TSTP Solution File: ITP027^2 by Leo-III---1.7.12
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%------------------------------------------------------------------------------
% File : Leo-III---1.7.12
% Problem : ITP027^2 : TPTP v8.2.0. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n011.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 22:22:12 EDT 2024
% Result : Theorem 259.52s 217.30s
% Output : Refutation 261.17s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 325
% Syntax : Number of formulae : 597 ( 42 unt; 55 typ; 0 def)
% Number of atoms : 3531 ( 401 equ; 0 cnn)
% Maximal formula atoms : 30 ( 6 avg)
% Number of connectives : 20351 ( 59 ~; 7 |; 54 &;18892 @)
% ( 0 <=>;1339 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 13 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 672 ( 672 >; 0 *; 0 +; 0 <<)
% Number of symbols : 56 ( 53 usr; 9 con; 0-11 aty)
% Number of variables : 2694 ( 118 ^2452 !; 26 ?;2694 :)
% ( 98 !>; 0 ?*; 0 @-; 0 @+)
% Comments :
%------------------------------------------------------------------------------
thf(product_unit_type,type,
product_unit: $tType ).
thf(nat_type,type,
nat: $tType ).
thf(e_type,type,
e: $tType ).
thf(d_type,type,
d: $tType ).
thf(c_type,type,
c: $tType ).
thf(b_type,type,
b: $tType ).
thf(a_type,type,
a: $tType ).
thf(plus_type,type,
plus:
!>[TA: $tType] : $o ).
thf(ord_type,type,
ord:
!>[TA: $tType] : $o ).
thf(group_add_type,type,
group_add:
!>[TA: $tType] : $o ).
thf(semigroup_add_type,type,
semigroup_add:
!>[TA: $tType] : $o ).
thf(comm_monoid_add_type,type,
comm_monoid_add:
!>[TA: $tType] : $o ).
thf(ab_semigroup_add_type,type,
ab_semigroup_add:
!>[TA: $tType] : $o ).
thf(cancel_semigroup_add_type,type,
cancel_semigroup_add:
!>[TA: $tType] : $o ).
thf(ordere779506340up_add_type,type,
ordere779506340up_add:
!>[TA: $tType] : $o ).
thf(ordere236663937imp_le_type,type,
ordere236663937imp_le:
!>[TA: $tType] : $o ).
thf(carrier_type,type,
carrier:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ TA ) > ( set @ TB ) ) ).
thf(j_rad_type,type,
j_rad:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) ) ).
thf(ring_type,type,
ring:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > $o ) ).
thf(tp_type,type,
tp:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > TB > TB > TB ) ).
thf(un_type,type,
un:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > TB ) ).
thf(sr_type,type,
sr:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) ) ).
thf(subring_type,type,
subring:
!>[TA: $tType,TB: $tType,TC: $tType] : ( ( carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TB ) ) ) > ( carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TA ) ) ) > $o ) ).
thf(unit_type,type,
unit:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > TB > $o ) ).
thf(aGroup_type,type,
aGroup:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) > $o ) ).
thf(mop_type,type,
mop:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) > TB > TB ) ).
thf(pop_type,type,
pop:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) > TB > TB > TB ) ).
thf(aset_sum_type,type,
aset_sum:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) > ( set @ TB ) > ( set @ TB ) > ( set @ TB ) ) ).
thf(carr_dsumag_type,type,
carr_dsumag:
!>[TA: $tType,TB: $tType,TC: $tType] : ( ( set @ TC ) > ( TC > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) ) > ( set @ ( TC > TB ) ) ) ).
thf(carr_prodag_type,type,
carr_prodag:
!>[TA: $tType,TB: $tType,TC: $tType] : ( ( set @ TC ) > ( TC > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) ) > ( set @ ( TC > TB ) ) ) ).
thf(local_ring_type,type,
local_ring:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > $o ) ).
thf(maximal_ideal_type,type,
maximal_ideal:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > $o ) ).
thf(mul_closed_set_type,type,
mul_closed_set:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > $o ) ).
thf(nscal_type,type,
nscal:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > TB > nat > TB ) ).
thf(primary_ideal_type,type,
primary_ideal:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > $o ) ).
thf(prod_mOp_type,type,
prod_mOp:
!>[TA: $tType,TB: $tType,TC: $tType] : ( ( set @ TC ) > ( TC > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) ) > ( TC > TB ) > TC > TB ) ).
thf(prod_pOp_type,type,
prod_pOp:
!>[TA: $tType,TB: $tType,TC: $tType] : ( ( set @ TC ) > ( TC > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) ) > ( TC > TB ) > ( TC > TB ) > TC > TB ) ).
thf(prod_zero_type,type,
prod_zero:
!>[TA: $tType,TB: $tType,TC: $tType] : ( ( set @ TC ) > ( TC > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) ) > TC > TB ) ).
thf(rInvim_type,type,
rInvim:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( TD > TB ) > ( set @ TB ) > ( set @ TD ) ) ).
thf(sr2_type,type,
sr2:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > $o ) ).
thf(zeroring_type,type,
zeroring:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > $o ) ).
thf(sum_mult_type,type,
sum_mult:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > ( set @ TB ) > ( set @ TB ) ) ).
thf(module_type,type,
module:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TC @ TB ) ) ) > ( carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TA ) ) ) > $o ) ).
thf(sprod_type,type,
sprod:
!>[TA: $tType,TB: $tType,TC: $tType] : ( ( carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TB @ TA ) ) ) > TB > TC > TC ) ).
thf(mHom_type,type,
mHom:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType] : ( ( carrier_ext @ TF @ ( aGroup_ext @ TF @ ( ring_ext @ TF @ TE ) ) ) > ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TF @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TF @ TA ) ) ) > ( set @ ( TD > TB ) ) ) ).
thf(algebr1000837334_prodM_type,type,
algebr1000837334_prodM:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType] : ( ( carrier_ext @ TE @ ( aGroup_ext @ TE @ ( ring_ext @ TE @ TD ) ) ) > ( set @ TC ) > ( TC > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TE @ TA ) ) ) ) > ( carrier_ext @ ( TC > TB ) @ ( aGroup_ext @ ( TC > TB ) @ ( module_ext @ ( TC > TB ) @ TE @ product_unit ) ) ) ) ).
thf(algebr142713364_sprod_type,type,
algebr142713364_sprod:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType] : ( ( carrier_ext @ TE @ ( aGroup_ext @ TE @ ( ring_ext @ TE @ TD ) ) ) > ( set @ TC ) > ( TC > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TE @ TA ) ) ) ) > TE > ( TC > TB ) > TC > TB ) ).
thf(pi_type,type,
pi:
!>[TA: $tType,TB: $tType] : ( ( set @ TB ) > ( TB > ( set @ TA ) ) > ( set @ ( TB > TA ) ) ) ).
thf(plus_plus_type,type,
plus_plus:
!>[TA: $tType] : ( TA > TA > TA ) ).
thf(ord_less_eq_type,type,
ord_less_eq:
!>[TA: $tType] : ( TA > TA > $o ) ).
thf(collect_type,type,
collect:
!>[TA: $tType] : ( ( TA > $o ) > ( set @ TA ) ) ).
thf(member_type,type,
member:
!>[TA: $tType] : ( TA > ( set @ TA ) > $o ) ).
thf(i_type,type,
i: set @ c ).
thf(m_type,type,
m: c > ( carrier_ext @ d @ ( aGroup_ext @ d @ ( module_ext @ d @ a @ e ) ) ) ).
thf(r_type,type,
r: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) ).
thf(201,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: nat,D: nat] :
( ( ring @ TB @ TA @ A )
=> ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( pop @ TB @ ( ring_ext @ TB @ TA ) @ A @ ( nscal @ TB @ TA @ A @ B @ C ) @ ( nscal @ TB @ TA @ A @ B @ D ) )
= ( nscal @ TB @ TA @ A @ B @ ( plus_plus @ nat @ C @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_Ring_OnsDistr) ).
thf(892,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: nat,D: nat] :
( ( ring @ TB @ TA @ A )
=> ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( pop @ TB @ ( ring_ext @ TB @ TA ) @ A @ ( nscal @ TB @ TA @ A @ B @ C ) @ ( nscal @ TB @ TA @ A @ B @ D ) )
= ( nscal @ TB @ TA @ A @ B @ ( plus_plus @ nat @ C @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).
thf(180,axiom,
! [A: a,B: nat] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( nscal @ a @ b @ r @ A @ B ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50_nsClose) ).
thf(809,plain,
! [A: a,B: nat] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( nscal @ a @ b @ r @ A @ B ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(243,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ B @ C )
= ( tp @ TA @ TB @ A @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B ) @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_Ring_Oring__inv1__3) ).
thf(1024,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ B @ C )
= ( tp @ TA @ TB @ A @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B ) @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).
thf(36,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) )] :
( ( subring @ TA @ TB @ TC @ A @ B )
=> ( ord_less_eq @ ( set @ TA ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_Subring__subset) ).
thf(365,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) )] :
( ( subring @ TA @ TB @ TC @ A @ B )
=> ( ord_less_eq @ ( set @ TA ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(190,axiom,
! [A: a,B: a,C: a,D: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ D @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ C @ D ) ) )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ D ) ) ) @ ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ C ) ) @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_152_ring__distrib4) ).
thf(846,plain,
! [A: a,B: a,C: a,D: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ D @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ C @ D ) ) )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ D ) ) ) @ ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ C ) ) @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(174,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( member @ ( TA > TA > TA ) @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A )
@ ( pi @ TA @ ( TA > TA ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
@ ^ [B: TA] :
( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
@ ^ [C: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_Ring_Opop__closed) ).
thf(777,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( member @ ( TA > TA > TA ) @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A )
@ ( pi @ TA @ ( TA > TA ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
@ ^ [B: TA] :
( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
@ ^ [C: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(270,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TD ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TC ) ),C: TA] :
( ( module @ TA @ TB @ TD @ TC @ A @ B )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TD ) ) @ A ) )
=> ( ( sprod @ TA @ TB @ TD @ A @ ( un @ TB @ TC @ B ) @ C )
= C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_Module_Osprod__one) ).
thf(1110,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TD ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TC ) ),C: TA] :
( ( module @ TA @ TB @ TD @ TC @ A @ B )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TD ) ) @ A ) )
=> ( ( sprod @ TA @ TB @ TD @ A @ ( un @ TB @ TC @ B ) @ C )
= C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).
thf(31,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( A
= ( plus_plus @ TA @ B @ C ) )
=> ( ( plus_plus @ TA @ A @ D )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ C @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_245_group__cancel_Oadd1) ).
thf(352,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( A
= ( plus_plus @ TA @ B @ C ) )
=> ( ( plus_plus @ TA @ A @ D )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ C @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(205,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB,C: TA > ( set @ TB )] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( member @ TB @ ( B @ D ) @ ( C @ D ) ) )
=> ( member @ ( TA > TB ) @ B @ ( pi @ TA @ TB @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_Pi__I_H) ).
thf(909,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB,C: TA > ( set @ TB )] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( member @ TB @ ( B @ D ) @ ( C @ D ) ) )
=> ( member @ ( TA > TB ) @ B @ ( pi @ TA @ TB @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(123,axiom,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) @ ( tp @ a @ b @ r @ A @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_66_ring__distrib1) ).
thf(628,plain,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) @ ( tp @ a @ b @ r @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(261,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) )
= ( tp @ TA @ TB @ A @ B @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_Ring_Oring__inv1__2) ).
thf(1083,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) )
= ( tp @ TA @ TB @ A @ B @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).
thf(224,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB,C: TA > TB,D: TA > ( set @ TB )] :
( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( ( B @ E )
= ( C @ E ) ) )
=> ( ( member @ ( TA > TB ) @ B @ ( pi @ TA @ TB @ A @ D ) )
= ( member @ ( TA > TB ) @ C @ ( pi @ TA @ TB @ A @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_Pi__cong) ).
thf(962,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB,C: TA > TB,D: TA > ( set @ TB )] :
( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( ( B @ E )
= ( C @ E ) ) )
=> ( ( member @ ( TA > TB ) @ B @ ( pi @ TA @ TB @ A @ D ) )
= ( member @ ( TA > TB ) @ C @ ( pi @ TA @ TB @ A @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).
thf(55,axiom,
ord @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oord_3) ).
thf(408,plain,
ord @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(140,axiom,
! [A: a,B: a,C: nat] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( A = B )
=> ( ( nscal @ a @ b @ r @ A @ C )
= ( nscal @ a @ b @ r @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_nsEqElm) ).
thf(683,plain,
! [A: a,B: a,C: nat] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( A = B )
=> ( ( nscal @ a @ b @ r @ A @ C )
= ( nscal @ a @ b @ r @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(3,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( A = B )
& ( C = D ) )
=> ( ( plus_plus @ TA @ A @ C )
= ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_add__mono__thms__linordered__semiring_I4_J) ).
thf(276,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( A = B )
& ( C = D ) )
=> ( ( plus_plus @ TA @ A @ C )
= ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(257,axiom,
! [TA: $tType,A: set @ TA] :
( member @ ( TA > TA )
@ ^ [B: TA] : B
@ ( pi @ TA @ TA @ A
@ ^ [B: TA] : A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_funcset__id) ).
thf(1073,plain,
! [TA: $tType,A: set @ TA] :
( member @ ( TA > TA )
@ ^ [B: TA] : B
@ ( pi @ TA @ TA @ A
@ ^ [B: TA] : A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).
thf(68,axiom,
! [TA: $tType,A: set @ TA,B: TA > $o] :
( ord_less_eq @ ( set @ TA )
@ ( collect @ TA
@ ^ [C: TA] :
( ( member @ TA @ C @ A )
& ( B @ C ) ) )
@ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_164_conditional__subset) ).
thf(449,plain,
! [TA: $tType,A: set @ TA,B: TA > $o] :
( ord_less_eq @ ( set @ TA )
@ ( collect @ TA
@ ^ [C: TA] :
( ( member @ TA @ C @ A )
& ( B @ C ) ) )
@ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(43,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( ( zeroring @ TA @ TB @ A )
=> ~ ? [B: set @ TA] : ( maximal_ideal @ TA @ TB @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_Ring_Ozeroring__no__maximal) ).
thf(379,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( ( zeroring @ TA @ TB @ A )
=> ~ ? [B: set @ TA] : ( maximal_ideal @ TA @ TB @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(66,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ a @ TC ) ) )] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( module @ TB @ a @ TC @ b @ ( B @ C ) @ r ) )
=> ( ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ TA @ TB @ TC @ r @ A @ B ) )
= ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ a @ TC ) @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5_prodM__carr) ).
thf(440,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ a @ TC ) ) )] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( module @ TB @ a @ TC @ b @ ( B @ C ) @ r ) )
=> ( ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ TA @ TB @ TC @ r @ A @ B ) )
= ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ a @ TC ) @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(118,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: TB,D: nat] :
( ( ring @ TB @ TA @ A )
=> ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( tp @ TB @ TA @ A @ ( nscal @ TB @ TA @ A @ C @ D ) @ B )
= ( nscal @ TB @ TA @ A @ ( tp @ TB @ TA @ A @ C @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_Ring_OnsMulDistrR) ).
thf(614,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: TB,D: nat] :
( ( ring @ TB @ TA @ A )
=> ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( tp @ TB @ TA @ A @ ( nscal @ TB @ TA @ A @ C @ D ) @ B )
= ( nscal @ TB @ TA @ A @ ( tp @ TB @ TA @ A @ C @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(195,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: set @ TA,D: TA,E: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ B )
=> ( ( member @ TA @ E @ C )
=> ( member @ TA @ ( tp @ TA @ TB @ A @ D @ E ) @ ( sum_mult @ TA @ TB @ A @ B @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_Ring_Otimes__mem__sum__mult) ).
thf(861,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: set @ TA,D: TA,E: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ B )
=> ( ( member @ TA @ E @ C )
=> ( member @ TA @ ( tp @ TA @ TB @ A @ D @ E ) @ ( sum_mult @ TA @ TB @ A @ B @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(259,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( subring @ TA @ TB @ TC @ A @ B )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
=> ( ( tp @ TA @ TC @ B @ C @ D )
= ( tp @ TA @ TB @ A @ C @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_126_Ring_OSubring__tOp__ring__tOp) ).
thf(1078,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( subring @ TA @ TB @ TC @ A @ B )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
=> ( ( tp @ TA @ TC @ B @ C @ D )
= ( tp @ TA @ TB @ A @ C @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).
thf(45,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TA,C: TA > ( set @ TB )] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ord_less_eq @ ( set @ ( TA > TB ) ) @ ( pi @ TA @ TB @ B @ C ) @ ( pi @ TA @ TB @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_167_Pi__anti__mono) ).
thf(385,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TA,C: TA > ( set @ TB )] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ord_less_eq @ ( set @ ( TA > TB ) ) @ ( pi @ TA @ TB @ B @ C ) @ ( pi @ TA @ TB @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(95,axiom,
! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: set @ TB,D: TA > TB,E: TA] :
( ( member @ ( TA > TB ) @ A
@ ( pi @ TA @ TB @ B
@ ^ [F: TA] : C ) )
=> ( ( member @ ( TA > TB ) @ D
@ ( pi @ TA @ TB @ B
@ ^ [F: TA] : C ) )
=> ( ( A = D )
=> ( ( member @ TA @ E @ B )
=> ( ( A @ E )
= ( D @ E ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_funcTr) ).
thf(533,plain,
! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: set @ TB,D: TA > TB,E: TA] :
( ( member @ ( TA > TB ) @ A
@ ( pi @ TA @ TB @ B
@ ^ [F: TA] : C ) )
=> ( ( member @ ( TA > TB ) @ D
@ ( pi @ TA @ TB @ B
@ ^ [F: TA] : C ) )
=> ( ( A = D )
=> ( ( member @ TA @ E @ B )
=> ( ( A @ E )
= ( D @ E ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(86,axiom,
! [A: set @ a,B: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( sum_mult @ a @ b @ r @ A @ B ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_195_sum__mult__subR) ).
thf(500,plain,
! [A: set @ a,B: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( sum_mult @ a @ b @ r @ A @ B ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(182,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( subring @ TA @ TB @ TC @ A @ B )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
=> ( ( pop @ TA @ ( ring_ext @ TA @ TC ) @ B @ C @ D )
= ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_Ring_OSubring__pOp__ring__pOp) ).
thf(814,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( subring @ TA @ TB @ TC @ A @ B )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
=> ( ( pop @ TA @ ( ring_ext @ TA @ TC ) @ B @ C @ D )
= ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(145,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
| ( ord_less_eq @ ( set @ TA ) @ B @ A ) )
=> ( ~ ( member @ TA @ C @ A )
=> ( ( member @ TA @ C @ B )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_159_sub__which2) ).
thf(698,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
| ( ord_less_eq @ ( set @ TA ) @ B @ A ) )
=> ( ~ ( member @ TA @ C @ A )
=> ( ( member @ TA @ C @ B )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(217,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( sr2 @ TA @ TB @ A @ B )
=> ( ( member @ TA @ C @ B )
=> ( ( member @ TA @ D @ B )
=> ( member @ TA @ ( tp @ TA @ TB @ A @ C @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_Ring_OSr__tOp__closed) ).
thf(945,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( sr2 @ TA @ TB @ A @ B )
=> ( ( member @ TA @ C @ B )
=> ( ( member @ TA @ D @ B )
=> ( member @ TA @ ( tp @ TA @ TB @ A @ C @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).
thf(27,axiom,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
= ( ord_less_eq @ TA @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_add__le__cancel__right) ).
thf(340,plain,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
= ( ord_less_eq @ TA @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(211,axiom,
! [A: set @ a] :
( ( mul_closed_set @ a @ b @ r @ A )
=> ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_143_mul__closed__set__sub) ).
thf(930,plain,
! [A: set @ a] :
( ( mul_closed_set @ a @ b @ r @ A )
=> ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(236,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ D ) @ E )
= ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ ( pop @ TA @ TB @ A @ D @ E ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_aGroup_OpOp__assocTr41) ).
thf(1001,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ D ) @ E )
= ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ ( pop @ TA @ TB @ A @ D @ E ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).
thf(149,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TF ) ),B: TA > TD,C: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ),D: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TC @ TB ) )] :
( ( ring @ TC @ TF @ A )
=> ( ( member @ ( TA > TD ) @ B @ ( mHom @ TC @ TF @ TA @ TE @ TD @ TB @ A @ C @ D ) )
=> ( member @ ( TA > TD ) @ B
@ ( pi @ TA @ TD @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ) @ C )
@ ^ [E: TA] : ( carrier @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TC @ TB ) ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_Ring_OmHom__func) ).
thf(709,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TF ) ),B: TA > TD,C: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ),D: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TC @ TB ) )] :
( ( ring @ TC @ TF @ A )
=> ( ( member @ ( TA > TD ) @ B @ ( mHom @ TC @ TF @ TA @ TE @ TD @ TB @ A @ C @ D ) )
=> ( member @ ( TA > TD ) @ B
@ ( pi @ TA @ TD @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ) @ C )
@ ^ [E: TA] : ( carrier @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TC @ TB ) ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(59,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( ( pop @ TA @ TB @ A @ B @ ( mop @ TA @ TB @ A @ C ) )
= D )
=> ( ( pop @ TA @ TB @ A @ C @ ( mop @ TA @ TB @ A @ B ) )
= ( mop @ TA @ TB @ A @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_aGroup_Oag__diff__minus) ).
thf(417,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( ( pop @ TA @ TB @ A @ B @ ( mop @ TA @ TB @ A @ C ) )
= D )
=> ( ( pop @ TA @ TB @ A @ C @ ( mop @ TA @ TB @ A @ B ) )
= ( mop @ TA @ TB @ A @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(227,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) )] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( aGroup @ TA @ ( module_ext @ TA @ TB @ TC ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_Module_Omodule__is__ag) ).
thf(973,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) )] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( aGroup @ TA @ ( module_ext @ TA @ TB @ TC ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).
thf(104,axiom,
! [A: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( un @ a @ b @ r ) @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_216_ring__l__one) ).
thf(565,plain,
! [A: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( un @ a @ b @ r ) @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(130,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( B != C )
=> ( ( mop @ TA @ TB @ A @ B )
!= ( mop @ TA @ TB @ A @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_171_aGroup_Oag__inv__inj) ).
thf(651,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( B != C )
=> ( ( mop @ TA @ TB @ A @ B )
!= ( mop @ TA @ TB @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(47,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( ( zeroring @ TA @ TB @ A )
=> ( ( j_rad @ TA @ TB @ A )
= ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_Ring_Ozeroring__J__rad__empty) ).
thf(389,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( ( zeroring @ TA @ TB @ A )
=> ( ( j_rad @ TA @ TB @ A )
= ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(155,axiom,
! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ),B: a] :
( ( subring @ a @ b @ TA @ r @ A )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
=> ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_mem__subring__mem__ring) ).
thf(724,plain,
! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ),B: a] :
( ( subring @ a @ b @ TA @ r @ A )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
=> ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(106,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ D )
= ( pop @ TA @ TB @ A @ B @ ( pop @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_aGroup_Oaassoc) ).
thf(570,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ D )
= ( pop @ TA @ TB @ A @ B @ ( pop @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(153,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( member @ TA @ ( tp @ TA @ TB @ A @ B @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_Ring_Oring__tOp__closed) ).
thf(719,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( member @ TA @ ( tp @ TA @ TB @ A @ B @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(41,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( ~ ( zeroring @ TA @ TB @ A )
=> ? [B: set @ TA] : ( maximal_ideal @ TA @ TB @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_Ring_Oid__maximal__Exist) ).
thf(372,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( ~ ( zeroring @ TA @ TB @ A )
=> ? [B: set @ TA] : ( maximal_ideal @ TA @ TB @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(72,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( aGroup @ TB @ TC @ ( B @ C ) ) )
=> ( member @ ( ( TA > TB ) > ( TA > TB ) > TA > TB ) @ ( prod_pOp @ TA @ TB @ TC @ A @ B )
@ ( pi @ ( TA > TB ) @ ( ( TA > TB ) > TA > TB ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B )
@ ^ [C: TA > TB] :
( pi @ ( TA > TB ) @ ( TA > TB ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B )
@ ^ [D: TA > TB] : ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_dsum__pOp__func) ).
thf(460,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( aGroup @ TB @ TC @ ( B @ C ) ) )
=> ( member @ ( ( TA > TB ) > ( TA > TB ) > TA > TB ) @ ( prod_pOp @ TA @ TB @ TC @ A @ B )
@ ( pi @ ( TA > TB ) @ ( ( TA > TB ) > TA > TB ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B )
@ ^ [C: TA > TB] :
( pi @ ( TA > TB ) @ ( TA > TB ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B )
@ ^ [D: TA > TB] : ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(111,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ~ ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ? [C: TA] :
( ( member @ TA @ C @ A )
& ~ ( member @ TA @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_162_not__subseteq) ).
thf(587,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ~ ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ? [C: TA] :
( ( member @ TA @ C @ A )
& ~ ( member @ TA @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(120,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ D ) @ E )
= ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ ( pop @ TA @ TB @ A @ C @ D ) ) @ E ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_aGroup_OpOp__assocTr42) ).
thf(619,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ D ) @ E )
= ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ ( pop @ TA @ TB @ A @ C @ D ) ) @ E ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(208,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: TB > TC > TA,B: set @ TB,C: set @ TC,D: set @ TA,E: TB,F: TC] :
( ( member @ ( TB > TC > TA ) @ A
@ ( pi @ TB @ ( TC > TA ) @ B
@ ^ [G: TB] :
( pi @ TC @ TA @ C
@ ^ [H: TC] : D ) ) )
=> ( ( member @ TB @ E @ B )
=> ( ( member @ TC @ F @ C )
=> ( member @ TA @ ( A @ E @ F ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_bivar__fun__mem) ).
thf(917,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: TB > TC > TA,B: set @ TB,C: set @ TC,D: set @ TA,E: TB,F: TC] :
( ( member @ ( TB > TC > TA ) @ A
@ ( pi @ TB @ ( TC > TA ) @ B
@ ^ [G: TB] :
( pi @ TC @ TA @ C
@ ^ [H: TC] : D ) ) )
=> ( ( member @ TB @ E @ B )
=> ( ( member @ TC @ F @ C )
=> ( member @ TA @ ( A @ E @ F ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(166,axiom,
! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) )] :
( ( subring @ a @ b @ TA @ r @ A )
=> ( ring @ a @ TA @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_subring__Ring) ).
thf(751,plain,
! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) )] :
( ( subring @ a @ b @ TA @ r @ A )
=> ( ring @ a @ TA @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(183,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB )] :
( ( aGroup @ TA @ TB @ A )
=> ( member @ ( TA > TA > TA ) @ ( pop @ TA @ TB @ A )
@ ( pi @ TA @ ( TA > TA ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A )
@ ^ [B: TA] :
( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A )
@ ^ [C: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_aGroup_Opop__closed) ).
thf(817,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB )] :
( ( aGroup @ TA @ TB @ A )
=> ( member @ ( TA > TA > TA ) @ ( pop @ TA @ TB @ A )
@ ( pi @ TA @ ( TA > TA ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A )
@ ^ [B: TA] :
( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A )
@ ^ [C: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(115,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ),C: TC,D: TC,E: TA] :
( ( module @ TA @ TC @ TB @ TD @ A @ B )
=> ( ( member @ TC @ C @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
=> ( ( member @ TC @ D @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
=> ( ( sprod @ TA @ TC @ TB @ A @ ( tp @ TC @ TD @ B @ C @ D ) @ E )
= ( sprod @ TA @ TC @ TB @ A @ C @ ( sprod @ TA @ TC @ TB @ A @ D @ E ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_Module_Osprod__assoc) ).
thf(606,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ),C: TC,D: TC,E: TA] :
( ( module @ TA @ TC @ TB @ TD @ A @ B )
=> ( ( member @ TC @ C @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
=> ( ( member @ TC @ D @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
=> ( ( sprod @ TA @ TC @ TB @ A @ ( tp @ TC @ TD @ B @ C @ D ) @ E )
= ( sprod @ TA @ TC @ TB @ A @ C @ ( sprod @ TA @ TC @ TB @ A @ D @ E ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(7,axiom,
cancel_semigroup_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__semigroup__add) ).
thf(289,plain,
cancel_semigroup_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(19,axiom,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_239_add_Oleft__commute) ).
thf(316,plain,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(267,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TE ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TF ) ),C: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TC ) ),D: TA,E: TA > TB] :
( ( module @ TA @ TD @ TE @ TF @ A @ B )
=> ( ( module @ TB @ TD @ TC @ TF @ C @ B )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TE ) ) @ A ) )
=> ( ( member @ ( TA > TB ) @ E @ ( mHom @ TD @ TF @ TA @ TE @ TB @ TC @ B @ A @ C ) )
=> ( ( E @ ( mop @ TA @ ( module_ext @ TA @ TD @ TE ) @ A @ D ) )
= ( mop @ TB @ ( module_ext @ TB @ TD @ TC ) @ C @ ( E @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_Module_OmHom__inv) ).
thf(1100,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TE ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TF ) ),C: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TC ) ),D: TA,E: TA > TB] :
( ( module @ TA @ TD @ TE @ TF @ A @ B )
=> ( ( module @ TB @ TD @ TC @ TF @ C @ B )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TE ) ) @ A ) )
=> ( ( member @ ( TA > TB ) @ E @ ( mHom @ TD @ TF @ TA @ TE @ TB @ TC @ B @ A @ C ) )
=> ( ( E @ ( mop @ TA @ ( module_ext @ TA @ TD @ TE ) @ A @ D ) )
= ( mop @ TB @ ( module_ext @ TB @ TD @ TC ) @ C @ ( E @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).
thf(60,axiom,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A @ B ) @ C )
= ( tp @ a @ b @ r @ A @ ( tp @ a @ b @ r @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_ring__tOp__assoc) ).
thf(421,plain,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A @ B ) @ C )
= ( tp @ a @ b @ r @ A @ ( tp @ a @ b @ r @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(198,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( B = C )
=> ( ( tp @ TA @ TB @ A @ B @ D )
= ( tp @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_105_Ring_OrEQMulR) ).
thf(870,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( B = C )
=> ( ( tp @ TA @ TB @ A @ B @ D )
= ( tp @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(5,axiom,
ordere236663937imp_le @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le) ).
thf(284,plain,
ordere236663937imp_le @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(172,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA,E: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ D @ E ) )
= ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ D ) @ ( tp @ TA @ TB @ A @ B @ E ) ) @ ( tp @ TA @ TB @ A @ C @ D ) ) @ ( tp @ TA @ TB @ A @ C @ E ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_Ring_Oring__distrib3) ).
thf(770,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA,E: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ D @ E ) )
= ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ D ) @ ( tp @ TA @ TB @ A @ B @ E ) ) @ ( tp @ TA @ TB @ A @ C @ D ) ) @ ( tp @ TA @ TB @ A @ C @ E ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(77,axiom,
! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ),B: a,C: a] :
( ( subring @ a @ b @ TA @ r @ A )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
=> ( ( tp @ a @ TA @ A @ B @ C )
= ( tp @ a @ b @ r @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_Subring__tOp__ring__tOp) ).
thf(475,plain,
! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ),B: a,C: a] :
( ( subring @ a @ b @ TA @ r @ A )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
=> ( ( tp @ a @ TA @ A @ B @ C )
= ( tp @ a @ b @ r @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(83,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ ( mop @ TA @ TB @ A @ C ) ) @ ( pop @ TA @ TB @ A @ C @ ( mop @ TA @ TB @ A @ D ) ) )
= ( pop @ TA @ TB @ A @ B @ ( mop @ TA @ TB @ A @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_182_aGroup_OpOp__assoc__cancel) ).
thf(492,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ ( mop @ TA @ TB @ A @ C ) ) @ ( pop @ TA @ TB @ A @ C @ ( mop @ TA @ TB @ A @ D ) ) )
= ( pop @ TA @ TB @ A @ B @ ( mop @ TA @ TB @ A @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(168,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A != B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
=> ? [C: TA] :
( ( member @ TA @ C @ A )
& ~ ( member @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_160_sets__not__eq) ).
thf(756,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A != B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
=> ? [C: TA] :
( ( member @ TA @ C @ A )
& ~ ( member @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(21,axiom,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
= ( ord_less_eq @ TA @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_221_add__le__cancel__left) ).
thf(321,plain,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
= ( ord_less_eq @ TA @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(129,axiom,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( A = B )
=> ( ( tp @ a @ b @ r @ A @ C )
= ( tp @ a @ b @ r @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_rEQMulR) ).
thf(646,plain,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( A = B )
=> ( ( tp @ a @ b @ r @ A @ C )
= ( tp @ a @ b @ r @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(90,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B )
= ( tp @ TA @ TB @ A @ B @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( un @ TA @ TB @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_249_Ring_Oring__times__minusr) ).
thf(514,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B )
= ( tp @ TA @ TB @ A @ B @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( un @ TA @ TB @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(250,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) )
= ( tp @ TA @ TB @ A @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_Ring_Oring__inv1__1) ).
thf(1051,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) )
= ( tp @ TA @ TB @ A @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).
thf(204,axiom,
! [A: a,B: a,C: nat] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( nscal @ a @ b @ r @ A @ C ) @ ( nscal @ a @ b @ r @ B @ C ) )
= ( nscal @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_nsDistrL) ).
thf(905,plain,
! [A: a,B: a,C: nat] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( nscal @ a @ b @ r @ A @ C ) @ ( nscal @ a @ b @ r @ B @ C ) )
= ( nscal @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(114,axiom,
! [A: a] :
( ~ ( zeroring @ a @ b @ r )
=> ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ~ ( unit @ a @ b @ r @ A )
=> ? [B: set @ a] :
( ( maximal_ideal @ a @ b @ r @ B )
& ( member @ a @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_nonunit__contained__maxid) ).
thf(602,plain,
! [A: a] :
( ~ ( zeroring @ a @ b @ r )
=> ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ~ ( unit @ a @ b @ r @ A )
=> ? [B: set @ a] :
( ( maximal_ideal @ a @ b @ r @ B )
& ( member @ a @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(4,axiom,
! [A: nat,B: nat] :
( ( A = B )
=> ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_225_nat__eq__le) ).
thf(280,plain,
! [A: nat,B: nat] :
( ( A = B )
=> ( ord_less_eq @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(143,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( aGroup @ TB @ TC @ ( B @ C ) ) )
=> ( member @ ( TA > TB ) @ ( prod_zero @ TA @ TB @ TC @ A @ B ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_114_dsum__zero__func) ).
thf(692,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( aGroup @ TB @ TC @ ( B @ C ) ) )
=> ( member @ ( TA > TB ) @ ( prod_zero @ TA @ TB @ TC @ A @ B ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(160,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( aGroup @ TB @ TC @ ( B @ C ) ) )
=> ( member @ ( TA > TB ) @ ( prod_zero @ TA @ TB @ TC @ A @ B ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_prod__zero__func) ).
thf(734,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( aGroup @ TB @ TC @ ( B @ C ) ) )
=> ( member @ ( TA > TB ) @ ( prod_zero @ TA @ TB @ TC @ A @ B ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(197,axiom,
! [A: a,B: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A @ B )
= ( tp @ a @ b @ r @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_ring__tOp__commute) ).
thf(866,plain,
! [A: a,B: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A @ B )
= ( tp @ a @ b @ r @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).
thf(49,axiom,
! [TA: $tType] : ( ord @ ( set @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oord_2) ).
thf(396,plain,
! [TA: $tType] : ( ord @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(235,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) @ B )
= ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ C @ B ) @ ( tp @ TA @ TB @ A @ D @ B ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_Ring_Oring__distrib2) ).
thf(998,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) @ B )
= ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ C @ B ) @ ( tp @ TA @ TB @ A @ D @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).
thf(232,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 )
=> ( A != B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_163_proper__subset) ).
thf(988,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 )
=> ( A != B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(88,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > ( set @ TB ),C: TA > ( set @ TB )] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ord_less_eq @ ( set @ TB ) @ ( B @ D ) @ ( C @ D ) ) )
=> ( ord_less_eq @ ( set @ ( TA > TB ) ) @ ( pi @ TA @ TB @ A @ B ) @ ( pi @ TA @ TB @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_166_Pi__mono) ).
thf(507,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > ( set @ TB ),C: TA > ( set @ TB )] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ord_less_eq @ ( set @ TB ) @ ( B @ D ) @ ( C @ D ) ) )
=> ( ord_less_eq @ ( set @ ( TA > TB ) ) @ ( pi @ TA @ TB @ A @ B ) @ ( pi @ TA @ TB @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(18,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( A
= ( plus_plus @ TA @ B @ C ) )
=> ( ( plus_plus @ TA @ D @ A )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ D @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_244_group__cancel_Oadd2) ).
thf(312,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( A
= ( plus_plus @ TA @ B @ C ) )
=> ( ( plus_plus @ TA @ D @ A )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ D @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(10,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( ring @ TA @ TB @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_Ring_ORing) ).
thf(293,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(222,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( ( pop @ TA @ TB @ A @ D @ B )
= ( pop @ TA @ TB @ A @ D @ C ) )
=> ( B = C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_aGroup_OpOp__cancel__l) ).
thf(956,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( ( pop @ TA @ TB @ A @ D @ B )
= ( pop @ TA @ TB @ A @ D @ C ) )
=> ( B = C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).
thf(15,axiom,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
= ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_ab__semigroup__add__class_Oadd__ac_I1_J) ).
thf(307,plain,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
= ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(9,axiom,
ordere779506340up_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add) ).
thf(292,plain,
ordere779506340up_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(134,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType] :
( ( rInvim @ TA @ TB @ TD @ TC )
= ( ^ [A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ),C: TA > TD,D: set @ TD] :
( collect @ TA
@ ^ [E: TA] :
( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
& ( member @ TD @ ( C @ E ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_rInvim__def) ).
thf(661,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType] :
( ( rInvim @ TA @ TB @ TD @ TC )
= ( ^ [A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ),C: TA > TD,D: set @ TD] :
( collect @ TA
@ ^ [E: TA] :
( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
& ( member @ TD @ ( C @ E ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(99,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB,C: set @ TB] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( member @ TB @ ( B @ D ) @ C ) )
=> ( member @ ( TA > TB ) @ B
@ ( pi @ TA @ TB @ A
@ ^ [D: TA] : C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8_funcsetI) ).
thf(546,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB,C: set @ TB] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( member @ TB @ ( B @ D ) @ C ) )
=> ( member @ ( TA > TB ) @ B
@ ( pi @ TA @ TB @ A
@ ^ [D: TA] : C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(169,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA,E: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ D @ E ) ) )
= ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ E ) ) ) @ ( tp @ TA @ TB @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ D ) ) @ E ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_Ring_Oring__distrib4) ).
thf(761,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA,E: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ D @ E ) ) )
= ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ E ) ) ) @ ( tp @ TA @ TB @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ D ) ) @ E ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(44,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_155_eqsets__sub) ).
thf(381,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)],[44]) ).
thf(255,axiom,
! [A: set @ a,B: set @ a,C: a,D: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( sum_mult @ a @ b @ r @ A @ B ) )
=> ( ( member @ a @ D @ ( sum_mult @ a @ b @ r @ A @ B ) )
=> ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ C @ D ) @ ( sum_mult @ a @ b @ r @ A @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194_sum__mult__pOp__closed) ).
thf(1067,plain,
! [A: set @ a,B: set @ a,C: a,D: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( sum_mult @ a @ b @ r @ A @ B ) )
=> ( ( member @ a @ D @ ( sum_mult @ a @ b @ r @ A @ B ) )
=> ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ C @ D ) @ ( sum_mult @ a @ b @ r @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).
thf(178,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( mul_closed_set @ TA @ TB @ A @ B )
=> ( ( member @ TA @ C @ B )
=> ( ( member @ TA @ D @ B )
=> ( member @ TA @ ( tp @ TA @ TB @ A @ C @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_Ring_Omul__closed__set__tOp__closed) ).
thf(798,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( mul_closed_set @ TA @ TB @ A @ B )
=> ( ( member @ TA @ C @ B )
=> ( ( member @ TA @ D @ B )
=> ( member @ TA @ ( tp @ TA @ TB @ A @ C @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(252,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( ( pop @ TA @ TB @ A @ B @ C )
= D )
=> ( C
= ( pop @ TA @ TB @ A @ ( mop @ TA @ TB @ A @ B ) @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_aGroup_Oag__eq__sol1) ).
thf(1057,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( ( pop @ TA @ TB @ A @ B @ C )
= D )
=> ( C
= ( pop @ TA @ TB @ A @ ( mop @ TA @ TB @ A @ B ) @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).
thf(25,axiom,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
= ( B = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_214_add__left__cancel) ).
thf(334,plain,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
= ( B = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(249,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( aGroup @ TB @ TC @ ( B @ D ) ) )
=> ( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ( B @ D )
= ( C @ D ) ) )
=> ( ( prod_zero @ TA @ TB @ TC @ A @ B )
= ( prod_zero @ TA @ TB @ TC @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_prodag__sameTr4) ).
thf(1042,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( aGroup @ TB @ TC @ ( B @ D ) ) )
=> ( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ( B @ D )
= ( C @ D ) ) )
=> ( ( prod_zero @ TA @ TB @ TC @ A @ B )
= ( prod_zero @ TA @ TB @ TC @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(30,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ? [C: nat] :
( B
= ( plus_plus @ nat @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_le__imp__add__int) ).
thf(349,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ? [C: nat] :
( B
= ( plus_plus @ nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(48,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
& ( C = D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_251_add__mono__thms__linordered__semiring_I3_J) ).
thf(392,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
& ( C = D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(12,axiom,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
= ( A = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_add_Oright__cancel) ).
thf(297,plain,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
= ( A = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(63,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) )
= ( tp @ TA @ TB @ A @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B ) @ C ) )
& ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) )
= ( tp @ TA @ TB @ A @ B @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_Ring_Oring__inv1) ).
thf(429,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) )
= ( tp @ TA @ TB @ A @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B ) @ C ) )
& ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) )
= ( tp @ TA @ TB @ A @ B @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(164,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ B @ C )
= ( pop @ TA @ TB @ A @ C @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_85_aGroup_Oag__pOp__commute) ).
thf(746,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ B @ C )
= ( pop @ TA @ TB @ A @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(264,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB] :
( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( aGroup @ TB @ TC @ ( B @ E ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( member @ ( TA > TB ) @ ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29_prod__pOp__mem) ).
thf(1091,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB] :
( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( aGroup @ TB @ TC @ ( B @ E ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( member @ ( TA > TB ) @ ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).
thf(75,axiom,
! [A: set @ a,B: a] :
( ( sr2 @ a @ b @ r @ A )
=> ( ( member @ a @ B @ A )
=> ( member @ a @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_147_Sr__mOp__closed) ).
thf(467,plain,
! [A: set @ a,B: a] :
( ( sr2 @ a @ b @ r @ A )
=> ( ( member @ a @ B @ A )
=> ( member @ a @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(80,axiom,
! [A: a,B: set @ a,C: a,D: set @ a] :
( ( member @ a @ A @ B )
=> ( ( member @ a @ C @ D )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ D @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ C ) @ ( aset_sum @ a @ ( ring_ext @ a @ b ) @ r @ B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_set__sum__mem) ).
thf(484,plain,
! [A: a,B: set @ a,C: a,D: set @ a] :
( ( member @ a @ A @ B )
=> ( ( member @ a @ C @ D )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ D @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ C ) @ ( aset_sum @ a @ ( ring_ext @ a @ b ) @ r @ B @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(34,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: set @ TA] :
( ( ring @ TA @ TB @ A )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ord_less_eq @ ( set @ TA ) @ ( sum_mult @ TA @ TB @ A @ B @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_Ring_Osum__mult__subR) ).
thf(361,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: set @ TA] :
( ( ring @ TA @ TB @ A )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ord_less_eq @ ( set @ TA ) @ ( sum_mult @ TA @ TB @ A @ B @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(122,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( mop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) )
= ( pop @ TA @ TB @ A @ ( mop @ TA @ TB @ A @ B ) @ ( mop @ TA @ TB @ A @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_aGroup_Oag__p__inv) ).
thf(625,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( mop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) )
= ( pop @ TA @ TB @ A @ ( mop @ TA @ TB @ A @ B ) @ ( mop @ TA @ TB @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(192,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ B @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) )
= ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ ( tp @ TA @ TB @ A @ B @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_Ring_Oring__distrib1) ).
thf(852,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ B @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) )
= ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ ( tp @ TA @ TB @ A @ B @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(244,axiom,
! [A: set @ a] :
( ( sr2 @ a @ b @ r @ A )
=> ( member @ a @ ( un @ a @ b @ r ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_212_Sr__one) ).
thf(1027,plain,
! [A: set @ a] :
( ( sr2 @ a @ b @ r @ A )
=> ( member @ a @ ( un @ a @ b @ r ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).
thf(22,axiom,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
= ( B = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_242_add_Oleft__cancel) ).
thf(324,plain,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
= ( B = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(239,axiom,
( ( zeroring @ a @ b @ r )
=> ( ( j_rad @ a @ b @ r )
= ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_zeroring__J__rad__empty) ).
thf(1012,plain,
( ( zeroring @ a @ b @ r )
=> ( ( j_rad @ a @ b @ r )
= ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).
thf(6,axiom,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A = B )
=> ( ( ord_less_eq @ TA @ A @ C )
=> ( ord_less_eq @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_157_le__convert) ).
thf(285,plain,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A = B )
=> ( ( ord_less_eq @ TA @ A @ C )
=> ( ord_less_eq @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(110,axiom,
! [A: set @ a] :
( ~ ( zeroring @ a @ b @ r )
=> ( ( maximal_ideal @ a @ b @ r @ A )
=> ( ! [B: a] :
( ( member @ a @ B @ A )
=> ( unit @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ ( un @ a @ b @ r ) ) ) )
=> ( local_ring @ a @ b @ r ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_222_localring__unit) ).
thf(583,plain,
! [A: set @ a] :
( ~ ( zeroring @ a @ b @ r )
=> ( ( maximal_ideal @ a @ b @ r @ A )
=> ( ! [B: a] :
( ( member @ a @ B @ A )
=> ( unit @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ ( un @ a @ b @ r ) ) ) )
=> ( local_ring @ a @ b @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(186,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ a @ TC ) ) ),C: a,D: TA > TB,E: TA] :
( ! [F: TA] :
( ( member @ TA @ F @ A )
=> ( module @ TB @ a @ TC @ b @ ( B @ F ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ a @ TC ) @ A @ B ) )
=> ( ( member @ TA @ E @ A )
=> ( ( algebr142713364_sprod @ a @ b @ TA @ TB @ TC @ r @ A @ B @ C @ D @ E )
= ( sprod @ TB @ a @ TC @ ( B @ E ) @ C @ ( D @ E ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_prodM__sprod__val) ).
thf(830,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ a @ TC ) ) ),C: a,D: TA > TB,E: TA] :
( ! [F: TA] :
( ( member @ TA @ F @ A )
=> ( module @ TB @ a @ TC @ b @ ( B @ F ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ a @ TC ) @ A @ B ) )
=> ( ( member @ TA @ E @ A )
=> ( ( algebr142713364_sprod @ a @ b @ TA @ TB @ TC @ r @ A @ B @ C @ D @ E )
= ( sprod @ TB @ a @ TC @ ( B @ E ) @ C @ ( D @ E ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(263,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: set @ TA,D: TA,E: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( sum_mult @ TA @ TB @ A @ B @ C ) )
=> ( ( member @ TA @ E @ ( sum_mult @ TA @ TB @ A @ B @ C ) )
=> ( member @ TA @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ D @ E ) @ ( sum_mult @ TA @ TB @ A @ B @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_Ring_Osum__mult__pOp__closed) ).
thf(1089,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: set @ TA,D: TA,E: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( sum_mult @ TA @ TB @ A @ B @ C ) )
=> ( ( member @ TA @ E @ ( sum_mult @ TA @ TB @ A @ B @ C ) )
=> ( member @ TA @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ D @ E ) @ ( sum_mult @ TA @ TB @ A @ B @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).
thf(199,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB] :
( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( aGroup @ TB @ TC @ ( B @ E ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( ( C @ E )
= ( D @ E ) ) )
=> ( C = D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_carr__prodag__mem__eq) ).
thf(874,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB] :
( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( aGroup @ TB @ TC @ ( B @ E ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( ( C @ E )
= ( D @ E ) ) )
=> ( C = D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(219,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB,E: TA > TB] :
( ! [F: TA] :
( ( member @ TA @ F @ A )
=> ( aGroup @ TB @ TC @ ( B @ F ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ ( TA > TB ) @ E @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( prod_pOp @ TA @ TB @ TC @ A @ B @ ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D ) @ E )
= ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ ( prod_pOp @ TA @ TB @ TC @ A @ B @ D @ E ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_prod__pOp__assoc) ).
thf(948,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB,E: TA > TB] :
( ! [F: TA] :
( ( member @ TA @ F @ A )
=> ( aGroup @ TB @ TC @ ( B @ F ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ ( TA > TB ) @ E @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( prod_pOp @ TA @ TB @ TC @ A @ B @ ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D ) @ E )
= ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ ( prod_pOp @ TA @ TB @ TC @ A @ B @ D @ E ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).
thf(254,axiom,
! [A: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A )
= ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_220_ring__times__minusl) ).
thf(1064,plain,
! [A: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A )
= ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).
thf(53,axiom,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
=> ( B = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_238_add__left__imp__eq) ).
thf(403,plain,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
=> ( B = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(32,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TC @ A )
=> ( ( subring @ TA @ TC @ TB @ A @ B )
=> ( ring @ TA @ TB @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_Ring_Osubring__Ring) ).
thf(356,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TC @ A )
=> ( ( subring @ TA @ TC @ TB @ A @ B )
=> ( ring @ TA @ TB @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(113,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( aGroup @ TB @ TC @ ( B @ D ) ) )
=> ( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ( B @ D )
= ( C @ D ) ) )
=> ( ( carr_prodag @ TA @ TB @ TC @ A @ B )
= ( carr_prodag @ TA @ TB @ TC @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_prodag__sameTr1) ).
thf(593,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( aGroup @ TB @ TC @ ( B @ D ) ) )
=> ( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ( B @ D )
= ( C @ D ) ) )
=> ( ( carr_prodag @ TA @ TB @ TC @ A @ B )
= ( carr_prodag @ TA @ TB @ TC @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(167,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ B @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) )
= ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ ( tp @ TA @ TB @ A @ B @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_Ring_Org__distrib) ).
thf(753,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ B @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) )
= ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ ( tp @ TA @ TB @ A @ B @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(96,axiom,
! [A: set @ a] :
( ( primary_ideal @ a @ b @ r @ A )
=> ~ ( member @ a @ ( un @ a @ b @ r ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_223_primary__ideal__proper1) ).
thf(537,plain,
! [A: set @ a] :
( ( primary_ideal @ a @ b @ r @ A )
=> ~ ( member @ a @ ( un @ a @ b @ r ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(258,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ ( un @ TA @ TB @ A ) @ B )
= B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_233_Ring_Oring__l__one) ).
thf(1075,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ ( un @ TA @ TB @ A ) @ B )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).
thf(112,axiom,
! [A: set @ a,B: set @ a,C: a,D: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ A )
=> ( ( member @ a @ D @ B )
=> ( member @ a @ ( tp @ a @ b @ r @ C @ D ) @ ( sum_mult @ a @ b @ r @ A @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_times__mem__sum__mult) ).
thf(590,plain,
! [A: set @ a,B: set @ a,C: a,D: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ A )
=> ( ( member @ a @ D @ B )
=> ( member @ a @ ( tp @ a @ b @ r @ C @ D ) @ ( sum_mult @ a @ b @ r @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(226,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ),C: TC,D: TC,E: TA] :
( ( module @ TA @ TC @ TB @ TD @ A @ B )
=> ( ( member @ TC @ C @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
=> ( ( member @ TC @ D @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
=> ( ( sprod @ TA @ TC @ TB @ A @ ( pop @ TC @ ( ring_ext @ TC @ TD ) @ B @ C @ D ) @ E )
= ( pop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ ( sprod @ TA @ TC @ TB @ A @ C @ E ) @ ( sprod @ TA @ TC @ TB @ A @ D @ E ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_Module_Osprod__l__distr) ).
thf(970,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ),C: TC,D: TC,E: TA] :
( ( module @ TA @ TC @ TB @ TD @ A @ B )
=> ( ( member @ TC @ C @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
=> ( ( member @ TC @ D @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
=> ( ( sprod @ TA @ TC @ TB @ A @ ( pop @ TC @ ( ring_ext @ TC @ TD ) @ B @ C @ D ) @ E )
= ( pop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ ( sprod @ TA @ TC @ TB @ A @ C @ E ) @ ( sprod @ TA @ TC @ TB @ A @ D @ E ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).
thf(148,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ~ ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ? [C: TA] :
( ( member @ TA @ C @ A )
& ~ ( member @ TA @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_154_not__sub) ).
thf(706,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ~ ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ? [C: TA] :
( ( member @ TA @ C @ A )
& ~ ( member @ TA @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(121,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) @ D )
= ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_Ring_Opop__aassoc) ).
thf(622,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) @ D )
= ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(146,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ),C: TC,D: TA] :
( ( module @ TA @ TC @ TB @ TD @ A @ B )
=> ( ( member @ TC @ C @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
=> ( ( mop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ ( sprod @ TA @ TC @ TB @ A @ C @ D ) )
= ( sprod @ TA @ TC @ TB @ A @ C @ ( mop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_Module_Osc__minus__am) ).
thf(701,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ),C: TC,D: TA] :
( ( module @ TA @ TC @ TB @ TD @ A @ B )
=> ( ( member @ TC @ C @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
=> ( ( mop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ ( sprod @ TA @ TC @ TB @ A @ C @ D ) )
= ( sprod @ TA @ TC @ TB @ A @ C @ ( mop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(131,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( sr2 @ TA @ TB @ A @ B )
=> ( ( member @ TA @ C @ B )
=> ( ( member @ TA @ D @ B )
=> ( member @ TA @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_Ring_OSr__pOp__closed) ).
thf(654,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( sr2 @ TA @ TB @ A @ B )
=> ( ( member @ TA @ C @ B )
=> ( ( member @ TA @ D @ B )
=> ( member @ TA @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(71,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( aGroup @ TB @ TC @ ( B @ C ) ) )
=> ( member @ ( ( TA > TB ) > TA > TB ) @ ( prod_mOp @ TA @ TB @ TC @ A @ B )
@ ( pi @ ( TA > TB ) @ ( TA > TB ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B )
@ ^ [C: TA > TB] : ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_prod__mOp__func) ).
thf(457,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( aGroup @ TB @ TC @ ( B @ C ) ) )
=> ( member @ ( ( TA > TB ) > TA > TB ) @ ( prod_mOp @ TA @ TB @ TC @ A @ B )
@ ( pi @ ( TA > TB ) @ ( TA > TB ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B )
@ ^ [C: TA > TB] : ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(231,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: TB,D: nat] :
( ( ring @ TB @ TA @ A )
=> ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( B = C )
=> ( ( nscal @ TB @ TA @ A @ B @ D )
= ( nscal @ TB @ TA @ A @ C @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_Ring_OnsEqElm) ).
thf(984,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: TB,D: nat] :
( ( ring @ TB @ TA @ A )
=> ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( B = C )
=> ( ( nscal @ TB @ TA @ A @ B @ D )
= ( nscal @ TB @ TA @ A @ C @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).
thf(154,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ a @ TC ) ) ),C: a,D: TA > TB] :
( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( module @ TB @ a @ TC @ b @ ( B @ E ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ a @ TC ) @ A @ B ) )
=> ( member @ ( TA > TB ) @ ( algebr142713364_sprod @ a @ b @ TA @ TB @ TC @ r @ A @ B @ C @ D ) @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ a @ TC ) @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4_prodM__sprod__mem) ).
thf(721,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ a @ TC ) ) ),C: a,D: TA > TB] :
( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( module @ TB @ a @ TC @ b @ ( B @ E ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ a @ TC ) @ A @ B ) )
=> ( member @ ( TA > TB ) @ ( algebr142713364_sprod @ a @ b @ TA @ TB @ TC @ r @ A @ B @ C @ D ) @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ a @ TC ) @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(175,axiom,
( ( member @ ( ( c > d ) > ( c > d ) > c > d ) @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
@ ( pi @ ( c > d ) @ ( ( c > d ) > c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
@ ^ [A: c > d] :
( pi @ ( c > d ) @ ( c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
@ ^ [B: c > d] : ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) ) )
& ! [A: c > d] :
( ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ! [B: c > d] :
( ( member @ ( c > d ) @ B @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ! [C: c > d] :
( ( member @ ( c > d ) @ C @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ B ) @ C )
= ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ B @ C ) ) ) ) ) )
& ! [A: c > d] :
( ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ! [B: c > d] :
( ( member @ ( c > d ) @ B @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ B )
= ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ B @ A ) ) ) )
& ( member @ ( ( c > d ) > c > d ) @ ( prod_mOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
@ ( pi @ ( c > d ) @ ( c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
@ ^ [A: c > d] : ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) )
& ! [A: c > d] :
( ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_mOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A ) @ A )
= ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) )
& ( member @ ( c > d ) @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
& ! [A: c > d] :
( ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) @ A )
= A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_1) ).
thf(779,plain,
( ( member @ ( ( c > d ) > ( c > d ) > c > d ) @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
@ ( pi @ ( c > d ) @ ( ( c > d ) > c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
@ ^ [A: c > d] :
( pi @ ( c > d ) @ ( c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
@ ^ [B: c > d] : ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) ) )
& ! [A: c > d] :
( ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ! [B: c > d] :
( ( member @ ( c > d ) @ B @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ! [C: c > d] :
( ( member @ ( c > d ) @ C @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ B ) @ C )
= ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ B @ C ) ) ) ) ) )
& ! [A: c > d] :
( ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ! [B: c > d] :
( ( member @ ( c > d ) @ B @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ B )
= ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ B @ A ) ) ) )
& ( member @ ( ( c > d ) > c > d ) @ ( prod_mOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
@ ( pi @ ( c > d ) @ ( c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
@ ^ [A: c > d] : ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) )
& ! [A: c > d] :
( ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_mOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A ) @ A )
= ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) )
& ( member @ ( c > d ) @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
& ! [A: c > d] :
( ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) @ A )
= A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(40,axiom,
ord @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oord_1) ).
thf(371,plain,
ord @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(87,axiom,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A @ ( tp @ a @ b @ r @ B @ C ) )
= ( tp @ a @ b @ r @ B @ ( tp @ a @ b @ r @ A @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_rMulLC) ).
thf(503,plain,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A @ ( tp @ a @ b @ r @ B @ C ) )
= ( tp @ a @ b @ r @ B @ ( tp @ a @ b @ r @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(28,axiom,
! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_161_subset__self) ).
thf(343,plain,
! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(17,axiom,
semigroup_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Osemigroup__add) ).
thf(311,plain,
semigroup_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(251,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TF ) ),C: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TC @ TD ) ),D: TA > TB,E: TA,F: TA] :
( ( module @ TA @ TC @ TE @ TF @ A @ B )
=> ( ( module @ TB @ TC @ TD @ TF @ C @ B )
=> ( ( member @ ( TA > TB ) @ D @ ( mHom @ TC @ TF @ TA @ TE @ TB @ TD @ B @ A @ C ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ) @ A ) )
=> ( ( member @ TA @ F @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ) @ A ) )
=> ( ( D @ ( pop @ TA @ ( module_ext @ TA @ TC @ TE ) @ A @ E @ F ) )
= ( pop @ TB @ ( module_ext @ TB @ TC @ TD ) @ C @ ( D @ E ) @ ( D @ F ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_Module_OmHom__add) ).
thf(1054,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TF ) ),C: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TC @ TD ) ),D: TA > TB,E: TA,F: TA] :
( ( module @ TA @ TC @ TE @ TF @ A @ B )
=> ( ( module @ TB @ TC @ TD @ TF @ C @ B )
=> ( ( member @ ( TA > TB ) @ D @ ( mHom @ TC @ TF @ TA @ TE @ TB @ TD @ B @ A @ C ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ) @ A ) )
=> ( ( member @ TA @ F @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ) @ A ) )
=> ( ( D @ ( pop @ TA @ ( module_ext @ TA @ TC @ TE ) @ A @ E @ F ) )
= ( pop @ TB @ ( module_ext @ TB @ TC @ TD ) @ C @ ( D @ E ) @ ( D @ F ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).
thf(220,axiom,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA] :
( ( member @ TA @ A
@ ( collect @ TA
@ ^ [C: TA] : ( ord_less_eq @ TA @ C @ B ) ) )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_165_Nset__le) ).
thf(953,plain,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA] :
( ( member @ TA @ A
@ ( collect @ TA
@ ^ [C: TA] : ( ord_less_eq @ TA @ C @ B ) ) )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(262,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ B @ ( tp @ TA @ TB @ A @ C @ D ) )
= ( tp @ TA @ TB @ A @ C @ ( tp @ TA @ TB @ A @ B @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_106_Ring_OrMulLC) ).
thf(1086,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ B @ ( tp @ TA @ TB @ A @ C @ D ) )
= ( tp @ TA @ TB @ A @ C @ ( tp @ TA @ TB @ A @ B @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).
thf(214,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: nat] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( member @ TA @ ( nscal @ TA @ TB @ A @ B @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_Ring_OnsClose) ).
thf(938,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: nat] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( member @ TA @ ( nscal @ TA @ TB @ A @ B @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(50,axiom,
! [TA: $tType,TB: $tType] :
( ( ord @ TA )
=> ( ord @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oord) ).
thf(398,plain,
! [TA: $tType,TB: $tType] :
( ( ord @ TA )
=> ( ord @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(247,axiom,
! [A: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A @ ( un @ a @ b @ r ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_217_ring__r__one) ).
thf(1035,plain,
! [A: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A @ ( un @ a @ b @ r ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).
thf(173,axiom,
! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: set @ TB,D: TA > TB,E: TA] :
( ( member @ ( TA > TB ) @ A
@ ( pi @ TA @ TB @ B
@ ^ [F: TA] : C ) )
=> ( ( member @ ( TA > TB ) @ D
@ ( pi @ TA @ TB @ B
@ ^ [F: TA] : C ) )
=> ( ( A = D )
=> ( ( member @ TA @ E @ B )
=> ( ( A @ E )
= ( D @ E ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_eq__funcs) ).
thf(773,plain,
! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: set @ TB,D: TA > TB,E: TA] :
( ( member @ ( TA > TB ) @ A
@ ( pi @ TA @ TB @ B
@ ^ [F: TA] : C ) )
=> ( ( member @ ( TA > TB ) @ D
@ ( pi @ TA @ TB @ B
@ ^ [F: TA] : C ) )
=> ( ( A = D )
=> ( ( member @ TA @ E @ B )
=> ( ( A @ E )
= ( D @ E ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(176,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: TC > TA > TB,B: set @ TC,C: set @ TA,D: set @ TB,E: TC] :
( ( member @ ( TC > TA > TB ) @ A
@ ( pi @ TC @ ( TA > TB ) @ B
@ ^ [F: TC] :
( pi @ TA @ TB @ C
@ ^ [G: TA] : D ) ) )
=> ( ( member @ TC @ E @ B )
=> ( member @ ( TA > TB ) @ ( A @ E )
@ ( pi @ TA @ TB @ C
@ ^ [F: TA] : D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1_bivar__fun) ).
thf(794,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: TC > TA > TB,B: set @ TC,C: set @ TA,D: set @ TB,E: TC] :
( ( member @ ( TC > TA > TB ) @ A
@ ( pi @ TC @ ( TA > TB ) @ B
@ ^ [F: TC] :
( pi @ TA @ TB @ C
@ ^ [G: TA] : D ) ) )
=> ( ( member @ TC @ E @ B )
=> ( member @ ( TA > TB ) @ ( A @ E )
@ ( pi @ TA @ TB @ C
@ ^ [F: TA] : D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(57,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( aGroup @ TA @ ( ring_ext @ TA @ TB ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_Ring_Oring__is__ag) ).
thf(412,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( aGroup @ TA @ ( ring_ext @ TA @ TB ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(61,axiom,
member @ a @ ( un @ a @ b @ r ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_un__closed) ).
thf(425,plain,
member @ a @ ( un @ a @ b @ r ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(268,axiom,
! [A: a,B: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ B )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_pop__commute) ).
thf(1103,plain,
! [A: a,B: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ B )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).
thf(107,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB,C: TA > ( set @ TB )] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( member @ TB @ ( B @ D ) @ ( C @ D ) ) )
=> ( member @ ( TA > TB ) @ B @ ( pi @ TA @ TB @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_Pi__I) ).
thf(573,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB,C: TA > ( set @ TB )] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( member @ TB @ ( B @ D ) @ ( C @ D ) ) )
=> ( member @ ( TA > TB ) @ B @ ( pi @ TA @ TB @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(142,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TE ) ),B: set @ TA,C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TC @ TD ) ) ),D: TC,E: TA > TB] :
( ( ring @ TC @ TE @ A )
=> ( ! [F: TA] :
( ( member @ TA @ F @ B )
=> ( module @ TB @ TC @ TD @ TE @ ( C @ F ) @ A ) )
=> ( ( member @ TC @ D @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TE ) ) @ A ) )
=> ( ( member @ ( TA > TB ) @ E @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ TC @ TD ) @ B @ C ) )
=> ( member @ ( TA > TB ) @ ( algebr142713364_sprod @ TC @ TE @ TA @ TB @ TD @ A @ B @ C @ D @ E ) @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ TC @ TD ) @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26_Ring_OprodM__sprod__mem) ).
thf(689,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TE ) ),B: set @ TA,C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TC @ TD ) ) ),D: TC,E: TA > TB] :
( ( ring @ TC @ TE @ A )
=> ( ! [F: TA] :
( ( member @ TA @ F @ B )
=> ( module @ TB @ TC @ TD @ TE @ ( C @ F ) @ A ) )
=> ( ( member @ TC @ D @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TE ) ) @ A ) )
=> ( ( member @ ( TA > TB ) @ E @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ TC @ TD ) @ B @ C ) )
=> ( member @ ( TA > TB ) @ ( algebr142713364_sprod @ TC @ TE @ TA @ TB @ TD @ A @ B @ C @ D @ E ) @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ TC @ TD ) @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(159,axiom,
( member @ ( a > a > a ) @ ( pop @ a @ ( ring_ext @ a @ b ) @ r )
@ ( pi @ a @ ( a > a ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
@ ^ [A: a] :
( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
@ ^ [B: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_pop__closed) ).
thf(733,plain,
( member @ ( a > a > a ) @ ( pop @ a @ ( ring_ext @ a @ b ) @ r )
@ ( pi @ a @ ( a > a ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
@ ^ [A: a] :
( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
@ ^ [B: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(184,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: carrier_ext @ TE @ ( aGroup_ext @ TE @ ( ring_ext @ TE @ TC ) ),B: set @ TA,C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TE @ TD ) ) ),D: TA > TB,E: TA > TB] :
( ( ring @ TE @ TC @ A )
=> ( ! [F: TA] :
( ( member @ TA @ F @ B )
=> ( module @ TB @ TE @ TD @ TC @ ( C @ F ) @ A ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ TE @ product_unit ) ) @ ( algebr1000837334_prodM @ TE @ TC @ TA @ TB @ TD @ A @ B @ C ) ) )
=> ( ( member @ ( TA > TB ) @ E @ ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ TE @ product_unit ) ) @ ( algebr1000837334_prodM @ TE @ TC @ TA @ TB @ TD @ A @ B @ C ) ) )
=> ( ! [F: TA] :
( ( member @ TA @ F @ B )
=> ( ( D @ F )
= ( E @ F ) ) )
=> ( D = E ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_Ring_OprodM__mem__eq) ).
thf(819,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: carrier_ext @ TE @ ( aGroup_ext @ TE @ ( ring_ext @ TE @ TC ) ),B: set @ TA,C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TE @ TD ) ) ),D: TA > TB,E: TA > TB] :
( ( ring @ TE @ TC @ A )
=> ( ! [F: TA] :
( ( member @ TA @ F @ B )
=> ( module @ TB @ TE @ TD @ TC @ ( C @ F ) @ A ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ TE @ product_unit ) ) @ ( algebr1000837334_prodM @ TE @ TC @ TA @ TB @ TD @ A @ B @ C ) ) )
=> ( ( member @ ( TA > TB ) @ E @ ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ TE @ product_unit ) ) @ ( algebr1000837334_prodM @ TE @ TC @ TA @ TB @ TD @ A @ B @ C ) ) )
=> ( ! [F: TA] :
( ( member @ TA @ F @ B )
=> ( ( D @ F )
= ( E @ F ) ) )
=> ( D = E ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(163,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( aGroup @ TB @ TC @ ( B @ C ) ) )
=> ( member @ ( ( TA > TB ) > TA > TB ) @ ( prod_mOp @ TA @ TB @ TC @ A @ B )
@ ( pi @ ( TA > TB ) @ ( TA > TB ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B )
@ ^ [C: TA > TB] : ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_dsum__iOp__func) ).
thf(743,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( aGroup @ TB @ TC @ ( B @ C ) ) )
=> ( member @ ( ( TA > TB ) > TA > TB ) @ ( prod_mOp @ TA @ TB @ TC @ A @ B )
@ ( pi @ ( TA > TB ) @ ( TA > TB ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B )
@ ^ [C: TA > TB] : ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(150,axiom,
! [A: set @ a] :
( ( maximal_ideal @ a @ b @ r @ A )
=> ~ ( member @ a @ ( un @ a @ b @ r ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_211_maximal__ideal__proper) ).
thf(711,plain,
! [A: set @ a] :
( ( maximal_ideal @ a @ b @ r @ A )
=> ~ ( member @ a @ ( un @ a @ b @ r ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(103,axiom,
! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) )] :
( ( subring @ a @ b @ TA @ r @ A )
=> ( ( un @ a @ TA @ A )
= ( un @ a @ b @ r ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_218_Subring__one__ring__one) ).
thf(562,plain,
! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) )] :
( ( subring @ a @ b @ TA @ r @ A )
=> ( ( un @ a @ TA @ A )
= ( un @ a @ b @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(228,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ D )
= ( tp @ TA @ TB @ A @ B @ ( tp @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_104_Ring_Otp__assoc) ).
thf(975,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ D )
= ( tp @ TA @ TB @ A @ B @ ( tp @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).
thf(127,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( member @ ( TA > TA ) @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A )
@ ( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
@ ^ [B: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_181_Ring_Omop__closed) ).
thf(641,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( member @ ( TA > TA ) @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A )
@ ( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
@ ^ [B: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(67,axiom,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A @ B ) @ C )
= ( tp @ a @ b @ r @ A @ ( tp @ a @ b @ r @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_tp__assoc) ).
thf(445,plain,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A @ B ) @ C )
= ( tp @ a @ b @ r @ A @ ( tp @ a @ b @ r @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(269,axiom,
! [A: a,B: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( tp @ a @ b @ r @ A @ B ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_ring__tOp__closed) ).
thf(1107,plain,
! [A: a,B: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( tp @ a @ b @ r @ A @ B ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).
thf(74,axiom,
ring @ a @ b @ r,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0_Ring__axioms) ).
thf(466,plain,
ring @ a @ b @ r,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(209,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( aGroup @ TB @ TC @ ( B @ D ) ) )
=> ( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ( B @ D )
= ( C @ D ) ) )
=> ( ( prod_mOp @ TA @ TB @ TC @ A @ B )
= ( prod_mOp @ TA @ TB @ TC @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_prodag__sameTr3) ).
thf(919,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( aGroup @ TB @ TC @ ( B @ D ) ) )
=> ( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ( B @ D )
= ( C @ D ) ) )
=> ( ( prod_mOp @ TA @ TB @ TC @ A @ B )
= ( prod_mOp @ TA @ TB @ TC @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(147,axiom,
( ~ ( zeroring @ a @ b @ r )
=> ? [A: set @ a] : ( maximal_ideal @ a @ b @ r @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_id__maximal__Exist) ).
thf(704,plain,
( ~ ( zeroring @ a @ b @ r )
=> ? [A: set @ a] : ( maximal_ideal @ a @ b @ r @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(218,axiom,
( member @ ( a > a ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r )
@ ( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
@ ^ [A: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_146_mop__closed) ).
thf(947,plain,
( member @ ( a > a ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r )
@ ( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
@ ^ [A: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(151,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TE ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TF ) ),C: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TC ) ),D: TA,E: TA > TB,F: TD] :
( ( module @ TA @ TD @ TE @ TF @ A @ B )
=> ( ( module @ TB @ TD @ TC @ TF @ C @ B )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TE ) ) @ A ) )
=> ( ( member @ ( TA > TB ) @ E @ ( mHom @ TD @ TF @ TA @ TE @ TB @ TC @ B @ A @ C ) )
=> ( ( member @ TD @ F @ ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TF ) ) @ B ) )
=> ( ( E @ ( sprod @ TA @ TD @ TE @ A @ F @ D ) )
= ( sprod @ TB @ TD @ TC @ C @ F @ ( E @ D ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_Module_OmHom__lin) ).
thf(713,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TE ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TF ) ),C: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TC ) ),D: TA,E: TA > TB,F: TD] :
( ( module @ TA @ TD @ TE @ TF @ A @ B )
=> ( ( module @ TB @ TD @ TC @ TF @ C @ B )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TE ) ) @ A ) )
=> ( ( member @ ( TA > TB ) @ E @ ( mHom @ TD @ TF @ TA @ TE @ TB @ TC @ B @ A @ C ) )
=> ( ( member @ TD @ F @ ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TF ) ) @ B ) )
=> ( ( E @ ( sprod @ TA @ TD @ TE @ A @ F @ D ) )
= ( sprod @ TB @ TD @ TC @ C @ F @ ( E @ D ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(85,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: set @ TA,D: TA,E: set @ TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ C )
=> ( ( member @ TA @ D @ E )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( ord_less_eq @ ( set @ TA ) @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( member @ TA @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ D ) @ ( aset_sum @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ E ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_Ring_Oset__sum__mem) ).
thf(498,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: set @ TA,D: TA,E: set @ TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ C )
=> ( ( member @ TA @ D @ E )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( ord_less_eq @ ( set @ TA ) @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( member @ TA @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ D ) @ ( aset_sum @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ E ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(170,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( subring @ TA @ TB @ TC @ A @ B )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
=> ( ( mop @ TA @ ( ring_ext @ TA @ TC ) @ B @ C )
= ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_180_Ring_OSubring__minus__ring__minus) ).
thf(764,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( subring @ TA @ TB @ TC @ A @ B )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
=> ( ( mop @ TA @ ( ring_ext @ TA @ TC ) @ B @ C )
= ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(54,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( ring @ TA @ TB @ A )
=> ( ( sr2 @ TA @ TB @ A @ B )
=> ( ring @ TA @ TB @ ( sr @ TA @ TB @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_84_Ring_OSr__ring) ).
thf(406,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( ring @ TA @ TB @ A )
=> ( ( sr2 @ TA @ TB @ A @ B )
=> ( ring @ TA @ TB @ ( sr @ TA @ TB @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(69,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ ( un @ TA @ TB @ A ) @ B )
= B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_234_Ring_Org__l__unit) ).
thf(451,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ ( un @ TA @ TB @ A ) @ B )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(161,axiom,
! [A: set @ a] :
( ( sr2 @ a @ b @ r @ A )
=> ( ring @ a @ b @ ( sr @ a @ b @ r @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_Sr__ring) ).
thf(737,plain,
! [A: set @ a] :
( ( sr2 @ a @ b @ r @ A )
=> ( ring @ a @ b @ ( sr @ a @ b @ r @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(64,axiom,
! [A: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( un @ a @ b @ r ) @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_215_rg__l__unit) ).
thf(434,plain,
! [A: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( un @ a @ b @ r ) @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(14,axiom,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A = B )
=> ( ( ord_less_eq @ TA @ C @ A )
=> ( ord_less_eq @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_156_ge__convert) ).
thf(303,plain,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A = B )
=> ( ( ord_less_eq @ TA @ C @ A )
=> ( ord_less_eq @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(196,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA,E: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ ( tp @ TA @ TB @ A @ D @ E ) )
= ( tp @ TA @ TB @ A @ ( tp @ TA @ TB @ A @ B @ D ) @ ( tp @ TA @ TB @ A @ C @ E ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_Ring_Oring__tOp__rel) ).
thf(863,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA,E: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ ( tp @ TA @ TB @ A @ D @ E ) )
= ( tp @ TA @ TB @ A @ ( tp @ TA @ TB @ A @ B @ D ) @ ( tp @ TA @ TB @ A @ C @ E ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(46,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( ring @ TA @ TB @ A )
=> ( ( mul_closed_set @ TA @ TB @ A @ B )
=> ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_Ring_Omul__closed__set__sub) ).
thf(387,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( ring @ TA @ TB @ A )
=> ( ( mul_closed_set @ TA @ TB @ A @ B )
=> ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(116,axiom,
! [TA: $tType,TB: $tType] :
( ( mul_closed_set @ TA @ TB )
= ( ^ [A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
& ! [C: TA] :
( ( member @ TA @ C @ B )
=> ! [D: TA] :
( ( member @ TA @ D @ B )
=> ( member @ TA @ ( tp @ TA @ TB @ A @ C @ D ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_mul__closed__set__def) ).
thf(609,plain,
! [TA: $tType,TB: $tType] :
( ( mul_closed_set @ TA @ TB )
= ( ^ [A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
& ! [C: TA] :
( ( member @ TA @ C @ B )
=> ! [D: TA] :
( ( member @ TA @ D @ B )
=> ( member @ TA @ ( tp @ TA @ TB @ A @ C @ D ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(156,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( j_rad @ TA @ TB @ A ) )
=> ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_Ring_OJ__rad__mem) ).
thf(726,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( j_rad @ TA @ TB @ A ) )
=> ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(92,axiom,
! [A: a,B: a,C: a,D: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ D @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ B ) @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ C @ D ) )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ C ) @ ( tp @ a @ b @ r @ A @ D ) ) @ ( tp @ a @ b @ r @ B @ C ) ) @ ( tp @ a @ b @ r @ B @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_68_ring__distrib3) ).
thf(520,plain,
! [A: a,B: a,C: a,D: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ D @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ B ) @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ C @ D ) )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ C ) @ ( tp @ a @ b @ r @ A @ D ) ) @ ( tp @ a @ b @ r @ B @ C ) ) @ ( tp @ a @ b @ r @ B @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(207,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: TA > TC,B: set @ TA,C: TA > ( carrier_ext @ TC @ ( aGroup_ext @ TC @ TB ) ),D: TA] :
( ( member @ ( TA > TC ) @ A @ ( carr_prodag @ TA @ TC @ TB @ B @ C ) )
=> ( ( member @ TA @ D @ B )
=> ( member @ TC @ ( A @ D ) @ ( carrier @ TC @ ( aGroup_ext @ TC @ TB ) @ ( C @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_prodag__comp__i) ).
thf(915,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: TA > TC,B: set @ TA,C: TA > ( carrier_ext @ TC @ ( aGroup_ext @ TC @ TB ) ),D: TA] :
( ( member @ ( TA > TC ) @ A @ ( carr_prodag @ TA @ TC @ TB @ B @ C ) )
=> ( ( member @ TA @ D @ B )
=> ( member @ TC @ ( A @ D ) @ ( carrier @ TC @ ( aGroup_ext @ TC @ TB ) @ ( C @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(136,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
| ( ord_less_eq @ ( set @ TA ) @ B @ A ) )
=> ( ( member @ TA @ C @ A )
=> ( ~ ( member @ TA @ C @ B )
=> ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_158_sub__which1) ).
thf(668,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
| ( ord_less_eq @ ( set @ TA ) @ B @ A ) )
=> ( ( member @ TA @ C @ A )
=> ( ~ ( member @ TA @ C @ B )
=> ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(24,axiom,
! [TA: $tType] :
( ( plus @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A = B )
=> ( ( plus_plus @ TA @ C @ A )
= ( plus_plus @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_add__bothl) ).
thf(330,plain,
! [TA: $tType] :
( ( plus @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A = B )
=> ( ( plus_plus @ TA @ C @ A )
= ( plus_plus @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(29,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( A = B )
& ( ord_less_eq @ TA @ C @ D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_252_add__mono__thms__linordered__semiring_I2_J) ).
thf(345,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( A = B )
& ( ord_less_eq @ TA @ C @ D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(194,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB] :
( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( aGroup @ TB @ TC @ ( B @ E ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) )
=> ( member @ ( TA > TB ) @ ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_112_dsum__pOp__mem) ).
thf(858,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB] :
( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( aGroup @ TB @ TC @ ( B @ E ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) )
=> ( member @ ( TA > TB ) @ ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(37,axiom,
ord @ product_unit,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Ounit___Orderings_Oord_4) ).
thf(367,plain,
ord @ product_unit,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(101,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TD ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TE ) ),C: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TC @ TF ) ),D: TA > TB,E: TA > TB] :
( ( module @ TA @ TC @ TD @ TE @ A @ B )
=> ( ( module @ TB @ TC @ TF @ TE @ C @ B )
=> ( ( member @ ( TA > TB ) @ D @ ( mHom @ TC @ TE @ TA @ TD @ TB @ TF @ B @ A @ C ) )
=> ( ( member @ ( TA > TB ) @ E @ ( mHom @ TC @ TE @ TA @ TD @ TB @ TF @ B @ A @ C ) )
=> ( ! [F: TA] :
( ( member @ TA @ F @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TD ) ) @ A ) )
=> ( ( D @ F )
= ( E @ F ) ) )
=> ( D = E ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_Module_OmHom__eq) ).
thf(552,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TD ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TE ) ),C: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TC @ TF ) ),D: TA > TB,E: TA > TB] :
( ( module @ TA @ TC @ TD @ TE @ A @ B )
=> ( ( module @ TB @ TC @ TF @ TE @ C @ B )
=> ( ( member @ ( TA > TB ) @ D @ ( mHom @ TC @ TE @ TA @ TD @ TB @ TF @ B @ A @ C ) )
=> ( ( member @ ( TA > TB ) @ E @ ( mHom @ TC @ TE @ TA @ TD @ TB @ TF @ B @ A @ C ) )
=> ( ! [F: TA] :
( ( member @ TA @ F @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TD ) ) @ A ) )
=> ( ( D @ F )
= ( E @ F ) ) )
=> ( D = E ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(245,axiom,
! [A: a,B: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A @ B )
= ( tp @ a @ b @ r @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_tp__commute) ).
thf(1029,plain,
! [A: a,B: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A @ B )
= ( tp @ a @ b @ r @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(202,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( B = E )
=> ( ( C = D )
=> ( ( pop @ TA @ TB @ A @ B @ C )
= ( pop @ TA @ TB @ A @ D @ E ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_aGroup_OgEQAddcross) ).
thf(895,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( B = E )
=> ( ( C = D )
=> ( ( pop @ TA @ TB @ A @ B @ C )
= ( pop @ TA @ TB @ A @ D @ E ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).
thf(216,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( ring @ TA @ TB @ A )
=> ( ( sr2 @ TA @ TB @ A @ B )
=> ( member @ TA @ ( un @ TA @ TB @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_231_Ring_OSr__one) ).
thf(943,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( ring @ TA @ TB @ A )
=> ( ( sr2 @ TA @ TB @ A @ B )
=> ( member @ TA @ ( un @ TA @ TB @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(242,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( member @ ( TA > TA > TA ) @ ( tp @ TA @ TB @ A )
@ ( pi @ TA @ ( TA > TA ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
@ ^ [B: TA] :
( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
@ ^ [C: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_Ring_Otp__closed) ).
thf(1022,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( member @ ( TA > TA > TA ) @ ( tp @ TA @ TB @ A )
@ ( pi @ TA @ ( TA > TA ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
@ ^ [B: TA] :
( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
@ ^ [C: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).
thf(212,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ B @ C )
= ( tp @ TA @ TB @ A @ C @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_103_Ring_Otp__commute) ).
thf(932,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ B @ C )
= ( tp @ TA @ TB @ A @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(253,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ B @ C )
= ( tp @ TA @ TB @ A @ C @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_Ring_Oring__tOp__commute) ).
thf(1061,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ B @ C )
= ( tp @ TA @ TB @ A @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(206,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ B @ ( un @ TA @ TB @ A ) )
= B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_232_Ring_Oring__r__one) ).
thf(912,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ B @ ( un @ TA @ TB @ A ) )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(223,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C )
= ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_Ring_Opop__commute) ).
thf(959,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C )
= ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).
thf(144,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( aGroup @ TB @ TC @ ( B @ C ) ) )
=> ( member @ ( ( TA > TB ) > ( TA > TB ) > TA > TB ) @ ( prod_pOp @ TA @ TB @ TC @ A @ B )
@ ( pi @ ( TA > TB ) @ ( ( TA > TB ) > TA > TB ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B )
@ ^ [C: TA > TB] :
( pi @ ( TA > TB ) @ ( TA > TB ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B )
@ ^ [D: TA > TB] : ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_prod__pOp__func) ).
thf(695,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( aGroup @ TB @ TC @ ( B @ C ) ) )
=> ( member @ ( ( TA > TB ) > ( TA > TB ) > TA > TB ) @ ( prod_pOp @ TA @ TB @ TC @ A @ B )
@ ( pi @ ( TA > TB ) @ ( ( TA > TB ) > TA > TB ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B )
@ ^ [C: TA > TB] :
( pi @ ( TA > TB ) @ ( TA > TB ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B )
@ ^ [D: TA > TB] : ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(200,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( aGroup @ TB @ TC @ ( B @ D ) ) )
=> ( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ( B @ D )
= ( C @ D ) ) )
=> ( ( prod_pOp @ TA @ TB @ TC @ A @ B )
= ( prod_pOp @ TA @ TB @ TC @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_prodag__sameTr2) ).
thf(883,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( aGroup @ TB @ TC @ ( B @ D ) ) )
=> ( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ( B @ D )
= ( C @ D ) ) )
=> ( ( prod_pOp @ TA @ TB @ TC @ A @ B )
= ( prod_pOp @ TA @ TB @ TC @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(240,axiom,
! [A: a,B: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) )
= ( tp @ a @ b @ r @ A @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_150_ring__inv1__2) ).
thf(1015,plain,
! [A: a,B: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) )
= ( tp @ a @ b @ r @ A @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).
thf(271,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: TB,D: nat] :
( ( ring @ TB @ TA @ A )
=> ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( tp @ TB @ TA @ A @ B @ ( nscal @ TB @ TA @ A @ C @ D ) )
= ( nscal @ TB @ TA @ A @ ( tp @ TB @ TA @ A @ B @ C ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_Ring_OnsMulDistrL) ).
thf(1113,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: TB,D: nat] :
( ( ring @ TB @ TA @ A )
=> ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( tp @ TB @ TA @ A @ B @ ( nscal @ TB @ TA @ A @ C @ D ) )
= ( nscal @ TB @ TA @ A @ ( tp @ TB @ TA @ A @ B @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).
thf(62,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ ( pop @ TA @ TB @ A @ D @ E ) )
= ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ ( pop @ TA @ TB @ A @ C @ D ) ) @ E ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_87_aGroup_OpOp__assocTr43) ).
thf(426,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ ( pop @ TA @ TB @ A @ D @ E ) )
= ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ ( pop @ TA @ TB @ A @ C @ D ) ) @ E ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(139,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( ( pop @ TA @ TB @ A @ B @ D )
= ( pop @ TA @ TB @ A @ C @ D ) )
=> ( B = C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_aGroup_OpOp__cancel__r) ).
thf(680,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( ( pop @ TA @ TB @ A @ B @ D )
= ( pop @ TA @ TB @ A @ C @ D ) )
=> ( B = C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(42,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( plus_plus @ nat @ A @ C )
= ( plus_plus @ nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_add__both) ).
thf(374,plain,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( plus_plus @ nat @ A @ C )
= ( plus_plus @ nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(81,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( aGroup @ TB @ TC @ ( B @ C ) ) )
=> ( ord_less_eq @ ( set @ ( TA > TB ) ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_dsumag__sub__prodag) ).
thf(487,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( aGroup @ TB @ TC @ ( B @ C ) ) )
=> ( ord_less_eq @ ( set @ ( TA > TB ) ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(181,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( sr2 @ TA @ TB @ A @ B )
=> ( ( member @ TA @ C @ B )
=> ( member @ TA @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_Ring_OSr__mOp__closed) ).
thf(812,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( sr2 @ TA @ TB @ A @ B )
=> ( ( member @ TA @ C @ B )
=> ( member @ TA @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(119,axiom,
! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: TA > ( set @ TB ),D: TA] :
( ( member @ ( TA > TB ) @ A @ ( pi @ TA @ TB @ B @ C ) )
=> ( ~ ( member @ TB @ ( A @ D ) @ ( C @ D ) )
=> ~ ( member @ TA @ D @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_PiE) ).
thf(617,plain,
! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: TA > ( set @ TB ),D: TA] :
( ( member @ ( TA > TB ) @ A @ ( pi @ TA @ TB @ B @ C ) )
=> ( ~ ( member @ TB @ ( A @ D ) @ ( C @ D ) )
=> ~ ( member @ TA @ D @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(82,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( member @ nat @ A
@ ( collect @ nat
@ ^ [C: nat] : ( ord_less_eq @ nat @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_227_mem__of__Nset) ).
thf(490,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( member @ nat @ A
@ ( collect @ nat
@ ^ [C: nat] : ( ord_less_eq @ nat @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(138,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TE ) ),B: set @ TA,C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TC ) ) ),D: TD,E: TA > TB,F: TA] :
( ( ring @ TD @ TE @ A )
=> ( ! [G: TA] :
( ( member @ TA @ G @ B )
=> ( module @ TB @ TD @ TC @ TE @ ( C @ G ) @ A ) )
=> ( ( member @ TD @ D @ ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TE ) ) @ A ) )
=> ( ( member @ ( TA > TB ) @ E @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ TD @ TC ) @ B @ C ) )
=> ( ( member @ TA @ F @ B )
=> ( ( algebr142713364_sprod @ TD @ TE @ TA @ TB @ TC @ A @ B @ C @ D @ E @ F )
= ( sprod @ TB @ TD @ TC @ ( C @ F ) @ D @ ( E @ F ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_Ring_OprodM__sprod__val) ).
thf(675,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TE ) ),B: set @ TA,C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TC ) ) ),D: TD,E: TA > TB,F: TA] :
( ( ring @ TD @ TE @ A )
=> ( ! [G: TA] :
( ( member @ TA @ G @ B )
=> ( module @ TB @ TD @ TC @ TE @ ( C @ G ) @ A ) )
=> ( ( member @ TD @ D @ ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TE ) ) @ A ) )
=> ( ( member @ ( TA > TB ) @ E @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ TD @ TC ) @ B @ C ) )
=> ( ( member @ TA @ F @ B )
=> ( ( algebr142713364_sprod @ TD @ TE @ TA @ TB @ TC @ A @ B @ C @ D @ E @ F )
= ( sprod @ TB @ TD @ TC @ ( C @ F ) @ D @ ( E @ F ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(260,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: TA > TC,B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ a @ TD ) ),C: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ a @ TB ) )] :
( ( member @ ( TA > TC ) @ A @ ( mHom @ a @ b @ TA @ TD @ TC @ TB @ r @ B @ C ) )
=> ( member @ ( TA > TC ) @ A
@ ( pi @ TA @ TC @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ a @ TD ) ) @ B )
@ ^ [D: TA] : ( carrier @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ a @ TB ) ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_mHom__func) ).
thf(1081,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: TA > TC,B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ a @ TD ) ),C: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ a @ TB ) )] :
( ( member @ ( TA > TC ) @ A @ ( mHom @ a @ b @ TA @ TD @ TC @ TB @ r @ B @ C ) )
=> ( member @ ( TA > TC ) @ A
@ ( pi @ TA @ TC @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ a @ TD ) ) @ B )
@ ^ [D: TA] : ( carrier @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ a @ TB ) ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).
thf(165,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TB @ TF ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TE ) ),C: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),D: TD > TA,E: TD] :
( ( module @ TD @ TB @ TF @ TE @ A @ B )
=> ( ( module @ TA @ TB @ TC @ TE @ C @ B )
=> ( ( member @ ( TD > TA ) @ D @ ( mHom @ TB @ TE @ TD @ TF @ TA @ TC @ B @ A @ C ) )
=> ( ( member @ TD @ E @ ( carrier @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TB @ TF ) ) @ A ) )
=> ( member @ TA @ ( D @ E ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_141_Module_OmHom__mem) ).
thf(749,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TB @ TF ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TE ) ),C: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),D: TD > TA,E: TD] :
( ( module @ TD @ TB @ TF @ TE @ A @ B )
=> ( ( module @ TA @ TB @ TC @ TE @ C @ B )
=> ( ( member @ ( TD > TA ) @ D @ ( mHom @ TB @ TE @ TD @ TF @ TA @ TC @ B @ A @ C ) )
=> ( ( member @ TD @ E @ ( carrier @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TB @ TF ) ) @ A ) )
=> ( member @ TA @ ( D @ E ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(102,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: set @ TA,C: set @ TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( aset_sum @ TA @ TB @ A @ B @ C ) )
=> ? [E: TA] :
( ( member @ TA @ E @ B )
& ? [F: TA] :
( ( member @ TA @ F @ C )
& ( D
= ( pop @ TA @ TB @ A @ E @ F ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_aGroup_Omem__set__sum) ).
thf(557,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: set @ TA,C: set @ TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( aset_sum @ TA @ TB @ A @ B @ C ) )
=> ? [E: TA] :
( ( member @ TA @ E @ B )
& ? [F: TA] :
( ( member @ TA @ F @ C )
& ( D
= ( pop @ TA @ TB @ A @ E @ F ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(193,axiom,
! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ),B: a,C: a] :
( ( subring @ a @ b @ TA @ r @ A )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
=> ( ( pop @ a @ ( ring_ext @ a @ TA ) @ A @ B @ C )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_Subring__pOp__ring__pOp) ).
thf(855,plain,
! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ),B: a,C: a] :
( ( subring @ a @ b @ TA @ r @ A )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
=> ( ( pop @ a @ ( ring_ext @ a @ TA ) @ A @ B @ C )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).
thf(256,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ ( pop @ TA @ TB @ A @ D @ E ) )
= ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ D ) @ ( pop @ TA @ TB @ A @ C @ E ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_97_aGroup_Oag__add4__rel) ).
thf(1070,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ ( pop @ TA @ TB @ A @ D @ E ) )
= ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ D ) @ ( pop @ TA @ TB @ A @ C @ E ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).
thf(58,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ B @ C )
= ( pop @ TA @ TB @ A @ C @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_86_aGroup_Oag__add__commute) ).
thf(414,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ B @ C )
= ( pop @ TA @ TB @ A @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(23,axiom,
! [TA: $tType] :
( ( semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
= ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_243_add_Oassoc) ).
thf(327,plain,
! [TA: $tType] :
( ( semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
= ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(162,axiom,
! [A: a,B: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A @ B )
= ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_151_ring__inv1__3) ).
thf(739,plain,
! [A: a,B: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A @ B )
= ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(56,axiom,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
= ( A = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_213_add__right__cancel) ).
thf(409,plain,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
= ( A = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(16,axiom,
plus @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oplus) ).
thf(310,plain,
plus @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(73,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: TB,D: nat] :
( ( ring @ TB @ TA @ A )
=> ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( pop @ TB @ ( ring_ext @ TB @ TA ) @ A @ ( nscal @ TB @ TA @ A @ B @ D ) @ ( nscal @ TB @ TA @ A @ C @ D ) )
= ( nscal @ TB @ TA @ A @ ( pop @ TB @ ( ring_ext @ TB @ TA ) @ A @ B @ C ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_83_Ring_OnsDistrL) ).
thf(463,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: TB,D: nat] :
( ( ring @ TB @ TA @ A )
=> ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
=> ( ( pop @ TB @ ( ring_ext @ TB @ TA ) @ A @ ( nscal @ TB @ TA @ A @ B @ D ) @ ( nscal @ TB @ TA @ A @ C @ D ) )
= ( nscal @ TB @ TA @ A @ ( pop @ TB @ ( ring_ext @ TB @ TA ) @ A @ B @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(128,axiom,
! [A: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A )
= ( tp @ a @ b @ r @ A @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_ring__times__minusr) ).
thf(643,plain,
! [A: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A )
= ( tp @ a @ b @ r @ A @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(248,axiom,
! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: set @ TB,D: TA > TB] :
( ( member @ ( TA > TB ) @ A
@ ( pi @ TA @ TB @ B
@ ^ [E: TA] : C ) )
=> ( ( A = D )
=> ( member @ ( TA > TB ) @ D
@ ( pi @ TA @ TB @ B
@ ^ [E: TA] : C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_eq__fun) ).
thf(1038,plain,
! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: set @ TB,D: TA > TB] :
( ( member @ ( TA > TB ) @ A
@ ( pi @ TA @ TB @ B
@ ^ [E: TA] : C ) )
=> ( ( A = D )
=> ( member @ ( TA > TB ) @ D
@ ( pi @ TA @ TB @ B
@ ^ [E: TA] : C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).
thf(39,axiom,
ab_semigroup_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oab__semigroup__add) ).
thf(370,plain,
ab_semigroup_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(76,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB,E: TA] :
( ! [F: TA] :
( ( member @ TA @ F @ A )
=> ( aGroup @ TB @ TC @ ( B @ F ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ TA @ E @ A )
=> ( ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D @ E )
= ( pop @ TB @ TC @ ( B @ E ) @ ( C @ E ) @ ( D @ E ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_111_prod__pOp__mem__i) ).
thf(470,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB,E: TA] :
( ! [F: TA] :
( ( member @ TA @ F @ A )
=> ( aGroup @ TB @ TC @ ( B @ F ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ TA @ E @ A )
=> ( ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D @ E )
= ( pop @ TB @ TC @ ( B @ E ) @ ( C @ E ) @ ( D @ E ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(108,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB] :
( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( aGroup @ TB @ TC @ ( B @ E ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D )
= ( prod_pOp @ TA @ TB @ TC @ A @ B @ D @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_prod__pOp__commute) ).
thf(576,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB] :
( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( aGroup @ TB @ TC @ ( B @ E ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D )
= ( prod_pOp @ TA @ TB @ TC @ A @ B @ D @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(132,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ D )
= ( pop @ TA @ TB @ A @ B @ ( pop @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_aGroup_Oag__pOp__assoc) ).
thf(656,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ D )
= ( pop @ TA @ TB @ A @ B @ ( pop @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(238,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( B = C )
=> ( ( pop @ TA @ TB @ A @ B @ D )
= ( pop @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_aGroup_Oag__pOp__add__r) ).
thf(1008,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( B = C )
=> ( ( pop @ TA @ TB @ A @ B @ D )
= ( pop @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).
thf(117,axiom,
! [TA: $tType,TB: $tType,A: TB > TA,B: set @ TB,C: set @ TA,D: TB] :
( ( member @ ( TB > TA ) @ A
@ ( pi @ TB @ TA @ B
@ ^ [E: TB] : C ) )
=> ( ( member @ TB @ D @ B )
=> ( member @ TA @ ( A @ D ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_Algebra1_Ofuncset__mem) ).
thf(612,plain,
! [TA: $tType,TB: $tType,A: TB > TA,B: set @ TB,C: set @ TA,D: TB] :
( ( member @ ( TB > TA ) @ A
@ ( pi @ TB @ TA @ B
@ ^ [E: TB] : C ) )
=> ( ( member @ TB @ D @ B )
=> ( member @ TA @ ( A @ D ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(265,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ~ ( zeroring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ~ ( unit @ TA @ TB @ A @ B )
=> ? [C: set @ TA] :
( ( maximal_ideal @ TA @ TB @ A @ C )
& ( member @ TA @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_Ring_Ononunit__contained__maxid) ).
thf(1094,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ~ ( zeroring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ~ ( unit @ TA @ TB @ A @ B )
=> ? [C: set @ TA] :
( ( maximal_ideal @ TA @ TB @ A @ C )
& ( member @ TA @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).
thf(229,axiom,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) @ A )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ B @ A ) @ ( tp @ a @ b @ r @ C @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_67_ring__distrib2) ).
thf(978,plain,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) @ A )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ B @ A ) @ ( tp @ a @ b @ r @ C @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).
thf(35,axiom,
! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_224_self__le) ).
thf(363,plain,
! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(97,axiom,
! [A: a,B: nat,C: nat] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( nscal @ a @ b @ r @ A @ B ) @ ( nscal @ a @ b @ r @ A @ C ) )
= ( nscal @ a @ b @ r @ A @ ( plus_plus @ nat @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_nsDistr) ).
thf(539,plain,
! [A: a,B: nat,C: nat] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( nscal @ a @ b @ r @ A @ B ) @ ( nscal @ a @ b @ r @ A @ C ) )
= ( nscal @ a @ b @ r @ A @ ( plus_plus @ nat @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(79,axiom,
( ( zeroring @ a @ b @ r )
=> ~ ? [A: set @ a] : ( maximal_ideal @ a @ b @ r @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_144_zeroring__no__maximal) ).
thf(482,plain,
( ( zeroring @ a @ b @ r )
=> ~ ? [A: set @ a] : ( maximal_ideal @ a @ b @ r @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(137,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( B = C )
=> ( ( pop @ TA @ TB @ A @ D @ B )
= ( pop @ TA @ TB @ A @ D @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_aGroup_Oag__pOp__add__l) ).
thf(671,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( B = C )
=> ( ( pop @ TA @ TB @ A @ D @ B )
= ( pop @ TA @ TB @ A @ D @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(141,axiom,
( member @ ( a > a > a ) @ ( tp @ a @ b @ r )
@ ( pi @ a @ ( a > a ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
@ ^ [A: a] :
( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
@ ^ [B: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_tp__closed) ).
thf(688,plain,
( member @ ( a > a > a ) @ ( tp @ a @ b @ r )
@ ( pi @ a @ ( a > a ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
@ ^ [A: a] :
( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
@ ^ [B: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(84,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(495,plain,
! [TA: $tType,A: set @ TA] :
( ( collect @ TA
@ ^ [B: TA] : ( member @ TA @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(93,axiom,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) @ ( tp @ a @ b @ r @ A @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_rg__distrib) ).
thf(524,plain,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) @ ( tp @ a @ b @ r @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(109,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB )] :
( ( aGroup @ TA @ TB @ A )
=> ( member @ ( TA > TA ) @ ( mop @ TA @ TB @ A )
@ ( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A )
@ ^ [B: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_aGroup_Omop__closed) ).
thf(581,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB )] :
( ( aGroup @ TA @ TB @ A )
=> ( member @ ( TA > TA ) @ ( mop @ TA @ TB @ A )
@ ( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A )
@ ^ [B: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(38,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TA @ TD ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( module @ TC @ TA @ TD @ TB @ A @ B )
=> ( ring @ TA @ TB @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_Module_Osc__Ring) ).
thf(368,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TA @ TD ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( module @ TC @ TA @ TD @ TB @ A @ B )
=> ( ring @ TA @ TB @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(189,axiom,
! [A: nat] :
( member @ nat @ A
@ ( collect @ nat
@ ^ [B: nat] : ( ord_less_eq @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_226_n__in__Nsetn) ).
thf(844,plain,
! [A: nat] :
( member @ nat @ A
@ ( collect @ nat
@ ^ [B: nat] : ( ord_less_eq @ nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(91,axiom,
! [TA: $tType,TB: $tType] :
( ( sr2 @ TA @ TB )
= ( ^ [A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
& ( member @ TA @ ( un @ TA @ TB @ A ) @ B )
& ! [C: TA] :
( ( member @ TA @ C @ B )
=> ! [D: TA] :
( ( member @ TA @ D @ B )
=> ( ( member @ TA @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ D ) ) @ B )
& ( member @ TA @ ( tp @ TA @ TB @ A @ C @ D ) @ B ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_250_sr__def) ).
thf(517,plain,
! [TA: $tType,TB: $tType] :
( ( sr2 @ TA @ TB )
= ( ^ [A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
& ( member @ TA @ ( un @ TA @ TB @ A ) @ B )
& ! [C: TA] :
( ( member @ TA @ C @ B )
=> ! [D: TA] :
( ( member @ TA @ D @ B )
=> ( ( member @ TA @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ D ) ) @ B )
& ( member @ TA @ ( tp @ TA @ TB @ A @ C @ D ) @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(20,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
& ( ord_less_eq @ TA @ C @ D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_253_add__mono__thms__linordered__semiring_I1_J) ).
thf(319,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
& ( ord_less_eq @ TA @ C @ D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(124,axiom,
! [A: a] :
( ~ ( zeroring @ a @ b @ r )
=> ( ( member @ a @ A @ ( j_rad @ a @ b @ r ) )
=> ! [B: a] :
( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( unit @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) @ ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_J__rad__unit) ).
thf(632,plain,
! [A: a] :
( ~ ( zeroring @ a @ b @ r )
=> ( ( member @ a @ A @ ( j_rad @ a @ b @ r ) )
=> ! [B: a] :
( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( unit @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) @ ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(52,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ C @ A ) @ ( plus_plus @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_255_add__left__mono) ).
thf(401,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ C @ A ) @ ( plus_plus @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(158,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( ring @ TA @ TB @ A )
=> ( ( maximal_ideal @ TA @ TB @ A @ B )
=> ~ ( member @ TA @ ( un @ TA @ TB @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_229_Ring_Omaximal__ideal__proper) ).
thf(731,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
( ( ring @ TA @ TB @ A )
=> ( ( maximal_ideal @ TA @ TB @ A @ B )
=> ~ ( member @ TA @ ( un @ TA @ TB @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(234,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( mop @ TA @ TB @ A @ ( mop @ TA @ TB @ A @ B ) )
= B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_aGroup_Oag__inv__inv) ).
thf(995,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( mop @ TA @ TB @ A @ ( mop @ TA @ TB @ A @ B ) )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).
thf(105,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( member @ TA @ ( mop @ TA @ TB @ A @ B ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_aGroup_Oag__mOp__closed) ).
thf(568,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( member @ TA @ ( mop @ TA @ TB @ A @ B ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(8,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_254_add__mono) ).
thf(290,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(89,axiom,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ B ) @ C )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_pop__aassoc) ).
thf(510,plain,
! [A: a,B: a,C: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ B ) @ C )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(78,axiom,
! [A: a,B: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) )
= ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_149_ring__inv1__1) ).
thf(478,plain,
! [A: a,B: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) )
= ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(1,conjecture,
! [A: c > d] :
( ~ ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
| ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_2) ).
thf(2,negated_conjecture,
~ ! [A: c > d] :
( ~ ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
| ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
= A ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(272,plain,
~ ! [A: c > d] :
( ~ ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
| ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(203,axiom,
! [A: a,B: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) )
= ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A ) @ B ) )
& ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) )
= ( tp @ a @ b @ r @ A @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_148_ring__inv1) ).
thf(899,plain,
! [A: a,B: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) )
= ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A ) @ B ) )
& ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) )
= ( tp @ a @ b @ r @ A @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(100,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( aGroup @ TB @ TC @ ( B @ D ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( member @ ( TA > TB ) @ ( prod_mOp @ TA @ TB @ TC @ A @ B @ C ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_prod__mOp__mem) ).
thf(549,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( aGroup @ TB @ TC @ ( B @ D ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( member @ ( TA > TB ) @ ( prod_mOp @ TA @ TB @ TC @ A @ B @ C ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(152,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TB ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ),C: TD,D: TA] :
( ( module @ TA @ TD @ TB @ TC @ A @ B )
=> ( ( member @ TD @ C @ ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) @ B ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TB ) ) @ A ) )
=> ( ( mop @ TA @ ( module_ext @ TA @ TD @ TB ) @ A @ ( sprod @ TA @ TD @ TB @ A @ C @ D ) )
= ( sprod @ TA @ TD @ TB @ A @ ( mop @ TD @ ( ring_ext @ TD @ TC ) @ B @ C ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_Module_Osc__minus__am1) ).
thf(716,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TB ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ),C: TD,D: TA] :
( ( module @ TA @ TD @ TB @ TC @ A @ B )
=> ( ( member @ TD @ C @ ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) @ B ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TB ) ) @ A ) )
=> ( ( mop @ TA @ ( module_ext @ TA @ TD @ TB ) @ A @ ( sprod @ TA @ TD @ TB @ A @ C @ D ) )
= ( sprod @ TA @ TD @ TB @ A @ ( mop @ TD @ ( ring_ext @ TD @ TC ) @ B @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(179,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ a @ TC ) ) ),C: TA > TB,D: TA > TB] :
( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( module @ TB @ a @ TC @ b @ ( B @ E ) @ r ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ TA @ TB @ TC @ r @ A @ B ) ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ TA @ TB @ TC @ r @ A @ B ) ) )
=> ( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( ( C @ E )
= ( D @ E ) ) )
=> ( C = D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_prodM__mem__eq) ).
thf(800,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ a @ TC ) ) ),C: TA > TB,D: TA > TB] :
( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( module @ TB @ a @ TC @ b @ ( B @ E ) @ r ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ TA @ TB @ TC @ r @ A @ B ) ) )
=> ( ( member @ ( TA > TB ) @ D @ ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ TA @ TB @ TC @ r @ A @ B ) ) )
=> ( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( ( C @ E )
= ( D @ E ) ) )
=> ( C = D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(13,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(300,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)],[13]) ).
thf(225,axiom,
! [A: set @ a,B: a,C: a] :
( ( mul_closed_set @ a @ b @ r @ A )
=> ( ( member @ a @ B @ A )
=> ( ( member @ a @ C @ A )
=> ( member @ a @ ( tp @ a @ b @ r @ B @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_74_mul__closed__set__tOp__closed) ).
thf(967,plain,
! [A: set @ a,B: a,C: a] :
( ( mul_closed_set @ a @ b @ r @ A )
=> ( ( member @ a @ B @ A )
=> ( ( member @ a @ C @ A )
=> ( member @ a @ ( tp @ a @ b @ r @ B @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(266,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ),C: TC,D: TA,E: TA] :
( ( module @ TA @ TC @ TB @ TD @ A @ B )
=> ( ( member @ TC @ C @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
=> ( ( sprod @ TA @ TC @ TB @ A @ C @ ( pop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ D @ E ) )
= ( pop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ ( sprod @ TA @ TC @ TB @ A @ C @ D ) @ ( sprod @ TA @ TC @ TB @ A @ C @ E ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_Module_Osprod__r__distr) ).
thf(1097,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ),C: TC,D: TA,E: TA] :
( ( module @ TA @ TC @ TB @ TD @ A @ B )
=> ( ( member @ TC @ C @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
=> ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
=> ( ( sprod @ TA @ TC @ TB @ A @ C @ ( pop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ D @ E ) )
= ( pop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ ( sprod @ TA @ TC @ TB @ A @ C @ D ) @ ( sprod @ TA @ TC @ TB @ A @ C @ E ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).
thf(187,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( ( pop @ TA @ TB @ A @ C @ B )
= D )
=> ( C
= ( pop @ TA @ TB @ A @ D @ ( mop @ TA @ TB @ A @ B ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_184_aGroup_Oag__eq__sol2) ).
thf(835,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( ( pop @ TA @ TB @ A @ C @ B )
= D )
=> ( C
= ( pop @ TA @ TB @ A @ D @ ( mop @ TA @ TB @ A @ B ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(11,axiom,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
=> ( A = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_237_add__right__imp__eq) ).
thf(294,plain,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
=> ( A = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(33,axiom,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ( ( plus_plus @ TA )
= ( ^ [A: TA,B: TA] : ( plus_plus @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_240_add_Ocommute) ).
thf(358,plain,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ( ( plus_plus @ TA )
= ( ^ [A: TA,B: TA] : ( plus_plus @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(65,axiom,
! [TA: $tType,TB: $tType] :
( ( unit @ TA @ TB )
= ( ^ [A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
& ? [C: TA] :
( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
& ( ( tp @ TA @ TB @ A @ B @ C )
= ( un @ TA @ TB @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_236_Unit__def) ).
thf(437,plain,
! [TA: $tType,TB: $tType] :
( ( unit @ TA @ TB )
= ( ^ [A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
& ? [C: TA] :
( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
& ( ( tp @ TA @ TB @ A @ B @ C )
= ( un @ TA @ TB @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(171,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( aGroup @ TB @ TC @ ( B @ D ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) )
=> ( member @ ( TA > TB ) @ ( prod_mOp @ TA @ TB @ TC @ A @ B @ C ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_dsum__iOp__mem) ).
thf(767,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( aGroup @ TB @ TC @ ( B @ D ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) )
=> ( member @ ( TA > TB ) @ ( prod_mOp @ TA @ TB @ TC @ A @ B @ C ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(135,axiom,
! [A: a,B: a,C: nat] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( nscal @ a @ b @ r @ B @ C ) @ A )
= ( nscal @ a @ b @ r @ ( tp @ a @ b @ r @ B @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_nsMulDistrR) ).
thf(664,plain,
! [A: a,B: a,C: nat] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( nscal @ a @ b @ r @ B @ C ) @ A )
= ( nscal @ a @ b @ r @ ( tp @ a @ b @ r @ B @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(185,axiom,
! [A: c] :
( ( member @ c @ A @ i )
=> ( module @ d @ a @ e @ b @ ( m @ A ) @ r ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).
thf(828,plain,
! [A: c] :
( ( member @ c @ A @ i )
=> ( module @ d @ a @ e @ b @ ( m @ A ) @ r ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(125,axiom,
! [A: a,B: a,C: a,D: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ D @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A @ B ) @ ( tp @ a @ b @ r @ C @ D ) )
= ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A @ C ) @ ( tp @ a @ b @ r @ B @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_ring__tOp__rel) ).
thf(635,plain,
! [A: a,B: a,C: a,D: a] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ D @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A @ B ) @ ( tp @ a @ b @ r @ C @ D ) )
= ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A @ C ) @ ( tp @ a @ b @ r @ B @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(221,axiom,
aGroup @ a @ ( ring_ext @ a @ b ) @ r,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_ring__is__ag) ).
thf(955,plain,
aGroup @ a @ ( ring_ext @ a @ b ) @ r,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).
thf(177,axiom,
! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: TA > ( set @ TB ),D: TA] :
( ( member @ ( TA > TB ) @ A @ ( pi @ TA @ TB @ B @ C ) )
=> ( ( member @ TA @ D @ B )
=> ( member @ TB @ ( A @ D ) @ ( C @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_Pi__mem) ).
thf(796,plain,
! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: TA > ( set @ TB ),D: TA] :
( ( member @ ( TA > TB ) @ A @ ( pi @ TA @ TB @ B @ C ) )
=> ( ( member @ TA @ D @ B )
=> ( member @ TB @ ( A @ D ) @ ( C @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(210,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: TB,D: TA] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ) @ B ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) )
=> ( member @ TA @ ( sprod @ TA @ TB @ TC @ A @ C @ D ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_127_Module_Osprod__closed) ).
thf(928,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: TB,D: TA] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ) @ B ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) )
=> ( member @ TA @ ( sprod @ TA @ TB @ TC @ A @ C @ D ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).
thf(246,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( subring @ TA @ TB @ TC @ A @ B )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
=> ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_142_Ring_Omem__subring__mem__ring) ).
thf(1033,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),C: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( subring @ TA @ TB @ TC @ A @ B )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
=> ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).
thf(51,axiom,
comm_monoid_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__add) ).
thf(400,plain,
comm_monoid_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(188,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA] :
( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( aGroup @ TB @ TC @ ( B @ E ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ TA @ D @ A )
=> ( ( prod_mOp @ TA @ TB @ TC @ A @ B @ C @ D )
= ( mop @ TB @ TC @ ( B @ D ) @ ( C @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_prod__mOp__mem__i) ).
thf(839,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA] :
( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( aGroup @ TB @ TC @ ( B @ E ) ) )
=> ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
=> ( ( member @ TA @ D @ A )
=> ( ( prod_mOp @ TA @ TB @ TC @ A @ B @ C @ D )
= ( mop @ TB @ TC @ ( B @ D ) @ ( C @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(157,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ D )
= ( tp @ TA @ TB @ A @ B @ ( tp @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_101_Ring_Oring__tOp__assoc) ).
thf(728,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( tp @ TA @ TB @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ D )
= ( tp @ TA @ TB @ A @ B @ ( tp @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(241,axiom,
! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ),B: a] :
( ( subring @ a @ b @ TA @ r @ A )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
=> ( ( mop @ a @ ( ring_ext @ a @ TA ) @ A @ B )
= ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_Subring__minus__ring__minus) ).
thf(1019,plain,
! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ),B: a] :
( ( subring @ a @ b @ TA @ r @ A )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
=> ( ( mop @ a @ ( ring_ext @ a @ TA ) @ A @ B )
= ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).
thf(94,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TE ) ),B: set @ TA,C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TC @ TD ) ) )] :
( ( ring @ TC @ TE @ A )
=> ( ! [D: TA] :
( ( member @ TA @ D @ B )
=> ( module @ TB @ TC @ TD @ TE @ ( C @ D ) @ A ) )
=> ( ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ TC @ product_unit ) ) @ ( algebr1000837334_prodM @ TC @ TE @ TA @ TB @ TD @ A @ B @ C ) )
= ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ TC @ TD ) @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_Ring_OprodM__carr) ).
thf(528,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TE ) ),B: set @ TA,C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TC @ TD ) ) )] :
( ( ring @ TC @ TE @ A )
=> ( ! [D: TA] :
( ( member @ TA @ D @ B )
=> ( module @ TB @ TC @ TD @ TE @ ( C @ D ) @ A ) )
=> ( ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ TC @ product_unit ) ) @ ( algebr1000837334_prodM @ TC @ TE @ TA @ TB @ TD @ A @ B @ C ) )
= ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ TC @ TD ) @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(233,axiom,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_mem__Collect__eq) ).
thf(992,plain,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).
thf(191,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( member @ TA @ ( un @ TA @ TB @ A ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_228_Ring_Oun__closed) ).
thf(850,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( ring @ TA @ TB @ A )
=> ( member @ TA @ ( un @ TA @ TB @ A ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(230,axiom,
! [A: a] :
( ( member @ a @ A @ ( j_rad @ a @ b @ r ) )
=> ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_J__rad__mem) ).
thf(982,plain,
! [A: a] :
( ( member @ a @ A @ ( j_rad @ a @ b @ r ) )
=> ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).
thf(237,axiom,
! [A: a,B: a,C: nat] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A @ ( nscal @ a @ b @ r @ B @ C ) )
= ( nscal @ a @ b @ r @ ( tp @ a @ b @ r @ A @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_nsMulDistrL) ).
thf(1004,plain,
! [A: a,B: a,C: nat] :
( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A @ ( nscal @ a @ b @ r @ B @ C ) )
= ( nscal @ a @ b @ r @ ( tp @ a @ b @ r @ A @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).
thf(26,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) )] :
( ( ring @ TA @ TB @ A )
=> ( ( subring @ TA @ TB @ TC @ A @ B )
=> ( ( un @ TA @ TC @ B )
= ( un @ TA @ TB @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_230_Ring_OSubring__one__ring__one) ).
thf(337,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) )] :
( ( ring @ TA @ TB @ A )
=> ( ( subring @ TA @ TB @ TC @ A @ B )
=> ( ( un @ TA @ TC @ B )
= ( un @ TA @ TB @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(215,axiom,
! [A: set @ a,B: a,C: a] :
( ( sr2 @ a @ b @ r @ A )
=> ( ( member @ a @ B @ A )
=> ( ( member @ a @ C @ A )
=> ( member @ a @ ( tp @ a @ b @ r @ B @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_Sr__tOp__closed) ).
thf(940,plain,
! [A: set @ a,B: a,C: a] :
( ( sr2 @ a @ b @ r @ A )
=> ( ( member @ a @ B @ A )
=> ( ( member @ a @ C @ A )
=> ( member @ a @ ( tp @ a @ b @ r @ B @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(213,axiom,
! [A: set @ a,B: a,C: a] :
( ( sr2 @ a @ b @ r @ A )
=> ( ( member @ a @ B @ A )
=> ( ( member @ a @ C @ A )
=> ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_Sr__pOp__closed) ).
thf(935,plain,
! [A: set @ a,B: a,C: a] :
( ( sr2 @ a @ b @ r @ A )
=> ( ( member @ a @ B @ A )
=> ( ( member @ a @ C @ A )
=> ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(133,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( member @ TA @ ( pop @ TA @ TB @ A @ B @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_aGroup_Oag__pOp__closed) ).
thf(659,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
( ( aGroup @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
=> ( member @ TA @ ( pop @ TA @ TB @ A @ B @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(98,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B )
= ( tp @ TA @ TB @ A @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( un @ TA @ TB @ A ) ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_248_Ring_Oring__times__minusl) ).
thf(543,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
( ( ring @ TA @ TB @ A )
=> ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
=> ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B )
= ( tp @ TA @ TB @ A @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( un @ TA @ TB @ A ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(126,axiom,
! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: set @ TB,D: set @ TB] :
( ( member @ ( TA > TB ) @ A
@ ( pi @ TA @ TB @ B
@ ^ [E: TA] : C ) )
=> ( ( ord_less_eq @ ( set @ TB ) @ C @ D )
=> ( member @ ( TA > TB ) @ A
@ ( pi @ TA @ TB @ B
@ ^ [E: TA] : D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_168_extend__fun) ).
thf(639,plain,
! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: set @ TB,D: set @ TB] :
( ( member @ ( TA > TB ) @ A
@ ( pi @ TA @ TB @ B
@ ^ [E: TA] : C ) )
=> ( ( ord_less_eq @ ( set @ TB ) @ C @ D )
=> ( member @ ( TA > TB ) @ A
@ ( pi @ TA @ TB @ B
@ ^ [E: TA] : D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(70,axiom,
! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: TA > ( set @ TB )] :
( ( member @ ( TA > TB ) @ A @ ( pi @ TA @ TB @ B @ C ) )
= ( ! [D: TA] :
( ( member @ TA @ D @ B )
=> ( member @ TB @ ( A @ D ) @ ( C @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_Pi__iff) ).
thf(454,plain,
! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: TA > ( set @ TB )] :
( ( member @ ( TA > TB ) @ A @ ( pi @ TA @ TB @ B @ C ) )
= ( ! [D: TA] :
( ( member @ TA @ D @ B )
=> ( member @ TB @ ( A @ D ) @ ( C @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(1117,plain,
$false,
inference(cvc4,[status(thm)],[892,809,1024,365,846,777,1110,352,909,628,1083,962,408,683,276,1073,449,379,440,614,861,1078,385,533,500,814,698,945,340,930,1001,709,417,973,565,651,389,724,570,719,372,460,587,619,917,751,817,606,289,316,1100,421,870,284,770,475,492,756,321,646,514,1051,905,602,280,692,734,866,396,998,988,507,312,293,956,307,292,661,546,761,381,1067,798,1057,334,1042,349,392,297,429,746,1091,467,484,361,625,852,1027,324,1012,285,583,830,1089,874,948,1064,403,356,593,753,537,1075,590,970,706,622,701,654,457,984,721,779,371,503,343,311,1054,953,1086,938,398,1035,773,794,412,425,1103,573,689,733,819,743,711,562,975,641,445,1107,466,919,704,947,713,498,764,406,451,737,434,303,863,387,609,726,520,915,668,330,345,858,367,552,1029,895,943,1022,932,1061,912,959,695,883,1015,1113,426,680,374,487,812,617,490,675,1081,749,557,855,1070,414,327,739,409,310,463,643,1038,370,470,576,656,1008,612,1094,978,363,539,482,671,688,495,524,581,368,844,517,319,632,401,731,995,568,290,510,478,272,899,549,716,800,300,967,1097,835,294,358,437,767,664,828,635,955,796,928,1033,400,839,728,1019,528,992,850,982,1004,337,940,935,659,543,639,454]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.13 % Problem : ITP027^2 : TPTP v8.2.0. Released v7.5.0.
% 0.08/0.16 % Command : run_Leo-III %s %d
% 0.15/0.37 % Computer : n011.cluster.edu
% 0.15/0.37 % Model : x86_64 x86_64
% 0.15/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.37 % Memory : 8042.1875MB
% 0.15/0.37 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.37 % CPULimit : 300
% 0.15/0.37 % WCLimit : 300
% 0.15/0.37 % DateTime : Sat May 18 17:02:38 EDT 2024
% 0.15/0.37 % CPUTime :
% 1.20/1.16 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 2.71/1.65 % [INFO] Parsing done (486ms).
% 2.71/1.66 % [INFO] Running in sequential loop mode.
% 4.28/2.09 % [INFO] eprover registered as external prover.
% 4.28/2.09 % [INFO] cvc4 registered as external prover.
% 4.28/2.10 % [INFO] Scanning for conjecture ...
% 4.83/2.36 % [INFO] Found a conjecture (or negated_conjecture) and 270 axioms. Running axiom selection ...
% 5.58/2.56 % [INFO] Axiom selection finished. Selected 269 axioms (removed 1 axioms).
% 7.76/3.14 % [INFO] Problem is higher-order (TPTP THF).
% 7.76/3.19 % [INFO] Type checking passed.
% 8.17/3.19 % [CONFIG] Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>. Searching for refutation ...
% 259.52/217.29 % External prover 'cvc4' found a proof!
% 259.52/217.29 % [INFO] Killing All external provers ...
% 259.52/217.29 % Time passed: 216752ms (effective reasoning time: 215622ms)
% 259.52/217.29 % 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)>
% 259.52/217.30 % Axioms used in derivation (269): fact_124_Ring_OmHom__func, fact_246_add__mono__thms__linordered__semiring_I4_J, fact_13_ring__is__ag, fact_228_Ring_Oun__closed, fact_150_ring__inv1__2, fact_41_Sr__ring, fact_240_add_Ocommute, fact_119_Ring_OprodM__sprod__val, fact_217_ring__r__one, fact_244_group__cancel_Oadd2, fact_132_Module_Osprod__r__distr, fact_71_nsDistrL, fact_30_prod__mOp__mem, fact_198_Module_OmHom__inv, fact_80_Ring_Org__distrib, fact_191_Ring_Oring__distrib4, fact_173_Subring__subset, fact_42_mHom__func, fact_128_nsDistr, fact_145_id__maximal__Exist, fact_65_rg__distrib, fact_200_Module_Osc__minus__am, fact_12_prod__pOp__func, fact_235_Module_Osprod__one, fact_75_Ring_OSr__tOp__closed, fact_131_Module_OmHom__add, fact_64_Sr__pOp__closed, fact_26_Ring_OprodM__sprod__mem, fact_112_dsum__pOp__mem, fact_221_add__le__cancel__left, fact_227_mem__of__Nset, fact_118_Ring_Opop__closed, fact_202_Module_Osc__minus__am1, fact_122_Module_OmHom__lin, fact_212_Sr__one, fact_109_Ring_OnsEqElm, fact_2_bivar__fun__mem, fact_34_Pi__mem, fact_79_Ring_Oring__distrib1, fact_197_le__imp__add__int, fact_154_not__sub, fact_83_Ring_OnsDistrL, tcon_Nat_Onat___Orderings_Oord_1, fact_167_Pi__anti__mono, fact_59_tp__assoc, tcon_fun___Orderings_Oord, fact_209_add__le__cancel__right, fact_229_Ring_Omaximal__ideal__proper, fact_165_Nset__le, fact_176_Ring_Oring__inv1__1, fact_126_Ring_OSubring__tOp__ring__tOp, fact_172_Ring_OSr__mOp__closed, fact_186_aGroup_Oag__p__inv, fact_98_aGroup_Oaassoc, fact_31_prod__zero__func, fact_233_Ring_Oring__l__one, fact_93_aGroup_Oag__pOp__assoc, fact_169_aGroup_Oag__mOp__closed, fact_210_un__closed, fact_84_Ring_OSr__ring, fact_96_aGroup_OgEQAddcross, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, fact_54_rMulLC, fact_137_Ring_Ozeroring__J__rad__empty, fact_0_Ring__axioms, fact_153_Subring__minus__ring__minus, fact_14_prodM__mem__eq, fact_245_group__cancel_Oadd1, fact_43_rInvim__def, fact_44_dsum__pOp__func, fact_220_ring__times__minusl, tcon_Nat_Onat___Groups_Ocomm__monoid__add, fact_238_add__left__imp__eq, fact_77_Ring_Oring__distrib3, fact_248_Ring_Oring__times__minusl, fact_74_mul__closed__set__tOp__closed, fact_66_ring__distrib1, fact_62_pop__commute, fact_215_rg__l__unit, tcon_Product__Type_Ounit___Orderings_Oord_4, fact_52_pop__closed, fact_63_Sr__tOp__closed, fact_178_Ring_Oid__maximal__Exist, fact_113_dsum__iOp__mem, tcon_Nat_Onat___Groups_Ocancel__semigroup__add, fact_129_add__bothl, fact_214_add__left__cancel, fact_46_Collect__mem__eq, fact_72_Subring__tOp__ring__tOp, fact_23_prodag__sameTr3, fact_87_aGroup_OpOp__assocTr43, fact_35_Pi__iff, tcon_Nat_Onat___Groups_Oplus, fact_57_ring__tOp__commute, fact_40_prodM__sprod__val, fact_133_Module_Omodule__is__ag, fact_237_add__right__imp__eq, fact_103_Ring_Otp__commute, fact_130_add__both, tcon_Set_Oset___Orderings_Oord_2, fact_146_mop__closed, fact_187_Ring_Omul__closed__set__sub, fact_182_aGroup_OpOp__assoc__cancel, fact_158_sub__which1, fact_234_Ring_Org__l__unit, fact_161_subset__self, fact_16_Ring_ORing, fact_149_ring__inv1__1, fact_199_dsumag__sub__prodag, fact_51_tp__closed, fact_4_prodM__sprod__mem, fact_32_prod__mOp__func, fact_243_add_Oassoc, fact_49_nsEqElm, fact_219_ring__times__minusr, fact_155_eqsets__sub, fact_183_aGroup_Oag__diff__minus, fact_218_Subring__one__ring__one, fact_53_rEQMulR, fact_151_ring__inv1__3, fact_82_Ring_OnsMulDistrL, fact_104_Ring_Otp__assoc, fact_69_nsMulDistrL, fact_206_Ring_Ononunit__contained__maxid, fact_144_zeroring__no__maximal, fact_204_Ring_Osum__mult__pOp__closed, fact_205_Ring_Oset__sum__mem, fact_110_Ring_OnsClose, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, fact_136_Ring_Omul__closed__set__tOp__closed, fact_177_Ring_Oring__inv1, fact_89_aGroup_OpOp__assocTr41, fact_38_mem__subring__mem__ring, fact_10_funcset__id, fact_192_nonunit__contained__maxid, fact_135_Ring_OnsDistr, fact_115_aGroup_Opop__closed, fact_88_aGroup_OpOp__assocTr42, fact_45_mem__Collect__eq, fact_152_ring__distrib4, fact_67_ring__distrib2, fact_90_aGroup_Oag__pOp__closed, fact_226_n__in__Nsetn, fact_81_Ring_OnsMulDistrR, fact_143_mul__closed__set__sub, fact_5_prodM__carr, fact_239_add_Oleft__commute, fact_201_Ring_Osum__mult__subR, fact_7_funcTr, fact_120_dsum__iOp__func, fact_170_aGroup_Oag__inv__inv, fact_106_Ring_OrMulLC, fact_162_not__subseteq, fact_252_add__mono__thms__linordered__semiring_I2_J, fact_166_Pi__mono, fact_24_prodag__sameTr4, fact_156_ge__convert, fact_56_ring__tOp__closed, fact_255_add__left__mono, fact_163_proper__subset, fact_127_Module_Osprod__closed, fact_116_Ring_OJ__rad__mem, fact_6_eq__fun, fact_225_nat__eq__le, fact_15_J__rad__mem, fact_37_PiE, fact_18_Ring_OprodM__mem__eq, fact_95_aGroup_Oag__pOp__add__l, fact_39_subring__Ring, fact_47_Collect__cong, fact_36_Pi__I_H, fact_33_Pi__cong, fact_193_set__sum__mem, fact_76_Ring_OSr__pOp__closed, fact_142_Ring_Omem__subring__mem__ring, fact_117_Ring_Otp__closed, fact_159_sub__which2, fact_223_primary__ideal__proper1, fact_171_aGroup_Oag__inv__inj, fact_55_ring__tOp__assoc, fact_50_nsClose, fact_27_prod__pOp__commute, fact_28_prod__pOp__assoc, fact_231_Ring_OSr__one, fact_222_localring__unit, fact_101_Ring_Oring__tOp__assoc, fact_203_Ring_Otimes__mem__sum__mult, fact_160_sets__not__eq, fact_134_Module_Osprod__l__distr, fact_179_Ring_Ozeroring__no__maximal, fact_174_Ring_Oring__inv1__3, fact_11_Algebra1_Ofuncset__mem, fact_194_sum__mult__pOp__closed, fact_121_zeroring__J__rad__empty, fact_157_le__convert, fact_188_mul__closed__set__def, fact_207_aGroup_Omem__set__sum, fact_92_aGroup_OpOp__cancel__l, fact_180_Ring_OSubring__minus__ring__minus, fact_17_Ring_Oring__is__ag, fact_94_aGroup_Oag__pOp__add__r, fact_175_Ring_Oring__inv1__2, fact_251_add__mono__thms__linordered__semiring_I3_J, fact_108_Ring_Opop__aassoc, conj_1, fact_184_aGroup_Oag__eq__sol2, fact_181_Ring_Omop__closed, fact_29_prod__pOp__mem, fact_99_Ring_Oring__tOp__commute, fact_216_ring__l__one, fact_20_prodag__sameTr1, fact_139_Ring_Osubring__Ring, fact_85_aGroup_Oag__pOp__commute, fact_208_J__rad__unit, fact_1_bivar__fun, fact_21_prodag__comp__i, fact_190_prod__mOp__mem__i, fact_211_maximal__ideal__proper, tcon_HOL_Obool___Orderings_Oord_3, fact_196_times__mem__sum__mult, fact_61_pop__aassoc, fact_70_nsMulDistrR, fact_195_sum__mult__subR, fact_247_ab__semigroup__add__class_Oadd__ac_I1_J, tcon_Nat_Onat___Groups_Osemigroup__add, fact_58_ring__tOp__rel, fact_141_Module_OmHom__mem, fact_8_funcsetI, fact_232_Ring_Oring__r__one, fact_78_Ring_Oring__distrib2, fact_253_add__mono__thms__linordered__semiring_I1_J, fact_25_Ring_OprodM__carr, fact_125_Ring_OSubring__pOp__ring__pOp, fact_213_add__right__cancel, fact_105_Ring_OrEQMulR, fact_73_Subring__pOp__ring__pOp, fact_224_self__le, fact_22_prodag__sameTr2, fact_102_Ring_Oring__tOp__rel, fact_138_Module_Osc__Ring, fact_60_tp__commute, fact_249_Ring_Oring__times__minusr, fact_97_aGroup_Oag__add4__rel, tcon_Nat_Onat___Groups_Oab__semigroup__add, fact_91_aGroup_OpOp__cancel__r, fact_241_add_Oright__cancel, fact_250_sr__def, conj_0, fact_147_Sr__mOp__closed, fact_9_eq__funcs, fact_111_prod__pOp__mem__i, fact_185_aGroup_Oag__eq__sol1, fact_230_Ring_OSubring__one__ring__one, fact_236_Unit__def, fact_189_aGroup_Omop__closed, fact_164_conditional__subset, fact_19_carr__prodag__mem__eq, fact_242_add_Oleft__cancel, fact_168_extend__fun, fact_107_Ring_Opop__commute, fact_254_add__mono, fact_68_ring__distrib3, fact_123_Module_Osprod__assoc, fact_3_Pi__I, fact_86_aGroup_Oag__add__commute, fact_148_ring__inv1, fact_114_dsum__zero__func, fact_100_Ring_Oring__tOp__closed, fact_140_Module_OmHom__eq
% 259.52/217.30 % No. of inferences in proof: 542
% 259.52/217.30 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 216752 ms resp. 215622 ms w/o parsing
% 261.17/217.63 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 261.17/217.63 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------