TSTP Solution File: ITP107^2 by Leo-III---1.7.12
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%------------------------------------------------------------------------------
% File : Leo-III---1.7.12
% Problem : ITP107^2 : TPTP v8.2.0. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n032.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Mon May 20 22:22:58 EDT 2024
% Result : Theorem 60.01s 44.98s
% Output : Refutation 60.60s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 303
% Syntax : Number of formulae : 561 ( 143 unt; 47 typ; 0 def)
% Number of atoms : 2977 ( 504 equ; 2 cnn)
% Maximal formula atoms : 24 ( 5 avg)
% Number of connectives : 13939 ( 116 ~; 10 |; 66 &;13038 @)
% ( 0 <=>; 709 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 10 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 98 ( 98 >; 0 *; 0 +; 0 <<)
% Number of symbols : 52 ( 48 usr; 16 con; 0-6 aty)
% Number of variables : 1859 ( 20 ^1742 !; 30 ?;1859 :)
% ( 67 !>; 0 ?*; 0 @-; 0 @+)
% Comments :
%------------------------------------------------------------------------------
thf(product_unit_type,type,
product_unit: $tType ).
thf(nat_type,type,
nat: $tType ).
thf(int_type,type,
int: $tType ).
thf(b_type,type,
b: $tType ).
thf(a_type,type,
a: $tType ).
thf(a_RCOSETS_type,type,
a_RCOSETS:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > ( set @ ( set @ TB ) ) ) ).
thf(a_l_coset_type,type,
a_l_coset:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > TB > ( set @ TB ) > ( set @ TB ) ) ).
thf(a_r_coset_type,type,
a_r_coset:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > TB > ( set @ TB ) ) ).
thf(additive_subgroup_type,type,
additive_subgroup:
!>[TA: $tType,TB: $tType] : ( ( set @ TB ) > ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > $o ) ).
thf(set_add_type,type,
set_add:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > ( set @ TB ) > ( set @ TB ) ) ).
thf(equivalence_type,type,
equivalence:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( eq_eq_object_ext @ TB @ TA ) ) > $o ) ).
thf(partial_carrier_type,type,
partial_carrier:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ TA ) > ( set @ TB ) ) ).
thf(units_type,type,
units:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ TA ) ) > ( set @ TB ) ) ).
thf(group_type,type,
group:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ TA ) ) > $o ) ).
thf(monoid_type,type,
monoid:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ TA ) ) > $o ) ).
thf(mult_type,type,
mult:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ TA ) ) > TB > TB > TB ) ).
thf(one_type,type,
one:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ TA ) ) > TB ) ).
thf(units_of_type,type,
units_of:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ TA ) ) > ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ product_unit ) ) ) ).
thf(minus_minus_type,type,
minus_minus:
!>[TA: $tType] : ( TA > TA > TA ) ).
thf(locali1641774487f_frac_type,type,
locali1641774487f_frac:
!>[TA: $tType,TB: $tType,TC: $tType] : ( ( partia1265057091ct_ext @ ( product_prod @ TC @ TB ) @ ( eq_eq_object_ext @ ( product_prod @ TC @ TB ) @ TA ) ) > TC > TB > ( set @ ( product_prod @ TC @ TB ) ) ) ).
thf(locali1648044335f_frac_type,type,
locali1648044335f_frac:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > $o ) ).
thf(locali1768110497f_frac_type,type,
locali1768110497f_frac:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > ( partia1265057091ct_ext @ ( set @ ( product_prod @ TB @ TB ) ) @ ( monoid_ext @ ( set @ ( product_prod @ TB @ TB ) ) @ product_unit ) ) ) ).
thf(locali7914339f_crng_type,type,
locali7914339f_crng:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > $o ) ).
thf(locali1402578320of_rng_type,type,
locali1402578320of_rng:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > $o ) ).
thf(locali990921413f_frac_type,type,
locali990921413f_frac:
!>[TA: $tType,TB: $tType,TC: $tType] : ( ( partia1265057091ct_ext @ ( product_prod @ TC @ TB ) @ ( eq_eq_object_ext @ ( product_prod @ TC @ TB ) @ TA ) ) > ( set @ ( set @ ( product_prod @ TC @ TB ) ) ) ) ).
thf(locali1441642228monoid_type,type,
locali1441642228monoid:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ TA ) ) > ( set @ TB ) > $o ) ).
thf(bot_bot_type,type,
bot_bot:
!>[TA: $tType] : TA ).
thf(ord_less_eq_type,type,
ord_less_eq:
!>[TA: $tType] : ( TA > TA > $o ) ).
thf(product_Pair_type,type,
product_Pair:
!>[TA: $tType,TB: $tType] : ( TB > TA > ( product_prod @ TB @ TA ) ) ).
thf(product_rec_prod_type,type,
product_rec_prod:
!>[TA: $tType,TB: $tType,TC: $tType] : ( ( TC > TB > TA ) > ( product_prod @ TC @ TB ) > TA ) ).
thf(abelian_monoid_type,type,
abelian_monoid:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > $o ) ).
thf(add_pow_type,type,
add_pow:
!>[TA: $tType,TB: $tType,TC: $tType] : ( ( partia1265057091ct_ext @ TC @ ( monoid_ext @ TC @ ( ring_ext @ TC @ TB ) ) ) > TA > TC > TC ) ).
thf(field_type,type,
field:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > $o ) ).
thf(ring_type,type,
ring:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > $o ) ).
thf(zero_type,type,
zero:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > TB ) ).
thf(semiring_type,type,
semiring:
!>[TA: $tType,TB: $tType] : ( ( partia1265057091ct_ext @ TB @ ( monoid_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > $o ) ).
thf(collect_type,type,
collect:
!>[TA: $tType] : ( ( TA > $o ) > ( set @ TA ) ) ).
thf(pow_type,type,
pow:
!>[TA: $tType] : ( ( set @ TA ) > ( set @ ( set @ TA ) ) ) ).
thf(insert_type,type,
insert:
!>[TA: $tType] : ( TA > ( set @ TA ) > ( set @ TA ) ) ).
thf(member_type,type,
member:
!>[TA: $tType] : ( TA > ( set @ TA ) > $o ) ).
thf(r_type,type,
r: partia1265057091ct_ext @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) ).
thf(s_type,type,
s: set @ a ).
thf(r2_type,type,
r2: a ).
thf(r3_type,type,
r3: a ).
thf(rel_type,type,
rel: partia1265057091ct_ext @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) ).
thf(s2_type,type,
s2: a ).
thf(s3_type,type,
s3: a ).
thf(44,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ TA] :
( ( insert @ TA @ A @ ( insert @ TA @ B @ C ) )
= ( insert @ TA @ B @ ( insert @ TA @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_insert__commute) ).
thf(365,plain,
! [TA: $tType,A: TA,B: TA,C: set @ TA] :
( ( insert @ TA @ A @ ( insert @ TA @ B @ C ) )
= ( insert @ TA @ B @ ( insert @ TA @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(184,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA] :
( ( group @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( B
= ( mult @ TA @ TB @ A @ B @ C ) )
= ( C
= ( one @ TA @ TB @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_group_Ol__cancel__one_H) ).
thf(760,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA] :
( ( group @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( B
= ( mult @ TA @ TB @ A @ B @ C ) )
= ( C
= ( one @ TA @ TB @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(38,axiom,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_old_Oprod_Oinject) ).
thf(347,plain,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(210,axiom,
! [A: a,B: a,C: a,D: a] :
( ( member @ a @ ( zero @ a @ b @ r ) @ s )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ D ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B )
= ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ C @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_zero__in__mult__submonoid) ).
thf(846,plain,
! [A: a,B: a,C: a,D: a] :
( ( member @ a @ ( zero @ a @ b @ r ) @ s )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ D ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B )
= ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ C @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).
thf(83,axiom,
monoid @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_monoid__rng__of__frac) ).
thf(481,plain,
monoid @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(231,axiom,
! [A: a,B: a,C: a,D: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ D ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ C @ D ) )
= ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A @ C ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ B @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_mult__rng__of__frac__fundamental__lemma) ).
thf(909,plain,
! [A: a,B: a,C: a,D: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ D ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ C @ D ) )
= ( locali1641774487f_frac @ a @ a @ product_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)],[231]) ).
thf(33,axiom,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
=> ~ ( ( A = C )
=> ( B != D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_Pair__inject) ).
thf(333,plain,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
=> ~ ( ( A = C )
=> ( B != D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(249,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA] :
( ( group @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ? [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
& ( ( mult @ TA @ TB @ A @ C @ B )
= ( one @ TA @ TB @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_144_group_Ol__inv__ex) ).
thf(977,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA] :
( ( group @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ? [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
& ( ( mult @ TA @ TB @ A @ C @ B )
= ( one @ TA @ TB @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(119,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( semiring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) @ D )
= ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_semiring_Osemiring__simprules_I8_J) ).
thf(582,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( semiring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) @ D )
= ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(10,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( group @ TA @ TB @ A )
=> ( ( units @ TA @ TB @ A )
= ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_group_OUnits__eq) ).
thf(276,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( group @ TA @ TB @ A )
=> ( ( units @ TA @ TB @ A )
= ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(255,axiom,
! [A: set @ a,B: set @ a] :
( ( additive_subgroup @ a @ b @ A @ r )
=> ( ( additive_subgroup @ a @ b @ B @ r )
=> ( additive_subgroup @ a @ b @ ( set_add @ a @ b @ r @ A @ B ) @ r ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_add__additive__subgroups) ).
thf(994,plain,
! [A: set @ a,B: set @ a] :
( ( additive_subgroup @ a @ b @ A @ r )
=> ( ( additive_subgroup @ a @ b @ B @ r )
=> ( additive_subgroup @ a @ b @ ( set_add @ a @ b @ r @ A @ B ) @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).
thf(23,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( field @ TA @ TB @ A )
=> ( ( one @ TA @ ( ring_ext @ TA @ TB ) @ A )
!= ( zero @ TA @ TB @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_one__not__zero) ).
thf(308,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( field @ TA @ TB @ A )
=> ( ( one @ TA @ ( ring_ext @ TA @ TB ) @ A )
!= ( zero @ TA @ TB @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(50,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( abelian_monoid @ TA @ TB @ A )
=> ( member @ TA @ ( zero @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_66_abelian__monoid_Ozero__closed) ).
thf(379,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( abelian_monoid @ TA @ TB @ A )
=> ( member @ TA @ ( zero @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(220,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_equalityD1) ).
thf(878,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(145,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ ( insert @ TA @ A @ B ) @ C )
= ( ( member @ TA @ A @ C )
& ( ord_less_eq @ ( set @ TA ) @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_insert__subset) ).
thf(655,plain,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ ( insert @ TA @ A @ B ) @ C )
= ( ( member @ TA @ A @ C )
& ( ord_less_eq @ ( set @ TA ) @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(131,axiom,
! [TA: $tType] :
( ( pow @ TA @ ( bot_bot @ ( set @ TA ) ) )
= ( insert @ ( set @ TA ) @ ( bot_bot @ ( set @ TA ) ) @ ( bot_bot @ ( set @ ( set @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_255_Pow__empty) ).
thf(614,plain,
! [TA: $tType] :
( ( pow @ TA @ ( bot_bot @ ( set @ TA ) ) )
= ( insert @ ( set @ TA ) @ ( bot_bot @ ( set @ TA ) ) @ ( bot_bot @ ( set @ ( set @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(7,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( field @ TA @ TB @ A )
=> ( ring @ TA @ TB @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_field_Ois__ring) ).
thf(269,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( field @ TA @ TB @ A )
=> ( ring @ TA @ TB @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(154,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ! [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ B @ C )
= C ) )
=> ( B
= ( one @ TA @ TB @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_monoid_Oone__unique) ).
thf(677,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ! [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ B @ C )
= C ) )
=> ( B
= ( one @ TA @ TB @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(53,axiom,
locali1441642228monoid @ a @ ( ring_ext @ a @ b ) @ r @ s,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8_submonoid__axioms) ).
thf(385,plain,
locali1441642228monoid @ a @ ( ring_ext @ a @ b ) @ r @ s,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(107,axiom,
( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
!= ( bot_bot @ ( set @ a ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_carrier__not__empty) ).
thf(550,plain,
( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
!= ( bot_bot @ ( set @ a ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(167,axiom,
member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ s,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_one__closed) ).
thf(715,plain,
member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ s,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(199,axiom,
! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_subset__refl) ).
thf(814,plain,
! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(161,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( member @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ C ) )
= ( ( member @ TA @ A @ B )
& ~ ( member @ TA @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_228_Diff__iff) ).
thf(698,plain,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( member @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ C ) )
= ( ( member @ TA @ A @ B )
& ~ ( member @ TA @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(232,axiom,
! [TA: $tType,TB: $tType] :
( ( locali1768110497f_frac @ TA @ TB )
= ( locali1768110497f_frac @ TA @ TB ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_eq__obj__rng__of__frac_Orec__monoid__rng__of__frac_Ocong) ).
thf(913,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(35,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( monoid @ TA @ TB @ A )
=> ( member @ TA @ ( one @ TA @ TB @ A ) @ ( units @ TA @ TB @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_monoid_OUnits__one__closed) ).
thf(340,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( monoid @ TA @ TB @ A )
=> ( member @ TA @ ( one @ TA @ TB @ A ) @ ( units @ TA @ TB @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(103,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA] :
( ( group @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( B
= ( mult @ TA @ TB @ A @ C @ B ) )
= ( C
= ( one @ TA @ TB @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_group_Or__cancel__one_H) ).
thf(538,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA] :
( ( group @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( B
= ( mult @ TA @ TB @ A @ C @ B ) )
= ( C
= ( one @ TA @ TB @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(165,axiom,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_49_m__lcomm) ).
thf(709,plain,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[165]) ).
thf(65,axiom,
! [TA: $tType,A: set @ TA] :
( ! [B: TA] :
~ ( member @ TA @ B @ A )
=> ( A
= ( bot_bot @ ( set @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_164_equals0I) ).
thf(417,plain,
! [TA: $tType,A: set @ TA] :
( ! [B: TA] :
~ ( member @ TA @ B @ A )
=> ( A
= ( bot_bot @ ( set @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(222,axiom,
! [TA: $tType,A: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ ( bot_bot @ ( set @ TA ) ) @ A )
= ( bot_bot @ ( set @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_231_empty__Diff) ).
thf(885,plain,
! [TA: $tType,A: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ ( bot_bot @ ( set @ TA ) ) @ A )
= ( bot_bot @ ( set @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).
thf(21,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ TA] :
( ( member @ TA @ A @ ( insert @ TA @ B @ C ) )
= ( ( A = B )
| ( member @ TA @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_insert__iff) ).
thf(301,plain,
! [TA: $tType,A: TA,B: TA,C: set @ TA] :
( ( member @ TA @ A @ ( insert @ TA @ B @ C ) )
= ( ( A = B )
| ( member @ TA @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(72,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ! [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( member @ TA @ ( mult @ TA @ TB @ A @ B @ C ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ) )
=> ( ( member @ TA @ ( one @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ! [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ! [D: TA] :
( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( mult @ TA @ TB @ A @ B @ C ) @ D )
= ( mult @ TA @ TB @ A @ B @ ( mult @ TA @ TB @ A @ C @ D ) ) ) ) ) )
=> ( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( one @ TA @ TB @ A ) @ B )
= B ) )
=> ( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ? [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
& ( ( mult @ TA @ TB @ A @ C @ B )
= ( one @ TA @ TB @ A ) ) ) )
=> ( group @ TA @ TB @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_146_groupI) ).
thf(436,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ! [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( member @ TA @ ( mult @ TA @ TB @ A @ B @ C ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ) )
=> ( ( member @ TA @ ( one @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ! [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ! [D: TA] :
( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( mult @ TA @ TB @ A @ B @ C ) @ D )
= ( mult @ TA @ TB @ A @ B @ ( mult @ TA @ TB @ A @ C @ D ) ) ) ) ) )
=> ( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( one @ TA @ TB @ A ) @ B )
= B ) )
=> ( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ? [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
& ( ( mult @ TA @ TB @ A @ C @ B )
= ( one @ TA @ TB @ A ) ) ) )
=> ( group @ TA @ TB @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(169,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) )] :
~ ! [B: TE,C: TD,D: TC,E: TB,F: TA] :
( A
!= ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ B @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ C @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ D @ ( product_Pair @ TB @ TA @ E @ F ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_prod__cases5) ).
thf(719,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) )] :
~ ! [B: TE,C: TD,D: TC,E: TB,F: TA] :
( A
!= ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ B @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ C @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ D @ ( product_Pair @ TB @ TA @ E @ F ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(181,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( member @ TA @ A @ B )
=> ( ( minus_minus @ ( set @ TA ) @ ( insert @ TA @ A @ C ) @ B )
= ( minus_minus @ ( set @ TA ) @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_233_insert__Diff1) ).
thf(751,plain,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( member @ TA @ A @ B )
=> ( ( minus_minus @ ( set @ TA ) @ ( insert @ TA @ A @ C ) @ B )
= ( minus_minus @ ( set @ TA ) @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(194,axiom,
! [TA: $tType,A: set @ TA,B: TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) @ C )
=> ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_250_Diff__single__insert) ).
thf(802,plain,
! [TA: $tType,A: set @ TA,B: TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) @ C )
=> ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(240,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( semiring @ TA @ TB @ A )
=> ( ( ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
= ( insert @ TA @ ( zero @ TA @ TB @ A ) @ ( bot_bot @ ( set @ TA ) ) ) )
=> ( ( one @ TA @ ( ring_ext @ TA @ TB ) @ A )
= ( zero @ TA @ TB @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_223_semiring_Oone__zeroI) ).
thf(934,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( semiring @ TA @ TB @ A )
=> ( ( ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
= ( insert @ TA @ ( zero @ TA @ TB @ A ) @ ( bot_bot @ ( set @ TA ) ) ) )
=> ( ( one @ TA @ ( ring_ext @ TA @ TB ) @ A )
= ( zero @ TA @ TB @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).
thf(41,axiom,
! [TA: $tType,A: TA,B: set @ TA] :
( ( member @ TA @ A @ B )
=> ? [C: set @ TA] :
( ( B
= ( insert @ TA @ A @ C ) )
& ~ ( member @ TA @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_mk__disjoint__insert) ).
thf(357,plain,
! [TA: $tType,A: TA,B: set @ TA] :
( ( member @ TA @ A @ B )
=> ? [C: set @ TA] :
( ( B
= ( insert @ TA @ A @ C ) )
& ~ ( member @ TA @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(228,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( locali1648044335f_frac @ TA @ TB @ A @ B )
=> ( monoid @ ( set @ ( product_prod @ TA @ TA ) ) @ product_unit @ ( locali1768110497f_frac @ TA @ TB @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_eq__obj__rng__of__frac_Omonoid__rng__of__frac) ).
thf(902,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( locali1648044335f_frac @ TA @ TB @ A @ B )
=> ( monoid @ ( set @ ( product_prod @ TA @ TA ) ) @ product_unit @ ( locali1768110497f_frac @ TA @ TB @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).
thf(110,axiom,
! [TA: $tType,A: TA] :
~ ( member @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_154_empty__iff) ).
thf(559,plain,
! [TA: $tType,A: TA] :
~ ( member @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(128,axiom,
! [A: set @ a,B: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( set_add @ a @ b @ r @ A @ B ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_151_set__add__closed) ).
thf(606,plain,
! [A: set @ a,B: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( set_add @ a @ b @ r @ A @ B ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(188,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ! [B: TA,C: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( member @ TA @ ( mult @ TA @ TB @ A @ B @ C ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ) )
=> ( ! [B: TA,C: TA,D: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( mult @ TA @ TB @ A @ B @ C ) @ D )
= ( mult @ TA @ TB @ A @ B @ ( mult @ TA @ TB @ A @ C @ D ) ) ) ) ) )
=> ( ( member @ TA @ ( one @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( one @ TA @ TB @ A ) @ B )
= B ) )
=> ( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ B @ ( one @ TA @ TB @ A ) )
= B ) )
=> ( monoid @ TA @ TB @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_Group_Omonoid_Ointro) ).
thf(770,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ! [B: TA,C: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( member @ TA @ ( mult @ TA @ TB @ A @ B @ C ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ) )
=> ( ! [B: TA,C: TA,D: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( mult @ TA @ TB @ A @ B @ C ) @ D )
= ( mult @ TA @ TB @ A @ B @ ( mult @ TA @ TB @ A @ C @ D ) ) ) ) ) )
=> ( ( member @ TA @ ( one @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( one @ TA @ TB @ A ) @ B )
= B ) )
=> ( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ B @ ( one @ TA @ TB @ A ) )
= B ) )
=> ( monoid @ TA @ TB @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(26,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( group @ TA @ TB @ A )
=> ( group @ TA @ TB @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_group_Ois__group) ).
thf(316,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(139,axiom,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_43_m__assoc) ).
thf(638,plain,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[139]) ).
thf(212,axiom,
! [A: int] :
( ( add_pow @ a @ b @ int @ r @ A @ ( zero @ a @ b @ r ) )
= ( zero @ a @ b @ r ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_add_Oint__pow__one) ).
thf(853,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)],[212]) ).
thf(190,axiom,
! [A: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( zero @ a @ b @ r ) @ A )
= ( zero @ a @ b @ r ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_l__null) ).
thf(789,plain,
! [A: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[190]) ).
thf(123,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( semiring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( zero @ TA @ TB @ A ) @ B )
= ( zero @ TA @ TB @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_semiring_Ol__null) ).
thf(591,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( semiring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( zero @ TA @ TB @ A ) @ B )
= ( zero @ TA @ TB @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(29,axiom,
! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA] :
~ ! [B: TB,C: TA] :
( A
!= ( product_Pair @ TB @ TA @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_old_Oprod_Oexhaust) ).
thf(321,plain,
! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA] :
~ ! [B: TB,C: TA] :
( A
!= ( product_Pair @ TB @ TA @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(48,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( locali1648044335f_frac @ TA @ TB @ A @ B )
=> ( locali7914339f_crng @ TA @ TB @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_eq__obj__rng__of__frac_Oaxioms_I2_J) ).
thf(376,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( locali1648044335f_frac @ TA @ TB @ A @ B )
=> ( locali7914339f_crng @ TA @ TB @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(206,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,A: product_prod @ TG @ ( product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) )] :
~ ! [B: TG,C: TF,D: TE,E: TD,F: TC,G: TB,H: TA] :
( A
!= ( product_Pair @ TG @ ( product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) ) @ B @ ( product_Pair @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) @ C @ ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ D @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ E @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ F @ ( product_Pair @ TB @ TA @ G @ H ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_prod__cases7) ).
thf(834,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,A: product_prod @ TG @ ( product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) )] :
~ ! [B: TG,C: TF,D: TE,E: TD,F: TC,G: TB,H: TA] :
( A
!= ( product_Pair @ TG @ ( product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) ) @ B @ ( product_Pair @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) @ C @ ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ D @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ E @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ F @ ( product_Pair @ TB @ TA @ G @ H ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(241,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( units @ TA @ product_unit @ ( units_of @ TA @ TB @ A ) )
= ( units @ TA @ TB @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_units__of__units) ).
thf(937,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( units @ TA @ product_unit @ ( units_of @ TA @ TB @ A ) )
= ( units @ TA @ TB @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).
thf(245,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( units @ TA @ TB @ A ) )
=> ( ( member @ TA @ C @ ( units @ TA @ TB @ A ) )
=> ( member @ TA @ ( mult @ TA @ TB @ A @ B @ C ) @ ( units @ TA @ TB @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_monoid_OUnits__m__closed) ).
thf(966,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( units @ TA @ TB @ A ) )
=> ( ( member @ TA @ C @ ( units @ TA @ TB @ A ) )
=> ( member @ TA @ ( mult @ TA @ TB @ A @ B @ C ) @ ( units @ TA @ TB @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(159,axiom,
! [TA: $tType,A: TA,B: set @ TA] :
( ( insert @ TA @ A @ B )
!= ( bot_bot @ ( set @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_213_insert__not__empty) ).
thf(692,plain,
! [TA: $tType,A: TA,B: set @ TA] :
( ( insert @ TA @ A @ B )
!= ( bot_bot @ ( set @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(174,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_166_in__mono) ).
thf(734,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(58,axiom,
( ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_195_carrier__one__not__zero) ).
thf(396,plain,
( ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[58]) ).
thf(63,axiom,
! [A: set @ a,B: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( a_l_coset @ a @ b @ r @ B @ A ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_160_a__l__coset__subset__G) ).
thf(411,plain,
! [A: set @ a,B: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( a_l_coset @ a @ b @ r @ B @ A ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(5,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( member @ TA @ ( zero @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_ring_Oring__simprules_I2_J) ).
thf(265,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( member @ TA @ ( zero @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(250,axiom,
! [TA: $tType] :
( ( (=) @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
& ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_set__eq__subset) ).
thf(981,plain,
! [TA: $tType] :
( ( (=) @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
& ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).
thf(186,axiom,
! [TA: $tType,A: TA,B: set @ TA] :
( ( member @ TA @ A @ B )
=> ( ( insert @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_240_insert__Diff) ).
thf(766,plain,
! [TA: $tType,A: TA,B: set @ TA] :
( ( member @ TA @ A @ B )
=> ( ( insert @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(17,axiom,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_prod_Oinject) ).
thf(292,plain,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(147,axiom,
member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ r3 @ s3 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_assms_I2_J) ).
thf(661,plain,
member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ r3 @ s3 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(237,axiom,
! [A: a,B: a,C: a,D: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ D ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( member @ ( set @ ( product_prod @ a @ a ) ) @ ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ C @ D ) ) @ ( locali990921413f_frac @ a @ a @ product_unit @ rel ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_closed__mult__rng__of__frac) ).
thf(925,plain,
! [A: a,B: a,C: a,D: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ D ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( member @ ( set @ ( product_prod @ a @ a ) ) @ ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ C @ D ) ) @ ( locali990921413f_frac @ a @ a @ product_unit @ rel ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).
thf(197,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( member @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ C ) )
=> ~ ( member @ TA @ A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_248_DiffD2) ).
thf(810,plain,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( member @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ C ) )
=> ~ ( member @ TA @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).
thf(51,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( locali1402578320of_rng @ TA @ TB @ A @ B )
=> ( locali1441642228monoid @ TA @ ( ring_ext @ TA @ TB ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_mult__submonoid__of__rng_Oaxioms_I2_J) ).
thf(381,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( locali1402578320of_rng @ TA @ TB @ A @ B )
=> ( locali1441642228monoid @ TA @ ( ring_ext @ TA @ TB ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(225,axiom,
! [A: a,B: int] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( add_pow @ a @ b @ int @ r @ B @ A ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_106_add_Oint__pow__closed) ).
thf(894,plain,
! [A: a,B: int] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( add_pow @ a @ b @ int @ r @ B @ A ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(179,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: int] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( add_pow @ TA @ TB @ int @ A @ D @ C ) )
= ( add_pow @ TA @ TB @ int @ A @ D @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_112_ring_Oadd__pow__rdistr__int) ).
thf(746,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: int] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( add_pow @ TA @ TB @ int @ A @ D @ C ) )
= ( add_pow @ TA @ TB @ int @ A @ D @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(125,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/sandbox/benchmark/theBenchmark.p',fact_56_Units__m__closed) ).
thf(598,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)],[125]) ).
thf(201,axiom,
! [A: set @ a,B: set @ a,C: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_182_a__setmult__rcos__assoc) ).
thf(820,plain,
! [A: set @ a,B: set @ a,C: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[201]) ).
thf(111,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( one @ TA @ product_unit @ ( units_of @ TA @ TB @ A ) )
= ( one @ TA @ TB @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_units__of__one) ).
thf(561,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( one @ TA @ product_unit @ ( units_of @ TA @ TB @ A ) )
= ( one @ TA @ TB @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(19,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( ring @ TA @ TB @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_ring_Ois__ring) ).
thf(297,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(136,axiom,
! [TA: $tType,A: set @ TA,B: TA] : ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_subset__insertI) ).
thf(630,plain,
! [TA: $tType,A: set @ TA,B: TA] : ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(69,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA,D: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ D )
=> ( ord_less_eq @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ D ) @ ( minus_minus @ ( set @ TA ) @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_245_Diff__mono) ).
thf(429,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA,D: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ D )
=> ( ord_less_eq @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ D ) @ ( minus_minus @ ( set @ TA ) @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(3,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( semiring @ TA @ TB @ A )
=> ( monoid @ TA @ ( ring_ext @ TA @ TB ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_68_semiring_Oaxioms_I2_J) ).
thf(260,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( semiring @ TA @ TB @ A )
=> ( monoid @ TA @ ( ring_ext @ TA @ TB ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(216,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( mult @ TA @ product_unit @ ( units_of @ TA @ TB @ A ) )
= ( mult @ TA @ TB @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_units__of__mult) ).
thf(862,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( mult @ TA @ product_unit @ ( units_of @ TA @ TB @ A ) )
= ( mult @ TA @ TB @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(42,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( abelian_monoid @ TA @ TB @ A )
=> ( member @ TA @ ( zero @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_abelian__monoidE_I2_J) ).
thf(361,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( abelian_monoid @ TA @ TB @ A )
=> ( member @ TA @ ( zero @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(71,axiom,
group @ a @ product_unit @ ( units_of @ a @ ( ring_ext @ a @ b ) @ r ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_units__group) ).
thf(435,plain,
group @ a @ product_unit @ ( units_of @ a @ ( ring_ext @ a @ b ) @ r ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(98,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( units @ TA @ TB @ A ) )
=> ? [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
& ( ( mult @ TA @ TB @ A @ B @ C )
= ( one @ TA @ TB @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_101_monoid_OUnits__r__inv__ex) ).
thf(522,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( units @ TA @ TB @ A ) )
=> ? [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
& ( ( mult @ TA @ TB @ A @ B @ C )
= ( one @ TA @ TB @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(76,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/sandbox/benchmark/theBenchmark.p',fact_9_m__closed) ).
thf(461,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)],[76]) ).
thf(87,axiom,
! [A: a,B: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B ) @ ( one @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) ) )
= ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_right__unit__mult__rng__of__frac) ).
thf(493,plain,
! [A: a,B: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B ) @ ( one @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) ) )
= ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(183,axiom,
! [A: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( a_l_coset @ a @ b @ r @ ( zero @ a @ b @ r ) @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_150_a__lcos__mult__one) ).
thf(757,plain,
! [A: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( a_l_coset @ a @ b @ r @ ( zero @ a @ b @ r ) @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(155,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( group @ TA @ TB @ A )
=> ( ord_less_eq @ ( set @ TA ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) @ ( units @ TA @ TB @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_group_OUnits) ).
thf(682,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( group @ TA @ TB @ A )
=> ( ord_less_eq @ ( set @ TA ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) @ ( units @ TA @ TB @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(14,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: set @ TA] :
( ( locali1441642228monoid @ TA @ TB @ A @ B )
=> ( member @ TA @ ( one @ TA @ TB @ A ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Oone__closed) ).
thf(285,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: set @ TA] :
( ( locali1441642228monoid @ TA @ TB @ A @ B )
=> ( member @ TA @ ( one @ TA @ TB @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(234,axiom,
! [A: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( a_r_coset @ a @ b @ r @ A @ ( zero @ a @ b @ r ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_184_a__coset__add__zero) ).
thf(916,plain,
! [A: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( a_r_coset @ a @ b @ r @ A @ ( zero @ a @ b @ r ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).
thf(78,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ~ ( member @ TA @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ ( insert @ TA @ A @ C ) )
= ( ord_less_eq @ ( set @ TA ) @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_218_subset__insert) ).
thf(467,plain,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ~ ( member @ TA @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ ( insert @ TA @ A @ C ) )
= ( ord_less_eq @ ( set @ TA ) @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(30,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
= ( B @ C ) )
=> ( ( collect @ TA @ A )
= ( collect @ TA @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_47_Collect__cong) ).
thf(324,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
= ( B @ C ) )
=> ( ( collect @ TA @ A )
= ( collect @ TA @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(248,axiom,
( ( ( units @ a @ ( ring_ext @ a @ b ) @ r )
= ( minus_minus @ ( set @ a ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) @ ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) ) )
=> ( field @ a @ b @ r ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_226_cring__fieldI) ).
thf(974,plain,
( ( ( units @ a @ ( ring_ext @ a @ b ) @ r )
= ( minus_minus @ ( set @ a ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[248]) ).
thf(27,axiom,
! [TA: $tType,TB: $tType,A: ( product_prod @ TA @ TB ) > $o,B: product_prod @ TA @ TB] :
( ! [C: TA,D: TB] : ( A @ ( product_Pair @ TA @ TB @ C @ D ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_prod__cases) ).
thf(317,plain,
! [TA: $tType,TB: $tType,A: ( product_prod @ TA @ TB ) > $o,B: product_prod @ TA @ TB] :
( ! [C: TA,D: TB] : ( A @ ( product_Pair @ TA @ TB @ C @ D ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(143,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( monoid @ TA @ TB @ A )
=> ( group @ TA @ product_unit @ ( units_of @ TA @ TB @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_monoid_Ounits__group) ).
thf(650,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( monoid @ TA @ TB @ A )
=> ( group @ TA @ product_unit @ ( units_of @ TA @ TB @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(171,axiom,
! [TA: $tType,A: set @ TA] :
( ( ? [B: TA] : ( member @ TA @ B @ A ) )
= ( A
!= ( bot_bot @ ( set @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_165_ex__in__conv) ).
thf(725,plain,
! [TA: $tType,A: set @ TA] :
( ( ? [B: TA] : ( member @ TA @ B @ A ) )
= ( A
!= ( bot_bot @ ( set @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(55,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: set @ TA] :
( ( locali1441642228monoid @ TA @ TB @ A @ B )
=> ( ord_less_eq @ ( set @ TA ) @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Osubset) ).
thf(388,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: set @ TA] :
( ( locali1441642228monoid @ TA @ TB @ A @ B )
=> ( ord_less_eq @ ( set @ TA ) @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(66,axiom,
! [A: set @ a,B: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ ( set @ a ) @ ( a_r_coset @ a @ b @ r @ A @ B ) @ ( a_RCOSETS @ a @ b @ r @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_a__rcosetsI) ).
thf(420,plain,
! [A: set @ a,B: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[66]) ).
thf(89,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ s2 ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1_f4) ).
thf(499,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)],[89]) ).
thf(253,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( monoid @ TA @ TB @ A )
=> ( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ? [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
& ( ( mult @ TA @ TB @ A @ C @ B )
= ( one @ TA @ TB @ A ) ) ) )
=> ( group @ TA @ TB @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_147_monoid_Ogroup__l__invI) ).
thf(989,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( monoid @ TA @ TB @ A )
=> ( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ? [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
& ( ( mult @ TA @ TB @ A @ C @ B )
= ( one @ TA @ TB @ A ) ) ) )
=> ( group @ TA @ TB @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(214,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ TA] :
( ( ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) )
= ( insert @ TA @ B @ C ) )
= ( ( B = A )
& ( ord_less_eq @ ( set @ TA ) @ C @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_singleton__insert__inj__eq) ).
thf(857,plain,
! [TA: $tType,A: TA,B: TA,C: set @ TA] :
( ( ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) )
= ( insert @ TA @ B @ C ) )
= ( ( B = A )
& ( ord_less_eq @ ( set @ TA ) @ C @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(97,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( zero @ TA @ TB @ A ) )
= ( zero @ TA @ TB @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_ring_Oring__simprules_I25_J) ).
thf(519,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( zero @ TA @ TB @ A ) )
= ( zero @ TA @ TB @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(177,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) )] :
( ! [C: TA,D: TB,E: TC,F: TD] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ TD ) @ D @ ( product_Pair @ TC @ TD @ E @ F ) ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_86_prod__induct4) ).
thf(742,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) )] :
( ! [C: TA,D: TB,E: TC,F: TD] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ TD ) @ D @ ( product_Pair @ TC @ TD @ E @ F ) ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(140,axiom,
member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ r2 @ s2 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_assms_I1_J) ).
thf(642,plain,
member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ r2 @ s2 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(99,axiom,
( ( ( zero @ a @ b @ r )
!= ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( ! [A: a] :
( ( member @ a @ A @ ( minus_minus @ ( set @ a ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_225_field__intro2) ).
thf(526,plain,
( ( ( zero @ a @ b @ r )
!= ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( ! [A: a] :
( ( member @ a @ A @ ( minus_minus @ ( set @ a ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[99]) ).
thf(86,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA] :
( ( group @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ? [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
& ( ( mult @ TA @ TB @ A @ B @ C )
= ( one @ TA @ TB @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_143_group_Or__inv__ex) ).
thf(489,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA] :
( ( group @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ? [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
& ( ( mult @ TA @ TB @ A @ B @ C )
= ( one @ TA @ TB @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(122,axiom,
abelian_monoid @ a @ b @ r,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_abelian__monoid__axioms) ).
thf(590,plain,
abelian_monoid @ a @ b @ r,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(164,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( ( mult @ TA @ TB @ A @ B @ C )
= ( one @ TA @ TB @ A ) )
=> ( ( ( mult @ TA @ TB @ A @ C @ D )
= ( one @ TA @ TB @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( B = D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_monoid_Oinv__unique) ).
thf(706,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( ( mult @ TA @ TB @ A @ B @ C )
= ( one @ TA @ TB @ A ) )
=> ( ( ( mult @ TA @ TB @ A @ C @ D )
= ( one @ TA @ TB @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( B = D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(157,axiom,
! [TA: $tType,A: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ A @ ( bot_bot @ ( set @ TA ) ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_230_Diff__empty) ).
thf(686,plain,
! [TA: $tType,A: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ A @ ( bot_bot @ ( set @ TA ) ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(82,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( ( member @ TA @ A @ B )
=> ( ( minus_minus @ ( set @ TA ) @ ( insert @ TA @ A @ C ) @ B )
= ( minus_minus @ ( set @ TA ) @ C @ B ) ) )
& ( ~ ( member @ TA @ A @ B )
=> ( ( minus_minus @ ( set @ TA ) @ ( insert @ TA @ A @ C ) @ B )
= ( insert @ TA @ A @ ( minus_minus @ ( set @ TA ) @ C @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_237_insert__Diff__if) ).
thf(476,plain,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( ( member @ TA @ A @ B )
=> ( ( minus_minus @ ( set @ TA ) @ ( insert @ TA @ A @ C ) @ B )
= ( minus_minus @ ( set @ TA ) @ C @ B ) ) )
& ( ~ ( member @ TA @ A @ B )
=> ( ( minus_minus @ ( set @ TA ) @ ( insert @ TA @ A @ C ) @ B )
= ( insert @ TA @ A @ ( minus_minus @ ( set @ TA ) @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(207,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) @ D )
= ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_74_ring_Oring__simprules_I11_J) ).
thf(837,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) @ D )
= ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(57,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA] :
( ( group @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( ( mult @ TA @ TB @ A @ C @ B )
= B )
= ( C
= ( one @ TA @ TB @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_141_group_Or__cancel__one) ).
thf(393,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA] :
( ( group @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( ( mult @ TA @ TB @ A @ C @ B )
= B )
= ( C
= ( one @ TA @ TB @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(162,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_220_subset__insertI2) ).
thf(701,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(75,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ( ord_less_eq @ ( set @ TA ) @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_171_equalityD2) ).
thf(457,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ( ord_less_eq @ ( set @ TA ) @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(251,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ TE ) @ E @ ( product_Pair @ TD @ TE @ F @ G ) ) ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_85_prod__induct5) ).
thf(984,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ TE ) @ E @ ( product_Pair @ TD @ TE @ F @ G ) ) ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).
thf(46,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( group @ TA @ TB @ A )
=> ( monoid @ TA @ TB @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_group_Ois__monoid) ).
thf(371,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( group @ TA @ TB @ A )
=> ( monoid @ TA @ TB @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(91,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ B ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_244_Diff__subset) ).
thf(503,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ B ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(142,axiom,
! [A: set @ a,B: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( a_r_coset @ a @ b @ r @ A @ B ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_181_a__r__coset__subset__G) ).
thf(647,plain,
! [A: set @ a,B: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( a_r_coset @ a @ b @ r @ A @ B ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(256,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_158_subsetI) ).
thf(997,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).
thf(191,axiom,
! [A: a] :
( ( member @ a @ A @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ? [B: a] :
( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_51_Units__l__inv__ex) ).
thf(792,plain,
! [A: a] :
( ( member @ a @ A @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ? [B: a] :
( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ 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)],[191]) ).
thf(93,axiom,
( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) )
= ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ s2 ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4_f2) ).
thf(508,plain,
( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) )
= ( locali1641774487f_frac @ a @ a @ product_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)],[93]) ).
thf(24,axiom,
! [TA: $tType,A: TA,B: set @ TA] :
( ( member @ TA @ A @ B )
=> ( ( insert @ TA @ A @ B )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_insert__absorb) ).
thf(311,plain,
! [TA: $tType,A: TA,B: set @ TA] :
( ( member @ TA @ A @ B )
=> ( ( insert @ TA @ A @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(104,axiom,
! [TA: $tType,A: TA > $o] :
( ( ( collect @ TA @ A )
= ( bot_bot @ ( set @ TA ) ) )
= ( ! [B: TA] :
~ ( A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_156_Collect__empty__eq) ).
thf(541,plain,
! [TA: $tType,A: TA > $o] :
( ( ( collect @ TA @ A )
= ( bot_bot @ ( set @ TA ) ) )
= ( ! [B: TA] :
~ ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(152,axiom,
member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ ( units @ a @ ( ring_ext @ a @ b ) @ r ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_Units__one__closed) ).
thf(674,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)],[152]) ).
thf(121,axiom,
! [A: a] :
( ( member @ a @ A @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ? [B: a] :
( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
& ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A @ B )
= ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_Units__r__inv__ex) ).
thf(586,plain,
! [A: a] :
( ( member @ a @ A @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ? [B: a] :
( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[121]) ).
thf(148,axiom,
! [A: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ! [B: a] :
( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A @ B )
= B ) )
=> ( A
= ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_one__unique) ).
thf(662,plain,
! [A: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ! [B: a] :
( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[148]) ).
thf(223,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( semiring @ TA @ TB @ A )
=> ( ( ( one @ TA @ ( ring_ext @ TA @ TB ) @ A )
= ( zero @ TA @ TB @ A ) )
=> ( ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
= ( insert @ TA @ ( zero @ TA @ TB @ A ) @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_224_semiring_Oone__zeroD) ).
thf(888,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( semiring @ TA @ TB @ A )
=> ( ( ( one @ TA @ ( ring_ext @ TA @ TB ) @ A )
= ( zero @ TA @ TB @ A ) )
=> ( ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
= ( insert @ TA @ ( zero @ TA @ TB @ A ) @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).
thf(20,axiom,
! [TA: $tType,A: TA,B: set @ TA] :
( ( insert @ TA @ A @ ( insert @ TA @ A @ B ) )
= ( insert @ TA @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_insert__absorb2) ).
thf(298,plain,
! [TA: $tType,A: TA,B: set @ TA] :
( ( insert @ TA @ A @ ( insert @ TA @ A @ B ) )
= ( insert @ TA @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(95,axiom,
! [TA: $tType,A: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ ( bot_bot @ ( set @ TA ) ) )
= ( A
= ( bot_bot @ ( set @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_subset__empty) ).
thf(513,plain,
! [TA: $tType,A: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ ( bot_bot @ ( set @ TA ) ) )
= ( A
= ( bot_bot @ ( set @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(202,axiom,
! [A: a,B: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A @ B )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_m__comm) ).
thf(824,plain,
! [A: a,B: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[202]) ).
thf(213,axiom,
ring @ a @ b @ r,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_local_Oring__axioms) ).
thf(856,plain,
ring @ a @ b @ r,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(11,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( semiring @ TA @ TB @ A )
=> ( member @ TA @ ( one @ TA @ ( ring_ext @ TA @ TB ) @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_semiring_Osemiring__simprules_I4_J) ).
thf(279,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( semiring @ TA @ TB @ A )
=> ( member @ TA @ ( one @ TA @ ( ring_ext @ TA @ TB ) @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(115,axiom,
! [TA: $tType,A: TA,B: set @ TA] :
( ~ ( member @ TA @ A @ B )
=> ( ( minus_minus @ ( set @ TA ) @ ( insert @ TA @ A @ B ) @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_238_Diff__insert__absorb) ).
thf(573,plain,
! [TA: $tType,A: TA,B: set @ TA] :
( ~ ( member @ TA @ A @ B )
=> ( ( minus_minus @ ( set @ TA ) @ ( insert @ TA @ A @ B ) @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(158,axiom,
! [A: a,B: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( one @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B ) )
= ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_left__unit__mult__rng__of__frac) ).
thf(689,plain,
! [A: a,B: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( one @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B ) )
= ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(126,axiom,
! [A: set @ a] :
( ( additive_subgroup @ a @ b @ A @ r )
=> ( ord_less_eq @ ( set @ ( set @ a ) ) @ ( a_RCOSETS @ a @ b @ r @ A ) @ ( pow @ a @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_252_rcosets__subset__PowG) ).
thf(601,plain,
! [A: set @ a] :
( ( additive_subgroup @ a @ b @ A @ r )
=> ( ord_less_eq @ ( set @ ( set @ a ) ) @ ( a_RCOSETS @ a @ b @ r @ A ) @ ( pow @ a @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(229,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ C )
=> ( ( minus_minus @ ( set @ TA ) @ B @ ( minus_minus @ ( set @ TA ) @ C @ A ) )
= A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_243_double__diff) ).
thf(904,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ C )
=> ( ( minus_minus @ ( set @ TA ) @ B @ ( minus_minus @ ( set @ TA ) @ C @ A ) )
= A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).
thf(67,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( ord_less_eq @ ( set @ TA ) @ ( collect @ TA @ A ) @ ( collect @ TA @ B ) )
= ( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_Collect__mono__iff) ).
thf(423,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( ord_less_eq @ ( set @ TA ) @ ( collect @ TA @ A ) @ ( collect @ TA @ B ) )
= ( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(60,axiom,
! [A: set @ a,B: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( set_add @ a @ b @ r @ A @ B ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_152_setadd__subset__G) ).
thf(402,plain,
! [A: set @ a,B: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( set_add @ a @ b @ r @ A @ B ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(209,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ~ ( member @ TA @ A @ B )
=> ( ( minus_minus @ ( set @ TA ) @ B @ ( insert @ TA @ A @ C ) )
= ( minus_minus @ ( set @ TA ) @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_234_Diff__insert0) ).
thf(843,plain,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ~ ( member @ TA @ A @ B )
=> ( ( minus_minus @ ( set @ TA ) @ B @ ( insert @ TA @ A @ C ) )
= ( minus_minus @ ( set @ TA ) @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(12,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: set @ TA] :
( ( locali1441642228monoid @ TA @ TB @ A @ B )
=> ( monoid @ TA @ TB @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_submonoid_Oaxioms_I1_J) ).
thf(281,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: set @ TA] :
( ( locali1441642228monoid @ TA @ TB @ A @ B )
=> ( monoid @ TA @ TB @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(219,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( units @ TA @ TB @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( ( mult @ TA @ TB @ A @ B @ C )
= ( mult @ TA @ TB @ A @ B @ D ) )
= ( C = D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_monoid_OUnits__l__cancel) ).
thf(875,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( units @ TA @ TB @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( ( mult @ TA @ TB @ A @ B @ C )
= ( mult @ TA @ TB @ A @ B @ D ) )
= ( C = D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).
thf(117,axiom,
! [TA: $tType] :
( ( ord_less_eq @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_subset__eq) ).
thf(577,plain,
! [TA: $tType] :
( ( ord_less_eq @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(130,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( field @ TA @ TB @ A )
=> ( ( units @ TA @ ( ring_ext @ TA @ TB ) @ A )
= ( minus_minus @ ( set @ TA ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) @ ( insert @ TA @ ( zero @ TA @ TB @ A ) @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_251_field_Ofield__Units) ).
thf(611,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( field @ TA @ TB @ A )
=> ( ( units @ TA @ ( ring_ext @ TA @ TB ) @ A )
= ( minus_minus @ ( set @ TA ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) @ ( insert @ TA @ ( zero @ TA @ TB @ A ) @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(166,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ TC ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ TC )] :
( ! [C: TA,D: TB,E: TC] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ TC ) @ C @ ( product_Pair @ TB @ TC @ D @ E ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_87_prod__induct3) ).
thf(713,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ TC ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ TC )] :
( ! [C: TA,D: TB,E: TC] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ TC ) @ C @ ( product_Pair @ TB @ TC @ D @ E ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(243,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ! [B: TA,C: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( member @ TA @ ( mult @ TA @ TB @ A @ B @ C ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ) )
=> ( ( member @ TA @ ( one @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ! [B: TA,C: TA,D: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( mult @ TA @ TB @ A @ B @ C ) @ D )
= ( mult @ TA @ TB @ A @ B @ ( mult @ TA @ TB @ A @ C @ D ) ) ) ) ) )
=> ( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( one @ TA @ TB @ A ) @ B )
= B ) )
=> ( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ B @ ( one @ TA @ TB @ A ) )
= B ) )
=> ( monoid @ TA @ TB @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_monoidI) ).
thf(947,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ! [B: TA,C: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( member @ TA @ ( mult @ TA @ TB @ A @ B @ C ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ) )
=> ( ( member @ TA @ ( one @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ! [B: TA,C: TA,D: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( mult @ TA @ TB @ A @ B @ C ) @ D )
= ( mult @ TA @ TB @ A @ B @ ( mult @ TA @ TB @ A @ C @ D ) ) ) ) ) )
=> ( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( one @ TA @ TB @ A ) @ B )
= B ) )
=> ( ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ B @ ( one @ TA @ TB @ A ) )
= B ) )
=> ( monoid @ TA @ TB @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).
thf(39,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: TA] :
( ( ~ ( member @ TA @ A @ B )
=> ( A = C ) )
=> ( member @ TA @ A @ ( insert @ TA @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194_insertCI) ).
thf(350,plain,
! [TA: $tType,A: TA,B: set @ TA,C: TA] :
( ( ~ ( member @ TA @ A @ B )
=> ( A = C ) )
=> ( member @ TA @ A @ ( insert @ TA @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(235,axiom,
( ( ( one @ a @ ( ring_ext @ a @ b ) @ r )
= ( zero @ a @ b @ r ) )
=> ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
= ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_one__zeroD) ).
thf(919,plain,
( ( ( one @ a @ ( ring_ext @ a @ b ) @ r )
= ( zero @ a @ b @ r ) )
=> ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
= ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).
thf(195,axiom,
! [TA: $tType,A: set @ TA,B: TA] :
( ( A
= ( bot_bot @ ( set @ TA ) ) )
=> ~ ( member @ TA @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_163_equals0D) ).
thf(804,plain,
! [TA: $tType,A: set @ TA,B: TA] :
( ( A
= ( bot_bot @ ( set @ TA ) ) )
=> ~ ( member @ TA @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(175,axiom,
! [TA: $tType,A: TA,B: TA,C: TA,D: TA] :
( ( ( insert @ TA @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
= ( insert @ TA @ C @ ( insert @ TA @ D @ ( bot_bot @ ( set @ TA ) ) ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_214_doubleton__eq__iff) ).
thf(736,plain,
! [TA: $tType,A: TA,B: TA,C: TA,D: TA] :
( ( ( insert @ TA @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
= ( insert @ TA @ C @ ( insert @ TA @ D @ ( bot_bot @ ( set @ TA ) ) ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(73,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: nat] :
( ( semiring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( add_pow @ TA @ TB @ nat @ A @ D @ B ) @ C )
= ( add_pow @ TA @ TB @ nat @ A @ D @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_149_semiring_Oadd__pow__ldistr) ).
thf(451,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: nat] :
( ( semiring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( add_pow @ TA @ TB @ nat @ A @ D @ B ) @ C )
= ( add_pow @ TA @ TB @ nat @ A @ D @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(8,axiom,
! [TA: $tType,A: TA,B: set @ TA] : ( member @ TA @ A @ ( insert @ TA @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_insertI1) ).
thf(271,plain,
! [TA: $tType,A: TA,B: set @ TA] : ( member @ TA @ A @ ( insert @ TA @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(37,axiom,
! [TA: $tType,TB: $tType,A: ( product_prod @ TA @ TB ) > $o,B: product_prod @ TA @ TB] :
( ! [C: TA,D: TB] : ( A @ ( product_Pair @ TA @ TB @ C @ D ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_old_Oprod_Oinducts) ).
thf(345,plain,
! [TA: $tType,TB: $tType,A: ( product_prod @ TA @ TB ) > $o,B: product_prod @ TA @ TB] :
( ! [C: TA,D: TB] : ( A @ ( product_Pair @ TA @ TB @ C @ D ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(102,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( semiring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( one @ TA @ ( ring_ext @ TA @ TB ) @ A ) @ B )
= B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_semiring_Osemiring__simprules_I9_J) ).
thf(535,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( semiring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( one @ TA @ ( ring_ext @ TA @ TB ) @ A ) @ B )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(200,axiom,
( ! [A: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ? [B: a] :
( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_129_group__l__invI) ).
thf(816,plain,
( ! [A: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ? [B: a] :
( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[200]) ).
thf(129,axiom,
! [TA: $tType,A: TA] :
~ ( member @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_162_emptyE) ).
thf(609,plain,
! [TA: $tType,A: TA] :
~ ( member @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(124,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( units @ TA @ TB @ A ) )
=> ? [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
& ( ( mult @ TA @ TB @ A @ C @ B )
= ( one @ TA @ TB @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_monoid_OUnits__l__inv__ex) ).
thf(594,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( units @ TA @ TB @ A ) )
=> ? [C: TA] :
( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
& ( ( mult @ TA @ TB @ A @ C @ B )
= ( one @ TA @ TB @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(193,axiom,
! [TA: $tType,A: set @ TA,B: TA,C: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ A @ ( insert @ TA @ B @ C ) )
= ( minus_minus @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ C ) @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_Diff__insert) ).
thf(799,plain,
! [TA: $tType,A: set @ TA,B: TA,C: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ A @ ( insert @ TA @ B @ C ) )
= ( minus_minus @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ C ) @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).
thf(187,axiom,
member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_semiring__simprules_I4_J) ).
thf(769,plain,
member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(150,axiom,
! [A: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_r__one) ).
thf(668,plain,
! [A: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[150]) ).
thf(32,axiom,
! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA] :
? [B: TB,C: TA] :
( A
= ( product_Pair @ TB @ TA @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_surj__pair) ).
thf(330,plain,
! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA] :
? [B: TB,C: TA] :
( A
= ( product_Pair @ TB @ TA @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(239,axiom,
( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) )
= ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_f1) ).
thf(932,plain,
( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) )
= ( locali1641774487f_frac @ a @ a @ product_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)],[239]) ).
thf(59,axiom,
( ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_190_one__zeroI) ).
thf(399,plain,
( ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[59]) ).
thf(160,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) )
=> ( ord_less_eq @ ( set @ TA ) @ ( collect @ TA @ A ) @ ( collect @ TA @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_Collect__mono) ).
thf(695,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) )
=> ( ord_less_eq @ ( set @ TA ) @ ( collect @ TA @ A ) @ ( collect @ TA @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(218,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ~ ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ~ ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_168_equalityE) ).
thf(868,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ~ ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ~ ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(252,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA,D: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ ( minus_minus @ ( set @ TA ) @ B @ ( insert @ TA @ C @ D ) ) )
= ( ( ord_less_eq @ ( set @ TA ) @ A @ ( minus_minus @ ( set @ TA ) @ B @ D ) )
& ~ ( member @ TA @ C @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_242_subset__Diff__insert) ).
thf(986,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA,D: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ ( minus_minus @ ( set @ TA ) @ B @ ( insert @ TA @ C @ D ) ) )
= ( ( ord_less_eq @ ( set @ TA ) @ A @ ( minus_minus @ ( set @ TA ) @ B @ D ) )
& ~ ( member @ TA @ C @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).
thf(257,axiom,
! [TA: $tType,A: set @ TA,B: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
= ( ( A
= ( bot_bot @ ( set @ TA ) ) )
| ( A
= ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_subset__singleton__iff) ).
thf(1000,plain,
! [TA: $tType,A: set @ TA,B: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
= ( ( A
= ( bot_bot @ ( set @ TA ) ) )
| ( A
= ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).
thf(6,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( ring @ TA @ TB @ A )
=> ( ( locali1441642228monoid @ TA @ ( ring_ext @ TA @ TB ) @ A @ B )
=> ( locali1402578320of_rng @ TA @ TB @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_mult__submonoid__of__rng_Ointro) ).
thf(267,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( ring @ TA @ TB @ A )
=> ( ( locali1441642228monoid @ TA @ ( ring_ext @ TA @ TB ) @ A @ B )
=> ( locali1402578320of_rng @ TA @ TB @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(92,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_159_subset__antisym) ).
thf(505,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(70,axiom,
! [A: a,B: a,C: nat] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_126_add__pow__rdistr) ).
thf(431,plain,
! [A: a,B: a,C: nat] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[70]) ).
thf(236,axiom,
! [TA: $tType,A: set @ TA,B: TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ B @ C ) )
= ( ( ( member @ TA @ B @ A )
=> ( ord_less_eq @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) @ C ) )
& ( ~ ( member @ TA @ B @ A )
=> ( ord_less_eq @ ( set @ TA ) @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_249_subset__insert__iff) ).
thf(922,plain,
! [TA: $tType,A: set @ TA,B: TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ B @ C ) )
= ( ( ( member @ TA @ B @ A )
=> ( ord_less_eq @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) @ C ) )
& ( ~ ( member @ TA @ B @ A )
=> ( ord_less_eq @ ( set @ TA ) @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).
thf(146,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/sandbox/benchmark/theBenchmark.p',fact_38_Units__inv__comm) ).
thf(658,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)],[146]) ).
thf(80,axiom,
locali1648044335f_frac @ a @ b @ r @ s,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_eq__obj__rng__of__frac__axioms) ).
thf(473,plain,
locali1648044335f_frac @ a @ b @ r @ s,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(170,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: int] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( add_pow @ TA @ TB @ int @ A @ D @ B ) @ C )
= ( add_pow @ TA @ TB @ int @ A @ D @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_111_ring_Oadd__pow__ldistr__int) ).
thf(722,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: int] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( add_pow @ TA @ TB @ int @ A @ D @ B ) @ C )
= ( add_pow @ TA @ TB @ int @ A @ D @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(153,axiom,
( ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_196_carrier__one__zero) ).
thf(675,plain,
( ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[153]) ).
thf(68,axiom,
! [TA: $tType,A: set @ TA,B: TA,C: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ A @ ( insert @ TA @ B @ C ) )
= ( minus_minus @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_239_Diff__insert2) ).
thf(426,plain,
! [TA: $tType,A: set @ TA,B: TA,C: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ A @ ( insert @ TA @ B @ C ) )
= ( minus_minus @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(100,axiom,
semiring @ a @ b @ r,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_local_Osemiring__axioms) ).
thf(531,plain,
semiring @ a @ b @ r,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(106,axiom,
! [TA: $tType,A: TA] : ( member @ TA @ A @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_singletonI) ).
thf(548,plain,
! [TA: $tType,A: TA] : ( member @ TA @ A @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(198,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ C )
=> ( ord_less_eq @ ( set @ TA ) @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_subset__trans) ).
thf(812,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ C )
=> ( ord_less_eq @ ( set @ TA ) @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(180,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( member @ TA @ A @ B )
=> ( ~ ( member @ TA @ A @ C )
=> ( member @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_227_DiffI) ).
thf(749,plain,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( member @ TA @ A @ B )
=> ( ~ ( member @ TA @ A @ C )
=> ( member @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(36,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ TA] :
( ( member @ TA @ A @ ( insert @ TA @ B @ C ) )
=> ( ( A != B )
=> ( member @ TA @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_211_insertE) ).
thf(342,plain,
! [TA: $tType,A: TA,B: TA,C: set @ TA] :
( ( member @ TA @ A @ ( insert @ TA @ B @ C ) )
=> ( ( A != B )
=> ( member @ TA @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(40,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( member @ TA @ ( one @ TA @ ( ring_ext @ TA @ TB ) @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_ring_Oring__simprules_I6_J) ).
thf(355,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( member @ TA @ ( one @ TA @ ( ring_ext @ TA @ TB ) @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(182,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( group @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( ( mult @ TA @ TB @ A @ C @ B )
= ( mult @ TA @ TB @ A @ D @ B ) )
= ( C = D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_Group_Ogroup_Oright__cancel) ).
thf(754,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( group @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( ( mult @ TA @ TB @ A @ C @ B )
= ( mult @ TA @ TB @ A @ D @ B ) )
= ( C = D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(108,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA] :
( ( group @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( ( mult @ TA @ TB @ A @ B @ C )
= B )
= ( C
= ( one @ TA @ TB @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_142_group_Ol__cancel__one) ).
thf(553,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA] :
( ( group @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( ( mult @ TA @ TB @ A @ B @ C )
= B )
= ( C
= ( one @ TA @ TB @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(246,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( zero @ TA @ TB @ A ) @ B )
= ( zero @ TA @ TB @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_ring_Oring__simprules_I24_J) ).
thf(968,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( zero @ TA @ TB @ A ) @ B )
= ( zero @ TA @ TB @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).
thf(132,axiom,
! [TA: $tType,A: TA,B: TA] :
( ( ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) )
= ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
=> ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_212_singleton__inject) ).
thf(617,plain,
! [TA: $tType,A: TA,B: TA] :
( ( ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) )
= ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(96,axiom,
! [TA: $tType,A: TA,B: set @ TA] :
( ( insert @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) )
= ( insert @ TA @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_236_insert__Diff__single) ).
thf(516,plain,
! [TA: $tType,A: TA,B: set @ TA] :
( ( insert @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) )
= ( insert @ TA @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(118,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE,H: TF,I: TG] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) @ E @ ( product_Pair @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) @ F @ ( product_Pair @ TE @ ( product_prod @ TF @ TG ) @ G @ ( product_Pair @ TF @ TG @ H @ I ) ) ) ) ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_83_prod__induct7) ).
thf(580,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE,H: TF,I: TG] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) @ E @ ( product_Pair @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) @ F @ ( product_Pair @ TE @ ( product_prod @ TF @ TG ) @ G @ ( product_Pair @ TF @ TG @ H @ I ) ) ) ) ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(18,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( monoid @ TA @ ( ring_ext @ TA @ TB ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_67_ring_Ois__monoid) ).
thf(295,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( monoid @ TA @ ( ring_ext @ TA @ TB ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(31,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: TC > TA > TB,B: TC,C: TA] :
( ( product_rec_prod @ TC @ TA @ TB @ A @ ( product_Pair @ TC @ TA @ B @ C ) )
= ( A @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_old_Oprod_Orec) ).
thf(327,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: TC > TA > TB,B: TC,C: TA] :
( ( product_rec_prod @ TC @ TA @ TB @ A @ ( product_Pair @ TC @ TA @ B @ C ) )
= ( A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(176,axiom,
! [A: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_l__one) ).
thf(739,plain,
! [A: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[176]) ).
thf(62,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: set @ TA,C: TA,D: TA] :
( ( locali1441642228monoid @ TA @ TB @ A @ B )
=> ( ( member @ TA @ C @ B )
=> ( ( member @ TA @ D @ B )
=> ( member @ TA @ ( mult @ TA @ TB @ A @ C @ D ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Om__closed) ).
thf(409,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: set @ TA,C: TA,D: TA] :
( ( locali1441642228monoid @ TA @ TB @ A @ B )
=> ( ( member @ TA @ C @ B )
=> ( ( member @ TA @ D @ B )
=> ( member @ TA @ ( mult @ TA @ TB @ A @ C @ D ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(254,axiom,
locali7914339f_crng @ a @ b @ r @ s,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5_mult__submonoid__of__crng__axioms) ).
thf(993,plain,
locali7914339f_crng @ a @ b @ r @ s,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).
thf(64,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ( minus_minus @ ( set @ TA ) @ A @ B )
= ( bot_bot @ ( set @ TA ) ) )
= ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_Diff__eq__empty__iff) ).
thf(414,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ( minus_minus @ ( set @ TA ) @ A @ B )
= ( bot_bot @ ( set @ TA ) ) )
= ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(247,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) )] :
~ ! [B: TD,C: TC,D: TB,E: TA] :
( A
!= ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ B @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ C @ ( product_Pair @ TB @ TA @ D @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_prod__cases4) ).
thf(971,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) )] :
~ ! [B: TD,C: TC,D: TB,E: TA] :
( A
!= ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ B @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ C @ ( product_Pair @ TB @ TA @ D @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).
thf(120,axiom,
equivalence @ ( product_prod @ a @ a ) @ product_unit @ rel,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_equiv__obj__rng__of__frac) ).
thf(585,plain,
equivalence @ ( product_prod @ a @ a ) @ product_unit @ rel,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(208,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( mult @ TA @ TB @ A @ B @ C ) @ D )
= ( mult @ TA @ TB @ A @ B @ ( mult @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_114_monoid_Om__assoc) ).
thf(840,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( mult @ TA @ TB @ A @ B @ C ) @ D )
= ( mult @ TA @ TB @ A @ B @ ( mult @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(34,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( locali1402578320of_rng @ TA @ TB @ A @ B )
=> ( ring @ TA @ TB @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_mult__submonoid__of__rng_Oaxioms_I1_J) ).
thf(338,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( locali1402578320of_rng @ TA @ TB @ A @ B )
=> ( ring @ TA @ TB @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(196,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE,H: TF] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) @ E @ ( product_Pair @ TD @ ( product_prod @ TE @ TF ) @ F @ ( product_Pair @ TE @ TF @ G @ H ) ) ) ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_84_prod__induct6) ).
thf(808,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE,H: TF] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) @ E @ ( product_Pair @ TD @ ( product_prod @ TE @ TF ) @ F @ ( product_Pair @ TE @ TF @ G @ H ) ) ) ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(141,axiom,
! [A: a,B: a,C: int] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_104_add__pow__ldistr__int) ).
thf(643,plain,
! [A: a,B: a,C: int] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[141]) ).
thf(79,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( semiring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( zero @ TA @ TB @ A ) )
= ( zero @ TA @ TB @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_semiring_Or__null) ).
thf(470,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( semiring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( zero @ TA @ TB @ A ) )
= ( zero @ TA @ TB @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(116,axiom,
ord_less_eq @ ( set @ a ) @ s @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_subset) ).
thf(576,plain,
ord_less_eq @ ( set @ a ) @ s @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(105,axiom,
! [A: a,B: a,C: a,D: a,E: a,F: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ D ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ E @ F ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ C @ D ) ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ E @ F ) )
= ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B ) @ ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ C @ D ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ E @ F ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_assoc__mult__rng__of__frac) ).
thf(544,plain,
! [A: a,B: a,C: a,D: a,E: a,F: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ C @ D ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ E @ F ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ C @ D ) ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ E @ F ) )
= ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B ) @ ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ C @ D ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ E @ F ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(230,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( member @ TA @ ( mult @ TA @ TB @ A @ B @ C ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_monoid_Om__closed) ).
thf(907,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( member @ TA @ ( mult @ TA @ TB @ A @ B @ C ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).
thf(178,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ord_less_eq @ ( set @ TA ) @ ( insert @ TA @ C @ A ) @ ( insert @ TA @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_217_insert__mono) ).
thf(744,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ord_less_eq @ ( set @ TA ) @ ( insert @ TA @ C @ A ) @ ( insert @ TA @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(163,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ B ) @ B )
= ( minus_minus @ ( set @ TA ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_229_Diff__idemp) ).
thf(703,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ B ) @ B )
= ( minus_minus @ ( set @ TA ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(54,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( semiring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( member @ TA @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_semiring_Osemiring__simprules_I3_J) ).
thf(386,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( semiring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( member @ TA @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(25,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( semiring @ TA @ TB @ A )
=> ( member @ TA @ ( zero @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_semiring_Osemiring__simprules_I2_J) ).
thf(314,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( semiring @ TA @ TB @ A )
=> ( member @ TA @ ( zero @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(114,axiom,
! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ ( bot_bot @ ( set @ TA ) ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_161_empty__subsetI) ).
thf(571,plain,
! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ ( bot_bot @ ( set @ TA ) ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(135,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( ( mult @ TA @ TB @ A @ B @ C )
= ( one @ TA @ TB @ A ) )
=> ( ( member @ TA @ B @ ( units @ TA @ TB @ A ) )
=> ( ( member @ TA @ C @ ( units @ TA @ TB @ A ) )
=> ( ( mult @ TA @ TB @ A @ C @ B )
= ( one @ TA @ TB @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_103_monoid_OUnits__inv__comm) ).
thf(627,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( ( mult @ TA @ TB @ A @ B @ C )
= ( one @ TA @ TB @ A ) )
=> ( ( member @ TA @ B @ ( units @ TA @ TB @ A ) )
=> ( ( member @ TA @ C @ ( units @ TA @ TB @ A ) )
=> ( ( mult @ TA @ TB @ A @ C @ B )
= ( one @ TA @ TB @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(84,axiom,
! [A: set @ a,B: set @ a,C: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_180_a__rcos__assoc__lcos) ).
thf(482,plain,
! [A: set @ a,B: set @ a,C: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[84]) ).
thf(151,axiom,
! [TA: $tType,TB: $tType] :
( ( monoid @ TA @ TB )
= ( ^ [A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ! [B: TA,C: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( member @ TA @ ( mult @ TA @ TB @ A @ B @ C ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ) )
& ! [B: TA,C: TA,D: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( mult @ TA @ TB @ A @ B @ C ) @ D )
= ( mult @ TA @ TB @ A @ B @ ( mult @ TA @ TB @ A @ C @ D ) ) ) ) ) )
& ( member @ TA @ ( one @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
& ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( one @ TA @ TB @ A ) @ B )
= B ) )
& ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ B @ ( one @ TA @ TB @ A ) )
= B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_Group_Omonoid__def) ).
thf(671,plain,
! [TA: $tType,TB: $tType] :
( ( monoid @ TA @ TB )
= ( ^ [A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ! [B: TA,C: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( member @ TA @ ( mult @ TA @ TB @ A @ B @ C ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ) )
& ! [B: TA,C: TA,D: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( mult @ TA @ TB @ A @ B @ C ) @ D )
= ( mult @ TA @ TB @ A @ B @ ( mult @ TA @ TB @ A @ C @ D ) ) ) ) ) )
& ( member @ TA @ ( one @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
& ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( one @ TA @ TB @ A ) @ B )
= B ) )
& ! [B: TA] :
( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ B @ ( one @ TA @ TB @ A ) )
= B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(224,axiom,
! [A: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A @ ( zero @ a @ b @ r ) )
= ( zero @ a @ b @ r ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_r__null) ).
thf(891,plain,
! [A: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[224]) ).
thf(45,axiom,
! [TA: $tType,A: set @ TA] :
( ( collect @ TA
@ ^ [B: TA] : ( member @ TA @ B @ A ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_Collect__mem__eq) ).
thf(368,plain,
! [TA: $tType,A: set @ TA] :
( ( collect @ TA
@ ^ [B: TA] : ( member @ TA @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(43,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( locali7914339f_crng @ TA @ TB @ A @ B )
=> ( locali1402578320of_rng @ TA @ TB @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_mult__submonoid__of__crng_Oaxioms_I2_J) ).
thf(363,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( locali7914339f_crng @ TA @ TB @ A @ B )
=> ( locali1402578320of_rng @ TA @ TB @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(109,axiom,
! [TA: $tType,A: set @ TA,B: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
=> ( ( A
= ( bot_bot @ ( set @ TA ) ) )
| ( A
= ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_subset__singletonD) ).
thf(556,plain,
! [TA: $tType,A: set @ TA,B: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
=> ( ( A
= ( bot_bot @ ( set @ TA ) ) )
| ( A
= ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(244,axiom,
! [A: a,B: nat] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( add_pow @ a @ b @ nat @ r @ B @ A ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_add_Onat__pow__closed) ).
thf(963,plain,
! [A: a,B: nat] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( add_pow @ a @ b @ nat @ r @ B @ A ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).
thf(28,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( units @ TA @ TB @ A ) )
=> ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_monoid_OUnits__closed) ).
thf(319,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( units @ TA @ TB @ A ) )
=> ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(22,axiom,
! [TA: $tType,A: TA,B: set @ TA] :
( ( member @ TA @ A @ B )
=> ~ ! [C: set @ TA] :
( ( B
= ( insert @ TA @ A @ C ) )
=> ( member @ TA @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_Set_Oset__insert) ).
thf(304,plain,
! [TA: $tType,A: TA,B: set @ TA] :
( ( member @ TA @ A @ B )
=> ~ ! [C: set @ TA] :
( ( B
= ( insert @ TA @ A @ C ) )
=> ( member @ TA @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(1,conjecture,
( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) )
= ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).
thf(2,negated_conjecture,
( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) )
!= ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(258,plain,
( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) )
!= ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(144,axiom,
! [TA: $tType,A: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ A @ A )
= ( bot_bot @ ( set @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_232_Diff__cancel) ).
thf(652,plain,
! [TA: $tType,A: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ A @ A )
= ( bot_bot @ ( set @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(56,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( partial_carrier @ TA @ ( monoid_ext @ TA @ product_unit ) @ ( units_of @ TA @ TB @ A ) )
= ( units @ TA @ TB @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_units__of__carrier) ).
thf(390,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( partial_carrier @ TA @ ( monoid_ext @ TA @ product_unit ) @ ( units_of @ TA @ TB @ A ) )
= ( units @ TA @ TB @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(137,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( one @ TA @ ( ring_ext @ TA @ TB ) @ A ) @ B )
= B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_ring_Oring__simprules_I12_J) ).
thf(632,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( one @ TA @ ( ring_ext @ TA @ TB ) @ A ) @ B )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(173,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( field @ TA @ TB @ A )
=> ( ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C )
= ( zero @ TA @ TB @ A ) )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( B
= ( zero @ TA @ TB @ A ) )
| ( C
= ( zero @ TA @ TB @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_integral) ).
thf(731,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( field @ TA @ TB @ A )
=> ( ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C )
= ( zero @ TA @ TB @ A ) )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( B
= ( zero @ TA @ TB @ A ) )
| ( C
= ( zero @ TA @ TB @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(189,axiom,
! [TA: $tType,A: TA,B: TA] :
( ( member @ TA @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
= ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_215_singleton__iff) ).
thf(786,plain,
! [TA: $tType,A: TA,B: TA] :
( ( member @ TA @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
= ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(113,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( one @ TA @ TB @ A ) @ B )
= B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_monoid_Ol__one) ).
thf(568,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ ( one @ TA @ TB @ A ) @ B )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(94,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA] :
( ( group @ TA @ TB @ A )
=> ( ( ( mult @ TA @ TB @ A @ B @ C )
= ( one @ TA @ TB @ A ) )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ C @ B )
= ( one @ TA @ TB @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_group_Oinv__comm) ).
thf(510,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA,C: TA] :
( ( group @ TA @ TB @ A )
=> ( ( ( mult @ TA @ TB @ A @ B @ C )
= ( one @ TA @ TB @ A ) )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ C @ B )
= ( one @ TA @ TB @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(149,axiom,
monoid @ a @ ( ring_ext @ a @ b ) @ r,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_monoid__axioms) ).
thf(667,plain,
monoid @ a @ ( ring_ext @ a @ b ) @ r,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(185,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( monoid @ TA @ TB @ A )
=> ( ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A )
!= ( bot_bot @ ( set @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_monoid_Ocarrier__not__empty) ).
thf(763,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( monoid @ TA @ TB @ A )
=> ( ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A )
!= ( bot_bot @ ( set @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(52,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: TA] :
( ( member @ TA @ A @ B )
=> ( member @ TA @ A @ ( insert @ TA @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_insertI2) ).
thf(383,plain,
! [TA: $tType,A: TA,B: set @ TA,C: TA] :
( ( member @ TA @ A @ B )
=> ( member @ TA @ A @ ( insert @ TA @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(227,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: product_prod @ TC @ ( product_prod @ TB @ TA )] :
~ ! [B: TC,C: TB,D: TA] :
( A
!= ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ B @ ( product_Pair @ TB @ TA @ C @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_prod__cases3) ).
thf(899,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: product_prod @ TC @ ( product_prod @ TB @ TA )] :
~ ! [B: TC,C: TB,D: TA] :
( A
!= ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ B @ ( product_Pair @ TB @ TA @ C @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).
thf(16,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( semiring @ TA @ TB @ A )
=> ( abelian_monoid @ TA @ TB @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_semiring_Oaxioms_I1_J) ).
thf(290,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( semiring @ TA @ TB @ A )
=> ( abelian_monoid @ TA @ TB @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(233,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( member @ TA @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_ring_Oring__simprules_I5_J) ).
thf(914,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( member @ TA @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).
thf(49,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: set @ TA] :
( ( locali1441642228monoid @ TA @ TB @ A @ B )
=> ( locali1441642228monoid @ TA @ TB @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Ois__submonoid) ).
thf(378,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(204,axiom,
locali1402578320of_rng @ a @ b @ r @ s,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_mult__submonoid__of__rng__axioms) ).
thf(832,plain,
locali1402578320of_rng @ a @ b @ r @ s,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(156,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_167_subsetD) ).
thf(684,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(168,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) )] :
~ ! [B: TF,C: TE,D: TD,E: TC,F: TB,G: TA] :
( A
!= ( product_Pair @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) @ B @ ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ C @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ E @ ( product_Pair @ TB @ TA @ F @ G ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_prod__cases6) ).
thf(716,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) )] :
~ ! [B: TF,C: TE,D: TD,E: TC,F: TB,G: TA] :
( A
!= ( product_Pair @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) @ B @ ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ C @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ E @ ( product_Pair @ TB @ TA @ F @ G ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(15,axiom,
! [TA: $tType,TB: $tType] :
( ( locali1402578320of_rng @ TA @ TB )
= ( ^ [A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( ring @ TA @ TB @ A )
& ( locali1441642228monoid @ TA @ ( ring_ext @ TA @ TB ) @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_mult__submonoid__of__rng__def) ).
thf(287,plain,
! [TA: $tType,TB: $tType] :
( ( locali1402578320of_rng @ TA @ TB )
= ( ^ [A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( ring @ TA @ TB @ A )
& ( locali1441642228monoid @ TA @ ( ring_ext @ TA @ TB ) @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(133,axiom,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_62_Units__l__cancel) ).
thf(620,plain,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[133]) ).
thf(205,axiom,
member @ a @ ( zero @ a @ b @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_zero__closed) ).
thf(833,plain,
member @ a @ ( zero @ a @ b @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(4,axiom,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_mem__Collect__eq) ).
thf(262,plain,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(61,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( semiring @ TA @ TB @ A )
=> ( ( ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
!= ( insert @ TA @ ( zero @ TA @ TB @ A ) @ ( bot_bot @ ( set @ TA ) ) ) )
= ( ( one @ TA @ ( ring_ext @ TA @ TB ) @ A )
!= ( zero @ TA @ TB @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_221_semiring_Ocarrier__one__not__zero) ).
thf(405,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( semiring @ TA @ TB @ A )
=> ( ( ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
!= ( insert @ TA @ ( zero @ TA @ TB @ A ) @ ( bot_bot @ ( set @ TA ) ) ) )
= ( ( one @ TA @ ( ring_ext @ TA @ TB ) @ A )
!= ( zero @ TA @ TB @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(9,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ~ ( member @ TA @ A @ B )
=> ( ~ ( member @ TA @ A @ C )
=> ( ( ( insert @ TA @ A @ B )
= ( insert @ TA @ A @ C ) )
= ( B = C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_insert__ident) ).
thf(273,plain,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ~ ( member @ TA @ A @ B )
=> ( ~ ( member @ TA @ A @ C )
=> ( ( ( insert @ TA @ A @ B )
= ( insert @ TA @ A @ C ) )
= ( B = C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(217,axiom,
! [A: nat] :
( ( add_pow @ a @ b @ nat @ r @ A @ ( zero @ a @ b @ r ) )
= ( zero @ a @ b @ r ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_add_Onat__pow__one) ).
thf(865,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)],[217]) ).
thf(47,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: TA,D: set @ TA] :
( ~ ( member @ TA @ A @ B )
=> ( ~ ( member @ TA @ C @ D )
=> ( ( ( insert @ TA @ A @ B )
= ( insert @ TA @ C @ D ) )
= ( ( ( A = C )
=> ( B = D ) )
& ( ( A != C )
=> ? [E: set @ TA] :
( ( B
= ( insert @ TA @ C @ E ) )
& ~ ( member @ TA @ C @ E )
& ( D
= ( insert @ TA @ A @ E ) )
& ~ ( member @ TA @ A @ E ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_insert__eq__iff) ).
thf(373,plain,
! [TA: $tType,A: TA,B: set @ TA,C: TA,D: set @ TA] :
( ~ ( member @ TA @ A @ B )
=> ( ~ ( member @ TA @ C @ D )
=> ( ( ( insert @ TA @ A @ B )
= ( insert @ TA @ C @ D ) )
= ( ( ( A = C )
=> ( B = D ) )
& ( ( A != C )
=> ? [E: set @ TA] :
( ( B
= ( insert @ TA @ C @ E ) )
& ~ ( member @ TA @ C @ E )
& ( D
= ( insert @ TA @ A @ E ) )
& ~ ( member @ TA @ A @ E ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(127,axiom,
! [A: a,B: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B )
!= ( bot_bot @ ( set @ ( product_prod @ a @ a ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_non__empty__class) ).
thf(603,plain,
! [A: a,B: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ A @ B ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ A @ B )
!= ( bot_bot @ ( set @ ( product_prod @ a @ a ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(112,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 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( A = C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_inv__unique) ).
thf(564,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 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( A = C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(203,axiom,
! [A: a,B: a,C: nat] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_127_add__pow__ldistr) ).
thf(828,plain,
! [A: a,B: a,C: nat] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[203]) ).
thf(138,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: nat] :
( ( semiring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( add_pow @ TA @ TB @ nat @ A @ D @ C ) )
= ( add_pow @ TA @ TB @ nat @ A @ D @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_148_semiring_Oadd__pow__rdistr) ).
thf(635,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: nat] :
( ( semiring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( add_pow @ TA @ TB @ nat @ A @ D @ C ) )
= ( add_pow @ TA @ TB @ nat @ A @ D @ ( mult @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(101,axiom,
! [TA: $tType,A: TA > $o] :
( ( ( bot_bot @ ( set @ TA ) )
= ( collect @ TA @ A ) )
= ( ! [B: TA] :
~ ( A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_157_empty__Collect__eq) ).
thf(532,plain,
! [TA: $tType,A: TA > $o] :
( ( ( bot_bot @ ( set @ TA ) )
= ( collect @ TA @ A ) )
= ( ! [B: TA] :
~ ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(238,axiom,
! [A: a,B: a,C: int] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_105_add__pow__rdistr__int) ).
thf(928,plain,
! [A: a,B: a,C: int] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[238]) ).
thf(226,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r3 )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r2 ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0_f3) ).
thf(897,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)],[226]) ).
thf(81,axiom,
! [A: a] :
( ( member @ a @ A @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50_Units__closed) ).
thf(474,plain,
! [A: a] :
( ( member @ a @ A @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(192,axiom,
! [TA: $tType,A: set @ TA] :
( ( ! [B: TA] :
~ ( member @ TA @ B @ A ) )
= ( A
= ( bot_bot @ ( set @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_155_all__not__in__conv) ).
thf(796,plain,
! [TA: $tType,A: set @ TA] :
( ( ! [B: TA] :
~ ( member @ TA @ B @ A ) )
= ( A
= ( bot_bot @ ( set @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(13,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( monoid @ TA @ TB @ A )
=> ( member @ TA @ ( one @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_monoid_Oone__closed) ).
thf(283,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB )] :
( ( monoid @ TA @ TB @ A )
=> ( member @ TA @ ( one @ TA @ TB @ A ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(134,axiom,
! [A: a,B: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A @ B ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_semiring__simprules_I3_J) ).
thf(624,plain,
! [A: a,B: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A @ B ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(172,axiom,
! [TA: $tType,A: TA,B: TA] :
( ( member @ TA @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
=> ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_216_singletonD) ).
thf(728,plain,
! [TA: $tType,A: TA,B: TA] :
( ( member @ TA @ A @ ( insert @ TA @ B @ ( bot_bot @ ( set @ TA ) ) ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(77,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( semiring @ TA @ TB @ A )
=> ( ( ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
= ( insert @ TA @ ( zero @ TA @ TB @ A ) @ ( bot_bot @ ( set @ TA ) ) ) )
= ( ( one @ TA @ ( ring_ext @ TA @ TB ) @ A )
= ( zero @ TA @ TB @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_222_semiring_Ocarrier__one__zero) ).
thf(464,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( semiring @ TA @ TB @ A )
=> ( ( ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
= ( insert @ TA @ ( zero @ TA @ TB @ A ) @ ( bot_bot @ ( set @ TA ) ) ) )
= ( ( one @ TA @ ( ring_ext @ TA @ TB ) @ A )
= ( zero @ TA @ TB @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(215,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: set @ TA] :
( ( abelian_monoid @ TA @ TB @ A )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ord_less_eq @ ( set @ TA ) @ ( set_add @ TA @ TB @ A @ B @ C ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_253_abelian__monoid_Oset__add__closed) ).
thf(860,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: set @ TA] :
( ( abelian_monoid @ TA @ TB @ A )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ord_less_eq @ ( set @ TA ) @ ( set_add @ TA @ TB @ A @ B @ C ) @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(211,axiom,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ B @ ( one @ TA @ TB @ A ) )
= B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_monoid_Or__one) ).
thf(850,plain,
! [TA: $tType,TB: $tType,A: partia1265057091ct_ext @ TA @ ( monoid_ext @ TA @ TB ),B: TA] :
( ( monoid @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( partial_carrier @ TA @ ( monoid_ext @ TA @ TB ) @ A ) )
=> ( ( mult @ TA @ TB @ A @ B @ ( one @ TA @ TB @ A ) )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(90,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( member @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ C ) )
=> ( member @ TA @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_DiffD1) ).
thf(501,plain,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( member @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ C ) )
=> ( member @ TA @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(85,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ( pow @ TA @ A )
= ( insert @ ( set @ TA ) @ B @ ( bot_bot @ ( set @ ( set @ TA ) ) ) ) )
= ( ( A
= ( bot_bot @ ( set @ TA ) ) )
& ( B
= ( bot_bot @ ( set @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_254_Pow__singleton__iff) ).
thf(486,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ( pow @ TA @ A )
= ( insert @ ( set @ TA ) @ B @ ( bot_bot @ ( set @ ( set @ TA ) ) ) ) )
= ( ( A
= ( bot_bot @ ( set @ TA ) ) )
& ( B
= ( bot_bot @ ( set @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(242,axiom,
( ( ( zero @ a @ b @ r )
!= ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( ! [A: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( A
!= ( zero @ a @ b @ r ) )
=> ? [B: a] :
( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_97_cring__fieldI2) ).
thf(940,plain,
( ( ( zero @ a @ b @ r )
!= ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( ! [A: a] :
( ( member @ a @ A @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( A
!= ( zero @ a @ b @ r ) )
=> ? [B: a] :
( ( member @ a @ B @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ 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)],[242]) ).
thf(221,axiom,
! [TA: $tType] :
( ( ord_less_eq @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_subset__iff) ).
thf(882,plain,
! [TA: $tType] :
( ( ord_less_eq @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).
thf(88,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: TA] :
( ( ( insert @ TA @ A @ B )
= ( insert @ TA @ C @ ( bot_bot @ ( set @ TA ) ) ) )
= ( ( A = C )
& ( ord_less_eq @ ( set @ TA ) @ B @ ( insert @ TA @ C @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_singleton__insert__inj__eq_H) ).
thf(496,plain,
! [TA: $tType,A: TA,B: set @ TA,C: TA] :
( ( ( insert @ TA @ A @ B )
= ( insert @ TA @ C @ ( bot_bot @ ( set @ TA ) ) ) )
= ( ( A = C )
& ( ord_less_eq @ ( set @ TA ) @ B @ ( insert @ TA @ C @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(74,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( member @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ C ) )
=> ~ ( ( member @ TA @ A @ B )
=> ( member @ TA @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_DiffE) ).
thf(454,plain,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( member @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ C ) )
=> ~ ( ( member @ TA @ A @ B )
=> ( member @ TA @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(1003,plain,
$false,
inference(cvc4,[status(thm)],[365,760,347,846,481,909,333,977,582,276,994,308,379,878,655,614,269,677,385,550,715,814,698,913,340,538,709,417,885,301,436,719,751,802,934,357,902,559,606,770,316,638,853,789,591,321,376,834,937,966,692,734,396,411,265,981,766,292,661,925,810,381,894,746,598,820,561,297,630,429,260,862,361,435,522,461,493,757,682,285,916,467,324,974,317,650,725,388,420,499,989,857,519,742,642,526,489,590,706,686,476,837,393,701,457,984,371,503,647,997,792,508,311,541,674,586,662,888,298,513,824,856,279,573,689,601,904,423,402,843,281,875,577,611,713,947,350,919,804,736,451,271,345,535,816,609,594,799,769,668,330,932,399,695,868,986,1000,267,505,431,922,658,473,722,675,426,531,548,812,749,342,355,754,553,968,617,516,580,295,327,739,409,993,414,971,585,840,338,808,643,470,576,544,907,744,703,386,314,571,627,482,671,891,368,363,556,963,319,304,258,652,390,632,731,786,568,510,667,763,383,899,290,914,378,832,684,716,287,620,833,262,405,273,865,373,603,564,828,635,532,928,897,474,796,283,624,728,464,860,850,501,486,940,882,496,454]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.09 % Problem : ITP107^2 : TPTP v8.2.0. Released v7.5.0.
% 0.03/0.12 % Command : run_Leo-III %s %d
% 0.11/0.31 % Computer : n032.cluster.edu
% 0.11/0.31 % Model : x86_64 x86_64
% 0.11/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.31 % Memory : 8042.1875MB
% 0.11/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.31 % CPULimit : 300
% 0.11/0.31 % WCLimit : 300
% 0.11/0.31 % DateTime : Sat May 18 18:31:54 EDT 2024
% 0.11/0.31 % CPUTime :
% 0.72/0.76 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.38/1.00 % [INFO] Parsing done (246ms).
% 1.68/1.01 % [INFO] Running in sequential loop mode.
% 2.37/1.26 % [INFO] eprover registered as external prover.
% 2.37/1.26 % [INFO] cvc4 registered as external prover.
% 2.37/1.27 % [INFO] Scanning for conjecture ...
% 2.81/1.39 % [INFO] Found a conjecture (or negated_conjecture) and 256 axioms. Running axiom selection ...
% 3.14/1.49 % [INFO] Axiom selection finished. Selected 255 axioms (removed 1 axioms).
% 4.03/1.73 % [INFO] Problem is higher-order (TPTP THF).
% 4.03/1.75 % [INFO] Type checking passed.
% 4.03/1.75 % [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 ...
% 60.01/44.98 % External prover 'cvc4' found a proof!
% 60.01/44.98 % [INFO] Killing All external provers ...
% 60.01/44.98 % Time passed: 44553ms (effective reasoning time: 43965ms)
% 60.01/44.98 % 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)>
% 60.01/44.98 % Axioms used in derivation (255): fact_186_monoid_Ounits__group, fact_195_carrier__one__not__zero, fact_145_group_Oinv__comm, fact_165_ex__in__conv, fact_1_f4, fact_142_group_Ol__cancel__one, fact_162_emptyE, fact_25_mult__submonoid__of__crng_Oaxioms_I2_J, fact_147_monoid_Ogroup__l__invI, fact_154_empty__iff, fact_121_Group_Omonoid_Ointro, fact_99_monoid_Ol__one, fact_237_insert__Diff__if, fact_66_abelian__monoid_Ozero__closed, fact_198_insert__subset, fact_132_add_Onat__pow__one, fact_75_semiring_Osemiring__simprules_I3_J, fact_143_group_Or__inv__ex, fact_247_DiffD1, fact_12_one__closed, fact_175_subset__trans, fact_38_Units__inv__comm, fact_69_ring_Oring__simprules_I24_J, fact_216_singletonD, fact_82_old_Oprod_Oexhaust, fact_168_equalityE, fact_130_carrier__not__empty, fact_189_a__rcosetsI, fact_98_monoid_Or__one, fact_225_field__intro2, fact_124_monoid_OUnits__l__cancel, fact_68_semiring_Oaxioms_I2_J, fact_29_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Ois__submonoid, fact_32_right__unit__mult__rng__of__frac, fact_242_subset__Diff__insert, fact_157_empty__Collect__eq, fact_231_empty__Diff, fact_16_assoc__mult__rng__of__frac, fact_236_insert__Diff__single, fact_70_ring_Oring__simprules_I25_J, fact_114_monoid_Om__assoc, fact_210_insertI1, fact_182_a__setmult__rcos__assoc, fact_74_ring_Oring__simprules_I11_J, fact_215_singleton__iff, fact_33_left__unit__mult__rng__of__frac, fact_212_singleton__inject, fact_30_eq__obj__rng__of__frac_Omonoid__rng__of__frac, fact_39_prod_Oinject, fact_135_group_Ois__monoid, fact_50_Units__closed, fact_196_carrier__one__zero, fact_86_prod__induct4, fact_223_semiring_Oone__zeroI, fact_136_Group_Ogroup_Oright__cancel, fact_137_group_OUnits__eq, fact_155_all__not__in__conv, fact_163_equals0D, fact_187_units__of__mult, fact_174_Collect__mono, fact_140_group_Ol__cancel__one_H, fact_226_cring__fieldI, fact_192_insert__absorb2, fact_43_m__assoc, fact_190_one__zeroI, fact_183_units__of__units, fact_191_one__zeroD, fact_67_ring_Ois__monoid, fact_138_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Osubset, fact_202_subset__singletonD, fact_0_f3, fact_160_a__l__coset__subset__G, fact_156_Collect__empty__eq, fact_46_Collect__mem__eq, fact_90_prod__cases5, fact_56_Units__m__closed, fact_246_DiffE, fact_201_subset__singleton__iff, fact_185_units__of__carrier, fact_21_submonoid_Oaxioms_I1_J, fact_118_monoid_OUnits__one__closed, fact_107_add_Oint__pow__one, fact_173_subset__refl, fact_10_monoid__rng__of__frac, fact_40_old_Oprod_Oinject, fact_164_equals0I, fact_200_singleton__insert__inj__eq, fact_177_Collect__mono__iff, fact_84_prod__induct6, fact_19_local_Oring__axioms, fact_181_a__r__coset__subset__G, fact_106_add_Oint__pow__closed, fact_8_submonoid__axioms, fact_184_a__coset__add__zero, fact_144_group_Ol__inv__ex, fact_207_insert__ident, fact_89_prod__cases6, fact_250_Diff__single__insert, fact_72_semiring_Ol__null, fact_206_insert__absorb, fact_241_Diff__insert, fact_123_monoid_Oone__unique, fact_188_units__of__one, fact_42_monoid__axioms, fact_220_subset__insertI2, fact_209_insertI2, fact_228_Diff__iff, fact_109_one__not__zero, fact_22_eq__obj__rng__of__frac_Oaxioms_I2_J, fact_96_ring_Ois__ring, fact_2_eq__obj__rng__of__frac__axioms, fact_217_insert__mono, fact_85_prod__induct5, fact_111_ring_Oadd__pow__ldistr__int, fact_7_eq__obj__rng__of__frac_Orec__monoid__rng__of__frac_Ocong, fact_254_Pow__singleton__iff, fact_120_Group_Omonoid__def, fact_52_Units__r__inv__ex, fact_199_singleton__insert__inj__eq_H, fact_57_Units__one__closed, fact_170_equalityD1, fact_102_monoid_OUnits__l__inv__ex, fact_238_Diff__insert__absorb, fact_83_prod__induct7, fact_244_Diff__subset, fact_169_subset__eq, fact_95_surj__pair, fact_49_m__lcomm, fact_35_non__empty__class, fact_150_a__lcos__mult__one, fact_151_set__add__closed, fact_45_mem__Collect__eq, fact_167_subsetD, fact_248_DiffD2, fact_218_subset__insert, fact_179_add__additive__subgroups, fact_129_group__l__invI, fact_193_insert__iff, fact_51_Units__l__inv__ex, fact_64_ring_Oring__simprules_I2_J, fact_214_doubleton__eq__iff, fact_235_Diff__eq__empty__iff, fact_161_empty__subsetI, fact_149_semiring_Oadd__pow__ldistr, fact_79_ring_Oring__simprules_I12_J, fact_71_semiring_Or__null, fact_222_semiring_Ocarrier__one__zero, fact_62_Units__l__cancel, fact_243_double__diff, fact_253_abelian__monoid_Oset__add__closed, fact_37_inv__unique, fact_77_ring_Oring__simprules_I6_J, fact_119_monoidI, fact_208_Set_Oset__insert, fact_233_insert__Diff1, fact_73_ring_Oring__simprules_I5_J, fact_58_l__null, fact_76_semiring_Osemiring__simprules_I8_J, fact_47_Collect__cong, fact_61_r__one, fact_41_semiring_Oaxioms_I1_J, fact_126_add__pow__rdistr, fact_159_subset__antisym, fact_180_a__rcos__assoc__lcos, fact_230_Diff__empty, fact_18_mult__rng__of__frac__fundamental__lemma, fact_148_semiring_Oadd__pow__rdistr, fact_34_zero__in__mult__submonoid, fact_63_abelian__monoidE_I2_J, fact_211_insertE, fact_252_rcosets__subset__PowG, fact_23_mult__submonoid__of__rng_Oaxioms_I2_J, fact_122_monoid_Oinv__unique, fact_31_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Om__closed, fact_127_add__pow__ldistr, fact_166_in__mono, fact_112_ring_Oadd__pow__rdistr__int, fact_87_prod__induct3, fact_255_Pow__empty, fact_104_add__pow__ldistr__int, fact_92_prod__cases3, fact_88_prod__cases7, fact_13_equiv__obj__rng__of__frac, fact_97_cring__fieldI2, fact_113_monoid_Ocarrier__not__empty, fact_20_closed__mult__rng__of__frac, fact_172_subset__iff, fact_78_semiring_Osemiring__simprules_I4_J, fact_232_Diff__cancel, fact_171_equalityD2, fact_105_add__pow__rdistr__int, fact_176_set__eq__subset, fact_3_f1, fact_227_DiffI, fact_134_group_OUnits, fact_27_mult__submonoid__of__rng__def, fact_213_insert__not__empty, fact_152_setadd__subset__G, fact_139_group_Or__cancel__one_H, fact_60_l__one, fact_9_m__closed, fact_205_insert__eq__iff, fact_14_assms_I2_J, fact_65_semiring_Osemiring__simprules_I2_J, fact_245_Diff__mono, fact_239_Diff__insert2, fact_110_integral, fact_229_Diff__idemp, fact_5_mult__submonoid__of__crng__axioms, fact_93_Pair__inject, fact_117_monoid_OUnits__m__closed, fact_81_old_Oprod_Oinducts, fact_219_subset__insertI, fact_91_prod__cases4, fact_100_monoid_Oone__closed, fact_234_Diff__insert0, fact_15_assms_I1_J, fact_251_field_Ofield__Units, fact_55_semiring__simprules_I4_J, fact_240_insert__Diff, fact_94_prod__cases, fact_103_monoid_OUnits__inv__comm, fact_194_insertCI, fact_54_semiring__simprules_I3_J, fact_28_mult__submonoid__of__rng_Ointro, fact_249_subset__insert__iff, fact_11_local_Osemiring__axioms, fact_17_abelian__monoid__axioms, fact_141_group_Or__cancel__one, fact_108_field_Ois__ring, fact_53_zero__closed, fact_158_subsetI, fact_36_one__unique, fact_59_r__null, fact_131_add_Onat__pow__closed, fact_204_insert__commute, fact_80_semiring_Osemiring__simprules_I9_J, fact_6_mult__submonoid__of__rng__axioms, fact_178_units__group, fact_133_group_Ois__group, fact_125_subset, fact_197_singletonI, fact_24_mult__submonoid__of__rng_Oaxioms_I1_J, fact_116_monoid_OUnits__closed, fact_153_subset__empty, fact_101_monoid_OUnits__r__inv__ex, fact_146_groupI, fact_221_semiring_Ocarrier__one__not__zero, fact_26_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Oone__closed, fact_4_f2, fact_115_monoid_Om__closed, fact_44_m__comm, fact_203_mk__disjoint__insert, fact_224_semiring_Oone__zeroD, fact_128_old_Oprod_Orec
% 60.01/44.98 % No. of inferences in proof: 514
% 60.01/44.98 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 44553 ms resp. 43965 ms w/o parsing
% 60.60/45.13 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 60.60/45.13 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------