one @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).
thf(418,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ A )
= B ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_589_diff__add__inverse) ).
thf(2514,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ A )
= B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[418]) ).
thf(860,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_641_trans__less__add2) ).
thf(4032,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[860]) ).
thf(1051,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_438_power__le__imp__le__exp) ).
thf(4709,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1051]) ).
thf(1040,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_681_add__mult__distrib) ).
thf(4605,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1040]) ).
thf(121,axiom,
ordere779506340up_add @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add) ).
thf(1535,plain,
ordere779506340up_add @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(564,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ min ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ min ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_297_rel__simps_I30_J) ).
thf(2991,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ min ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ min ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ min ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ min ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[564]) ).
thf(66,axiom,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_897_mult__left_Odiff) ).
thf(1403,plain,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(695,axiom,
! [A: $i] :
( ( semiri456707255roduct @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ E ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
<=> ( ( ( ti @ A @ B )
= ( ti @ A @ D ) )
| ( ( ti @ A @ C )
= ( ti @ A @ E ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_243_crossproduct__eq) ).
thf(3458,plain,
! [A: $i] :
( ( semiri456707255roduct @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ E ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
=> ( ( ( ti @ A @ B )
= ( ti @ A @ D ) )
| ( ( ti @ A @ C )
= ( ti @ A @ E ) ) ) )
& ( ( ( ( ti @ A @ B )
= ( ti @ A @ D ) )
| ( ( ti @ A @ C )
= ( ti @ A @ E ) ) )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ E ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[695]) ).
thf(738,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
!= ( zero_zero @ nat ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ C ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_868_pow__divides__eq__int) ).
thf(3590,plain,
! [A: $i,B: $i,C: $i] :
( ( C
!= ( zero_zero @ nat ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ C ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ C ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[738]) ).
thf(947,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ ( multInv @ B @ A ) ) @ A ) @ ( one_one @ int ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_850_MultInv__prop2a) ).
thf(4328,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ ( multInv @ B @ A ) ) @ A ) @ ( one_one @ int ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[947]) ).
thf(1035,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C ) ) @ ( zero_zero @ int ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( zero_zero @ int ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_264_q__neg__lemma) ).
thf(4590,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C ) ) @ ( zero_zero @ int ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( zero_zero @ int ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1035]) ).
thf(1135,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_zdvd__not__zless) ).
thf(4975,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1135]) ).
thf(114,axiom,
! [A: $i] :
( ( ( number @ A )
& ( ring @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ ( number_number_of @ A @ D ) )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( number_number_of @ A @ D ) ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ ( number_number_of @ A @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_481_left__diff__distrib__number__of) ).
thf(1520,plain,
! [A: $i] :
( ( ( number @ A )
& ( ring @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ ( number_number_of @ A @ D ) )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( number_number_of @ A @ D ) ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ ( number_number_of @ A @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(509,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( zero_zero @ A ) ) )
<=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_105_sum__power2__le__zero__iff) ).
thf(2817,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( zero_zero @ A ) ) )
=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) )
& ( ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[509]) ).
thf(1066,axiom,
ordered_ring @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oordered__ring) ).
thf(4758,plain,
ordered_ring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1066]) ).
thf(1072,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( multInv @ D @ C ) ) @ C ) ) @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( multInv @ D @ C ) ) @ B ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( multInv @ D @ C ) ) ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_844_aux______4) ).
thf(4769,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( multInv @ D @ C ) ) @ C ) ) @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( multInv @ D @ C ) ) @ B ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( multInv @ D @ C ) ) ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1072]) ).
thf(481,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ A @ A @ ( div_mod @ A @ D ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ F ) @ E ) )
=> ( ( ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ E ) )
=> ( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ B ) ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ F ) @ C ) ) @ E ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_923_mod__add__cong) ).
thf(2695,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ A @ A @ ( div_mod @ A @ D ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ F ) @ E ) )
=> ( ( ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ E ) )
=> ( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ B ) ) @ E )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ F ) @ C ) ) @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[481]) ).
thf(1150,axiom,
! [A: $i,B: $i] :
( ( ( bit0 @ A )
= ( bit0 @ B ) )
<=> ( ( ti @ int @ A )
= ( ti @ int @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_rel__simps_I48_J) ).
thf(5022,plain,
! [A: $i,B: $i] :
( ( ( ( bit0 @ A )
= ( bit0 @ B ) )
=> ( ( ti @ int @ A )
= ( ti @ int @ B ) ) )
& ( ( ( ti @ int @ A )
= ( ti @ int @ B ) )
=> ( ( bit0 @ A )
= ( bit0 @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1150]) ).
thf(98,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( times_times @ A @ D ) @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_213_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) ).
thf(1488,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( times_times @ A @ D ) @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(322,axiom,
! [A: $i] :
( ( comm_ring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ D ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_470_dvd__diff) ).
thf(2161,plain,
! [A: $i] :
( ( comm_ring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ D ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[322]) ).
thf(787,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( bit1 @ A ) ) @ B )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( bit0 @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_112_mult__Bit1) ).
thf(3796,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( bit1 @ A ) ) @ B )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( bit0 @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[787]) ).
thf(753,axiom,
comm_semiring_1 @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Ocomm__semiring__1) ).
thf(3651,plain,
comm_semiring_1 @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[753]) ).
thf(400,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( number_number_of @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_152_number__of__add) ).
thf(2442,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( number_number_of @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[400]) ).
thf(324,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_584_nat__add__left__commute) ).
thf(2166,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[324]) ).
thf(3,axiom,
! [A: $i] :
( ( ( number @ A )
& ( ring @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_480_right__diff__distrib__number__of) ).
thf(1258,plain,
! [A: $i] :
( ( ( number @ A )
& ( ring @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(647,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B )
= ( one_one @ int ) )
=> ( ( ( ti @ int @ A )
= ( one_one @ int ) )
| ( ( ti @ int @ A )
= ( number_number_of @ int @ min ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_305_pos__zmult__eq__1__iff__lemma) ).
thf(3302,plain,
! [A: $i,B: $i] :
( ( ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B )
= ( one_one @ int ) )
=> ( ( ( ti @ int @ A )
= ( one_one @ int ) )
| ( ( ti @ int @ A )
= ( number_number_of @ int @ min ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[647]) ).
thf(921,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_926_mod__add__eq) ).
thf(4245,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[921]) ).
thf(1153,axiom,
! [A: $i] :
( ( semiring_1 @ A )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ ( one_one @ A ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( one_one @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_one__power2) ).
thf(5039,plain,
! [A: $i] :
( ( semiring_1 @ A )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ ( one_one @ A ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( one_one @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1153]) ).
thf(1027,axiom,
! [A: $i] :
( ( zero_neq_one @ A )
=> ( ( one_one @ A )
!= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_321_one__neq__zero) ).
thf(4558,plain,
! [A: $i] :
( ( zero_neq_one @ A )
=> ( ( one_one @ A )
!= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1027]) ).
thf(173,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ C ) @ B ) ) )
=> ( ( B
!= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_525_divides__rev) ).
thf(1669,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ C ) @ B ) ) )
=> ( ( B
!= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(599,axiom,
ordere216010020id_add @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oordered__comm__monoid__add) ).
thf(3118,plain,
ordere216010020id_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[599]) ).
thf(889,axiom,
! [A: $i,B: $i] :
( ( ( B
= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ A )
= ( zero_zero @ nat ) ) )
& ( ( B
!= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ A )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ ( one_one @ nat ) ) ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_517_mult__eq__if) ).
thf(4127,plain,
! [A: $i,B: $i] :
( ( ( B
= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ A )
= ( zero_zero @ nat ) ) )
& ( ( B
!= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ A )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ ( one_one @ nat ) ) ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[889]) ).
thf(183,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( number_number_of @ A @ ( bit1 @ B ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( number_number_of @ A @ B ) ) ) @ ( number_number_of @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_number__of__Bit1) ).
thf(1707,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( number_number_of @ A @ ( bit1 @ B ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( number_number_of @ A @ B ) ) ) @ ( number_number_of @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(1176,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( ( ti @ int @ C )
= ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( one_one @ int ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_self__quotient__aux1) ).
thf(5118,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( ( ti @ int @ C )
= ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( one_one @ int ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1176]) ).
thf(1250,axiom,
ordere216010020id_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oordered__comm__monoid__add) ).
thf(5363,plain,
ordere216010020id_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1250]) ).
thf(1145,axiom,
semiring @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Osemiring) ).
thf(5011,plain,
semiring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1145]) ).
thf(52,axiom,
ordered_ring @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Oordered__ring) ).
thf(1371,plain,
ordered_ring @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(751,axiom,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_893_mult__left_Oadd) ).
thf(3646,plain,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[751]) ).
thf(883,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) )
= ( number_number_of @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_arith__simps_I32_J) ).
thf(4112,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) )
= ( number_number_of @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[883]) ).
thf(1224,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_803_add__increasing2) ).
thf(5282,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1224]) ).
thf(759,axiom,
! [A: $i] :
( ( number_semiring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_semiring__mult__2) ).
thf(3663,plain,
! [A: $i] :
( ( number_semiring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[759]) ).
thf(230,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) ).
thf(1865,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).
thf(163,axiom,
! [A: $i] :
( ( ordere1490568538miring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_367_comm__mult__left__mono) ).
thf(1635,plain,
! [A: $i] :
( ( ordere1490568538miring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(543,axiom,
ab_group_add @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oab__group__add) ).
thf(2935,plain,
ab_group_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[543]) ).
thf(1030,axiom,
! [A: $i,B: $i] :
( ( A
= ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) )
=> ( ( B
= ( one_one @ nat ) )
| ( A
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_708_mult__eq__self__implies__10) ).
thf(4573,plain,
! [A: $i,B: $i] :
( ( A
= ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) )
=> ( ( B
= ( one_one @ nat ) )
| ( A
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1030]) ).
thf(1168,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ A ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_655_dvd__reduce) ).
thf(5086,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1168]) ).
thf(1185,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( B
= ( zero_zero @ nat ) )
| ( B = A )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_511_divides__cases) ).
thf(5148,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( B
= ( zero_zero @ nat ) )
| ( B = A )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1185]) ).
thf(35,axiom,
linord1278240602ring_1 @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Olinordered__semiring__1) ).
thf(1332,plain,
linord1278240602ring_1 @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(767,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( number_number_of @ int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ B @ C ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ ( inv @ C @ A ) ) @ ( wset @ B @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_827_wset__mem__inv__mem) ).
thf(3700,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( number_number_of @ int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ B @ C ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ ( inv @ C @ A ) ) @ ( wset @ B @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[767]) ).
thf(869,axiom,
( ( one_one @ nat )
= ( number_number_of @ nat @ ( bit1 @ pls ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_Numeral1__eq1__nat) ).
thf(4060,plain,
( ( one_one @ nat )
= ( number_number_of @ nat @ ( bit1 @ pls ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[869]) ).
thf(381,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ ( one_one @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_350_zcong__1) ).
thf(2358,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ ( one_one @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[381]) ).
thf(436,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( multInv @ D @ A ) ) @ A ) ) @ B ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( multInv @ D @ A ) ) @ C ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_838_aux______3) ).
thf(2573,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( multInv @ D @ A ) ) @ A ) ) @ B ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( multInv @ D @ A ) ) @ C ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[436]) ).
thf(773,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ( ( number_number_of @ A @ pls )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_number__of__Pls) ).
thf(3719,plain,
! [A: $i] :
( ( number_ring @ A )
=> ( ( number_number_of @ A @ pls )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[773]) ).
thf(294,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ min ) @ ( bit0 @ A ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_303_rel__simps_I25_J) ).
thf(2079,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ min ) @ ( bit0 @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ min ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ min ) @ ( bit0 @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).
thf(592,axiom,
! [A: $i] :
( ( monoid_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_767_add__0__left) ).
thf(3101,plain,
! [A: $i] :
( ( monoid_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[592]) ).
thf(662,axiom,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( ( zero_zero @ A )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) )
<=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_769_double__zero__sym) ).
thf(3352,plain,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( ( ( zero_zero @ A )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) )
=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) )
& ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
=> ( ( zero_zero @ A )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[662]) ).
thf(1234,axiom,
! [A: $i] :
( ( ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ C ) @ E ) ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ D ) ) @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_693_mult__diff__mult) ).
thf(5325,plain,
! [A: $i] :
( ( ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ C ) @ E ) ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ D ) ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1234]) ).
thf(641,axiom,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ E ) ) @ C ) ) @ D ) ) @ F ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_491_less__add__iff1) ).
thf(3270,plain,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ E ) ) @ C ) ) @ D ) ) @ F ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ E ) ) @ C ) ) @ D ) ) @ F ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[641]) ).
thf(368,axiom,
semiring_1 @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Osemiring__1) ).
thf(2299,plain,
semiring_1 @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[368]) ).
thf(305,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_659_le__add1) ).
thf(2116,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).
thf(506,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( one_one @ nat ) ) )
<=> ( A
= ( one_one @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_876_gcd__lcm__complete__lattice__nat_Obot__unique) ).
thf(2802,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( one_one @ nat ) ) )
=> ( A
= ( one_one @ nat ) ) )
& ( ( A
= ( one_one @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( one_one @ nat ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[506]) ).
thf(671,axiom,
monoid_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Omonoid__add) ).
thf(3384,plain,
monoid_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[671]) ).
thf(139,axiom,
comm_semiring_1 @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Ocomm__semiring__1) ).
thf(1571,plain,
comm_semiring_1 @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(969,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B )
= A )
=> ( B
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_624_add__eq__self__zero) ).
thf(4388,plain,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B )
= A )
=> ( B
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[969]) ).
thf(952,axiom,
comm_ring @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Ocomm__ring) ).
thf(4337,plain,
comm_ring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[952]) ).
thf(476,axiom,
! [A: $i,B: $i] :
( ( ( B
= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B )
= ( one_one @ nat ) ) )
& ( ( B
!= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ ( one_one @ nat ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_518_power__eq__if) ).
thf(2679,plain,
! [A: $i,B: $i] :
( ( ( B
= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B )
= ( one_one @ nat ) ) )
& ( ( B
!= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ ( one_one @ nat ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[476]) ).
thf(916,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) )
<=> ( ( B
= ( zero_zero @ nat ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_521_nat__mult__dvd__cancel__disj_H) ).
thf(4228,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) )
=> ( ( B
= ( zero_zero @ nat ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ C ) ) ) )
& ( ( ( B
= ( zero_zero @ nat ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[916]) ).
thf(747,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
<=> ( B = A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_575_dvd_Oantisym__conv) ).
thf(3631,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
=> ( B = A ) )
& ( ( B = A )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[747]) ).
thf(425,axiom,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( zero_zero @ A ) ) @ B )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_887_mult__left_Ozero) ).
thf(2538,plain,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( zero_zero @ A ) ) @ B )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[425]) ).
thf(342,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_658_le__add2) ).
thf(2229,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[342]) ).
thf(792,axiom,
! [A: $i,B: $i] :
( ( A = B )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_598_eq__imp__le) ).
thf(3815,plain,
! [A: $i,B: $i] :
( ( A = B )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[792]) ).
thf(295,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( number_number_of @ A @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ C ) )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_469_number__of__diff) ).
thf(2084,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( number_number_of @ A @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ C ) )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( number_number_of @ A @ B ) ) @ ( number_number_of @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).
thf(385,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ int @ ( power_power @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) @ B ) ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_zadd__power2) ).
thf(2383,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ int @ ( power_power @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) @ B ) ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[385]) ).
thf(457,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( standardRes @ B @ A ) ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_991_StandardRes__ubound) ).
thf(2624,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( standardRes @ B @ A ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[457]) ).
thf(393,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ B ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) )
=> ( ( ti @ int @ C )
= ( ti @ int @ B ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_415_zcong__less__eq) ).
thf(2415,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ B ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) )
=> ( ( ti @ int @ C )
= ( ti @ int @ B ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[393]) ).
thf(562,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_666_add__leD2) ).
thf(2987,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[562]) ).
thf(116,axiom,
comm_ring @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Ocomm__ring) ).
thf(1524,plain,
comm_ring @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(1089,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B )
= A ) )
& ( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( div_mod @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_984_mod__if) ).
thf(4828,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B )
= A ) )
& ( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( div_mod @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1089]) ).
thf(149,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) )
<=> ( B = C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_586_nat__add__left__cancel) ).
thf(1589,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) )
=> ( B = C ) )
& ( ( B = C )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(778,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_720_nat__less__add__iff2) ).
thf(3732,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ C ) ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[778]) ).
thf(594,axiom,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_366_mult__right__mono__neg) ).
thf(3106,plain,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[594]) ).
thf(696,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) @ ( zero_zero @ int ) ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_452_zcong__zmult__prop3) ).
thf(3468,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) @ ( zero_zero @ int ) ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[696]) ).
thf(186,axiom,
linorder @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Orderings_Olinorder) ).
thf(1722,plain,
linorder @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(1158,axiom,
? [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ s1 @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) )
& ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ s1 @ B ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) ) )
=> ( ( ti @ int @ B )
= ( ti @ int @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1) ).
thf(5054,plain,
? [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ s1 @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) )
& ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ s1 @ B ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) ) )
=> ( ( ti @ int @ B )
= ( ti @ int @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1158]) ).
thf(892,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_600_le__refl) ).
thf(4139,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[892]) ).
thf(546,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_106_sum__power2__ge__zero) ).
thf(2943,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[546]) ).
thf(547,axiom,
! [A: $i] :
( ( ordere779506340up_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_776_add__mono) ).
thf(2945,plain,
! [A: $i] :
( ( ordere779506340up_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[547]) ).
thf(938,axiom,
! [A: $i] :
( ( ti @ int @ ( bit1 @ A ) )
= ( bit1 @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Int_OBit1_res) ).
thf(4300,plain,
! [A: $i] :
( ( ti @ int @ ( bit1 @ A ) )
= ( bit1 @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[938]) ).
thf(1254,axiom,
ordere779506340up_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oordered__ab__semigroup__add) ).
thf(5381,plain,
ordere779506340up_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1254]) ).
thf(881,axiom,
( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( quadRes @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ ( number_number_of @ int @ min ) ) )
=> ( ( legendre @ ( number_number_of @ int @ min ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) )
!= ( one_one @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_467__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N) ).
thf(4107,plain,
( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( quadRes @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ ( number_number_of @ int @ min ) ) )
=> ( ( legendre @ ( number_number_of @ int @ min ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) )
!= ( one_one @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[881]) ).
thf(609,axiom,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( number_number_of @ A @ B ) ) @ ( one_one @ A ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( bit1 @ pls ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_le__special_I4_J) ).
thf(3160,plain,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( number_number_of @ A @ B ) ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( bit1 @ pls ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( bit1 @ pls ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( number_number_of @ A @ B ) ) @ ( one_one @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[609]) ).
thf(26,axiom,
! [A: $i,B: $i] :
( ( ab_semigroup_mult @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( times_times @ B @ A ) )
= ( times_times @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_0_res) ).
thf(1307,plain,
! [A: $i,B: $i] :
( ( ab_semigroup_mult @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( times_times @ B @ A ) )
= ( times_times @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(694,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_684_le__cube) ).
thf(3456,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[694]) ).
thf(627,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_569_dvd_Oantisym) ).
thf(3219,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[627]) ).
thf(1252,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( ( legendre @ A @ B )
= ( zero_zero @ int ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( quadRes @ B ) @ A ) )
=> ( ( legendre @ A @ B )
= ( one_one @ int ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( quadRes @ B ) @ A ) )
=> ( ( legendre @ A @ B )
= ( number_number_of @ int @ min ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_508_Legendre__def) ).
thf(5367,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( ( legendre @ A @ B )
= ( zero_zero @ int ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( quadRes @ B ) @ A ) )
=> ( ( legendre @ A @ B )
= ( one_one @ int ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( quadRes @ B ) @ A ) )
=> ( ( legendre @ A @ B )
= ( number_number_of @ int @ min ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1252]) ).
thf(973,axiom,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_365_mult__left__mono__neg) ).
thf(4401,plain,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[973]) ).
thf(1022,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_zadd__zmult__distrib2) ).
thf(4541,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1022]) ).
thf(1167,axiom,
! [A: $i,B: $i] :
( ( bit0 @ A )
!= ( bit1 @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_rel__simps_I49_J) ).
thf(5082,plain,
! [A: $i,B: $i] :
( ( bit0 @ A )
!= ( bit1 @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1167]) ).
thf(1058,axiom,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( number_number_of @ A @ B ) ) @ ( zero_zero @ A ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ pls ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_47_le__special_I3_J) ).
thf(4731,plain,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( number_number_of @ A @ B ) ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ pls ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ pls ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( number_number_of @ A @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1058]) ).
thf(839,axiom,
monoid_add @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Omonoid__add) ).
thf(3977,plain,
monoid_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[839]) ).
thf(1096,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( zero_zero @ int ) ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( one_one @ int ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
= ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( one_one @ int ) ) ) @ B ) ) ) @ ( one_one @ int ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_899_neg__zmod__mult__2) ).
thf(4850,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( zero_zero @ int ) ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( one_one @ int ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
= ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( one_one @ int ) ) ) @ B ) ) ) @ ( one_one @ int ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1096]) ).
thf(359,axiom,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_809_add__less__le__mono) ).
thf(2278,plain,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[359]) ).
thf(57,axiom,
! [A: $i] :
( ( ring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_950_mod__diff__eq) ).
thf(1381,plain,
! [A: $i] :
( ( ring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(486,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ ( hAPP @ real @ real @ ( times_times @ real @ A ) @ C ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ B ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_703_real__mult__le__cancel__iff1) ).
thf(2710,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ ( hAPP @ real @ real @ ( times_times @ real @ A ) @ C ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ B ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ ( hAPP @ real @ real @ ( times_times @ real @ A ) @ C ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[486]) ).
thf(234,axiom,
comm_monoid_mult @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Ocomm__monoid__mult) ).
thf(1877,plain,
comm_monoid_mult @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).
thf(1202,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_342_le__imp__power__dvd) ).
thf(5202,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1202]) ).
thf(780,axiom,
! [A: $i] :
( ( comm_monoid_mult @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_326_power__mult__distrib) ).
thf(3747,plain,
! [A: $i] :
( ( comm_monoid_mult @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[780]) ).
thf(276,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ C ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_461_zprime__zdvd__zmult) ).
thf(2025,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ C ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).
thf(508,axiom,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ E ) @ B ) ) @ C ) ) @ F ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_490_less__add__iff2) ).
thf(2812,plain,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ E ) @ B ) ) @ C ) ) @ F ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ E ) @ B ) ) @ C ) ) @ F ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[508]) ).
thf(460,axiom,
! [A: $i] :
( ( number_semiring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ B ) )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ C ) ) @ ( number_number_of @ A @ B ) )
= ( number_number_of @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_85_semiring__add__number__of) ).
thf(2632,plain,
! [A: $i] :
( ( number_semiring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ B ) )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ C ) ) @ ( number_number_of @ A @ B ) )
= ( number_number_of @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[460]) ).
thf(989,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ ( number_number_of @ int @ ( bit0 @ A ) ) ) @ ( number_number_of @ int @ ( bit0 @ B ) ) )
= ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( hAPP @ int @ int @ ( div_mod @ int @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_964_zmod__number__of__Bit0) ).
thf(4445,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ ( number_number_of @ int @ ( bit0 @ A ) ) ) @ ( number_number_of @ int @ ( bit0 @ B ) ) )
= ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( hAPP @ int @ int @ ( div_mod @ int @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[989]) ).
thf(450,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hAPP @ A @ B @ ( combs @ A @ C @ B @ D @ E ) @ F )
= ( hAPP @ C @ B @ ( hAPP @ A @ ( fun @ C @ B ) @ D @ F ) @ ( hAPP @ A @ C @ E @ F ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_COMBS_1_1_U) ).
thf(2605,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hAPP @ A @ B @ ( combs @ A @ C @ B @ D @ E ) @ F )
= ( hAPP @ C @ B @ ( hAPP @ A @ ( fun @ C @ B ) @ D @ F ) @ ( hAPP @ A @ C @ E @ F ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[450]) ).
thf(1106,axiom,
zero_neq_one @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Ozero__neq__one) ).
thf(4882,plain,
zero_neq_one @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1106]) ).
thf(582,axiom,
! [A: $i,B: $i] :
( ( ti @ ( fun @ int @ bool ) @ ( zcong @ A @ B ) )
= ( zcong @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_IntPrimes_Ozcong_res) ).
thf(3055,plain,
! [A: $i,B: $i] :
( ( ti @ ( fun @ int @ bool ) @ ( zcong @ A @ B ) )
= ( zcong @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[582]) ).
thf(849,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( multInv @ C @ B ) @ ( multInv @ C @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ A ) @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_841_MultInv__prop5) ).
thf(4009,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( multInv @ C @ B ) @ ( multInv @ C @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ A ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[849]) ).
thf(354,axiom,
! [A: $i] :
( ( linordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_426_mult__right__less__imp__less) ).
thf(2261,plain,
! [A: $i] :
( ( linordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[354]) ).
thf(191,axiom,
ordere779506340up_add @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oordered__ab__semigroup__add) ).
thf(1731,plain,
ordere779506340up_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(732,axiom,
! [A: $i] :
( ( linordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_424_mult__left__less__imp__less) ).
thf(3572,plain,
! [A: $i] :
( ( linordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[732]) ).
thf(1137,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( one_one @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( one_one @ A ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_408_one__le__power) ).
thf(4985,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( one_one @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( one_one @ A ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1137]) ).
thf(610,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ int @ ( power_power @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_zpower__zpower) ).
thf(3165,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ int @ ( power_power @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[610]) ).
thf(229,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_439_power__increasing__iff) ).
thf(1862,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ D ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).
thf(699,axiom,
! [A: $i] :
( ( ( plus @ A )
& ( linorder @ A )
& ( dvd @ A ) )
=> ! [B: $i,C: $i] :
? [D: $i] :
! [E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ E ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_865_minf_I9_J) ).
thf(3483,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[699]) ).
thf(41,axiom,
ab_group_add @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Oab__group__add) ).
thf(1344,plain,
ab_group_add @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(1139,axiom,
! [A: $i] :
( ( ab_group_add @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
<=> ( ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_787_eq__iff__diff__eq__0) ).
thf(4990,plain,
! [A: $i] :
( ( ab_group_add @ A )
=> ! [B: $i,C: $i] :
( ( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
=> ( ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C )
= ( zero_zero @ A ) ) )
& ( ( ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C )
= ( zero_zero @ A ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1139]) ).
thf(438,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_938_pos__mod__bound) ).
thf(2578,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[438]) ).
thf(390,axiom,
! [A: $i] :
( ( ( bit0 @ A )
= pls )
<=> ( ( ti @ int @ A )
= pls ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_rel__simps_I44_J) ).
thf(2401,plain,
! [A: $i] :
( ( ( ( bit0 @ A )
= pls )
=> ( ( ti @ int @ A )
= pls ) )
& ( ( ( ti @ int @ A )
= pls )
=> ( ( bit0 @ A )
= pls ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).
thf(337,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ B ) ) @ ( number_number_of @ nat @ A ) )
= ( zero_zero @ nat ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ B ) ) @ ( number_number_of @ nat @ A ) )
= ( number_number_of @ nat @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_230_mult__nat__number__of) ).
thf(2212,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ B ) ) @ ( number_number_of @ nat @ A ) )
= ( zero_zero @ nat ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ B ) ) @ ( number_number_of @ nat @ A ) )
= ( number_number_of @ nat @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[337]) ).
thf(1253,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( minus_minus @ int @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_853_int__less__induct) ).
thf(5377,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( minus_minus @ int @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1253]) ).
thf(118,axiom,
dvd @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Odvd) ).
thf(1528,plain,
dvd @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(1010,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ t ) @ ( zero_zero @ int ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0__096t_A_060_A0_096) ).
thf(4502,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ t ) @ ( zero_zero @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1010]) ).
thf(314,axiom,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_614_not__less0) ).
thf(2139,plain,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).
thf(961,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_610_diff__mult__distrib2) ).
thf(4362,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[961]) ).
thf(504,axiom,
! [A: $i,B: $i,C: $i] :
( ( B = C )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_572_dvd_Oord__eq__le__trans) ).
thf(2797,plain,
! [A: $i,B: $i,C: $i] :
( ( B = C )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[504]) ).
thf(79,axiom,
! [A: $i,B: $i] :
( ( standardRes @ A @ ( ti @ int @ B ) )
= ( standardRes @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Residues_OStandardRes_arg2) ).
thf(1435,plain,
! [A: $i,B: $i] :
( ( standardRes @ A @ ( ti @ int @ B ) )
= ( standardRes @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(275,axiom,
! [A: $i,B: $i] :
( ( collect @ A @ ( ti @ ( fun @ A @ bool ) @ B ) )
= ( collect @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Set_OCollect_arg1) ).
thf(2022,plain,
! [A: $i,B: $i] :
( ( collect @ A @ ( ti @ ( fun @ A @ bool ) @ B ) )
= ( collect @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).
thf(424,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ pls ) ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_147_add__numeral__0) ).
thf(2535,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ pls ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[424]) ).
thf(1008,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ A ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_354_zcong__scalar) ).
thf(4497,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ A ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1008]) ).
thf(291,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( multInv @ C @ D ) ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ D ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( multInv @ C @ D ) ) ) @ D ) ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_837_aux______1) ).
thf(2069,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( multInv @ C @ D ) ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ D ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( multInv @ C @ D ) ) ) @ D ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).
thf(415,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_625_diff__add__0) ).
thf(2506,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[415]) ).
thf(394,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) )
=> ( ( ( ti @ int @ B )
!= ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( one_one @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( one_one @ int ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_493_Euler_Oaux1) ).
thf(2418,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) )
=> ( ( ( ti @ int @ B )
!= ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( one_one @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( one_one @ int ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[394]) ).
thf(1094,axiom,
! [A: $i] :
( ( bit0 @ ( ti @ int @ A ) )
= ( bit0 @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Int_OBit0_arg1) ).
thf(4844,plain,
! [A: $i] :
( ( bit0 @ ( ti @ int @ A ) )
= ( bit0 @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1094]) ).
thf(225,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_less__number__of__int__code) ).
thf(1841,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(63,axiom,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_895_mult__right_Odiff) ).
thf(1396,plain,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(1200,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_609_nat__mult__assoc) ).
thf(5197,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1200]) ).
thf(285,axiom,
! [A: $i,B: $i] :
( ( standardRes @ A @ B )
= ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_979_StandardRes__def) ).
thf(2051,plain,
! [A: $i,B: $i] :
( ( standardRes @ A @ B )
= ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).
thf(1154,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( multInv @ B @ ( multInv @ B @ A ) ) @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_842_MultInv__prop4) ).
thf(5042,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( multInv @ B @ ( multInv @ B @ A ) ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1154]) ).
thf(363,axiom,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_890_mult_Oadd__right) ).
thf(2288,plain,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[363]) ).
thf(335,axiom,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_891_mult__right_Oadd) ).
thf(2207,plain,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[335]) ).
thf(1108,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ ( one_one @ A ) ) ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_253_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) ).
thf(4890,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ ( one_one @ A ) ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1108]) ).
thf(911,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B )
= ( zero_zero @ nat ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_628_diff__is__0__eq) ).
thf(4209,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B )
= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[911]) ).
thf(1002,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_402_less__1__mult) ).
thf(4481,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1002]) ).
thf(898,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ C ) @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_910_zcong__zmod) ).
thf(4158,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ C ) @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ C ) @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[898]) ).
thf(212,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( combs @ A @ B @ C @ ( ti @ ( fun @ A @ ( fun @ B @ C ) ) @ D ) @ E )
= ( combs @ A @ B @ C @ D @ E ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_COMBS_arg1) ).
thf(1805,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( combs @ A @ B @ C @ ( ti @ ( fun @ A @ ( fun @ B @ C ) ) @ D ) @ E )
= ( combs @ A @ B @ C @ D @ E ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(1060,axiom,
ordere236663937imp_le @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le) ).
thf(4741,plain,
ordere236663937imp_le @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1060]) ).
thf(1046,axiom,
! [A: $i] :
( ( ( plus @ A )
& ( linorder @ A )
& ( dvd @ A ) )
=> ! [B: $i,C: $i] :
? [D: $i] :
! [E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ E ) @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ C ) ) )
<=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_866_minf_I10_J) ).
thf(4690,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1046]) ).
thf(826,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( multInv @ B @ A ) ) @ ( one_one @ int ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_851_MultInv__prop2) ).
thf(3936,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( multInv @ B @ A ) ) @ ( one_one @ int ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[826]) ).
thf(925,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_237_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) ).
thf(4260,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[925]) ).
thf(14,axiom,
! [A: $i,B: $i] :
( ( wset @ ( ti @ int @ A ) @ B )
= ( wset @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_WilsonRuss_Owset_arg1) ).
thf(1277,plain,
! [A: $i,B: $i] :
( ( wset @ ( ti @ int @ A ) @ B )
= ( wset @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(604,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( div_mod @ A @ D ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_921_dvd__mod) ).
thf(3146,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( div_mod @ A @ D ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[604]) ).
thf(109,axiom,
! [A: $i,B: $i] :
( ( semiring_div @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( div_mod @ B @ A ) )
= ( div_mod @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Divides_Odiv__class_Omod_res) ).
thf(1509,plain,
! [A: $i,B: $i] :
( ( semiring_div @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( div_mod @ B @ A ) )
= ( div_mod @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(822,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( minus_minus @ int @ ( bit1 @ A ) ) @ ( bit0 @ B ) )
= ( bit1 @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_482_diff__bin__simps_I9_J) ).
thf(3921,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( minus_minus @ int @ ( bit1 @ A ) ) @ ( bit0 @ B ) )
= ( bit1 @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[822]) ).
thf(92,axiom,
! [A: $i] :
( ( quadRes @ ( ti @ int @ A ) )
= ( quadRes @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Residues_OQuadRes_arg1) ).
thf(1472,plain,
! [A: $i] :
( ( quadRes @ ( ti @ int @ A ) )
= ( quadRes @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(447,axiom,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ B )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_788_diff__self) ).
thf(2597,plain,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ B )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[447]) ).
thf(959,axiom,
linord893533164strict @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Olinordered__comm__semiring__strict) ).
thf(4355,plain,
linord893533164strict @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[959]) ).
thf(268,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_687_mult__le__mono) ).
thf(2005,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).
thf(70,axiom,
! [A: $i] :
( ( ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) )
<=> ( ( ti @ A @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ E ) @ B ) ) @ C ) ) @ F ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_478_eq__add__iff2) ).
thf(1413,plain,
! [A: $i] :
( ( ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) )
=> ( ( ti @ A @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ E ) @ B ) ) @ C ) ) @ F ) ) )
& ( ( ( ti @ A @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ E ) @ B ) ) @ C ) ) @ F ) )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) @ F ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(716,axiom,
( ( zero_zero @ int )
!= ( one_one @ int ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_int__0__neq__1) ).
thf(3522,plain,
( ( zero_zero @ int )
!= ( one_one @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[716]) ).
thf(949,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_597_le__trans) ).
thf(4331,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[949]) ).
thf(286,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_645_add__lessD1) ).
thf(2054,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).
thf(910,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_516_divides__exp) ).
thf(4207,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[910]) ).
thf(214,axiom,
cancel146912293up_add @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Ocancel__ab__semigroup__add) ).
thf(1809,plain,
cancel146912293up_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(551,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_387_zero__less__mult__pos) ).
thf(2952,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[551]) ).
thf(171,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_814_add__nonneg__pos) ).
thf(1662,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(971,axiom,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( number_number_of @ A @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_less__special_I1_J) ).
thf(4394,plain,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( number_number_of @ A @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( number_number_of @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[971]) ).
thf(299,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
| ( B
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_882_divides__le) ).
thf(2096,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
| ( B
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).
thf(25,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ C ) ) @ C )
= ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_912_mod__mod__trivial) ).
thf(1304,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ C ) ) @ C )
= ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(232,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_399_not__one__less__zero) ).
thf(1873,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(686,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( one_one @ nat ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_690_nat__mult__1__right) ).
thf(3427,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( one_one @ nat ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[686]) ).
thf(995,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( multInv @ B @ A ) @ ( zero_zero @ int ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_843_MultInv__prop3) ).
thf(4458,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( multInv @ B @ A ) @ ( zero_zero @ int ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[995]) ).
thf(470,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( one_one @ nat ) ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_878_gcd__lcm__complete__lattice__nat_Obot__least) ).
thf(2665,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( one_one @ nat ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[470]) ).
thf(373,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_455_power__strict__decreasing) ).
thf(2318,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[373]) ).
thf(501,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_883_mult__left__cancel) ).
thf(2784,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[501]) ).
thf(994,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_386_zero__less__mult__pos2) ).
thf(4456,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[994]) ).
thf(1038,axiom,
~ ! [A: $i] :
~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of @ int @ min ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_279__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_) ).
thf(4598,plain,
~ ! [A: $i] :
~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of @ int @ min ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1038]) ).
thf(465,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_553_dvd_Oless__asym) ).
thf(2652,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[465]) ).
thf(646,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ 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_691_nat__mult__eq__1__iff) ).
thf(3293,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( one_one @ nat ) )
=> ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) ) )
& ( ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( one_one @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[646]) ).
thf(241,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
<=> ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_512_divides__antisym) ).
thf(1905,plain,
! [A: $i,B: $i] :
( ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ( A = B ) )
& ( ( A = B )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).
thf(71,axiom,
! [A: $i,B: $i] :
( ( ti @ int @ ( legendre @ A @ B ) )
= ( legendre @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Residues_OLegendre_res) ).
thf(1420,plain,
! [A: $i,B: $i] :
( ( ti @ int @ ( legendre @ A @ B ) )
= ( legendre @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(842,axiom,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( one_one @ A ) ) @ ( number_number_of @ A @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ pls ) ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_le__special_I2_J) ).
thf(3981,plain,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( one_one @ A ) ) @ ( number_number_of @ A @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ pls ) ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ pls ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( one_one @ A ) ) @ ( number_number_of @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[842]) ).
thf(414,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ A ) ) @ ( zero_zero @ int ) ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ A )
= ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_963_mod__pos__neg__trivial) ).
thf(2503,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ A ) ) @ ( zero_zero @ int ) ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ A )
= ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[414]) ).
thf(902,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_647_less__diff__conv) ).
thf(4177,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[902]) ).
thf(541,axiom,
! [A: $i] :
( ( pls
= ( bit0 @ A ) )
<=> ( pls
= ( ti @ int @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_rel__simps_I38_J) ).
thf(2927,plain,
! [A: $i] :
( ( ( pls
= ( bit0 @ A ) )
=> ( pls
= ( ti @ int @ A ) ) )
& ( ( pls
= ( ti @ int @ A ) )
=> ( pls
= ( bit0 @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[541]) ).
thf(459,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ C )
= ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_930_mod__add__self2) ).
thf(2629,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ C )
= ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[459]) ).
thf(265,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ C ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_670_le__diff__conv) ).
thf(1990,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ B ) ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).
thf(1099,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ C @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_830_wset__subset) ).
thf(4859,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1099]) ).
thf(334,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ ( bit1 @ A ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_rel__simps_I22_J) ).
thf(2202,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ ( bit1 @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ ( bit1 @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).
thf(1045,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_804_add__nonpos__nonpos) ).
thf(4688,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1045]) ).
thf(376,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_854_int__ge__induct) ).
thf(2333,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[376]) ).
thf(350,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ A ) @ ( hAPP @ nat @ int @ ( power_power @ int @ C ) @ A ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_359_zcong__zpower) ).
thf(2256,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ A ) @ ( hAPP @ nat @ int @ ( power_power @ int @ C ) @ A ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[350]) ).
thf(129,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) @ C ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_906_mod__mult__left__eq) ).
thf(1551,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) @ C ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(1143,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) )
= ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_502_zspecial__product) ).
thf(5003,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) )
= ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1143]) ).
thf(865,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ ( bit0 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_127_rel__simps_I14_J) ).
thf(4045,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ ( bit0 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ ( bit0 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[865]) ).
thf(844,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ A ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ C ) )
& ( ( hAPP @ nat @ real @ ( power_power @ real @ C ) @ B )
= A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_530_realpow__pos__nth) ).
thf(3988,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ A ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ C ) )
& ( ( hAPP @ nat @ real @ ( power_power @ real @ C ) @ B )
= A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[844]) ).
thf(209,axiom,
order @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Orderings_Oorder) ).
thf(1798,plain,
order @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(819,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
| ( ( ti @ int @ A )
= ( ti @ int @ B ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_zless__linear) ).
thf(3913,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
| ( ( ti @ int @ A )
= ( ti @ int @ B ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[819]) ).
thf(927,axiom,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_373_mult__nonneg__nonpos) ).
thf(4272,plain,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[927]) ).
thf(1129,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) @ D ) )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) @ B ) ) @ D ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_727_left__add__mult__distrib) ).
thf(4950,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) @ D ) )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) @ B ) ) @ D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1129]) ).
thf(166,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ B ) )
<=> ( A = C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_587_nat__add__right__cancel) ).
thf(1647,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ B ) )
=> ( A = C ) )
& ( ( A = C )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(830,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ A ) @ B ) )
<=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) )
& ( A != B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_542_real__less__def) ).
thf(3949,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) )
& ( A != B ) ) )
& ( ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ B ) )
& ( A != B ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[830]) ).
thf(1166,axiom,
( ( zero_zero @ real )
!= ( one_one @ real ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_540_real__zero__not__eq__one) ).
thf(5079,plain,
( ( zero_zero @ real )
!= ( one_one @ real ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1166]) ).
thf(528,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of @ int @ min ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_270_s) ).
thf(2882,plain,
hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of @ int @ min ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[528]) ).
thf(522,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_101_power2__eq__imp__eq) ).
thf(2863,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[522]) ).
thf(650,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ pls ) @ A )
= pls ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_mult__Pls) ).
thf(3315,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ pls ) @ A )
= pls ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[650]) ).
thf(799,axiom,
ordered_semiring @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Oordered__semiring) ).
thf(3842,plain,
ordered_semiring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[799]) ).
thf(1005,axiom,
( ( zero_zero @ nat )
= ( number_number_of @ nat @ pls ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_semiring__norm_I113_J) ).
thf(4490,plain,
( ( zero_zero @ nat )
= ( number_number_of @ nat @ pls ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1005]) ).
thf(597,axiom,
( ( t
= ( zero_zero @ int ) )
=> ( ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( one_one @ int ) )
= ( zero_zero @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4_calculation_I2_J) ).
thf(3112,plain,
( ( t
= ( zero_zero @ int ) )
=> ( ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( one_one @ int ) )
= ( zero_zero @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[597]) ).
thf(1123,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ ( bit0 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_154_less__int__code_I15_J) ).
thf(4937,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ ( bit0 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ ( bit0 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1123]) ).
thf(153,axiom,
( ( legendre @ ( number_number_of @ int @ min ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) )
= ( one_one @ int ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_280__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_096) ).
thf(1611,plain,
( ( legendre @ ( number_number_of @ int @ min ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) )
= ( one_one @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(136,axiom,
ring_1 @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Oring__1) ).
thf(1566,plain,
ring_1 @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(816,axiom,
! [A: $i] :
( ( zero @ A )
=> ( ( ti @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Ozero__class_Ozero_res) ).
thf(3904,plain,
! [A: $i] :
( ( zero @ A )
=> ( ( ti @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[816]) ).
thf(630,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B )
= ( zero_zero @ A ) )
<=> ( ( ( ti @ A @ C )
= ( zero_zero @ A ) )
& ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_801_add__nonneg__eq__0__iff) ).
thf(3228,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( ( ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B )
= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ C )
= ( zero_zero @ A ) )
& ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) )
& ( ( ( ( ti @ A @ C )
= ( zero_zero @ A ) )
& ( ( ti @ A @ B )
= ( zero_zero @ A ) ) )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B )
= ( zero_zero @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[630]) ).
thf(327,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( bit1 @ A ) ) @ ( bit0 @ B ) )
= ( bit1 @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_161_add__Bit1__Bit0) ).
thf(2175,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( bit1 @ A ) ) @ ( bit0 @ B ) )
= ( bit1 @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[327]) ).
thf(8,axiom,
! [A: $i,B: $i] :
( ( zcong @ ( ti @ int @ A ) @ B )
= ( zcong @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_IntPrimes_Ozcong_arg1) ).
thf(1267,plain,
! [A: $i,B: $i] :
( ( zcong @ ( ti @ int @ A ) @ B )
= ( zcong @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(583,axiom,
( ( ti @ ( fun @ int @ bool ) @ twoSqu658283162sum2sq )
= twoSqu658283162sum2sq ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_TwoSquares__Mirabelle__lldtdwpyrp_Ois__sum2sq_res) ).
thf(3058,plain,
( ( ti @ ( fun @ int @ bool ) @ twoSqu658283162sum2sq )
= twoSqu658283162sum2sq ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[583]) ).
thf(386,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ ( zero_zero @ nat ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_622_Nat_Oadd__0__right) ).
thf(2386,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ ( zero_zero @ nat ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).
thf(490,axiom,
! [A: $i] :
( ( zero @ A )
=> ! [B: $i] :
( ( ( zero_zero @ A )
= ( ti @ A @ B ) )
<=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_757_zero__reorient) ).
thf(2720,plain,
! [A: $i] :
( ( zero @ A )
=> ! [B: $i] :
( ( ( ( zero_zero @ A )
= ( ti @ A @ B ) )
=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) )
& ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
=> ( ( zero_zero @ A )
= ( ti @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[490]) ).
thf(966,axiom,
dvd @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Odvd) ).
thf(4377,plain,
dvd @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[966]) ).
thf(518,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_816_add__strict__increasing2) ).
thf(2850,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[518]) ).
thf(96,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_217_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) ).
thf(1484,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(512,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( minus_minus @ int @ pls ) @ ( bit0 @ A ) )
= ( bit0 @ ( hAPP @ int @ int @ ( minus_minus @ int @ pls ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_484_diff__bin__simps_I3_J) ).
thf(2831,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( minus_minus @ int @ pls ) @ ( bit0 @ A ) )
= ( bit0 @ ( hAPP @ int @ int @ ( minus_minus @ int @ pls ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[512]) ).
thf(478,axiom,
! [A: $i] :
( ( semiri456707255roduct @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( ti @ A @ F )
!= ( zero_zero @ A ) )
=> ( ( ( ( ti @ A @ D )
= ( ti @ A @ E ) )
& ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ ( hAPP @ A @ A @ ( times_times @ A @ F ) @ B ) )
!= ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ ( hAPP @ A @ A @ ( times_times @ A @ F ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_251_add__scale__eq__noteq) ).
thf(2689,plain,
! [A: $i] :
( ( semiri456707255roduct @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( ti @ A @ F )
!= ( zero_zero @ A ) )
=> ( ( ( ( ti @ A @ D )
= ( ti @ A @ E ) )
& ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ ( hAPP @ A @ A @ ( times_times @ A @ F ) @ B ) )
!= ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ ( hAPP @ A @ A @ ( times_times @ A @ F ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[478]) ).
thf(103,axiom,
linord893533164strict @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Olinordered__comm__semiring__strict) ).
thf(1499,plain,
linord893533164strict @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(976,axiom,
( ( number_number_of @ nat @ pls )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_nat__number__of__Pls) ).
thf(4409,plain,
( ( number_number_of @ nat @ pls )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[976]) ).
thf(1111,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_591_Nat_Odiff__cancel) ).
thf(4905,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1111]) ).
thf(76,axiom,
ab_semigroup_add @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Oab__semigroup__add) ).
thf(1432,plain,
ab_semigroup_add @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(774,axiom,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_789_diff__0__right) ).
thf(3722,plain,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[774]) ).
thf(725,axiom,
power @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Power_Opower) ).
thf(3554,plain,
power @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[725]) ).
thf(1071,axiom,
comm_monoid_add @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Ocomm__monoid__add) ).
thf(4768,plain,
comm_monoid_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1071]) ).
thf(377,axiom,
ab_semigroup_mult @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oab__semigroup__mult) ).
thf(2337,plain,
ab_semigroup_mult @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[377]) ).
thf(338,axiom,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_892_mult_Oadd__left) ).
thf(2219,plain,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[338]) ).
thf(795,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( number_number_of @ int @ pls ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_747_number__of2) ).
thf(3828,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( number_number_of @ int @ pls ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[795]) ).
thf(919,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_555_dvd_Oless__asym_H) ).
thf(4241,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[919]) ).
thf(37,axiom,
comm_ring_1 @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Ocomm__ring__1) ).
thf(1336,plain,
comm_ring_1 @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(1017,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ A )
= ( zero_zero @ int ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_917_zmod__self) ).
thf(4522,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ A )
= ( zero_zero @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1017]) ).
thf(1217,axiom,
! [A: $i,B: $i] :
( ( product_Pair @ int @ int @ ( ti @ int @ A ) @ B )
= ( product_Pair @ int @ int @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Product__Type_OPair_arg1) ).
thf(5262,plain,
! [A: $i,B: $i] :
( ( product_Pair @ int @ int @ ( ti @ int @ A ) @ B )
= ( product_Pair @ int @ int @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1217]) ).
thf(749,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ real @ ( power_power @ real @ ( hAPP @ real @ real @ ( plus_plus @ real @ A ) @ B ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ real @ real @ ( plus_plus @ real @ ( hAPP @ real @ real @ ( plus_plus @ real @ ( hAPP @ nat @ real @ ( power_power @ real @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ real @ ( power_power @ real @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ ( hAPP @ real @ real @ ( times_times @ real @ ( number_number_of @ real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_510_real__sum__squared__expand) ).
thf(3642,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ real @ ( power_power @ real @ ( hAPP @ real @ real @ ( plus_plus @ real @ A ) @ B ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ real @ real @ ( plus_plus @ real @ ( hAPP @ real @ real @ ( plus_plus @ real @ ( hAPP @ nat @ real @ ( power_power @ real @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ real @ ( power_power @ real @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ ( hAPP @ real @ real @ ( times_times @ real @ ( number_number_of @ real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[749]) ).
thf(1028,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ ( zero_zero @ int ) ) )
<=> ( ( ti @ int @ A )
= ( ti @ int @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_349_IntPrimes_Ozcong__zero) ).
thf(4561,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ ( zero_zero @ int ) ) )
=> ( ( ti @ int @ A )
= ( ti @ int @ B ) ) )
& ( ( ( ti @ int @ A )
= ( ti @ int @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ ( zero_zero @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1028]) ).
thf(956,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_662_trans__le__add1) ).
thf(4346,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[956]) ).
thf(206,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( combb @ A @ B @ C @ ( ti @ ( fun @ A @ B ) @ D ) @ E )
= ( combb @ A @ B @ C @ D @ E ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_COMBB_arg1) ).
thf(1790,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( combb @ A @ B @ C @ ( ti @ ( fun @ A @ B ) @ D ) @ E )
= ( combb @ A @ B @ C @ D @ E ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(340,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_444_power__less__imp__less__exp) ).
thf(2224,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[340]) ).
thf(216,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ ( bit0 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_less__eq__int__code_I13_J) ).
thf(1813,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ ( bit0 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ ( bit0 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(23,axiom,
comm_monoid_add @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Ocomm__monoid__add) ).
thf(1300,plain,
comm_monoid_add @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(765,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ twoSqu658283162sum2sq @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ twoSqu658283162sum2sq @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ twoSqu658283162sum2sq @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_is__mult__sum2sq) ).
thf(3691,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ twoSqu658283162sum2sq @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ twoSqu658283162sum2sq @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ twoSqu658283162sum2sq @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[765]) ).
thf(164,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( A != B )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_651_le__neq__implies__less) ).
thf(1637,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( A != B )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(658,axiom,
semiring_div @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Divides_Osemiring__div) ).
thf(3344,plain,
semiring_div @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[658]) ).
thf(1024,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( zero_zero @ nat ) ) @ A )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_536_diff__0__eq__0) ).
thf(4549,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( zero_zero @ nat ) ) @ A )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1024]) ).
thf(623,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( div_mod @ nat @ B ) @ A )
= ( hAPP @ nat @ nat @ ( div_mod @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ A ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_981_le__mod__geq) ).
thf(3202,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( div_mod @ nat @ B ) @ A )
= ( hAPP @ nat @ nat @ ( div_mod @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ A ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[623]) ).
thf(915,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ A ) ) @ B ) )
<=> ( A
= ( one_one @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_717_dvd__mult__cancel1) ).
thf(4222,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ A ) ) @ B ) )
=> ( A
= ( one_one @ nat ) ) )
& ( ( A
= ( one_one @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ A ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[915]) ).
thf(1222,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( minus_minus @ int @ D ) @ C ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( minus_minus @ int @ D ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_748_decr__mult__lemma) ).
thf(5276,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( minus_minus @ int @ D ) @ C ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( minus_minus @ int @ D ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1222]) ).
thf(204,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_656_dvd__diffD) ).
thf(1785,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(756,axiom,
~ ! [A: $i] :
~ ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ s1 @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_195__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_) ).
thf(3655,plain,
~ ! [A: $i] :
~ ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ s1 @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[756]) ).
thf(5,axiom,
linordered_semidom @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Olinordered__semidom) ).
thf(1262,plain,
linordered_semidom @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(219,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_913_mod__mod__is__mod) ).
thf(1830,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).
thf(602,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_236_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) ).
thf(3127,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[602]) ).
thf(1080,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( sr @ B ) ) )
=> ( ( standardRes @ B @ A )
= ( ti @ int @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_992_StandardRes__SR__prop) ).
thf(4800,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( sr @ B ) ) )
=> ( ( standardRes @ B @ A )
= ( ti @ int @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1080]) ).
thf(744,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_916_zdvd__zmod__imp__zdvd) ).
thf(3623,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[744]) ).
thf(177,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) ) )
<=> ( ( hAPP @ int @ int @ ( div_mod @ int @ ( number_number_of @ int @ B ) ) @ ( number_number_of @ int @ A ) )
= ( zero_zero @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_953_zdvd__iff__zmod__eq__0__number__of) ).
thf(1679,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ ( number_number_of @ int @ B ) ) @ ( number_number_of @ int @ A ) )
= ( zero_zero @ int ) ) )
& ( ( ( hAPP @ int @ int @ ( div_mod @ int @ ( number_number_of @ int @ B ) ) @ ( number_number_of @ int @ A ) )
= ( zero_zero @ int ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(680,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_565_dvd_Ole__less__trans) ).
thf(3412,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[680]) ).
thf(729,axiom,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_361_split__mult__neg__le) ).
thf(3564,plain,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[729]) ).
thf(663,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( ( inv @ B @ A )
!= ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_822_inv__not__p__minus__1) ).
thf(3359,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( ( inv @ B @ A )
!= ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[663]) ).
thf(281,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ ( number_number_of @ A @ min ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
= ( one_one @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_360_power__m1__even) ).
thf(2044,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ ( number_number_of @ A @ min ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
= ( one_one @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).
thf(18,axiom,
! [A: $i,B: $i] :
( ( hAPP @ real @ real @ ( times_times @ real @ A ) @ B )
= ( hAPP @ real @ real @ ( times_times @ real @ B ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_594_real__mult__commute) ).
thf(1285,plain,
! [A: $i,B: $i] :
( ( hAPP @ real @ real @ ( times_times @ real @ A ) @ B )
= ( hAPP @ real @ real @ ( times_times @ real @ B ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(194,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ ( zero_zero @ int ) ) @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ ( multInv @ D @ A ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ ( multInv @ D @ B ) ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ D ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_847_MultInv__zcong__prop3) ).
thf(1738,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ ( zero_zero @ int ) ) @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ ( multInv @ D @ A ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ ( multInv @ D @ B ) ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ D ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(803,axiom,
! [A: $i] :
( ~ ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ A ) @ A ) ) )
<=> ( A
= ( zero_zero @ real ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_468_not__real__square__gt__zero) ).
thf(3853,plain,
! [A: $i] :
( ( ~ ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ A ) @ A ) ) )
=> ( A
= ( zero_zero @ real ) ) )
& ( ( A
= ( zero_zero @ real ) )
=> ~ ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ A ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[803]) ).
thf(287,axiom,
! [A: $i] :
~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_582_dvd_Oless__irrefl) ).
thf(2056,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).
thf(247,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B )
= ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_zadd__commute) ).
thf(1934,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B )
= ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).
thf(125,axiom,
! [A: $i,B: $i] :
( ( ab_semigroup_mult @ B )
=> ( ( times_times @ B @ ( ti @ B @ A ) )
= ( times_times @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_0_arg1) ).
thf(1543,plain,
! [A: $i,B: $i] :
( ( ab_semigroup_mult @ B )
=> ( ( times_times @ B @ ( ti @ B @ A ) )
= ( times_times @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(1132,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ) @ D ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ E ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ F ) ) ) @ G ) )
= ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ D ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ E ) @ G ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ D ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ F ) @ G ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_486_xzgcda__linear__aux1) ).
thf(4959,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ) @ D ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ E ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ F ) ) ) @ G ) )
= ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ D ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ E ) @ G ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ D ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ F ) @ G ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1132]) ).
thf(180,axiom,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ min ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_299_rel__simps_I20_J) ).
thf(1702,plain,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ min ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(1242,axiom,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_784_add__less__imp__less__right) ).
thf(5347,plain,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1242]) ).
thf(17,axiom,
zero_neq_one @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Ozero__neq__one) ).
thf(1284,plain,
zero_neq_one @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(374,axiom,
! [A: $i] :
( ( ( hAPP @ nat @ int @ ( power_power @ int @ ( number_number_of @ int @ min ) ) @ A )
= ( one_one @ int ) )
| ( ( hAPP @ nat @ int @ ( power_power @ int @ ( number_number_of @ int @ min ) ) @ A )
= ( number_number_of @ int @ min ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_506_neg__one__power) ).
thf(2320,plain,
! [A: $i] :
( ( ( hAPP @ nat @ int @ ( power_power @ int @ ( number_number_of @ int @ min ) ) @ A )
= ( one_one @ int ) )
| ( ( hAPP @ nat @ int @ ( power_power @ int @ ( number_number_of @ int @ min ) ) @ A )
= ( number_number_of @ int @ min ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[374]) ).
thf(106,axiom,
linordered_ring @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Olinordered__ring) ).
thf(1504,plain,
linordered_ring @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(278,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_568_dvd_Oorder__trans) ).
thf(2030,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).
thf(570,axiom,
! [A: $i] :
( ( semiring_1 @ A )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ ( zero_zero @ A ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_zero__power2) ).
thf(3007,plain,
! [A: $i] :
( ( semiring_1 @ A )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ ( zero_zero @ A ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[570]) ).
thf(441,axiom,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ C )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_797_diff__add__cancel) ).
thf(2584,plain,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ C )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[441]) ).
thf(978,axiom,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( number_number_of @ A @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_48_le__special_I1_J) ).
thf(4414,plain,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( number_number_of @ A @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( number_number_of @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[978]) ).
thf(913,axiom,
! [A: $i] :
( ( ordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_363_mult__mono) ).
thf(4217,plain,
! [A: $i] :
( ( ordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[913]) ).
thf(238,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_701_mult__less__mono1) ).
thf(1898,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).
thf(227,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_336_dvd__add) ).
thf(1849,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).
thf(754,axiom,
( ( bit0 @ pls )
= pls ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_Bit0__Pls) ).
thf(3652,plain,
( ( bit0 @ pls )
= pls ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[754]) ).
thf(1,conjecture,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( one_one @ int ) ) ) @ ( zero_zero @ int ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).
thf(2,negated_conjecture,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( one_one @ int ) ) ) @ ( zero_zero @ int ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(1257,plain,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( one_one @ int ) ) ) @ ( zero_zero @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(836,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_513_divides__add__revr) ).
thf(3970,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[836]) ).
thf(681,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ D ) ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_994_mod__lemma) ).
thf(3415,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ D ) ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[681]) ).
thf(796,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_722_real__le__refl) ).
thf(3829,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less_eq @ real ) @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[796]) ).
thf(1207,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ min ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ min ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_300_rel__simps_I28_J) ).
thf(5223,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ min ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ min ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ min ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit0 @ A ) ) @ min ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1207]) ).
thf(122,axiom,
! [A: $i,B: $i] :
( ( cancel_semigroup_add @ B )
=> ( ( plus_plus @ B @ ( ti @ B @ A ) )
= ( plus_plus @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Oplus__class_Oplus_2_arg1) ).
thf(1536,plain,
! [A: $i,B: $i] :
( ( cancel_semigroup_add @ B )
=> ( ( plus_plus @ B @ ( ti @ B @ A ) )
= ( plus_plus @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(261,axiom,
! [A: $i,B: $i] :
( ( power @ B )
=> ( ( ti @ ( fun @ nat @ B ) @ ( power_power @ B @ A ) )
= ( power_power @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Power_Opower__class_Opower_res) ).
thf(1981,plain,
! [A: $i,B: $i] :
( ( power @ B )
=> ( ( ti @ ( fun @ nat @ B ) @ ( power_power @ B @ A ) )
= ( power_power @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).
thf(211,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_583_nat__add__commute) ).
thf(1802,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(704,axiom,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ ( zero_zero @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_159_bin__less__0__simps_I1_J) ).
thf(3493,plain,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ ( zero_zero @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[704]) ).
thf(197,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ ( bit1 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_less__int__code_I16_J) ).
thf(1749,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ ( bit1 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ ( bit1 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).
thf(1121,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B ) ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_986_mod__less__divisor) ).
thf(4932,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1121]) ).
thf(1020,axiom,
monoid_mult @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Omonoid__mult) ).
thf(4538,plain,
monoid_mult @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1020]) ).
thf(74,axiom,
( ( ti @ int @ min )
= min ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Int_OMin_res) ).
thf(1427,plain,
( ( ti @ int @ min )
= min ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(908,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B )
= ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) ).
thf(4202,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B )
= ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[908]) ).
thf(147,axiom,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_783_add__strict__mono) ).
thf(1585,plain,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(417,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( zero_zero @ A ) ) @ B )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_234_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) ).
thf(2511,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( zero_zero @ A ) ) @ B )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[417]) ).
thf(521,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ( ti @ int @ A )
= ( zero_zero @ int ) )
| ( ( ti @ int @ A )
= ( one_one @ int ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_184_int__pos__lt__two__imp__zero__or__one) ).
thf(2860,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ( ti @ int @ A )
= ( zero_zero @ int ) )
| ( ( ti @ int @ A )
= ( one_one @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[521]) ).
thf(1251,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_728_nat__mult__less__cancel1) ).
thf(5364,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1251]) ).
thf(549,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( number_number_of @ int @ min ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_463__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_N_A_N1_096) ).
thf(2949,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( number_number_of @ int @ min ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[549]) ).
thf(53,axiom,
! [A: $i] :
( ( ring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) @ C ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_951_mod__diff__left__eq) ).
thf(1372,plain,
! [A: $i] :
( ( ring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) @ C ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(1091,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ s1 ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of @ int @ min ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_s1) ).
thf(4838,plain,
hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ s1 ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of @ int @ min ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1091]) ).
thf(534,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_260_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) ).
thf(2892,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[534]) ).
thf(611,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) )
=> ( ( ( ti @ int @ A )
!= ( zero_zero @ int ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_zdvd__mult__cancel) ).
thf(3168,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) )
=> ( ( ( ti @ int @ A )
!= ( zero_zero @ int ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[611]) ).
thf(692,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_640_trans__less__add1) ).
thf(3452,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[692]) ).
thf(1056,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ A ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_604_diff__diff__cancel) ).
thf(4727,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1056]) ).
thf(370,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ real @ real @ ( times_times @ real @ ( hAPP @ real @ real @ ( plus_plus @ real @ A ) @ B ) ) @ C )
= ( hAPP @ real @ real @ ( plus_plus @ real @ ( hAPP @ real @ real @ ( times_times @ real @ A ) @ C ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_692_real__add__mult__distrib) ).
thf(2310,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ real @ real @ ( times_times @ real @ ( hAPP @ real @ real @ ( plus_plus @ real @ A ) @ B ) ) @ C )
= ( hAPP @ real @ real @ ( plus_plus @ real @ ( hAPP @ real @ real @ ( times_times @ real @ A ) @ C ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[370]) ).
thf(1140,axiom,
comm_ring_1 @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Ocomm__ring__1) ).
thf(4997,plain,
comm_ring_1 @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1140]) ).
thf(113,axiom,
! [A: $i,B: $i] :
( ( no_zero_divisors @ B )
=> ( ( times_times @ B @ ( ti @ B @ A ) )
= ( times_times @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_5_arg1) ).
thf(1517,plain,
! [A: $i,B: $i] :
( ( no_zero_divisors @ B )
=> ( ( times_times @ B @ ( ti @ B @ A ) )
= ( times_times @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(453,axiom,
! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ B ) ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_351_zcong__zmult__self) ).
thf(2611,plain,
! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ B ) ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[453]) ).
thf(202,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_389_mult__pos__neg) ).
thf(1781,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).
thf(331,axiom,
! [A: $i,B: $i] :
( ( ( one_one @ nat )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) )
<=> ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_689_nat__1__eq__mult__iff) ).
thf(2187,plain,
! [A: $i,B: $i] :
( ( ( ( one_one @ nat )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) )
=> ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) ) )
& ( ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) )
=> ( ( one_one @ nat )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).
thf(341,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) )
= ( number_number_of @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_67_plus__numeral__code_I9_J) ).
thf(2226,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) )
= ( number_number_of @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).
thf(1057,axiom,
ordere223160158up_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add) ).
thf(4730,plain,
ordere223160158up_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1057]) ).
thf(267,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( zero_zero @ nat ) ) )
<=> ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_627_le__0__eq) ).
thf(1998,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( zero_zero @ nat ) ) )
=> ( A
= ( zero_zero @ nat ) ) )
& ( ( A
= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( zero_zero @ nat ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).
thf(272,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( ( ti @ int @ C )
= ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( one_one @ int ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_self__quotient__aux2) ).
thf(2013,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( ( ti @ int @ C )
= ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( one_one @ int ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).
thf(676,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_570_dvd__antisym) ).
thf(3400,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[676]) ).
thf(829,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ A @ B @ ( ti @ ( fun @ A @ B ) @ C ) @ D )
= ( hAPP @ A @ B @ C @ D ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_hAPP_arg1) ).
thf(3946,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ A @ B @ ( ti @ ( fun @ A @ B ) @ C ) @ D )
= ( hAPP @ A @ B @ C @ D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[829]) ).
thf(933,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_477_zdvd__zdiffD) ).
thf(4289,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[933]) ).
thf(736,axiom,
! [A: $i] :
( ( ( number @ A )
& ( semiring @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( number_number_of @ A @ D ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( number_number_of @ A @ D ) ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ ( number_number_of @ A @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_left__distrib__number__of) ).
thf(3584,plain,
! [A: $i] :
( ( ( number @ A )
& ( semiring @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( number_number_of @ A @ D ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( number_number_of @ A @ D ) ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ ( number_number_of @ A @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[736]) ).
thf(743,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) @ A )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_678_add__diff__assoc2) ).
thf(3620,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) @ A )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[743]) ).
thf(458,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( div_mod @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_978_mod__mult__distrib) ).
thf(2626,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( div_mod @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[458]) ).
thf(982,axiom,
! [A: $i] :
( ( ti @ ( fun @ int @ bool ) @ ( sr @ A ) )
= ( sr @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Residues_OSR_res) ).
thf(4424,plain,
! [A: $i] :
( ( ti @ ( fun @ int @ bool ) @ ( sr @ A ) )
= ( sr @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[982]) ).
thf(1048,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B )
= ( zero_zero @ nat ) )
<=> ? [C: $i] :
( A
= ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_985_mod__eq__0__iff) ).
thf(4698,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B )
= ( zero_zero @ nat ) )
=> ? [C: $i] :
( A
= ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ C ) ) )
& ( ? [C: $i] :
( A
= ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1048]) ).
thf(1194,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) @ E ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ D ) ) @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_353_zcong__zmult__prop2) ).
thf(5179,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ D ) @ E ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) @ E ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ D ) ) @ E ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ D ) ) @ E ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1194]) ).
thf(87,axiom,
! [A: $i,B: $i] :
( ( group_add @ B )
=> ( ( minus_minus @ B @ ( ti @ B @ A ) )
= ( minus_minus @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Ominus__class_Ominus_0_arg1) ).
thf(1459,plain,
! [A: $i,B: $i] :
( ( group_add @ B )
=> ( ( minus_minus @ B @ ( ti @ B @ A ) )
= ( minus_minus @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(1034,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ A @ B @ C @ ( ti @ A @ D ) )
= ( hAPP @ A @ B @ C @ D ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_hAPP_arg2) ).
thf(4587,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ A @ B @ C @ ( ti @ A @ D ) )
= ( hAPP @ A @ B @ C @ D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1034]) ).
thf(1069,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_669_diff__diff__right) ).
thf(4763,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1069]) ).
thf(900,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ B ) @ ( wset @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_829_wset__mem__mem) ).
thf(4165,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ B ) @ ( wset @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[900]) ).
thf(511,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( number_number_of @ int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( ( inv @ B @ ( inv @ B @ A ) )
= ( ti @ int @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_819_inv__inv) ).
thf(2828,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( number_number_of @ int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( ( inv @ B @ ( inv @ B @ A ) )
= ( ti @ int @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[511]) ).
thf(612,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( number_number_of @ int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ ( inv @ C @ A ) ) @ ( wset @ B @ C ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ B @ C ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_828_wset__inv__mem__mem) ).
thf(3171,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( number_number_of @ int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ ( inv @ C @ A ) ) @ ( wset @ B @ C ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ B @ C ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[612]) ).
thf(554,axiom,
linord219039673up_add @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Olinordered__ab__group__add) ).
thf(2960,plain,
linord219039673up_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[554]) ).
thf(1104,axiom,
! [A: $i] :
( ( one @ A )
=> ! [B: $i] :
( ( ( one_one @ A )
= ( ti @ A @ B ) )
<=> ( ( ti @ A @ B )
= ( one_one @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_765_one__reorient) ).
thf(4873,plain,
! [A: $i] :
( ( one @ A )
=> ! [B: $i] :
( ( ( ( one_one @ A )
= ( ti @ A @ B ) )
=> ( ( ti @ A @ B )
= ( one_one @ A ) ) )
& ( ( ( ti @ A @ B )
= ( one_one @ A ) )
=> ( ( one_one @ A )
= ( ti @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1104]) ).
thf(293,axiom,
ab_semigroup_mult @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oab__semigroup__mult) ).
thf(2078,plain,
ab_semigroup_mult @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).
thf(979,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_308_dvd__trans) ).
thf(4419,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[979]) ).
thf(720,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
!= ( zero_zero @ nat ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ B ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_881_pow__divides__eq__nat) ).
thf(3533,plain,
! [A: $i,B: $i,C: $i] :
( ( C
!= ( zero_zero @ nat ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ B ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[720]) ).
thf(691,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ ( bit0 @ A ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_rel__simps_I21_J) ).
thf(3447,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ ( bit0 @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ ( bit0 @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[691]) ).
thf(167,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_335_dvd__mult__right) ).
thf(1654,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(69,axiom,
cancel_semigroup_add @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Ocancel__semigroup__add) ).
thf(1412,plain,
cancel_semigroup_add @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(745,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ B ) ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_554_dvd_Oless__trans) ).
thf(3625,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ B ) ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ C ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[745]) ).
thf(775,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) @ ( number_number_of @ A @ B ) )
= ( number_number_of @ A @ ( bit0 @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_double__number__of__Bit0) ).
thf(3725,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) @ ( number_number_of @ A @ B ) )
= ( number_number_of @ A @ ( bit0 @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[775]) ).
thf(303,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) )
<=> ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ C )
= ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_956_zcong__zmod__eq) ).
thf(2110,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ C )
= ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) ) )
& ( ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ C )
= ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).
thf(60,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ E ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_214_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) ).
thf(1390,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(1178,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_919_dvd__mod__iff) ).
thf(5124,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1178]) ).
thf(141,axiom,
ring @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Oring) ).
thf(1575,plain,
ring @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(292,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( number_number_of @ int @ B ) ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ ( number_number_of @ int @ ( bit1 @ A ) ) ) @ ( number_number_of @ int @ ( bit0 @ B ) ) )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( hAPP @ int @ int @ ( div_mod @ int @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) ) ) ) @ ( one_one @ int ) ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( number_number_of @ int @ B ) ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ ( number_number_of @ int @ ( bit1 @ A ) ) ) @ ( number_number_of @ int @ ( bit0 @ B ) ) )
= ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( number_number_of @ int @ A ) ) @ ( one_one @ int ) ) ) @ ( number_number_of @ int @ B ) ) ) ) @ ( one_one @ int ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_898_zmod__number__of__Bit1) ).
thf(2071,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( number_number_of @ int @ B ) ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ ( number_number_of @ int @ ( bit1 @ A ) ) ) @ ( number_number_of @ int @ ( bit0 @ B ) ) )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( hAPP @ int @ int @ ( div_mod @ int @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) ) ) ) @ ( one_one @ int ) ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( number_number_of @ int @ B ) ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ ( number_number_of @ int @ ( bit1 @ A ) ) ) @ ( number_number_of @ int @ ( bit0 @ B ) ) )
= ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( number_number_of @ int @ A ) ) @ ( one_one @ int ) ) ) @ ( number_number_of @ int @ B ) ) ) ) @ ( one_one @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).
thf(825,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
<=> ( ( ti @ A @ B )
!= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_zero__less__power2) ).
thf(3929,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
=> ( ( ti @ A @ B )
!= ( zero_zero @ A ) ) )
& ( ( ( ti @ A @ B )
!= ( zero_zero @ A ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[825]) ).
thf(1163,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
& ( A != B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_648_nat__less__le) ).
thf(5068,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
& ( A != B ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
& ( A != B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1163]) ).
thf(503,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_418_zcong__eq__zdvd__prop) ).
thf(2792,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[503]) ).
thf(1196,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ C ) )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_737_nat__diff__add__eq2) ).
thf(5184,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ C ) )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1196]) ).
thf(538,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_423_mult__less__imp__less__left) ).
thf(2907,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[538]) ).
thf(1231,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_514_divides__mul__r) ).
thf(5315,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1231]) ).
thf(445,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_390_mult__pos__pos) ).
thf(2594,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[445]) ).
thf(358,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ ( bit0 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_rel__simps_I33_J) ).
thf(2273,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ ( bit0 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( bit1 @ A ) ) @ ( bit0 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).
thf(700,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_343_nat__power__less__imp__less) ).
thf(3484,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[700]) ).
thf(520,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_661_nat__add__left__cancel__le) ).
thf(2855,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[520]) ).
thf(348,axiom,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( one_one @ int ) ) @ t ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5__096_126_A1_A_060_061_At_096) ).
thf(2251,plain,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( one_one @ int ) ) @ t ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[348]) ).
thf(240,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_873_gcd__lcm__complete__lattice__nat_Otop__greatest) ).
thf(1903,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).
thf(1255,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_power2__ge__self) ).
thf(5382,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1255]) ).
thf(95,axiom,
! [A: $i,B: $i] :
( ( semiring @ B )
=> ( ( times_times @ B @ ( ti @ B @ A ) )
= ( times_times @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_3_arg1) ).
thf(1481,plain,
! [A: $i,B: $i] :
( ( semiring @ B )
=> ( ( times_times @ B @ ( ti @ B @ A ) )
= ( times_times @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(1244,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_711_dvd__imp__le) ).
thf(5351,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1244]) ).
thf(654,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) ) ) @ C )
= ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_949_zdiff__zmod__right) ).
thf(3336,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) ) ) @ C )
= ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[654]) ).
thf(864,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ D ) @ E ) )
=> ( ( ( ti @ int @ D )
= ( ti @ int @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ A ) @ E ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ A ) @ E ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_311_zcong__eq__trans) ).
thf(4042,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ D ) @ E ) )
=> ( ( ( ti @ int @ D )
= ( ti @ int @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ A ) @ E ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ A ) @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[864]) ).
thf(1208,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( one_one @ int ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( one_one @ int ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_973_pos__zmod__mult__2) ).
thf(5228,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( one_one @ int ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( one_one @ int ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1208]) ).
thf(953,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) @ C ) ) @ B )
= ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ C ) ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_948_zdiff__zmod__left) ).
thf(4338,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B ) ) @ C ) ) @ B )
= ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ C ) ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[953]) ).
thf(568,axiom,
mult_zero @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Omult__zero) ).
thf(3003,plain,
mult_zero @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[568]) ).
thf(1088,axiom,
! [A: $i] :
( ( ordered_ab_group_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( minus_minus @ A @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_791_diff__eq__diff__less) ).
thf(4823,plain,
! [A: $i] :
( ( ordered_ab_group_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( minus_minus @ A @ D ) @ E ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1088]) ).
thf(389,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
<=> ( ( standardRes @ B @ A )
!= ( zero_zero @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_989_StandardRes__prop3) ).
thf(2394,plain,
! [A: $i,B: $i] :
( ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( ( standardRes @ B @ A )
!= ( zero_zero @ int ) ) )
& ( ( ( standardRes @ B @ A )
!= ( zero_zero @ int ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[389]) ).
thf(13,axiom,
number @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Int_Onumber) ).
thf(1276,plain,
number @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(444,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_608_nat__mult__commute) ).
thf(2591,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ B ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[444]) ).
thf(33,axiom,
! [A: $i,B: $i] :
( ( inv @ A @ ( ti @ int @ B ) )
= ( inv @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_WilsonRuss_Oinv_arg2) ).
thf(1326,plain,
! [A: $i,B: $i] :
( ( inv @ A @ ( ti @ int @ B ) )
= ( inv @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(777,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ A ) )
=> ( ( ( ti @ int @ A )
!= ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) )
=> ( ( ( ti @ int @ A )
!= ( number_number_of @ int @ ( bit1 @ ( bit1 @ pls ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( number_number_of @ int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_273_prime__g__5) ).
thf(3729,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ A ) )
=> ( ( ( ti @ int @ A )
!= ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) )
=> ( ( ( ti @ int @ A )
!= ( number_number_of @ int @ ( bit1 @ ( bit1 @ pls ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( number_number_of @ int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[777]) ).
thf(485,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ A ) )
=> ( ( ti @ int @ B )
= ( zero_zero @ int ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_448_zcong__zless__0) ).
thf(2707,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ A ) )
=> ( ( ti @ int @ B )
= ( zero_zero @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[485]) ).
thf(1114,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ B ) )
<=> ( A
= ( one_one @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_531_dvd__mult__cancel2) ).
thf(4913,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ B ) )
=> ( A
= ( one_one @ nat ) ) )
& ( ( A
= ( one_one @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1114]) ).
thf(361,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_457_one__less__power) ).
thf(2283,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[361]) ).
thf(387,axiom,
! [A: $i,B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_271_Euler_Oaux____1) ).
thf(2389,plain,
! [A: $i,B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[387]) ).
thf(1082,axiom,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_370_mult__nonpos__nonpos) ).
thf(4806,plain,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1082]) ).
thf(423,axiom,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ ( one_one @ A ) ) @ B )
= ( one_one @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_337_power__one) ).
thf(2532,plain,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ ( one_one @ A ) ) @ B )
= ( one_one @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[423]) ).
thf(108,axiom,
comm_semiring @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Ocomm__semiring) ).
thf(1508,plain,
comm_semiring @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(160,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( number_number_of @ int @ min ) ) @ B ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( one_one @ int ) ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_313_zcong__neg__1__impl__ne__1) ).
thf(1629,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( number_number_of @ int @ min ) ) @ B ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( one_one @ int ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(593,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_339_dvd__power__same) ).
thf(3104,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[593]) ).
thf(728,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ ( zero_zero @ nat ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_537_minus__nat_Odiff__0) ).
thf(3561,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ ( zero_zero @ nat ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[728]) ).
thf(451,axiom,
linordered_semiring @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Olinordered__semiring) ).
thf(2608,plain,
linordered_semiring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[451]) ).
thf(631,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_808_add__neg__neg) ).
thf(3235,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[631]) ).
thf(713,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ nat @ int @ ( power_power @ int @ D ) @ A ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ nat @ int @ ( power_power @ int @ D ) @ A ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_zprime__power__zdvd__cancel__right) ).
thf(3516,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ nat @ int @ ( power_power @ int @ D ) @ A ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ nat @ int @ ( power_power @ int @ D ) @ A ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[713]) ).
thf(1007,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_450_zpower__zdvd__prop1) ).
thf(4495,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1007]) ).
thf(1212,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ( B != A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_559_dvd_Oless__imp__not__eq2) ).
thf(5243,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
=> ( B != A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1212]) ).
thf(115,axiom,
monoid_mult @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Omonoid__mult) ).
thf(1523,plain,
monoid_mult @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(1049,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_585_nat__add__assoc) ).
thf(4705,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1049]) ).
thf(311,axiom,
( ( one_one @ int )
= ( number_number_of @ int @ ( bit1 @ pls ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_111_one__is__num__one) ).
thf(2130,plain,
( ( one_one @ int )
= ( number_number_of @ int @ ( bit1 @ pls ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).
thf(653,axiom,
! [A: $i] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ! [E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ E ) @ ( hAPP @ A @ A @ ( times_times @ A @ F ) @ D ) ) ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_745_inf__period_I3_J) ).
thf(3331,plain,
! [A: $i] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ! [E: $i,F: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ E ) @ ( hAPP @ A @ A @ ( times_times @ A @ F ) @ D ) ) ) @ B ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ E ) @ ( hAPP @ A @ A @ ( times_times @ A @ F ) @ D ) ) ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[653]) ).
thf(666,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_382_mult__strict__right__mono) ).
thf(3368,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[666]) ).
thf(758,axiom,
zero @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ozero) ).
thf(3662,plain,
zero @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[758]) ).
thf(420,axiom,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) )
= ( hAPP @ nat @ A @ ( power_power @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_232_power__even__eq) ).
thf(2526,plain,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) )
= ( hAPP @ nat @ A @ ( power_power @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[420]) ).
thf(132,axiom,
ring_11004092258visors @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors) ).
thf(1558,plain,
ring_11004092258visors @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(542,axiom,
! [A: $i] :
( ( ( plus @ A )
& ( linorder @ A )
& ( dvd @ A ) )
=> ! [B: $i,C: $i] :
? [D: $i] :
! [E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_864_pinf_I9_J) ).
thf(2934,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[542]) ).
thf(319,axiom,
comm_semiring @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Ocomm__semiring) ).
thf(2152,plain,
comm_semiring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).
thf(903,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( one_one @ int ) ) ) @ B ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_83_add1__zle__eq) ).
thf(4182,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( one_one @ int ) ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( one_one @ int ) ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[903]) ).
thf(99,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ C )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_945_mod__mult__self2__is__0) ).
thf(1491,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ C )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(840,axiom,
no_zero_divisors @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Ono__zero__divisors) ).
thf(3978,plain,
no_zero_divisors @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[840]) ).
thf(1067,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ B ) ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ ( number_number_of @ A @ C ) ) @ D ) )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( number_number_of @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_492_add__number__of__diff1) ).
thf(4759,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( number_number_of @ A @ B ) ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ ( number_number_of @ A @ C ) ) @ D ) )
= ( hAPP @ A @ A @ ( minus_minus @ A @ ( number_number_of @ A @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1067]) ).
thf(290,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( d22set @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_860_d22set__g__1) ).
thf(2066,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( d22set @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).
thf(182,axiom,
ring_char_0 @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Int_Oring__char__0) ).
thf(1706,plain,
ring_char_0 @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(111,axiom,
idom @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Oidom) ).
thf(1513,plain,
idom @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(1237,axiom,
( ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( number_number_of @ int @ min ) )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( one_one @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_466__096s_A_094_A2_A_N_A_N1_A_061_As_A_094_A2_A_L_A1_096) ).
thf(5336,plain,
( ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( number_number_of @ int @ min ) )
= ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( one_one @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1237]) ).
thf(408,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ ( one_one @ int ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_163_int__0__less__1) ).
thf(2475,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ ( one_one @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[408]) ).
thf(1039,axiom,
! [A: $i] :
( ( zfact @ ( ti @ int @ A ) )
= ( zfact @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_IntFact_Ozfact_arg1) ).
thf(4602,plain,
! [A: $i] :
( ( zfact @ ( ti @ int @ A ) )
= ( zfact @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1039]) ).
thf(1191,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( one_one @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_396_zero__le__one) ).
thf(5164,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( one_one @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1191]) ).
thf(1190,axiom,
( pls
= ( zero_zero @ int ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_Pls__def) ).
thf(5162,plain,
( pls
= ( zero_zero @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1190]) ).
thf(935,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_552_diff__less__mono2) ).
thf(4293,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[935]) ).
thf(706,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_332_mult__dvd__mono) ).
thf(3497,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ E ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[706]) ).
thf(808,axiom,
number @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Int_Onumber) ).
thf(3879,plain,
number @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[808]) ).
thf(464,axiom,
! [A: $i,B: $i] :
( ( A = B )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_576_dvd_Oeq__refl) ).
thf(2648,plain,
! [A: $i,B: $i] :
( ( A = B )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[464]) ).
thf(687,axiom,
! [A: $i,B: $i] :
( ( ( bit1 @ A )
= ( bit1 @ B ) )
<=> ( ( ti @ int @ A )
= ( ti @ int @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_rel__simps_I51_J) ).
thf(3430,plain,
! [A: $i,B: $i] :
( ( ( ( bit1 @ A )
= ( bit1 @ B ) )
=> ( ( ti @ int @ A )
= ( ti @ int @ B ) ) )
& ( ( ( ti @ int @ A )
= ( ti @ int @ B ) )
=> ( ( bit1 @ A )
= ( bit1 @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[687]) ).
thf(640,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) )
= ( number_number_of @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_times__numeral__code_I5_J) ).
thf(3267,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ A ) ) @ ( number_number_of @ int @ B ) )
= ( number_number_of @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[640]) ).
thf(1074,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ A @ C ) @ B ) ) )
=> ( ( ( B = C )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ A @ C ) @ B ) ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ A @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ A @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_550_nat__less__cases) ).
thf(4774,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ A @ C ) @ B ) ) )
=> ( ( ( B = C )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ A @ C ) @ B ) ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ A @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ A @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1074]) ).
thf(21,axiom,
! [A: $i,B: $i] :
( ( semiring @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( times_times @ B @ A ) )
= ( times_times @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_3_res) ).
thf(1294,plain,
! [A: $i,B: $i] :
( ( semiring @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( times_times @ B @ A ) )
= ( times_times @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(366,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_252_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) ).
thf(2294,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).
thf(750,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( one_one @ int ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_181__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_L_A1_096) ).
thf(3645,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ m ) ) @ ( one_one @ int ) ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ s ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( one_one @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[750]) ).
thf(878,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) ) ) @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_909_zmod__zmult1__eq) ).
thf(4098,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) ) ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[878]) ).
thf(412,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( zero_zero @ nat ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ B @ A ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ A ) )
& ! [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ B @ D ) ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ ( one_one @ nat ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_719_ex__least__nat__less) ).
thf(2494,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( zero_zero @ nat ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ B @ A ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ A ) )
& ! [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ B @ D ) ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ ( one_one @ nat ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[412]) ).
thf(82,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( combc @ A @ B @ C @ D @ ( ti @ B @ E ) )
= ( combc @ A @ B @ C @ D @ E ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_COMBC_arg2) ).
thf(1444,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( combc @ A @ B @ C @ D @ ( ti @ B @ E ) )
= ( combc @ A @ B @ C @ D @ E ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(940,axiom,
! [A: $i] :
( ( number_semiring @ A )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
= ( number_number_of @ A @ ( bit0 @ ( bit1 @ pls ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_semiring__one__add__one__is__two) ).
thf(4306,plain,
! [A: $i] :
( ( number_semiring @ A )
=> ( ( hAPP @ A @ A @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
= ( number_number_of @ A @ ( bit0 @ ( bit1 @ pls ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[940]) ).
thf(1199,axiom,
! [A: $i] :
( ( linordered_ring @ A )
=> ! [B: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_394_not__square__less__zero) ).
thf(5194,plain,
! [A: $i] :
( ( linordered_ring @ A )
=> ! [B: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) @ ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1199]) ).
thf(683,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ pls ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_rel__simps_I12_J) ).
thf(3420,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ pls ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ pls ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[683]) ).
thf(395,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) )
= ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) ).
thf(2421,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) )
= ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[395]) ).
thf(301,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ A ) @ B ) )
=> ( ( ti @ int @ C )
= ( ti @ int @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_449_zcong__zless__imp__eq) ).
thf(2102,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ A ) @ B ) )
=> ( ( ti @ int @ C )
= ( ti @ int @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).
thf(110,axiom,
linorder @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Orderings_Olinorder) ).
thf(1512,plain,
linorder @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(419,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ 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_522_nat__mult__eq__one) ).
thf(2517,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( one_one @ nat ) )
=> ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) ) )
& ( ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( one_one @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[419]) ).
thf(804,axiom,
! [A: $i] :
( ( ordere779506340up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_774_add__right__mono) ).
thf(3860,plain,
! [A: $i] :
( ( ordere779506340up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[804]) ).
thf(967,axiom,
! [A: $i] :
( ( ( bit1 @ A )
= min )
<=> ( ( ti @ int @ A )
= min ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_281_rel__simps_I47_J) ).
thf(4378,plain,
! [A: $i] :
( ( ( ( bit1 @ A )
= min )
=> ( ( ti @ int @ A )
= min ) )
& ( ( ( ti @ int @ A )
= min )
=> ( ( bit1 @ A )
= min ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[967]) ).
thf(813,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_zadd__assoc) ).
thf(3892,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[813]) ).
thf(126,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) @ C )
= ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_942_mod__mult__self2) ).
thf(1546,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) @ C )
= ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(158,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_zprime__zdvd__power) ).
thf(1625,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(638,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( one_one @ int ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_257_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J) ).
thf(3264,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( one_one @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[638]) ).
thf(1165,axiom,
hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_274_pos2) ).
thf(5078,plain,
hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1165]) ).
thf(715,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ pls ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_rel__simps_I19_J) ).
thf(3521,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ pls ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[715]) ).
thf(175,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ C ) )
=> ( ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ B )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_941_dvd__imp__mod__0) ).
thf(1673,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ C ) )
=> ( ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ B )
= ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(284,axiom,
hBOOL @ ( hAPP @ int @ bool @ zprime @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_zprime__2) ).
thf(2050,plain,
hBOOL @ ( hAPP @ int @ bool @ zprime @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).
thf(199,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ A ) )
<=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
& ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) ) )
=> ( ( ( ti @ int @ B )
= ( one_one @ int ) )
| ( ( ti @ int @ B )
= ( ti @ int @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_272_zprime__def) ).
thf(1757,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
& ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) ) )
=> ( ( ( ti @ int @ B )
= ( one_one @ int ) )
| ( ( ti @ int @ B )
= ( ti @ int @ A ) ) ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
& ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
& ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) ) )
=> ( ( ( ti @ int @ B )
= ( one_one @ int ) )
| ( ( ti @ int @ B )
= ( ti @ int @ A ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ zprime @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(1147,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_223_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) ).
thf(5014,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1147]) ).
thf(1213,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) @ ( zero_zero @ A ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_143_even__less__0__iff) ).
thf(5247,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1213]) ).
thf(963,axiom,
! [A: $i] :
( ( comm_monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( one_one @ A ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_795_mult_Ocomm__neutral) ).
thf(4367,plain,
! [A: $i] :
( ( comm_monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( one_one @ A ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[963]) ).
thf(845,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( number_number_of @ nat @ B ) ) @ ( number_number_of @ nat @ A ) )
= ( number_number_of @ nat @ A ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ pls ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( number_number_of @ nat @ B ) ) @ ( number_number_of @ nat @ A ) )
= ( number_number_of @ nat @ B ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( number_number_of @ nat @ B ) ) @ ( number_number_of @ nat @ A ) )
= ( number_number_of @ nat @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_86_add__nat__number__of) ).
thf(3992,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( number_number_of @ nat @ B ) ) @ ( number_number_of @ nat @ A ) )
= ( number_number_of @ nat @ A ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ pls ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( number_number_of @ nat @ B ) ) @ ( number_number_of @ nat @ A ) )
= ( number_number_of @ nat @ B ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( number_number_of @ nat @ B ) ) @ ( number_number_of @ nat @ A ) )
= ( number_number_of @ nat @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[845]) ).
thf(315,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ ( inv @ B @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_821_inv__g__1) ).
thf(2142,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ B ) @ ( one_one @ int ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ ( inv @ B @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).
thf(669,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( multInv @ D @ C ) ) ) @ C ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) @ B ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_845_aux______2) ).
thf(3379,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ ( multInv @ D @ C ) ) ) @ C ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[669]) ).
thf(308,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_599_nat__le__linear) ).
thf(2123,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).
thf(1197,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_605_diff__le__mono) ).
thf(5187,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1197]) ).
thf(351,axiom,
semiring_1 @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Osemiring__1) ).
thf(2258,plain,
semiring_1 @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[351]) ).
thf(1157,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( minus_minus @ int @ min ) @ ( bit0 @ A ) )
= ( bit1 @ ( hAPP @ int @ int @ ( minus_minus @ int @ min ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_496_diff__bin__simps_I5_J) ).
thf(5051,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( minus_minus @ int @ min ) @ ( bit0 @ A ) )
= ( bit1 @ ( hAPP @ int @ int @ ( minus_minus @ int @ min ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1157]) ).
thf(1105,axiom,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_810_add__le__less__mono) ).
thf(4880,plain,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ E ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1105]) ).
thf(43,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ real @ real @ ( times_times @ real @ ( hAPP @ real @ real @ ( times_times @ real @ A ) @ B ) ) @ C )
= ( hAPP @ real @ real @ ( times_times @ real @ A ) @ ( hAPP @ real @ real @ ( times_times @ real @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_593_real__mult__assoc) ).
thf(1348,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ real @ real @ ( times_times @ real @ ( hAPP @ real @ real @ ( times_times @ real @ A ) @ B ) ) @ C )
= ( hAPP @ real @ real @ ( times_times @ real @ A ) @ ( hAPP @ real @ real @ ( times_times @ real @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(770,axiom,
! [A: $i] :
( ( ordere779506340up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_775_add__left__mono) ).
thf(3705,plain,
! [A: $i] :
( ( ordere779506340up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[770]) ).
thf(15,axiom,
! [A: $i,B: $i] :
( ( wset @ A @ ( ti @ int @ B ) )
= ( wset @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_WilsonRuss_Owset_arg2) ).
thf(1280,plain,
! [A: $i,B: $i] :
( ( wset @ A @ ( ti @ int @ B ) )
= ( wset @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(852,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ C ) ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_535_diff__commute) ).
thf(4014,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ C ) ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[852]) ).
thf(998,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( zero_zero @ int ) ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_416_zdvd__bounds) ).
thf(4472,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( zero_zero @ int ) ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[998]) ).
thf(972,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( one_one @ int ) ) ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_zless__imp__add1__zle) ).
thf(4399,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( one_one @ int ) ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[972]) ).
thf(960,axiom,
! [A: $i] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [B: $i,C: $i] :
( ? [D: $i] : ( hBOOL @ ( hAPP @ A @ bool @ B @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) )
<=> ? [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ ( zero_zero @ A ) ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ B @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_744_unity__coeff__ex) ).
thf(4356,plain,
! [A: $i] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [B: $i,C: $i] :
( ( ? [D: $i] : ( hBOOL @ ( hAPP @ A @ bool @ B @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) )
=> ? [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ ( zero_zero @ A ) ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ B @ D ) ) ) )
& ( ? [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ ( zero_zero @ A ) ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ B @ D ) ) )
=> ? [D: $i] : ( hBOOL @ ( hAPP @ A @ bool @ B @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[960]) ).
thf(817,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_not__sum__power2__lt__zero) ).
thf(3907,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ C ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[817]) ).
thf(1084,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) )
=> ( ( ti @ int @ B )
= ( ti @ int @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_zdvd__antisym__nonneg) ).
thf(4811,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) )
=> ( ( ti @ int @ B )
= ( ti @ int @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1084]) ).
thf(442,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( combb @ A @ B @ C @ D @ ( ti @ ( fun @ C @ A ) @ E ) )
= ( combb @ A @ B @ C @ D @ E ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_COMBB_arg2) ).
thf(2587,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( combb @ A @ B @ C @ D @ ( ti @ ( fun @ C @ A ) @ E ) )
= ( combb @ A @ B @ C @ D @ E ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[442]) ).
thf(1219,axiom,
semiring_0 @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Osemiring__0) ).
thf(5268,plain,
semiring_0 @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1219]) ).
thf(344,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( hAPP @ real @ real @ ( times_times @ real @ C ) @ A ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_705_real__mult__less__mono2) ).
thf(2233,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( zero_zero @ real ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ real @ bool @ ( hAPP @ real @ ( fun @ real @ bool ) @ ( ord_less @ real ) @ ( hAPP @ real @ real @ ( times_times @ real @ C ) @ A ) ) @ ( hAPP @ real @ real @ ( times_times @ real @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[344]) ).
thf(668,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ D ) )
=> ( ( ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B )
= ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) )
<=> ( B = C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_409_power__inject__exp) ).
thf(3373,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ D ) )
=> ( ( ( ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B )
= ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) )
=> ( B = C ) )
& ( ( B = C )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B )
= ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[668]) ).
thf(1247,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_385_mult__less__cancel__left__neg) ).
thf(5356,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( zero_zero @ A ) ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1247]) ).
thf(270,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_683_le__square) ).
thf(2008,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).
thf(1093,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ B )
= ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_929_mod__add__self1) ).
thf(4841,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ B )
= ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1093]) ).
thf(263,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ C ) @ A ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_358_zcong__shift) ).
thf(1986,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ A ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ C ) @ A ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).
thf(496,axiom,
! [A: $i,B: $i] :
( ( collect @ A @ B )
= ( ti @ ( fun @ A @ bool ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_Collect__def) ).
thf(2751,plain,
! [A: $i,B: $i] :
( ( collect @ A @ B )
= ( ti @ ( fun @ A @ bool ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[496]) ).
thf(652,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_224_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) ).
thf(3328,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[652]) ).
thf(727,axiom,
! [A: $i] :
( ( ti @ ( fun @ nat @ nat ) @ ( minus_minus @ nat @ A ) )
= ( minus_minus @ nat @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Ominus__class_Ominus_1_res) ).
thf(3558,plain,
! [A: $i] :
( ( ti @ ( fun @ nat @ nat ) @ ( minus_minus @ nat @ A ) )
= ( minus_minus @ nat @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[727]) ).
thf(1205,axiom,
! [A: $i] :
( ( ordered_ab_group_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( minus_minus @ A @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_790_diff__eq__diff__less__eq) ).
thf(5215,plain,
! [A: $i] :
( ( ordered_ab_group_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( minus_minus @ A @ D ) @ E ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1205]) ).
thf(379,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) ) )
<=> ~ ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( zero_zero @ nat ) ) ) )
| ? [D: $i] :
( ( B
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ D ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ A @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_696_nat__diff__split__asm) ).
thf(2340,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) ) )
=> ~ ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( zero_zero @ nat ) ) ) )
| ? [D: $i] :
( ( B
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ D ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ A @ D ) ) ) ) )
& ( ~ ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( zero_zero @ nat ) ) ) )
| ? [D: $i] :
( ( B
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ D ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ A @ D ) ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).
thf(1233,axiom,
! [A: $i,B: $i] :
( ( monoid_add @ B )
=> ( ( plus_plus @ B @ ( ti @ B @ A ) )
= ( plus_plus @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Oplus__class_Oplus_1_arg1) ).
thf(5322,plain,
! [A: $i,B: $i] :
( ( monoid_add @ B )
=> ( ( plus_plus @ B @ ( ti @ B @ A ) )
= ( plus_plus @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1233]) ).
thf(939,axiom,
! [A: $i] :
( ( hAPP @ nat @ int @ ( power_power @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_269_quartic__square__square) ).
thf(4303,plain,
! [A: $i] :
( ( hAPP @ nat @ int @ ( power_power @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[939]) ).
thf(867,axiom,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ D ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_412_power__add) ).
thf(4051,plain,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ D ) )
= ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[867]) ).
thf(1226,axiom,
linordered_semidom @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__semidom) ).
thf(5300,plain,
linordered_semidom @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1226]) ).
thf(48,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( times_times @ A @ C ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_218_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) ).
thf(1363,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( times_times @ A @ C ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(29,axiom,
ring_n68954251visors @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Oring__no__zero__divisors) ).
thf(1314,plain,
ring_n68954251visors @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(1189,axiom,
! [A: $i] :
( ( order @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( ord_less @ A ) )
= ( ord_less @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Orderings_Oord__class_Oless_res) ).
thf(5159,plain,
! [A: $i] :
( ( order @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( ord_less @ A ) )
= ( ord_less @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1189]) ).
thf(101,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) ).
thf(1495,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C )
= ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(169,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_425_mult__less__imp__less__right) ).
thf(1657,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ D ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(1170,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( bit0 @ A ) ) @ ( bit1 @ B ) )
= ( bit1 @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_162_add__Bit0__Bit1) ).
thf(5092,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( bit0 @ A ) ) @ ( bit1 @ B ) )
= ( bit1 @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1170]) ).
thf(1110,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) )
<=> ( ( A
= ( zero_zero @ nat ) )
| ( B = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_726_nat__mult__eq__cancel__disj) ).
thf(4895,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) )
=> ( ( A
= ( zero_zero @ nat ) )
| ( B = C ) ) )
& ( ( ( A
= ( zero_zero @ nat ) )
| ( B = C ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1110]) ).
thf(930,axiom,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ pls ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_rel__simps_I2_J) ).
thf(4281,plain,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ pls ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[930]) ).
thf(660,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_602_Nat_Odiff__diff__eq) ).
thf(3347,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[660]) ).
thf(932,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( number_number_of @ A @ pls ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_148_add__numeral__0__right) ).
thf(4286,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( number_number_of @ A @ pls ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[932]) ).
thf(983,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_606_diff__le__mono2) ).
thf(4427,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[983]) ).
thf(999,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B ) ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_987_mod__less__eq__dividend) ).
thf(4474,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[999]) ).
thf(848,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_702_mult__less__mono2) ).
thf(4007,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[848]) ).
thf(1173,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ ( bit1 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_rel__simps_I15_J) ).
thf(5105,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ ( bit1 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit0 @ A ) ) @ ( bit1 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1173]) ).
thf(494,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_732_nat__mult__le__cancel1) ).
thf(2746,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[494]) ).
thf(897,axiom,
monoid_mult @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Omonoid__mult) ).
thf(4157,plain,
monoid_mult @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[897]) ).
thf(179,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ ( number_number_of @ nat @ B ) ) ) )
<=> ( ( ( number_number_of @ nat @ B )
= ( zero_zero @ nat ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_227_zero__less__power__nat__eq__number__of) ).
thf(1693,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ ( number_number_of @ nat @ B ) ) ) )
=> ( ( ( number_number_of @ nat @ B )
= ( zero_zero @ nat ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ) )
& ( ( ( ( number_number_of @ nat @ B )
= ( zero_zero @ nat ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( power_power @ nat @ A ) @ ( number_number_of @ nat @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(1149,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_222_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) ).
thf(5019,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1149]) ).
thf(435,axiom,
ordere1490568538miring @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oordered__comm__semiring) ).
thf(2572,plain,
ordere1490568538miring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[435]) ).
thf(555,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_813_add__pos__nonneg) ).
thf(2961,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[555]) ).
thf(19,axiom,
! [A: $i,B: $i] :
( ( standardRes @ ( ti @ int @ A ) @ B )
= ( standardRes @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Residues_OStandardRes_arg1) ).
thf(1288,plain,
! [A: $i,B: $i] :
( ( standardRes @ ( ti @ int @ A ) @ B )
= ( standardRes @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(184,axiom,
! [A: $i] :
( ( ( power @ A )
& ( mult_zero @ A )
& ( no_zero_divisors @ A )
& ( zero_neq_one @ A ) )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C )
= ( zero_zero @ A ) )
<=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( C
!= ( zero_zero @ nat ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_410_power__eq__0__iff) ).
thf(1710,plain,
! [A: $i] :
( ( ( power @ A )
& ( mult_zero @ A )
& ( no_zero_divisors @ A )
& ( zero_neq_one @ A ) )
=> ! [B: $i,C: $i] :
( ( ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C )
= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( C
!= ( zero_zero @ nat ) ) ) )
& ( ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( C
!= ( zero_zero @ nat ) ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(488,axiom,
! [A: $i] :
( ( comm_monoid_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_771_add_Ocomm__neutral) ).
thf(2714,plain,
! [A: $i] :
( ( comm_monoid_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[488]) ).
thf(621,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( twoSqu1929807760sum2sq @ ( product_Pair @ int @ int @ A @ B ) ) ) @ ( twoSqu1929807760sum2sq @ ( product_Pair @ int @ int @ C @ D ) ) )
= ( twoSqu1929807760sum2sq @ ( product_Pair @ int @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ D ) ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ D ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_752_mult__sum2sq) ).
thf(3196,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( twoSqu1929807760sum2sq @ ( product_Pair @ int @ int @ A @ B ) ) ) @ ( twoSqu1929807760sum2sq @ ( product_Pair @ int @ int @ C @ D ) ) )
= ( twoSqu1929807760sum2sq @ ( product_Pair @ int @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ C ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ D ) ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ D ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[621]) ).
thf(685,axiom,
ring_1 @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oring__1) ).
thf(3426,plain,
ring_1 @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[685]) ).
thf(1152,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_733_nat__le__add__iff1) ).
thf(5036,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1152]) ).
thf(578,axiom,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( zero_zero @ int ) ) )
=> ( ( zfact @ A )
= ( one_one @ int ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( zero_zero @ int ) ) )
=> ( ( zfact @ A )
= ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( zfact @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( one_one @ int ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_818_zfact_Osimps) ).
thf(3030,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( zero_zero @ int ) ) )
=> ( ( zfact @ A )
= ( one_one @ int ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( zero_zero @ int ) ) )
=> ( ( zfact @ A )
= ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( zfact @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ ( one_one @ int ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[578]) ).
thf(861,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_zdvd__imp__le) ).
thf(4034,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[861]) ).
thf(581,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ 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_632_mult__is__0) ).
thf(3045,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( zero_zero @ nat ) )
=> ( ( A
= ( zero_zero @ nat ) )
| ( B
= ( zero_zero @ nat ) ) ) )
& ( ( ( A
= ( zero_zero @ nat ) )
| ( B
= ( zero_zero @ nat ) ) )
=> ( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[581]) ).
thf(709,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B )
= ( hAPP @ int @ int @ ( div_mod @ int @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ C ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_914_Residues_Oaux) ).
thf(3506,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B )
= ( hAPP @ int @ int @ ( div_mod @ int @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[709]) ).
thf(800,axiom,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) @ ( zero_zero @ A ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_799_double__add__le__zero__iff__single__add__le__zero) ).
thf(3843,plain,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[800]) ).
thf(1201,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_643_add__less__mono) ).
thf(5200,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1201]) ).
thf(1174,axiom,
! [A: $i] :
( ( ab_group_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ C ) @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_613_add__diff__add) ).
thf(5110,plain,
! [A: $i] :
( ( ab_group_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( minus_minus @ A @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ E ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ C ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1174]) ).
thf(790,axiom,
! [A: $i] :
( ( ( power @ A )
& ( semiring_0 @ A ) )
=> ! [B: $i] :
( ( ( ( number_number_of @ nat @ B )
= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ ( zero_zero @ A ) ) @ ( number_number_of @ nat @ B ) )
= ( one_one @ A ) ) )
& ( ( ( number_number_of @ nat @ B )
!= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ ( zero_zero @ A ) ) @ ( number_number_of @ nat @ B ) )
= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_261_power__0__left__number__of) ).
thf(3805,plain,
! [A: $i] :
( ( ( power @ A )
& ( semiring_0 @ A ) )
=> ! [B: $i] :
( ( ( ( number_number_of @ nat @ B )
= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ ( zero_zero @ A ) ) @ ( number_number_of @ nat @ B ) )
= ( one_one @ A ) ) )
& ( ( ( number_number_of @ nat @ B )
!= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ ( zero_zero @ A ) ) @ ( number_number_of @ nat @ B ) )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[790]) ).
thf(944,axiom,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_780_add__less__cancel__left) ).
thf(4318,plain,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[944]) ).
thf(429,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ pls )
= ( ti @ int @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_471_diff__bin__simps_I1_J) ).
thf(2549,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ pls )
= ( ti @ int @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[429]) ).
thf(1117,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_331_dvd__mult) ).
thf(4923,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ C ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1117]) ).
thf(310,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) ) ) @ C )
= ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_908_zmod__simps_I3_J) ).
thf(2127,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ C ) ) ) @ C )
= ( hAPP @ int @ int @ ( div_mod @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).
thf(664,axiom,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_888_mult_Ozero__right) ).
thf(3362,plain,
! [A: $i] :
( ( real_normed_algebra @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[664]) ).
thf(1203,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( multInv @ C @ A ) @ ( multInv @ C @ B ) ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_839_MultInv__prop1) ).
thf(5204,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( multInv @ C @ A ) @ ( multInv @ C @ B ) ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1203]) ).
thf(1186,axiom,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_782_add__strict__left__mono) ).
thf(5151,plain,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1186]) ).
thf(56,axiom,
! [A: $i,B: $i] :
( ( legendre @ A @ ( ti @ int @ B ) )
= ( legendre @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Residues_OLegendre_arg2) ).
thf(1378,plain,
! [A: $i,B: $i] :
( ( legendre @ A @ ( ti @ int @ B ) )
= ( legendre @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(702,axiom,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_785_add__less__imp__less__left) ).
thf(3489,plain,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[702]) ).
thf(918,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ A ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_346_Euler_Oaux2) ).
thf(4239,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ A ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[918]) ).
thf(477,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_667_add__leD1) ).
thf(2687,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[477]) ).
thf(838,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ int @ ( power_power @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ B ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ A ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_505_zdiff__power3) ).
thf(3974,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ int @ ( power_power @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ B ) ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( minus_minus @ int @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ B ) ) ) @ ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ A ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[838]) ).
thf(575,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_zmult__assoc) ).
thf(3025,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( times_times @ int @ ( hAPP @ int @ int @ ( times_times @ int @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( times_times @ int @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[575]) ).
thf(218,axiom,
! [A: $i] :
( ( ( number @ A )
& ( semiring @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_right__distrib__number__of) ).
thf(1827,plain,
! [A: $i] :
( ( ( number @ A )
& ( semiring @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ D ) )
= ( hAPP @ A @ A @ ( plus_plus @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ B ) ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(4,axiom,
power @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Power_Opower) ).
thf(1261,plain,
power @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(557,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) )
<=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_392_mult__less__cancel__left__disj) ).
thf(2966,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ C ) ) ) ) )
& ( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[557]) ).
thf(398,axiom,
! [A: $i] :
( ( hAPP @ real @ real @ ( times_times @ real @ ( number_number_of @ real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ real @ ( power_power @ real @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( hAPP @ nat @ real @ ( power_power @ real @ ( hAPP @ real @ real @ ( times_times @ real @ ( number_number_of @ real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_446_four__x__squared) ).
thf(2430,plain,
! [A: $i] :
( ( hAPP @ real @ real @ ( times_times @ real @ ( number_number_of @ real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( hAPP @ nat @ real @ ( power_power @ real @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( hAPP @ nat @ real @ ( power_power @ real @ ( hAPP @ real @ real @ ( times_times @ real @ ( number_number_of @ real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[398]) ).
thf(723,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_699_mult__less__cancel1) ).
thf(3543,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[723]) ).
thf(120,axiom,
! [A: $i,B: $i] :
( ( dvd @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( times_times @ B @ A ) )
= ( times_times @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_2_res) ).
thf(1532,plain,
! [A: $i,B: $i] :
( ( dvd @ B )
=> ( ( ti @ ( fun @ B @ B ) @ ( times_times @ B @ A ) )
= ( times_times @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(517,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( minus_minus @ int @ min ) @ ( bit1 @ A ) )
= ( bit0 @ ( hAPP @ int @ int @ ( minus_minus @ int @ min ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_497_diff__bin__simps_I6_J) ).
thf(2847,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( minus_minus @ int @ min ) @ ( bit1 @ A ) )
= ( bit0 @ ( hAPP @ int @ int @ ( minus_minus @ int @ min ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[517]) ).
thf(448,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( one_one @ nat ) ) @ A )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_688_nat__mult__1) ).
thf(2600,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ ( one_one @ nat ) ) @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).
thf(731,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_384_mult__neg__pos) ).
thf(3570,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[731]) ).
thf(58,axiom,
! [A: $i,B: $i] :
( ( mult_zero @ B )
=> ( ( times_times @ B @ ( ti @ B @ A ) )
= ( times_times @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_4_arg1) ).
thf(1384,plain,
! [A: $i,B: $i] :
( ( mult_zero @ B )
=> ( ( times_times @ B @ ( ti @ B @ A ) )
= ( times_times @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(679,axiom,
mult_zero @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Omult__zero) ).
thf(3411,plain,
mult_zero @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[679]) ).
thf(410,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A )
= ( one_one @ int ) )
<=> ( ( ( ti @ int @ B )
= ( one_one @ int ) )
& ( ( ti @ int @ A )
= ( one_one @ int ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_pos__zmult__eq__1__iff) ).
thf(2485,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( ( ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A )
= ( one_one @ int ) )
=> ( ( ( ti @ int @ B )
= ( one_one @ int ) )
& ( ( ti @ int @ A )
= ( one_one @ int ) ) ) )
& ( ( ( ( ti @ int @ B )
= ( one_one @ int ) )
& ( ( ti @ int @ A )
= ( one_one @ int ) ) )
=> ( ( hAPP @ int @ int @ ( times_times @ int @ B ) @ A )
= ( one_one @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[410]) ).
thf(1134,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ A ) ) @ A )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_675_le__add__diff__inverse2) ).
thf(4972,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ B ) @ A ) ) @ A )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1134]) ).
thf(356,axiom,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_772_add__le__cancel__right) ).
thf(2265,plain,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[356]) ).
thf(885,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_578_dvd_Oless__le__not__le) ).
thf(4118,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[885]) ).
thf(1160,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_815_add__strict__increasing) ).
thf(5061,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1160]) ).
thf(128,axiom,
ordered_ab_group_add @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Groups_Oordered__ab__group__add) ).
thf(1550,plain,
ordered_ab_group_add @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(1138,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ A )
= ( ti @ int @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_959_mod__pos__pos__trivial) ).
thf(4987,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ B ) @ A )
= ( ti @ int @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1138]) ).
thf(46,axiom,
! [A: $i,B: $i] :
( ( multInv @ A @ ( ti @ int @ B ) )
= ( multInv @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Int2_OMultInv_arg2) ).
thf(1357,plain,
! [A: $i,B: $i] :
( ( multInv @ A @ ( ti @ int @ B ) )
= ( multInv @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(1012,axiom,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( number_number_of @ A @ B ) ) @ ( zero_zero @ A ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ pls ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_less__special_I3_J) ).
thf(4506,plain,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( number_number_of @ A @ B ) ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ pls ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ pls ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( number_number_of @ A @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1012]) ).
thf(440,axiom,
semiring_div @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Divides_Osemiring__div) ).
thf(2583,plain,
semiring_div @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[440]) ).
thf(636,axiom,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( one_one @ nat ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_347_power__one__right) ).
thf(3259,plain,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( one_one @ nat ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[636]) ).
thf(104,axiom,
semiring_1 @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef_Oreal___Rings_Osemiring__1) ).
thf(1500,plain,
semiring_1 @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(245,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( minus_minus @ int @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_836_int__le__induct) ).
thf(1923,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( minus_minus @ int @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(1159,axiom,
comm_monoid_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ocomm__monoid__add) ).
thf(5060,plain,
comm_monoid_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1159]) ).
thf(1156,axiom,
! [A: $i] :
( ( ordered_ab_group_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_811_le__iff__diff__le__0) ).
thf(5046,plain,
! [A: $i] :
( ( ordered_ab_group_add @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1156]) ).
thf(721,axiom,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_374_mult__nonneg__nonneg) ).
thf(3538,plain,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[721]) ).
thf(811,axiom,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_142_power2__eq__square) ).
thf(3886,plain,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[811]) ).
thf(1210,axiom,
! [A: $i] :
( ( linordered_ring @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_377_zero__le__square) ).
thf(5236,plain,
! [A: $i] :
( ( linordered_ring @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1210]) ).
thf(948,axiom,
semiring_0 @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Osemiring__0) ).
thf(4330,plain,
semiring_0 @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[948]) ).
thf(1246,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( zero_zero @ nat ) ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_626_less__eq__nat_Osimps_I1_J) ).
thf(5354,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( zero_zero @ nat ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1246]) ).
thf(236,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ B )
<=> ( hBOOL @ A ) ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
& ( hBOOL @ B ) )
<=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
& ( hBOOL @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_741_conj__le__cong) ).
thf(1885,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( ( hBOOL @ B )
=> ( hBOOL @ A ) )
& ( ( hBOOL @ A )
=> ( hBOOL @ B ) ) ) )
=> ( ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
& ( hBOOL @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
& ( hBOOL @ A ) ) )
& ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
& ( hBOOL @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
& ( hBOOL @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).
thf(1047,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B )
= ( hAPP @ int @ int @ ( div_mod @ int @ C ) @ B ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_934_zmod__eq__dvd__iff) ).
thf(4691,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B )
= ( hAPP @ int @ int @ ( div_mod @ int @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ C ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ A ) @ C ) ) )
=> ( ( hAPP @ int @ int @ ( div_mod @ int @ A ) @ B )
= ( hAPP @ int @ int @ ( div_mod @ int @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1047]) ).
thf(791,axiom,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_616_less__nat__zero__code) ).
thf(3812,plain,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[791]) ).
thf(107,axiom,
! [A: $i] :
( ( ring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_952_mod__diff__right__eq) ).
thf(1505,plain,
! [A: $i] :
( ( ring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( minus_minus @ A @ B ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(64,axiom,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_905_mod__mult__eq) ).
thf(1399,plain,
! [A: $i] :
( ( semiring_div @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( div_mod @ A @ ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( div_mod @ A @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( div_mod @ A @ C ) @ D ) ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(357,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ B ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ C ) @ B ) ) )
=> ( ( B
!= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_869_pow__divides__pow__int) ).
thf(2270,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ ( hAPP @ nat @ int @ ( power_power @ int @ A ) @ B ) ) @ ( hAPP @ nat @ int @ ( power_power @ int @ C ) @ B ) ) )
=> ( ( B
!= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[357]) ).
thf(527,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_228_nat__mult__2__right) ).
thf(2879,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( times_times @ nat @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[527]) ).
thf(734,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ B @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_834_wset__mem) ).
thf(3580,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ zprime @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ ( hAPP @ int @ int @ ( minus_minus @ int @ C ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( member @ int @ A ) @ ( wset @ B @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[734]) ).
thf(928,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
!= ( zero_zero @ real ) )
=> ( ( ( hAPP @ real @ real @ ( times_times @ real @ A ) @ C )
= ( hAPP @ real @ real @ ( times_times @ real @ B ) @ C ) )
<=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_636_real__mult__right__cancel) ).
thf(4274,plain,
! [A: $i,B: $i,C: $i] :
( ( C
!= ( zero_zero @ real ) )
=> ( ( ( ( hAPP @ real @ real @ ( times_times @ real @ A ) @ C )
= ( hAPP @ real @ real @ ( times_times @ real @ B ) @ C ) )
=> ( A = B ) )
& ( ( A = B )
=> ( ( hAPP @ real @ real @ ( times_times @ real @ A ) @ C )
= ( hAPP @ real @ real @ ( times_times @ real @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[928]) ).
thf(50,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_216_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) ).
thf(1367,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( hAPP @ A @ A @ ( times_times @ A @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( times_times @ A @ B ) @ ( hAPP @ A @ A @ ( times_times @ A @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(254,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_980_mod__less) ).
thf(1962,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( div_mod @ nat @ A ) @ B )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).
thf(289,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( one_one @ nat ) ) )
<=> ( A
= ( one_one @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_682_nat__dvd__1__iff__1) ).
thf(2059,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( one_one @ nat ) ) )
=> ( A
= ( one_one @ nat ) ) )
& ( ( A
= ( one_one @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ ( one_one @ nat ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).
thf(296,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_mult__2) ).
thf(2087,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( times_times @ A @ ( number_number_of @ A @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ B )
= ( hAPP @ A @ A @ ( plus_plus @ A @ B ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).
thf(782,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ A ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_zadd__strict__right__mono) ).
thf(3753,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ B ) @ A ) ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[782]) ).
thf(832,axiom,
! [A: $i] :
( ( bit1 @ ( ti @ int @ A ) )
= ( bit1 @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Int_OBit1_arg1) ).
thf(3960,plain,
! [A: $i] :
( ( bit1 @ ( ti @ int @ A ) )
= ( bit1 @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[832]) ).
thf(718,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ E ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_341_dvd__power__le) ).
thf(3528,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ D ) @ B ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ E ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[718]) ).
thf(497,axiom,
! [A: $i] :
( ( ( power @ A )
& ( mult_zero @ A )
& ( no_zero_divisors @ A )
& ( zero_neq_one @ A ) )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ C ) )
= ( zero_zero @ A ) )
<=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( number_number_of @ nat @ C )
!= ( zero_zero @ nat ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_255_power__eq__0__iff__number__of) ).
thf(2754,plain,
! [A: $i] :
( ( ( power @ A )
& ( mult_zero @ A )
& ( no_zero_divisors @ A )
& ( zero_neq_one @ A ) )
=> ! [B: $i,C: $i] :
( ( ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ C ) )
= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( number_number_of @ nat @ C )
!= ( zero_zero @ nat ) ) ) )
& ( ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( number_number_of @ nat @ C )
!= ( zero_zero @ nat ) ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ C ) )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[497]) ).
thf(688,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_694_nat__dvd__not__less) ).
thf(3437,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[688]) ).
thf(1151,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ twoSqu658283162sum2sq @ A ) )
<=> ? [B: $i,C: $i] :
( ( twoSqu1929807760sum2sq @ ( product_Pair @ int @ int @ B @ C ) )
= ( ti @ int @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_753_is__sum2sq__def) ).
thf(5029,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ twoSqu658283162sum2sq @ A ) )
=> ? [B: $i,C: $i] :
( ( twoSqu1929807760sum2sq @ ( product_Pair @ int @ int @ B @ C ) )
= ( ti @ int @ A ) ) )
& ( ? [B: $i,C: $i] :
( ( twoSqu1929807760sum2sq @ ( product_Pair @ int @ int @ B @ C ) )
= ( ti @ int @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ twoSqu658283162sum2sq @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1151]) ).
thf(823,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_306_linorder__neqE__linordered__idom) ).
thf(3924,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[823]) ).
thf(1230,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_677_diff__add__assoc) ).
thf(5312,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ C ) ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1230]) ).
thf(250,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_756_add__nonpos__neg) ).
thf(1945,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( plus_plus @ A @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).
thf(626,axiom,
! [A: $i] :
( ( ring_11004092258visors @ A )
=> ! [B: $i] :
( ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( zero_zero @ A ) )
<=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_zero__eq__power2) ).
thf(3212,plain,
! [A: $i] :
( ( ring_11004092258visors @ A )
=> ! [B: $i] :
( ( ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( zero_zero @ A ) )
=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) )
& ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[626]) ).
thf(1054,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_348_zcong__id) ).
thf(4723,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1054]) ).
thf(1026,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ B )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_588_diff__add__inverse2) ).
thf(4555,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ B )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1026]) ).
thf(1075,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ A ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ A ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_312_zcong__trans) ).
thf(4784,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ C @ A ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ B @ A ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1075]) ).
thf(193,axiom,
ordere453448008miring @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Oordered__cancel__semiring) ).
thf(1737,plain,
ordere453448008miring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).
thf(532,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_278_dvd__0__right) ).
thf(2887,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( dvd_dvd @ A ) @ B ) @ ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[532]) ).
thf(1128,axiom,
linord626643107strict @ int,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__semiring__1__strict) ).
thf(4949,plain,
linord626643107strict @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1128]) ).
thf(89,axiom,
! [A: $i] :
( ( ti @ A @ ( undefined @ A ) )
= ( undefined @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_HOL_Oundefined_res) ).
thf(1463,plain,
! [A: $i] :
( ( ti @ A @ ( undefined @ A ) )
= ( undefined @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(566,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ A )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_538_diff__self__eq__0) ).
thf(2998,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ A )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[566]) ).
thf(274,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_549_less__not__refl3) ).
thf(2018,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).
thf(523,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_590_diff__diff__left) ).
thf(2866,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( minus_minus @ nat @ A ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[523]) ).
thf(1061,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_437_power__decreasing) ).
thf(4742,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ D ) ) @ ( hAPP @ nat @ A @ ( power_power @ A @ B ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1061]) ).
thf(367,axiom,
( ( zero_zero @ int )
= ( number_number_of @ int @ pls ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_zero__is__num__zero) ).
thf(2297,plain,
( ( zero_zero @ int )
= ( number_number_of @ int @ pls ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[367]) ).
thf(943,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( A != B )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_573_dvd_Ole__neq__trans) ).
thf(4313,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
=> ( ( A != B )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ A ) @ B ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( dvd_dvd @ nat ) @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[943]) ).
thf(1215,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ ( zero_zero @ int ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( zero_zero @ int ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_156_bin__less__0__simps_I4_J) ).
thf(5255,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( zero_zero @ int ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( bit1 @ A ) ) @ ( zero_zero @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1215]) ).
thf(455,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) )
<=> ? [D: $i] :
( ( ti @ int @ B )
= ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_417_zcong__iff__lin) ).
thf(2615,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) )
=> ? [D: $i] :
( ( ti @ int @ B )
= ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ D ) ) ) )
& ( ? [D: $i] :
( ( ti @ int @ B )
= ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ ( hAPP @ int @ int @ ( times_times @ int @ C ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[455]) ).
thf(1136,axiom,
! [A: $i] :
( ( ( power @ A )
& ( semiring_0 @ A ) )
=> ! [B: $i] :
( ( ( B
= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ ( zero_zero @ A ) ) @ B )
= ( one_one @ A ) ) )
& ( ( B
!= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ ( zero_zero @ A ) ) @ B )
= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_442_power__0__left) ).
thf(4977,plain,
! [A: $i] :
( ( ( power @ A )
& ( semiring_0 @ A ) )
=> ! [B: $i] :
( ( ( B
= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ ( zero_zero @ A ) ) @ B )
= ( one_one @ A ) ) )
& ( ( B
!= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ A @ ( power_power @ A @ ( zero_zero @ A ) ) @ B )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1136]) ).
thf(981,axiom,
power @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Power_Opower) ).
thf(4423,plain,
power @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[981]) ).
thf(1112,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( zero_zero @ nat ) ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_534_le0) ).
thf(4908,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( zero_zero @ nat ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1112]) ).
thf(243,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( bit0 @ A ) ) @ ( bit0 @ B ) )
= ( bit0 @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_add__Bit0__Bit0) ).
thf(1917,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( plus_plus @ int @ ( bit0 @ A ) ) @ ( bit0 @ B ) )
= ( bit0 @ ( hAPP @ int @ int @ ( plus_plus @ int @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).
thf(748,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( one_one @ int ) ) @ A ) ) @ A ) ) @ ( zero_zero @ int ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( zero_zero @ int ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_odd__less__0) ).
thf(3637,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( one_one @ int ) ) @ A ) ) @ A ) ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( zero_zero @ int ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( hAPP @ int @ int @ ( plus_plus @ int @ ( one_one @ int ) ) @ A ) ) @ A ) ) @ ( zero_zero @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[748]) ).
thf(1101,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_679_diff__add__assoc2) ).
thf(4863,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( minus_minus @ nat @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ C ) @ B ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1101]) ).
thf(513,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_736_nat__le__add__iff2) ).
thf(2834,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ C ) ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ ( hAPP @ nat @ nat @ ( minus_minus @ nat @ E ) @ D ) ) @ A ) ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ D ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ ( hAPP @ nat @ nat @ ( times_times @ nat @ E ) @ A ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[513]) ).
thf(905,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_419_zcong__zero__equiv__div) ).
thf(4189,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( dvd_dvd @ int ) @ B ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ A @ ( zero_zero @ int ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[905]) ).
thf(1025,axiom,
! [A: $i,B: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_637_not__add__less1) ).
thf(4552,plain,
! [A: $i,B: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ A ) @ B ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1025]) ).
thf(937,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ C ) @ ( one_one @ int ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) @ ( one_one @ int ) ) @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_451_zcong__zpower__zmult) ).
thf(4298,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ C ) @ ( one_one @ int ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( zcong @ ( hAPP @ nat @ int @ ( power_power @ int @ B ) @ ( hAPP @ nat @ nat @ ( times_times @ nat @ C ) @ A ) ) @ ( one_one @ int ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[937]) ).
thf(846,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_665_add__le__mono) ).
thf(4002,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( plus_plus @ nat @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[846]) ).
thf(5543,plain,
$false,
inference(cvc4,[status(thm)],[2163,1501,1337,3944,3883,1469,1559,2412,3979,1840,2427,2544,3927,4682,1818,1686,2199,3021,1322,4835,2889,4101,4450,3345,5260,3661,4062,3399,1995,2263,3666,1729,1315,2787,1950,1549,2280,1554,2210,3549,2590,3098,3108,4711,2099,2837,4479,2976,4376,2940,1596,1406,2231,2622,4311,4515,5000,1808,2659,3417,3509,1618,3040,4391,2947,2248,2671,2676,3910,4326,5265,2380,1971,1793,4487,4592,3848,3530,5361,1310,4026,2316,2448,1275,4856,3693,2476,1988,1342,2844,1354,2575,4403,3439,3004,2035,4839,4011,3649,3365,2918,4110,4528,2184,1640,2580,3895,3717,2360,4762,1458,3370,3350,5252,3766,2763,1576,1441,3513,1613,2886,5301,3898,4467,4511,2957,4792,1835,2010,4803,4124,4030,4137,1724,1475,2173,1366,1529,1967,5269,3880,3688,2869,2654,4439,2603,3275,5284,3342,3495,3574,3821,1313,4568,2356,3248,1409,4539,3442,3938,3029,4348,3425,5306,4866,2698,4956,4070,2667,1264,4334,1323,3148,5130,1577,3280,5182,2562,4023,3089,1832,2897,4766,1462,5279,4243,2703,5231,5338,3385,4353,1704,4141,2158,3835,2989,5091,1875,4253,4595,2713,4119,2169,1514,4194,5127,2552,4492,4928,1572,3919,2635,3470,3312,1568,4460,1494,2852,4771,5077,1664,3199,2996,4454,2259,2451,3702,4847,5156,2291,4871,1283,3851,4432,2456,2007,2178,2222,4411,2950,4205,3966,2027,4443,1450,4578,2105,4018,2090,1632,2469,5333,4820,3525,4050,1960,5135,2555,3290,4115,1581,2777,3152,3389,3986,3189,5220,5384,1447,3685,2424,5343,2809,1433,5137,3606,5328,3406,4718,4751,1377,5353,1796,2530,1394,2772,3157,1351,5169,1345,4038,3184,2982,2884,4483,3510,4065,2954,3567,1438,3265,5017,2118,5142,4576,4910,4925,3037,3151,3097,4853,3582,5012,3503,4199,4216,4893,4323,4962,4753,2047,3318,1915,2093,1783,4370,2737,2905,1868,3735,2705,2541,1580,4434,2147,3022,1487,3239,4499,2132,1560,1274,5206,1387,1900,1723,1343,3972,2596,3867,4609,5044,5349,1676,4344,4091,2150,1370,2032,5102,3207,3682,4021,3339,3889,1423,2670,3877,1457,2609,4746,1656,3028,1291,1740,3799,3958,2445,4295,3191,4280,1837,1602,4608,1920,4585,2433,3403,4725,1968,2565,2663,1555,2826,4155,4947,3499,3065,3750,2872,4385,4187,5341,1787,5113,2181,4942,2313,3763,2492,4104,3595,4027,1402,5145,2801,3862,1375,3381,1878,4930,3119,4883,4012,2638,3001,4808,1360,2245,5189,2300,2498,3615,3831,2840,2115,1270,1851,1772,3122,5238,4512,5006,5345,4248,4789,4544,4429,2692,1498,1587,2694,4525,2509,1672,1540,3518,3963,3916,3237,3205,4219,3060,3176,3018,4817,4263,2877,5166,4167,3728,2196,3194,3671,4736,1567,4291,4333,1329,1942,1659,2323,3179,4421,1424,3010,4406,1584,4513,4744,2613,1937,1810,4861,4054,3555,4814,2978,2472,1478,1297,1539,1974,2465,1799,1947,1984,1599,4953,4086,1393,3703,5063,2285,5317,5273,1263,3486,2125,3654,4068,2153,3819,4237,3222,4934,3540,3628,3115,1429,3967,4135,4152,1720,3109,1979,3074,1627,2370,1525,3755,3225,2546,1779,4081,3587,1563,2963,1466,4448,2408,4036,1895,2645,2529,1732,2909,4163,3454,3476,1333,4148,2749,1434,3802,5310,5153,2883,3173,4476,2057,3306,1846,4004,2338,4921,2936,2391,2556,2253,5359,3491,1273,1927,3262,4797,4503,5095,4017,1631,2121,2235,2144,3707,1727,4919,3130,2677,4708,4685,1754,2292,3257,2717,4256,1747,1301,2734,2894,4365,3396,4453,3343,2260,1833,2048,2657,3086,4122,3604,1965,4946,2134,4786,4943,4998,3187,4341,2727,4283,1615,4077,2981,5067,4309,1620,5121,3081,3758,3283,2016,2514,4032,4709,4605,1535,2991,1403,3458,3590,4328,4590,4975,1520,2817,4758,4769,2695,5022,1488,2161,3796,3651,2442,2166,1258,3302,4245,5039,4558,1669,3118,4127,1707,5118,5363,5011,1371,3646,4112,5282,3663,1865,1635,2935,4573,5086,5148,1332,3700,4060,2358,2573,3719,2079,3101,3352,5325,3270,2299,2116,2802,3384,1571,4388,4337,2679,4228,3631,2538,2229,3815,2084,2383,2624,2415,2987,1524,4828,1589,3732,3106,3468,1722,5054,4139,2943,2945,4300,5381,4107,3160,1307,3456,3219,5367,4401,4541,5082,4731,3977,4850,2278,1381,2710,1877,5202,3747,2025,2812,2632,4445,2605,4882,3055,4009,2261,1731,3572,4985,3165,1862,3483,1344,4990,2578,2401,2212,5377,1528,4502,2139,4362,2797,1435,2022,2535,4497,2069,2506,2418,4844,1841,1396,5197,2051,5042,2288,2207,4890,4209,4481,4158,1805,4741,4690,3936,4260,1277,3146,1509,3921,1472,2597,4355,2005,1413,3522,4331,2054,4207,1809,2952,1662,4394,2096,1304,1873,3427,4458,2665,2318,2784,4456,4598,2652,3293,1905,1420,3981,2503,4177,2927,2629,1990,4859,2202,4688,2333,2256,1551,5003,4045,3988,1798,3913,4272,4950,1647,3949,5079,2882,2863,3315,3842,4490,3112,4937,1611,1566,3904,3228,2175,1267,3058,2386,2720,4377,2850,1484,2831,2689,1499,4409,4905,1432,3722,3554,4768,2337,2219,3828,4241,1336,4522,5262,3642,4561,4346,1790,2224,1813,1300,3691,1637,3344,4549,3202,4222,5276,1785,3655,1262,1830,3127,4800,3623,1679,3412,3564,3359,2044,1285,1738,3853,2056,1934,1543,4959,1702,5347,1284,2320,1504,2030,3007,2584,4414,4217,1898,1849,3652,1257,3970,3415,3829,5223,1536,1981,1802,3493,1749,4932,4538,1427,4202,1585,2511,2860,5364,2949,1372,4838,2892,3168,3452,4727,2310,4997,1517,2611,1781,2187,2226,4730,1998,2013,3400,3946,4289,3584,3620,2626,4424,4698,5179,1459,4587,4763,4165,2828,3171,2960,4873,2078,4419,3533,3447,1654,1412,3625,3725,2110,1390,5124,1575,2071,3929,5068,2792,5184,2907,5315,2594,2273,3484,2855,2251,1903,5382,1481,5351,3336,4042,5228,4338,3003,4823,2394,1276,2591,1326,3729,2707,4913,2283,2389,4806,2532,1508,1629,3104,3561,2608,3235,3516,4495,5243,1523,4705,2130,3331,3368,3662,2526,1558,2934,2152,4182,1491,3978,4759,2066,1706,1513,5336,2475,4602,5164,5162,4293,3497,3879,2648,3430,3267,4774,1294,2294,3645,4098,2494,1444,4306,5194,3420,2421,2102,1512,2517,3860,4378,3892,1546,1625,3264,5078,3521,1673,2050,1757,5014,5247,4367,3992,2142,3379,2123,5187,2258,5051,4880,1348,3705,1280,4014,4472,4399,4356,3907,4811,2587,5268,2233,3373,5356,2008,4841,1986,2751,3328,3558,5215,2340,5322,4303,4051,5300,1363,1314,5159,1495,1657,5092,4895,4281,3347,4286,4427,4474,4007,5105,2746,4157,1693,5019,2572,2961,1288,1710,2714,3196,3426,5036,3030,4034,3045,3506,3843,5200,5110,3805,4318,2549,4923,2127,3362,5204,5151,1378,3489,4239,2687,3974,3025,1827,1261,2966,2430,3543,1532,2847,2600,3570,1384,3411,2485,4972,2265,4118,5061,1550,4987,1357,4506,2583,3259,1500,1923,5060,5046,3538,3886,5236,4330,5354,1885,4691,3812,1505,1399,2270,2879,3580,4274,1367,1962,2059,2087,3753,3960,3528,2754,3437,5029,3924,5312,1945,3212,4723,4555,4784,1737,2887,4949,1463,2998,2018,2866,4742,2297,4313,5255,2615,4977,4423,4908,1917,3637,4863,2834,4189,4552,4298,4002]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : NUM924+7 : TPTP v8.1.2. Released v5.3.0.
% 0.11/0.15 % Command : run_Leo-III %s %d
% 0.14/0.36 % Computer : n027.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Thu May 18 17:05:51 EDT 2023
% 0.14/0.36 % CPUTime :
% 0.86/0.85 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.89/1.14 % [INFO] Parsing done (287ms).
% 1.89/1.15 % [INFO] Running in sequential loop mode.
% 2.49/1.34 % [INFO] eprover registered as external prover.
% 2.49/1.34 % [INFO] cvc4 registered as external prover.
% 2.49/1.34 % [INFO] Scanning for conjecture ...
% 3.27/1.53 % [INFO] Found a conjecture and 1254 axioms. Running axiom selection ...
% 4.47/1.80 % [INFO] Axiom selection finished. Selected 1254 axioms (removed 0 axioms).
% 6.25/2.37 % [INFO] Problem is first-order (TPTP FOF).
% 6.25/2.41 % [INFO] Type checking passed.
% 6.25/2.41 % [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 ...
% 31.28/9.30 % External prover 'cvc4' found a proof!
% 31.28/9.30 % [INFO] Killing All external provers ...
% 31.28/9.30 % Time passed: 8773ms (effective reasoning time: 8149ms)
% 31.28/9.30 % 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)>
% 31.28/9.31 % Axioms used in derivation (1254): fact_218_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J, fact_528_realpow__two__sum__zero__iff, fact_757_zero__reorient, fact_564_dvd_Oless__imp__neq, fact_859_d22set__le, fact_210_is__mult__sum2sq, fact_17_t__l__p, fact_989_StandardRes__prop3, fact_506_neg__one__power, fact_570_dvd__antisym, fact_442_power__0__left, fact_844_aux______4, fact_617_gr__implies__not0, fact_197_zero__less__power__nat__eq, tsy_c_Groups_Oplus__class_Oplus_2_arg1, arity_RealDef_Oreal___Groups_Oordered__ab__group__add, fact_718_realpow__num__eq__if, tsy_c_fconj_res, fact_820_inv__less__p__minus__1, arity_RealDef_Oreal___Groups_Oplus, fact_703_real__mult__le__cancel__iff1, fact_579_dvd_Oless__le, fact_803_add__increasing2, tsy_c_COMBB_arg2, arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add, fact_693_mult__diff__mult, fact_307_dvd__refl, fact_509_QuadRes__def, arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add, fact_189_self__quotient__aux2, arity_HOL_Obool___Orderings_Oorder, arity_Nat_Onat___Groups_Omonoid__mult, fact_112_mult__Bit1, arity_Int_Oint___Rings_Oordered__semiring, fact_152_number__of__add, help_COMBS_1_1_U, fact_380_comm__mult__strict__left__mono, fact_979_StandardRes__def, fact_701_mult__less__mono1, fact_841_MultInv__prop5, fact_702_mult__less__mono2, arity_Int_Oint___Groups_Oone, fact_677_diff__add__assoc, arity_Nat_Onat___Groups_Oab__semigroup__mult, fact_686_mult__le__mono2, fact_95_right__distrib__number__of, arity_Nat_Onat___Rings_Oordered__semiring, fact_54_rel__simps_I27_J, fact_956_zcong__zmod__eq, fact_349_IntPrimes_Ozcong__zero, fact_256_pos__zmult__pos, fact_302_rel__simps_I11_J, fact_655_dvd__reduce, fact_815_add__strict__increasing, fact_544_less__not__refl, fact_514_divides__mul__r, fact_171_semiring__numeral__1__eq__1, fact_633_mult__cancel1, tsy_c_Product__Type_OPair_arg2, fact_631_mult__0__right, fact_937_zmod__le__nonneg__dividend, arity_Int_Oint___Groups_Oordered__comm__monoid__add, fact_685_mult__le__mono1, fact_606_diff__le__mono2, fact_224_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J, fact_491_less__add__iff1, fact_612_real__add__left__mono, fact_387_zero__less__mult__pos, arity_Int_Oint___Rings_Ocomm__ring, fact_167_power2__eq__square__number__of, fact_525_divides__rev, arity_Int_Oint___Rings_Olinordered__ring__strict, arity_Int_Oint___Rings_Oordered__cancel__semiring, help_fconj_3_1_U, fact_591_Nat_Odiff__cancel, arity_Nat_Onat___Rings_Osemiring__1, fact_259_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J, fact_59_le__special_I4_J, fact_931_mod__self, fact_608_nat__mult__commute, fact_668_add__leE, fact_988_StandardRes__eq__zcong, fact_426_mult__right__less__imp__less, fact_943_mod__mult__self1, fact_225_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J, fact_457_one__less__power, fact_42_mult__number__of__left, arity_Int_Oint___Groups_Oordered__ab__semigroup__add, tsy_c_Product__Type_OPair_res, arity_Int_Oint___Rings_Olinordered__semiring, fact_35_less__eq__int__code_I13_J, fact_392_mult__less__cancel__left__disj, arity_Int_Oint___Rings_Olinordered__idom, fact_550_nat__less__cases, fact_545_nat__neq__iff, fact_908_zmod__simps_I3_J, fact_881_pow__divides__eq__nat, fact_521_nat__mult__dvd__cancel__disj_H, fact_623_add__is__0, fact_513_divides__add__revr, fact_984_mod__if, fact_413_zcong__not, fact_707_real__mult__less__iff1, fact_542_real__less__def, fact_125_rel__simps_I2_J, fact_728_nat__mult__less__cancel1, fact_31_zpower__zadd__distrib, fact_50_le__number__of__eq__not__less, fact_182_zadd__power3, fact_25_times__numeral__code_I5_J, fact_358_zcong__shift, arity_Int_Oint___Orderings_Oorder, arity_RealDef_Oreal___Rings_Olinordered__ring__strict, fact_599_nat__le__linear, fact_13_zero__power2, arity_Int_Oint___Int_Oring__char__0, fact_106_sum__power2__ge__zero, fact_237_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J, fact_87_le__imp__0__less, fact_992_StandardRes__SR__prop, tsy_v_s_____res, fact_917_zmod__self, fact_515_divides__mul__l, tsy_c_Int_OPls_res, fact_651_le__neq__implies__less, fact_554_dvd_Oless__trans, fact_192_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J, fact_957_pos__mod__sign, fact_503_power2__diff, fact_918_zmod__zero, fact_870_gcd__lcm__complete__lattice__nat_Otop__le, fact_69_zadd__zmult__distrib2, fact_504_zdiff__power2, fact_723_real__le__linear, arity_Nat_Onat___Rings_Olinordered__semiring, fact_580_dvd_Ole__less, tsy_c_Residues_OStandardRes_arg1, fact_901_mod__mult__cong, fact_616_less__nat__zero__code, fact_565_dvd_Ole__less__trans, fact_180_less__special_I2_J, fact_587_nat__add__right__cancel, fact_676_le__imp__diff__is__add, tsy_c_WilsonRuss_Owset_arg2, fact_420_zprime__zdvd__zmult__better, fact_58_semiring__norm_I113_J, fact_173_semiring__norm_I110_J, fact_670_le__diff__conv, tsy_c_Groups_Otimes__class_Otimes_5_res, fact_754_norR__mem__unique__aux, fact_634_mult__cancel2, arity_RealDef_Oreal___Groups_Oordered__cancel__ab__semigroup__add, tsy_c_member_arg1, fact_296_rel__simps_I26_J, fact_642_add__less__mono1, tsy_c_member_res, fact_265_unique__quotient__lemma, fact_289_rel__simps_I24_J, fact_787_eq__iff__diff__eq__0, tsy_c_Groups_Otimes__class_Otimes_3_arg1, arity_Int_Oint___Rings_Olinordered__ring, arity_Int_Oint___Groups_Ocomm__monoid__add, tsy_c_Groups_Oplus__class_Oplus_0_res, fact_552_diff__less__mono2, fact_231_nat__number__of__mult__left, fact_249_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J, tsy_c_Groups_Ominus__class_Ominus_0_arg1, fact_473_zdiff__zmult__distrib, fact_549_less__not__refl3, fact_359_zcong__zpower, fact_814_add__nonneg__pos, fact_384_mult__neg__pos, fact_798_zero__le__double__add__iff__zero__le__single__add, fact_605_diff__le__mono, fact_761_add__right__cancel, fact_255_power__eq__0__iff__number__of, fact_84_zle__add1__eq__le, arity_RealDef_Oreal___Groups_Oordered__comm__monoid__add, fact_273_prime__g__5, fact_325_dvd__0__left, fact_905_mod__mult__eq, fact_377_zero__le__square, fact_796_add__diff__cancel, tsy_c_Groups_Otimes__class_Otimes_0_res, arity_RealDef_Oreal___Rings_Osemiring__0, fact_958_pos__mod__conj, fact_206_zdvd__reduce, fact_63_mult__Bit0, fact_777_add__le__imp__le__right, fact_134_zadd__0, fact_86_add__nat__number__of, fact_709_real__two__squares__add__zero__iff, arity_RealDef_Oreal___Groups_Ozero, tsy_c_WilsonRuss_Owset_res, fact_588_diff__add__inverse2, fact_33_rel__simps_I34_J, fact_959_mod__pos__pos__trivial, fact_812_less__iff__diff__less__0, tsy_c_Groups_Oplus__class_Oplus_3_arg1, fact_621_plus__nat_Oadd__0, fact_244_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J, fact_28_zmult__assoc, arity_Int_Oint___Rings_Oordered__ring, fact_750_zcong__zless__unique, fact_77_rel__simps_I15_J, fact_898_zmod__number__of__Bit1, fact_673_add__diff__assoc, fact_367_comm__mult__left__mono, arity_Nat_Onat___Rings_Oordered__comm__semiring, fact_123_less__int__code_I16_J, fact_566_dvd_Oord__eq__less__trans, arity_Nat_Onat___Int_Onumber__semiring, fact_598_eq__imp__le, fact_364_mult__mono_H, fact_148_add__numeral__0__right, fact_810_add__le__less__mono, fact_214_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J, fact_991_StandardRes__ubound, fact_563_dvd_Oless__not__sym, fact_926_mod__add__eq, fact_951_mod__diff__left__eq, fact_183_zadd__power2, fact_440_power__less__power__Suc, arity_RealDef_Oreal___Int_Onumber__semiring, fact_356_zcong__zmult, fact_913_mod__mod__is__mod, fact_760_add__left__cancel, fact_638_not__add__less2, fact_722_real__le__refl, arity_RealDef_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossp, fact_115_rel__simps_I46_J, fact_857_d22set__eq__wset, fact_448_zcong__zless__0, fact_328_dvd__triv__left, fact_191_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J, tsy_c_Int_OMin_res, fact_386_zero__less__mult__pos2, fact_972_zmod__minus1, fact_632_mult__is__0, 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fact_880_dvd__pos__nat, fact_235_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J, fact_789_diff__0__right, fact_370_mult__nonpos__nonpos, fact_470_dvd__diff, fact_809_add__less__le__mono, fact_38_zadd__left__mono, fact_443_power__strict__increasing, fact_251_add__scale__eq__noteq, fact_153_rel__simps_I12_J, fact_682_nat__dvd__1__iff__1, fact_128_int__0__neq__1, tsy_c_TwoSquares__Mirabelle__lldtdwpyrp_Osum2sq_arg1, fact_900_dvd__eq__mod__eq__0__number__of, fact_355_zcong__scalar2, tsy_c_hAPP_arg2, fact_476_zcong__zdiff, fact_910_zcong__zmod, arity_RealDef_Oreal___Rings_Osemiring__1, fact_40_eq__0__number__of, fact_335_dvd__mult__right, fact_76_Collect__def, fact_906_mod__mult__left__eq, fact_846_MultInv__zcong__prop2, fact_129_add__Pls__right, fact_939_neg__mod__bound, tsy_c_IntFact_Od22set_res, fact_561_dvd_Oless__imp__not__less, tsy_c_Groups_Ominus__class_Ominus_0_res, help_fconj_2_1_U, tsy_v_m_res, fact_987_mod__less__eq__dividend, fact_196_s1, arity_Nat_Onat___Rings_Osemiring__0, fact_619_diff__less, arity_RealDef_Oreal___Rings_Ocomm__semiring, tsy_c_COMBB_res, tsy_c_Groups_Ominus__class_Ominus_1_arg1, fact_627_le__0__eq, fact_903_mod__mult__mult2, fact_802_add__increasing, arity_Int_Oint___Divides_Oring__div, fact_3_t, fact_487_zcong__def, fact_904_mod__mult__mult1, fact_223_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J, fact_743_zdvd__mono, fact_10_power2__less__0, fact_826_inv__is__inv, fact_56_zadd__zless__mono, fact_947_mod__diff__cong, fact_343_nat__power__less__imp__less, fact_672_le__add__diff__inverse, fact_575_dvd_Oantisym__conv, fact_324_field__power__not__zero, fact_774_add__right__mono, fact_166_zless__add1__eq, fact_886_mult_Ozero__left, fact_446_four__x__squared, tsy_c_Groups_Otimes__class_Otimes_1_arg1, tsy_c_Divides_Odiv__class_Omod_arg1, tsy_c_Int_OBit1_res, fact_450_zpower__zdvd__prop1, fact_856_mono__nat__linear__lb, fact_518_power__eq__if, fact_105_sum__power2__le__zero__iff, fact_314_order__le__neq__implies__less, fact_847_MultInv__zcong__prop3, fact_466__096s_A_094_A2_A_N_A_N1_A_061_As_A_094_A2_A_L_A1_096, fact_653_diff__less__mono, fact_318_mult__zero__right, arity_Int_Oint___Groups_Omonoid__mult, fact_453_convex__bound__le, fact_228_nat__mult__2__right, fact_263_q__pos__lemma, fact_664_add__le__mono1, fact_219_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J, fact_422_mult__right__le__imp__le, arity_RealDef_Oreal___Int_Onumber__ring, fact_635_real__mult__left__cancel, fact_949_zdiff__zmod__right, fact_896_mult_Odiff__left, fact_543_less__eq__real__def, fact_993_split__mod, fact_751__096sum2sq_A_Is_M_A1_J_A_061_A_I4_A_K_Am_A_L_A1_J_A_K_At_096, fact_113_pos__zmult__eq__1__iff, fact_779_add__less__cancel__right, fact_853_int__less__induct, fact_505_zdiff__power3, fact_5__096_126_A1_A_060_061_At_096, fact_428_mult__less__le__imp__less, fact_805_zero__less__double__add__iff__zero__less__single__add, fact_716_two__realpow__ge__one, arity_RealDef_Oreal___Rings_Oring, fact_395_pos__add__strict, fact_732_nat__mult__le__cancel1, fact_784_add__less__imp__less__right, fact_818_zfact_Osimps, fact_596_le__antisym, tsy_c_HOL_Oundefined_res, arity_RealDef_Oreal___Rings_Oordered__semiring, tsy_c_Residues_OLegendre_res, arity_RealDef_Oreal___Rings_Olinordered__idom, fact_162_add__Bit0__Bit1, fact_462__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096, tsy_c_IntPrimes_Ozcong_arg2, fact_410_power__eq__0__iff, fact_114_double__eq__0__iff, fact_867_pinf_I10_J, fact_390_mult__pos__pos, fact_346_Euler_Oaux2, fact_389_mult__pos__neg, fact_230_mult__nat__number__of, fact_666_add__leD2, fact_860_d22set__g__1, fact_181__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_L_A1_096, fact_661_nat__add__left__cancel__le, fact_641_trans__less__add2, fact_329_dvd__triv__right, fact_807_add__pos__pos, fact_405_dvd__mult__cancel__left, fact_806_double__add__less__zero__iff__single__add__less__zero, fact_694_nat__dvd__not__less, fact_429_mult__strict__mono_H, fact_765_one__reorient, fact_680_add__mult__distrib2, arity_Int_Oint___Rings_Oring__no__zero__divisors, fact_669_diff__diff__right, fact_643_add__less__mono, fact_762_add__left__imp__eq, tsy_c_IntPrimes_Ozcong_res, fact_471_diff__bin__simps_I1_J, fact_203_zdvd__not__zless, fact_342_le__imp__power__dvd, fact_884_mult_Oprod__diff__prod, fact_660_le__iff__add, help_ti_idem, fact_212_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J, fact_499_power__dvd__imp__le, arity_Nat_Onat___Rings_Olinordered__semidom, fact_78_less__nat__number__of, fact_613_add__diff__add, fact_865_minf_I9_J, fact_456_power__eq__imp__eq__base, fact_311_zcong__eq__trans, fact_79_int__one__le__iff__zero__less, fact_696_nat__diff__split__asm, fact_940_dvd__eq__mod__eq__0, arity_RealDef_Oreal___Rings_Oring__1, fact_714_mult__le__cancel2, fact_360_power__m1__even, fact_109_mult__numeral__1, arity_Int_Oint___Orderings_Olinorder, fact_83_add1__zle__eq, fact_496_diff__bin__simps_I5_J, fact_849_Int2_Oaux____1, fact_938_pos__mod__bound, fact_766_diff__eq__diff__eq, fact_254_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J, fact_502_zspecial__product, fact_775_add__left__mono, arity_Nat_Onat___Rings_Ocomm__semiring, fact_7_not__sum__power2__lt__zero, fact_404_less__add__one, fact_862_ex__least__nat__le, fact_344_nat__zero__less__power__iff, fact_568_dvd_Oorder__trans, fact_104_power2__less__imp__less, fact_67_plus__numeral__code_I9_J, fact_271_Euler_Oaux____1, fact_899_neg__zmod__mult__2, fact_952_mod__diff__right__eq, fact_582_dvd_Oless__irrefl, fact_48_le__special_I1_J, fact_785_add__less__imp__less__left, fact_930_mod__add__self2, fact_215_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J, fact_155_rel__simps_I16_J, fact_832_wset__less, arity_RealDef_Oreal___Rings_Oidom, fact_969_zmult2__lemma__aux3, fact_770_add__0__right, fact_888_mult_Ozero__right, fact_416_zdvd__bounds, fact_286_rel__simps_I45_J, fact_887_mult__left_Ozero, fact_27_le__number__of, arity_Int_Oint___Rings_Osemiring, fact_968_zmult2__lemma__aux4, fact_175_less__special_I3_J, fact_625_diff__add__0, fact_925_zmod__simps_I2_J, fact_361_split__mult__neg__le, fact_474_zdiff__zmult__distrib2, arity_Nat_Onat___Rings_Ocomm__semiring__1, fact_793_mult__1, fact_657_dvd__diffD1, fact_914_Residues_Oaux, fact_194__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1, fact_851_MultInv__prop2, arity_RealDef_Oreal___Rings_Olinordered__semidom, fact_89_number__of__reorient, fact_681_add__mult__distrib, fact_353_zcong__zmult__prop2, fact_226_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J, fact_691_nat__mult__eq__1__iff, fact_478_eq__add__iff2, fact_737_nat__diff__add__eq2, fact_144_semiring__numeral__0__eq__0, fact_279__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_, fact_205_zdvd__mult__cancel, fact_601_le__diff__iff, fact_935_zmod__eq__0__iff, fact_946_mod__mult__self1__is__0, fact_726_nat__mult__eq__cancel__disj, fact_280__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_096, fact_729_nat__mult__eq__cancel1, arity_Nat_Onat___Rings_Oordered__cancel__semiring, fact_734_nat__diff__add__eq1, fact_551_less__imp__diff__less, fact_967_zmult2__lemma__aux1, fact_6_p0, tsy_c_COMBC_arg2, fact_738_nat__eq__add__iff2, arity_RealDef_Oreal___Groups_Oab__group__add, fact_20_qf1pt, tsy_c_Residues_OLegendre_arg1, fact_858_d22set__le__swap, fact_207_zdvd__period, fact_135_zadd__strict__right__mono, tsy_c_Groups_Oplus__class_Oplus_2_res, arity_RealDef_Oreal___Groups_Oab__semigroup__add, fact_510_real__sum__squared__expand, fact_140_semiring__mult__2__right, fact_524_divides__exp2, fact_110_mult__numeral__1__right, fact_577_dvd_Oneq__le__trans, arity_Int_Oint___Groups_Omonoid__add, fact_719_ex__least__nat__less, fact_99_zero__is__num__zero, fact_498_inv__not__p__minus__1__aux, fact_301_eq__number__of__Pls__Min, fact_133_zadd__0__right, tsy_c_Int2_OMultInv_arg1, fact_363_mult__mono, arity_Nat_Onat___Groups_Ozero, fact_432_mult__le__cancel__left__pos, fact_973_pos__zmod__mult__2, fact_674_le__diff__conv2, fact_955_zmod__zdvd__zmod, fact_278_dvd__0__right, fact_406_dvd__mult__cancel__right, fact_507_Legendre__1mod4, fact_845_aux______2, arity_Nat_Onat___Groups_Oordered__comm__monoid__add, fact_2__096_I4_A_K_Am_A_L_A1_J_A_K_At_A_060_A_I4_A_K_Am_A_L_A1_J_A_K_A0_096, fact_96_zadd__assoc, tsy_c_Residues_OQuadRes_arg1, fact_970_split__zmod, fact_445_power__strict__increasing__iff, fact_352_zcong__zmult__prop1, fact_912_mod__mod__trivial, arity_RealDef_Oreal___Rings_Omult__zero, arity_RealDef_Oreal___Int_Oring__char__0, fact_834_wset__mem, fact_287_rel__simps_I42_J, fact_711_dvd__imp__le, fact_610_diff__mult__distrib2, fact_341_dvd__power__le, fact_727_left__add__mult__distrib, fact_68_zadd__zmult__distrib, fact_327_power__commutes, fact_147_add__numeral__0, arity_RealDef_Oreal___Rings_Osemiring, fact_966_zmult2__lemma__aux2, arity_RealDef_Oreal___Groups_Ocancel__semigroup__add, fact_340_power__le__dvd, fact_519_diff__square, fact_252_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J, fact_288_rel__simps_I7_J, fact_982_mod__mult__self3, arity_Nat_Onat___Groups_Ocomm__monoid__mult, fact_248_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J, fact_71_rel__simps_I5_J, fact_264_q__neg__lemma, fact_837_aux______1, fact_326_power__mult__distrib, fact_923_mod__add__cong, fact_330_dvd__mult2, arity_Nat_Onat___Groups_Oordered__ab__semigroup__add, fact_840_MultInv__zcong__prop1, fact_452_zcong__zmult__prop3, fact_274_pos2, fact_70_rel__simps_I29_J, fact_835_MultInvPair__distinct, tsy_c_Residues_OStandardRes_arg2, tsy_c_WilsonRuss_Oinv_arg2, tsy_c_Power_Opower__class_Opower_arg1, fact_692_real__add__mult__distrib, fact_29_zle__trans, fact_186_cube__square, fact_725_real__le__antisym, fact_178_one__add__one__is__two, arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc, tsy_c_Power_Opower__class_Opower_res, fact_679_diff__add__assoc2, fact_174_odd__less__0, fact_117_rel__simps_I50_J, fact_376_zero__le__mult__iff, fact_481_left__diff__distrib__number__of, fact_522_nat__mult__eq__one, arity_Int_Oint___Rings_Olinordered__comm__semiring__strict, fact_336_dvd__add, fact_934_zmod__eq__dvd__iff, fact_216_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J, fact_994_mod__lemma, fact_864_pinf_I9_J, fact_882_divides__le, tsy_c_Groups_Otimes__class_Otimes_4_arg1, arity_RealDef_Oreal___Orderings_Oorder, tsy_c_IntFact_Od22set_arg1, fact_535_diff__commute, fact_705_real__mult__less__mono2, fact_689_nat__1__eq__mult__iff, arity_RealDef_Oreal___Rings_Ocomm__ring__1, arity_RealDef_Oreal___Rings_Ono__zero__divisors, fact_739_number__of1, fact_753_is__sum2sq__def, fact_827_wset__mem__inv__mem, arity_Int_Oint___Groups_Ogroup__add, fact_98_zadd__commute, arity_RealDef_Oreal___Groups_Ocomm__monoid__add, fact_322_combine__common__factor, fact_721_nat__less__add__iff1, fact_469_number__of__diff, fact_142_power2__eq__square, arity_Int_Oint___Power_Opower, fact_9_sum__power2__eq__zero__iff, arity_RealDef_Oreal___Orderings_Olinorder, fact_733_nat__le__add__iff1, arity_Int_Oint___Rings_Oidom, arity_RealDef_Oreal___Groups_Olinordered__ab__group__add, tsy_c_Int2_OMultInv_arg2, fact_8_sum__power2__gt__zero__iff, fact_417_zcong__iff__lin, fact_362_split__mult__pos__le, fact_30_zle__antisym, fact_520_divides__ge, fact_790_diff__eq__diff__less__eq, fact_758_ab__semigroup__mult__class_Omult__ac_I1_J, fact_749_incr__mult__lemma, fact_584_nat__add__left__commute, fact_866_minf_I10_J, arity_RealDef_Oreal___Rings_Olinordered__semiring__1__strict, fact_615_neq0__conv, arity_Int_Oint___Groups_Ocancel__semigroup__add, fact_920_mod__mod__cancel, fact_268_zdiv__mono2__neg__lemma, help_COMBC_1_1_U, fact_188_self__quotient__aux1, fact_338_one__dvd, fact_485_less__bin__lemma, fact_32_less__eq__int__code_I16_J, fact_385_mult__less__cancel__left__neg, fact_421_mult__left__le__imp__le, fact_759_ab__semigroup__add__class_Oadd__ac_I1_J, fact_911_zpower__zmod, arity_Int_Oint___Rings_Osemiring__0, fact_492_add__number__of__diff1, fact_41_semiring__mult__number__of, fact_139_mult__2, fact_687_mult__le__mono, fact_101_power2__eq__imp__eq, fact_348_zcong__id, fact_393_mult__less__cancel__right__disj, fact_607_Nat_Odiff__le__self, fact_688_nat__mult__1, fact_671_le__add__diff, arity_Int_Oint___Rings_Oordered__comm__semiring, fact_875_gcd__lcm__complete__lattice__nat_Ole__bot, fact_409_power__inject__exp, fact_855_int__gr__induct, fact_783_add__strict__mono, fact_243_crossproduct__eq, fact_861_d22set__mem, tsy_c_Groups_Ominus__class_Ominus_1_res, arity_RealDef_Oreal___Rings_Ozero__neq__one, arity_Int_Oint___Rings_Ocomm__semiring__1, fact_500_zcong__square__zless, fact_819_inv__inv, fact_397_not__one__le__zero, arity_RealDef_Oreal___Rings_Oordered__cancel__semiring, arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc, arity_RealDef_Oreal___Groups_Oone, fact_768_add__0, arity_Nat_Onat___Int_Onumber, fact_665_add__le__mono, tsy_c_IntPrimes_Ozprime_res, fact_894_mult_Odiff__right, fact_339_dvd__power__same, fact_26_less__eq__number__of__int__code, fact_494_zle__diff1__eq, tsy_c_Residues_OSR_arg1, fact_333_dvdI, arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add, fact_240_crossproduct__noteq, fact_1_calculation_I1_J, fact_590_diff__diff__left, fact_546_linorder__neqE__nat, fact_295_bin__less__0__simps_I2_J, fact_157_rel__simps_I10_J, fact_332_mult__dvd__mono, fact_430_mult__strict__mono, fact_907_mod__mult__right__eq, fact_297_rel__simps_I30_J, arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors, fact_975_StandardRes__prop4, fact_695_add__gr__0, arity_RealDef_Oreal___RealVector_Oreal__normed__algebra, fact_978_mod__mult__distrib, fact_764_add__right__imp__eq, fact_320_zero__neq__one, fact_526_exp__eq__1, fact_463__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_N_A_N1_096, fact_163_int__0__less__1, fact_158_rel__simps_I4_J, fact_12_one__power2, fact_512_divides__antisym, fact_36_rel__simps_I31_J, fact_479_eq__add__iff1, fact_465_one__not__neg__one__mod__m, fact_246_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J, fact_168_power2__sum, fact_60_le__special_I2_J, fact_874_gcd__lcm__complete__lattice__nat_Onot__top__less, tsy_c_IntPrimes_Ozcong_arg1, fact_585_nat__add__assoc, fact_531_dvd__mult__cancel2, fact_236_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J, fact_258_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J, fact_46_sum__squares__ge__zero, fact_415_zcong__less__eq, fact_649_le__eq__less__or__eq, fact_731_nat__mult__dvd__cancel1, fact_778_add__le__imp__le__left, arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le, tsy_c_Residues_OSR_res, help_fconj_1_1_U, fact_282_rel__simps_I43_J, fact_118_rel__simps_I49_J, fact_990_StandardRes__prop1, fact_850_MultInv__prop2a, fact_493_Euler_Oaux1, fact_480_right__diff__distrib__number__of, tsy_c_Groups_Ozero__class_Ozero_res, fact_185_s0p, fact_490_less__add__iff2, arity_RealDef_Oreal___Rings_Olinordered__semiring__1, fact_305_pos__zmult__eq__1__iff__lemma, fact_497_diff__bin__simps_I6_J, arity_Int_Oint___Rings_Ozero__neq__one, fact_960_neg__mod__sign, fact_488_le__add__iff1, tsy_c_Groups_Otimes__class_Otimes_3_res, fact_102_power2__le__imp__le, arity_Nat_Onat___Groups_Oone, fact_373_mult__nonneg__nonpos, fact_484_diff__bin__simps_I3_J, fact_23_zmult__commute, fact_65_zmult__1__right, fact_871_gcd__lcm__complete__lattice__nat_Otop__unique, fact_624_add__eq__self__zero, fact_281_rel__simps_I47_J, fact_556_dvd_Oless__le__trans, fact_190_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J, fact_22_number__of__is__id, fact_909_zmod__zmult1__eq, arity_RealDef_Oreal___Groups_Omonoid__mult, fact_961_neg__mod__conj, tsy_c_Int_Onumber__class_Onumber__of_arg1, fact_319_mult__zero__left, fact_883_mult__left__cancel, fact_345_power__mult, fact_782_add__strict__left__mono, arity_Int_Oint___Groups_Oordered__ab__group__add, tsy_c_COMBC_arg1, fact_107_sum__squares__gt__zero__iff, fact_143_even__less__0__iff, fact_872_gcd__lcm__complete__lattice__nat_Oless__top, fact_91_rel__simps_I48_J, fact_458_dvd__power, tsy_c_Int_OBit0_res, fact_892_mult_Oadd__left, arity_Int_Oint___Groups_Oab__semigroup__add, fact_208_zprime__2, fact_611_diff__mult__distrib, fact_121_Bit0__Pls, fact_441_power__gt1__lemma, tsy_c_Int2_OMultInv_res, fact_283_Bit1__Min, tsy_c_Groups_Otimes__class_Otimes_2_arg1, fact_199_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J, fact_644_less__add__eq__less, fact_495_diff__bin__simps_I4_J, fact_400_power__mono, fact_122_Pls__def, fact_460_zcong__zprime__prod__zero__contra, arity_Int_Oint___Rings_Ocomm__ring__1, fact_527_divides__div__not, fact_124_rel__simps_I17_J, fact_303_rel__simps_I25_J, arity_Nat_Onat___Orderings_Oorder, fact_922_dvd__mod__imp__dvd, fact_893_mult__left_Oadd, fact_572_dvd_Oord__eq__le__trans, fact_44_number__of__mult, fact_713_mult__le__cancel1, fact_593_real__mult__assoc, arity_Nat_Onat___Rings_Osemiring, tsy_c_Groups_Oone__class_Oone_res, fact_126_less__int__code_I13_J, fact_316_no__zero__divisors, fact_161_add__Bit1__Bit0, fact_539_diffs0__imp__equal, fact_971_neq__one__mod__two, fact_626_less__eq__nat_Osimps_I1_J, fact_331_dvd__mult, fact_80_nat__numeral__1__eq__1, fact_863_less__imp__add__positive, fact_486_xzgcda__linear__aux1, fact_388_mult__pos__neg2, fact_569_dvd_Oantisym, fact_776_add__mono, fact_559_dvd_Oless__imp__not__eq2, fact_720_nat__less__add__iff2, arity_RealDef_Oreal___Power_Opower, fact_136_double__number__of__Bit0, tsy_c_Set_OCollect_arg1, fact_891_mult__right_Oadd, fact_111_one__is__num__one, fact_700_mult__less__cancel2, fact_291_rel__simps_I9_J, fact_179_less__special_I4_J, fact_562_dvd_Oless__imp__le, fact_658_le__add2, fact_555_dvd_Oless__asym_H, tsy_c_WilsonRuss_Owset_arg1, fact_622_Nat_Oadd__0__right, fact_895_mult__right_Odiff, fact_763_add__imp__eq, arity_Nat_Onat___Rings_Odvd, fact_156_bin__less__0__simps_I4_J, fact_977_mod__mult__distrib2, fact_57_nat__number__of__Pls, fact_401_zero__le__power, fact_477_zdvd__zdiffD, fact_130_add__Pls, fact_184_int__pos__lt__two__imp__zero__or__one, fact_437_power__decreasing, arity_Nat_Onat___Rings_Ono__zero__divisors, tsy_v_t_____res, fact_15_add__special_I2_J, arity_Nat_Onat___Rings_Olinordered__comm__semiring__strict, fact_62_mult__Pls, fact_557_dvd_Oord__less__eq__trans, arity_Int_Oint___Groups_Oab__semigroup__mult, fact_241_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J, fact_424_mult__left__less__imp__less, fact_929_mod__add__self1, fact_604_diff__diff__cancel, tsy_c_Groups_Oplus__class_Oplus_3_res, fact_253_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J, fact_391_mult__less__cancel__left__pos, fact_586_nat__add__left__cancel, tsy_c_WilsonRuss_Oinv_arg1, fact_137_power3__eq__cube, fact_103_zero__le__power2, arity_RealDef_Oreal___Rings_Oordered__ring, fact_275_power__strict__mono, fact_354_zcong__scalar, arity_Int_Oint___Int_Onumber, fact_944_mod__by__1, fact_794_mult__1__right, fact_735_nat__eq__add__iff1, fact_897_mult__left_Odiff, fact_823_inv__not__1, fact_508_Legendre__def, fact_221_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J, fact_786_right__minus__eq, fact_293_rel__simps_I6_J, fact_55_rel__simps_I21_J, fact_547_less__irrefl__nat, arity_Nat_Onat___Groups_Omonoid__add, arity_RealDef_Oreal___Int_Onumber, tsy_c_Int_OBit0_arg1, fact_24_zle__linear, arity_Int_Oint___Divides_Osemiring__div, fact_266_zdiv__mono2__lemma, fact_592_diff__cancel2, arity_RealDef_Oreal___Groups_Oab__semigroup__mult, fact_885_SR__def, fact_927_mod__add__left__eq, fact_120_rel__simps_I38_J, fact_154_less__int__code_I15_J, arity_Int_Oint___Groups_Olinordered__ab__group__add, fact_200_zprime__power__zdvd__cancel__right, arity_Nat_Onat___Groups_Oab__semigroup__add, tsy_c_TwoSquares__Mirabelle__lldtdwpyrp_Osum2sq_res, fact_767_add__0__left, fact_119_rel__simps_I44_J, fact_351_zcong__zmult__self, fact_438_power__le__imp__le__exp, fact_202_zpower__zpower, fact_45_sum__squares__le__zero__iff, arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add, fact_66_zmult__1, fact_932_mod__by__0, fact_449_zcong__zless__imp__eq, fact_312_zcong__trans, fact_350_zcong__1, fact_902_zmod__simps_I4_J, arity_Int_Oint___Rings_Olinordered__semidom, fact_571_dvd_Oord__le__eq__trans, fact_953_zdvd__iff__zmod__eq__0__number__of, fact_647_less__diff__conv, fact_294_rel__simps_I8_J, fact_277_zpower__zdvd__prop2, arity_Int_Oint___Groups_Ozero, fact_667_add__leD1, fact_93_sum__squares__eq__zero__iff, fact_640_trans__less__add1, tsy_c_Int_OBit1_arg1, arity_Int_Oint___Rings_Omult__zero, fact_127_rel__simps_I14_J, fact_683_le__square, fact_209_zdvd__imp__le, fact_769_double__zero__sym, tsy_c_Rings_Odvd__class_Odvd_res, arity_Int_Oint___Rings_Odvd, fact_399_not__one__less__zero, arity_RealDef_Oreal___Rings_Olinordered__semiring, fact_382_mult__strict__right__mono, fact_403_zero__less__power, fact_648_nat__less__le, fact_534_le0, fact_748_decr__mult__lemma, fact_741_conj__le__cong, fact_533_less__zeroE, fact_467__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N, fact_511_divides__cases, fact_131_add__Bit0__Bit0, fact_90_rel__simps_I51_J, fact_472_diff__bin__simps_I7_J, arity_Int_Oint___Rings_Olinordered__semiring__1__strict, fact_578_dvd_Oless__le__not__le, fact_942_mod__mult__self2, fact_646_add__diff__inverse, fact_425_mult__less__imp__less__right, fact_394_not__square__less__zero, fact_357_zcong__zadd, fact_43_arith__simps_I32_J, fact_334_dvd__mult__left, fact_877_gcd__lcm__complete__lattice__nat_Obot__less, fact_310_zcong__refl, arity_Nat_Onat___Rings_Ozero__neq__one, fact_708_mult__eq__self__implies__10, fact_369_mult__right__mono, fact_412_power__add, fact_780_add__less__cancel__left, fact_375_mult__le__0__iff, fact_964_zmod__number__of__Bit0, arity_Int_Oint___Groups_Ocomm__monoid__mult, fact_165_odd__nonzero, fact_414_zcong__not__zero, fact_435_zero__less__two, fact_304_zmult__eq__1__iff, fact_822_inv__not__p__minus__1, fact_447_Int2_Ozcong__zero, fact_269_quartic__square__square, fact_573_dvd_Ole__neq__trans, fact_250_zero__le__even__power_H, arity_RealDef_Oreal___Groups_Ogroup__add, fact_489_le__add__iff2, fact_788_diff__self, fact_854_int__ge__induct, fact_276_convex__bound__lt, fact_159_bin__less__0__simps_I1_J, fact_724_real__le__trans, fact_82_zless__imp__add1__zle, tsy_c_Orderings_Oord__class_Oless_res, arity_Nat_Onat___Groups_Ocancel__semigroup__add, fact_146_semiring__norm_I112_J, fact_16_add__special_I3_J, fact_213_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J, tsy_c_TwoSquares__Mirabelle__lldtdwpyrp_Ois__sum2sq_res, fact_825_inv__not__0, fact_53_rel__simps_I32_J, fact_169_number__of__Bit0, fact_292_rel__simps_I3_J, fact_594_real__mult__commute, tsy_c_WilsonRuss_Oinv_res, fact_368_mult__left__mono, fact_629_diff__is__0__eq_H, fact_980_mod__less, fact_781_add__strict__right__mono, fact_455_power__strict__decreasing, fact_374_mult__nonneg__nonneg, fact_195__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_, fact_434_mult__right__le__one__le, fact_962_mod__neg__neg__trivial, fact_659_le__add1, fact_347_power__one__right, fact_876_gcd__lcm__complete__lattice__nat_Obot__unique, fact_321_one__neq__zero
% 31.28/9.31 % No. of inferences in proof: 2512
% 31.28/9.31 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 8773 ms resp. 8149 ms w/o parsing
% 32.91/9.86 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 32.91/9.87 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------