%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : ITP107^1 : TPTP v9.3.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% Computer : n019.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 08:06:47 AM UTC 2026
% Result : Theorem 53.28s 19.74s
% Output : Refutation 54.29s
% Verified :
% SZS Type : Refutation
% Derivation depth : 18
% Number of leaves : 317
% Syntax : Number of formulae : 782 ( 270 unt; 0 typ; 3 def)
% Number of atoms : 2479 ( 865 equ; 6 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 8146 ( 337 ~; 111 |; 124 &;6502 @)
% ( 0 <=>;1072 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 49 ( 49 >; 0 *; 0 +; 0 <<)
% Number of symbols : 78 ( 74 usr; 17 con; 0-3 aty)
% Number of variables : 1711 ( 132 ^;1504 !; 72 ?;1711 :)
% ( 0 !>; 0 ?*; 0 @-; 3 @+)
% Comments :
%------------------------------------------------------------------------------
thf(partia539144763t_unit_type,type,
partia539144763t_unit: $tType ).
thf(partia1993116613t_unit_type,type,
partia1993116613t_unit: $tType ).
thf(partia1833973666xt_a_b_type,type,
partia1833973666xt_a_b: $tType ).
thf(partia96731725t_unit_type,type,
partia96731725t_unit: $tType ).
thf(set_se1596668135od_a_a_type,type,
set_se1596668135od_a_a: $tType ).
thf(set_Product_prod_a_a_type,type,
set_Product_prod_a_a: $tType ).
thf(product_prod_a_a_type,type,
product_prod_a_a: $tType ).
thf(set_set_a_type,type,
set_set_a: $tType ).
thf(set_a_type,type,
set_a: $tType ).
thf(nat_type,type,
nat: $tType ).
thf(int_type,type,
int: $tType ).
thf(a_type,type,
a: $tType ).
thf(a_RCOSETS_a_b_decl,type,
a_RCOSETS_a_b: partia1833973666xt_a_b > set_a > set_set_a ).
thf(a_l_coset_a_b_decl,type,
a_l_coset_a_b: partia1833973666xt_a_b > a > set_a > set_a ).
thf(a_r_coset_a_b_decl,type,
a_r_coset_a_b: partia1833973666xt_a_b > set_a > a > set_a ).
thf(additi2104487374up_a_b_decl,type,
additi2104487374up_a_b: set_a > partia1833973666xt_a_b > $o ).
thf(set_add_a_b_decl,type,
set_add_a_b: partia1833973666xt_a_b > set_a > set_a > set_a ).
thf(equiva2136214035t_unit_decl,type,
equiva2136214035t_unit: partia1993116613t_unit > $o ).
thf(partia206007992t_unit_decl,type,
partia206007992t_unit: partia1993116613t_unit > set_Product_prod_a_a ).
thf(partia616129906t_unit_decl,type,
partia616129906t_unit: partia539144763t_unit > set_se1596668135od_a_a ).
thf(partia1955795460t_unit_decl,type,
partia1955795460t_unit: partia96731725t_unit > set_a ).
thf(partia1066395285xt_a_b_decl,type,
partia1066395285xt_a_b: partia1833973666xt_a_b > set_a ).
thf(units_1734235042t_unit_decl,type,
units_1734235042t_unit: partia539144763t_unit > set_se1596668135od_a_a ).
thf(units_a_Product_unit_decl,type,
units_a_Product_unit: partia96731725t_unit > set_a ).
thf(units_a_ring_ext_a_b_decl,type,
units_a_ring_ext_a_b: partia1833973666xt_a_b > set_a ).
thf(group_52730354t_unit_decl,type,
group_52730354t_unit: partia539144763t_unit > $o ).
thf(group_a_Product_unit_decl,type,
group_a_Product_unit: partia96731725t_unit > $o ).
thf(group_a_ring_ext_a_b_decl,type,
group_a_ring_ext_a_b: partia1833973666xt_a_b > $o ).
thf(monoid1242292471t_unit_decl,type,
monoid1242292471t_unit: partia539144763t_unit > $o ).
thf(monoid1447254976t_unit_decl,type,
monoid1447254976t_unit: partia96731725t_unit > $o ).
thf(monoid1022105667xt_a_b_decl,type,
monoid1022105667xt_a_b: partia1833973666xt_a_b > $o ).
thf(mult_s977248631t_unit_decl,type,
mult_s977248631t_unit: partia539144763t_unit > set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(mult_a_Product_unit_decl,type,
mult_a_Product_unit: partia96731725t_unit > a > a > a ).
thf(mult_a_ring_ext_a_b_decl,type,
mult_a_ring_ext_a_b: partia1833973666xt_a_b > a > a > a ).
thf(one_se1488596747t_unit_decl,type,
one_se1488596747t_unit: partia539144763t_unit > set_Product_prod_a_a ).
thf(one_a_Product_unit_decl,type,
one_a_Product_unit: partia96731725t_unit > a ).
thf(one_a_ring_ext_a_b_decl,type,
one_a_ring_ext_a_b: partia1833973666xt_a_b > a ).
thf(units_390215097t_unit_decl,type,
units_390215097t_unit: partia539144763t_unit > partia539144763t_unit ).
thf(units_1411277569xt_a_b_decl,type,
units_1411277569xt_a_b: partia1833973666xt_a_b > partia96731725t_unit ).
thf(minus_minus_set_a_decl,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(locali1170246543t_unit_decl,type,
locali1170246543t_unit: partia1993116613t_unit > a > a > set_Product_prod_a_a ).
thf(locali1648887798ac_a_b_decl,type,
locali1648887798ac_a_b: partia1833973666xt_a_b > set_a > $o ).
thf(locali1944243560ac_a_b_decl,type,
locali1944243560ac_a_b: partia1833973666xt_a_b > set_a > partia539144763t_unit ).
thf(locali807230110ng_a_b_decl,type,
locali807230110ng_a_b: partia1833973666xt_a_b > set_a > $o ).
thf(locali880295127ng_a_b_decl,type,
locali880295127ng_a_b: partia1833973666xt_a_b > set_a > $o ).
thf(locali1538946157t_unit_decl,type,
locali1538946157t_unit: partia1993116613t_unit > set_se1596668135od_a_a ).
thf(locali1488796916t_unit_decl,type,
locali1488796916t_unit: partia539144763t_unit > set_se1596668135od_a_a > $o ).
thf(locali425460998xt_a_b_decl,type,
locali425460998xt_a_b: partia1833973666xt_a_b > set_a > $o ).
thf(bot_bo2131659635od_a_a_decl,type,
bot_bo2131659635od_a_a: set_Product_prod_a_a ).
thf(bot_bo1424703315od_a_a_decl,type,
bot_bo1424703315od_a_a: set_se1596668135od_a_a ).
thf(bot_bot_set_a_decl,type,
bot_bot_set_a: set_a ).
thf(ord_le318720350_set_a_decl,type,
ord_le318720350_set_a: set_set_a > set_set_a > $o ).
thf(ord_less_eq_set_a_decl,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(product_Pair_a_a_decl,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(abelian_monoid_a_b_decl,type,
abelian_monoid_a_b: partia1833973666xt_a_b > $o ).
thf(add_pow_a_b_int_decl,type,
add_pow_a_b_int: partia1833973666xt_a_b > int > a > a ).
thf(add_pow_a_b_nat_decl,type,
add_pow_a_b_nat: partia1833973666xt_a_b > nat > a > a ).
thf(field_a_b_decl,type,
field_a_b: partia1833973666xt_a_b > $o ).
thf(ring_a_b_decl,type,
ring_a_b: partia1833973666xt_a_b > $o ).
thf(zero_a_b_decl,type,
zero_a_b: partia1833973666xt_a_b > a ).
thf(semiring_a_b_decl,type,
semiring_a_b: partia1833973666xt_a_b > $o ).
thf(collec645855634od_a_a_decl,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(collec183727474od_a_a_decl,type,
collec183727474od_a_a: ( set_Product_prod_a_a > $o ) > set_se1596668135od_a_a ).
thf(collect_set_a_decl,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(collect_a_decl,type,
collect_a: ( a > $o ) > set_a ).
thf(pow_a_decl,type,
pow_a: set_a > set_set_a ).
thf(insert1116662519od_a_a_decl,type,
insert1116662519od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(insert_set_a_decl,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(insert_a_decl,type,
insert_a: a > set_a > set_a ).
thf(member449909584od_a_a_decl,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(member1838126896od_a_a_decl,type,
member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o ).
thf(member_set_a_decl,type,
member_set_a: set_a > set_set_a > $o ).
thf(member_a_decl,type,
member_a: a > set_a > $o ).
thf(r_decl,type,
r: partia1833973666xt_a_b ).
thf(s_decl,type,
s: set_a ).
thf(r2_decl,type,
r2: a ).
thf(r3_decl,type,
r3: a ).
thf(rel_decl,type,
rel: partia1993116613t_unit ).
thf(s2_decl,type,
s2: a ).
thf(s3_decl,type,
s3: a ).
thf(sk1_decl,type,
sk1: a > partia96731725t_unit > a ).
thf(sk2_decl,type,
sk2: a > partia96731725t_unit > a ).
thf(sk107_decl,type,
sk107: a ).
thf(sk111_decl,type,
sk111: a ).
thf(sk112_decl,type,
sk112: a ).
thf(sk113_decl,type,
sk113: a ).
thf(sk1_def,definition,
( sk1
= ( ^ [A: a,B: partia96731725t_unit] :
@+[C: a] :
( ( member_a @ C @ ( partia1955795460t_unit @ B ) )
& ( ( mult_a_Product_unit @ B @ C @ A )
= ( one_a_Product_unit @ B ) ) ) ) ) ).
thf(sk2_def,definition,
( sk2
= ( ^ [A: a,B: partia96731725t_unit] :
@+[C: a] :
( ( member_a @ C @ ( partia1955795460t_unit @ B ) )
& ( ( mult_a_Product_unit @ B @ A @ C )
= ( one_a_Product_unit @ B ) ) ) ) ) ).
thf(sk107_def,definition,
( sk107
= ( @+[A: a] :
~ ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ? [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ B @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
thf(299,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ s,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_12_one__closed) ).
thf(1374,plain,
member_a @ ( one_a_ring_ext_a_b @ r ) @ s,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).
thf(3,axiom,
! [A: a,B: a > $o] :
( ( member_a @ A @ ( collect_a @ B ) )
= ( B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_48_mem__Collect__eq) ).
thf(318,plain,
! [A: a,B: a > $o] :
( ( member_a @ A @ ( collect_a @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(319,plain,
! [B: a > $o,A: a] :
( ( member_a @ A @ ( collect_a @ B ) )
= ( B @ A ) ),
inference(cnf,[status(esa)],[318]) ).
thf(320,plain,
! [B: a > $o,A: a] :
( ( member_a @ A @ ( collect_a @ B ) )
= ( B @ A ) ),
inference(lifteq,[status(thm)],[319]) ).
thf(11,axiom,
! [A: set_se1596668135od_a_a,B: set_Product_prod_a_a] :
( ( A = bot_bo1424703315od_a_a )
=> ~ ( member1838126896od_a_a @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_221_equals0D) ).
thf(353,plain,
! [A: set_se1596668135od_a_a,B: set_Product_prod_a_a] :
( ( A = bot_bo1424703315od_a_a )
=> ~ ( member1838126896od_a_a @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(354,plain,
! [A: set_se1596668135od_a_a] :
( ( A = bot_bo1424703315od_a_a )
=> ~ ? [B: set_Product_prod_a_a] : ( member1838126896od_a_a @ B @ A ) ),
inference(miniscope,[status(thm)],[353]) ).
thf(355,plain,
! [B: set_Product_prod_a_a,A: set_se1596668135od_a_a] :
( ( A != bot_bo1424703315od_a_a )
| ~ ( member1838126896od_a_a @ B @ A ) ),
inference(cnf,[status(esa)],[354]) ).
thf(356,plain,
! [B: set_Product_prod_a_a,A: set_se1596668135od_a_a] :
( ( A != bot_bo1424703315od_a_a )
| ~ ( member1838126896od_a_a @ B @ A ) ),
inference(lifteq,[status(thm)],[355]) ).
thf(357,plain,
! [A: set_Product_prod_a_a] :
~ ( member1838126896od_a_a @ A @ bot_bo1424703315od_a_a ),
inference(simp,[status(thm)],[356]) ).
thf(1450,plain,
! [C: set_Product_prod_a_a,B: a > $o,A: a] :
( ~ ( member_a @ A @ ( collect_a @ B ) )
| ( ( B @ A )
!= ( member1838126896od_a_a @ C @ bot_bo1424703315od_a_a ) ) ),
inference(paramod_ordered,[status(thm)],[320,357]) ).
thf(1464,plain,
! [B: a > set_Product_prod_a_a,A: a] :
~ ( member_a @ A
@ ( collect_a
@ ^ [C: a] : ( member1838126896od_a_a @ ( B @ C ) @ bot_bo1424703315od_a_a ) ) ),
inference(pre_uni,[status(thm)],[1450:[bind(A,$thf( A )),bind(B,$thf( ^ [E: a] : ( member1838126896od_a_a @ ( D @ E ) @ bot_bo1424703315od_a_a ) )),bind(C,$thf( D @ A ))]]) ).
thf(1472,plain,
! [B: a > set_Product_prod_a_a,A: a] :
~ ( member_a @ A
@ ( collect_a
@ ^ [C: a] : ( member1838126896od_a_a @ ( B @ C ) @ bot_bo1424703315od_a_a ) ) ),
inference(simp,[status(thm)],[1464]) ).
thf(2119,plain,
! [A: a] :
~ ( member_a @ A
@ ( collect_a
@ ^ [B: a] : $false ) ),
inference(rewrite,[status(thm)],[1472,357]) ).
thf(2140,plain,
! [A: a] :
( ~ $true
| ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ s )
!= ( member_a @ A
@ ( collect_a
@ ^ [B: a] : $false ) ) ) ),
inference(paramod_ordered,[status(thm)],[1374,2119]) ).
thf(2141,plain,
! [A: a] :
( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ s )
!= ( member_a @ A
@ ( collect_a
@ ^ [B: a] : $false ) ) ),
inference(simp,[status(thm)],[2140]) ).
thf(2149,plain,
! [A: a] :
( ( ( one_a_ring_ext_a_b @ r )
!= A )
| ( ( collect_a
@ ^ [B: a] : $false )
!= s ) ),
inference(simp,[status(thm)],[2141]) ).
thf(2163,plain,
( ( collect_a
@ ^ [A: a] : $false )
!= s ),
inference(simp,[status(thm)],[2149]) ).
thf(287,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a] :
( ( group_52730354t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ? [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
& ( ( mult_s977248631t_unit @ A @ B @ C )
= ( one_se1488596747t_unit @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_173_group_Or__inv__ex) ).
thf(1322,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a] :
( ( group_52730354t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ? [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
& ( ( mult_s977248631t_unit @ A @ B @ C )
= ( one_se1488596747t_unit @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).
thf(141,axiom,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali880295127ng_a_b @ A @ B )
=> ( locali425460998xt_a_b @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_24_mult__submonoid__of__rng_Oaxioms_I2_J) ).
thf(809,plain,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali880295127ng_a_b @ A @ B )
=> ( locali425460998xt_a_b @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(307,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ! [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ B @ C )
= C ) )
=> ( B
= ( one_se1488596747t_unit @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_136_monoid_Oone__unique) ).
thf(1411,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ! [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ B @ C )
= C ) )
=> ( B
= ( one_se1488596747t_unit @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).
thf(89,axiom,
! [A: set_a > $o,B: set_a > $o] :
( ! [C: set_a] :
( ( A @ C )
=> ( B @ C ) )
=> ( ord_le318720350_set_a @ ( collect_set_a @ A ) @ ( collect_set_a @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_258_Collect__mono) ).
thf(629,plain,
! [A: set_a > $o,B: set_a > $o] :
( ! [C: set_a] :
( ( A @ C )
=> ( B @ C ) )
=> ( ord_le318720350_set_a @ ( collect_set_a @ A ) @ ( collect_set_a @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(261,axiom,
ord_less_eq_set_a @ s @ ( partia1066395285xt_a_b @ r ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_140_subset) ).
thf(1237,plain,
ord_less_eq_set_a @ s @ ( partia1066395285xt_a_b @ r ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).
thf(6,axiom,
! [A: set_a,B: set_set_a,C: set_a] :
( ( ~ ( member_set_a @ A @ B )
=> ( A = C ) )
=> ( member_set_a @ A @ ( insert_set_a @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_290_insertCI) ).
thf(331,plain,
! [A: set_a,B: set_set_a,C: set_a] :
( ( ~ ( member_set_a @ A @ B )
=> ( A = C ) )
=> ( member_set_a @ A @ ( insert_set_a @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(332,plain,
! [C: set_a,B: set_set_a,A: set_a] :
( ( ~ ( member_set_a @ A @ B )
| ( member_set_a @ A @ ( insert_set_a @ C @ B ) ) )
& ( ( A != C )
| ( member_set_a @ A @ ( insert_set_a @ C @ B ) ) ) ),
inference(cnf,[status(esa)],[331]) ).
thf(333,plain,
! [C: set_a,B: set_set_a,A: set_a] :
( ~ ( member_set_a @ A @ B )
| ( member_set_a @ A @ ( insert_set_a @ C @ B ) ) ),
inference(cnfConj,[status(thm)],[332]) ).
thf(39,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a] :
( ( insert1116662519od_a_a @ A @ B )
!= bot_bo2131659635od_a_a ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_348_insert__not__empty) ).
thf(465,plain,
! [A: product_prod_a_a,B: set_Product_prod_a_a] :
( ( insert1116662519od_a_a @ A @ B )
!= bot_bo2131659635od_a_a ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(466,plain,
~ ? [A: product_prod_a_a,B: set_Product_prod_a_a] :
( ( insert1116662519od_a_a @ A @ B )
= bot_bo2131659635od_a_a ),
inference(miniscope,[status(thm)],[465]) ).
thf(467,plain,
! [B: set_Product_prod_a_a,A: product_prod_a_a] :
( ( insert1116662519od_a_a @ A @ B )
!= bot_bo2131659635od_a_a ),
inference(cnf,[status(esa)],[466]) ).
thf(468,plain,
! [B: set_Product_prod_a_a,A: product_prod_a_a] :
( ( insert1116662519od_a_a @ A @ B )
!= bot_bo2131659635od_a_a ),
inference(lifteq,[status(thm)],[467]) ).
thf(64,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_347_singleton__inject) ).
thf(555,plain,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(280,axiom,
! [A: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ B @ A ) @ ( partia1066395285xt_a_b @ r ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_213_a__l__coset__subset__G) ).
thf(1295,plain,
! [A: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ B @ A ) @ ( partia1066395285xt_a_b @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).
thf(1545,plain,
! [B: a > $o,A: a] :
( ( member_a @ A @ ( collect_a @ B ) )
| ( ( B @ A )
!= ( member_a @ ( one_a_ring_ext_a_b @ r ) @ s ) ) ),
inference(paramod_ordered,[status(thm)],[320,1374]) ).
thf(1549,plain,
( member_a @ ( one_a_ring_ext_a_b @ r )
@ ( collect_a
@ ^ [A: a] : ( member_a @ A @ s ) ) ),
inference(pre_uni,[status(thm)],[1545:[bind(A,$thf( one_a_ring_ext_a_b @ r )),bind(B,$thf( ^ [C: a] : ( member_a @ C @ s ) ))]]) ).
thf(43,axiom,
! [A: partia96731725t_unit,B: a,C: a] :
( ( group_a_Product_unit @ A )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ( ( ( mult_a_Product_unit @ A @ B @ C )
= B )
= ( C
= ( one_a_Product_unit @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_169_group_Ol__cancel__one) ).
thf(481,plain,
! [A: partia96731725t_unit,B: a,C: a] :
( ( group_a_Product_unit @ A )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ( ( ( mult_a_Product_unit @ A @ B @ C )
= B )
= ( C
= ( one_a_Product_unit @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(105,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_121_monoid_Om__closed) ).
thf(683,plain,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(210,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_42_prod_Oinject) ).
thf(1068,plain,
! [A: a,B: a,C: a,D: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).
thf(190,axiom,
! [A: partia539144763t_unit,B: set_se1596668135od_a_a] :
( ( locali1488796916t_unit @ A @ B )
=> ( monoid1242292471t_unit @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_21_submonoid_Oaxioms_I1_J) ).
thf(994,plain,
! [A: partia539144763t_unit,B: set_se1596668135od_a_a] :
( ( locali1488796916t_unit @ A @ B )
=> ( monoid1242292471t_unit @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(303,axiom,
( ! [A: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ? [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ B @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) )
=> ( group_a_ring_ext_a_b @ r ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_143_group__l__invI) ).
thf(1399,plain,
( ! [A: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ? [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ B @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) )
=> ( group_a_ring_ext_a_b @ r ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).
thf(1400,plain,
! [A: a] :
( ( ( member_a @ sk107 @ ( partia1066395285xt_a_b @ r ) )
| ( group_a_ring_ext_a_b @ r ) )
& ( ~ ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
| ( ( mult_a_ring_ext_a_b @ r @ A @ sk107 )
!= ( one_a_ring_ext_a_b @ r ) )
| ( group_a_ring_ext_a_b @ r ) ) ),
inference(cnf,[status(esa)],[1399]) ).
thf(1401,plain,
( ( member_a @ sk107 @ ( partia1066395285xt_a_b @ r ) )
| ( group_a_ring_ext_a_b @ r ) ),
inference(cnfConj,[status(thm)],[1400]) ).
thf(309,axiom,
locali880295127ng_a_b @ r @ s,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_6_mult__submonoid__of__rng__axioms) ).
thf(1422,plain,
locali880295127ng_a_b @ r @ s,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).
thf(231,axiom,
monoid1022105667xt_a_b @ r,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_45_monoid__axioms) ).
thf(1139,plain,
monoid1022105667xt_a_b @ r,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).
thf(1454,plain,
! [B: a > $o,A: a] :
( ( member_a @ A @ ( collect_a @ B ) )
| ( ( monoid1022105667xt_a_b @ r )
!= ( B @ A ) ) ),
inference(paramod_ordered,[status(thm)],[1139,320]) ).
thf(1461,plain,
! [A: a] :
( member_a @ A
@ ( collect_a
@ ^ [B: a] : ( monoid1022105667xt_a_b @ r ) ) ),
inference(pre_uni,[status(thm)],[1454:[bind(A,$thf( A )),bind(B,$thf( ^ [C: a] : ( monoid1022105667xt_a_b @ r ) ))]]) ).
thf(2078,plain,
! [A: a] :
( member_a @ A
@ ( collect_a
@ ^ [B: a] : $true ) ),
inference(rewrite,[status(thm)],[1461,1139]) ).
thf(63,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_198_empty__iff) ).
thf(552,plain,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(553,plain,
~ ? [A: a] : ( member_a @ A @ bot_bot_set_a ),
inference(miniscope,[status(thm)],[552]) ).
thf(554,plain,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ),
inference(cnf,[status(esa)],[553]) ).
thf(2092,plain,
! [B: a,A: a] :
( ~ $true
| ( ( member_a @ A
@ ( collect_a
@ ^ [C: a] : $true ) )
!= ( member_a @ B @ bot_bot_set_a ) ) ),
inference(paramod_ordered,[status(thm)],[2078,554]) ).
thf(2093,plain,
! [B: a,A: a] :
( ( member_a @ A
@ ( collect_a
@ ^ [C: a] : $true ) )
!= ( member_a @ B @ bot_bot_set_a ) ),
inference(simp,[status(thm)],[2092]) ).
thf(2105,plain,
! [B: a,A: a] :
( ( A != B )
| ( ( collect_a
@ ^ [C: a] : $true )
!= bot_bot_set_a ) ),
inference(simp,[status(thm)],[2093]) ).
thf(2114,plain,
( ( collect_a
@ ^ [A: a] : $true )
!= bot_bot_set_a ),
inference(simp,[status(thm)],[2105]) ).
thf(97,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le318720350_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_236_in__mono) ).
thf(655,plain,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le318720350_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(314,axiom,
locali807230110ng_a_b @ r @ s,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_5_mult__submonoid__of__crng__axioms) ).
thf(1437,plain,
locali807230110ng_a_b @ r @ s,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).
thf(155,axiom,
! [A: partia539144763t_unit] :
( ( group_52730354t_unit @ A )
=> ( monoid1242292471t_unit @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_152_group_Ois__monoid) ).
thf(861,plain,
! [A: partia539144763t_unit] :
( ( group_52730354t_unit @ A )
=> ( monoid1242292471t_unit @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(252,axiom,
! [A: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ? [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ B @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_58_Units__l__inv__ex) ).
thf(1205,plain,
! [A: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ? [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ B @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).
thf(38,axiom,
! [A: partia96731725t_unit] :
( ! [B: a] :
( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ! [C: a] :
( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ( member_a @ ( mult_a_Product_unit @ A @ B @ C ) @ ( partia1955795460t_unit @ A ) ) ) )
=> ( ( member_a @ ( one_a_Product_unit @ A ) @ ( partia1955795460t_unit @ A ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ! [C: a] :
( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ! [D: a] :
( ( member_a @ D @ ( partia1955795460t_unit @ A ) )
=> ( ( mult_a_Product_unit @ A @ ( mult_a_Product_unit @ A @ B @ C ) @ D )
= ( mult_a_Product_unit @ A @ B @ ( mult_a_Product_unit @ A @ C @ D ) ) ) ) ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ( ( mult_a_Product_unit @ A @ ( one_a_Product_unit @ A ) @ B )
= B ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
& ( ( mult_a_Product_unit @ A @ C @ B )
= ( one_a_Product_unit @ A ) ) ) )
=> ( group_a_Product_unit @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_181_groupI) ).
thf(449,plain,
! [A: partia96731725t_unit] :
( ! [B: a] :
( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ! [C: a] :
( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ( member_a @ ( mult_a_Product_unit @ A @ B @ C ) @ ( partia1955795460t_unit @ A ) ) ) )
=> ( ( member_a @ ( one_a_Product_unit @ A ) @ ( partia1955795460t_unit @ A ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ! [C: a] :
( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ! [D: a] :
( ( member_a @ D @ ( partia1955795460t_unit @ A ) )
=> ( ( mult_a_Product_unit @ A @ ( mult_a_Product_unit @ A @ B @ C ) @ D )
= ( mult_a_Product_unit @ A @ B @ ( mult_a_Product_unit @ A @ C @ D ) ) ) ) ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ( ( mult_a_Product_unit @ A @ ( one_a_Product_unit @ A ) @ B )
= B ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
& ( ( mult_a_Product_unit @ A @ C @ B )
= ( one_a_Product_unit @ A ) ) ) )
=> ( group_a_Product_unit @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(213,axiom,
! [A: a,B: set_a,C: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_334_insertI2) ).
thf(1078,plain,
! [A: a,B: set_a,C: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(230,axiom,
( ( mult_a_ring_ext_a_b @ r @ r2 @ r3 )
= ( mult_a_ring_ext_a_b @ r @ r3 @ r2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_0_f3) ).
thf(1137,plain,
( ( mult_a_ring_ext_a_b @ r @ r2 @ r3 )
= ( mult_a_ring_ext_a_b @ r @ r3 @ r2 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).
thf(57,axiom,
! [A: partia1833973666xt_a_b] :
( ( ring_a_b @ A )
=> ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_84_ring_Oring__simprules_I6_J) ).
thf(533,plain,
! [A: partia1833973666xt_a_b] :
( ( ring_a_b @ A )
=> ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(260,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( group_52730354t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( B
= ( mult_s977248631t_unit @ A @ B @ C ) )
= ( C
= ( one_se1488596747t_unit @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_164_group_Ol__cancel__one_H) ).
thf(1233,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( group_52730354t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( B
= ( mult_s977248631t_unit @ A @ B @ C ) )
= ( C
= ( one_se1488596747t_unit @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).
thf(7,axiom,
! [A: partia96731725t_unit,B: a] :
( ( group_a_Product_unit @ A )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
& ( ( mult_a_Product_unit @ A @ B @ C )
= ( one_a_Product_unit @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_172_group_Or__inv__ex) ).
thf(337,plain,
! [A: partia96731725t_unit,B: a] :
( ( group_a_Product_unit @ A )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
& ( ( mult_a_Product_unit @ A @ B @ C )
= ( one_a_Product_unit @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(338,plain,
! [A: partia96731725t_unit] :
( ( group_a_Product_unit @ A )
=> ! [B: a] :
( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
& ( ( mult_a_Product_unit @ A @ B @ C )
= ( one_a_Product_unit @ A ) ) ) ) ),
inference(miniscope,[status(thm)],[337]) ).
thf(339,plain,
! [B: a,A: partia96731725t_unit] :
( ( ~ ( group_a_Product_unit @ A )
| ~ ( member_a @ B @ ( partia1955795460t_unit @ A ) )
| ( member_a @ ( sk2 @ B @ A ) @ ( partia1955795460t_unit @ A ) ) )
& ( ~ ( group_a_Product_unit @ A )
| ~ ( member_a @ B @ ( partia1955795460t_unit @ A ) )
| ( ( mult_a_Product_unit @ A @ B @ ( sk2 @ B @ A ) )
= ( one_a_Product_unit @ A ) ) ) ),
inference(cnf,[status(esa)],[338]) ).
thf(340,plain,
! [B: a,A: partia96731725t_unit] :
( ~ ( group_a_Product_unit @ A )
| ~ ( member_a @ B @ ( partia1955795460t_unit @ A ) )
| ( member_a @ ( sk2 @ B @ A ) @ ( partia1955795460t_unit @ A ) ) ),
inference(cnfConj,[status(thm)],[339]) ).
thf(59,axiom,
! [A: a > $o,B: a > $o] :
( ! [C: a] :
( ( A @ C )
=> ( B @ C ) )
=> ( ord_less_eq_set_a @ ( collect_a @ A ) @ ( collect_a @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_257_Collect__mono) ).
thf(538,plain,
! [A: a > $o,B: a > $o] :
( ! [C: a] :
( ( A @ C )
=> ( B @ C ) )
=> ( ord_less_eq_set_a @ ( collect_a @ A ) @ ( collect_a @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(101,axiom,
( monoid1022105667xt_a_b
= ( ^ [A: partia1833973666xt_a_b] :
( ! [B: a,C: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) )
& ! [B: a,C: a,D: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
= ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) )
& ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) )
& ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
= B ) )
& ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ ( one_a_ring_ext_a_b @ A ) )
= B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_131_Group_Omonoid__def) ).
thf(670,plain,
( monoid1022105667xt_a_b
= ( ^ [A: partia1833973666xt_a_b] :
( ! [B: a,C: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) )
& ! [B: a,C: a,D: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
= ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) )
& ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) )
& ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
= B ) )
& ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ ( one_a_ring_ext_a_b @ A ) )
= B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(269,axiom,
! [A: nat] :
( ( add_pow_a_b_nat @ r @ A @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_146_add_Onat__pow__one) ).
thf(1260,plain,
! [A: nat] :
( ( add_pow_a_b_nat @ r @ A @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).
thf(110,axiom,
( ( (=) @ set_a )
= ( ^ [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_261_set__eq__subset) ).
thf(698,plain,
( ( (=) @ set_a )
= ( ^ [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(167,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
( ( semiring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
= ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_83_semiring_Osemiring__simprules_I8_J) ).
thf(898,plain,
! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
( ( semiring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
= ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(264,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ A )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_189_a__lcos__mult__one) ).
thf(1243,plain,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).
thf(114,axiom,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_338_insertI1) ).
thf(709,plain,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(164,axiom,
! [A: partia1833973666xt_a_b,B: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
& ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= ( one_a_ring_ext_a_b @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_102_monoid_OUnits__r__inv__ex) ).
thf(885,plain,
! [A: partia1833973666xt_a_b,B: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
& ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= ( one_a_ring_ext_a_b @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(96,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= ( one_a_ring_ext_a_b @ A ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ A ) )
=> ( ( member_a @ C @ ( units_a_ring_ext_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ C @ B )
= ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_106_monoid_OUnits__inv__comm) ).
thf(651,plain,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= ( one_a_ring_ext_a_b @ A ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ A ) )
=> ( ( member_a @ C @ ( units_a_ring_ext_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ C @ B )
= ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(2126,plain,
! [A: a] :
( ~ $true
| ( ( member_a @ ( one_a_ring_ext_a_b @ r )
@ ( collect_a
@ ^ [B: a] : ( member_a @ B @ s ) ) )
!= ( member_a @ A
@ ( collect_a
@ ^ [B: a] : $false ) ) ) ),
inference(paramod_ordered,[status(thm)],[1549,2119]) ).
thf(2127,plain,
! [A: a] :
( ( member_a @ ( one_a_ring_ext_a_b @ r )
@ ( collect_a
@ ^ [B: a] : ( member_a @ B @ s ) ) )
!= ( member_a @ A
@ ( collect_a
@ ^ [B: a] : $false ) ) ),
inference(simp,[status(thm)],[2126]) ).
thf(2159,plain,
! [A: a] :
( ( ( one_a_ring_ext_a_b @ r )
!= A )
| ( ( collect_a
@ ^ [B: a] : ( member_a @ B @ s ) )
!= ( collect_a
@ ^ [B: a] : $false ) ) ),
inference(simp,[status(thm)],[2127]) ).
thf(2173,plain,
( ( collect_a
@ ^ [A: a] : ( member_a @ A @ s ) )
!= ( collect_a
@ ^ [A: a] : $false ) ),
inference(simp,[status(thm)],[2159]) ).
thf(120,axiom,
! [A: partia539144763t_unit] :
( ( monoid1242292471t_unit @ A )
=> ( ( partia616129906t_unit @ A )
!= bot_bo1424703315od_a_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_116_monoid_Ocarrier__not__empty) ).
thf(724,plain,
! [A: partia539144763t_unit] :
( ( monoid1242292471t_unit @ A )
=> ( ( partia616129906t_unit @ A )
!= bot_bo1424703315od_a_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(278,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( mult_a_ring_ext_a_b @ r @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_47_m__comm) ).
thf(1287,plain,
! [A: a,B: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( mult_a_ring_ext_a_b @ r @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).
thf(281,axiom,
( ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) )
=> ( ( partia1066395285xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_281_one__zeroD) ).
thf(1298,plain,
( ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) )
=> ( ( partia1066395285xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).
thf(118,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_259_subset__trans) ).
thf(719,plain,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(284,axiom,
! [A: a,B: a,C: a,D: a] :
( ( member_a @ ( zero_a_b @ r ) @ s )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ C @ D ) @ ( partia206007992t_unit @ rel ) )
=> ( ( locali1170246543t_unit @ rel @ A @ B )
= ( locali1170246543t_unit @ rel @ C @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_37_zero__in__mult__submonoid) ).
thf(1313,plain,
! [A: a,B: a,C: a,D: a] :
( ( member_a @ ( zero_a_b @ r ) @ s )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ C @ D ) @ ( partia206007992t_unit @ rel ) )
=> ( ( locali1170246543t_unit @ rel @ A @ B )
= ( locali1170246543t_unit @ rel @ C @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).
thf(60,axiom,
! [A: partia1833973666xt_a_b] :
( ( monoid1022105667xt_a_b @ A )
=> ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_100_monoid_Oone__closed) ).
thf(542,plain,
! [A: partia1833973666xt_a_b] :
( ( monoid1022105667xt_a_b @ A )
=> ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(74,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ A ) )
=> ( ( member_a @ C @ ( units_a_ring_ext_a_b @ A ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( units_a_ring_ext_a_b @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_125_monoid_OUnits__m__closed) ).
thf(587,plain,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ A ) )
=> ( ( member_a @ C @ ( units_a_ring_ext_a_b @ A ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( units_a_ring_ext_a_b @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(85,axiom,
! [A: partia1833973666xt_a_b] :
( ( semiring_a_b @ A )
=> ( abelian_monoid_a_b @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_44_semiring_Oaxioms_I1_J) ).
thf(619,plain,
! [A: partia1833973666xt_a_b] :
( ( semiring_a_b @ A )
=> ( abelian_monoid_a_b @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(172,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_211_subset__antisym) ).
thf(934,plain,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(168,axiom,
! [A: partia1833973666xt_a_b] :
( ! [B: a,C: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) )
=> ( ! [B: a,C: a,D: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
= ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
= B ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ ( one_a_ring_ext_a_b @ A ) )
= B ) )
=> ( monoid1022105667xt_a_b @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_133_Group_Omonoid_Ointro) ).
thf(902,plain,
! [A: partia1833973666xt_a_b] :
( ! [B: a,C: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) )
=> ( ! [B: a,C: a,D: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
= ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
= B ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ ( one_a_ring_ext_a_b @ A ) )
= B ) )
=> ( monoid1022105667xt_a_b @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(153,axiom,
! [A: product_prod_a_a > $o,B: product_prod_a_a] :
( ! [C: a,D: a] : ( A @ ( product_Pair_a_a @ C @ D ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_88_old_Oprod_Oinducts) ).
thf(853,plain,
! [A: product_prod_a_a > $o,B: product_prod_a_a] :
( ! [C: a,D: a] : ( A @ ( product_Pair_a_a @ C @ D ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(301,axiom,
! [A: partia539144763t_unit] :
( ! [B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ A @ B @ C ) @ ( partia616129906t_unit @ A ) ) ) )
=> ( ! [B: set_Product_prod_a_a,C: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ D @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( mult_s977248631t_unit @ A @ B @ C ) @ D )
= ( mult_s977248631t_unit @ A @ B @ ( mult_s977248631t_unit @ A @ C @ D ) ) ) ) ) )
=> ( ( member1838126896od_a_a @ ( one_se1488596747t_unit @ A ) @ ( partia616129906t_unit @ A ) )
=> ( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( one_se1488596747t_unit @ A ) @ B )
= B ) )
=> ( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ B @ ( one_se1488596747t_unit @ A ) )
= B ) )
=> ( monoid1242292471t_unit @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_132_Group_Omonoid_Ointro) ).
thf(1379,plain,
! [A: partia539144763t_unit] :
( ! [B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ A @ B @ C ) @ ( partia616129906t_unit @ A ) ) ) )
=> ( ! [B: set_Product_prod_a_a,C: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ D @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( mult_s977248631t_unit @ A @ B @ C ) @ D )
= ( mult_s977248631t_unit @ A @ B @ ( mult_s977248631t_unit @ A @ C @ D ) ) ) ) ) )
=> ( ( member1838126896od_a_a @ ( one_se1488596747t_unit @ A ) @ ( partia616129906t_unit @ A ) )
=> ( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( one_se1488596747t_unit @ A ) @ B )
= B ) )
=> ( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ B @ ( one_se1488596747t_unit @ A ) )
= B ) )
=> ( monoid1242292471t_unit @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).
thf(218,axiom,
equiva2136214035t_unit @ rel,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_13_equiv__obj__rng__of__frac) ).
thf(1087,plain,
equiva2136214035t_unit @ rel,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(235,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( units_1734235042t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( units_1734235042t_unit @ A ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ A @ B @ C ) @ ( units_1734235042t_unit @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_124_monoid_OUnits__m__closed) ).
thf(1147,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( units_1734235042t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( units_1734235042t_unit @ A ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ A @ B @ C ) @ ( units_1734235042t_unit @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).
thf(30,axiom,
! [A: set_Product_prod_a_a] :
~ ( member1838126896od_a_a @ A @ bot_bo1424703315od_a_a ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_217_emptyE) ).
thf(421,plain,
! [A: set_Product_prod_a_a] :
~ ( member1838126896od_a_a @ A @ bot_bo1424703315od_a_a ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(130,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_249_equalityD2) ).
thf(770,plain,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(1,conjecture,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
= ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).
thf(2,negated_conjecture,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
!= ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(316,plain,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
!= ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(80,axiom,
! [A: set_set_a] : ( ord_le318720350_set_a @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_256_subset__refl) ).
thf(606,plain,
! [A: set_set_a] : ( ord_le318720350_set_a @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(295,axiom,
! [A: partia539144763t_unit] :
( ! [B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ A @ B @ C ) @ ( partia616129906t_unit @ A ) ) ) )
=> ( ( member1838126896od_a_a @ ( one_se1488596747t_unit @ A ) @ ( partia616129906t_unit @ A ) )
=> ( ! [B: set_Product_prod_a_a,C: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ D @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( mult_s977248631t_unit @ A @ B @ C ) @ D )
= ( mult_s977248631t_unit @ A @ B @ ( mult_s977248631t_unit @ A @ C @ D ) ) ) ) ) )
=> ( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( one_se1488596747t_unit @ A ) @ B )
= B ) )
=> ( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ B @ ( one_se1488596747t_unit @ A ) )
= B ) )
=> ( monoid1242292471t_unit @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_128_monoidI) ).
thf(1347,plain,
! [A: partia539144763t_unit] :
( ! [B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ A @ B @ C ) @ ( partia616129906t_unit @ A ) ) ) )
=> ( ( member1838126896od_a_a @ ( one_se1488596747t_unit @ A ) @ ( partia616129906t_unit @ A ) )
=> ( ! [B: set_Product_prod_a_a,C: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ D @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( mult_s977248631t_unit @ A @ B @ C ) @ D )
= ( mult_s977248631t_unit @ A @ B @ ( mult_s977248631t_unit @ A @ C @ D ) ) ) ) ) )
=> ( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( one_se1488596747t_unit @ A ) @ B )
= B ) )
=> ( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ B @ ( one_se1488596747t_unit @ A ) )
= B ) )
=> ( monoid1242292471t_unit @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).
thf(4,axiom,
! [A: partia96731725t_unit,B: a] :
( ( group_a_Product_unit @ A )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
& ( ( mult_a_Product_unit @ A @ C @ B )
= ( one_a_Product_unit @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_175_group_Ol__inv__ex) ).
thf(321,plain,
! [A: partia96731725t_unit,B: a] :
( ( group_a_Product_unit @ A )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
& ( ( mult_a_Product_unit @ A @ C @ B )
= ( one_a_Product_unit @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(220,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia1066395285xt_a_b @ r ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_62_semiring__simprules_I4_J) ).
thf(1089,plain,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia1066395285xt_a_b @ r ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(1613,plain,
! [B: a > $o,A: a] :
( ( member_a @ A @ ( collect_a @ B ) )
| ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) )
!= ( B @ A ) ) ),
inference(paramod_ordered,[status(thm)],[1089,320]) ).
thf(1622,plain,
( member_a @ ( one_a_ring_ext_a_b @ r )
@ ( collect_a
@ ^ [A: a] : ( member_a @ A @ ( partia1066395285xt_a_b @ r ) ) ) ),
inference(pre_uni,[status(thm)],[1613:[bind(A,$thf( one_a_ring_ext_a_b @ r )),bind(B,$thf( ^ [C: a] : ( member_a @ C @ ( partia1066395285xt_a_b @ r ) ) ))]]) ).
thf(2130,plain,
! [A: a] :
( ~ $true
| ( ( member_a @ ( one_a_ring_ext_a_b @ r )
@ ( collect_a
@ ^ [B: a] : ( member_a @ B @ ( partia1066395285xt_a_b @ r ) ) ) )
!= ( member_a @ A
@ ( collect_a
@ ^ [B: a] : $false ) ) ) ),
inference(paramod_ordered,[status(thm)],[1622,2119]) ).
thf(2131,plain,
! [A: a] :
( ( member_a @ ( one_a_ring_ext_a_b @ r )
@ ( collect_a
@ ^ [B: a] : ( member_a @ B @ ( partia1066395285xt_a_b @ r ) ) ) )
!= ( member_a @ A
@ ( collect_a
@ ^ [B: a] : $false ) ) ),
inference(simp,[status(thm)],[2130]) ).
thf(2148,plain,
! [A: a] :
( ( ( one_a_ring_ext_a_b @ r )
!= A )
| ( ( collect_a
@ ^ [B: a] : ( member_a @ B @ ( partia1066395285xt_a_b @ r ) ) )
!= ( collect_a
@ ^ [B: a] : $false ) ) ),
inference(simp,[status(thm)],[2131]) ).
thf(2162,plain,
( ( collect_a
@ ^ [A: a] : ( member_a @ A @ ( partia1066395285xt_a_b @ r ) ) )
!= ( collect_a
@ ^ [A: a] : $false ) ),
inference(simp,[status(thm)],[2148]) ).
thf(2259,plain,
( ( ^ [A: a] : ( member_a @ A @ ( partia1066395285xt_a_b @ r ) ) )
!= ( ^ [A: a] : $false ) ),
inference(simp,[status(thm)],[2162]) ).
thf(1805,plain,
! [A: a] :
( ~ $true
| ( ( member_a @ ( one_a_ring_ext_a_b @ r )
@ ( collect_a
@ ^ [B: a] : ( member_a @ B @ s ) ) )
!= ( member_a @ A @ bot_bot_set_a ) ) ),
inference(paramod_ordered,[status(thm)],[1549,554]) ).
thf(1806,plain,
! [A: a] :
( ( member_a @ ( one_a_ring_ext_a_b @ r )
@ ( collect_a
@ ^ [B: a] : ( member_a @ B @ s ) ) )
!= ( member_a @ A @ bot_bot_set_a ) ),
inference(simp,[status(thm)],[1805]) ).
thf(1821,plain,
! [A: a] :
( ( ( one_a_ring_ext_a_b @ r )
!= A )
| ( ( collect_a
@ ^ [B: a] : ( member_a @ B @ s ) )
!= bot_bot_set_a ) ),
inference(simp,[status(thm)],[1806]) ).
thf(1828,plain,
( ( collect_a
@ ^ [A: a] : ( member_a @ A @ s ) )
!= bot_bot_set_a ),
inference(simp,[status(thm)],[1821]) ).
thf(51,axiom,
! [A: a,B: set_a] :
( ( member_a @ A @ B )
=> ? [C: set_a] :
( ( B
= ( insert_a @ A @ C ) )
& ~ ( member_a @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_313_mk__disjoint__insert) ).
thf(514,plain,
! [A: a,B: set_a] :
( ( member_a @ A @ B )
=> ? [C: set_a] :
( ( B
= ( insert_a @ A @ C ) )
& ~ ( member_a @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(173,axiom,
! [A: partia1833973666xt_a_b,B: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
& ( ( mult_a_ring_ext_a_b @ A @ C @ B )
= ( one_a_ring_ext_a_b @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_104_monoid_OUnits__l__inv__ex) ).
thf(937,plain,
! [A: partia1833973666xt_a_b,B: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
& ( ( mult_a_ring_ext_a_b @ A @ C @ B )
= ( one_a_ring_ext_a_b @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(277,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) )
= ( mult_a_ring_ext_a_b @ r @ B @ ( mult_a_ring_ext_a_b @ r @ A @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_56_m__lcomm) ).
thf(1283,plain,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) )
= ( mult_a_ring_ext_a_b @ r @ B @ ( mult_a_ring_ext_a_b @ r @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).
thf(204,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_235_in__mono) ).
thf(1051,plain,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(308,axiom,
! [A: a,B: a,C: int] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ C @ A ) @ B )
= ( add_pow_a_b_int @ r @ C @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_107_add__pow__ldistr__int) ).
thf(1418,plain,
! [A: a,B: a,C: int] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ C @ A ) @ B )
= ( add_pow_a_b_int @ r @ C @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).
thf(87,axiom,
! [A: a,B: set_a,C: a,D: set_a] :
( ~ ( member_a @ A @ B )
=> ( ~ ( member_a @ C @ D )
=> ( ( ( insert_a @ A @ B )
= ( insert_a @ C @ D ) )
= ( ( ( A = C )
=> ( B = D ) )
& ( ( A != C )
=> ? [E: set_a] :
( ( B
= ( insert_a @ C @ E ) )
& ~ ( member_a @ C @ E )
& ( D
= ( insert_a @ A @ E ) )
& ~ ( member_a @ A @ E ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_318_insert__eq__iff) ).
thf(623,plain,
! [A: a,B: set_a,C: a,D: set_a] :
( ~ ( member_a @ A @ B )
=> ( ~ ( member_a @ C @ D )
=> ( ( ( insert_a @ A @ B )
= ( insert_a @ C @ D ) )
= ( ( ( A = C )
=> ( B = D ) )
& ( ( A != C )
=> ? [E: set_a] :
( ( B
= ( insert_a @ C @ E ) )
& ~ ( member_a @ C @ E )
& ( D
= ( insert_a @ A @ E ) )
& ~ ( member_a @ A @ E ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(44,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ~ ( member449909584od_a_a @ A @ B )
=> ( A = C ) )
=> ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_288_insertCI) ).
thf(485,plain,
! [A: product_prod_a_a,B: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ~ ( member449909584od_a_a @ A @ B )
=> ( A = C ) )
=> ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(145,axiom,
! [A: partia1833973666xt_a_b] :
( ( semiring_a_b @ A )
=> ( member_a @ ( zero_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_72_semiring_Osemiring__simprules_I2_J) ).
thf(834,plain,
! [A: partia1833973666xt_a_b] :
( ( semiring_a_b @ A )
=> ( member_a @ ( zero_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(70,axiom,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali880295127ng_a_b @ A @ B )
=> ( ring_a_b @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_25_mult__submonoid__of__rng_Oaxioms_I1_J) ).
thf(574,plain,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali880295127ng_a_b @ A @ B )
=> ( ring_a_b @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(196,axiom,
! [A: set_Product_prod_a_a] :
( ( ? [B: product_prod_a_a] : ( member449909584od_a_a @ B @ A ) )
= ( A != bot_bo2131659635od_a_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_231_ex__in__conv) ).
thf(1020,plain,
! [A: set_Product_prod_a_a] :
( ( ? [B: product_prod_a_a] : ( member449909584od_a_a @ B @ A ) )
= ( A != bot_bo2131659635od_a_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(10,axiom,
! [A: set_a,B: set_set_a] :
( ( member_set_a @ A @ B )
=> ~ ! [C: set_set_a] :
( ( B
= ( insert_set_a @ A @ C ) )
=> ( member_set_a @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_333_Set_Oset__insert) ).
thf(348,plain,
! [A: set_a,B: set_set_a] :
( ( member_set_a @ A @ B )
=> ~ ! [C: set_set_a] :
( ( B
= ( insert_set_a @ A @ C ) )
=> ( member_set_a @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(181,axiom,
! [A: set_a] :
( ! [B: a] :
~ ( member_a @ B @ A )
=> ( A = bot_bot_set_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_228_equals0I) ).
thf(966,plain,
! [A: set_a] :
( ! [B: a] :
~ ( member_a @ B @ A )
=> ( A = bot_bot_set_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(124,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [C: set_a] :
( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) )
=> ( ord_le318720350_set_a @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_210_subsetI) ).
thf(734,plain,
! [A: set_set_a,B: set_set_a] :
( ! [C: set_a] :
( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) )
=> ( ord_le318720350_set_a @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(317,plain,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) )
!= ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) ) ),
inference(lifteq,[status(thm)],[316]) ).
thf(1444,plain,
( ( ( locali1944243560ac_a_b @ r @ s )
!= ( locali1944243560ac_a_b @ r @ s ) )
| ( ( locali1170246543t_unit @ rel @ r3 @ s3 )
!= ( locali1170246543t_unit @ rel @ r2 @ s2 ) )
| ( ( locali1170246543t_unit @ rel @ r3 @ s3 )
!= ( locali1170246543t_unit @ rel @ r2 @ s2 ) ) ),
inference(simp,[status(thm)],[317]) ).
thf(1445,plain,
( ( locali1170246543t_unit @ rel @ r3 @ s3 )
!= ( locali1170246543t_unit @ rel @ r2 @ s2 ) ),
inference(simp,[status(thm)],[1444]) ).
thf(1446,plain,
( ( rel != rel )
| ( r3 != r2 )
| ( s3 != s2 ) ),
inference(simp,[status(thm)],[1445]) ).
thf(1447,plain,
( ( r3 != r2 )
| ( s3 != s2 ) ),
inference(simp,[status(thm)],[1446]) ).
thf(312,axiom,
monoid1242292471t_unit @ ( locali1944243560ac_a_b @ r @ s ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_10_monoid__rng__of__frac) ).
thf(1433,plain,
monoid1242292471t_unit @ ( locali1944243560ac_a_b @ r @ s ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).
thf(305,axiom,
( ( partia1066395285xt_a_b @ r )
!= bot_bot_set_a ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_144_carrier__not__empty) ).
thf(1405,plain,
( ( partia1066395285xt_a_b @ r )
!= bot_bot_set_a ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).
thf(224,axiom,
! [A: partia539144763t_unit] :
( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ! [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ A @ B @ C ) @ ( partia616129906t_unit @ A ) ) ) )
=> ( ( member1838126896od_a_a @ ( one_se1488596747t_unit @ A ) @ ( partia616129906t_unit @ A ) )
=> ( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ! [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ! [D: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ D @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( mult_s977248631t_unit @ A @ B @ C ) @ D )
= ( mult_s977248631t_unit @ A @ B @ ( mult_s977248631t_unit @ A @ C @ D ) ) ) ) ) )
=> ( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( one_se1488596747t_unit @ A ) @ B )
= B ) )
=> ( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ? [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
& ( ( mult_s977248631t_unit @ A @ C @ B )
= ( one_se1488596747t_unit @ A ) ) ) )
=> ( group_52730354t_unit @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_182_groupI) ).
thf(1104,plain,
! [A: partia539144763t_unit] :
( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ! [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ A @ B @ C ) @ ( partia616129906t_unit @ A ) ) ) )
=> ( ( member1838126896od_a_a @ ( one_se1488596747t_unit @ A ) @ ( partia616129906t_unit @ A ) )
=> ( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ! [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ! [D: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ D @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( mult_s977248631t_unit @ A @ B @ C ) @ D )
= ( mult_s977248631t_unit @ A @ B @ ( mult_s977248631t_unit @ A @ C @ D ) ) ) ) ) )
=> ( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( one_se1488596747t_unit @ A ) @ B )
= B ) )
=> ( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ? [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
& ( ( mult_s977248631t_unit @ A @ C @ B )
= ( one_se1488596747t_unit @ A ) ) ) )
=> ( group_52730354t_unit @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).
thf(131,axiom,
! [A: product_prod_a_a] :
~ ! [B: a,C: a] :
( A
!= ( product_Pair_a_a @ B @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_89_old_Oprod_Oexhaust) ).
thf(774,plain,
! [A: product_prod_a_a] :
~ ! [B: a,C: a] :
( A
!= ( product_Pair_a_a @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(188,axiom,
! [A: set_a,B: set_set_a,C: set_set_a] :
( ( ord_le318720350_set_a @ ( insert_set_a @ A @ B ) @ C )
= ( ( member_set_a @ A @ C )
& ( ord_le318720350_set_a @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_300_insert__subset) ).
thf(988,plain,
! [A: set_a,B: set_set_a,C: set_set_a] :
( ( ord_le318720350_set_a @ ( insert_set_a @ A @ B ) @ C )
= ( ( member_set_a @ A @ C )
& ( ord_le318720350_set_a @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(2257,plain,
( ( ^ [A: a] : ( member_a @ A @ s ) )
!= ( ^ [A: a] : $false ) ),
inference(simp,[status(thm)],[2173]) ).
thf(2258,plain,
member_a @ sk111 @ s,
inference(func_ext,[status(esa)],[2257]) ).
thf(2265,plain,
! [B: a > $o,A: a] :
( ( member_a @ A @ ( collect_a @ B ) )
| ( ( member_a @ sk111 @ s )
!= ( B @ A ) ) ),
inference(paramod_ordered,[status(thm)],[2258,320]) ).
thf(2286,plain,
( member_a @ sk111
@ ( collect_a
@ ^ [A: a] : ( member_a @ A @ s ) ) ),
inference(pre_uni,[status(thm)],[2265:[bind(A,$thf( sk111 )),bind(B,$thf( ^ [C: a] : ( member_a @ C @ s ) ))]]) ).
thf(169,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_241_equalityE) ).
thf(920,plain,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(18,axiom,
! [A: set_a,B: set_a,C: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ C ) )
= ( ( A = B )
| ( member_set_a @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_286_insert__iff) ).
thf(380,plain,
! [A: set_a,B: set_a,C: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ C ) )
= ( ( A = B )
| ( member_set_a @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(50,axiom,
( locali880295127ng_a_b
= ( ^ [A: partia1833973666xt_a_b,B: set_a] :
( ( ring_a_b @ A )
& ( locali425460998xt_a_b @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_29_mult__submonoid__of__rng__def) ).
thf(512,plain,
( locali880295127ng_a_b
= ( ^ [A: partia1833973666xt_a_b,B: set_a] :
( ( ring_a_b @ A )
& ( locali425460998xt_a_b @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(104,axiom,
! [A: a,B: set_a] :
( ( member_a @ A @ B )
=> ~ ! [C: set_a] :
( ( B
= ( insert_a @ A @ C ) )
=> ( member_a @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_330_Set_Oset__insert) ).
thf(678,plain,
! [A: a,B: set_a] :
( ( member_a @ A @ B )
=> ~ ! [C: set_a] :
( ( B
= ( insert_a @ A @ C ) )
=> ( member_a @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(315,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( units_1734235042t_unit @ A ) )
=> ? [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
& ( ( mult_s977248631t_unit @ A @ C @ B )
= ( one_se1488596747t_unit @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_103_monoid_OUnits__l__inv__ex) ).
thf(1438,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( units_1734235042t_unit @ A ) )
=> ? [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
& ( ( mult_s977248631t_unit @ A @ C @ B )
= ( one_se1488596747t_unit @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).
thf(292,axiom,
! [A: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ A )
= ( zero_a_b @ r ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_65_l__null) ).
thf(1338,plain,
! [A: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ A )
= ( zero_a_b @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).
thf(14,axiom,
! [A: set_se1596668135od_a_a] :
( ( ! [B: set_Product_prod_a_a] :
~ ( member1838126896od_a_a @ B @ A ) )
= ( A = bot_bo1424703315od_a_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_199_all__not__in__conv) ).
thf(366,plain,
! [A: set_se1596668135od_a_a] :
( ( ! [B: set_Product_prod_a_a] :
~ ( member1838126896od_a_a @ B @ A ) )
= ( A = bot_bo1424703315od_a_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(209,axiom,
! [A: partia1833973666xt_a_b] :
( ( group_a_ring_ext_a_b @ A )
=> ( group_a_ring_ext_a_b @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_147_group_Ois__group) ).
thf(1067,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(137,axiom,
! [A: product_prod_a_a] : ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_295_singletonI) ).
thf(798,plain,
! [A: product_prod_a_a] : ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(250,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ A @ B @ C ) @ ( partia616129906t_unit @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_120_monoid_Om__closed) ).
thf(1199,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ A @ B @ C ) @ ( partia616129906t_unit @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).
thf(288,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia1066395285xt_a_b @ r ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_60_zero__closed) ).
thf(1328,plain,
member_a @ ( zero_a_b @ r ) @ ( partia1066395285xt_a_b @ r ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).
thf(2136,plain,
! [A: a] :
( ~ $true
| ( ( member_a @ ( zero_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) )
!= ( member_a @ A
@ ( collect_a
@ ^ [B: a] : $false ) ) ) ),
inference(paramod_ordered,[status(thm)],[1328,2119]) ).
thf(2137,plain,
! [A: a] :
( ( member_a @ ( zero_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) )
!= ( member_a @ A
@ ( collect_a
@ ^ [B: a] : $false ) ) ),
inference(simp,[status(thm)],[2136]) ).
thf(2150,plain,
! [A: a] :
( ( ( zero_a_b @ r )
!= A )
| ( ( collect_a
@ ^ [B: a] : $false )
!= ( partia1066395285xt_a_b @ r ) ) ),
inference(simp,[status(thm)],[2137]) ).
thf(2164,plain,
( ( collect_a
@ ^ [A: a] : $false )
!= ( partia1066395285xt_a_b @ r ) ),
inference(simp,[status(thm)],[2150]) ).
thf(178,axiom,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( ring_a_b @ A )
=> ( ( locali425460998xt_a_b @ A @ B )
=> ( locali880295127ng_a_b @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_30_mult__submonoid__of__rng_Ointro) ).
thf(957,plain,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( ring_a_b @ A )
=> ( ( locali425460998xt_a_b @ A @ B )
=> ( locali880295127ng_a_b @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(132,axiom,
! [A: a,B: a,C: set_a] :
( ( member_a @ A @ ( insert_a @ B @ C ) )
= ( ( A = B )
| ( member_a @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_283_insert__iff) ).
thf(778,plain,
! [A: a,B: a,C: set_a] :
( ( member_a @ A @ ( insert_a @ B @ C ) )
= ( ( A = B )
| ( member_a @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(226,axiom,
locali1944243560ac_a_b = locali1944243560ac_a_b,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_7_eq__obj__rng__of__frac_Orec__monoid__rng__of__frac_Ocong) ).
thf(1123,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).
thf(77,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_351_doubleton__eq__iff) ).
thf(598,plain,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(66,axiom,
! [A: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ A ) )
= ( ! [B: a] :
~ ( A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_206_empty__Collect__eq) ).
thf(561,plain,
! [A: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ A ) )
= ( ! [B: a] :
~ ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(31,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a,C: product_prod_a_a] :
( ( member449909584od_a_a @ A @ B )
=> ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_335_insertI2) ).
thf(424,plain,
! [A: product_prod_a_a,B: set_Product_prod_a_a,C: product_prod_a_a] :
( ( member449909584od_a_a @ A @ B )
=> ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(109,axiom,
! [A: product_prod_a_a > $o,B: product_prod_a_a] :
( ! [C: a,D: a] : ( A @ ( product_Pair_a_a @ C @ D ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_91_prod__cases) ).
thf(693,plain,
! [A: product_prod_a_a > $o,B: product_prod_a_a] :
( ! [C: a,D: a] : ( A @ ( product_Pair_a_a @ C @ D ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(298,axiom,
! [A: a,B: a,C: int] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ C @ B ) )
= ( add_pow_a_b_int @ r @ C @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_108_add__pow__rdistr__int) ).
thf(1370,plain,
! [A: a,B: a,C: int] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ C @ B ) )
= ( add_pow_a_b_int @ r @ C @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).
thf(122,axiom,
! [A: product_prod_a_a,B: product_prod_a_a > $o] :
( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ B ) )
= ( B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_49_mem__Collect__eq) ).
thf(729,plain,
! [A: product_prod_a_a,B: product_prod_a_a > $o] :
( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(262,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A @ B ) @ ( partia1066395285xt_a_b @ r ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_190_set__add__closed) ).
thf(1238,plain,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A @ B ) @ ( partia1066395285xt_a_b @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).
thf(310,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( units_1734235042t_unit @ A ) )
=> ? [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
& ( ( mult_s977248631t_unit @ A @ B @ C )
= ( one_se1488596747t_unit @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_101_monoid_OUnits__r__inv__ex) ).
thf(1423,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( units_1734235042t_unit @ A ) )
=> ? [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
& ( ( mult_s977248631t_unit @ A @ B @ C )
= ( one_se1488596747t_unit @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).
thf(285,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_64_Units__one__closed) ).
thf(1317,plain,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).
thf(1691,plain,
! [B: a > $o,A: a] :
( ( member_a @ A @ ( collect_a @ B ) )
| ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) )
!= ( B @ A ) ) ),
inference(paramod_ordered,[status(thm)],[1317,320]) ).
thf(1707,plain,
( member_a @ ( one_a_ring_ext_a_b @ r )
@ ( collect_a
@ ^ [A: a] : ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ),
inference(pre_uni,[status(thm)],[1691:[bind(A,$thf( one_a_ring_ext_a_b @ r )),bind(B,$thf( ^ [C: a] : ( member_a @ C @ ( units_a_ring_ext_a_b @ r ) ) ))]]) ).
thf(1917,plain,
! [A: a] :
( ~ $true
| ( ( member_a @ ( one_a_ring_ext_a_b @ r )
@ ( collect_a
@ ^ [B: a] : ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) )
!= ( member_a @ A @ bot_bot_set_a ) ) ),
inference(paramod_ordered,[status(thm)],[1707,554]) ).
thf(1918,plain,
! [A: a] :
( ( member_a @ ( one_a_ring_ext_a_b @ r )
@ ( collect_a
@ ^ [B: a] : ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) )
!= ( member_a @ A @ bot_bot_set_a ) ),
inference(simp,[status(thm)],[1917]) ).
thf(1932,plain,
! [A: a] :
( ( ( one_a_ring_ext_a_b @ r )
!= A )
| ( ( collect_a
@ ^ [B: a] : ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) )
!= bot_bot_set_a ) ),
inference(simp,[status(thm)],[1918]) ).
thf(1938,plain,
( ( collect_a
@ ^ [A: a] : ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) )
!= bot_bot_set_a ),
inference(simp,[status(thm)],[1932]) ).
thf(244,axiom,
! [A: a,B: a,C: nat] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_nat @ r @ C @ A ) @ B )
= ( add_pow_a_b_nat @ r @ C @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_142_add__pow__ldistr) ).
thf(1178,plain,
! [A: a,B: a,C: nat] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_nat @ r @ C @ A ) @ B )
= ( add_pow_a_b_nat @ r @ C @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).
thf(239,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( partia1066395285xt_a_b @ r ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_61_semiring__simprules_I3_J) ).
thf(1159,plain,
! [A: a,B: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( partia1066395285xt_a_b @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).
thf(253,axiom,
! [A: a,B: nat] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( member_a @ ( add_pow_a_b_nat @ r @ B @ A ) @ ( partia1066395285xt_a_b @ r ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_145_add_Onat__pow__closed) ).
thf(1210,plain,
! [A: a,B: nat] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( member_a @ ( add_pow_a_b_nat @ r @ B @ A ) @ ( partia1066395285xt_a_b @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(279,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( group_52730354t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( ( mult_s977248631t_unit @ A @ C @ B )
= B )
= ( C
= ( one_se1488596747t_unit @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_167_group_Or__cancel__one) ).
thf(1291,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( group_52730354t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( ( mult_s977248631t_unit @ A @ C @ B )
= B )
= ( C
= ( one_se1488596747t_unit @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).
thf(148,axiom,
! [A: partia96731725t_unit] :
( ( group_a_Product_unit @ A )
=> ( group_a_Product_unit @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_148_group_Ois__group) ).
thf(842,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(13,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ B )
=> ~ ! [C: set_Product_prod_a_a] :
( ( B
= ( insert1116662519od_a_a @ A @ C ) )
=> ( member449909584od_a_a @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_331_Set_Oset__insert) ).
thf(361,plain,
! [A: product_prod_a_a,B: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ B )
=> ~ ! [C: set_Product_prod_a_a] :
( ( B
= ( insert1116662519od_a_a @ A @ C ) )
=> ( member449909584od_a_a @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(296,axiom,
! [A: a,B: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( partia206007992t_unit @ rel ) )
=> ( ( locali1170246543t_unit @ rel @ A @ B )
!= bot_bo2131659635od_a_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_38_non__empty__class) ).
thf(1365,plain,
! [A: a,B: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( partia206007992t_unit @ rel ) )
=> ( ( locali1170246543t_unit @ rel @ A @ B )
!= bot_bo2131659635od_a_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).
thf(249,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ B @ ( one_se1488596747t_unit @ A ) )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_95_monoid_Or__one) ).
thf(1195,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ B @ ( one_se1488596747t_unit @ A ) )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(1406,plain,
( ( partia1066395285xt_a_b @ r )
!= bot_bot_set_a ),
inference(polarity_switch,[status(thm)],[1405]) ).
thf(1407,plain,
( ( partia1066395285xt_a_b @ r )
!= bot_bot_set_a ),
inference(lifteq,[status(thm)],[1406]) ).
thf(322,plain,
! [A: partia96731725t_unit] :
( ( group_a_Product_unit @ A )
=> ! [B: a] :
( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
& ( ( mult_a_Product_unit @ A @ C @ B )
= ( one_a_Product_unit @ A ) ) ) ) ),
inference(miniscope,[status(thm)],[321]) ).
thf(323,plain,
! [B: a,A: partia96731725t_unit] :
( ( ~ ( group_a_Product_unit @ A )
| ~ ( member_a @ B @ ( partia1955795460t_unit @ A ) )
| ( member_a @ ( sk1 @ B @ A ) @ ( partia1955795460t_unit @ A ) ) )
& ( ~ ( group_a_Product_unit @ A )
| ~ ( member_a @ B @ ( partia1955795460t_unit @ A ) )
| ( ( mult_a_Product_unit @ A @ ( sk1 @ B @ A ) @ B )
= ( one_a_Product_unit @ A ) ) ) ),
inference(cnf,[status(esa)],[322]) ).
thf(324,plain,
! [B: a,A: partia96731725t_unit] :
( ~ ( group_a_Product_unit @ A )
| ~ ( member_a @ B @ ( partia1955795460t_unit @ A ) )
| ( member_a @ ( sk1 @ B @ A ) @ ( partia1955795460t_unit @ A ) ) ),
inference(cnfConj,[status(thm)],[323]) ).
thf(183,axiom,
! [A: partia1833973666xt_a_b,B: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ ( one_a_ring_ext_a_b @ A ) )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_96_monoid_Or__one) ).
thf(974,plain,
! [A: partia1833973666xt_a_b,B: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ ( one_a_ring_ext_a_b @ A ) )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(160,axiom,
! [A: partia539144763t_unit,B: set_se1596668135od_a_a] :
( ( locali1488796916t_unit @ A @ B )
=> ( member1838126896od_a_a @ ( one_se1488596747t_unit @ A ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_27_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Oone__closed) ).
thf(874,plain,
! [A: partia539144763t_unit,B: set_se1596668135od_a_a] :
( ( locali1488796916t_unit @ A @ B )
=> ( member1838126896od_a_a @ ( one_se1488596747t_unit @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(300,axiom,
! [A: a,B: a,C: a,D: a,E: a,F: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ C @ D ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ E @ F ) @ ( partia206007992t_unit @ rel ) )
=> ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ A @ B ) @ ( locali1170246543t_unit @ rel @ C @ D ) ) @ ( locali1170246543t_unit @ rel @ E @ F ) )
= ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ A @ B ) @ ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ C @ D ) @ ( locali1170246543t_unit @ rel @ E @ F ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_16_assoc__mult__rng__of__frac) ).
thf(1375,plain,
! [A: a,B: a,C: a,D: a,E: a,F: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ C @ D ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ E @ F ) @ ( partia206007992t_unit @ rel ) )
=> ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ A @ B ) @ ( locali1170246543t_unit @ rel @ C @ D ) ) @ ( locali1170246543t_unit @ rel @ E @ F ) )
= ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ A @ B ) @ ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ C @ D ) @ ( locali1170246543t_unit @ rel @ E @ F ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).
thf(24,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,C: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ C ) )
= ( ( A = B )
| ( member449909584od_a_a @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_284_insert__iff) ).
thf(403,plain,
! [A: product_prod_a_a,B: product_prod_a_a,C: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ C ) )
= ( ( A = B )
| ( member449909584od_a_a @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(20,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ B )
=> ( ( insert1116662519od_a_a @ A @ B )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_323_insert__absorb) ).
thf(388,plain,
! [A: product_prod_a_a,B: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ B )
=> ( ( insert1116662519od_a_a @ A @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(82,axiom,
! [A: partia1833973666xt_a_b,B: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ A ) )
=> ( member_a @ B @ ( partia1066395285xt_a_b @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_123_monoid_OUnits__closed) ).
thf(610,plain,
! [A: partia1833973666xt_a_b,B: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ A ) )
=> ( member_a @ B @ ( partia1066395285xt_a_b @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(302,axiom,
( ( mult_a_ring_ext_a_b @ r @ s2 @ s3 )
= ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1_f4) ).
thf(1397,plain,
( ( mult_a_ring_ext_a_b @ r @ s2 @ s3 )
= ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).
thf(52,axiom,
! [A: partia1833973666xt_a_b] :
( ( group_a_ring_ext_a_b @ A )
=> ( ord_less_eq_set_a @ ( partia1066395285xt_a_b @ A ) @ ( units_a_ring_ext_a_b @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_150_group_OUnits) ).
thf(519,plain,
! [A: partia1833973666xt_a_b] :
( ( group_a_ring_ext_a_b @ A )
=> ( ord_less_eq_set_a @ ( partia1066395285xt_a_b @ A ) @ ( units_a_ring_ext_a_b @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(126,axiom,
! [A: partia1833973666xt_a_b,B: a] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
& ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= ( one_a_ring_ext_a_b @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_174_group_Or__inv__ex) ).
thf(742,plain,
! [A: partia1833973666xt_a_b,B: a] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
& ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= ( one_a_ring_ext_a_b @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(272,axiom,
! [A: a,B: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( partia206007992t_unit @ rel ) )
=> ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( one_se1488596747t_unit @ ( locali1944243560ac_a_b @ r @ s ) ) @ ( locali1170246543t_unit @ rel @ A @ B ) )
= ( locali1170246543t_unit @ rel @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_36_left__unit__mult__rng__of__frac) ).
thf(1270,plain,
! [A: a,B: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( partia206007992t_unit @ rel ) )
=> ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( one_se1488596747t_unit @ ( locali1944243560ac_a_b @ r @ s ) ) @ ( locali1170246543t_unit @ rel @ A @ B ) )
= ( locali1170246543t_unit @ rel @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).
thf(1138,plain,
( ( mult_a_ring_ext_a_b @ r @ r3 @ r2 )
= ( mult_a_ring_ext_a_b @ r @ r2 @ r3 ) ),
inference(lifteq,[status(thm)],[1137]) ).
thf(1694,plain,
! [A: a] :
( ~ $true
| ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) )
!= ( member_a @ A @ bot_bot_set_a ) ) ),
inference(paramod_ordered,[status(thm)],[1317,554]) ).
thf(1695,plain,
! [A: a] :
( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) )
!= ( member_a @ A @ bot_bot_set_a ) ),
inference(simp,[status(thm)],[1694]) ).
thf(1709,plain,
! [A: a] :
( ( ( one_a_ring_ext_a_b @ r )
!= A )
| ( ( units_a_ring_ext_a_b @ r )
!= bot_bot_set_a ) ),
inference(simp,[status(thm)],[1695]) ).
thf(1719,plain,
( ( units_a_ring_ext_a_b @ r )
!= bot_bot_set_a ),
inference(simp,[status(thm)],[1709]) ).
thf(289,axiom,
! [A: a,B: a] :
( ( member_a @ A @ s )
=> ( ( member_a @ B @ s )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ s ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_9_m__closed) ).
thf(1329,plain,
! [A: a,B: a] :
( ( member_a @ A @ s )
=> ( ( member_a @ B @ s )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ s ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).
thf(215,axiom,
! [A: a,B: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( partia206007992t_unit @ rel ) )
=> ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ A @ B ) @ ( one_se1488596747t_unit @ ( locali1944243560ac_a_b @ r @ s ) ) )
= ( locali1170246543t_unit @ rel @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_35_right__unit__mult__rng__of__frac) ).
thf(1082,plain,
! [A: a,B: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( partia206007992t_unit @ rel ) )
=> ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ A @ B ) @ ( one_se1488596747t_unit @ ( locali1944243560ac_a_b @ r @ s ) ) )
= ( locali1170246543t_unit @ rel @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(245,axiom,
! [A: a,B: a,C: nat] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_nat @ r @ C @ B ) )
= ( add_pow_a_b_nat @ r @ C @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_141_add__pow__rdistr) ).
thf(1182,plain,
! [A: a,B: a,C: nat] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_nat @ r @ C @ B ) )
= ( add_pow_a_b_nat @ r @ C @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(55,axiom,
! [A: partia1833973666xt_a_b,B: a] :
( ( ring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ ( zero_a_b @ A ) )
= ( zero_a_b @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_77_ring_Oring__simprules_I25_J) ).
thf(526,plain,
! [A: partia1833973666xt_a_b,B: a] :
( ( ring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ ( zero_a_b @ A ) )
= ( zero_a_b @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(91,axiom,
! [A: product_prod_a_a] :
~ ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_219_emptyE) ).
thf(637,plain,
! [A: product_prod_a_a] :
~ ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(182,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
( ( ring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
= ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_81_ring_Oring__simprules_I11_J) ).
thf(970,plain,
! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
( ( ring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
= ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(1723,plain,
! [B: a > $o,A: a] :
( ( member_a @ A @ ( collect_a @ B ) )
| ( ( member_a @ ( zero_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) )
!= ( B @ A ) ) ),
inference(paramod_ordered,[status(thm)],[1328,320]) ).
thf(1739,plain,
( member_a @ ( zero_a_b @ r )
@ ( collect_a
@ ^ [A: a] : ( member_a @ A @ ( partia1066395285xt_a_b @ r ) ) ) ),
inference(pre_uni,[status(thm)],[1723:[bind(A,$thf( zero_a_b @ r )),bind(B,$thf( ^ [C: a] : ( member_a @ C @ ( partia1066395285xt_a_b @ r ) ) ))]]) ).
thf(58,axiom,
! [A: partia1833973666xt_a_b] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( partia1066395285xt_a_b @ A )
!= bot_bot_set_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_117_monoid_Ocarrier__not__empty) ).
thf(535,plain,
! [A: partia1833973666xt_a_b] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( partia1066395285xt_a_b @ A )
!= bot_bot_set_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(536,plain,
! [A: partia1833973666xt_a_b] :
( ~ ( monoid1022105667xt_a_b @ A )
| ( ( partia1066395285xt_a_b @ A )
!= bot_bot_set_a ) ),
inference(cnf,[status(esa)],[535]) ).
thf(537,plain,
! [A: partia1833973666xt_a_b] :
( ( ( partia1066395285xt_a_b @ A )
!= bot_bot_set_a )
| ~ ( monoid1022105667xt_a_b @ A ) ),
inference(lifteq,[status(thm)],[536]) ).
thf(106,axiom,
! [A: partia1833973666xt_a_b] :
( ( mult_a_Product_unit @ ( units_1411277569xt_a_b @ A ) )
= ( mult_a_ring_ext_a_b @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_275_units__of__mult) ).
thf(686,plain,
! [A: partia1833973666xt_a_b] :
( ( mult_a_Product_unit @ ( units_1411277569xt_a_b @ A ) )
= ( mult_a_ring_ext_a_b @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(212,axiom,
! [A: partia1833973666xt_a_b] :
( ( units_a_Product_unit @ ( units_1411277569xt_a_b @ A ) )
= ( units_a_ring_ext_a_b @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_270_units__of__units) ).
thf(1075,plain,
! [A: partia1833973666xt_a_b] :
( ( units_a_Product_unit @ ( units_1411277569xt_a_b @ A ) )
= ( units_a_ring_ext_a_b @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(195,axiom,
! [A: partia1833973666xt_a_b,B: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_98_monoid_Ol__one) ).
thf(1016,plain,
! [A: partia1833973666xt_a_b,B: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(75,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= ( one_a_ring_ext_a_b @ A ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ C @ B )
= ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_180_group_Oinv__comm) ).
thf(590,plain,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= ( one_a_ring_ext_a_b @ A ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ C @ B )
= ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(187,axiom,
! [A: product_prod_a_a > $o] :
( ( bot_bo2131659635od_a_a
= ( collec645855634od_a_a @ A ) )
= ( ! [B: product_prod_a_a] :
~ ( A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_205_empty__Collect__eq) ).
thf(985,plain,
! [A: product_prod_a_a > $o] :
( ( bot_bo2131659635od_a_a
= ( collec645855634od_a_a @ A ) )
= ( ! [B: product_prod_a_a] :
~ ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(236,axiom,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali1648887798ac_a_b @ A @ B )
=> ( monoid1242292471t_unit @ ( locali1944243560ac_a_b @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_32_eq__obj__rng__of__frac_Omonoid__rng__of__frac) ).
thf(1150,plain,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali1648887798ac_a_b @ A @ B )
=> ( monoid1242292471t_unit @ ( locali1944243560ac_a_b @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).
thf(186,axiom,
! [A: partia1833973666xt_a_b] :
( ( ring_a_b @ A )
=> ( ring_a_b @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_93_ring_Ois__ring) ).
thf(984,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(311,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ A @ ( a_r_coset_a_b @ r @ B @ C ) )
= ( a_r_coset_a_b @ r @ ( set_add_a_b @ r @ A @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_269_a__setmult__rcos__assoc) ).
thf(1429,plain,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ A @ ( a_r_coset_a_b @ r @ B @ C ) )
= ( a_r_coset_a_b @ r @ ( set_add_a_b @ r @ A @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).
thf(265,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( units_1734235042t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ D @ ( partia616129906t_unit @ A ) )
=> ( ( ( mult_s977248631t_unit @ A @ B @ C )
= ( mult_s977248631t_unit @ A @ B @ D ) )
= ( C = D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_138_monoid_OUnits__l__cancel) ).
thf(1246,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( units_1734235042t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ D @ ( partia616129906t_unit @ A ) )
=> ( ( ( mult_s977248631t_unit @ A @ B @ C )
= ( mult_s977248631t_unit @ A @ B @ D ) )
= ( C = D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).
thf(94,axiom,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali425460998xt_a_b @ A @ B )
=> ( locali425460998xt_a_b @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_31_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Ois__submonoid) ).
thf(647,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(270,axiom,
! [A: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ? [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( one_a_ring_ext_a_b @ r ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_59_Units__r__inv__ex) ).
thf(1263,plain,
! [A: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ? [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( one_a_ring_ext_a_b @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).
thf(191,axiom,
! [A: partia1833973666xt_a_b] :
( ( field_a_b @ A )
=> ( ( one_a_ring_ext_a_b @ A )
!= ( zero_a_b @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_112_one__not__zero) ).
thf(997,plain,
! [A: partia1833973666xt_a_b] :
( ( field_a_b @ A )
=> ( ( one_a_ring_ext_a_b @ A )
!= ( zero_a_b @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(125,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= ( one_a_ring_ext_a_b @ A ) )
=> ( ( ( mult_a_ring_ext_a_b @ A @ C @ D )
= ( one_a_ring_ext_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( B = D ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_135_monoid_Oinv__unique) ).
thf(738,plain,
! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= ( one_a_ring_ext_a_b @ A ) )
=> ( ( ( mult_a_ring_ext_a_b @ A @ C @ D )
= ( one_a_ring_ext_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( B = D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(2304,plain,
member_a @ sk112 @ ( partia1066395285xt_a_b @ r ),
inference(func_ext,[status(esa)],[2259]) ).
thf(313,axiom,
! [A: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ A )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_67_l__one) ).
thf(1434,plain,
! [A: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).
thf(251,axiom,
! [A: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_r_coset_a_b @ r @ A @ B ) @ ( partia1066395285xt_a_b @ r ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_268_a__r__coset__subset__G) ).
thf(1202,plain,
! [A: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_r_coset_a_b @ r @ A @ B ) @ ( partia1066395285xt_a_b @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).
thf(217,axiom,
group_a_Product_unit @ ( units_1411277569xt_a_b @ r ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_265_units__group) ).
thf(1086,plain,
group_a_Product_unit @ ( units_1411277569xt_a_b @ r ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).
thf(263,axiom,
( monoid1242292471t_unit
= ( ^ [A: partia539144763t_unit] :
( ! [B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ A @ B @ C ) @ ( partia616129906t_unit @ A ) ) ) )
& ! [B: set_Product_prod_a_a,C: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ D @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( mult_s977248631t_unit @ A @ B @ C ) @ D )
= ( mult_s977248631t_unit @ A @ B @ ( mult_s977248631t_unit @ A @ C @ D ) ) ) ) ) )
& ( member1838126896od_a_a @ ( one_se1488596747t_unit @ A ) @ ( partia616129906t_unit @ A ) )
& ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( one_se1488596747t_unit @ A ) @ B )
= B ) )
& ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ B @ ( one_se1488596747t_unit @ A ) )
= B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_130_Group_Omonoid__def) ).
thf(1241,plain,
( monoid1242292471t_unit
= ( ^ [A: partia539144763t_unit] :
( ! [B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ A @ B @ C ) @ ( partia616129906t_unit @ A ) ) ) )
& ! [B: set_Product_prod_a_a,C: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ D @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( mult_s977248631t_unit @ A @ B @ C ) @ D )
= ( mult_s977248631t_unit @ A @ B @ ( mult_s977248631t_unit @ A @ C @ D ) ) ) ) ) )
& ( member1838126896od_a_a @ ( one_se1488596747t_unit @ A ) @ ( partia616129906t_unit @ A ) )
& ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( one_se1488596747t_unit @ A ) @ B )
= B ) )
& ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ B @ ( one_se1488596747t_unit @ A ) )
= B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).
thf(90,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le318720350_set_a @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_248_equalityD1) ).
thf(633,plain,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le318720350_set_a @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(205,axiom,
! [A: partia96731725t_unit] :
( ( group_a_Product_unit @ A )
=> ( ord_less_eq_set_a @ ( partia1955795460t_unit @ A ) @ ( units_a_Product_unit @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_149_group_OUnits) ).
thf(1054,plain,
! [A: partia96731725t_unit] :
( ( group_a_Product_unit @ A )
=> ( ord_less_eq_set_a @ ( partia1955795460t_unit @ A ) @ ( units_a_Product_unit @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(242,axiom,
! [A: partia539144763t_unit] :
( ( monoid1242292471t_unit @ A )
=> ( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ? [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
& ( ( mult_s977248631t_unit @ A @ C @ B )
= ( one_se1488596747t_unit @ A ) ) ) )
=> ( group_52730354t_unit @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_185_monoid_Ogroup__l__invI) ).
thf(1170,plain,
! [A: partia539144763t_unit] :
( ( monoid1242292471t_unit @ A )
=> ( ! [B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ? [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
& ( ( mult_s977248631t_unit @ A @ C @ B )
= ( one_se1488596747t_unit @ A ) ) ) )
=> ( group_52730354t_unit @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).
thf(200,axiom,
( ord_less_eq_set_a
= ( ^ [A: set_a,B: set_a] :
! [C: a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_245_subset__eq) ).
thf(1035,plain,
( ord_less_eq_set_a
= ( ^ [A: set_a,B: set_a] :
! [C: a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(56,axiom,
! [A: set_se1596668135od_a_a] :
( ( collec183727474od_a_a
@ ^ [B: set_Product_prod_a_a] : ( member1838126896od_a_a @ B @ A ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_54_Collect__mem__eq) ).
thf(530,plain,
! [A: set_se1596668135od_a_a] :
( ( collec183727474od_a_a
@ ^ [B: set_Product_prod_a_a] : ( member1838126896od_a_a @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(8,axiom,
! [A: partia96731725t_unit] :
( ( group_a_Product_unit @ A )
=> ( ( units_a_Product_unit @ A )
= ( partia1955795460t_unit @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_157_group_OUnits__eq) ).
thf(343,plain,
! [A: partia96731725t_unit] :
( ( group_a_Product_unit @ A )
=> ( ( units_a_Product_unit @ A )
= ( partia1955795460t_unit @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(286,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ D @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( mult_s977248631t_unit @ A @ B @ C ) @ D )
= ( mult_s977248631t_unit @ A @ B @ ( mult_s977248631t_unit @ A @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_118_monoid_Om__assoc) ).
thf(1318,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ D @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( mult_s977248631t_unit @ A @ B @ C ) @ D )
= ( mult_s977248631t_unit @ A @ B @ ( mult_s977248631t_unit @ A @ C @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).
thf(65,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( semiring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_82_semiring_Osemiring__simprules_I3_J) ).
thf(558,plain,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( semiring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(211,axiom,
! [A: a,B: set_a,C: set_a] :
( ~ ( member_a @ A @ B )
=> ( ~ ( member_a @ A @ C )
=> ( ( ( insert_a @ A @ B )
= ( insert_a @ A @ C ) )
= ( B = C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_326_insert__ident) ).
thf(1071,plain,
! [A: a,B: set_a,C: set_a] :
( ~ ( member_a @ A @ B )
=> ( ~ ( member_a @ A @ C )
=> ( ( ( insert_a @ A @ B )
= ( insert_a @ A @ C ) )
= ( B = C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(771,plain,
! [B: set_a,A: set_a] :
( ( A != B )
| ( ord_less_eq_set_a @ B @ A ) ),
inference(cnf,[status(esa)],[770]) ).
thf(772,plain,
! [B: set_a,A: set_a] :
( ( A != B )
| ( ord_less_eq_set_a @ B @ A ) ),
inference(lifteq,[status(thm)],[771]) ).
thf(773,plain,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ),
inference(simp,[status(thm)],[772]) ).
thf(136,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( ( mult_a_ring_ext_a_b @ A @ C @ B )
= ( mult_a_ring_ext_a_b @ A @ D @ B ) )
= ( C = D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_156_Group_Ogroup_Oright__cancel) ).
thf(794,plain,
! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( ( mult_a_ring_ext_a_b @ A @ C @ B )
= ( mult_a_ring_ext_a_b @ A @ D @ B ) )
= ( C = D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(275,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( group_52730354t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( ( mult_s977248631t_unit @ A @ B @ C )
= B )
= ( C
= ( one_se1488596747t_unit @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_170_group_Ol__cancel__one) ).
thf(1278,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( group_52730354t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( ( mult_s977248631t_unit @ A @ B @ C )
= B )
= ( C
= ( one_se1488596747t_unit @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).
thf(513,plain,
( locali880295127ng_a_b
= ( ^ [A: partia1833973666xt_a_b,B: set_a] :
( ( ring_a_b @ A )
& ( locali425460998xt_a_b @ A @ B ) ) ) ),
inference(lifteq,[status(thm)],[512]) ).
thf(273,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( group_52730354t_unit @ A )
=> ( ( ( mult_s977248631t_unit @ A @ B @ C )
= ( one_se1488596747t_unit @ A ) )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ C @ B )
= ( one_se1488596747t_unit @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_179_group_Oinv__comm) ).
thf(1273,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( group_52730354t_unit @ A )
=> ( ( ( mult_s977248631t_unit @ A @ B @ C )
= ( one_se1488596747t_unit @ A ) )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ C @ B )
= ( one_se1488596747t_unit @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).
thf(282,axiom,
! [A: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= B ) )
=> ( A
= ( one_a_ring_ext_a_b @ r ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_39_one__unique) ).
thf(1301,plain,
! [A: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= B ) )
=> ( A
= ( one_a_ring_ext_a_b @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).
thf(107,axiom,
! [A: partia1833973666xt_a_b] :
( ( ring_a_b @ A )
=> ( member_a @ ( zero_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_71_ring_Oring__simprules_I2_J) ).
thf(689,plain,
! [A: partia1833973666xt_a_b] :
( ( ring_a_b @ A )
=> ( member_a @ ( zero_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(78,axiom,
! [A: product_prod_a_a] :
? [B: a,C: a] :
( A
= ( product_Pair_a_a @ B @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_92_surj__pair) ).
thf(601,plain,
! [A: product_prod_a_a] :
? [B: a,C: a] :
( A
= ( product_Pair_a_a @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(123,axiom,
! [A: partia1833973666xt_a_b] :
( ( semiring_a_b @ A )
=> ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_85_semiring_Osemiring__simprules_I4_J) ).
thf(732,plain,
! [A: partia1833973666xt_a_b] :
( ( semiring_a_b @ A )
=> ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(149,axiom,
! [A: partia1833973666xt_a_b] :
( ( abelian_monoid_a_b @ A )
=> ( member_a @ ( zero_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_73_abelian__monoid_Ozero__closed) ).
thf(843,plain,
! [A: partia1833973666xt_a_b] :
( ( abelian_monoid_a_b @ A )
=> ( member_a @ ( zero_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(2142,plain,
! [A: a] :
( ~ $true
| ( ( member_a @ ( one_a_ring_ext_a_b @ r )
@ ( collect_a
@ ^ [B: a] : ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) )
!= ( member_a @ A
@ ( collect_a
@ ^ [B: a] : $false ) ) ) ),
inference(paramod_ordered,[status(thm)],[1707,2119]) ).
thf(2143,plain,
! [A: a] :
( ( member_a @ ( one_a_ring_ext_a_b @ r )
@ ( collect_a
@ ^ [B: a] : ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) )
!= ( member_a @ A
@ ( collect_a
@ ^ [B: a] : $false ) ) ),
inference(simp,[status(thm)],[2142]) ).
thf(2156,plain,
! [A: a] :
( ( ( one_a_ring_ext_a_b @ r )
!= A )
| ( ( collect_a
@ ^ [B: a] : ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) )
!= ( collect_a
@ ^ [B: a] : $false ) ) ),
inference(simp,[status(thm)],[2143]) ).
thf(2170,plain,
( ( collect_a
@ ^ [A: a] : ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) )
!= ( collect_a
@ ^ [A: a] : $false ) ),
inference(simp,[status(thm)],[2156]) ).
thf(2398,plain,
( ( ^ [A: a] : ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) )
!= ( ^ [A: a] : $false ) ),
inference(simp,[status(thm)],[2170]) ).
thf(227,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia1066395285xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) )
=> ( field_a_b @ r ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_352_field__intro2) ).
thf(1124,plain,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia1066395285xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) )
=> ( field_a_b @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).
thf(27,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a] : ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_339_insertI1) ).
thf(413,plain,
! [A: product_prod_a_a,B: set_Product_prod_a_a] : ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(71,axiom,
! [A: partia1833973666xt_a_b,B: a] :
( ( semiring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ ( zero_a_b @ A ) )
= ( zero_a_b @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_78_semiring_Or__null) ).
thf(577,plain,
! [A: partia1833973666xt_a_b,B: a] :
( ( semiring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ ( zero_a_b @ A ) )
= ( zero_a_b @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(115,axiom,
! [A: partia1833973666xt_a_b] :
( ( abelian_monoid_a_b @ A )
=> ( member_a @ ( zero_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_70_abelian__monoidE_I2_J) ).
thf(711,plain,
! [A: partia1833973666xt_a_b] :
( ( abelian_monoid_a_b @ A )
=> ( member_a @ ( zero_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(268,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( mult_a_ring_ext_a_b @ r @ A @ C ) )
= ( B = C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_69_Units__l__cancel) ).
thf(1256,plain,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( mult_a_ring_ext_a_b @ r @ A @ C ) )
= ( B = C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).
thf(258,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_63_Units__m__closed) ).
thf(1226,plain,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).
thf(290,axiom,
! [A: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_66_r__null) ).
thf(1332,plain,
! [A: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).
thf(139,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le318720350_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_240_subsetD) ).
thf(804,plain,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le318720350_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(112,axiom,
! [A: a > $o,B: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ A ) @ ( collect_a @ B ) )
= ( ! [C: a] :
( ( A @ C )
=> ( B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_263_Collect__mono__iff) ).
thf(704,plain,
! [A: a > $o,B: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ A ) @ ( collect_a @ B ) )
= ( ! [C: a] :
( ( A @ C )
=> ( B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(257,axiom,
( ( ( partia1066395285xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_292_carrier__one__zero) ).
thf(1224,plain,
( ( ( partia1066395285xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).
thf(102,axiom,
! [A: partia1833973666xt_a_b,B: set_a,C: a,D: a] :
( ( locali425460998xt_a_b @ A @ B )
=> ( ( member_a @ C @ B )
=> ( ( member_a @ D @ B )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ C @ D ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_34_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Om__closed) ).
thf(672,plain,
! [A: partia1833973666xt_a_b,B: set_a,C: a,D: a] :
( ( locali425460998xt_a_b @ A @ B )
=> ( ( member_a @ C @ B )
=> ( ( member_a @ D @ B )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ C @ D ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(21,axiom,
! [A: set_a,B: set_set_a,C: set_a,D: set_set_a] :
( ~ ( member_set_a @ A @ B )
=> ( ~ ( member_set_a @ C @ D )
=> ( ( ( insert_set_a @ A @ B )
= ( insert_set_a @ C @ D ) )
= ( ( ( A = C )
=> ( B = D ) )
& ( ( A != C )
=> ? [E: set_set_a] :
( ( B
= ( insert_set_a @ C @ E ) )
& ~ ( member_set_a @ C @ E )
& ( D
= ( insert_set_a @ A @ E ) )
& ~ ( member_set_a @ A @ E ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_321_insert__eq__iff) ).
thf(391,plain,
! [A: set_a,B: set_set_a,C: set_a,D: set_set_a] :
( ~ ( member_set_a @ A @ B )
=> ( ~ ( member_set_a @ C @ D )
=> ( ( ( insert_set_a @ A @ B )
= ( insert_set_a @ C @ D ) )
= ( ( ( A = C )
=> ( B = D ) )
& ( ( A != C )
=> ? [E: set_set_a] :
( ( B
= ( insert_set_a @ C @ E ) )
& ~ ( member_set_a @ C @ E )
& ( D
= ( insert_set_a @ A @ E ) )
& ~ ( member_set_a @ A @ E ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(116,axiom,
! [A: a,B: a,C: set_a] :
( ( member_a @ A @ ( insert_a @ B @ C ) )
=> ( ( A != B )
=> ( member_a @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_342_insertE) ).
thf(713,plain,
! [A: a,B: a,C: set_a] :
( ( member_a @ A @ ( insert_a @ B @ C ) )
=> ( ( A != B )
=> ( member_a @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(47,axiom,
! [A: partia1833973666xt_a_b,B: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ! [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= C ) )
=> ( B
= ( one_a_ring_ext_a_b @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_137_monoid_Oone__unique) ).
thf(498,plain,
! [A: partia1833973666xt_a_b,B: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ! [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= C ) )
=> ( B
= ( one_a_ring_ext_a_b @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(214,axiom,
member449909584od_a_a @ ( product_Pair_a_a @ r2 @ s2 ) @ ( partia206007992t_unit @ rel ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_15_assms_I1_J) ).
thf(1081,plain,
member449909584od_a_a @ ( product_Pair_a_a @ r2 @ s2 ) @ ( partia206007992t_unit @ rel ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(638,plain,
~ ? [A: product_prod_a_a] : ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ),
inference(miniscope,[status(thm)],[637]) ).
thf(639,plain,
! [A: product_prod_a_a] :
~ ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ),
inference(cnf,[status(esa)],[638]) ).
thf(1756,plain,
! [A: product_prod_a_a] :
( ~ $true
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ r2 @ s2 ) @ ( partia206007992t_unit @ rel ) )
!= ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ) ) ),
inference(paramod_ordered,[status(thm)],[1081,639]) ).
thf(1757,plain,
! [A: product_prod_a_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ r2 @ s2 ) @ ( partia206007992t_unit @ rel ) )
!= ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ) ),
inference(simp,[status(thm)],[1756]) ).
thf(1764,plain,
! [A: product_prod_a_a] :
( ( ( product_Pair_a_a @ r2 @ s2 )
!= A )
| ( ( partia206007992t_unit @ rel )
!= bot_bo2131659635od_a_a ) ),
inference(simp,[status(thm)],[1757]) ).
thf(1765,plain,
( ( partia206007992t_unit @ rel )
!= bot_bo2131659635od_a_a ),
inference(simp,[status(thm)],[1764]) ).
thf(221,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( A
!= ( zero_a_b @ r ) )
=> ? [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_94_cring__fieldI2) ).
thf(1090,plain,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( A
!= ( zero_a_b @ r ) )
=> ? [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).
thf(25,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ~ ( member449909584od_a_a @ A @ B )
=> ( ~ ( member449909584od_a_a @ A @ C )
=> ( ( ( insert1116662519od_a_a @ A @ B )
= ( insert1116662519od_a_a @ A @ C ) )
= ( B = C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_327_insert__ident) ).
thf(406,plain,
! [A: product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ~ ( member449909584od_a_a @ A @ B )
=> ( ~ ( member449909584od_a_a @ A @ C )
=> ( ( ( insert1116662519od_a_a @ A @ B )
= ( insert1116662519od_a_a @ A @ C ) )
= ( B = C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(306,axiom,
! [A: a,B: a,C: a,D: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ C @ D ) @ ( partia206007992t_unit @ rel ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ A @ B ) @ ( locali1170246543t_unit @ rel @ C @ D ) ) @ ( locali1538946157t_unit @ rel ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_20_closed__mult__rng__of__frac) ).
thf(1408,plain,
! [A: a,B: a,C: a,D: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ C @ D ) @ ( partia206007992t_unit @ rel ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ A @ B ) @ ( locali1170246543t_unit @ rel @ C @ D ) ) @ ( locali1538946157t_unit @ rel ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).
thf(276,axiom,
ring_a_b @ r,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_19_local_Oring__axioms) ).
thf(1282,plain,
ring_a_b @ r,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).
thf(134,axiom,
! [A: partia1833973666xt_a_b,B: a] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
& ( ( mult_a_ring_ext_a_b @ A @ C @ B )
= ( one_a_ring_ext_a_b @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_177_group_Ol__inv__ex) ).
thf(784,plain,
! [A: partia1833973666xt_a_b,B: a] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
& ( ( mult_a_ring_ext_a_b @ A @ C @ B )
= ( one_a_ring_ext_a_b @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(83,axiom,
! [A: a,B: set_a] :
( ( insert_a @ A @ B )
!= bot_bot_set_a ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_349_insert__not__empty) ).
thf(613,plain,
! [A: a,B: set_a] :
( ( insert_a @ A @ B )
!= bot_bot_set_a ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(614,plain,
~ ? [A: a,B: set_a] :
( ( insert_a @ A @ B )
= bot_bot_set_a ),
inference(miniscope,[status(thm)],[613]) ).
thf(615,plain,
! [B: set_a,A: a] :
( ( insert_a @ A @ B )
!= bot_bot_set_a ),
inference(cnf,[status(esa)],[614]) ).
thf(616,plain,
! [B: set_a,A: a] :
( ( insert_a @ A @ B )
!= bot_bot_set_a ),
inference(lifteq,[status(thm)],[615]) ).
thf(1562,plain,
! [A: a] :
( ~ $true
| ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ s )
!= ( member_a @ A @ bot_bot_set_a ) ) ),
inference(paramod_ordered,[status(thm)],[1374,554]) ).
thf(1563,plain,
! [A: a] :
( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ s )
!= ( member_a @ A @ bot_bot_set_a ) ),
inference(simp,[status(thm)],[1562]) ).
thf(1567,plain,
! [A: a] :
( ( ( one_a_ring_ext_a_b @ r )
!= A )
| ( s != bot_bot_set_a ) ),
inference(simp,[status(thm)],[1563]) ).
thf(1571,plain,
s != bot_bot_set_a,
inference(simp,[status(thm)],[1567]) ).
thf(283,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a] :
( ( group_52730354t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ? [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
& ( ( mult_s977248631t_unit @ A @ C @ B )
= ( one_se1488596747t_unit @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_176_group_Ol__inv__ex) ).
thf(1307,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a] :
( ( group_52730354t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ? [C: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
& ( ( mult_s977248631t_unit @ A @ C @ B )
= ( one_se1488596747t_unit @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).
thf(128,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_239_subsetD) ).
thf(764,plain,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(79,axiom,
! [A: partia539144763t_unit] :
( ( monoid1242292471t_unit @ A )
=> ( member1838126896od_a_a @ ( one_se1488596747t_unit @ A ) @ ( units_1734235042t_unit @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_127_monoid_OUnits__one__closed) ).
thf(604,plain,
! [A: partia539144763t_unit] :
( ( monoid1242292471t_unit @ A )
=> ( member1838126896od_a_a @ ( one_se1488596747t_unit @ A ) @ ( units_1734235042t_unit @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(17,axiom,
! [A: set_a,B: set_set_a,C: set_a] :
( ( member_set_a @ A @ B )
=> ( member_set_a @ A @ ( insert_set_a @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_337_insertI2) ).
thf(377,plain,
! [A: set_a,B: set_set_a,C: set_a] :
( ( member_set_a @ A @ B )
=> ( member_set_a @ A @ ( insert_set_a @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(49,axiom,
! [A: a,B: set_a,C: a] :
( ( ( insert_a @ A @ B )
= ( insert_a @ C @ bot_bot_set_a ) )
= ( ( A = C )
& ( ord_less_eq_set_a @ B @ ( insert_a @ C @ bot_bot_set_a ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_302_singleton__insert__inj__eq_H) ).
thf(509,plain,
! [A: a,B: set_a,C: a] :
( ( ( insert_a @ A @ B )
= ( insert_a @ C @ bot_bot_set_a ) )
= ( ( A = C )
& ( ord_less_eq_set_a @ B @ ( insert_a @ C @ bot_bot_set_a ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(162,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_296_singletonI) ).
thf(880,plain,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(34,axiom,
! [A: set_se1596668135od_a_a] :
( ( ? [B: set_Product_prod_a_a] : ( member1838126896od_a_a @ B @ A ) )
= ( A != bot_bo1424703315od_a_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_229_ex__in__conv) ).
thf(434,plain,
! [A: set_se1596668135od_a_a] :
( ( ? [B: set_Product_prod_a_a] : ( member1838126896od_a_a @ B @ A ) )
= ( A != bot_bo1424703315od_a_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(156,axiom,
( ord_le318720350_set_a
= ( ^ [A: set_set_a,B: set_set_a] :
! [C: set_a] :
( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_254_subset__iff) ).
thf(863,plain,
( ord_le318720350_set_a
= ( ^ [A: set_set_a,B: set_set_a] :
! [C: set_a] :
( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(243,axiom,
! [A: partia539144763t_unit,B: set_se1596668135od_a_a,C: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( locali1488796916t_unit @ A @ B )
=> ( ( member1838126896od_a_a @ C @ B )
=> ( ( member1838126896od_a_a @ D @ B )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ A @ C @ D ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_33_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Om__closed) ).
thf(1175,plain,
! [A: partia539144763t_unit,B: set_se1596668135od_a_a,C: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( locali1488796916t_unit @ A @ B )
=> ( ( member1838126896od_a_a @ C @ B )
=> ( ( member1838126896od_a_a @ D @ B )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ A @ C @ D ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).
thf(142,axiom,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali1648887798ac_a_b @ A @ B )
=> ( locali807230110ng_a_b @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_23_eq__obj__rng__of__frac_Oaxioms_I2_J) ).
thf(811,plain,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali1648887798ac_a_b @ A @ B )
=> ( locali807230110ng_a_b @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(144,axiom,
! [A: partia1833973666xt_a_b] :
( ! [B: a,C: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) )
=> ( ! [B: a,C: a,D: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
= ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
= B ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ ( one_a_ring_ext_a_b @ A ) )
= B ) )
=> ( monoid1022105667xt_a_b @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_129_monoidI) ).
thf(816,plain,
! [A: partia1833973666xt_a_b] :
( ! [B: a,C: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) )
=> ( ! [B: a,C: a,D: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
= ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
= B ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ ( one_a_ring_ext_a_b @ A ) )
= B ) )
=> ( monoid1022105667xt_a_b @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(76,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a,D: int] :
( ( ring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( add_pow_a_b_int @ A @ D @ B ) @ C )
= ( add_pow_a_b_int @ A @ D @ ( mult_a_ring_ext_a_b @ A @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_114_ring_Oadd__pow__ldistr__int) ).
thf(594,plain,
! [A: partia1833973666xt_a_b,B: a,C: a,D: int] :
( ( ring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( add_pow_a_b_int @ A @ D @ B ) @ C )
= ( add_pow_a_b_int @ A @ D @ ( mult_a_ring_ext_a_b @ A @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(86,axiom,
! [A: partia1833973666xt_a_b] :
( ( field_a_b @ A )
=> ( ring_a_b @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_111_field_Ois__ring) ).
thf(621,plain,
! [A: partia1833973666xt_a_b] :
( ( field_a_b @ A )
=> ( ring_a_b @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(166,axiom,
! [A: a,B: set_a] :
( ( insert_a @ A @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_282_insert__absorb2) ).
thf(895,plain,
! [A: a,B: set_a] :
( ( insert_a @ A @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(294,axiom,
! [A: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( one_a_ring_ext_a_b @ r ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_68_r__one) ).
thf(1344,plain,
! [A: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( one_a_ring_ext_a_b @ r ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).
thf(174,axiom,
! [A: product_prod_a_a > $o] :
( ( ( collec645855634od_a_a @ A )
= bot_bo2131659635od_a_a )
= ( ! [B: product_prod_a_a] :
~ ( A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_203_Collect__empty__eq) ).
thf(943,plain,
! [A: product_prod_a_a > $o] :
( ( ( collec645855634od_a_a @ A )
= bot_bo2131659635od_a_a )
= ( ! [B: product_prod_a_a] :
~ ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(5,axiom,
! [A: partia96731725t_unit,B: a,C: a] :
( ( group_a_Product_unit @ A )
=> ( ( ( mult_a_Product_unit @ A @ B @ C )
= ( one_a_Product_unit @ A ) )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ( ( mult_a_Product_unit @ A @ C @ B )
= ( one_a_Product_unit @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_178_group_Oinv__comm) ).
thf(327,plain,
! [A: partia96731725t_unit,B: a,C: a] :
( ( group_a_Product_unit @ A )
=> ( ( ( mult_a_Product_unit @ A @ B @ C )
= ( one_a_Product_unit @ A ) )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ( ( mult_a_Product_unit @ A @ C @ B )
= ( one_a_Product_unit @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(328,plain,
! [A: partia96731725t_unit] :
( ( group_a_Product_unit @ A )
=> ! [B: a,C: a] :
( ( ( mult_a_Product_unit @ A @ B @ C )
= ( one_a_Product_unit @ A ) )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ( ( mult_a_Product_unit @ A @ C @ B )
= ( one_a_Product_unit @ A ) ) ) ) ) ),
inference(miniscope,[status(thm)],[327]) ).
thf(329,plain,
! [C: a,B: a,A: partia96731725t_unit] :
( ~ ( group_a_Product_unit @ A )
| ( ( mult_a_Product_unit @ A @ B @ C )
!= ( one_a_Product_unit @ A ) )
| ~ ( member_a @ B @ ( partia1955795460t_unit @ A ) )
| ~ ( member_a @ C @ ( partia1955795460t_unit @ A ) )
| ( ( mult_a_Product_unit @ A @ C @ B )
= ( one_a_Product_unit @ A ) ) ),
inference(cnf,[status(esa)],[328]) ).
thf(330,plain,
! [C: a,B: a,A: partia96731725t_unit] :
( ( ( mult_a_Product_unit @ A @ B @ C )
!= ( one_a_Product_unit @ A ) )
| ( ( mult_a_Product_unit @ A @ C @ B )
= ( one_a_Product_unit @ A ) )
| ~ ( group_a_Product_unit @ A )
| ~ ( member_a @ B @ ( partia1955795460t_unit @ A ) )
| ~ ( member_a @ C @ ( partia1955795460t_unit @ A ) ) ),
inference(lifteq,[status(thm)],[329]) ).
thf(344,plain,
! [A: partia96731725t_unit] :
( ~ ( group_a_Product_unit @ A )
| ( ( units_a_Product_unit @ A )
= ( partia1955795460t_unit @ A ) ) ),
inference(cnf,[status(esa)],[343]) ).
thf(345,plain,
! [A: partia96731725t_unit] :
( ( ( units_a_Product_unit @ A )
= ( partia1955795460t_unit @ A ) )
| ~ ( group_a_Product_unit @ A ) ),
inference(lifteq,[status(thm)],[344]) ).
thf(154,axiom,
! [A: a,B: set_a] :
( ( member_a @ A @ B )
=> ( ( insert_a @ A @ B )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_322_insert__absorb) ).
thf(858,plain,
! [A: a,B: set_a] :
( ( member_a @ A @ B )
=> ( ( insert_a @ A @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(240,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( group_52730354t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ D @ ( partia616129906t_unit @ A ) )
=> ( ( ( mult_s977248631t_unit @ A @ C @ B )
= ( mult_s977248631t_unit @ A @ D @ B ) )
= ( C = D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_155_Group_Ogroup_Oright__cancel) ).
thf(1162,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( group_52730354t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ D @ ( partia616129906t_unit @ A ) )
=> ( ( ( mult_s977248631t_unit @ A @ C @ B )
= ( mult_s977248631t_unit @ A @ D @ B ) )
= ( C = D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).
thf(725,plain,
! [A: partia539144763t_unit] :
( ~ ( monoid1242292471t_unit @ A )
| ( ( partia616129906t_unit @ A )
!= bot_bo1424703315od_a_a ) ),
inference(cnf,[status(esa)],[724]) ).
thf(726,plain,
! [A: partia539144763t_unit] :
( ( ( partia616129906t_unit @ A )
!= bot_bo1424703315od_a_a )
| ~ ( monoid1242292471t_unit @ A ) ),
inference(lifteq,[status(thm)],[725]) ).
thf(151,axiom,
! [A: set_Product_prod_a_a] :
( ( ! [B: product_prod_a_a] :
~ ( member449909584od_a_a @ B @ A ) )
= ( A = bot_bo2131659635od_a_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_201_all__not__in__conv) ).
thf(848,plain,
! [A: set_Product_prod_a_a] :
( ( ! [B: product_prod_a_a] :
~ ( member449909584od_a_a @ B @ A ) )
= ( A = bot_bo2131659635od_a_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(111,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a,D: int] :
( ( ring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ ( add_pow_a_b_int @ A @ D @ C ) )
= ( add_pow_a_b_int @ A @ D @ ( mult_a_ring_ext_a_b @ A @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_115_ring_Oadd__pow__rdistr__int) ).
thf(700,plain,
! [A: partia1833973666xt_a_b,B: a,C: a,D: int] :
( ( ring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ ( add_pow_a_b_int @ A @ D @ C ) )
= ( add_pow_a_b_int @ A @ D @ ( mult_a_ring_ext_a_b @ A @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(53,axiom,
! [A: a > $o] :
( ( ( collect_a @ A )
= bot_bot_set_a )
= ( ! [B: a] :
~ ( A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_204_Collect__empty__eq) ).
thf(521,plain,
! [A: a > $o] :
( ( ( collect_a @ A )
= bot_bot_set_a )
= ( ! [B: a] :
~ ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(185,axiom,
( ( (=) @ set_set_a )
= ( ^ [A: set_set_a,B: set_set_a] :
( ( ord_le318720350_set_a @ A @ B )
& ( ord_le318720350_set_a @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_262_set__eq__subset) ).
thf(982,plain,
( ( (=) @ set_set_a )
= ( ^ [A: set_set_a,B: set_set_a] :
( ( ord_le318720350_set_a @ A @ B )
& ( ord_le318720350_set_a @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(983,plain,
( ( ^ [A: set_set_a,B: set_set_a] :
( ( ord_le318720350_set_a @ A @ B )
& ( ord_le318720350_set_a @ B @ A ) ) )
= ( (=) @ set_set_a ) ),
inference(lifteq,[status(thm)],[982]) ).
thf(23,axiom,
! [A: partia96731725t_unit,B: a,C: a,D: a] :
( ( group_a_Product_unit @ A )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ( ( member_a @ D @ ( partia1955795460t_unit @ A ) )
=> ( ( ( mult_a_Product_unit @ A @ C @ B )
= ( mult_a_Product_unit @ A @ D @ B ) )
= ( C = D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_154_Group_Ogroup_Oright__cancel) ).
thf(399,plain,
! [A: partia96731725t_unit,B: a,C: a,D: a] :
( ( group_a_Product_unit @ A )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ( ( member_a @ D @ ( partia1955795460t_unit @ A ) )
=> ( ( ( mult_a_Product_unit @ A @ C @ B )
= ( mult_a_Product_unit @ A @ D @ B ) )
= ( C = D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(2144,plain,
! [A: a] :
( ~ $true
| ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) )
!= ( member_a @ A
@ ( collect_a
@ ^ [B: a] : $false ) ) ) ),
inference(paramod_ordered,[status(thm)],[1317,2119]) ).
thf(2145,plain,
! [A: a] :
( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) )
!= ( member_a @ A
@ ( collect_a
@ ^ [B: a] : $false ) ) ),
inference(simp,[status(thm)],[2144]) ).
thf(2157,plain,
! [A: a] :
( ( ( one_a_ring_ext_a_b @ r )
!= A )
| ( ( collect_a
@ ^ [B: a] : $false )
!= ( units_a_ring_ext_a_b @ r ) ) ),
inference(simp,[status(thm)],[2145]) ).
thf(2171,plain,
( ( collect_a
@ ^ [A: a] : $false )
!= ( units_a_ring_ext_a_b @ r ) ),
inference(simp,[status(thm)],[2157]) ).
thf(254,axiom,
( ( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia1066395285xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( field_a_b @ r ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_353_cring__fieldI) ).
thf(1213,plain,
( ( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia1066395285xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( field_a_b @ r ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).
thf(81,axiom,
! [A: partia1833973666xt_a_b] :
( ( ring_a_b @ A )
=> ( monoid1022105667xt_a_b @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_74_ring_Ois__monoid) ).
thf(608,plain,
! [A: partia1833973666xt_a_b] :
( ( ring_a_b @ A )
=> ( monoid1022105667xt_a_b @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(192,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( B
= ( mult_a_ring_ext_a_b @ A @ B @ C ) )
= ( C
= ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_165_group_Ol__cancel__one_H) ).
thf(1000,plain,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( B
= ( mult_a_ring_ext_a_b @ A @ B @ C ) )
= ( C
= ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(266,axiom,
! [A: a,B: int] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( member_a @ ( add_pow_a_b_int @ r @ B @ A ) @ ( partia1066395285xt_a_b @ r ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_109_add_Oint__pow__closed) ).
thf(1250,plain,
! [A: a,B: int] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( member_a @ ( add_pow_a_b_int @ r @ B @ A ) @ ( partia1066395285xt_a_b @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).
thf(152,axiom,
! [A: partia539144763t_unit] :
( ( monoid1242292471t_unit @ A )
=> ( member1838126896od_a_a @ ( one_se1488596747t_unit @ A ) @ ( partia616129906t_unit @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_99_monoid_Oone__closed) ).
thf(851,plain,
! [A: partia539144763t_unit] :
( ( monoid1242292471t_unit @ A )
=> ( member1838126896od_a_a @ ( one_se1488596747t_unit @ A ) @ ( partia616129906t_unit @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(48,axiom,
! [A: partia1833973666xt_a_b,B: a] :
( ( semiring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_87_semiring_Osemiring__simprules_I9_J) ).
thf(505,plain,
! [A: partia1833973666xt_a_b,B: a] :
( ( semiring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(33,axiom,
! [A: product_prod_a_a,B: product_prod_a_a] :
( ( ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a )
= ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) )
=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_346_singleton__inject) ).
thf(431,plain,
! [A: product_prod_a_a,B: product_prod_a_a] :
( ( ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a )
= ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(98,axiom,
! [A: partia1833973666xt_a_b] :
( ( monoid1022105667xt_a_b @ A )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
& ( ( mult_a_ring_ext_a_b @ A @ C @ B )
= ( one_a_ring_ext_a_b @ A ) ) ) )
=> ( group_a_ring_ext_a_b @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_186_monoid_Ogroup__l__invI) ).
thf(658,plain,
! [A: partia1833973666xt_a_b] :
( ( monoid1022105667xt_a_b @ A )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
& ( ( mult_a_ring_ext_a_b @ A @ C @ B )
= ( one_a_ring_ext_a_b @ A ) ) ) )
=> ( group_a_ring_ext_a_b @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(158,axiom,
! [A: set_a > $o,B: set_a > $o] :
( ( ord_le318720350_set_a @ ( collect_set_a @ A ) @ ( collect_set_a @ B ) )
= ( ! [C: set_a] :
( ( A @ C )
=> ( B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_264_Collect__mono__iff) ).
thf(868,plain,
! [A: set_a > $o,B: set_a > $o] :
( ( ord_le318720350_set_a @ ( collect_set_a @ A ) @ ( collect_set_a @ B ) )
= ( ! [C: set_a] :
( ( A @ C )
=> ( B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(41,axiom,
! [A: partia96731725t_unit] :
( ( monoid1447254976t_unit @ A )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
& ( ( mult_a_Product_unit @ A @ C @ B )
= ( one_a_Product_unit @ A ) ) ) )
=> ( group_a_Product_unit @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_184_monoid_Ogroup__l__invI) ).
thf(473,plain,
! [A: partia96731725t_unit] :
( ( monoid1447254976t_unit @ A )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
& ( ( mult_a_Product_unit @ A @ C @ B )
= ( one_a_Product_unit @ A ) ) ) )
=> ( group_a_Product_unit @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(119,axiom,
! [A: partia539144763t_unit] :
( ( monoid1242292471t_unit @ A )
=> ( group_52730354t_unit @ ( units_390215097t_unit @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_273_monoid_Ounits__group) ).
thf(722,plain,
! [A: partia539144763t_unit] :
( ( monoid1242292471t_unit @ A )
=> ( group_52730354t_unit @ ( units_390215097t_unit @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(103,axiom,
! [A: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ B @ bot_bot_set_a ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_311_subset__singletonD) ).
thf(675,plain,
! [A: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ B @ bot_bot_set_a ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(241,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ ( a_r_coset_a_b @ r @ A @ C ) @ B )
= ( set_add_a_b @ r @ A @ ( a_l_coset_a_b @ r @ C @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_267_a__rcos__assoc__lcos) ).
thf(1166,plain,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ ( a_r_coset_a_b @ r @ A @ C ) @ B )
= ( set_add_a_b @ r @ A @ ( a_l_coset_a_b @ r @ C @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).
thf(274,axiom,
semiring_a_b @ r,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_11_local_Osemiring__axioms) ).
thf(1277,plain,
semiring_a_b @ r,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).
thf(84,axiom,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali425460998xt_a_b @ A @ B )
=> ( ord_less_eq_set_a @ B @ ( partia1066395285xt_a_b @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_159_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Osubset) ).
thf(617,plain,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali425460998xt_a_b @ A @ B )
=> ( ord_less_eq_set_a @ B @ ( partia1066395285xt_a_b @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(341,plain,
! [B: a,A: partia96731725t_unit] :
( ~ ( group_a_Product_unit @ A )
| ~ ( member_a @ B @ ( partia1955795460t_unit @ A ) )
| ( ( mult_a_Product_unit @ A @ B @ ( sk2 @ B @ A ) )
= ( one_a_Product_unit @ A ) ) ),
inference(cnfConj,[status(thm)],[339]) ).
thf(342,plain,
! [B: a,A: partia96731725t_unit] :
( ( ( mult_a_Product_unit @ A @ B @ ( sk2 @ B @ A ) )
= ( one_a_Product_unit @ A ) )
| ~ ( group_a_Product_unit @ A )
| ~ ( member_a @ B @ ( partia1955795460t_unit @ A ) ) ),
inference(lifteq,[status(thm)],[341]) ).
thf(234,axiom,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
= ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r2 @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_3_f1) ).
thf(1145,plain,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
= ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r2 @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).
thf(146,axiom,
! [A: set_a,B: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ B ) )
= ( B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_51_mem__Collect__eq) ).
thf(836,plain,
! [A: set_a,B: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(203,axiom,
! [A: partia1833973666xt_a_b] :
( ( group_a_ring_ext_a_b @ A )
=> ( monoid1022105667xt_a_b @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_153_group_Ois__monoid) ).
thf(1049,plain,
! [A: partia1833973666xt_a_b] :
( ( group_a_ring_ext_a_b @ A )
=> ( monoid1022105667xt_a_b @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(99,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( field_a_b @ A )
=> ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= ( zero_a_b @ A ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( B
= ( zero_a_b @ A ) )
| ( C
= ( zero_a_b @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_113_integral) ).
thf(663,plain,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( field_a_b @ A )
=> ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= ( zero_a_b @ A ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( B
= ( zero_a_b @ A ) )
| ( C
= ( zero_a_b @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(62,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( ( mult_a_ring_ext_a_b @ A @ C @ B )
= B )
= ( C
= ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_168_group_Or__cancel__one) ).
thf(548,plain,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( ( mult_a_ring_ext_a_b @ A @ C @ B )
= B )
= ( C
= ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(229,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A @ B ) @ ( partia1066395285xt_a_b @ r ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_191_setadd__subset__G) ).
thf(1134,plain,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A @ B ) @ ( partia1066395285xt_a_b @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).
thf(95,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le318720350_set_a @ A @ B )
=> ( ( ord_le318720350_set_a @ B @ C )
=> ( ord_le318720350_set_a @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_260_subset__trans) ).
thf(648,plain,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le318720350_set_a @ A @ B )
=> ( ( ord_le318720350_set_a @ B @ C )
=> ( ord_le318720350_set_a @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(54,axiom,
( ord_less_eq_set_a
= ( ^ [A: set_a,B: set_a] :
! [C: a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_253_subset__iff) ).
thf(524,plain,
( ord_less_eq_set_a
= ( ^ [A: set_a,B: set_a] :
! [C: a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(525,plain,
( ord_less_eq_set_a
= ( ^ [A: set_a,B: set_a] :
! [C: a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ),
inference(lifteq,[status(thm)],[524]) ).
thf(92,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le318720350_set_a @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_250_equalityD2) ).
thf(640,plain,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le318720350_set_a @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(198,axiom,
! [A: set_Product_prod_a_a,B: product_prod_a_a] :
( ( A = bot_bo2131659635od_a_a )
=> ~ ( member449909584od_a_a @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_223_equals0D) ).
thf(1025,plain,
! [A: set_Product_prod_a_a,B: product_prod_a_a] :
( ( A = bot_bo2131659635od_a_a )
=> ~ ( member449909584od_a_a @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(238,axiom,
! [A: set_a,B: set_a] :
( ( additi2104487374up_a_b @ A @ r )
=> ( ( additi2104487374up_a_b @ B @ r )
=> ( additi2104487374up_a_b @ ( set_add_a_b @ r @ A @ B ) @ r ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_266_add__additive__subgroups) ).
thf(1156,plain,
! [A: set_a,B: set_a] :
( ( additi2104487374up_a_b @ A @ r )
=> ( ( additi2104487374up_a_b @ B @ r )
=> ( additi2104487374up_a_b @ ( set_add_a_b @ r @ A @ B ) @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).
thf(113,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_215_empty__subsetI) ).
thf(707,plain,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(177,axiom,
! [A: set_a] :
( ( ! [B: a] :
~ ( member_a @ B @ A ) )
= ( A = bot_bot_set_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_202_all__not__in__conv) ).
thf(954,plain,
! [A: set_a] :
( ( ! [B: a] :
~ ( member_a @ B @ A ) )
= ( A = bot_bot_set_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(222,axiom,
( ( ( partia1066395285xt_a_b @ r )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_291_carrier__one__not__zero) ).
thf(1098,plain,
( ( ( partia1066395285xt_a_b @ r )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).
thf(297,axiom,
! [A: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ A @ ( partia1066395285xt_a_b @ r ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_57_Units__closed) ).
thf(1368,plain,
! [A: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ A @ ( partia1066395285xt_a_b @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).
thf(2146,plain,
! [B: a,A: a] :
( ~ $true
| ( ( member_a @ A
@ ( collect_a
@ ^ [C: a] : $true ) )
!= ( member_a @ B
@ ( collect_a
@ ^ [C: a] : $false ) ) ) ),
inference(paramod_ordered,[status(thm)],[2078,2119]) ).
thf(2147,plain,
! [B: a,A: a] :
( ( member_a @ A
@ ( collect_a
@ ^ [C: a] : $true ) )
!= ( member_a @ B
@ ( collect_a
@ ^ [C: a] : $false ) ) ),
inference(simp,[status(thm)],[2146]) ).
thf(2161,plain,
! [B: a,A: a] :
( ( A != B )
| ( ( collect_a
@ ^ [C: a] : $false )
!= ( collect_a
@ ^ [C: a] : $true ) ) ),
inference(simp,[status(thm)],[2147]) ).
thf(2175,plain,
( ( collect_a
@ ^ [A: a] : $false )
!= ( collect_a
@ ^ [A: a] : $true ) ),
inference(simp,[status(thm)],[2161]) ).
thf(73,axiom,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali425460998xt_a_b @ A @ B )
=> ( member_a @ ( one_a_ring_ext_a_b @ A ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_28_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Oone__closed) ).
thf(585,plain,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali425460998xt_a_b @ A @ B )
=> ( member_a @ ( one_a_ring_ext_a_b @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(216,axiom,
member449909584od_a_a @ ( product_Pair_a_a @ r3 @ s3 ) @ ( partia206007992t_unit @ rel ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_14_assms_I2_J) ).
thf(1085,plain,
member449909584od_a_a @ ( product_Pair_a_a @ r3 @ s3 ) @ ( partia206007992t_unit @ rel ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(228,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ C )
= ( mult_a_ring_ext_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_46_m__assoc) ).
thf(1130,plain,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ C )
= ( mult_a_ring_ext_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).
thf(256,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( group_52730354t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( B
= ( mult_s977248631t_unit @ A @ C @ B ) )
= ( C
= ( one_se1488596747t_unit @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_161_group_Or__cancel__one_H) ).
thf(1220,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( group_52730354t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( B
= ( mult_s977248631t_unit @ A @ C @ B ) )
= ( C
= ( one_se1488596747t_unit @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).
thf(61,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a,D: nat] :
( ( semiring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ ( add_pow_a_b_nat @ A @ D @ C ) )
= ( add_pow_a_b_nat @ A @ D @ ( mult_a_ring_ext_a_b @ A @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_187_semiring_Oadd__pow__rdistr) ).
thf(544,plain,
! [A: partia1833973666xt_a_b,B: a,C: a,D: nat] :
( ( semiring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ B @ ( add_pow_a_b_nat @ A @ D @ C ) )
= ( add_pow_a_b_nat @ A @ D @ ( mult_a_ring_ext_a_b @ A @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(246,axiom,
! [A: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ B @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_41_Units__inv__comm) ).
thf(1186,plain,
! [A: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ B @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).
thf(304,axiom,
locali1648887798ac_a_b @ r @ s,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_2_eq__obj__rng__of__frac__axioms) ).
thf(1404,plain,
locali1648887798ac_a_b @ r @ s,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).
thf(334,plain,
! [C: set_a,B: set_set_a,A: set_a] :
( ( A != C )
| ( member_set_a @ A @ ( insert_set_a @ C @ B ) ) ),
inference(cnfConj,[status(thm)],[332]) ).
thf(335,plain,
! [C: set_a,B: set_set_a,A: set_a] :
( ( A != C )
| ( member_set_a @ A @ ( insert_set_a @ C @ B ) ) ),
inference(lifteq,[status(thm)],[334]) ).
thf(336,plain,
! [B: set_a,A: set_set_a] : ( member_set_a @ B @ ( insert_set_a @ B @ A ) ),
inference(simp,[status(thm)],[335]) ).
thf(233,axiom,
! [A: a,B: a,C: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B @ C )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
=> ( A = C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_40_inv__unique) ).
thf(1141,plain,
! [A: a,B: a,C: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B @ C )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
=> ( A = C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).
thf(197,axiom,
! [A: partia1833973666xt_a_b] :
( ( monoid1022105667xt_a_b @ A )
=> ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( units_a_ring_ext_a_b @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_126_monoid_OUnits__one__closed) ).
thf(1023,plain,
! [A: partia1833973666xt_a_b] :
( ( monoid1022105667xt_a_b @ A )
=> ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( units_a_ring_ext_a_b @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).
thf(864,plain,
( ord_le318720350_set_a
= ( ^ [A: set_set_a,B: set_set_a] :
! [C: set_a] :
( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ),
inference(lifteq,[status(thm)],[863]) ).
thf(121,axiom,
! [A: partia1833973666xt_a_b] :
( ( monoid1022105667xt_a_b @ A )
=> ( group_a_Product_unit @ ( units_1411277569xt_a_b @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_274_monoid_Ounits__group) ).
thf(727,plain,
! [A: partia1833973666xt_a_b] :
( ( monoid1022105667xt_a_b @ A )
=> ( group_a_Product_unit @ ( units_1411277569xt_a_b @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(161,axiom,
! [A: partia1833973666xt_a_b,B: a] :
( ( ring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_86_ring_Oring__simprules_I12_J) ).
thf(876,plain,
! [A: partia1833973666xt_a_b,B: a] :
( ( ring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(189,axiom,
! [A: set_Product_prod_a_a] :
( ( collec645855634od_a_a
@ ^ [B: product_prod_a_a] : ( member449909584od_a_a @ B @ A ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_53_Collect__mem__eq) ).
thf(991,plain,
! [A: set_Product_prod_a_a] :
( ( collec645855634od_a_a
@ ^ [B: product_prod_a_a] : ( member449909584od_a_a @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(184,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
= ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_119_monoid_Om__assoc) ).
thf(978,plain,
! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
= ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(232,axiom,
abelian_monoid_a_b @ r,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_17_abelian__monoid__axioms) ).
thf(1140,plain,
abelian_monoid_a_b @ r,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(37,axiom,
! [A: set_Product_prod_a_a] :
~ ( member1838126896od_a_a @ A @ bot_bo1424703315od_a_a ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_195_empty__iff) ).
thf(446,plain,
! [A: set_Product_prod_a_a] :
~ ( member1838126896od_a_a @ A @ bot_bo1424703315od_a_a ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(69,axiom,
! [A: a,B: a,C: set_a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ C ) )
= ( ( B = A )
& ( ord_less_eq_set_a @ C @ ( insert_a @ A @ bot_bot_set_a ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_305_singleton__insert__inj__eq) ).
thf(571,plain,
! [A: a,B: a,C: set_a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ C ) )
= ( ( B = A )
& ( ord_less_eq_set_a @ C @ ( insert_a @ A @ bot_bot_set_a ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(150,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( ring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_80_ring_Oring__simprules_I5_J) ).
thf(845,plain,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( ring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(88,axiom,
! [A: partia96731725t_unit] :
( ( group_a_Product_unit @ A )
=> ( monoid1447254976t_unit @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_151_group_Ois__monoid) ).
thf(627,plain,
! [A: partia96731725t_unit] :
( ( group_a_Product_unit @ A )
=> ( monoid1447254976t_unit @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(2350,plain,
! [A: partia96731725t_unit] :
( ( ( units_a_Product_unit @ A )
= ( partia1955795460t_unit @ A ) )
| ~ $true
| ( ( group_a_Product_unit @ ( units_1411277569xt_a_b @ r ) )
!= ( group_a_Product_unit @ A ) ) ),
inference(paramod_ordered,[status(thm)],[1086,345]) ).
thf(2351,plain,
! [A: partia96731725t_unit] :
( ( ( units_a_Product_unit @ A )
= ( partia1955795460t_unit @ A ) )
| ( ( group_a_Product_unit @ ( units_1411277569xt_a_b @ r ) )
!= ( group_a_Product_unit @ A ) ) ),
inference(simp,[status(thm)],[2350]) ).
thf(2352,plain,
( ( units_a_Product_unit @ ( units_1411277569xt_a_b @ r ) )
= ( partia1955795460t_unit @ ( units_1411277569xt_a_b @ r ) ) ),
inference(pattern_uni,[status(thm)],[2351:[bind(A,$thf( units_1411277569xt_a_b @ r ))]]) ).
thf(171,axiom,
! [A: a,B: a,C: set_a] :
( ( insert_a @ A @ ( insert_a @ B @ C ) )
= ( insert_a @ B @ ( insert_a @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_317_insert__commute) ).
thf(931,plain,
! [A: a,B: a,C: set_a] :
( ( insert_a @ A @ ( insert_a @ B @ C ) )
= ( insert_a @ B @ ( insert_a @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(1947,plain,
! [A: partia539144763t_unit] :
( ( ( partia616129906t_unit @ A )
!= bot_bo1424703315od_a_a )
| ~ $true
| ( ( monoid1242292471t_unit @ ( locali1944243560ac_a_b @ r @ s ) )
!= ( monoid1242292471t_unit @ A ) ) ),
inference(paramod_ordered,[status(thm)],[1433,726]) ).
thf(1948,plain,
! [A: partia539144763t_unit] :
( ( ( partia616129906t_unit @ A )
!= bot_bo1424703315od_a_a )
| ( ( monoid1242292471t_unit @ ( locali1944243560ac_a_b @ r @ s ) )
!= ( monoid1242292471t_unit @ A ) ) ),
inference(simp,[status(thm)],[1947]) ).
thf(1949,plain,
( ( partia616129906t_unit @ ( locali1944243560ac_a_b @ r @ s ) )
!= bot_bo1424703315od_a_a ),
inference(pattern_uni,[status(thm)],[1948:[bind(A,$thf( locali1944243560ac_a_b @ r @ s ))]]) ).
thf(46,axiom,
! [A: partia1833973666xt_a_b] :
( ( one_a_Product_unit @ ( units_1411277569xt_a_b @ A ) )
= ( one_a_ring_ext_a_b @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_277_units__of__one) ).
thf(495,plain,
! [A: partia1833973666xt_a_b] :
( ( one_a_Product_unit @ ( units_1411277569xt_a_b @ A ) )
= ( one_a_ring_ext_a_b @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(247,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( ( mult_s977248631t_unit @ A @ B @ C )
= ( one_se1488596747t_unit @ A ) )
=> ( ( ( mult_s977248631t_unit @ A @ C @ D )
= ( one_se1488596747t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ D @ ( partia616129906t_unit @ A ) )
=> ( B = D ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_134_monoid_Oinv__unique) ).
thf(1189,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( ( mult_s977248631t_unit @ A @ B @ C )
= ( one_se1488596747t_unit @ A ) )
=> ( ( ( mult_s977248631t_unit @ A @ C @ D )
= ( one_se1488596747t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( member1838126896od_a_a @ D @ ( partia616129906t_unit @ A ) )
=> ( B = D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).
thf(72,axiom,
! [A: partia1833973666xt_a_b,B: a] :
( ( ring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( zero_a_b @ A ) @ B )
= ( zero_a_b @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_76_ring_Oring__simprules_I24_J) ).
thf(581,plain,
! [A: partia1833973666xt_a_b,B: a] :
( ( ring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( zero_a_b @ A ) @ B )
= ( zero_a_b @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(255,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( ( mult_s977248631t_unit @ A @ B @ C )
= ( one_se1488596747t_unit @ A ) )
=> ( ( member1838126896od_a_a @ B @ ( units_1734235042t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( units_1734235042t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ C @ B )
= ( one_se1488596747t_unit @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_105_monoid_OUnits__inv__comm) ).
thf(1216,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( ( mult_s977248631t_unit @ A @ B @ C )
= ( one_se1488596747t_unit @ A ) )
=> ( ( member1838126896od_a_a @ B @ ( units_1734235042t_unit @ A ) )
=> ( ( member1838126896od_a_a @ C @ ( units_1734235042t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ C @ B )
= ( one_se1488596747t_unit @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).
thf(165,axiom,
! [A: set_Product_prod_a_a] :
( ! [B: product_prod_a_a] :
~ ( member449909584od_a_a @ B @ A )
=> ( A = bot_bo2131659635od_a_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_227_equals0I) ).
thf(891,plain,
! [A: set_Product_prod_a_a] :
( ! [B: product_prod_a_a] :
~ ( member449909584od_a_a @ B @ A )
=> ( A = bot_bo2131659635od_a_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(143,axiom,
! [A: partia1833973666xt_a_b] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( units_a_ring_ext_a_b @ A )
= ( partia1066395285xt_a_b @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_158_group_OUnits__eq) ).
thf(813,plain,
! [A: partia1833973666xt_a_b] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( units_a_ring_ext_a_b @ A )
= ( partia1066395285xt_a_b @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(180,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [B: set_a] : ( member_set_a @ B @ A ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_55_Collect__mem__eq) ).
thf(963,plain,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [B: set_a] : ( member_set_a @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(207,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a > $o] :
( ( member1838126896od_a_a @ A @ ( collec183727474od_a_a @ B ) )
= ( B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_50_mem__Collect__eq) ).
thf(1062,plain,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a > $o] :
( ( member1838126896od_a_a @ A @ ( collec183727474od_a_a @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(1836,plain,
! [A: a] :
( ~ $true
| ( ( member_a @ ( one_a_ring_ext_a_b @ r )
@ ( collect_a
@ ^ [B: a] : ( member_a @ B @ ( partia1066395285xt_a_b @ r ) ) ) )
!= ( member_a @ A @ bot_bot_set_a ) ) ),
inference(paramod_ordered,[status(thm)],[1622,554]) ).
thf(1837,plain,
! [A: a] :
( ( member_a @ ( one_a_ring_ext_a_b @ r )
@ ( collect_a
@ ^ [B: a] : ( member_a @ B @ ( partia1066395285xt_a_b @ r ) ) ) )
!= ( member_a @ A @ bot_bot_set_a ) ),
inference(simp,[status(thm)],[1836]) ).
thf(1849,plain,
! [A: a] :
( ( ( one_a_ring_ext_a_b @ r )
!= A )
| ( ( collect_a
@ ^ [B: a] : ( member_a @ B @ ( partia1066395285xt_a_b @ r ) ) )
!= bot_bot_set_a ) ),
inference(simp,[status(thm)],[1837]) ).
thf(1854,plain,
( ( collect_a
@ ^ [A: a] : ( member_a @ A @ ( partia1066395285xt_a_b @ r ) ) )
!= bot_bot_set_a ),
inference(simp,[status(thm)],[1849]) ).
thf(2399,plain,
member_a @ sk113 @ ( units_a_ring_ext_a_b @ r ),
inference(func_ext,[status(esa)],[2398]) ).
thf(2408,plain,
! [B: a > $o,A: a] :
( ( member_a @ A @ ( collect_a @ B ) )
| ( ( B @ A )
!= ( member_a @ sk113 @ ( units_a_ring_ext_a_b @ r ) ) ) ),
inference(paramod_ordered,[status(thm)],[320,2399]) ).
thf(2431,plain,
( member_a @ sk113
@ ( collect_a
@ ^ [A: a] : ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ),
inference(pre_uni,[status(thm)],[2408:[bind(A,$thf( sk113 )),bind(B,$thf( ^ [C: a] : ( member_a @ C @ ( units_a_ring_ext_a_b @ r ) ) ))]]) ).
thf(29,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,C: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ C ) )
=> ( ( A != B )
=> ( member449909584od_a_a @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_343_insertE) ).
thf(418,plain,
! [A: product_prod_a_a,B: product_prod_a_a,C: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ C ) )
=> ( ( A != B )
=> ( member449909584od_a_a @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(133,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_220_emptyE) ).
thf(781,plain,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(267,axiom,
! [A: partia539144763t_unit] :
( ( mult_s977248631t_unit @ ( units_390215097t_unit @ A ) )
= ( mult_s977248631t_unit @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_276_units__of__mult) ).
thf(1253,plain,
! [A: partia539144763t_unit] :
( ( mult_s977248631t_unit @ ( units_390215097t_unit @ A ) )
= ( mult_s977248631t_unit @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).
thf(699,plain,
( ( ^ [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ B @ A ) ) )
= ( (=) @ set_a ) ),
inference(lifteq,[status(thm)],[698]) ).
thf(293,axiom,
( ( ( partia1066395285xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_280_one__zeroI) ).
thf(1341,plain,
( ( ( partia1066395285xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).
thf(127,axiom,
! [A: partia1833973666xt_a_b] :
( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ! [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ! [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ! [D: a] :
( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
= ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
= B ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
& ( ( mult_a_ring_ext_a_b @ A @ C @ B )
= ( one_a_ring_ext_a_b @ A ) ) ) )
=> ( group_a_ring_ext_a_b @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_183_groupI) ).
thf(748,plain,
! [A: partia1833973666xt_a_b] :
( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ! [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ! [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ! [D: a] :
( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
= ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
= B ) )
=> ( ! [B: a] :
( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ? [C: a] :
( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
& ( ( mult_a_ring_ext_a_b @ A @ C @ B )
= ( one_a_ring_ext_a_b @ A ) ) ) )
=> ( group_a_ring_ext_a_b @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(175,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= B )
= ( C
= ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_171_group_Ol__cancel__one) ).
thf(946,plain,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= B )
= ( C
= ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(223,axiom,
! [A: int] :
( ( add_pow_a_b_int @ r @ A @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_110_add_Oint__pow__one) ).
thf(1101,plain,
! [A: int] :
( ( add_pow_a_b_int @ r @ A @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).
thf(199,axiom,
! [A: set_a,B: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_224_equals0D) ).
thf(1030,plain,
! [A: set_a,B: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(68,axiom,
! [A: a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ A @ B ) @ C )
= ( ( member_a @ A @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_299_insert__subset) ).
thf(568,plain,
! [A: a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ A @ B ) @ C )
= ( ( member_a @ A @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(100,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le318720350_set_a @ A @ B )
=> ( ( ord_le318720350_set_a @ B @ A )
=> ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_212_subset__antisym) ).
thf(667,plain,
! [A: set_set_a,B: set_set_a] :
( ( ord_le318720350_set_a @ A @ B )
=> ( ( ord_le318720350_set_a @ B @ A )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(2312,plain,
! [B: a > $o,A: a] :
( ( member_a @ A @ ( collect_a @ B ) )
| ( ( B @ A )
!= ( member_a @ sk112 @ ( partia1066395285xt_a_b @ r ) ) ) ),
inference(paramod_ordered,[status(thm)],[320,2304]) ).
thf(2330,plain,
( member_a @ sk112
@ ( collect_a
@ ^ [A: a] : ( member_a @ A @ ( partia1066395285xt_a_b @ r ) ) ) ),
inference(pre_uni,[status(thm)],[2312:[bind(A,$thf( sk112 )),bind(B,$thf( ^ [C: a] : ( member_a @ C @ ( partia1066395285xt_a_b @ r ) ) ))]]) ).
thf(248,axiom,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) )
= ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r3 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_4_f2) ).
thf(1193,plain,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) )
= ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r3 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).
thf(19,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ B )
=> ? [C: set_Product_prod_a_a] :
( ( B
= ( insert1116662519od_a_a @ A @ C ) )
& ~ ( member449909584od_a_a @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_314_mk__disjoint__insert) ).
thf(383,plain,
! [A: product_prod_a_a,B: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ B )
=> ? [C: set_Product_prod_a_a] :
( ( B
= ( insert1116662519od_a_a @ A @ C ) )
& ~ ( member449909584od_a_a @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(9,axiom,
! [A: set_a,B: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_341_insertI1) ).
thf(346,plain,
! [A: set_a,B: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(202,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_247_equalityD1) ).
thf(1045,plain,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).
thf(42,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [B: a] : ( member_a @ B @ A ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_52_Collect__mem__eq) ).
thf(478,plain,
! [A: set_a] :
( ( collect_a
@ ^ [B: a] : ( member_a @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(117,axiom,
! [A: partia539144763t_unit] :
( ( one_se1488596747t_unit @ ( units_390215097t_unit @ A ) )
= ( one_se1488596747t_unit @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_278_units__of__one) ).
thf(716,plain,
! [A: partia539144763t_unit] :
( ( one_se1488596747t_unit @ ( units_390215097t_unit @ A ) )
= ( one_se1488596747t_unit @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(138,axiom,
! [A: set_a,B: set_a] :
( ! [C: a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_209_subsetI) ).
thf(800,plain,
! [A: set_a,B: set_a] :
( ! [C: a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(26,axiom,
! [A: set_a,B: set_set_a] :
( ( member_set_a @ A @ B )
=> ( ( insert_set_a @ A @ B )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_325_insert__absorb) ).
thf(410,plain,
! [A: set_a,B: set_set_a] :
( ( member_set_a @ A @ B )
=> ( ( insert_set_a @ A @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(176,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= ( mult_a_ring_ext_a_b @ A @ B @ D ) )
= ( C = D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_139_monoid_OUnits__l__cancel) ).
thf(950,plain,
! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
( ( monoid1022105667xt_a_b @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
=> ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
= ( mult_a_ring_ext_a_b @ A @ B @ D ) )
= ( C = D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(259,axiom,
! [A: a,B: a,C: a,D: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ C @ D ) @ ( partia206007992t_unit @ rel ) )
=> ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ A @ B ) @ ( locali1170246543t_unit @ rel @ C @ D ) )
= ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_18_mult__rng__of__frac__fundamental__lemma) ).
thf(1229,plain,
! [A: a,B: a,C: a,D: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ C @ D ) @ ( partia206007992t_unit @ rel ) )
=> ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ A @ B ) @ ( locali1170246543t_unit @ rel @ C @ D ) )
= ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).
thf(135,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( B
= ( mult_a_ring_ext_a_b @ A @ C @ B ) )
= ( C
= ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_162_group_Or__cancel__one_H) ).
thf(790,plain,
! [A: partia1833973666xt_a_b,B: a,C: a] :
( ( group_a_ring_ext_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( B
= ( mult_a_ring_ext_a_b @ A @ C @ B ) )
= ( C
= ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(157,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( units_1734235042t_unit @ A ) )
=> ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_122_monoid_OUnits__closed) ).
thf(865,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( units_1734235042t_unit @ A ) )
=> ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(16,axiom,
! [A: partia96731725t_unit,B: a,C: a] :
( ( group_a_Product_unit @ A )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ( ( B
= ( mult_a_Product_unit @ A @ B @ C ) )
= ( C
= ( one_a_Product_unit @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_163_group_Ol__cancel__one_H) ).
thf(373,plain,
! [A: partia96731725t_unit,B: a,C: a] :
( ( group_a_Product_unit @ A )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ( ( B
= ( mult_a_Product_unit @ A @ B @ C ) )
= ( C
= ( one_a_Product_unit @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(67,axiom,
! [A: partia1833973666xt_a_b,B: a,C: a,D: nat] :
( ( semiring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( add_pow_a_b_nat @ A @ D @ B ) @ C )
= ( add_pow_a_b_nat @ A @ D @ ( mult_a_ring_ext_a_b @ A @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_188_semiring_Oadd__pow__ldistr) ).
thf(564,plain,
! [A: partia1833973666xt_a_b,B: a,C: a,D: nat] :
( ( semiring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( add_pow_a_b_nat @ A @ D @ B ) @ C )
= ( add_pow_a_b_nat @ A @ D @ ( mult_a_ring_ext_a_b @ A @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(325,plain,
! [B: a,A: partia96731725t_unit] :
( ~ ( group_a_Product_unit @ A )
| ~ ( member_a @ B @ ( partia1955795460t_unit @ A ) )
| ( ( mult_a_Product_unit @ A @ ( sk1 @ B @ A ) @ B )
= ( one_a_Product_unit @ A ) ) ),
inference(cnfConj,[status(thm)],[323]) ).
thf(326,plain,
! [B: a,A: partia96731725t_unit] :
( ( ( mult_a_Product_unit @ A @ ( sk1 @ B @ A ) @ B )
= ( one_a_Product_unit @ A ) )
| ~ ( group_a_Product_unit @ A )
| ~ ( member_a @ B @ ( partia1955795460t_unit @ A ) ) ),
inference(lifteq,[status(thm)],[325]) ).
thf(129,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_43_old_Oprod_Oinject) ).
thf(767,plain,
! [A: a,B: a,C: a,D: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(206,axiom,
! [A: a,B: set_a,C: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = C ) )
=> ( member_a @ A @ ( insert_a @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_287_insertCI) ).
thf(1056,plain,
! [A: a,B: set_a,C: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = C ) )
=> ( member_a @ A @ ( insert_a @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(179,axiom,
! [A: set_a] :
( ( ? [B: a] : ( member_a @ B @ A ) )
= ( A != bot_bot_set_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_232_ex__in__conv) ).
thf(960,plain,
! [A: set_a] :
( ( ? [B: a] : ( member_a @ B @ A ) )
= ( A != bot_bot_set_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(208,axiom,
( ord_le318720350_set_a
= ( ^ [A: set_set_a,B: set_set_a] :
! [C: set_a] :
( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_246_subset__eq) ).
thf(1065,plain,
( ord_le318720350_set_a
= ( ^ [A: set_set_a,B: set_set_a] :
! [C: set_a] :
( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(12,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,C: product_prod_a_a,D: product_prod_a_a] :
( ( ( insert1116662519od_a_a @ A @ ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) )
= ( insert1116662519od_a_a @ C @ ( insert1116662519od_a_a @ D @ bot_bo2131659635od_a_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_350_doubleton__eq__iff) ).
thf(358,plain,
! [A: product_prod_a_a,B: product_prod_a_a,C: product_prod_a_a,D: product_prod_a_a] :
( ( ( insert1116662519od_a_a @ A @ ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) )
= ( insert1116662519od_a_a @ C @ ( insert1116662519od_a_a @ D @ bot_bo2131659635od_a_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(140,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_255_subset__refl) ).
thf(807,plain,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(35,axiom,
! [A: set_a,B: set_set_a] :
( ( member_set_a @ A @ B )
=> ? [C: set_set_a] :
( ( B
= ( insert_set_a @ A @ C ) )
& ~ ( member_set_a @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_316_mk__disjoint__insert) ).
thf(437,plain,
! [A: set_a,B: set_set_a] :
( ( member_set_a @ A @ B )
=> ? [C: set_set_a] :
( ( B
= ( insert_set_a @ A @ C ) )
& ~ ( member_set_a @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(291,axiom,
! [A: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( member_set_a @ ( a_r_coset_a_b @ r @ A @ B ) @ ( a_RCOSETS_a_b @ r @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_279_a__rcosetsI) ).
thf(1335,plain,
! [A: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
=> ( member_set_a @ ( a_r_coset_a_b @ r @ A @ B ) @ ( a_RCOSETS_a_b @ r @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).
thf(170,axiom,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali425460998xt_a_b @ A @ B )
=> ( monoid1022105667xt_a_b @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_22_submonoid_Oaxioms_I1_J) ).
thf(928,plain,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali425460998xt_a_b @ A @ B )
=> ( monoid1022105667xt_a_b @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(28,axiom,
! [A: set_a,B: set_a,C: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ C ) )
=> ( ( A != B )
=> ( member_set_a @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_345_insertE) ).
thf(415,plain,
! [A: set_a,B: set_a,C: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ C ) )
=> ( ( A != B )
=> ( member_set_a @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(271,axiom,
! [A: set_a] :
( ( additi2104487374up_a_b @ A @ r )
=> ( ord_le318720350_set_a @ ( a_RCOSETS_a_b @ r @ A ) @ ( pow_a @ ( partia1066395285xt_a_b @ r ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_354_rcosets__subset__PowG) ).
thf(1268,plain,
! [A: set_a] :
( ( additi2104487374up_a_b @ A @ r )
=> ( ord_le318720350_set_a @ ( a_RCOSETS_a_b @ r @ A ) @ ( pow_a @ ( partia1066395285xt_a_b @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).
thf(93,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_193_subset__empty) ).
thf(644,plain,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(108,axiom,
! [A: partia1833973666xt_a_b] :
( ( semiring_a_b @ A )
=> ( monoid1022105667xt_a_b @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_75_semiring_Oaxioms_I2_J) ).
thf(691,plain,
! [A: partia1833973666xt_a_b] :
( ( semiring_a_b @ A )
=> ( monoid1022105667xt_a_b @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(36,axiom,
! [A: partia96731725t_unit,B: a,C: a] :
( ( group_a_Product_unit @ A )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ( ( ( mult_a_Product_unit @ A @ C @ B )
= B )
= ( C
= ( one_a_Product_unit @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_166_group_Or__cancel__one) ).
thf(442,plain,
! [A: partia96731725t_unit,B: a,C: a] :
( ( group_a_Product_unit @ A )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ( ( ( mult_a_Product_unit @ A @ C @ B )
= B )
= ( C
= ( one_a_Product_unit @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(159,axiom,
! [A: product_prod_a_a] :
~ ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_197_empty__iff) ).
thf(871,plain,
! [A: product_prod_a_a] :
~ ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(225,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ A @ ( zero_a_b @ r ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_271_a__coset__add__zero) ).
thf(1120,plain,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ A @ ( zero_a_b @ r ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(22,axiom,
! [A: set_a,B: set_set_a,C: set_set_a] :
( ~ ( member_set_a @ A @ B )
=> ( ~ ( member_set_a @ A @ C )
=> ( ( ( insert_set_a @ A @ B )
= ( insert_set_a @ A @ C ) )
= ( B = C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_329_insert__ident) ).
thf(395,plain,
! [A: set_a,B: set_set_a,C: set_set_a] :
( ~ ( member_set_a @ A @ B )
=> ( ~ ( member_set_a @ A @ C )
=> ( ( ( insert_set_a @ A @ B )
= ( insert_set_a @ A @ C ) )
= ( B = C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(147,axiom,
! [A: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ B @ bot_bot_set_a ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_308_subset__singleton__iff) ).
thf(839,plain,
! [A: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ B @ bot_bot_set_a ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(237,axiom,
! [A: partia539144763t_unit,B: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( one_se1488596747t_unit @ A ) @ B )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_97_monoid_Ol__one) ).
thf(1152,plain,
! [A: partia539144763t_unit,B: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ A )
=> ( ( member1838126896od_a_a @ B @ ( partia616129906t_unit @ A ) )
=> ( ( mult_s977248631t_unit @ A @ ( one_se1488596747t_unit @ A ) @ B )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).
thf(32,axiom,
! [A: set_se1596668135od_a_a] :
( ! [B: set_Product_prod_a_a] :
~ ( member1838126896od_a_a @ B @ A )
=> ( A = bot_bo1424703315od_a_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_225_equals0I) ).
thf(427,plain,
! [A: set_se1596668135od_a_a] :
( ! [B: set_Product_prod_a_a] :
~ ( member1838126896od_a_a @ B @ A )
=> ( A = bot_bo1424703315od_a_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(193,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ C @ D ) )
=> ~ ( ( A = C )
=> ( B != D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_90_Pair__inject) ).
thf(1004,plain,
! [A: a,B: a,C: a,D: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ C @ D ) )
=> ~ ( ( A = C )
=> ( B != D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).
thf(15,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a,C: product_prod_a_a,D: set_Product_prod_a_a] :
( ~ ( member449909584od_a_a @ A @ B )
=> ( ~ ( member449909584od_a_a @ C @ D )
=> ( ( ( insert1116662519od_a_a @ A @ B )
= ( insert1116662519od_a_a @ C @ D ) )
= ( ( ( A = C )
=> ( B = D ) )
& ( ( A != C )
=> ? [E: set_Product_prod_a_a] :
( ( B
= ( insert1116662519od_a_a @ C @ E ) )
& ~ ( member449909584od_a_a @ C @ E )
& ( D
= ( insert1116662519od_a_a @ A @ E ) )
& ~ ( member449909584od_a_a @ A @ E ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_319_insert__eq__iff) ).
thf(369,plain,
! [A: product_prod_a_a,B: set_Product_prod_a_a,C: product_prod_a_a,D: set_Product_prod_a_a] :
( ~ ( member449909584od_a_a @ A @ B )
=> ( ~ ( member449909584od_a_a @ C @ D )
=> ( ( ( insert1116662519od_a_a @ A @ B )
= ( insert1116662519od_a_a @ C @ D ) )
= ( ( ( A = C )
=> ( B = D ) )
& ( ( A != C )
=> ? [E: set_Product_prod_a_a] :
( ( B
= ( insert1116662519od_a_a @ C @ E ) )
& ~ ( member449909584od_a_a @ C @ E )
& ( D
= ( insert1116662519od_a_a @ A @ E ) )
& ~ ( member449909584od_a_a @ A @ E ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(607,plain,
! [A: set_set_a] : ( ord_le318720350_set_a @ A @ A ),
inference(cnf,[status(esa)],[606]) ).
thf(40,axiom,
! [A: partia96731725t_unit,B: a,C: a] :
( ( group_a_Product_unit @ A )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ( ( B
= ( mult_a_Product_unit @ A @ C @ B ) )
= ( C
= ( one_a_Product_unit @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_160_group_Or__cancel__one_H) ).
thf(469,plain,
! [A: partia96731725t_unit,B: a,C: a] :
( ( group_a_Product_unit @ A )
=> ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
=> ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
=> ( ( B
= ( mult_a_Product_unit @ A @ C @ B ) )
= ( C
= ( one_a_Product_unit @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(201,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ~ ( ( ord_le318720350_set_a @ A @ B )
=> ~ ( ord_le318720350_set_a @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_242_equalityE) ).
thf(1037,plain,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ~ ( ( ord_le318720350_set_a @ A @ B )
=> ~ ( ord_le318720350_set_a @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).
thf(708,plain,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ),
inference(cnf,[status(esa)],[707]) ).
thf(163,axiom,
! [A: partia1833973666xt_a_b] :
( ( partia1955795460t_unit @ ( units_1411277569xt_a_b @ A ) )
= ( units_a_ring_ext_a_b @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_272_units__of__carrier) ).
thf(882,plain,
! [A: partia1833973666xt_a_b] :
( ( partia1955795460t_unit @ ( units_1411277569xt_a_b @ A ) )
= ( units_a_ring_ext_a_b @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(194,axiom,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali807230110ng_a_b @ A @ B )
=> ( locali880295127ng_a_b @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_26_mult__submonoid__of__crng_Oaxioms_I2_J) ).
thf(1014,plain,
! [A: partia1833973666xt_a_b,B: set_a] :
( ( locali807230110ng_a_b @ A @ B )
=> ( locali880295127ng_a_b @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(45,axiom,
! [A: partia1833973666xt_a_b,B: a] :
( ( semiring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( zero_a_b @ A ) @ B )
= ( zero_a_b @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_79_semiring_Ol__null) ).
thf(491,plain,
! [A: partia1833973666xt_a_b,B: a] :
( ( semiring_a_b @ A )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
=> ( ( mult_a_ring_ext_a_b @ A @ ( zero_a_b @ A ) @ B )
= ( zero_a_b @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(219,axiom,
locali425460998xt_a_b @ r @ s,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_8_submonoid__axioms) ).
thf(1088,plain,
locali425460998xt_a_b @ r @ s,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).
thf(3007,plain,
$false,
inference(e,[status(thm)],[2163,1322,809,1411,629,1237,333,468,555,1295,1549,481,683,1068,994,1401,1422,2114,655,1437,861,1205,449,1078,1137,533,1233,340,538,670,1260,698,898,1243,709,885,651,320,2173,724,1287,1298,719,1313,542,587,619,934,357,902,853,1379,1087,1147,421,770,316,606,1347,321,2259,1828,514,937,1283,1051,1418,623,485,834,574,1020,348,353,966,734,1447,1433,1405,1104,1622,774,988,2286,920,380,512,678,1438,1338,366,1067,798,1199,2164,957,778,1123,598,561,424,693,1370,729,1238,1423,1938,1178,1159,1210,1291,842,361,1365,1195,1407,324,974,1089,874,1328,1375,403,388,610,1397,519,742,1270,317,1138,1719,1329,1082,554,1182,526,637,970,1739,537,686,1075,1016,590,985,1150,984,1429,1246,647,1263,997,738,2304,1434,1202,1086,1241,633,1054,1170,1035,530,343,1318,558,1071,773,794,1278,513,1273,1301,689,1374,601,732,843,2398,1124,413,577,711,1256,1226,1707,1332,804,704,1224,672,391,1139,713,498,1765,1090,2119,406,1408,1282,784,318,616,1571,1307,764,604,377,2257,509,880,434,863,1175,811,535,816,594,621,895,1344,943,330,345,858,552,1162,726,848,700,521,983,399,2171,1213,608,1000,1250,851,505,431,658,1445,868,473,722,675,1166,1277,1081,617,342,1145,836,1049,663,548,1134,648,525,640,1025,1156,707,327,954,1098,1368,2175,585,1317,1085,1130,1220,544,331,1186,1404,336,1141,1023,613,864,727,876,991,978,1140,446,571,845,465,627,2352,931,1949,2078,495,1189,524,581,1216,891,813,963,1062,1854,2431,418,781,1253,699,1341,748,946,1101,1030,568,2399,667,2330,1193,383,346,1045,478,2162,716,800,2258,410,950,1229,790,865,373,564,326,767,1056,960,1065,358,807,437,1335,928,415,1268,644,691,442,871,1399,1120,395,839,1152,427,982,1004,369,607,337,469,1037,639,708,882,1014,2170,491,1088]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : ITP107^1 : TPTP v9.3.1. Released v7.5.0.
% 0.00/0.08 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.09/0.23 % Computer : n019.cluster.edu
% 0.09/0.23 % Model : x86_64 x86_64
% 0.09/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.23 % Memory : 8046.5625MB
% 0.09/0.23 % OS : Linux 6.8.0-71-generic
% 0.09/0.23 % CPULimit : 300
% 0.09/0.23 % WCLimit : 300
% 0.09/0.23 % DateTime : Tue Sep 29 19:54:32 UTC 2026
% 0.09/0.23 % CPUTime :
% 0.09/0.23 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.85/0.79 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 2.09/1.18 % [INFO] Parsing done (386ms).
% 2.09/1.19 % [INFO] Running in sequential loop mode.
% 3.32/1.58 % [INFO] eprover registered as external prover.
% 3.32/1.59 % [INFO] Scanning for conjecture ...
% 3.79/1.74 % [INFO] Found a conjecture (or negated_conjecture) and 355 axioms. Running axiom selection ...
% 4.49/1.90 % [INFO] Axiom selection finished. Selected 313 axioms (removed 42 axioms).
% 5.50/2.16 % [INFO] Problem is higher-order (TPTP THF).
% 5.50/2.18 % [INFO] Type checking passed.
% 5.50/2.19 % [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 ...
% 53.28/19.73 % External prover 'e' found a proof!
% 53.28/19.73 % [INFO] Killing All external provers ...
% 53.28/19.73 % Time passed: 19364ms (effective reasoning time: 18531ms)
% 53.28/19.73 % 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)>
% 53.28/19.74 % Axioms used in derivation (313): fact_269_a__setmult__rcos__assoc, fact_54_Collect__mem__eq, fact_149_group_OUnits, fact_317_insert__commute, fact_220_emptyE, fact_256_subset__refl, fact_46_m__assoc, fact_225_equals0I, fact_113_integral, fact_350_doubleton__eq__iff, fact_180_group_Oinv__comm, fact_164_group_Ol__cancel__one_H, fact_1_f4, fact_185_monoid_Ogroup__l__invI, fact_280_one__zeroI, fact_305_singleton__insert__inj__eq, fact_163_group_Ol__cancel__one_H, fact_36_left__unit__mult__rng__of__frac, fact_311_subset__singletonD, fact_299_insert__subset, fact_246_subset__eq, fact_270_units__of__units, fact_126_monoid_OUnits__one__closed, fact_221_equals0D, fact_284_insert__iff, fact_128_monoidI, fact_172_group_Or__inv__ex, fact_349_insert__not__empty, fact_343_insertE, fact_150_group_OUnits, fact_12_one__closed, fact_93_ring_Ois__ring, fact_258_Collect__mono, fact_68_r__one, fact_84_ring_Oring__simprules_I6_J, fact_335_insertI2, fact_66_r__null, fact_247_equalityD1, fact_138_monoid_OUnits__l__cancel, fact_257_Collect__mono, fact_37_zero__in__mult__submonoid, fact_202_all__not__in__conv, fact_215_empty__subsetI, fact_189_a__lcos__mult__one, fact_346_singleton__inject, fact_151_group_Ois__monoid, fact_345_insertE, fact_160_group_Or__cancel__one_H, fact_290_insertCI, fact_121_monoid_Om__closed, fact_352_field__intro2, fact_71_ring_Oring__simprules_I2_J, fact_157_group_OUnits__eq, fact_27_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Oone__closed, fact_16_assoc__mult__rng__of__frac, fact_137_monoid_Oone__unique, fact_329_insert__ident, fact_348_insert__not__empty, fact_227_equals0I, fact_224_equals0D, fact_331_Set_Oset__insert, fact_144_carrier__not__empty, fact_187_semiring_Oadd__pow__rdistr, fact_146_add_Onat__pow__one, fact_313_mk__disjoint__insert, fact_145_add_Onat__pow__closed, fact_267_a__rcos__assoc__lcos, fact_98_monoid_Ol__one, fact_287_insertCI, fact_156_Group_Ogroup_Oright__cancel, fact_308_subset__singleton__iff, fact_232_ex__in__conv, fact_236_in__mono, fact_44_semiring_Oaxioms_I1_J, fact_264_Collect__mono__iff, fact_195_empty__iff, fact_277_units__of__one, fact_52_Collect__mem__eq, fact_184_monoid_Ogroup__l__invI, fact_120_monoid_Om__closed, fact_338_insertI1, fact_80_ring_Oring__simprules_I5_J, fact_105_monoid_OUnits__inv__comm, fact_213_a__l__coset__subset__G, fact_275_units__of__mult, fact_197_empty__iff, fact_108_add__pow__rdistr__int, fact_110_add_Oint__pow__one, fact_64_Units__one__closed, fact_86_ring_Oring__simprules_I12_J, fact_75_semiring_Oaxioms_I2_J, fact_109_add_Oint__pow__closed, fact_173_group_Or__inv__ex, fact_72_semiring_Osemiring__simprules_I2_J, fact_99_monoid_Oone__closed, fact_318_insert__eq__iff, fact_182_groupI, fact_0_f3, fact_135_monoid_Oinv__unique, fact_158_group_OUnits__eq, fact_165_group_Ol__cancel__one_H, fact_70_abelian__monoidE_I2_J, fact_103_monoid_OUnits__l__inv__ex, fact_114_ring_Oadd__pow__ldistr__int, fact_124_monoid_OUnits__m__closed, fact_55_Collect__mem__eq, fact_94_cring__fieldI2, fact_235_in__mono, fact_97_monoid_Ol__one, fact_316_mk__disjoint__insert, fact_21_submonoid_Oaxioms_I1_J, fact_292_carrier__one__zero, fact_10_monoid__rng__of__frac, fact_125_monoid_OUnits__m__closed, fact_19_local_Oring__axioms, fact_229_ex__in__conv, fact_56_m__lcomm, fact_123_monoid_OUnits__closed, fact_296_singletonI, fact_82_semiring_Osemiring__simprules_I3_J, fact_8_submonoid__axioms, fact_53_Collect__mem__eq, fact_262_set__eq__subset, fact_89_old_Oprod_Oexhaust, fact_152_group_Ois__monoid, fact_183_groupI, fact_254_subset__iff, fact_261_set__eq__subset, fact_354_rcosets__subset__PowG, fact_26_mult__submonoid__of__crng_Oaxioms_I2_J, fact_342_insertE, fact_231_ex__in__conv, fact_188_semiring_Oadd__pow__ldistr, fact_337_insertI2, fact_334_insertI2, fact_300_insert__subset, fact_327_insert__ident, fact_2_eq__obj__rng__of__frac__axioms, fact_168_group_Or__cancel__one, fact_319_insert__eq__iff, fact_7_eq__obj__rng__of__frac_Orec__monoid__rng__of__frac_Ocong, fact_127_monoid_OUnits__one__closed, fact_283_insert__iff, fact_106_monoid_OUnits__inv__comm, fact_353_cring__fieldI, fact_169_group_Ol__cancel__one, fact_45_monoid__axioms, fact_286_insert__iff, fact_161_group_Or__cancel__one_H, fact_29_mult__submonoid__of__rng__def, fact_23_eq__obj__rng__of__frac_Oaxioms_I2_J, fact_74_ring_Ois__monoid, fact_178_group_Oinv__comm, fact_193_subset__empty, fact_41_Units__inv__comm, fact_153_group_Ois__monoid, fact_42_prod_Oinject, fact_323_insert__absorb, fact_263_Collect__mono__iff, fact_133_Group_Omonoid_Ointro, fact_199_all__not__in__conv, fact_190_set__add__closed, fact_170_group_Ol__cancel__one, fact_176_group_Ol__inv__ex, fact_65_l__null, fact_209_subsetI, fact_248_equalityD1, fact_51_mem__Collect__eq, fact_112_one__not__zero, fact_107_add__pow__ldistr__int, fact_73_abelian__monoid_Ozero__closed, fact_217_emptyE, fact_33_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Om__closed, fact_206_empty__Collect__eq, fact_32_eq__obj__rng__of__frac_Omonoid__rng__of__frac, fact_147_group_Ois__group, fact_255_subset__refl, fact_278_units__of__one, fact_134_monoid_Oinv__unique, fact_167_group_Or__cancel__one, fact_24_mult__submonoid__of__rng_Oaxioms_I2_J, fact_115_ring_Oadd__pow__rdistr__int, fact_67_l__one, fact_272_units__of__carrier, fact_130_Group_Omonoid__def, fact_155_Group_Ogroup_Oright__cancel, fact_212_subset__antisym, fact_57_Units__closed, fact_249_equalityD2, fact_118_monoid_Om__assoc, fact_148_group_Ois__group, fact_223_equals0D, fact_266_add__additive__subgroups, fact_62_semiring__simprules_I4_J, fact_40_inv__unique, fact_219_emptyE, fact_326_insert__ident, fact_122_monoid_OUnits__closed, fact_351_doubleton__eq__iff, fact_88_old_Oprod_Oinducts, fact_18_mult__rng__of__frac__fundamental__lemma, fact_83_semiring_Osemiring__simprules_I8_J, fact_279_a__rcosetsI, fact_47_m__comm, fact_131_Group_Omonoid__def, fact_140_subset, fact_31_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Ois__submonoid, fact_76_ring_Oring__simprules_I24_J, fact_260_subset__trans, fact_49_mem__Collect__eq, fact_81_ring_Oring__simprules_I11_J, fact_25_mult__submonoid__of__rng_Oaxioms_I1_J, fact_271_a__coset__add__zero, fact_48_mem__Collect__eq, fact_265_units__group, fact_58_Units__l__inv__ex, fact_13_equiv__obj__rng__of__frac, fact_77_ring_Oring__simprules_I25_J, fact_129_monoidI, fact_20_closed__mult__rng__of__frac, fact_90_Pair__inject, fact_166_group_Or__cancel__one, fact_288_insertCI, fact_210_subsetI, fact_139_monoid_OUnits__l__cancel, fact_141_add__pow__rdistr, fact_191_setadd__subset__G, fact_341_insertI1, fact_333_Set_Oset__insert, fact_102_monoid_OUnits__r__inv__ex, fact_239_subsetD, fact_174_group_Or__inv__ex, fact_92_surj__pair, fact_132_Group_Omonoid_Ointro, fact_79_semiring_Ol__null, fact_63_Units__m__closed, fact_3_f1, fact_159_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Osubset, fact_38_non__empty__class, fact_245_subset__eq, fact_9_m__closed, fact_282_insert__absorb2, fact_268_a__r__coset__subset__G, fact_179_group_Oinv__comm, fact_39_one__unique, fact_61_semiring__simprules_I3_J, fact_203_Collect__empty__eq, fact_91_prod__cases, fact_274_monoid_Ounits__group, fact_281_one__zeroD, fact_302_singleton__insert__inj__eq_H, fact_211_subset__antisym, fact_104_monoid_OUnits__l__inv__ex, fact_204_Collect__empty__eq, fact_14_assms_I2_J, fact_119_monoid_Om__assoc, fact_22_submonoid_Oaxioms_I1_J, fact_250_equalityD2, fact_43_old_Oprod_Oinject, fact_295_singletonI, fact_59_Units__r__inv__ex, fact_5_mult__submonoid__of__crng__axioms, fact_347_singleton__inject, fact_85_semiring_Osemiring__simprules_I4_J, fact_100_monoid_Oone__closed, fact_116_monoid_Ocarrier__not__empty, fact_15_assms_I1_J, fact_60_zero__closed, fact_162_group_Or__cancel__one_H, fact_322_insert__absorb, fact_291_carrier__one__not__zero, fact_95_monoid_Or__one, fact_228_equals0I, fact_177_group_Ol__inv__ex, fact_17_abelian__monoid__axioms, fact_242_equalityE, fact_240_subsetD, fact_205_empty__Collect__eq, fact_339_insertI1, fact_11_local_Osemiring__axioms, fact_171_group_Ol__cancel__one, fact_276_units__of__mult, fact_325_insert__absorb, fact_96_monoid_Or__one, fact_273_monoid_Ounits__group, fact_69_Units__l__cancel, fact_117_monoid_Ocarrier__not__empty, fact_28_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Oone__closed, fact_201_all__not__in__conv, fact_142_add__pow__ldistr, fact_198_empty__iff, fact_6_mult__submonoid__of__rng__axioms, fact_330_Set_Oset__insert, fact_181_groupI, fact_34_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Om__closed, fact_314_mk__disjoint__insert, fact_101_monoid_OUnits__r__inv__ex, fact_78_semiring_Or__null, fact_50_mem__Collect__eq, fact_143_group__l__invI, fact_253_subset__iff, fact_35_right__unit__mult__rng__of__frac, fact_175_group_Ol__inv__ex, fact_30_mult__submonoid__of__rng_Ointro, fact_241_equalityE, fact_4_f2, fact_321_insert__eq__iff, fact_154_Group_Ogroup_Oright__cancel, fact_111_field_Ois__ring, fact_87_semiring_Osemiring__simprules_I9_J, fact_186_monoid_Ogroup__l__invI, fact_136_monoid_Oone__unique, fact_259_subset__trans
% 53.28/19.74 % No. of inferences in proof: 779
% 53.28/19.74 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 19364 ms resp. 18531 ms w/o parsing
% 54.29/20.03 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 54.29/20.04 % [INFO] Killing All external provers ...
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