%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : ITP094^2 : TPTP v9.3.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% Computer : n014.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Sep 30 08:06:44 AM UTC 2026
% Result : Theorem 50.66s 32.00s
% Output : Refutation 51.75s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 316
% Syntax : Number of formulae : 634 ( 198 unt; 0 typ; 0 def)
% Number of atoms : 1878 ( 593 equ; 8 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 5892 ( 262 ~; 22 |; 163 &;4665 @)
% ( 0 <=>; 780 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 110 ( 110 >; 0 *; 0 +; 0 <<)
% Number of symbols : 65 ( 61 usr; 8 con; 0-7 aty)
% Number of variables : 1280 ( 67 ^;1145 !; 68 ?;1280 :)
% Comments :
%------------------------------------------------------------------------------
thf(real_type,type,
real: $tType ).
thf(nat_type,type,
nat: $tType ).
thf(one_decl,type,
one:
!>[TA: $tType] : $o ).
thf(idom_decl,type,
idom:
!>[TA: $tType] : $o ).
thf(zero_decl,type,
zero:
!>[TA: $tType] : $o ).
thf(field_decl,type,
field:
!>[TA: $tType] : $o ).
thf(ring_1_decl,type,
ring_1:
!>[TA: $tType] : $o ).
thf(ring_char_0_decl,type,
ring_char_0:
!>[TA: $tType] : $o ).
thf(order_decl,type,
order:
!>[TA: $tType] : $o ).
thf(finite_finite_decl,type,
finite_finite:
!>[TA: $tType] : $o ).
thf(comm_ring_1_decl,type,
comm_ring_1:
!>[TA: $tType] : $o ).
thf(idom_divide_decl,type,
idom_divide:
!>[TA: $tType] : $o ).
thf(zero_neq_one_decl,type,
zero_neq_one:
!>[TA: $tType] : $o ).
thf(field_char_0_decl,type,
field_char_0:
!>[TA: $tType] : $o ).
thf(semiring_char_0_decl,type,
semiring_char_0:
!>[TA: $tType] : $o ).
thf(wellorder_decl,type,
wellorder:
!>[TA: $tType] : $o ).
thf(zero_less_one_decl,type,
zero_less_one:
!>[TA: $tType] : $o ).
thf(division_ring_decl,type,
division_ring:
!>[TA: $tType] : $o ).
thf(comm_semiring_0_decl,type,
comm_semiring_0:
!>[TA: $tType] : $o ).
thf(comm_semiring_1_decl,type,
comm_semiring_1:
!>[TA: $tType] : $o ).
thf(linordered_idom_decl,type,
linordered_idom:
!>[TA: $tType] : $o ).
thf(linordered_field_decl,type,
linordered_field:
!>[TA: $tType] : $o ).
thf(comm_monoid_mult_decl,type,
comm_monoid_mult:
!>[TA: $tType] : $o ).
thf(algebraic_semidom_decl,type,
algebraic_semidom:
!>[TA: $tType] : $o ).
thf(ring_n68954251visors_decl,type,
ring_n68954251visors:
!>[TA: $tType] : $o ).
thf(semiri1193490041visors_decl,type,
semiri1193490041visors:
!>[TA: $tType] : $o ).
thf(linord1659791738miring_decl,type,
linord1659791738miring:
!>[TA: $tType] : $o ).
thf(semiri134348788visors_decl,type,
semiri134348788visors:
!>[TA: $tType] : $o ).
thf(canoni770627133id_add_decl,type,
canoni770627133id_add:
!>[TA: $tType] : $o ).
thf(inverse_inverse_decl,type,
inverse_inverse:
!>[TA: $tType] : ( TA > TA ) ).
thf(finite_finite2_decl,type,
finite_finite2:
!>[TA: $tType] : ( ( set @ TA ) > $o ) ).
thf(one_one_decl,type,
one_one:
!>[TA: $tType] : TA ).
thf(zero_zero_decl,type,
zero_zero:
!>[TA: $tType] : TA ).
thf(ring_1_Ints_decl,type,
ring_1_Ints:
!>[TA: $tType] : ( set @ TA ) ).
thf(suc_decl,type,
suc: nat > nat ).
thf(ord_less_decl,type,
ord_less:
!>[TA: $tType] : ( TA > TA > $o ) ).
thf(ord_less_eq_decl,type,
ord_less_eq:
!>[TA: $tType] : ( TA > TA > $o ) ).
thf(algebraic_decl,type,
algebraic:
!>[TA: $tType] : ( TA > $o ) ).
thf(algebraic_int_decl,type,
algebraic_int:
!>[TA: $tType] : ( TA > $o ) ).
thf(cr_poly_decl,type,
cr_poly:
!>[TA: $tType] : ( ( nat > TA ) > ( poly @ TA ) > $o ) ).
thf(degree_decl,type,
degree:
!>[TA: $tType] : ( ( poly @ TA ) > nat ) ).
thf(divide_poly_main_decl,type,
divide_poly_main:
!>[TA: $tType] : ( TA > ( poly @ TA ) > ( poly @ TA ) > ( poly @ TA ) > nat > nat > ( poly @ TA ) ) ).
thf(is_zero_decl,type,
is_zero:
!>[TA: $tType] : ( ( poly @ TA ) > $o ) ).
thf(order2_decl,type,
order2:
!>[TA: $tType] : ( TA > ( poly @ TA ) > nat ) ).
thf(pcr_poly_decl,type,
pcr_poly:
!>[TA: $tType,TB: $tType] : ( ( TB > TA > $o ) > ( nat > TB ) > ( poly @ TA ) > $o ) ).
thf(pderiv_decl,type,
pderiv:
!>[TA: $tType] : ( ( poly @ TA ) > ( poly @ TA ) ) ).
thf(poly2_decl,type,
poly2:
!>[TA: $tType] : ( ( poly @ TA ) > TA > TA ) ).
thf(coeff_decl,type,
coeff:
!>[TA: $tType] : ( ( poly @ TA ) > nat > TA ) ).
thf(poly_cutoff_decl,type,
poly_cutoff:
!>[TA: $tType] : ( nat > ( poly @ TA ) > ( poly @ TA ) ) ).
thf(poly_shift_decl,type,
poly_shift:
!>[TA: $tType] : ( nat > ( poly @ TA ) > ( poly @ TA ) ) ).
thf(pseudo_mod_decl,type,
pseudo_mod:
!>[TA: $tType] : ( ( poly @ TA ) > ( poly @ TA ) > ( poly @ TA ) ) ).
thf(reflect_poly_decl,type,
reflect_poly:
!>[TA: $tType] : ( ( poly @ TA ) > ( poly @ TA ) ) ).
thf(rsquarefree_decl,type,
rsquarefree:
!>[TA: $tType] : ( ( poly @ TA ) > $o ) ).
thf(synthetic_div_decl,type,
synthetic_div:
!>[TA: $tType] : ( ( poly @ TA ) > TA > ( poly @ TA ) ) ).
thf(field_char_0_Rats_decl,type,
field_char_0_Rats:
!>[TA: $tType] : ( set @ TA ) ).
thf(dvd_dvd_decl,type,
dvd_dvd:
!>[TA: $tType] : ( TA > TA > $o ) ).
thf(collect_decl,type,
collect:
!>[TA: $tType] : ( ( TA > $o ) > ( set @ TA ) ) ).
thf(member_decl,type,
member:
!>[TA: $tType] : ( TA > ( set @ TA ) > $o ) ).
thf(n_decl,type,
n: nat ).
thf(p_decl,type,
p: poly @ real ).
thf(x_decl,type,
x: real ).
thf(230,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( reflect_poly @ TA @ ( zero_zero @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_reflect__poly__0) ).
thf(1036,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( reflect_poly @ TA @ ( zero_zero @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).
thf(17,axiom,
canoni770627133id_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add) ).
thf(365,plain,
canoni770627133id_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(191,axiom,
! [TA: $tType] :
( ( division_ring @ TA )
=> ! [A: TA] :
( ( ( inverse_inverse @ TA @ A )
= ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_inverse__nonzero__iff__nonzero) ).
thf(893,plain,
! [TA: $tType] :
( ( division_ring @ TA )
=> ! [A: TA] :
( ( ( inverse_inverse @ TA @ A )
= ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(311,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat,B: poly @ TA] :
( ( ord_less @ nat @ A @ ( degree @ TA @ B ) )
=> ? [C: nat] :
( ( ord_less @ nat @ A @ C )
& ( ( coeff @ TA @ B @ C )
!= ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194_less__degree__imp) ).
thf(1295,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat,B: poly @ TA] :
( ( ord_less @ nat @ A @ ( degree @ TA @ B ) )
=> ? [C: nat] :
( ( ord_less @ nat @ A @ C )
& ( ( coeff @ TA @ B @ C )
!= ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).
thf(13,axiom,
( ( dvd_dvd @ nat )
= ( ^ [A: nat,B: nat] :
( ( ( dvd_dvd @ nat @ A @ B )
& ( A != B ) )
| ( A = B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_gcd__nat_Oorder__iff__strict) ).
thf(352,plain,
( ( dvd_dvd @ nat )
= ( ^ [A: nat,B: nat] :
( ( ( dvd_dvd @ nat @ A @ B )
& ( A != B ) )
| ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(31,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( suc @ A ) @ B )
=> ( ord_less @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_166_Suc__lessD) ).
thf(408,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( suc @ A ) @ B )
=> ( ord_less @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(90,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( linordered_idom @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Rings_Olinordered__idom_24) ).
thf(582,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( linordered_idom @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(127,axiom,
! [TA: $tType] :
( ( zero_less_one @ TA )
=> ( ord_less @ TA @ ( zero_zero @ TA ) @ ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_zero__less__one) ).
thf(683,plain,
! [TA: $tType] :
( ( zero_less_one @ TA )
=> ( ord_less @ TA @ ( zero_zero @ TA ) @ ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(144,axiom,
! [TA: $tType] :
( ( ( comm_ring_1 @ TA )
& ( ring_n68954251visors @ TA ) )
=> ! [A: poly @ TA] :
( ( A
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [B: TA] :
( ( poly2 @ TA @ A @ B )
= ( zero_zero @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0_poly__roots__finite) ).
thf(730,plain,
! [TA: $tType] :
( ( ( comm_ring_1 @ TA )
& ( ring_n68954251visors @ TA ) )
=> ! [A: poly @ TA] :
( ( A
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [B: TA] :
( ( poly2 @ TA @ A @ B )
= ( zero_zero @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(243,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
= ( B @ C ) )
=> ( ( collect @ TA @ A )
= ( collect @ TA @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_Collect__cong) ).
thf(1068,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
= ( B @ C ) )
=> ( ( collect @ TA @ A )
= ( collect @ TA @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).
thf(221,axiom,
! [TA: $tType] :
( ( idom @ TA )
=> ( ( rsquarefree @ TA )
= ( ^ [A: poly @ TA] :
( ( A
!= ( zero_zero @ ( poly @ TA ) ) )
& ! [B: TA] :
( ( ( order2 @ TA @ B @ A )
= ( zero_zero @ nat ) )
| ( ( order2 @ TA @ B @ A )
= ( one_one @ nat ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_rsquarefree__def) ).
thf(994,plain,
! [TA: $tType] :
( ( idom @ TA )
=> ( ( rsquarefree @ TA )
= ( ^ [A: poly @ TA] :
( ( A
!= ( zero_zero @ ( poly @ TA ) ) )
& ! [B: TA] :
( ( ( order2 @ TA @ B @ A )
= ( zero_zero @ nat ) )
| ( ( order2 @ TA @ B @ A )
= ( one_one @ nat ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).
thf(184,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_less__nat__zero__code) ).
thf(873,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(267,axiom,
linordered_field @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Fields_Olinordered__field) ).
thf(1142,plain,
linordered_field @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).
thf(125,axiom,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ! [A: poly @ TA,B: TA,C: nat] :
( ( ( pderiv @ TA @ A )
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( ( order2 @ TA @ B @ A )
!= ( zero_zero @ nat ) )
=> ( ( ( order2 @ TA @ B @ ( pderiv @ TA @ A ) )
= C )
= ( ( order2 @ TA @ B @ A )
= ( suc @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_order__pderiv2) ).
thf(677,plain,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ! [A: poly @ TA,B: TA,C: nat] :
( ( ( pderiv @ TA @ A )
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( ( order2 @ TA @ B @ A )
!= ( zero_zero @ nat ) )
=> ( ( ( order2 @ TA @ B @ ( pderiv @ TA @ A ) )
= C )
= ( ( order2 @ TA @ B @ A )
= ( suc @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(315,axiom,
! [A: nat] :
( ( suc @ A )
!= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_old_Onat_Odistinct_I2_J) ).
thf(1305,plain,
! [A: nat] :
( ( suc @ A )
!= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).
thf(279,axiom,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ( ( finite_finite2 @ TA )
= ( ^ [A: set @ TA] : $true ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_finite__code) ).
thf(1173,plain,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ( ( finite_finite2 @ TA )
= ( ^ [A: set @ TA] : $true ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).
thf(37,axiom,
( ( (=) @ nat )
= ( ^ [A: nat,B: nat] :
( ( dvd_dvd @ nat @ A @ B )
& ( dvd_dvd @ nat @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_gcd__nat_Oeq__iff) ).
thf(440,plain,
( ( (=) @ nat )
= ( ^ [A: nat,B: nat] :
( ( dvd_dvd @ nat @ A @ B )
& ( dvd_dvd @ nat @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(117,axiom,
! [TA: $tType] :
( ( idom @ TA )
=> ( idom @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Rings_Oidom_36) ).
thf(655,plain,
! [TA: $tType] :
( ( idom @ TA )
=> ( idom @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(246,axiom,
( ( pderiv @ real @ p )
!= ( zero_zero @ ( poly @ real ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2__092_060open_062pderiv_Ap_A_092_060noteq_062_A0_092_060close_062) ).
thf(1078,plain,
( ( pderiv @ real @ p )
!= ( zero_zero @ ( poly @ real ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).
thf(24,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ ( suc @ B ) )
=> ( ~ ( ord_less @ nat @ A @ B )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_163_less__SucE) ).
thf(385,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ ( suc @ B ) )
=> ( ~ ( ord_less @ nat @ A @ B )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(23,axiom,
zero @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ozero) ).
thf(384,plain,
zero @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(57,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd @ nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd @ nat @ A @ B )
& ( A != B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_gcd__nat_Ostrict__iff__order) ).
thf(500,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(276,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ! [A: nat] :
( ( ( A
= ( zero_zero @ nat ) )
=> ( ( poly_cutoff @ TA @ A @ ( one_one @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) )
& ( ( A
!= ( zero_zero @ nat ) )
=> ( ( poly_cutoff @ TA @ A @ ( one_one @ ( poly @ TA ) ) )
= ( one_one @ ( poly @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_poly__cutoff__1) ).
thf(1164,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ! [A: nat] :
( ( ( A
= ( zero_zero @ nat ) )
=> ( ( poly_cutoff @ TA @ A @ ( one_one @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) )
& ( ( A
!= ( zero_zero @ nat ) )
=> ( ( poly_cutoff @ TA @ A @ ( one_one @ ( poly @ TA ) ) )
= ( one_one @ ( poly @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).
thf(139,axiom,
idom_divide @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Rings_Oidom__divide) ).
thf(715,plain,
idom_divide @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(294,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( ( coeff @ TA @ A @ ( zero_zero @ nat ) )
!= ( zero_zero @ TA ) )
=> ( ( reflect_poly @ TA @ ( reflect_poly @ TA @ A ) )
= A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_reflect__poly__reflect__poly) ).
thf(1222,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( ( coeff @ TA @ A @ ( zero_zero @ nat ) )
!= ( zero_zero @ TA ) )
=> ( ( reflect_poly @ TA @ ( reflect_poly @ TA @ A ) )
= A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).
thf(170,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( coeff @ TA @ ( reflect_poly @ TA @ A ) @ ( zero_zero @ nat ) )
= ( coeff @ TA @ A @ ( degree @ TA @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_coeff__0__reflect__poly) ).
thf(829,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( coeff @ TA @ ( reflect_poly @ TA @ A ) @ ( zero_zero @ nat ) )
= ( coeff @ TA @ A @ ( degree @ TA @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(193,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: poly @ TA] :
( ( ( poly2 @ TA @ ( reflect_poly @ TA @ A ) @ ( zero_zero @ TA ) )
= ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ ( poly @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_reflect__poly__at__0__eq__0__iff) ).
thf(898,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: poly @ TA] :
( ( ( poly2 @ TA @ ( reflect_poly @ TA @ A ) @ ( zero_zero @ TA ) )
= ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ ( poly @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).
thf(188,axiom,
! [A: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [B: nat] : ( dvd_dvd @ nat @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_184_finite__divisors__nat) ).
thf(885,plain,
! [A: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [B: nat] : ( dvd_dvd @ nat @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(82,axiom,
! [A: nat] :
( A
!= ( suc @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_n__not__Suc__n) ).
thf(565,plain,
! [A: nat] :
( A
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(115,axiom,
! [A: nat] :
~ ( ( dvd_dvd @ nat @ A @ A )
& ( A != A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_gcd__nat_Oirrefl) ).
thf(651,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(301,axiom,
! [TA: $tType] :
( ( field @ TA )
=> ( ( dvd_dvd @ TA )
= ( ^ [A: TA,B: TA] :
( ( A
= ( zero_zero @ TA ) )
=> ( B
= ( zero_zero @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_226_dvd__field__iff) ).
thf(1243,plain,
! [TA: $tType] :
( ( field @ TA )
=> ( ( dvd_dvd @ TA )
= ( ^ [A: TA,B: TA] :
( ( A
= ( zero_zero @ TA ) )
=> ( B
= ( zero_zero @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).
thf(262,axiom,
semiri1193490041visors @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_8) ).
thf(1127,plain,
semiri1193490041visors @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).
thf(26,axiom,
! [TA: $tType] : ( order @ ( set @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oorder_3) ).
thf(389,plain,
! [TA: $tType] : ( order @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(237,axiom,
ring_n68954251visors @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Rings_Oring__no__zero__divisors) ).
thf(1055,plain,
ring_n68954251visors @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).
thf(141,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat,B: nat,C: poly @ TA] :
( ( ( ord_less @ nat @ A @ B )
=> ( ( coeff @ TA @ ( poly_cutoff @ TA @ B @ C ) @ A )
= ( coeff @ TA @ C @ A ) ) )
& ( ~ ( ord_less @ nat @ A @ B )
=> ( ( coeff @ TA @ ( poly_cutoff @ TA @ B @ C ) @ A )
= ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_coeff__poly__cutoff) ).
thf(719,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat,B: nat,C: poly @ TA] :
( ( ( ord_less @ nat @ A @ B )
=> ( ( coeff @ TA @ ( poly_cutoff @ TA @ B @ C ) @ A )
= ( coeff @ TA @ C @ A ) ) )
& ( ~ ( ord_less @ nat @ A @ B )
=> ( ( coeff @ TA @ ( poly_cutoff @ TA @ B @ C ) @ A )
= ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(60,axiom,
finite_finite @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Finite__Set_Ofinite_4) ).
thf(504,plain,
finite_finite @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(173,axiom,
! [TA: $tType] :
( ( ring_1 @ TA )
=> ( member @ TA @ ( zero_zero @ TA ) @ ( ring_1_Ints @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_Ints__0) ).
thf(838,plain,
! [TA: $tType] :
( ( ring_1 @ TA )
=> ( member @ TA @ ( zero_zero @ TA ) @ ( ring_1_Ints @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(200,axiom,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ( ( rsquarefree @ TA )
= ( ^ [A: poly @ TA] :
! [B: TA] :
~ ( ( ( poly2 @ TA @ A @ B )
= ( zero_zero @ TA ) )
& ( ( poly2 @ TA @ ( pderiv @ TA @ A ) @ B )
= ( zero_zero @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_rsquarefree__roots) ).
thf(917,plain,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ( ( rsquarefree @ TA )
= ( ^ [A: poly @ TA] :
! [B: TA] :
~ ( ( ( poly2 @ TA @ A @ B )
= ( zero_zero @ TA ) )
& ( ( poly2 @ TA @ ( pderiv @ TA @ A ) @ B )
= ( zero_zero @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(16,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd @ nat @ A @ B )
& ( A != B ) )
=> ( ( dvd_dvd @ nat @ B @ C )
=> ( ( dvd_dvd @ nat @ A @ C )
& ( A != C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_gcd__nat_Ostrict__trans2) ).
thf(357,plain,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd @ nat @ A @ B )
& ( A != B ) )
=> ( ( dvd_dvd @ nat @ B @ C )
=> ( ( dvd_dvd @ nat @ A @ C )
& ( A != C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(239,axiom,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( one_one @ TA ) @ ( inverse_inverse @ TA @ A ) )
= ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
& ( ord_less @ TA @ A @ ( one_one @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_one__less__inverse__iff) ).
thf(1059,plain,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( one_one @ TA ) @ ( inverse_inverse @ TA @ A ) )
= ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
& ( ord_less @ TA @ A @ ( one_one @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).
thf(163,axiom,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( inverse_inverse @ TA @ A ) @ ( inverse_inverse @ TA @ B ) )
=> ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
=> ( ord_less @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_234_inverse__less__imp__less__neg) ).
thf(802,plain,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( inverse_inverse @ TA @ A ) @ ( inverse_inverse @ TA @ B ) )
=> ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
=> ( ord_less @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(11,axiom,
! [TA: $tType] :
( ( ( comm_semiring_1 @ TA )
& ( semiri134348788visors @ TA ) )
=> ( semiri134348788visors @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Rings_Osemiring__1__no__zero__divisors_18) ).
thf(344,plain,
! [TA: $tType] :
( ( ( comm_semiring_1 @ TA )
& ( semiri134348788visors @ TA ) )
=> ( semiri134348788visors @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(80,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( pcr_poly @ TA @ TA @ ( (=) @ TA ) )
= ( cr_poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_poly_Opcr__cr__eq) ).
thf(559,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( pcr_poly @ TA @ TA @ ( (=) @ TA ) )
= ( cr_poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(183,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_251_le__zero__eq) ).
thf(870,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(98,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ ( suc @ B ) )
= ( ( ord_less @ nat @ A @ B )
| ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_160_less__Suc__eq) ).
thf(606,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ ( suc @ B ) )
= ( ( ord_less @ nat @ A @ B )
| ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(40,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ~ ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_linorder__neqE__nat) ).
thf(448,plain,
! [A: nat,B: nat] :
( ( A != B )
=> ( ~ ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(297,axiom,
! [TA: $tType,A: set @ TA] :
( ( collect @ TA
@ ^ [B: TA] : ( member @ TA @ B @ A ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_Collect__mem__eq) ).
thf(1232,plain,
! [TA: $tType,A: set @ TA] :
( ( collect @ TA
@ ^ [B: TA] : ( member @ TA @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).
thf(226,axiom,
! [TA: $tType,A: set @ TA,B: ( set @ TA ) > $o] :
( ( finite_finite2 @ TA @ A )
=> ( ! [C: set @ TA] :
( ( finite_finite2 @ TA @ C )
=> ( ! [D: set @ TA] :
( ( ord_less @ ( set @ TA ) @ D @ C )
=> ( B @ D ) )
=> ( B @ C ) ) )
=> ( B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_222_finite__psubset__induct) ).
thf(1013,plain,
! [TA: $tType,A: set @ TA,B: ( set @ TA ) > $o] :
( ( finite_finite2 @ TA @ A )
=> ( ! [C: set @ TA] :
( ( finite_finite2 @ TA @ C )
=> ( ! [D: set @ TA] :
( ( ord_less @ ( set @ TA ) @ D @ C )
=> ( B @ D ) )
=> ( B @ C ) ) )
=> ( B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).
thf(157,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( B
!= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_gr__implies__not__zero) ).
thf(770,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( B
!= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(256,axiom,
! [TA: $tType,TB: $tType,TC: $tType] :
( ( zero @ TB )
=> ! [A: ( TC > TB ) > TA,B: TA] :
( ( A
= ( ^ [C: TC > TB] : B ) )
=> ( ( A
@ ^ [C: TC] : ( zero_zero @ TB ) )
= B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_fun__cong__unused__0) ).
thf(1109,plain,
! [TA: $tType,TB: $tType,TC: $tType] :
( ( zero @ TB )
=> ! [A: ( TC > TB ) > TA,B: TA] :
( ( A
= ( ^ [C: TC > TB] : B ) )
=> ( ( A
@ ^ [C: TC] : ( zero_zero @ TB ) )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).
thf(105,axiom,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ! [A: TA] :
( ( member @ TA @ A @ ( field_char_0_Rats @ TA ) )
=> ( member @ TA @ ( inverse_inverse @ TA @ A ) @ ( field_char_0_Rats @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_168_Rats__inverse) ).
thf(623,plain,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ! [A: TA] :
( ( member @ TA @ A @ ( field_char_0_Rats @ TA ) )
=> ( member @ TA @ ( inverse_inverse @ TA @ A ) @ ( field_char_0_Rats @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(113,axiom,
! [TA: $tType] :
( ( ( comm_semiring_0 @ TA )
& ( semiri1193490041visors @ TA ) )
=> ( semiri1193490041visors @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Rings_Osemiring__no__zero__divisors_20) ).
thf(646,plain,
! [TA: $tType] :
( ( ( comm_semiring_0 @ TA )
& ( semiri1193490041visors @ TA ) )
=> ( semiri1193490041visors @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(265,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: poly @ TA] :
( ( poly2 @ TA @ ( reflect_poly @ TA @ A ) @ ( zero_zero @ TA ) )
= ( coeff @ TA @ A @ ( degree @ TA @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_poly__reflect__poly__0) ).
thf(1136,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: poly @ TA] :
( ( poly2 @ TA @ ( reflect_poly @ TA @ A ) @ ( zero_zero @ TA ) )
= ( coeff @ TA @ A @ ( degree @ TA @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).
thf(271,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_not__gr__zero) ).
thf(1151,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).
thf(212,axiom,
! [TA: $tType,A: TA > $o] :
( ~ ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
=> ? [B: TA] : ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_not__finite__existsD) ).
thf(966,plain,
! [TA: $tType,A: TA > $o] :
( ~ ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
=> ? [B: TA] : ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(50,axiom,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ( finite_finite @ ( set @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Finite__Set_Ofinite_2) ).
thf(480,plain,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ( finite_finite @ ( set @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(314,axiom,
( p
!= ( zero_zero @ ( poly @ real ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1_p_I2_J) ).
thf(1302,plain,
( p
!= ( zero_zero @ ( poly @ real ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).
thf(152,axiom,
! [TA: $tType] :
( ( ( comm_semiring_1 @ TA )
& ( semiri1193490041visors @ TA ) )
=> ( ( pderiv @ TA @ ( one_one @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_pderiv__1) ).
thf(756,plain,
! [TA: $tType] :
( ( ( comm_semiring_1 @ TA )
& ( semiri1193490041visors @ TA ) )
=> ( ( pderiv @ TA @ ( one_one @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(227,axiom,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ord_less @ TA @ ( inverse_inverse @ TA @ B ) @ ( inverse_inverse @ TA @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_233_less__imp__inverse__less) ).
thf(1020,plain,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ord_less @ TA @ ( inverse_inverse @ TA @ B ) @ ( inverse_inverse @ TA @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).
thf(85,axiom,
! [TA: $tType] :
( ( idom_divide @ TA )
=> ( idom_divide @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Rings_Oidom__divide_30) ).
thf(574,plain,
! [TA: $tType] :
( ( idom_divide @ TA )
=> ( idom_divide @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(235,axiom,
ring_char_0 @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Nat_Oring__char__0) ).
thf(1051,plain,
ring_char_0 @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).
thf(167,axiom,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_111_nat_Odistinct_I1_J) ).
thf(821,plain,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(216,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( A
!= ( zero_zero @ TA ) )
=> ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_142_gr__zeroI) ).
thf(981,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( A
!= ( zero_zero @ TA ) )
=> ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(129,axiom,
semiri134348788visors @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_6) ).
thf(687,plain,
semiri134348788visors @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(291,axiom,
! [TA: $tType] :
( ( ( ring_char_0 @ TA )
& ( comm_ring_1 @ TA )
& ( ring_n68954251visors @ TA ) )
=> ! [A: poly @ TA] :
( ( ! [B: TA] :
( ( poly2 @ TA @ A @ B )
= ( zero_zero @ TA ) ) )
= ( A
= ( zero_zero @ ( poly @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8_poly__all__0__iff__0) ).
thf(1215,plain,
! [TA: $tType] :
( ( ( ring_char_0 @ TA )
& ( comm_ring_1 @ TA )
& ( ring_n68954251visors @ TA ) )
=> ! [A: poly @ TA] :
( ( ! [B: TA] :
( ( poly2 @ TA @ A @ B )
= ( zero_zero @ TA ) ) )
= ( A
= ( zero_zero @ ( poly @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).
thf(208,axiom,
linord1659791738miring @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_7) ).
thf(956,plain,
linord1659791738miring @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(181,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( coeff @ TA @ ( zero_zero @ ( poly @ TA ) ) )
= ( ^ [A: nat] : ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_zero__poly_Orep__eq) ).
thf(866,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( coeff @ TA @ ( zero_zero @ ( poly @ TA ) ) )
= ( ^ [A: nat] : ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(158,axiom,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_103_Zero__neq__Suc) ).
thf(774,plain,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(29,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: nat > TA,B: nat,C: nat] :
( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) )
= ( ord_less @ nat @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_181_lift__Suc__mono__less__iff) ).
thf(396,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: nat > TA,B: nat,C: nat] :
( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) )
= ( ord_less @ nat @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(286,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat] :
( ( poly_shift @ TA @ A @ ( zero_zero @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_poly__shift__0) ).
thf(1201,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat] :
( ( poly_shift @ TA @ A @ ( zero_zero @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).
thf(62,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( zero_less_one @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Rings_Ozero__less__one_27) ).
thf(507,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( zero_less_one @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(69,axiom,
! [TA: $tType] :
( ( field @ TA )
=> ! [A: TA] :
( ( ( inverse_inverse @ TA @ A )
= ( one_one @ TA ) )
= ( A
= ( one_one @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_220_inverse__eq__1__iff) ).
thf(527,plain,
! [TA: $tType] :
( ( field @ TA )
=> ! [A: TA] :
( ( ( inverse_inverse @ TA @ A )
= ( one_one @ TA ) )
= ( A
= ( one_one @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(255,axiom,
! [A: nat > nat > $o,B: nat,C: nat] :
( ! [D: nat] : ( A @ D @ ( zero_zero @ nat ) )
=> ( ! [D: nat] : ( A @ ( zero_zero @ nat ) @ ( suc @ D ) )
=> ( ! [D: nat,E: nat] :
( ( A @ D @ E )
=> ( A @ ( suc @ D ) @ ( suc @ E ) ) )
=> ( A @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_106_diff__induct) ).
thf(1104,plain,
! [A: nat > nat > $o,B: nat,C: nat] :
( ! [D: nat] : ( A @ D @ ( zero_zero @ nat ) )
=> ( ! [D: nat] : ( A @ ( zero_zero @ nat ) @ ( suc @ D ) )
=> ( ! [D: nat,E: nat] :
( ( A @ D @ E )
=> ( A @ ( suc @ D ) @ ( suc @ E ) ) )
=> ( A @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).
thf(177,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
=> ( ( ord_less @ nat @ A @ B )
=> ~ ( dvd_dvd @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_nat__dvd__not__less) ).
thf(849,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
=> ( ( ord_less @ nat @ A @ B )
=> ~ ( dvd_dvd @ nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(245,axiom,
! [A: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A )
& ( ( zero_zero @ nat )
!= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_212_gcd__nat_Oextremum__strict) ).
thf(1074,plain,
! [A: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A )
& ( ( zero_zero @ nat )
!= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(288,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ ( suc @ B ) )
= ( ( A
= ( zero_zero @ nat ) )
| ? [C: nat] :
( ( A
= ( suc @ C ) )
& ( ord_less @ nat @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_less__Suc__eq__0__disj) ).
thf(1206,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ ( suc @ B ) )
= ( ( A
= ( zero_zero @ nat ) )
| ? [C: nat] :
( ( A
= ( suc @ C ) )
& ( ord_less @ nat @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).
thf(201,axiom,
division_ring @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Fields_Odivision__ring) ).
thf(920,plain,
division_ring @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).
thf(64,axiom,
order @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oorder_1) ).
thf(512,plain,
order @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(120,axiom,
! [TA: $tType] :
( ( field @ TA )
=> ! [A: TA] :
( ( member @ TA @ A @ ( ring_1_Ints @ TA ) )
=> ( algebraic_int @ TA @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_int__imp__algebraic__int) ).
thf(661,plain,
! [TA: $tType] :
( ( field @ TA )
=> ! [A: TA] :
( ( member @ TA @ A @ ( ring_1_Ints @ TA ) )
=> ( algebraic_int @ TA @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(154,axiom,
! [TA: $tType] :
( ( ( semiring_char_0 @ TA )
& ( comm_semiring_1 @ TA )
& ( semiri1193490041visors @ TA ) )
=> ! [A: poly @ TA] :
( ( dvd_dvd @ ( poly @ TA ) @ A @ ( pderiv @ TA @ A ) )
= ( ( degree @ TA @ A )
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_67_dvd__pderiv__iff) ).
thf(761,plain,
! [TA: $tType] :
( ( ( semiring_char_0 @ TA )
& ( comm_semiring_1 @ TA )
& ( semiri1193490041visors @ TA ) )
=> ! [A: poly @ TA] :
( ( dvd_dvd @ ( poly @ TA ) @ A @ ( pderiv @ TA @ A ) )
= ( ( degree @ TA @ A )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(166,axiom,
! [A: nat > $o,B: nat] :
( ( A @ ( zero_zero @ nat ) )
=> ( ! [C: nat] :
( ( A @ C )
=> ( A @ ( suc @ C ) ) )
=> ( A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_nat__induct) ).
thf(810,plain,
! [A: nat > $o,B: nat] :
( ( A @ ( zero_zero @ nat ) )
=> ( ! [C: nat] :
( ( A @ C )
=> ( A @ ( suc @ C ) ) )
=> ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(18,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( comm_monoid_mult @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Groups_Ocomm__monoid__mult_23) ).
thf(366,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( comm_monoid_mult @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(242,axiom,
comm_ring_1 @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Rings_Ocomm__ring__1) ).
thf(1067,plain,
comm_ring_1 @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).
thf(138,axiom,
comm_semiring_1 @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Rings_Ocomm__semiring__1_10) ).
thf(714,plain,
comm_semiring_1 @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(8,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( dvd_dvd @ nat @ A @ B )
=> ( ( dvd_dvd @ nat @ A @ B )
& ( A != B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_gcd__nat_Onot__eq__order__implies__strict) ).
thf(334,plain,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( dvd_dvd @ nat @ A @ B )
=> ( ( dvd_dvd @ nat @ A @ B )
& ( A != B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(209,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( ( coeff @ TA @ A @ ( zero_zero @ nat ) )
!= ( zero_zero @ TA ) )
=> ( ( degree @ TA @ ( reflect_poly @ TA @ A ) )
= ( degree @ TA @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_degree__reflect__poly__eq) ).
thf(957,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( ( coeff @ TA @ A @ ( zero_zero @ nat ) )
!= ( zero_zero @ TA ) )
=> ( ( degree @ TA @ ( reflect_poly @ TA @ A ) )
= ( degree @ TA @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(250,axiom,
( n
= ( degree @ real @ p ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_n__def) ).
thf(1091,plain,
( n
= ( degree @ real @ p ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).
thf(168,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less @ TA @ ( one_one @ TA ) @ ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_not__one__less__zero) ).
thf(825,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less @ TA @ ( one_one @ TA ) @ ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(159,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( A
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( coeff @ TA @ A @ ( degree @ TA @ A ) )
!= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_leading__coeff__neq__0) ).
thf(778,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( A
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( coeff @ TA @ A @ ( degree @ TA @ A ) )
!= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(179,axiom,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ( ( algebraic @ TA )
= ( ^ [A: TA] :
? [B: poly @ TA] :
( ! [C: nat] : ( member @ TA @ ( coeff @ TA @ B @ C ) @ ( field_char_0_Rats @ TA ) )
& ( B
!= ( zero_zero @ ( poly @ TA ) ) )
& ( ( poly2 @ TA @ B @ A )
= ( zero_zero @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_algebraic__altdef) ).
thf(862,plain,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ( ( algebraic @ TA )
= ( ^ [A: TA] :
? [B: poly @ TA] :
( ! [C: nat] : ( member @ TA @ ( coeff @ TA @ B @ C ) @ ( field_char_0_Rats @ TA ) )
& ( B
!= ( zero_zero @ ( poly @ TA ) ) )
& ( ( poly2 @ TA @ B @ A )
= ( zero_zero @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(96,axiom,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
=> ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_Suc__inject) ).
thf(598,plain,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(299,axiom,
! [TA: $tType] :
( ( ( semiring_char_0 @ TA )
& ( comm_semiring_1 @ TA )
& ( semiri1193490041visors @ TA ) )
=> ! [A: poly @ TA] :
( ( ( degree @ TA @ A )
!= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ ( poly @ TA ) @ A @ ( pderiv @ TA @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_not__dvd__pderiv) ).
thf(1238,plain,
! [TA: $tType] :
( ( ( semiring_char_0 @ TA )
& ( comm_semiring_1 @ TA )
& ( semiri1193490041visors @ TA ) )
=> ! [A: poly @ TA] :
( ( ( degree @ TA @ A )
!= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ ( poly @ TA ) @ A @ ( pderiv @ TA @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).
thf(176,axiom,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
=> ( ord_less @ TA @ ( inverse_inverse @ TA @ A ) @ ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_238_negative__imp__inverse__negative) ).
thf(847,plain,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
=> ( ord_less @ TA @ ( inverse_inverse @ TA @ A ) @ ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(106,axiom,
! [TA: $tType,TB: $tType] :
( ( wellorder @ TB )
=> ! [A: TA > TB,B: TA > $o,C: TA] :
( ! [D: TA] :
( ! [E: TA] :
( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
=> ( B @ E ) )
=> ( B @ D ) )
=> ( B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_measure__induct__rule) ).
thf(625,plain,
! [TA: $tType,TB: $tType] :
( ( wellorder @ TB )
=> ! [A: TA > TB,B: TA > $o,C: TA] :
( ! [D: TA] :
( ! [E: TA] :
( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
=> ( B @ E ) )
=> ( B @ D ) )
=> ( B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(186,axiom,
! [A: nat] :
( ( ord_less @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_less__Suc0) ).
thf(879,plain,
! [A: nat] :
( ( ord_less @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(178,axiom,
! [A: nat > $o] :
( ~ ( A @ ( zero_zero @ nat ) )
=> ( ? [B: nat] : ( A @ B )
=> ? [B: nat] :
( ~ ( A @ B )
& ( A @ ( suc @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_exists__least__lemma) ).
thf(852,plain,
! [A: nat > $o] :
( ~ ( A @ ( zero_zero @ nat ) )
=> ( ? [B: nat] : ( A @ B )
=> ? [B: nat] :
( ~ ( A @ B )
& ( A @ ( suc @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(6,axiom,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ! [A: TA] :
( ( algebraic_int @ TA @ A )
=> ( algebraic @ TA @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_algebraic__int__imp__algebraic) ).
thf(329,plain,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ! [A: TA] :
( ( algebraic_int @ TA @ A )
=> ( algebraic @ TA @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(45,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( comm_semiring_1 @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_25) ).
thf(461,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( comm_semiring_1 @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(54,axiom,
! [TA: $tType] :
( ( field @ TA )
=> ( algebraic_int @ TA @ ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_algebraic__int__1) ).
thf(493,plain,
! [TA: $tType] :
( ( field @ TA )
=> ( algebraic_int @ TA @ ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(284,axiom,
! [A: nat] :
( ( dvd_dvd @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
= ( A
= ( suc @ ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_dvd__1__iff__1) ).
thf(1195,plain,
! [A: nat] :
( ( dvd_dvd @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
= ( A
= ( suc @ ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).
thf(263,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB > $o] :
( ( finite_finite2 @ TA @ A )
=> ( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ? [D: TB] : ( B @ C @ D ) )
=> ? [C: TA > TB] :
! [D: TA] :
( ( member @ TA @ D @ A )
=> ( B @ D @ ( C @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_finite__set__choice) ).
thf(1128,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB > $o] :
( ( finite_finite2 @ TA @ A )
=> ( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ? [D: TB] : ( B @ C @ D ) )
=> ? [C: TA > TB] :
! [D: TA] :
( ( member @ TA @ D @ A )
=> ( B @ D @ ( C @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).
thf(137,axiom,
! [TA: $tType] :
( ( field @ TA )
=> ! [A: TA,B: poly @ TA,C: TA] :
( ( algebraic_int @ TA @ A )
=> ( ( ( poly2 @ TA @ B @ C )
= A )
=> ( ! [D: nat] : ( member @ TA @ ( coeff @ TA @ B @ D ) @ ( ring_1_Ints @ TA ) )
=> ( ( ( coeff @ TA @ B @ ( degree @ TA @ B ) )
= ( one_one @ TA ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( degree @ TA @ B ) )
=> ( algebraic_int @ TA @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_algebraic__int__root) ).
thf(710,plain,
! [TA: $tType] :
( ( field @ TA )
=> ! [A: TA,B: poly @ TA,C: TA] :
( ( algebraic_int @ TA @ A )
=> ( ( ( poly2 @ TA @ B @ C )
= A )
=> ( ! [D: nat] : ( member @ TA @ ( coeff @ TA @ B @ D ) @ ( ring_1_Ints @ TA ) )
=> ( ( ( coeff @ TA @ B @ ( degree @ TA @ B ) )
= ( one_one @ TA ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( degree @ TA @ B ) )
=> ( algebraic_int @ TA @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(88,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less @ nat @ A @ B ) )
= ( ord_less @ nat @ B @ ( suc @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_159_not__less__eq) ).
thf(578,plain,
! [A: nat,B: nat] :
( ( ~ ( ord_less @ nat @ A @ B ) )
= ( ord_less @ nat @ B @ ( suc @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(133,axiom,
! [TA: $tType] :
( ( ( comm_ring_1 @ TA )
& ( semiri134348788visors @ TA ) )
=> ! [A: poly @ TA,B: poly @ TA] :
( ( A
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( ( pseudo_mod @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) )
| ( ord_less @ nat @ ( degree @ TA @ ( pseudo_mod @ TA @ B @ A ) ) @ ( degree @ TA @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_249_pseudo__mod_I2_J) ).
thf(697,plain,
! [TA: $tType] :
( ( ( comm_ring_1 @ TA )
& ( semiri134348788visors @ TA ) )
=> ! [A: poly @ TA,B: poly @ TA] :
( ( A
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( ( pseudo_mod @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) )
| ( ord_less @ nat @ ( degree @ TA @ ( pseudo_mod @ TA @ B @ A ) ) @ ( degree @ TA @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(268,axiom,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ~ ( finite_finite2 @ TA @ ( field_char_0_Rats @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_Rats__infinite) ).
thf(1143,plain,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ~ ( finite_finite2 @ TA @ ( field_char_0_Rats @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).
thf(304,axiom,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
=> ( ( ord_less @ TA @ ( inverse_inverse @ TA @ A ) @ ( inverse_inverse @ TA @ B ) )
= ( ord_less @ TA @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_217_inverse__less__iff__less) ).
thf(1259,plain,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
=> ( ( ord_less @ TA @ ( inverse_inverse @ TA @ A ) @ ( inverse_inverse @ TA @ B ) )
= ( ord_less @ TA @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).
thf(102,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: nat > TA,B: nat,C: nat] :
( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less @ nat @ B @ C )
=> ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_182_lift__Suc__mono__less) ).
thf(615,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: nat > TA,B: nat,C: nat] :
( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less @ nat @ B @ C )
=> ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(142,axiom,
! [A: nat > $o,B: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [C: nat] :
( ( A @ C )
& ( ord_less @ nat @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_243_finite__M__bounded__by__nat) ).
thf(725,plain,
! [A: nat > $o,B: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [C: nat] :
( ( A @ C )
& ( ord_less @ nat @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(252,axiom,
! [TA: $tType] :
( ( ( semiring_char_0 @ TA )
& ( comm_semiring_1 @ TA )
& ( semiri1193490041visors @ TA ) )
=> ! [A: poly @ TA] :
( ( ( pderiv @ TA @ A )
= ( zero_zero @ ( poly @ TA ) ) )
= ( ( degree @ TA @ A )
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_pderiv__eq__0__iff) ).
thf(1096,plain,
! [TA: $tType] :
( ( ( semiring_char_0 @ TA )
& ( comm_semiring_1 @ TA )
& ( semiri1193490041visors @ TA ) )
=> ! [A: poly @ TA] :
( ( ( pderiv @ TA @ A )
= ( zero_zero @ ( poly @ TA ) ) )
= ( ( degree @ TA @ A )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).
thf(225,axiom,
! [A: real,B: real,C: poly @ real] :
( ( ord_less @ real @ A @ B )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ C @ A ) )
=> ( ( ord_less @ real @ ( poly2 @ real @ C @ B ) @ ( zero_zero @ real ) )
=> ? [D: real] :
( ( ord_less @ real @ A @ D )
& ( ord_less @ real @ D @ B )
& ( ( poly2 @ real @ C @ D )
= ( zero_zero @ real ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_225_poly__IVT__neg) ).
thf(1006,plain,
! [A: real,B: real,C: poly @ real] :
( ( ord_less @ real @ A @ B )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ C @ A ) )
=> ( ( ord_less @ real @ ( poly2 @ real @ C @ B ) @ ( zero_zero @ real ) )
=> ? [D: real] :
( ( ord_less @ real @ A @ D )
& ( ord_less @ real @ D @ B )
& ( ( poly2 @ real @ C @ D )
= ( zero_zero @ real ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(25,axiom,
algebraic_semidom @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Oalgebraic__semidom) ).
thf(388,plain,
algebraic_semidom @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(182,axiom,
idom @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Rings_Oidom) ).
thf(869,plain,
idom @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(259,axiom,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_old_Onat_Odistinct_I1_J) ).
thf(1118,plain,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).
thf(52,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: poly @ TA] :
( ( ( coeff @ TA @ A )
= ( coeff @ TA @ B ) )
= ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50_coeff__inject) ).
thf(488,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: poly @ TA] :
( ( ( coeff @ TA @ A )
= ( coeff @ TA @ B ) )
= ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(310,axiom,
! [TA: $tType] :
( ( idom_divide @ TA )
=> ! [A: poly @ TA,B: poly @ TA,C: nat,D: nat] :
( ( divide_poly_main @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ ( poly @ TA ) ) @ A @ B @ C @ D )
= ( zero_zero @ ( poly @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_divide__poly__main__0) ).
thf(1292,plain,
! [TA: $tType] :
( ( idom_divide @ TA )
=> ! [A: poly @ TA,B: poly @ TA,C: nat,D: nat] :
( ( divide_poly_main @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ ( poly @ TA ) ) @ A @ B @ C @ D )
= ( zero_zero @ ( poly @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).
thf(78,axiom,
! [TA: $tType,TB: $tType] :
( ( ( finite_finite @ TB )
& ( finite_finite @ TA ) )
=> ( finite_finite @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Finite__Set_Ofinite) ).
thf(554,plain,
! [TA: $tType,TB: $tType] :
( ( ( finite_finite @ TB )
& ( finite_finite @ TA ) )
=> ( finite_finite @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(73,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( linord1659791738miring @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Rings_Olinordered__nonzero__semiring_19) ).
thf(537,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( linord1659791738miring @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(194,axiom,
! [A: nat,B: nat > $o] :
( ( ? [C: nat] :
( ( ord_less @ nat @ C @ ( suc @ A ) )
& ( B @ C ) ) )
= ( ( B @ ( zero_zero @ nat ) )
| ? [C: nat] :
( ( ord_less @ nat @ C @ A )
& ( B @ ( suc @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_Ex__less__Suc2) ).
thf(901,plain,
! [A: nat,B: nat > $o] :
( ( ? [C: nat] :
( ( ord_less @ nat @ C @ ( suc @ A ) )
& ( B @ C ) ) )
= ( ( B @ ( zero_zero @ nat ) )
| ? [C: nat] :
( ( ord_less @ nat @ C @ A )
& ( B @ ( suc @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(112,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( B != A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_171_less__not__refl2) ).
thf(642,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( B != A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(68,axiom,
zero_neq_one @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Ozero__neq__one) ).
thf(526,plain,
zero_neq_one @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(110,axiom,
! [TA: $tType] :
( ( comm_monoid_mult @ TA )
=> ! [A: TA] : ( dvd_dvd @ TA @ ( one_one @ TA ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_one__dvd) ).
thf(637,plain,
! [TA: $tType] :
( ( comm_monoid_mult @ TA )
=> ! [A: TA] : ( dvd_dvd @ TA @ ( one_one @ TA ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(92,axiom,
! [TA: $tType,TB: $tType] :
( ( order @ TA )
=> ( order @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oorder) ).
thf(590,plain,
! [TA: $tType,TB: $tType] :
( ( order @ TA )
=> ( order @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(231,axiom,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_Zero__not__Suc) ).
thf(1039,plain,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).
thf(273,axiom,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ( member @ TA @ ( zero_zero @ TA ) @ ( field_char_0_Rats @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_Rats__0) ).
thf(1155,plain,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ( member @ TA @ ( zero_zero @ TA ) @ ( field_char_0_Rats @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).
thf(75,axiom,
! [TA: $tType] :
( ( comm_ring_1 @ TA )
=> ( comm_ring_1 @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Rings_Ocomm__ring__1_31) ).
thf(547,plain,
! [TA: $tType] :
( ( comm_ring_1 @ TA )
=> ( comm_ring_1 @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(28,axiom,
! [A: nat,B: nat > $o] :
( ( ? [C: nat] :
( ( ord_less @ nat @ C @ ( suc @ A ) )
& ( B @ C ) ) )
= ( ( B @ A )
| ? [C: nat] :
( ( ord_less @ nat @ C @ A )
& ( B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_161_Ex__less__Suc) ).
thf(393,plain,
! [A: nat,B: nat > $o] :
( ( ? [C: nat] :
( ( ord_less @ nat @ C @ ( suc @ A ) )
& ( B @ C ) ) )
= ( ( B @ A )
| ? [C: nat] :
( ( ord_less @ nat @ C @ A )
& ( B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(222,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_bot__nat__0_Onot__eq__extremum) ).
thf(997,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).
thf(253,axiom,
! [TA: $tType] :
( ( field @ TA )
=> ( algebraic_int @ TA @ ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_algebraic__int__0) ).
thf(1099,plain,
! [TA: $tType] :
( ( field @ TA )
=> ( algebraic_int @ TA @ ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(303,axiom,
! [TA: $tType] :
( ( field @ TA )
=> ! [A: TA,B: TA > $o] :
( ( algebraic_int @ TA @ A )
=> ( ! [C: poly @ TA,D: TA] :
( ( ( coeff @ TA @ C @ ( degree @ TA @ C ) )
= ( one_one @ TA ) )
=> ( ! [E: nat] : ( member @ TA @ ( coeff @ TA @ C @ E ) @ ( ring_1_Ints @ TA ) )
=> ( ( ( poly2 @ TA @ C @ D )
= ( zero_zero @ TA ) )
=> ( B @ D ) ) ) )
=> ( B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_112_algebraic__int_Oinducts) ).
thf(1249,plain,
! [TA: $tType] :
( ( field @ TA )
=> ! [A: TA,B: TA > $o] :
( ( algebraic_int @ TA @ A )
=> ( ! [C: poly @ TA,D: TA] :
( ( ( coeff @ TA @ C @ ( degree @ TA @ C ) )
= ( one_one @ TA ) )
=> ( ! [E: nat] : ( member @ TA @ ( coeff @ TA @ C @ E ) @ ( ring_1_Ints @ TA ) )
=> ( ( ( poly2 @ TA @ C @ D )
= ( zero_zero @ TA ) )
=> ( B @ D ) ) ) )
=> ( B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).
thf(283,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( poly_shift @ TA @ ( zero_zero @ nat ) )
= ( ^ [A: poly @ TA] : A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_poly__shift__id) ).
thf(1192,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( poly_shift @ TA @ ( zero_zero @ nat ) )
= ( ^ [A: poly @ TA] : A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).
thf(43,axiom,
! [TA: $tType] :
( ( ( ring_char_0 @ TA )
& ( comm_ring_1 @ TA ) )
=> ( ring_char_0 @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Nat_Oring__char__0_33) ).
thf(457,plain,
! [TA: $tType] :
( ( ( ring_char_0 @ TA )
& ( comm_ring_1 @ TA ) )
=> ( ring_char_0 @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(83,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) )
=> ( ord_less @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_155_Suc__less__SucD) ).
thf(569,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) )
=> ( ord_less @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(302,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ! [A: TA] :
( ( poly2 @ TA @ ( one_one @ ( poly @ TA ) ) @ A )
= ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_poly__1) ).
thf(1246,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ! [A: TA] :
( ( poly2 @ TA @ ( one_one @ ( poly @ TA ) ) @ A )
= ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).
thf(217,axiom,
! [TA: $tType] :
( ( field @ TA )
=> ( ( algebraic_int @ TA )
= ( ^ [A: TA] :
? [B: poly @ TA,C: TA] :
( ( A = C )
& ( ( coeff @ TA @ B @ ( degree @ TA @ B ) )
= ( one_one @ TA ) )
& ! [D: nat] : ( member @ TA @ ( coeff @ TA @ B @ D ) @ ( ring_1_Ints @ TA ) )
& ( ( poly2 @ TA @ B @ C )
= ( zero_zero @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_114_algebraic__int_Osimps) ).
thf(984,plain,
! [TA: $tType] :
( ( field @ TA )
=> ( ( algebraic_int @ TA )
= ( ^ [A: TA] :
? [B: poly @ TA,C: TA] :
( ( A = C )
& ( ( coeff @ TA @ B @ ( degree @ TA @ B ) )
= ( one_one @ TA ) )
& ! [D: nat] : ( member @ TA @ ( coeff @ TA @ B @ D ) @ ( ring_1_Ints @ TA ) )
& ( ( poly2 @ TA @ B @ C )
= ( zero_zero @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).
thf(198,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: nat] :
( ( ord_less @ nat @ ( degree @ TA @ A ) @ B )
=> ( ( coeff @ TA @ A @ B )
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_195_coeff__eq__0) ).
thf(911,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: nat] :
( ( ord_less @ nat @ ( degree @ TA @ A ) @ B )
=> ( ( coeff @ TA @ A @ B )
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(10,axiom,
comm_semiring_0 @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Ocomm__semiring__0) ).
thf(343,plain,
comm_semiring_0 @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(206,axiom,
! [A: nat > $o,B: nat] :
( ( A @ ( zero_zero @ nat ) )
=> ( ! [C: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
=> ( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less @ nat @ D @ C )
& ~ ( A @ D ) ) ) )
=> ( A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_148_infinite__descent0) ).
thf(938,plain,
! [A: nat > $o,B: nat] :
( ( A @ ( zero_zero @ nat ) )
=> ( ! [C: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
=> ( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less @ nat @ D @ C )
& ~ ( A @ D ) ) ) )
=> ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(300,axiom,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( inverse_inverse @ TA @ A ) @ ( inverse_inverse @ TA @ B ) )
=> ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ord_less @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_232_inverse__less__imp__less) ).
thf(1241,plain,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( inverse_inverse @ TA @ A ) @ ( inverse_inverse @ TA @ B ) )
=> ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ord_less @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).
thf(202,axiom,
order @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Orderings_Oorder_15) ).
thf(921,plain,
order @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).
thf(118,axiom,
! [TA: $tType] :
( ( ring_1 @ TA )
=> ( member @ TA @ ( one_one @ TA ) @ ( ring_1_Ints @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_Ints__1) ).
thf(657,plain,
! [TA: $tType] :
( ( ring_1 @ TA )
=> ( member @ TA @ ( one_one @ TA ) @ ( ring_1_Ints @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(244,axiom,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ! [A: poly @ TA,B: TA] :
( ! [C: nat] : ( member @ TA @ ( coeff @ TA @ A @ C ) @ ( ring_1_Ints @ TA ) )
=> ( ( A
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( ( poly2 @ TA @ A @ B )
= ( zero_zero @ TA ) )
=> ( algebraic @ TA @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_algebraicI) ).
thf(1071,plain,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ! [A: poly @ TA,B: TA] :
( ! [C: nat] : ( member @ TA @ ( coeff @ TA @ A @ C ) @ ( ring_1_Ints @ TA ) )
=> ( ( A
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( ( poly2 @ TA @ A @ B )
= ( zero_zero @ TA ) )
=> ( algebraic @ TA @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).
thf(124,axiom,
! [A: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
=> ? [B: nat] :
( A
= ( suc @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_gr0__implies__Suc) ).
thf(674,plain,
! [A: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
=> ? [B: nat] :
( A
= ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(70,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ A @ ( suc @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_162_less__SucI) ).
thf(530,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ A @ ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(308,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
| ( finite_finite2 @ TA @ ( collect @ TA @ B ) ) )
=> ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [C: TA] :
( ( A @ C )
& ( B @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5_finite__Collect__conjI) ).
thf(1278,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
| ( finite_finite2 @ TA @ ( collect @ TA @ B ) ) )
=> ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [C: TA] :
( ( A @ C )
& ( B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).
thf(149,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_83_One__nat__def) ).
thf(750,plain,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(313,axiom,
zero @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Groups_Ozero_16) ).
thf(1301,plain,
zero @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).
thf(145,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_149_bot__nat__0_Oextremum__strict) ).
thf(733,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(65,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd @ nat @ A @ B )
& ( A != B ) )
=> ( ( ( dvd_dvd @ nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd @ nat @ A @ C )
& ( A != C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_gcd__nat_Ostrict__trans) ).
thf(513,plain,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd @ nat @ A @ B )
& ( A != B ) )
=> ( ( ( dvd_dvd @ nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd @ nat @ A @ C )
& ( A != C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(33,axiom,
! [TA: $tType] :
( ( ( ring_char_0 @ TA )
& ( comm_ring_1 @ TA ) )
=> ( semiring_char_0 @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Nat_Osemiring__char__0_28) ).
thf(430,plain,
! [TA: $tType] :
( ( ( ring_char_0 @ TA )
& ( comm_ring_1 @ TA ) )
=> ( semiring_char_0 @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(132,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( degree @ TA @ ( zero_zero @ ( poly @ TA ) ) )
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_degree__0) ).
thf(694,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( degree @ TA @ ( zero_zero @ ( poly @ TA ) ) )
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(175,axiom,
! [A: real] :
? [B: real] :
( ( member @ real @ B @ ( field_char_0_Rats @ real ) )
& ( ord_less @ real @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_Rats__no__bot__less) ).
thf(843,plain,
! [A: real] :
? [B: real] :
( ( member @ real @ B @ ( field_char_0_Rats @ real ) )
& ( ord_less @ real @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(87,axiom,
zero_less_one @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Ozero__less__one) ).
thf(577,plain,
zero_less_one @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(261,axiom,
! [TA: $tType] :
( ( field @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( poly2 @ TA @ A @ B )
= ( zero_zero @ TA ) )
=> ( ! [C: nat] : ( member @ TA @ ( coeff @ TA @ A @ C ) @ ( ring_1_Ints @ TA ) )
=> ( ( ( coeff @ TA @ A @ ( zero_zero @ nat ) )
= ( one_one @ TA ) )
=> ( algebraic_int @ TA @ ( inverse_inverse @ TA @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_algebraic__int__inverse) ).
thf(1124,plain,
! [TA: $tType] :
( ( field @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( poly2 @ TA @ A @ B )
= ( zero_zero @ TA ) )
=> ( ! [C: nat] : ( member @ TA @ ( coeff @ TA @ A @ C ) @ ( ring_1_Ints @ TA ) )
=> ( ( ( coeff @ TA @ A @ ( zero_zero @ nat ) )
= ( one_one @ TA ) )
=> ( algebraic_int @ TA @ ( inverse_inverse @ TA @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).
thf(91,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ( B
!= ( suc @ A ) )
=> ~ ! [C: nat] :
( ( ord_less @ nat @ A @ C )
=> ( B
!= ( suc @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_167_Nat_OlessE) ).
thf(584,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ( B
!= ( suc @ A ) )
=> ~ ! [C: nat] :
( ( ord_less @ nat @ A @ C )
=> ( B
!= ( suc @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(272,axiom,
one @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Groups_Oone_17) ).
thf(1154,plain,
one @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).
thf(81,axiom,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
= ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_86_old_Onat_Oinject) ).
thf(562,plain,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
= ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(195,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ( dvd_dvd @ nat @ A @ ( zero_zero @ nat ) )
& ( A
!= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_127_gcd__nat_Onot__eq__extremum) ).
thf(904,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ( dvd_dvd @ nat @ A @ ( zero_zero @ nat ) )
& ( A
!= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(39,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
=> ( ~ ( ord_less @ TA @ A @ B )
=> ( ord_less @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_180_linorder__neqE__linordered__idom) ).
thf(445,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
=> ( ~ ( ord_less @ TA @ A @ B )
=> ( ord_less @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(22,axiom,
! [TA: $tType] :
( ( algebraic_semidom @ TA )
=> ! [A: TA,B: TA] :
( ( dvd_dvd @ TA @ A @ B )
=> ( ( dvd_dvd @ TA @ B @ ( one_one @ TA ) )
=> ( dvd_dvd @ TA @ A @ ( one_one @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_dvd__unit__imp__unit) ).
thf(382,plain,
! [TA: $tType] :
( ( algebraic_semidom @ TA )
=> ! [A: TA,B: TA] :
( ( dvd_dvd @ TA @ A @ B )
=> ( ( dvd_dvd @ TA @ B @ ( one_one @ TA ) )
=> ( dvd_dvd @ TA @ A @ ( one_one @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(307,axiom,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ! [A: TA] :
( ( algebraic @ TA @ A )
=> ~ ! [B: poly @ TA] :
( ! [C: nat] : ( member @ TA @ ( coeff @ TA @ B @ C ) @ ( ring_1_Ints @ TA ) )
=> ( ( B
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( poly2 @ TA @ B @ A )
!= ( zero_zero @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_algebraicE) ).
thf(1271,plain,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ! [A: TA] :
( ( algebraic @ TA @ A )
=> ~ ! [B: poly @ TA] :
( ! [C: nat] : ( member @ TA @ ( coeff @ TA @ B @ C ) @ ( ring_1_Ints @ TA ) )
=> ( ( B
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( poly2 @ TA @ B @ A )
!= ( zero_zero @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).
thf(317,axiom,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ord_less @ TA @ ( zero_zero @ TA ) @ ( inverse_inverse @ TA @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_239_positive__imp__inverse__positive) ).
thf(1312,plain,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ord_less @ TA @ ( zero_zero @ TA ) @ ( inverse_inverse @ TA @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).
thf(162,axiom,
! [A: nat,B: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
=> ( ( B @ ( one_one @ nat ) )
=> ( ! [C: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
=> ( ( B @ C )
=> ( B @ ( suc @ C ) ) ) )
=> ( B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_nat__induct__non__zero) ).
thf(787,plain,
! [A: nat,B: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
=> ( ( B @ ( one_one @ nat ) )
=> ( ! [C: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
=> ( ( B @ C )
=> ( B @ ( suc @ C ) ) ) )
=> ( B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(47,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less @ nat @ A @ B )
=> ( ( ord_less @ nat @ A @ ( suc @ B ) )
=> ( B = A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_156_less__antisym) ).
thf(466,plain,
! [A: nat,B: nat] :
( ~ ( ord_less @ nat @ A @ B )
=> ( ( ord_less @ nat @ A @ ( suc @ B ) )
=> ( B = A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(164,axiom,
! [TA: $tType] :
( ( idom @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( poly2 @ TA @ A @ B )
= ( zero_zero @ TA ) )
= ( ( A
= ( zero_zero @ ( poly @ TA ) ) )
| ( ( order2 @ TA @ B @ A )
!= ( zero_zero @ nat ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29_order__root) ).
thf(804,plain,
! [TA: $tType] :
( ( idom @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( poly2 @ TA @ A @ B )
= ( zero_zero @ TA ) )
= ( ( A
= ( zero_zero @ ( poly @ TA ) ) )
| ( ( order2 @ TA @ B @ A )
!= ( zero_zero @ nat ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(27,axiom,
! [TA: $tType] :
( ( algebraic_semidom @ TA )
=> ! [A: TA,B: TA] :
( ( dvd_dvd @ TA @ A @ ( one_one @ TA ) )
=> ( dvd_dvd @ TA @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_97_unit__imp__dvd) ).
thf(391,plain,
! [TA: $tType] :
( ( algebraic_semidom @ TA )
=> ! [A: TA,B: TA] :
( ( dvd_dvd @ TA @ A @ ( one_one @ TA ) )
=> ( dvd_dvd @ TA @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(266,axiom,
! [A: nat,B: nat > $o] :
( ( ! [C: nat] :
( ( ord_less @ nat @ C @ ( suc @ A ) )
=> ( B @ C ) ) )
= ( ( B @ ( zero_zero @ nat ) )
& ! [C: nat] :
( ( ord_less @ nat @ C @ A )
=> ( B @ ( suc @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_All__less__Suc2) ).
thf(1139,plain,
! [A: nat,B: nat > $o] :
( ( ! [C: nat] :
( ( ord_less @ nat @ C @ ( suc @ A ) )
=> ( B @ C ) ) )
= ( ( B @ ( zero_zero @ nat ) )
& ! [C: nat] :
( ( ord_less @ nat @ C @ A )
=> ( B @ ( suc @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).
thf(155,axiom,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( inverse_inverse @ TA @ A ) @ ( zero_zero @ TA ) )
=> ( ( A
!= ( zero_zero @ TA ) )
=> ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_236_inverse__negative__imp__negative) ).
thf(764,plain,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( inverse_inverse @ TA @ A ) @ ( zero_zero @ TA ) )
=> ( ( A
!= ( zero_zero @ TA ) )
=> ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(289,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ! [A: TA] :
( ( dvd_dvd @ TA @ ( zero_zero @ TA ) @ A )
=> ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_dvd__0__left) ).
thf(1209,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ! [A: TA] :
( ( dvd_dvd @ TA @ ( zero_zero @ TA ) @ A )
=> ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).
thf(309,axiom,
! [TA: $tType,A: TA > nat,B: TA > $o,C: TA] :
( ! [D: TA] :
( ( ( A @ D )
= ( zero_zero @ nat ) )
=> ( B @ D ) )
=> ( ! [D: TA] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( A @ D ) )
=> ( ~ ( B @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( A @ E ) @ ( A @ D ) )
& ~ ( B @ E ) ) ) )
=> ( B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_150_infinite__descent0__measure) ).
thf(1282,plain,
! [TA: $tType,A: TA > nat,B: TA > $o,C: TA] :
( ! [D: TA] :
( ( ( A @ D )
= ( zero_zero @ nat ) )
=> ( B @ D ) )
=> ( ! [D: TA] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( A @ D ) )
=> ( ~ ( B @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( A @ E ) @ ( A @ D ) )
& ~ ( B @ E ) ) ) )
=> ( B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).
thf(161,axiom,
! [A: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
= ( ? [B: nat] :
( A
= ( suc @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_gr0__conv__Suc) ).
thf(784,plain,
! [A: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
= ( ? [B: nat] :
( A
= ( suc @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(41,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_less__irrefl__nat) ).
thf(451,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(260,axiom,
! [TA: $tType] :
( ( algebraic_semidom @ TA )
=> ~ ( dvd_dvd @ TA @ ( zero_zero @ TA ) @ ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_84_not__is__unit__0) ).
thf(1122,plain,
! [TA: $tType] :
( ( algebraic_semidom @ TA )
=> ~ ( dvd_dvd @ TA @ ( zero_zero @ TA ) @ ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).
thf(234,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [C: TA] :
( ( A @ C )
| ( B @ C ) ) ) )
= ( ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
& ( finite_finite2 @ TA @ ( collect @ TA @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_finite__Collect__disjI) ).
thf(1048,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [C: TA] :
( ( A @ C )
| ( B @ C ) ) ) )
= ( ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
& ( finite_finite2 @ TA @ ( collect @ TA @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).
thf(146,axiom,
field_char_0 @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Fields_Ofield__char__0) ).
thf(736,plain,
field_char_0 @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(21,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( suc @ A ) @ B )
=> ~ ! [C: nat] :
( ( ord_less @ nat @ A @ C )
=> ( B
!= ( suc @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_165_Suc__lessE) ).
thf(377,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( suc @ A ) @ B )
=> ~ ! [C: nat] :
( ( ord_less @ nat @ A @ C )
=> ( B
!= ( suc @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(219,axiom,
! [A: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [B: nat] : ( ord_less @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_finite__Collect__less__nat) ).
thf(990,plain,
! [A: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [B: nat] : ( ord_less @ nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).
thf(147,axiom,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less @ TA @ A @ ( one_one @ TA ) )
=> ( ord_less @ TA @ ( one_one @ TA ) @ ( inverse_inverse @ TA @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_240_one__less__inverse) ).
thf(737,plain,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less @ TA @ A @ ( one_one @ TA ) )
=> ( ord_less @ TA @ ( one_one @ TA ) @ ( inverse_inverse @ TA @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(1,conjecture,
( finite_finite2 @ real
@ ( collect @ real
@ ^ [A: real] :
( ( poly2 @ real @ ( pderiv @ real @ p ) @ A )
= ( zero_zero @ real ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).
thf(2,negated_conjecture,
~ ( finite_finite2 @ real
@ ( collect @ real
@ ^ [A: real] :
( ( poly2 @ real @ ( pderiv @ real @ p ) @ A )
= ( zero_zero @ real ) ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(318,plain,
~ ( finite_finite2 @ real
@ ( collect @ real
@ ^ [A: real] :
( ( poly2 @ real @ ( pderiv @ real @ p ) @ A )
= ( zero_zero @ real ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(63,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd @ nat @ A @ B )
=> ( ( dvd_dvd @ nat @ B @ C )
=> ( dvd_dvd @ nat @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_gcd__nat_Otrans) ).
thf(509,plain,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd @ nat @ A @ B )
=> ( ( dvd_dvd @ nat @ B @ C )
=> ( dvd_dvd @ nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(232,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ! [A: TA] : ( dvd_dvd @ TA @ A @ ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_dvd__0__right) ).
thf(1043,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ! [A: TA] : ( dvd_dvd @ TA @ A @ ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(35,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_less__not__refl) ).
thf(434,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(190,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( is_zero @ TA )
= ( ^ [A: poly @ TA] :
( A
= ( zero_zero @ ( poly @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_is__zero__null) ).
thf(890,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( is_zero @ TA )
= ( ^ [A: poly @ TA] :
( A
= ( zero_zero @ ( poly @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(107,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd @ nat @ A @ B )
& ( A != B ) )
=> ~ ( ( dvd_dvd @ nat @ B @ A )
& ( B != A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_gcd__nat_Oasym) ).
thf(631,plain,
! [A: nat,B: nat] :
( ( ( dvd_dvd @ nat @ A @ B )
& ( A != B ) )
=> ~ ( ( dvd_dvd @ nat @ B @ A )
& ( B != A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(72,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ( zero @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Groups_Ozero_35) ).
thf(535,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ( zero @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(99,axiom,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
= ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_85_nat_Oinject) ).
thf(609,plain,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
= ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(4,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( cr_poly @ TA )
= ( ^ [A: nat > TA,B: poly @ TA] :
( A
= ( coeff @ TA @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_66_cr__poly__def) ).
thf(323,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( cr_poly @ TA )
= ( ^ [A: nat > TA,B: poly @ TA] :
( A
= ( coeff @ TA @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(94,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( order @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Orderings_Oorder_32) ).
thf(594,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( order @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(275,axiom,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
=> ( ord_less @ TA @ ( inverse_inverse @ TA @ B ) @ ( inverse_inverse @ TA @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_less__imp__inverse__less__neg) ).
thf(1162,plain,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
=> ( ord_less @ TA @ ( inverse_inverse @ TA @ B ) @ ( inverse_inverse @ TA @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).
thf(134,axiom,
! [A: nat,B: nat] :
( ( A
= ( suc @ B ) )
=> ( A
!= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_nat_OdiscI) ).
thf(700,plain,
! [A: nat,B: nat] :
( ( A
= ( suc @ B ) )
=> ( A
!= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(228,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( ( coeff @ TA @ A @ ( degree @ TA @ A ) )
= ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ ( poly @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_leading__coeff__0__iff) ).
thf(1022,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( ( coeff @ TA @ A @ ( degree @ TA @ A ) )
= ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ ( poly @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).
thf(285,axiom,
! [A: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_144_not__gr0) ).
thf(1198,plain,
! [A: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).
thf(30,axiom,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ! [D: nat] :
( ( ord_less @ nat @ D @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_nat__less__induct) ).
thf(399,plain,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ! [D: nat] :
( ( ord_less @ nat @ D @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(169,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ( pcr_poly @ TA @ TA @ ( (=) @ TA )
@ ^ [A: nat] : ( zero_zero @ TA )
@ ( zero_zero @ ( poly @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_zero__poly_Otransfer) ).
thf(827,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ( pcr_poly @ TA @ TA @ ( (=) @ TA )
@ ^ [A: nat] : ( zero_zero @ TA )
@ ( zero_zero @ ( poly @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(109,axiom,
semiri134348788visors @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors) ).
thf(636,plain,
semiri134348788visors @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(280,axiom,
! [A: real,B: real] :
( ( ord_less @ real @ A @ B )
=> ? [C: real] :
( ( member @ real @ C @ ( field_char_0_Rats @ real ) )
& ( ord_less @ real @ A @ C )
& ( ord_less @ real @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_Rats__dense__in__real) ).
thf(1176,plain,
! [A: real,B: real] :
( ( ord_less @ real @ A @ B )
=> ? [C: real] :
( ( member @ real @ C @ ( field_char_0_Rats @ real ) )
& ( ord_less @ real @ A @ C )
& ( ord_less @ real @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).
thf(257,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ? [B: nat] :
( A
= ( suc @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_not0__implies__Suc) ).
thf(1113,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ? [B: nat] :
( A
= ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).
thf(223,axiom,
! [A: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_213_gcd__nat_Oextremum__unique) ).
thf(1000,plain,
! [A: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).
thf(61,axiom,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ( member @ TA @ ( one_one @ TA ) @ ( field_char_0_Rats @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_Rats__1) ).
thf(505,plain,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ( member @ TA @ ( one_one @ TA ) @ ( field_char_0_Rats @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(203,axiom,
field @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Fields_Ofield) ).
thf(922,plain,
field @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(128,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_244_less__numeral__extra_I3_J) ).
thf(685,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(66,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: poly @ TA] :
( ! [C: nat] :
( ( coeff @ TA @ A @ C )
= ( coeff @ TA @ B @ C ) )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_48_poly__eqI) ).
thf(521,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: poly @ TA] :
( ! [C: nat] :
( ( coeff @ TA @ A @ C )
= ( coeff @ TA @ B @ C ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(277,axiom,
comm_semiring_0 @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Rings_Ocomm__semiring__0_11) ).
thf(1171,plain,
comm_semiring_0 @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).
thf(126,axiom,
! [TA: $tType] :
( ( division_ring @ TA )
=> ! [A: TA] :
( ( A
!= ( zero_zero @ TA ) )
=> ( ( inverse_inverse @ TA @ A )
!= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_231_nonzero__imp__inverse__nonzero) ).
thf(680,plain,
! [TA: $tType] :
( ( division_ring @ TA )
=> ! [A: TA] :
( ( A
!= ( zero_zero @ TA ) )
=> ( ( inverse_inverse @ TA @ A )
!= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(121,axiom,
linordered_idom @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Rings_Olinordered__idom) ).
thf(663,plain,
linordered_idom @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(114,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less @ nat @ A @ B )
=> ( ( ord_less @ nat @ A @ ( suc @ B ) )
= ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_151_not__less__less__Suc__eq) ).
thf(648,plain,
! [A: nat,B: nat] :
( ~ ( ord_less @ nat @ A @ B )
=> ( ( ord_less @ nat @ A @ ( suc @ B ) )
= ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(247,axiom,
! [TA: $tType] :
( ( division_ring @ TA )
=> ( ( inverse_inverse @ TA @ ( zero_zero @ TA ) )
= ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_218_inverse__zero) ).
thf(1081,plain,
! [TA: $tType] :
( ( division_ring @ TA )
=> ( ( inverse_inverse @ TA @ ( zero_zero @ TA ) )
= ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).
thf(103,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( ( reflect_poly @ TA @ ( one_one @ ( poly @ TA ) ) )
= ( one_one @ ( poly @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_68_reflect__poly__1) ).
thf(617,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( ( reflect_poly @ TA @ ( one_one @ ( poly @ TA ) ) )
= ( one_one @ ( poly @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(269,axiom,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( inverse_inverse @ TA @ A ) )
= ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_214_inverse__positive__iff__positive) ).
thf(1145,plain,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( inverse_inverse @ TA @ A ) )
= ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).
thf(15,axiom,
! [A: nat] : ( ord_less @ nat @ A @ ( suc @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_lessI) ).
thf(355,plain,
! [A: nat] : ( ord_less @ nat @ A @ ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(151,axiom,
~ ( member @ real @ x @ ( field_char_0_Rats @ real ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_irrationsl) ).
thf(754,plain,
~ ( member @ real @ x @ ( field_char_0_Rats @ real ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(77,axiom,
order @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oorder_5) ).
thf(553,plain,
order @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(172,axiom,
! [A: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_dvd__1__left) ).
thf(836,plain,
! [A: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(213,axiom,
! [TA: $tType] :
( ( idom_divide @ TA )
=> ! [A: TA,B: poly @ TA,C: poly @ TA,D: poly @ TA,E: nat] :
( ( divide_poly_main @ TA @ A @ B @ C @ D @ E @ ( zero_zero @ nat ) )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_divide__poly__main_Osimps_I2_J) ).
thf(968,plain,
! [TA: $tType] :
( ( idom_divide @ TA )
=> ! [A: TA,B: poly @ TA,C: poly @ TA,D: poly @ TA,E: nat] :
( ( divide_poly_main @ TA @ A @ B @ C @ D @ E @ ( zero_zero @ nat ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(281,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat] :
( ( poly_cutoff @ TA @ A @ ( zero_zero @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_poly__cutoff__0) ).
thf(1181,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat] :
( ( poly_cutoff @ TA @ A @ ( zero_zero @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).
thf(224,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( ( degree @ TA @ ( one_one @ ( poly @ TA ) ) )
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_degree__1) ).
thf(1003,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( ( degree @ TA @ ( one_one @ ( poly @ TA ) ) )
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).
thf(229,axiom,
! [A: nat > $o,B: nat] :
( ( A @ ( zero_zero @ nat ) )
=> ( ! [C: nat] :
( ( A @ C )
=> ( A @ ( suc @ C ) ) )
=> ( A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_old_Onat_Oinducts) ).
thf(1025,plain,
! [A: nat > $o,B: nat] :
( ( A @ ( zero_zero @ nat ) )
=> ( ! [C: nat] :
( ( A @ C )
=> ( A @ ( suc @ C ) ) )
=> ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).
thf(135,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_143_gr0I) ).
thf(705,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(295,axiom,
! [A: nat] :
( ( suc @ A )
!= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_104_Suc__neq__Zero) ).
thf(1225,plain,
! [A: nat] :
( ( suc @ A )
!= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).
thf(189,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_not__less0) ).
thf(887,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(214,axiom,
! [TA: $tType] :
( ( field @ TA )
=> ! [A: TA] :
( ( algebraic_int @ TA @ A )
=> ~ ! [B: poly @ TA] :
( ( ( coeff @ TA @ B @ ( degree @ TA @ B ) )
= ( one_one @ TA ) )
=> ( ! [C: nat] : ( member @ TA @ ( coeff @ TA @ B @ C ) @ ( ring_1_Ints @ TA ) )
=> ( ( poly2 @ TA @ B @ A )
!= ( zero_zero @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_algebraic__int_Ocases) ).
thf(971,plain,
! [TA: $tType] :
( ( field @ TA )
=> ! [A: TA] :
( ( algebraic_int @ TA @ A )
=> ~ ! [B: poly @ TA] :
( ( ( coeff @ TA @ B @ ( degree @ TA @ B ) )
= ( one_one @ TA ) )
=> ( ! [C: nat] : ( member @ TA @ ( coeff @ TA @ B @ C ) @ ( ring_1_Ints @ TA ) )
=> ( ( poly2 @ TA @ B @ A )
!= ( zero_zero @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(148,axiom,
! [A: nat > $o,B: nat] :
( ( A @ B )
=> ( ! [C: nat] :
( ( A @ ( suc @ C ) )
=> ( A @ C ) )
=> ( A @ ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_105_zero__induct) ).
thf(739,plain,
! [A: nat > $o,B: nat] :
( ( A @ B )
=> ( ! [C: nat] :
( ( A @ ( suc @ C ) )
=> ( A @ C ) )
=> ( A @ ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(312,axiom,
zero_less_one @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Rings_Ozero__less__one_12) ).
thf(1300,plain,
zero_less_one @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).
thf(7,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ( ord_less @ nat @ B @ C )
=> ( ord_less @ nat @ ( suc @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_154_less__trans__Suc) ).
thf(331,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ( ord_less @ nat @ B @ C )
=> ( ord_less @ nat @ ( suc @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(59,axiom,
! [TA: $tType] :
( ( comm_monoid_mult @ TA )
=> ! [A: TA] : ( dvd_dvd @ TA @ A @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_dvd__refl) ).
thf(502,plain,
! [TA: $tType] :
( ( comm_monoid_mult @ TA )
=> ! [A: TA] : ( dvd_dvd @ TA @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(174,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_neq0__conv) ).
thf(840,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(46,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( (=) @ ( poly @ TA ) )
= ( ^ [A: poly @ TA,B: poly @ TA] :
! [C: nat] :
( ( coeff @ TA @ A @ C )
= ( coeff @ TA @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_poly__eq__iff) ).
thf(463,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( (=) @ ( poly @ TA ) )
= ( ^ [A: poly @ TA,B: poly @ TA] :
! [C: nat] :
( ( coeff @ TA @ A @ C )
= ( coeff @ TA @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(86,axiom,
comm_monoid_mult @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__mult) ).
thf(576,plain,
comm_monoid_mult @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(292,axiom,
! [TA: $tType] :
( ( idom @ TA )
=> ! [A: TA] :
( ( order2 @ TA @ A @ ( one_one @ ( poly @ TA ) ) )
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_order__1__eq__0) ).
thf(1218,plain,
! [TA: $tType] :
( ( idom @ TA )
=> ! [A: TA] :
( ( order2 @ TA @ A @ ( one_one @ ( poly @ TA ) ) )
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).
thf(196,axiom,
comm_monoid_mult @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Groups_Ocomm__monoid__mult_9) ).
thf(907,plain,
comm_monoid_mult @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(305,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TB,C: TA > TB > $o] :
( ~ ( finite_finite2 @ TA @ A )
=> ( ( finite_finite2 @ TB @ B )
=> ( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ? [E: TB] :
( ( member @ TB @ E @ B )
& ( C @ D @ E ) ) )
=> ? [D: TB] :
( ( member @ TB @ D @ B )
& ~ ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [E: TA] :
( ( member @ TA @ E @ A )
& ( C @ E @ D ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_pigeonhole__infinite__rel) ).
thf(1262,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TB,C: TA > TB > $o] :
( ~ ( finite_finite2 @ TA @ A )
=> ( ( finite_finite2 @ TB @ B )
=> ( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ? [E: TB] :
( ( member @ TB @ E @ B )
& ( C @ D @ E ) ) )
=> ? [D: TB] :
( ( member @ TB @ D @ B )
& ~ ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [E: TA] :
( ( member @ TA @ E @ A )
& ( C @ E @ D ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).
thf(298,axiom,
! [TA: $tType] :
( ( field @ TA )
=> ( ( inverse_inverse @ TA @ ( zero_zero @ TA ) )
= ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_227_field__class_Ofield__inverse__zero) ).
thf(1235,plain,
! [TA: $tType] :
( ( field @ TA )
=> ( ( inverse_inverse @ TA @ ( zero_zero @ TA ) )
= ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).
thf(101,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( zero_neq_one @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Rings_Ozero__neq__one_29) ).
thf(613,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( zero_neq_one @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(293,axiom,
ring_1 @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Rings_Oring__1) ).
thf(1221,plain,
ring_1 @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).
thf(236,axiom,
! [A: nat] :
( ( ord_less @ nat @ A @ ( one_one @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_less__one) ).
thf(1052,plain,
! [A: nat] :
( ( ord_less @ nat @ A @ ( one_one @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).
thf(143,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat] :
( ( coeff @ TA @ ( zero_zero @ ( poly @ TA ) ) @ A )
= ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_coeff__0) ).
thf(727,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat] :
( ( coeff @ TA @ ( zero_zero @ ( poly @ TA ) ) @ A )
= ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(215,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ! [A: TA] :
( ( dvd_dvd @ TA @ ( zero_zero @ TA ) @ A )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_dvd__0__left__iff) ).
thf(978,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ! [A: TA] :
( ( dvd_dvd @ TA @ ( zero_zero @ TA ) @ A )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(84,axiom,
! [TA: $tType] :
( ( one @ TA )
=> ! [A: TA] :
( ( ( one_one @ TA )
= A )
= ( A
= ( one_one @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_74_one__reorient) ).
thf(571,plain,
! [TA: $tType] :
( ( one @ TA )
=> ! [A: TA] :
( ( ( one_one @ TA )
= A )
= ( A
= ( one_one @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(100,axiom,
semiri1193490041visors @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors) ).
thf(612,plain,
semiri1193490041visors @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(56,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) )
= ( ord_less @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_Suc__less__eq) ).
thf(497,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) )
= ( ord_less @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(185,axiom,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ( ( algebraic @ TA )
= ( ^ [A: TA] :
? [B: poly @ TA] :
( ! [C: nat] : ( member @ TA @ ( coeff @ TA @ B @ C ) @ ( ring_1_Ints @ TA ) )
& ( B
!= ( zero_zero @ ( poly @ TA ) ) )
& ( ( poly2 @ TA @ B @ A )
= ( zero_zero @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_algebraic__def) ).
thf(876,plain,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ( ( algebraic @ TA )
= ( ^ [A: TA] :
? [B: poly @ TA] :
( ! [C: nat] : ( member @ TA @ ( coeff @ TA @ B @ C ) @ ( ring_1_Ints @ TA ) )
& ( B
!= ( zero_zero @ ( poly @ TA ) ) )
& ( ( poly2 @ TA @ B @ A )
= ( zero_zero @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(248,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( B
!= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_147_gr__implies__not0) ).
thf(1084,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( B
!= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).
thf(79,axiom,
! [TA: $tType] :
( ( division_ring @ TA )
=> ( ( inverse_inverse @ TA @ ( one_one @ TA ) )
= ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_221_inverse__1) ).
thf(556,plain,
! [TA: $tType] :
( ( division_ring @ TA )
=> ( ( inverse_inverse @ TA @ ( one_one @ TA ) )
= ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(51,axiom,
! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
( ! [D: TA] :
( ~ ( A @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
& ~ ( A @ E ) ) )
=> ( A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_infinite__descent__measure) ).
thf(482,plain,
! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
( ! [D: TA] :
( ~ ( A @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
& ~ ( A @ E ) ) )
=> ( A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(123,axiom,
! [TA: $tType] :
( ( field @ TA )
=> ! [A: poly @ TA] :
( ( A
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( dvd_dvd @ ( poly @ TA ) @ A @ ( one_one @ ( poly @ TA ) ) )
= ( ( degree @ TA @ A )
= ( zero_zero @ nat ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_is__unit__iff__degree) ).
thf(671,plain,
! [TA: $tType] :
( ( field @ TA )
=> ! [A: poly @ TA] :
( ( A
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( dvd_dvd @ ( poly @ TA ) @ A @ ( one_one @ ( poly @ TA ) ) )
= ( ( degree @ TA @ A )
= ( zero_zero @ nat ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(274,axiom,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
=> ? [C: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
& ( ord_less @ TA @ C @ A )
& ( ord_less @ TA @ C @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_248_field__lbound__gt__zero) ).
thf(1157,plain,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
=> ? [C: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
& ( ord_less @ TA @ C @ A )
& ( ord_less @ TA @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).
thf(130,axiom,
! [A: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A )
=> ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_126_gcd__nat_Oextremum__uniqueI) ).
thf(688,plain,
! [A: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A )
=> ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(55,axiom,
! [TA: $tType] :
( ( idom @ TA )
=> ( ring_n68954251visors @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Rings_Oring__no__zero__divisors_21) ).
thf(495,plain,
! [TA: $tType] :
( ( idom @ TA )
=> ( ring_n68954251visors @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(153,axiom,
! [A: nat] : ( member @ real @ ( coeff @ real @ p @ A ) @ ( ring_1_Ints @ real ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_p_I1_J) ).
thf(759,plain,
! [A: nat] : ( member @ real @ ( coeff @ real @ p @ A ) @ ( ring_1_Ints @ real ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(19,axiom,
comm_semiring_1 @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Ocomm__semiring__1) ).
thf(368,plain,
comm_semiring_1 @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(205,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ! [A: nat] :
( ( ( A
= ( zero_zero @ nat ) )
=> ( ( poly_shift @ TA @ A @ ( one_one @ ( poly @ TA ) ) )
= ( one_one @ ( poly @ TA ) ) ) )
& ( ( A
!= ( zero_zero @ nat ) )
=> ( ( poly_shift @ TA @ A @ ( one_one @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_poly__shift__1) ).
thf(931,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ! [A: nat] :
( ( ( A
= ( zero_zero @ nat ) )
=> ( ( poly_shift @ TA @ A @ ( one_one @ ( poly @ TA ) ) )
= ( one_one @ ( poly @ TA ) ) ) )
& ( ( A
!= ( zero_zero @ nat ) )
=> ( ( poly_shift @ TA @ A @ ( one_one @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(74,axiom,
! [A: nat,B: nat,C: nat > nat > $o] :
( ( ord_less @ nat @ A @ B )
=> ( ! [D: nat] : ( C @ D @ ( suc @ D ) )
=> ( ! [D: nat,E: nat,F: nat] :
( ( ord_less @ nat @ D @ E )
=> ( ( ord_less @ nat @ E @ F )
=> ( ( C @ D @ E )
=> ( ( C @ E @ F )
=> ( C @ D @ F ) ) ) ) )
=> ( C @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_less__Suc__induct) ).
thf(539,plain,
! [A: nat,B: nat,C: nat > nat > $o] :
( ( ord_less @ nat @ A @ B )
=> ( ! [D: nat] : ( C @ D @ ( suc @ D ) )
=> ( ! [D: nat,E: nat,F: nat] :
( ( ord_less @ nat @ D @ E )
=> ( ( ord_less @ nat @ E @ F )
=> ( ( C @ D @ E )
=> ( ( C @ E @ F )
=> ( C @ D @ F ) ) ) ) )
=> ( C @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(67,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ( comm_semiring_0 @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0_26) ).
thf(524,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ( comm_semiring_0 @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(89,axiom,
wellorder @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Owellorder) ).
thf(581,plain,
wellorder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(211,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
=> ( ( dvd_dvd @ nat @ B @ A )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_dvd__pos__nat) ).
thf(963,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
=> ( ( dvd_dvd @ nat @ B @ A )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(3,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd @ nat @ A @ B )
& ( A != B ) )
=> ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_gcd__nat_Ostrict__implies__not__eq) ).
thf(319,plain,
! [A: nat,B: nat] :
( ( ( dvd_dvd @ nat @ A @ B )
& ( A != B ) )
=> ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(192,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ( ord_less @ TA @ ( zero_zero @ TA ) @ ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_242_less__numeral__extra_I1_J) ).
thf(896,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ( ord_less @ TA @ ( zero_zero @ TA ) @ ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(240,axiom,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( inverse_inverse @ TA @ A ) @ ( zero_zero @ TA ) )
= ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_215_inverse__negative__iff__negative) ).
thf(1062,plain,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( inverse_inverse @ TA @ A ) @ ( zero_zero @ TA ) )
= ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).
thf(254,axiom,
! [TA: $tType] :
( ( field @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( coeff @ TA @ A @ ( degree @ TA @ A ) )
= ( one_one @ TA ) )
=> ( ! [C: nat] : ( member @ TA @ ( coeff @ TA @ A @ C ) @ ( ring_1_Ints @ TA ) )
=> ( ( ( poly2 @ TA @ A @ B )
= ( zero_zero @ TA ) )
=> ( algebraic_int @ TA @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_algebraic__int_Ointros) ).
thf(1101,plain,
! [TA: $tType] :
( ( field @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( coeff @ TA @ A @ ( degree @ TA @ A ) )
= ( one_one @ TA ) )
=> ( ! [C: nat] : ( member @ TA @ ( coeff @ TA @ A @ C ) @ ( ring_1_Ints @ TA ) )
=> ( ( ( poly2 @ TA @ A @ B )
= ( zero_zero @ TA ) )
=> ( algebraic_int @ TA @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).
thf(116,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ( ( suc @ A )
!= B )
=> ( ord_less @ nat @ ( suc @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_164_Suc__lessI) ).
thf(652,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ( ( suc @ A )
!= B )
=> ( ord_less @ nat @ ( suc @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(160,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( ( coeff @ TA @ ( reflect_poly @ TA @ A ) @ ( zero_zero @ nat ) )
= ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ ( poly @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_coeff__0__reflect__poly__0__iff) ).
thf(781,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( ( coeff @ TA @ ( reflect_poly @ TA @ A ) @ ( zero_zero @ nat ) )
= ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ ( poly @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(199,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
= ( A
!= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_zero__less__iff__neq__zero) ).
thf(914,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
= ( A
!= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(49,axiom,
! [A: nat] : ( dvd_dvd @ nat @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_gcd__nat_Orefl) ).
thf(478,plain,
! [A: nat] : ( dvd_dvd @ nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(290,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( synthetic_div @ TA @ A @ B )
= ( zero_zero @ ( poly @ TA ) ) )
= ( ( degree @ TA @ A )
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_synthetic__div__eq__0__iff) ).
thf(1212,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( synthetic_div @ TA @ A @ B )
= ( zero_zero @ ( poly @ TA ) ) )
= ( ( degree @ TA @ A )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).
thf(171,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ~ ! [B: nat] :
( A
!= ( suc @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_101_old_Onat_Oexhaust) ).
thf(832,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ~ ! [B: nat] :
( A
!= ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(76,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_less__not__refl3) ).
thf(549,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(233,axiom,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_mem__Collect__eq) ).
thf(1045,plain,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).
thf(140,axiom,
! [TA: $tType] :
( ( division_ring @ TA )
=> ! [A: TA,B: TA] :
( ( ( inverse_inverse @ TA @ A )
= ( inverse_inverse @ TA @ B ) )
=> ( ( A
!= ( zero_zero @ TA ) )
=> ( ( B
!= ( zero_zero @ TA ) )
=> ( A = B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_229_nonzero__inverse__eq__imp__eq) ).
thf(716,plain,
! [TA: $tType] :
( ( division_ring @ TA )
=> ! [A: TA,B: TA] :
( ( ( inverse_inverse @ TA @ A )
= ( inverse_inverse @ TA @ B ) )
=> ( ( A
!= ( zero_zero @ TA ) )
=> ( ( B
!= ( zero_zero @ TA ) )
=> ( A = B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(12,axiom,
! [TA: $tType,TB: $tType] :
( ( wellorder @ TB )
=> ! [A: TA > TB,B: TA > $o,C: TA] :
( ! [D: TA] :
( ! [E: TA] :
( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
=> ( B @ E ) )
=> ( B @ D ) )
=> ( B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_measure__induct) ).
thf(346,plain,
! [TA: $tType,TB: $tType] :
( ( wellorder @ TB )
=> ! [A: TA > TB,B: TA > $o,C: TA] :
( ! [D: TA] :
( ! [E: TA] :
( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
=> ( B @ E ) )
=> ( B @ D ) )
=> ( B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(316,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA] :
( ( synthetic_div @ TA @ ( zero_zero @ ( poly @ TA ) ) @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_synthetic__div__0) ).
thf(1309,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA] :
( ( synthetic_div @ TA @ ( zero_zero @ ( poly @ TA ) ) @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).
thf(258,axiom,
! [A: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_zero__less__Suc) ).
thf(1116,plain,
! [A: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).
thf(264,axiom,
! [TA: $tType] :
( ( division_ring @ TA )
=> ! [A: TA] :
( ( A
!= ( zero_zero @ TA ) )
=> ( ( inverse_inverse @ TA @ ( inverse_inverse @ TA @ A ) )
= A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_230_nonzero__inverse__inverse__eq) ).
thf(1133,plain,
! [TA: $tType] :
( ( division_ring @ TA )
=> ! [A: TA] :
( ( A
!= ( zero_zero @ TA ) )
=> ( ( inverse_inverse @ TA @ ( inverse_inverse @ TA @ A ) )
= A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).
thf(104,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( ( coeff @ TA @ ( one_one @ ( poly @ TA ) ) @ ( degree @ TA @ ( one_one @ ( poly @ TA ) ) ) )
= ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_lead__coeff__1) ).
thf(620,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( ( coeff @ TA @ ( one_one @ ( poly @ TA ) ) @ ( degree @ TA @ ( one_one @ ( poly @ TA ) ) ) )
= ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(150,axiom,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ! [A: set @ TA] : ( finite_finite2 @ TA @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_finite) ).
thf(752,plain,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ! [A: set @ TA] : ( finite_finite2 @ TA @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(34,axiom,
! [TA: $tType] :
( ( comm_ring_1 @ TA )
=> ( ring_1 @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Rings_Oring__1_34) ).
thf(432,plain,
! [TA: $tType] :
( ( comm_ring_1 @ TA )
=> ( ring_1 @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(32,axiom,
! [A: nat,B: nat,C: nat > $o] :
( ( ord_less @ nat @ A @ B )
=> ( ! [D: nat] :
( ( B
= ( suc @ D ) )
=> ( C @ D ) )
=> ( ! [D: nat] :
( ( ord_less @ nat @ D @ B )
=> ( ( C @ ( suc @ D ) )
=> ( C @ D ) ) )
=> ( C @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_152_strict__inc__induct) ).
thf(410,plain,
! [A: nat,B: nat,C: nat > $o] :
( ( ord_less @ nat @ A @ B )
=> ( ! [D: nat] :
( ( B
= ( suc @ D ) )
=> ( C @ D ) )
=> ( ! [D: nat] :
( ( ord_less @ nat @ D @ B )
=> ( ( C @ ( suc @ D ) )
=> ( C @ D ) ) )
=> ( C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(296,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: poly @ TA] :
( ( poly2 @ TA @ A @ ( zero_zero @ TA ) )
= ( coeff @ TA @ A @ ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_poly__0__coeff__0) ).
thf(1229,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: poly @ TA] :
( ( poly2 @ TA @ A @ ( zero_zero @ TA ) )
= ( coeff @ TA @ A @ ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).
thf(270,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_146_less__zeroE) ).
thf(1148,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).
thf(180,axiom,
semiring_char_0 @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Nat_Osemiring__char__0_13) ).
thf(865,plain,
semiring_char_0 @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(97,axiom,
! [A: nat,B: nat > $o] :
( ( ! [C: nat] :
( ( ord_less @ nat @ C @ ( suc @ A ) )
=> ( B @ C ) ) )
= ( ( B @ A )
& ! [C: nat] :
( ( ord_less @ nat @ C @ A )
=> ( B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_158_All__less__Suc) ).
thf(603,plain,
! [A: nat,B: nat > $o] :
( ( ! [C: nat] :
( ( ord_less @ nat @ C @ ( suc @ A ) )
=> ( B @ C ) ) )
= ( ( B @ A )
& ! [C: nat] :
( ( ord_less @ nat @ C @ A )
=> ( B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(5,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( suc @ A ) @ B )
= ( ? [C: nat] :
( ( B
= ( suc @ C ) )
& ( ord_less @ nat @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_157_Suc__less__eq2) ).
thf(326,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( suc @ A ) @ B )
= ( ? [C: nat] :
( ( B
= ( suc @ C ) )
& ( ord_less @ nat @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(9,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( one @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Groups_Oone_37) ).
thf(341,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( one @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(165,axiom,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
=> ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
=> ( ( ord_less @ TA @ ( inverse_inverse @ TA @ A ) @ ( inverse_inverse @ TA @ B ) )
= ( ord_less @ TA @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_216_inverse__less__iff__less__neg) ).
thf(807,plain,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
=> ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
=> ( ( ord_less @ TA @ ( inverse_inverse @ TA @ A ) @ ( inverse_inverse @ TA @ B ) )
= ( ord_less @ TA @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(36,axiom,
! [A: nat,B: nat] :
( ( A != B )
= ( ( ord_less @ nat @ A @ B )
| ( ord_less @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_nat__neq__iff) ).
thf(437,plain,
! [A: nat,B: nat] :
( ( A != B )
= ( ( ord_less @ nat @ A @ B )
| ( ord_less @ nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(207,axiom,
zero_neq_one @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Rings_Ozero__neq__one_14) ).
thf(955,plain,
zero_neq_one @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(278,axiom,
algebraic @ real @ x,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_assms_I2_J) ).
thf(1172,plain,
algebraic @ real @ x,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).
thf(156,axiom,
! [TA: $tType] :
( ( division_ring @ TA )
=> ! [A: TA] :
( ( ( inverse_inverse @ TA @ A )
= ( zero_zero @ TA ) )
=> ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_228_inverse__zero__imp__zero) ).
thf(767,plain,
! [TA: $tType] :
( ( division_ring @ TA )
=> ! [A: TA] :
( ( ( inverse_inverse @ TA @ A )
= ( zero_zero @ TA ) )
=> ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(238,axiom,
! [TA: $tType] :
( ( zero_neq_one @ TA )
=> ( ( zero_zero @ TA )
!= ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_zero__neq__one) ).
thf(1056,plain,
! [TA: $tType] :
( ( zero_neq_one @ TA )
=> ( ( zero_zero @ TA )
!= ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).
thf(122,axiom,
! [A: real,B: real,C: poly @ real] :
( ( ord_less @ real @ A @ B )
=> ( ( ord_less @ real @ ( poly2 @ real @ C @ A ) @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ C @ B ) )
=> ? [D: real] :
( ( ord_less @ real @ A @ D )
& ( ord_less @ real @ D @ B )
& ( ( poly2 @ real @ C @ D )
= ( zero_zero @ real ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_224_poly__IVT__pos) ).
thf(664,plain,
! [A: real,B: real,C: poly @ real] :
( ( ord_less @ real @ A @ B )
=> ( ( ord_less @ real @ ( poly2 @ real @ C @ A ) @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ C @ B ) )
=> ? [D: real] :
( ( ord_less @ real @ A @ D )
& ( ord_less @ real @ D @ B )
& ( ( poly2 @ real @ C @ D )
= ( zero_zero @ real ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(210,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA] :
( ( ( zero_zero @ TA )
= A )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_zero__reorient) ).
thf(960,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA] :
( ( ( zero_zero @ TA )
= A )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).
thf(108,axiom,
one @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oone) ).
thf(635,plain,
one @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(71,axiom,
! [A: nat] :
( ( dvd_dvd @ nat @ A @ ( one_one @ nat ) )
= ( A
= ( one_one @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_87_nat__dvd__1__iff__1) ).
thf(532,plain,
! [A: nat] :
( ( dvd_dvd @ nat @ A @ ( one_one @ nat ) )
= ( A
= ( one_one @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(241,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
~ ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_141_not__less__zero) ).
thf(1065,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
~ ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).
thf(251,axiom,
! [TA: $tType] :
( ( ( comm_semiring_1 @ TA )
& ( semiri1193490041visors @ TA ) )
=> ( ( pderiv @ TA @ ( zero_zero @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_pderiv__0) ).
thf(1093,plain,
! [TA: $tType] :
( ( ( comm_semiring_1 @ TA )
& ( semiri1193490041visors @ TA ) )
=> ( ( pderiv @ TA @ ( zero_zero @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).
thf(218,axiom,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( pderiv @ TA @ A )
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( ( order2 @ TA @ B @ A )
!= ( zero_zero @ nat ) )
=> ( ( order2 @ TA @ B @ A )
= ( suc @ ( order2 @ TA @ B @ ( pderiv @ TA @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_order__pderiv) ).
thf(987,plain,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( pderiv @ TA @ A )
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( ( order2 @ TA @ B @ A )
!= ( zero_zero @ nat ) )
=> ( ( order2 @ TA @ B @ A )
= ( suc @ ( order2 @ TA @ B @ ( pderiv @ TA @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(306,axiom,
! [TA: $tType] :
( ( ( ring_char_0 @ TA )
& ( comm_ring_1 @ TA )
& ( ring_n68954251visors @ TA ) )
=> ! [A: poly @ TA,B: poly @ TA] :
( ( ( poly2 @ TA @ A )
= ( poly2 @ TA @ B ) )
= ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_poly__eq__poly__eq__iff) ).
thf(1268,plain,
! [TA: $tType] :
( ( ( ring_char_0 @ TA )
& ( comm_ring_1 @ TA )
& ( ring_n68954251visors @ TA ) )
=> ! [A: poly @ TA,B: poly @ TA] :
( ( ( poly2 @ TA @ A )
= ( poly2 @ TA @ B ) )
= ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).
thf(95,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_245_less__numeral__extra_I4_J) ).
thf(596,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(44,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_Suc__mono) ).
thf(459,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(131,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA] :
( ( poly2 @ TA @ ( zero_zero @ ( poly @ TA ) ) @ A )
= ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4_poly__0) ).
thf(691,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA] :
( ( poly2 @ TA @ ( zero_zero @ ( poly @ TA ) ) @ A )
= ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(38,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd @ nat @ A @ B )
=> ( ( dvd_dvd @ nat @ B @ A )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_dvd__antisym) ).
thf(442,plain,
! [A: nat,B: nat] :
( ( dvd_dvd @ nat @ A @ B )
=> ( ( dvd_dvd @ nat @ B @ A )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(58,axiom,
linord1659791738miring @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring) ).
thf(501,plain,
linord1659791738miring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(20,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd @ nat @ A @ B )
=> ( ( ( dvd_dvd @ nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd @ nat @ A @ C )
& ( A != C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_gcd__nat_Ostrict__trans1) ).
thf(369,plain,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd @ nat @ A @ B )
=> ( ( ( dvd_dvd @ nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd @ nat @ A @ C )
& ( A != C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(197,axiom,
! [TA: $tType] :
( ( comm_monoid_mult @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ ( set @ TA )
@ ( collect @ TA
@ ^ [C: TA] : ( dvd_dvd @ TA @ C @ A ) )
@ ( collect @ TA
@ ^ [C: TA] : ( dvd_dvd @ TA @ C @ B ) ) )
= ( ( dvd_dvd @ TA @ A @ B )
& ~ ( dvd_dvd @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_223_strict__subset__divisors__dvd) ).
thf(908,plain,
! [TA: $tType] :
( ( comm_monoid_mult @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ ( set @ TA )
@ ( collect @ TA
@ ^ [C: TA] : ( dvd_dvd @ TA @ C @ A ) )
@ ( collect @ TA
@ ^ [C: TA] : ( dvd_dvd @ TA @ C @ B ) ) )
= ( ( dvd_dvd @ TA @ A @ B )
& ~ ( dvd_dvd @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).
thf(220,axiom,
( ( poly2 @ real @ p @ x )
= ( zero_zero @ real ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_p_I3_J) ).
thf(992,plain,
( ( poly2 @ real @ p @ x )
= ( zero_zero @ real ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(48,axiom,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less @ nat @ D @ C )
& ~ ( A @ D ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_infinite__descent) ).
thf(469,plain,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less @ nat @ D @ C )
& ~ ( A @ D ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(14,axiom,
semiring_char_0 @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Nat_Osemiring__char__0) ).
thf(354,plain,
semiring_char_0 @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(204,axiom,
~ ! [A: poly @ real] :
( ! [B: nat] : ( member @ real @ ( coeff @ real @ A @ B ) @ ( ring_1_Ints @ real ) )
=> ( ( A
!= ( zero_zero @ ( poly @ real ) ) )
=> ( ( poly2 @ real @ A @ x )
!= ( zero_zero @ real ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26__092_060open_062_092_060And_062thesisa_O_A_I_092_060And_062p_O_A_092_060lbrakk_062_092_060And_062i_O_Acoeff_Ap_Ai_A_092_060in_062_A_092_060int_062_059_Ap_A_092_060noteq_062_A0_059_Apoly_Ap_Ax_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesisa_J_A_092_060Longrightarrow_062_Athesisa_092_060close_062) ).
thf(923,plain,
~ ! [A: poly @ real] :
( ! [B: nat] : ( member @ real @ ( coeff @ real @ A @ B ) @ ( ring_1_Ints @ real ) )
=> ( ( A
!= ( zero_zero @ ( poly @ real ) ) )
=> ( ( poly2 @ real @ A @ x )
!= ( zero_zero @ real ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(282,axiom,
! [TA: $tType,A: nat > TA > $o,B: nat > TA > TA > $o] :
( ? [C: TA] : ( A @ ( zero_zero @ nat ) @ C )
=> ( ! [C: TA,D: nat] :
( ( A @ D @ C )
=> ? [E: TA] :
( ( A @ ( suc @ D ) @ E )
& ( B @ D @ C @ E ) ) )
=> ? [C: nat > TA] :
! [D: nat] :
( ( A @ D @ ( C @ D ) )
& ( B @ D @ ( C @ D ) @ ( C @ ( suc @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_dependent__nat__choice) ).
thf(1184,plain,
! [TA: $tType,A: nat > TA > $o,B: nat > TA > TA > $o] :
( ? [C: TA] : ( A @ ( zero_zero @ nat ) @ C )
=> ( ! [C: TA,D: nat] :
( ( A @ D @ C )
=> ? [E: TA] :
( ( A @ ( suc @ D ) @ E )
& ( B @ D @ C @ E ) ) )
=> ? [C: nat > TA] :
! [D: nat] :
( ( A @ D @ ( C @ D ) )
& ( B @ D @ ( C @ D ) @ ( C @ ( suc @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).
thf(119,axiom,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ! [A: TA] :
( ( algebraic_int @ TA @ A )
=> ( ( member @ TA @ A @ ( field_char_0_Rats @ TA ) )
=> ( member @ TA @ A @ ( ring_1_Ints @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_rational__algebraic__int__is__int) ).
thf(659,plain,
! [TA: $tType] :
( ( field_char_0 @ TA )
=> ! [A: TA] :
( ( algebraic_int @ TA @ A )
=> ( ( member @ TA @ A @ ( field_char_0_Rats @ TA ) )
=> ( member @ TA @ A @ ( ring_1_Ints @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(111,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd @ nat @ A @ B )
=> ( ( dvd_dvd @ nat @ B @ A )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_gcd__nat_Oantisym) ).
thf(639,plain,
! [A: nat,B: nat] :
( ( dvd_dvd @ nat @ A @ B )
=> ( ( dvd_dvd @ nat @ B @ A )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(287,axiom,
! [A: nat] : ( dvd_dvd @ nat @ A @ ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_211_gcd__nat_Oextremum) ).
thf(1204,plain,
! [A: nat] : ( dvd_dvd @ nat @ A @ ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).
thf(93,axiom,
! [TA: $tType] :
( ( idom_divide @ TA )
=> ( algebraic_semidom @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Polynomial_Opoly___Rings_Oalgebraic__semidom_22) ).
thf(592,plain,
! [TA: $tType] :
( ( idom_divide @ TA )
=> ( algebraic_semidom @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(42,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd @ nat @ A @ B )
& ( A != B ) )
=> ( dvd_dvd @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_gcd__nat_Ostrict__implies__order) ).
thf(454,plain,
! [A: nat,B: nat] :
( ( ( dvd_dvd @ nat @ A @ B )
& ( A != B ) )
=> ( dvd_dvd @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(136,axiom,
! [A: poly @ real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( coeff @ real @ A @ ( degree @ real @ A ) ) )
=> ? [B: real] :
! [C: real] :
( ( ord_less_eq @ real @ B @ C )
=> ( ord_less_eq @ real @ ( coeff @ real @ A @ ( degree @ real @ A ) ) @ ( poly2 @ real @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_250_poly__pinfty__gt__lc) ).
thf(708,plain,
! [A: poly @ real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( coeff @ real @ A @ ( degree @ real @ A ) ) )
=> ? [B: real] :
! [C: real] :
( ( ord_less_eq @ real @ B @ C )
=> ( ord_less_eq @ real @ ( coeff @ real @ A @ ( degree @ real @ A ) ) @ ( poly2 @ real @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(187,axiom,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( inverse_inverse @ TA @ A ) )
=> ( ( A
!= ( zero_zero @ TA ) )
=> ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_237_inverse__positive__imp__positive) ).
thf(882,plain,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( inverse_inverse @ TA @ A ) )
=> ( ( A
!= ( zero_zero @ TA ) )
=> ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(53,axiom,
! [TA: $tType] :
( ( comm_monoid_mult @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( dvd_dvd @ TA @ A @ B )
=> ( ( dvd_dvd @ TA @ B @ C )
=> ( dvd_dvd @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_dvd__trans) ).
thf(491,plain,
! [TA: $tType] :
( ( comm_monoid_mult @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( dvd_dvd @ TA @ A @ B )
=> ( ( dvd_dvd @ TA @ B @ C )
=> ( dvd_dvd @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(249,axiom,
! [TA: $tType] :
( ( idom @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( poly2 @ TA @ A @ B )
!= ( zero_zero @ TA ) )
=> ( ( order2 @ TA @ B @ A )
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_order__0I) ).
thf(1088,plain,
! [TA: $tType] :
( ( idom @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( poly2 @ TA @ A @ B )
!= ( zero_zero @ TA ) )
=> ( ( order2 @ TA @ B @ A )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(1314,plain,
$false,
inference(e,[status(thm)],[1036,365,893,1295,352,408,582,683,730,1068,994,873,1142,677,1305,1173,440,655,1078,385,384,500,1164,715,1222,829,898,885,565,651,1243,1127,389,1055,719,504,838,917,357,1059,802,344,559,870,606,448,1232,1013,770,1109,623,646,1136,1151,966,480,1302,756,1020,574,1051,821,981,687,1215,956,866,774,396,1201,507,527,1104,849,1074,1206,920,512,661,761,810,366,1067,714,334,957,1091,825,778,862,598,1238,847,625,879,852,329,461,493,1195,1128,710,578,697,1143,1259,615,725,1096,1006,388,869,1118,488,1292,554,537,901,642,526,637,590,1039,1155,547,393,997,1099,1249,1192,457,569,1246,984,911,343,938,1241,921,657,1071,674,530,1278,750,1301,733,513,430,694,843,577,1124,584,1154,562,904,445,382,1271,1312,787,466,804,391,1139,764,1209,1282,784,451,1122,1048,736,377,990,737,318,509,1043,434,890,631,535,609,323,594,1162,700,1022,1198,399,827,636,1176,1113,1000,505,922,685,521,1171,680,663,648,1081,617,1145,355,754,553,836,968,1181,1003,1025,705,1225,887,971,739,1300,331,502,840,463,576,1218,907,1262,1235,613,1221,1052,727,978,571,612,497,876,1084,556,482,671,1157,688,495,759,368,931,539,524,581,963,319,896,1062,1101,652,781,914,478,1212,832,549,1045,716,346,1309,1116,1133,620,752,432,410,1229,1148,865,603,326,341,807,437,955,1172,767,1056,664,960,635,532,1065,1093,987,1268,596,459,691,442,501,369,908,992,469,354,923,1184,659,639,1204,592,454,708,882,491,1088]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : ITP094^2 : TPTP v9.3.1. Released v7.5.0.
% 0.00/0.08 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.11/0.21 % Computer : n014.cluster.edu
% 0.11/0.21 % Model : x86_64 x86_64
% 0.11/0.21 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.21 % Memory : 8046.5625MB
% 0.11/0.21 % OS : Linux 6.8.0-71-generic
% 0.11/0.21 % CPULimit : 300
% 0.11/0.21 % WCLimit : 300
% 0.11/0.21 % DateTime : Tue Sep 29 19:52:44 UTC 2026
% 0.11/0.21 % CPUTime :
% 0.11/0.21 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.86/0.79 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 2.35/1.20 % [INFO] Parsing done (410ms).
% 2.35/1.22 % [INFO] Running in sequential loop mode.
% 3.57/1.62 % [INFO] eprover registered as external prover.
% 3.57/1.63 % [INFO] Scanning for conjecture ...
% 3.74/1.79 % [INFO] Found a conjecture (or negated_conjecture) and 316 axioms. Running axiom selection ...
% 4.19/1.94 % [INFO] Axiom selection finished. Selected 315 axioms (removed 1 axioms).
% 5.19/2.13 % [INFO] Problem is higher-order (TPTP THF).
% 5.19/2.15 % [INFO] Type checking passed.
% 5.19/2.15 % [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 ...
% 50.66/32.00 % External prover 'e' found a proof!
% 50.66/32.00 % [INFO] Killing All external provers ...
% 50.66/32.00 % Time passed: 31639ms (effective reasoning time: 30772ms)
% 50.66/32.00 % 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)>
% 50.66/32.00 % Axioms used in derivation (315): fact_145_not__less0, fact_103_Zero__neq__Suc, fact_4_poly__0, fact_99_not0__implies__Suc, fact_132_neq0__conv, fact_217_inverse__less__iff__less, fact_153_less__Suc__induct, fact_200_gcd__nat_Oorder__iff__strict, tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_25, fact_86_old_Onat_Oinject, fact_111_nat_Odistinct_I1_J, fact_234_inverse__less__imp__less__neg, fact_7_finite__code, fact_58_leading__coeff__neq__0, fact_223_strict__subset__divisors__dvd, tcon_fun___Finite__Set_Ofinite, fact_27_synthetic__div__eq__0__iff, tcon_Polynomial_Opoly___Nat_Osemiring__char__0_28, fact_120_dvd__antisym, fact_89_dvd__1__left, fact_174_less__irrefl__nat, fact_138_less__one, fact_85_nat_Oinject, fact_246_Rats__dense__in__real, fact_128_finite__Collect__less__nat, fact_179_infinite__descent__measure, fact_114_algebraic__int_Osimps, tcon_Real_Oreal___Fields_Ofield__char__0, tcon_Polynomial_Opoly___Groups_Oone_37, fact_130_bot__nat__0_Onot__eq__extremum, fact_20_zero__reorient, fact_80_poly__shift__1, fact_51_algebraicE, fact_216_inverse__less__iff__less__neg, tcon_Nat_Onat___Rings_Ozero__neq__one, fact_221_inverse__1, fact_122_algebraic__int__imp__algebraic, fact_140_gr__implies__not__zero, fact_238_negative__imp__inverse__negative, fact_182_lift__Suc__mono__less, fact_113_algebraic__int_Ointros, fact_32_zero__poly_Otransfer, tcon_Nat_Onat___Rings_Oalgebraic__semidom, fact_184_finite__divisors__nat, tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors, tcon_HOL_Obool___Orderings_Oorder_5, fact_175_nat__less__induct, tcon_Set_Oset___Orderings_Oorder_3, tcon_Nat_Onat___Rings_Ozero__less__one, fact_137_less__Suc0, fact_178_measure__induct__rule, tcon_Nat_Onat___Groups_Ocomm__monoid__mult, fact_185_not__one__less__zero, fact_90_dvd__trans, fact_38_reflect__poly__reflect__poly, tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, fact_25_reflect__poly__at__0__eq__0__iff, fact_247_Rats__no__bot__less, fact_17_degree__0, fact_3_p_I3_J, tcon_Real_Oreal___Nat_Oring__char__0, fact_121_int__imp__algebraic__int, tcon_Polynomial_Opoly___Rings_Oidom_36, tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_6, tcon_Polynomial_Opoly___Rings_Oidom__divide_30, fact_127_gcd__nat_Onot__eq__extremum, fact_163_less__SucE, fact_101_old_Onat_Oexhaust, tcon_HOL_Obool___Finite__Set_Ofinite_4, fact_160_less__Suc__eq, fact_82_dvd__0__right, fact_165_Suc__lessE, fact_204_gcd__nat_Oantisym, fact_66_cr__poly__def, tcon_Real_Oreal___Groups_Oone_17, fact_63_poly__cutoff__1, tcon_Nat_Onat___Groups_Oone, tcon_Real_Oreal___Rings_Oring__1, fact_154_less__trans__Suc, fact_177_linorder__neqE__nat, fact_208_gcd__nat_Orefl, fact_13_poly__eq__poly__eq__iff, fact_59_Rats__infinite, fact_161_Ex__less__Suc, fact_48_poly__eqI, fact_213_gcd__nat_Oextremum__unique, fact_136_zero__less__Suc, tcon_Polynomial_Opoly___Rings_Oring__1_34, fact_81_dvd__0__left__iff, fact_237_inverse__positive__imp__positive, fact_219_inverse__nonzero__iff__nonzero, fact_115_algebraic__int_Ocases, fact_97_unit__imp__dvd, fact_226_dvd__field__iff, tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0_26, fact_187_less__Suc__eq__0__disj, fact_202_gcd__nat_Ostrict__trans1, fact_146_less__zeroE, fact_118_algebraic__int__0, tcon_Real_Oreal___Rings_Ocomm__semiring__1_10, tcon_Real_Oreal___Rings_Ocomm__ring__1, fact_77_is__unit__iff__degree, fact_79_poly_Opcr__cr__eq, fact_197_gcd__nat_Ostrict__implies__not__eq, fact_64_order__pderiv2, fact_218_inverse__zero, fact_67_dvd__pderiv__iff, fact_34_reflect__poly__0, fact_119_algebraic__int__1, fact_245_less__numeral__extra_I4_J, fact_211_gcd__nat_Oextremum, fact_133_lessI, fact_75_Ints__1, fact_198_gcd__nat_Ostrict__implies__order, fact_129_not__gr__zero, fact_42_poly__reflect__poly__0, fact_94_zero__neq__one, fact_24_poly__cutoff__0, fact_164_Suc__lessI, fact_251_le__zero__eq, fact_123_rational__algebraic__int__is__int, tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors, tcon_Polynomial_Opoly___Nat_Oring__char__0_33, fact_0_poly__roots__finite, fact_5_finite__Collect__conjI, fact_244_less__numeral__extra_I3_J, fact_22_finite__set__choice, fact_40_coeff__0__reflect__poly__0__iff, fact_16_irrationsl, tcon_Nat_Onat___Orderings_Owellorder, tcon_Polynomial_Opoly___Rings_Oring__no__zero__divisors_21, fact_23_is__zero__null, fact_157_Suc__less__eq2, fact_65_order__pderiv, tcon_Nat_Onat___Groups_Ozero, fact_36_poly__shift__id, fact_173_measure__induct, fact_166_Suc__lessD, fact_112_algebraic__int_Oinducts, fact_70_poly__1, fact_183_dvd__pos__nat, fact_214_inverse__positive__iff__positive, fact_26__092_060open_062_092_060And_062thesisa_O_A_I_092_060And_062p_O_A_092_060lbrakk_062_092_060And_062i_O_Acoeff_Ap_Ai_A_092_060in_062_A_092_060int_062_059_Ap_A_092_060noteq_062_A0_059_Apoly_Ap_Ax_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesisa_J_A_092_060Longrightarrow_062_Athesisa_092_060close_062, fact_152_strict__inc__induct, tcon_Real_Oreal___Rings_Oidom, fact_176_infinite__descent, fact_209_gcd__nat_Oasym, fact_229_nonzero__inverse__eq__imp__eq, fact_143_gr0I, fact_69_degree__1, tcon_Nat_Onat___Rings_Ocomm__semiring__0, fact_72_order__1__eq__0, fact_116_exists__least__lemma, fact_33_p_I1_J, tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_8, fact_21_finite, tcon_Real_Oreal___Nat_Osemiring__char__0_13, tcon_Polynomial_Opoly___Groups_Ocomm__monoid__mult_23, fact_91_dvd__refl, fact_135_Suc__less__eq, fact_193_nat__induct__non__zero, tcon_Real_Oreal___Rings_Oidom__divide, fact_106_diff__induct, fact_186_zero__less__one, fact_74_one__reorient, fact_235_less__imp__inverse__less__neg, fact_142_gr__zeroI, fact_2__092_060open_062pderiv_Ap_A_092_060noteq_062_A0_092_060close_062, fact_249_pseudo__mod_I2_J, tcon_Real_Oreal___Rings_Ocomm__semiring__0_11, fact_39_coeff__0__reflect__poly, fact_181_lift__Suc__mono__less__iff, fact_9_rsquarefree__roots, tcon_Real_Oreal___Rings_Oring__no__zero__divisors, fact_141_not__less__zero, fact_110_old_Onat_Odistinct_I2_J, fact_169_nat__neq__iff, tcon_Polynomial_Opoly___Rings_Olinordered__idom_24, tcon_Real_Oreal___Groups_Ocomm__monoid__mult_9, fact_50_coeff__inject, fact_233_less__imp__inverse__less, fact_207_gcd__nat_Otrans, fact_28_poly__shift__0, fact_239_positive__imp__inverse__positive, fact_57_order__0I, fact_54_algebraic__altdef, fact_12_fun__cong__unused__0, tcon_Polynomial_Opoly___Rings_Osemiring__no__zero__divisors_20, fact_250_poly__pinfty__gt__lc, fact_125_algebraic__int__root, tcon_Polynomial_Opoly___Orderings_Oorder_32, fact_203_gcd__nat_Ostrict__trans, fact_215_inverse__negative__iff__negative, fact_8_poly__all__0__iff__0, fact_131_less__nat__zero__code, fact_170_less__not__refl, fact_147_gr__implies__not0, fact_43_divide__poly__main_Osimps_I2_J, tcon_Polynomial_Opoly___Rings_Oalgebraic__semidom_22, fact_87_nat__dvd__1__iff__1, fact_56_poly__0__coeff__0, fact_124_algebraic__int__inverse, fact_68_reflect__poly__1, fact_49_poly__eq__iff, fact_240_one__less__inverse, fact_100_old_Onat_Oinducts, fact_191_Ex__less__Suc2, tcon_Polynomial_Opoly___Groups_Ozero_35, fact_236_inverse__negative__imp__negative, fact_230_nonzero__inverse__inverse__eq, fact_149_bot__nat__0_Oextremum__strict, fact_196_gcd__nat_Onot__eq__order__implies__strict, fact_227_field__class_Ofield__inverse__zero, fact_194_less__degree__imp, fact_126_gcd__nat_Oextremum__uniqueI, fact_95_dvd__0__left, fact_168_Rats__inverse, tcon_Nat_Onat___Rings_Ocomm__semiring__1, fact_37_coeff__0, tcon_Real_Oreal___Groups_Ozero_16, fact_30_leading__coeff__0__iff, fact_167_Nat_OlessE, tcon_Polynomial_Opoly___Rings_Ozero__less__one_27, fact_224_poly__IVT__pos, fact_45_Collect__mem__eq, fact_201_gcd__nat_Ostrict__trans2, fact_180_linorder__neqE__linordered__idom, tcon_Real_Oreal___Fields_Olinordered__field, fact_195_coeff__eq__0, fact_83_One__nat__def, tcon_Real_Oreal___Rings_Olinordered__idom, fact_61_Ints__0, fact_248_field__lbound__gt__zero, fact_60_Rats__0, fact_199_gcd__nat_Ostrict__iff__order, fact_134_Suc__mono, fact_155_Suc__less__SucD, fact_76_Rats__1, fact_108_nat_OdiscI, fact_190_gr0__conv__Suc, fact_105_zero__induct, fact_144_not__gr0, fact_78_not__dvd__pderiv, tcon_Polynomial_Opoly___Rings_Osemiring__1__no__zero__divisors_18, fact_53_algebraic__def, fact_52_algebraicI, fact_192_nat__dvd__not__less, fact_29_order__root, tcon_Polynomial_Opoly___Rings_Ocomm__ring__1_31, fact_96_dvd__unit__imp__unit, fact_41_degree__reflect__poly__eq, fact_188_gr0__implies__Suc, tcon_fun___Orderings_Oorder, tcon_Polynomial_Opoly___Rings_Olinordered__nonzero__semiring_19, fact_107_nat__induct, fact_139_zero__less__iff__neq__zero, fact_205_gcd__nat_Oirrefl, fact_73_lead__coeff__1, fact_102_Zero__not__Suc, fact_162_less__SucI, fact_241_one__less__inverse__iff, fact_109_old_Onat_Odistinct_I1_J, fact_14_n__def, tcon_Nat_Onat___Orderings_Oorder_1, fact_93_Suc__inject, fact_171_less__not__refl2, fact_104_Suc__neq__Zero, tcon_Real_Oreal___Rings_Ozero__neq__one_14, fact_206_gcd__nat_Oeq__iff, fact_228_inverse__zero__imp__zero, fact_242_less__numeral__extra_I1_J, fact_92_n__not__Suc__n, tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, fact_55_zero__poly_Orep__eq, fact_10_not__finite__existsD, fact_189_All__less__Suc2, fact_159_not__less__eq, fact_31_divide__poly__main__0, fact_148_infinite__descent0, fact_15_assms_I2_J, fact_156_less__antisym, fact_1_p_I2_J, fact_243_finite__M__bounded__by__nat, tcon_Set_Oset___Finite__Set_Ofinite_2, fact_151_not__less__less__Suc__eq, fact_88_dvd__1__iff__1, fact_11_pigeonhole__infinite__rel, fact_98_one__dvd, fact_117_dependent__nat__choice, tcon_Real_Oreal___Rings_Ozero__less__one_12, fact_19_pderiv__eq__0__iff, fact_172_less__not__refl3, fact_62_rsquarefree__def, fact_18_pderiv__0, fact_35_synthetic__div__0, tcon_Real_Oreal___Orderings_Oorder_15, fact_220_inverse__eq__1__iff, tcon_Nat_Onat___Nat_Osemiring__char__0, fact_84_not__is__unit__0, tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_7, fact_158_All__less__Suc, tcon_Real_Oreal___Fields_Ofield, fact_225_poly__IVT__neg, fact_232_inverse__less__imp__less, fact_212_gcd__nat_Oextremum__strict, fact_46_Collect__cong, fact_71_pderiv__1, fact_210_coeff__poly__cutoff, tcon_Polynomial_Opoly___Rings_Ozero__neq__one_29, fact_222_finite__psubset__induct, fact_231_nonzero__imp__inverse__nonzero, fact_6_finite__Collect__disjI, fact_44_mem__Collect__eq, fact_150_infinite__descent0__measure, tcon_Real_Oreal___Fields_Odivision__ring
% 50.66/32.00 % No. of inferences in proof: 634
% 50.66/32.00 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 31639 ms resp. 30772 ms w/o parsing
% 51.75/32.14 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 51.75/32.15 % [INFO] Killing All external provers ...
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