TSTP Solution File: ITP026^1 by Leo-III---1.7.10
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%------------------------------------------------------------------------------
% File : Leo-III---1.7.10
% Problem : ITP026^1 : TPTP v8.1.2. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n012.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 : Tue May 7 07:47:13 EDT 2024
% Result : Theorem 12.29s 3.55s
% Output : Refutation 13.18s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 398
% Syntax : Number of formulae : 709 ( 242 unt; 89 typ; 0 def)
% Number of atoms : 1790 ( 404 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 6411 ( 140 ~; 22 |; 92 &;5277 @)
% ( 0 <=>; 880 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 370 ( 370 >; 0 *; 0 +; 0 <<)
% Number of symbols : 72 ( 70 usr; 14 con; 0-5 aty)
% Number of variables : 1750 ( 272 ^1412 !; 66 ?;1750 :)
% Comments :
%------------------------------------------------------------------------------
thf(carrie377329159t_unit_type,type,
carrie377329159t_unit: $tType ).
thf(carrie292259835t_unit_type,type,
carrie292259835t_unit: $tType ).
thf(carrie1065537299t_unit_type,type,
carrie1065537299t_unit: $tType ).
thf(carrie980467975t_unit_type,type,
carrie980467975t_unit: $tType ).
thf(carrie1963041556t_unit_type,type,
carrie1963041556t_unit: $tType ).
thf(carrie1821755406_e_b_f_type,type,
carrie1821755406_e_b_f: $tType ).
thf(carrie722926983_a_b_c_type,type,
carrie722926983_a_b_c: $tType ).
thf(carrie1950868226xt_b_d_type,type,
carrie1950868226xt_b_d: $tType ).
thf(set_nat_e_type,type,
set_nat_e: $tType ).
thf(set_nat_b_type,type,
set_nat_b: $tType ).
thf(set_nat_a_type,type,
set_nat_a: $tType ).
thf(set_nat_type,type,
set_nat: $tType ).
thf(set_e_type,type,
set_e: $tType ).
thf(set_b_type,type,
set_b: $tType ).
thf(set_a_type,type,
set_a: $tType ).
thf(nat_type,type,
nat: $tType ).
thf(e_type,type,
e: $tType ).
thf(b_type,type,
b: $tType ).
thf(a_type,type,
a: $tType ).
thf(carrie1074654371t_unit_type,type,
carrie1074654371t_unit: carrie1963041556t_unit > set_a ).
thf(carrie2021454486_a_b_c_type,type,
carrie2021454486_a_b_c: carrie722926983_a_b_c > set_a ).
thf(carrie2079586589xt_b_d_type,type,
carrie2079586589xt_b_d: carrie1950868226xt_b_d > set_b ).
thf(ring_b_d_type,type,
ring_b_d: carrie1950868226xt_b_d > $o ).
thf(zero_a1261444626_a_b_c_type,type,
zero_a1261444626_a_b_c: carrie722926983_a_b_c > a ).
thf(ideal_b_d_type,type,
ideal_b_d: carrie1950868226xt_b_d > set_b > $o ).
thf(annihi259882159t_unit_type,type,
annihi259882159t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_b ).
thf(annihilator_b_d_a_c_type,type,
annihilator_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_b ).
thf(hOM_b_1103679019t_unit_type,type,
hOM_b_1103679019t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie1963041556t_unit > carrie980467975t_unit ).
thf(hOM_b_796081438it_a_c_type,type,
hOM_b_796081438it_a_c: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie722926983_a_b_c > carrie980467975t_unit ).
thf(hOM_b_717364638t_unit_type,type,
hOM_b_717364638t_unit: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie1963041556t_unit > carrie980467975t_unit ).
thf(hOM_b_d_a_c_a_c_type,type,
hOM_b_d_a_c_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie722926983_a_b_c > carrie980467975t_unit ).
thf(module589927589unit_d_type,type,
module589927589unit_d: carrie980467975t_unit > carrie1950868226xt_b_d > $o ).
thf(module1821517916unit_d_type,type,
module1821517916unit_d: carrie1963041556t_unit > carrie1950868226xt_b_d > $o ).
thf(module_a_b_c_d_type,type,
module_a_b_c_d: carrie722926983_a_b_c > carrie1950868226xt_b_d > $o ).
thf(free_g637607517t_unit_type,type,
free_g637607517t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > $o ).
thf(free_g1087686480_d_a_c_type,type,
free_g1087686480_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > $o ).
thf(genera1692266857t_unit_type,type,
genera1692266857t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > $o ).
thf(generator_b_d_a_c_type,type,
generator_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > $o ).
thf(l_comb1138968323t_unit_type,type,
l_comb1138968323t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > nat > ( nat > b ) > ( nat > a ) > a ).
thf(l_comb_b_d_a_c_type,type,
l_comb_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > nat > ( nat > b ) > ( nat > a ) > a ).
thf(linear_span_b_d_a_c_type,type,
linear_span_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_b > set_a > set_a ).
thf(mdl_a_b_c_type,type,
mdl_a_b_c: carrie722926983_a_b_c > set_a > carrie1963041556t_unit ).
thf(misomo493403663t_unit_type,type,
misomo493403663t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie1963041556t_unit > $o ).
thf(misomo1935876354it_a_c_type,type,
misomo1935876354it_a_c: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie722926983_a_b_c > $o ).
thf(misomo1857159554t_unit_type,type,
misomo1857159554t_unit: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie1963041556t_unit > $o ).
thf(misomo1282343797_c_a_c_type,type,
misomo1282343797_c_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie722926983_a_b_c > $o ).
thf(quotie1196826005t_unit_type,type,
quotie1196826005t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > set_a > set_b ).
thf(quotie1153752712_d_a_c_type,type,
quotie1153752712_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > set_a > set_b ).
thf(smodul132175289t_unit_type,type,
smodul132175289t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_b > set_a ).
thf(smodul818989740_d_a_c_type,type,
smodul818989740_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_b > set_a ).
thf(submod903911234t_unit_type,type,
submod903911234t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > $o ).
thf(submodule_b_d_a_c_type,type,
submodule_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > $o ).
thf(algebr1039611956t_unit_type,type,
algebr1039611956t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > set_a ).
thf(algebr1152250919_d_a_c_type,type,
algebr1152250919_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > set_a ).
thf(pi_nat_a_type,type,
pi_nat_a: set_nat > ( nat > set_a ) > set_nat_a ).
thf(pi_nat_b_type,type,
pi_nat_b: set_nat > ( nat > set_b ) > set_nat_b ).
thf(minus_minus_nat_type,type,
minus_minus_nat: nat > nat > nat ).
thf(minus_minus_set_a_type,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(zero_zero_nat_type,type,
zero_zero_nat: nat ).
thf(suc_type,type,
suc: nat > nat ).
thf(bot_bot_set_nat_a_type,type,
bot_bot_set_nat_a: set_nat_a ).
thf(bot_bot_set_nat_b_type,type,
bot_bot_set_nat_b: set_nat_b ).
thf(bot_bot_set_nat_type,type,
bot_bot_set_nat: set_nat ).
thf(bot_bot_set_a_type,type,
bot_bot_set_a: set_a ).
thf(ord_less_nat_type,type,
ord_less_nat: nat > nat > $o ).
thf(ord_less_set_a_type,type,
ord_less_set_a: set_a > set_a > $o ).
thf(ord_less_eq_nat_type,type,
ord_less_eq_nat: nat > nat > $o ).
thf(ord_le157368549_nat_a_type,type,
ord_le157368549_nat_a: set_nat_a > set_nat_a > $o ).
thf(ord_le1296932838_nat_b_type,type,
ord_le1296932838_nat_b: set_nat_b > set_nat_b > $o ).
thf(ord_less_eq_set_a_type,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(collect_nat_a_type,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(collect_nat_b_type,type,
collect_nat_b: ( ( nat > b ) > $o ) > set_nat_b ).
thf(collect_nat_type,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(collect_a_type,type,
collect_a: ( a > $o ) > set_a ).
thf(insert_a_type,type,
insert_a: a > set_a > set_a ).
thf(member_nat_a_type,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(member_nat_b_type,type,
member_nat_b: ( nat > b ) > set_nat_b > $o ).
thf(member_nat_type,type,
member_nat: nat > set_nat > $o ).
thf(member_a_type,type,
member_a: a > set_a > $o ).
thf(h_type,type,
h: set_a ).
thf(h1_type,type,
h1: set_a ).
thf(m_type,type,
m: carrie722926983_a_b_c ).
thf(r_type,type,
r: carrie1950868226xt_b_d ).
thf(g_type,type,
g: nat > a ).
thf(h2_type,type,
h2: a ).
thf(m2_type,type,
m2: nat ).
thf(t_type,type,
t: nat > b ).
thf(x_type,type,
x: a ).
thf(xa_type,type,
xa: nat ).
thf(100,axiom,
! [A: nat > set_a,B: nat,C: nat] :
( ! [D: nat] : ( ord_less_eq_set_a @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_set_a @ ( A @ B ) @ ( A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_233_lift__Suc__mono__le) ).
thf(645,plain,
! [A: nat > set_a,B: nat,C: nat] :
( ! [D: nat] : ( ord_less_eq_set_a @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_set_a @ ( A @ B ) @ ( A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(299,axiom,
member_a @ h2 @ h,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_14) ).
thf(1322,plain,
member_a @ h2 @ h,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).
thf(206,axiom,
! [A: nat] :
( ( suc @ A )
!= zero_zero_nat ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_180_old_Onat_Odistinct_I2_J) ).
thf(1031,plain,
! [A: nat] :
( ( suc @ A )
!= zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(226,axiom,
! [A: nat,B: nat] :
( ( A
= ( suc @ B ) )
=> ( A != zero_zero_nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_178_nat_OdiscI) ).
thf(1083,plain,
! [A: nat,B: nat] :
( ( A
= ( suc @ B ) )
=> ( A != zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).
thf(70,axiom,
! [A: set_a] :
( ( generator_b_d_a_c @ r @ m @ A )
=> ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_39_generator__sub__carrier) ).
thf(555,plain,
! [A: set_a] :
( ( generator_b_d_a_c @ r @ m @ A )
=> ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(50,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( A != B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_187_nat__less__le) ).
thf(481,plain,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( A != B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(13,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_222_less__imp__diff__less) ).
thf(352,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(260,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_81_diff__self__eq__0) ).
thf(1200,plain,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).
thf(63,axiom,
! [A: nat > a] :
~ ( member_nat_a @ A @ bot_bot_set_nat_a ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_287_empty__iff) ).
thf(523,plain,
! [A: nat > a] :
~ ( member_nat_a @ A @ bot_bot_set_nat_a ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(220,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a,D: set_a,E: set_a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( free_g637607517t_unit @ B @ A @ C )
=> ( ( ord_less_eq_set_a @ D @ E )
=> ( ( ord_less_eq_set_a @ E @ C )
=> ( ord_less_eq_set_a @ ( algebr1039611956t_unit @ B @ A @ D ) @ ( algebr1039611956t_unit @ B @ A @ E ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_23_Module_Ofgs__mono) ).
thf(1068,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a,D: set_a,E: set_a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( free_g637607517t_unit @ B @ A @ C )
=> ( ( ord_less_eq_set_a @ D @ E )
=> ( ( ord_less_eq_set_a @ E @ C )
=> ( ord_less_eq_set_a @ ( algebr1039611956t_unit @ B @ A @ D ) @ ( algebr1039611956t_unit @ B @ A @ E ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(164,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ A @ B ) )
= ( ord_less_nat @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_47_zero__less__diff) ).
thf(873,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ A @ B ) )
= ( ord_less_nat @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(91,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d] :
( ( module1821517916unit_d @ A @ B )
=> ( submod903911234t_unit @ B @ A @ ( carrie1074654371t_unit @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_262_Module_Osubmodule__whole) ).
thf(614,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d] :
( ( module1821517916unit_d @ A @ B )
=> ( submod903911234t_unit @ B @ A @ ( carrie1074654371t_unit @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(245,axiom,
ideal_b_d @ r @ ( carrie2079586589xt_b_d @ r ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_8) ).
thf(1142,plain,
ideal_b_d @ r @ ( carrie2079586589xt_b_d @ r ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(280,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_b,D: set_a,E: nat > b,F: nat,G: nat > a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ideal_b_d @ B @ C )
=> ( ( ord_less_eq_set_a @ D @ ( carrie1074654371t_unit @ A ) )
=> ( ( member_nat_b @ E
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ F ) )
@ ^ [H: nat] : C ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ F ) )
@ ^ [H: nat] : D ) )
=> ( member_a @ ( l_comb1138968323t_unit @ B @ A @ F @ E @ G ) @ ( carrie1074654371t_unit @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_43_Module_Ol__comb__mem) ).
thf(1269,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_b,D: set_a,E: nat > b,F: nat,G: nat > a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ideal_b_d @ B @ C )
=> ( ( ord_less_eq_set_a @ D @ ( carrie1074654371t_unit @ A ) )
=> ( ( member_nat_b @ E
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ F ) )
@ ^ [H: nat] : C ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ F ) )
@ ^ [H: nat] : D ) )
=> ( member_a @ ( l_comb1138968323t_unit @ B @ A @ F @ E @ G ) @ ( carrie1074654371t_unit @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).
thf(210,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( suc @ ( minus_minus_nat @ B @ ( suc @ A ) ) )
= ( minus_minus_nat @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_255_Suc__diff__Suc) ).
thf(1041,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( suc @ ( minus_minus_nat @ B @ ( suc @ A ) ) )
= ( minus_minus_nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).
thf(160,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ A @ B )
= zero_zero_nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_55_diff__is__0__eq_H) ).
thf(861,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ A @ B )
= zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(224,axiom,
! [A: a,B: a,C: set_a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ C ) )
= ( ( B = A )
& ( ord_less_eq_set_a @ C @ ( insert_a @ A @ bot_bot_set_a ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_274_singleton__insert__inj__eq) ).
thf(1078,plain,
! [A: a,B: a,C: set_a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ C ) )
= ( ( B = A )
& ( ord_less_eq_set_a @ C @ ( insert_a @ A @ bot_bot_set_a ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).
thf(295,axiom,
( ( ( g @ xa )
= h2 )
| ( member_a @ ( g @ xa ) @ h1 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_20) ).
thf(1310,plain,
( ( ( g @ xa )
= h2 )
| ( member_a @ ( g @ xa ) @ h1 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).
thf(118,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ C @ B ) @ ( minus_minus_nat @ C @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_212_diff__le__mono2) ).
thf(715,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ C @ B ) @ ( minus_minus_nat @ C @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(47,axiom,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat] : ( ord_less_nat @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( A @ B ) @ ( A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_235_lift__Suc__mono__less) ).
thf(472,plain,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat] : ( ord_less_nat @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( A @ B ) @ ( A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(278,axiom,
member_a @ ( zero_a1261444626_a_b_c @ m ) @ ( carrie2021454486_a_b_c @ m ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_278_module__inc__zero) ).
thf(1260,plain,
member_a @ ( zero_a1261444626_a_b_c @ m ) @ ( carrie2021454486_a_b_c @ m ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).
thf(147,axiom,
! [A: nat,B: nat] :
( ( ( minus_minus_nat @ A @ B )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ B @ A )
= zero_zero_nat )
=> ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_221_diffs0__imp__equal) ).
thf(814,plain,
! [A: nat,B: nat] :
( ( ( minus_minus_nat @ A @ B )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ B @ A )
= zero_zero_nat )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(302,axiom,
! [A: a,B: set_a,C: set_a] :
( ( member_a @ A @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ A @ C ) @ B )
= ( minus_minus_set_a @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_302_insert__Diff1) ).
thf(1327,plain,
! [A: a,B: set_a,C: set_a] :
( ( member_a @ A @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ A @ C ) @ B )
= ( minus_minus_set_a @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).
thf(168,axiom,
! [A: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_328_Diff__eq__empty__iff) ).
thf(898,plain,
! [A: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(116,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_326_diff__zero) ).
thf(709,plain,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(156,axiom,
! [A: nat > set_a,B: nat,C: nat] :
( ! [D: nat] : ( ord_less_eq_set_a @ ( A @ ( suc @ D ) ) @ ( A @ D ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_set_a @ ( A @ C ) @ ( A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_231_lift__Suc__antimono__le) ).
thf(841,plain,
! [A: nat > set_a,B: nat,C: nat] :
( ! [D: nat] : ( ord_less_eq_set_a @ ( A @ ( suc @ D ) ) @ ( A @ D ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_set_a @ ( A @ C ) @ ( A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(74,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_329_subset__empty) ).
thf(565,plain,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(290,axiom,
! [A: set_a] :
( ( submodule_b_d_a_c @ r @ m @ A )
=> ( member_a @ ( zero_a1261444626_a_b_c @ m ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_310_submodule__inc__0) ).
thf(1298,plain,
! [A: set_a] :
( ( submodule_b_d_a_c @ r @ m @ A )
=> ( member_a @ ( zero_a1261444626_a_b_c @ m ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).
thf(20,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( suc @ A ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_191_less__trans__Suc) ).
thf(372,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( suc @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(216,axiom,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat] : ( ord_less_eq_nat @ ( A @ ( suc @ D ) ) @ ( A @ D ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( A @ C ) @ ( A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_230_lift__Suc__antimono__le) ).
thf(1059,plain,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat] : ( ord_less_eq_nat @ ( A @ ( suc @ D ) ) @ ( A @ D ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( A @ C ) @ ( A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(184,axiom,
! [A: set_a,B: nat > b,C: nat,D: nat > a,E: nat > a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ( member_nat_b @ B
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : ( carrie2079586589xt_b_d @ r ) ) )
& ( member_nat_a @ D
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : A ) )
& ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : A ) )
& ! [F: nat] :
( ( member_nat @ F
@ ( collect_nat
@ ^ [G: nat] : ( ord_less_eq_nat @ G @ C ) ) )
=> ( ( D @ F )
= ( E @ F ) ) ) )
=> ( ( l_comb_b_d_a_c @ r @ m @ C @ B @ D )
= ( l_comb_b_d_a_c @ r @ m @ C @ B @ E ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_10_linear__comb__eqTr) ).
thf(949,plain,
! [A: set_a,B: nat > b,C: nat,D: nat > a,E: nat > a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ( member_nat_b @ B
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : ( carrie2079586589xt_b_d @ r ) ) )
& ( member_nat_a @ D
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : A ) )
& ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : A ) )
& ! [F: nat] :
( ( member_nat @ F
@ ( collect_nat
@ ^ [G: nat] : ( ord_less_eq_nat @ G @ C ) ) )
=> ( ( D @ F )
= ( E @ F ) ) ) )
=> ( ( l_comb_b_d_a_c @ r @ m @ C @ B @ D )
= ( l_comb_b_d_a_c @ r @ m @ C @ B @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(82,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_136_le__antisym) ).
thf(587,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(300,axiom,
! [A: set_b,B: set_a,C: nat > b,D: nat,E: nat > a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ( member_nat_b @ C
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : A ) )
& ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ B ) ) ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ D @ C @ E ) @ ( linear_span_b_d_a_c @ r @ m @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_275_l__span__closed2Tr) ).
thf(1323,plain,
! [A: set_b,B: set_a,C: nat > b,D: nat,E: nat > a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ( member_nat_b @ C
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : A ) )
& ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ B ) ) ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ D @ C @ E ) @ ( linear_span_b_d_a_c @ r @ m @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).
thf(173,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_279_psubsetI) ).
thf(917,plain,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(125,axiom,
! [A: set_nat_a] :
( ( ! [B: nat > a] :
~ ( member_nat_a @ B @ A ) )
= ( A = bot_bot_set_nat_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_290_all__not__in__conv) ).
thf(751,plain,
! [A: set_nat_a] :
( ( ! [B: nat > a] :
~ ( member_nat_a @ B @ A ) )
= ( A = bot_bot_set_nat_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(296,axiom,
ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ r @ m @ h1 ) @ ( carrie2021454486_a_b_c @ m ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_11) ).
thf(1313,plain,
ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ r @ m @ h1 ) @ ( carrie2021454486_a_b_c @ m ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).
thf(284,axiom,
~ ( member_a @ h2 @ h1 ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_15) ).
thf(1279,plain,
~ ( member_a @ h2 @ h1 ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).
thf(179,axiom,
free_g1087686480_d_a_c @ r @ m @ h,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_5) ).
thf(934,plain,
free_g1087686480_d_a_c @ r @ m @ h,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(257,axiom,
! [A: nat] :
( zero_zero_nat
!= ( suc @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_172_Zero__not__Suc) ).
thf(1191,plain,
! [A: nat] :
( zero_zero_nat
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).
thf(148,axiom,
! [A: set_b,B: set_a] :
( ( ideal_b_d @ r @ A )
=> ( ( generator_b_d_a_c @ r @ m @ B )
=> ( ( ( smodul818989740_d_a_c @ r @ m @ A )
= ( carrie2021454486_a_b_c @ m ) )
=> ( ( linear_span_b_d_a_c @ r @ m @ A @ B )
= ( carrie2021454486_a_b_c @ m ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_344_l__span__closed3) ).
thf(817,plain,
! [A: set_b,B: set_a] :
( ( ideal_b_d @ r @ A )
=> ( ( generator_b_d_a_c @ r @ m @ B )
=> ( ( ( smodul818989740_d_a_c @ r @ m @ A )
= ( carrie2021454486_a_b_c @ m ) )
=> ( ( linear_span_b_d_a_c @ r @ m @ A @ B )
= ( carrie2021454486_a_b_c @ m ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(274,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_b,D: a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ideal_b_d @ B @ C )
=> ( ( member_a @ D @ ( smodul132175289t_unit @ B @ A @ C ) )
=> ? [E: nat,F: nat > b] :
( ( member_nat_b @ F
@ ( pi_nat_b
@ ( collect_nat
@ ^ [G: nat] : ( ord_less_eq_nat @ G @ E ) )
@ ^ [G: nat] : C ) )
& ? [G: nat > a] :
( ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : ( carrie1074654371t_unit @ A ) ) )
& ( D
= ( l_comb1138968323t_unit @ B @ A @ E @ F @ G ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_228_Module_Omem__smodule__ideal__coeff) ).
thf(1247,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_b,D: a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ideal_b_d @ B @ C )
=> ( ( member_a @ D @ ( smodul132175289t_unit @ B @ A @ C ) )
=> ? [E: nat,F: nat > b] :
( ( member_nat_b @ F
@ ( pi_nat_b
@ ( collect_nat
@ ^ [G: nat] : ( ord_less_eq_nat @ G @ E ) )
@ ^ [G: nat] : C ) )
& ? [G: nat > a] :
( ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : ( carrie1074654371t_unit @ A ) ) )
& ( D
= ( l_comb1138968323t_unit @ B @ A @ E @ F @ G ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).
thf(190,axiom,
! [A: set_a,B: nat > b,C: nat,D: nat > a,E: nat > a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_nat_b @ B
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : ( carrie2079586589xt_b_d @ r ) ) )
=> ( ( member_nat_a @ D
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : A ) )
=> ( ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : A ) )
=> ( ! [F: nat] :
( ( member_nat @ F
@ ( collect_nat
@ ^ [G: nat] : ( ord_less_eq_nat @ G @ C ) ) )
=> ( ( D @ F )
= ( E @ F ) ) )
=> ( ( l_comb_b_d_a_c @ r @ m @ C @ B @ D )
= ( l_comb_b_d_a_c @ r @ m @ C @ B @ E ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_9_linear__comb__eq) ).
thf(969,plain,
! [A: set_a,B: nat > b,C: nat,D: nat > a,E: nat > a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_nat_b @ B
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : ( carrie2079586589xt_b_d @ r ) ) )
=> ( ( member_nat_a @ D
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : A ) )
=> ( ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : A ) )
=> ( ! [F: nat] :
( ( member_nat @ F
@ ( collect_nat
@ ^ [G: nat] : ( ord_less_eq_nat @ G @ C ) ) )
=> ( ( D @ F )
= ( E @ F ) ) )
=> ( ( l_comb_b_d_a_c @ r @ m @ C @ B @ D )
= ( l_comb_b_d_a_c @ r @ m @ C @ B @ E ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(237,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ B @ C ) )
= ( ord_less_eq_nat @ C @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_213_le__diff__iff_H) ).
thf(1115,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ B @ C ) )
= ( ord_less_eq_nat @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).
thf(4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( suc @ A ) @ ( suc @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_65_Suc__mono) ).
thf(316,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( suc @ A ) @ ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(232,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( generator_b_d_a_c @ B @ A @ C )
=> ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_129_Module_Ogenerator__sub__carrier) ).
thf(1100,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( generator_b_d_a_c @ B @ A @ C )
=> ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(270,axiom,
! [A: a,B: a,C: set_a] :
( ( member_a @ A @ ( insert_a @ B @ C ) )
= ( ( A = B )
| ( member_a @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_308_insert__iff) ).
thf(1232,plain,
! [A: a,B: a,C: set_a] :
( ( member_a @ A @ ( insert_a @ B @ C ) )
= ( ( A = B )
| ( member_a @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).
thf(48,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( suc @ A )
!= B )
=> ( ord_less_nat @ ( suc @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_201_Suc__lessI) ).
thf(475,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( suc @ A )
!= B )
=> ( ord_less_nat @ ( suc @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(140,axiom,
! [A: carrie1963041556t_unit] :
( ( module1821517916unit_d @ A @ r )
=> ( ( misomo1857159554t_unit @ r @ m @ A )
=> ( misomo1935876354it_a_c @ r @ A @ m ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_59_misom__sym) ).
thf(789,plain,
! [A: carrie1963041556t_unit] :
( ( module1821517916unit_d @ A @ r )
=> ( ( misomo1857159554t_unit @ r @ m @ A )
=> ( misomo1935876354it_a_c @ r @ A @ m ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(235,axiom,
! [A: nat > b,B: ( nat > b ) > $o] :
( ( member_nat_b @ A @ ( collect_nat_b @ B ) )
= ( B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_67_mem__Collect__eq) ).
thf(1109,plain,
! [A: nat > b,B: ( nat > b ) > $o] :
( ( member_nat_b @ A @ ( collect_nat_b @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).
thf(94,axiom,
! [A: nat > $o,B: nat] :
( ( A @ B )
=> ( ! [C: nat] :
( ( A @ ( suc @ C ) )
=> ( A @ C ) )
=> ( A @ zero_zero_nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_175_zero__induct) ).
thf(623,plain,
! [A: nat > $o,B: nat] :
( ( A @ B )
=> ( ! [C: nat] :
( ( A @ ( suc @ C ) )
=> ( A @ C ) )
=> ( A @ zero_zero_nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(304,axiom,
! [A: a,B: set_a,C: set_a] :
( ~ ( member_a @ A @ B )
=> ( ( minus_minus_set_a @ B @ ( insert_a @ A @ C ) )
= ( minus_minus_set_a @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_299_Diff__insert0) ).
thf(1334,plain,
! [A: a,B: set_a,C: set_a] :
( ~ ( member_a @ A @ B )
=> ( ( minus_minus_set_a @ B @ ( insert_a @ A @ C ) )
= ( minus_minus_set_a @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).
thf(61,axiom,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ! [D: nat] :
( ( ord_less_nat @ D @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_142_nat__less__induct) ).
thf(514,plain,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ! [D: nat] :
( ( ord_less_nat @ D @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(77,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( submodule_b_d_a_c @ r @ m @ ( algebr1152250919_d_a_c @ r @ m @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_8_fgs__submodule) ).
thf(574,plain,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( submodule_b_d_a_c @ r @ m @ ( algebr1152250919_d_a_c @ r @ m @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(144,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ r @ m @ A ) @ ( carrie2021454486_a_b_c @ m ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_6_fgs__sub__carrier) ).
thf(806,plain,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ r @ m @ A ) @ ( carrie2021454486_a_b_c @ m ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(189,axiom,
! [A: nat,B: nat > $o] :
( ( ? [C: nat] :
( ( ord_less_nat @ C @ ( suc @ A ) )
& ( B @ C ) ) )
= ( ( B @ zero_zero_nat )
| ? [C: nat] :
( ( ord_less_nat @ C @ A )
& ( B @ ( suc @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_250_Ex__less__Suc2) ).
thf(966,plain,
! [A: nat,B: nat > $o] :
( ( ? [C: nat] :
( ( ord_less_nat @ C @ ( suc @ A ) )
& ( B @ C ) ) )
= ( ( B @ zero_zero_nat )
| ? [C: nat] :
( ( ord_less_nat @ C @ A )
& ( B @ ( suc @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(149,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( genera1692266857t_unit @ B @ A @ C )
=> ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_130_Module_Ogenerator__sub__carrier) ).
thf(821,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( genera1692266857t_unit @ B @ A @ C )
=> ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(52,axiom,
! [A: nat,B: nat > $o] :
( ( ! [C: nat] :
( ( ord_less_nat @ C @ ( suc @ A ) )
=> ( B @ C ) ) )
= ( ( B @ A )
& ! [C: nat] :
( ( ord_less_nat @ C @ A )
=> ( B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_195_All__less__Suc) ).
thf(485,plain,
! [A: nat,B: nat > $o] :
( ( ! [C: nat] :
( ( ord_less_nat @ C @ ( suc @ A ) )
=> ( B @ C ) ) )
= ( ( B @ A )
& ! [C: nat] :
( ( ord_less_nat @ C @ A )
=> ( B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(243,axiom,
! [A: carrie722926983_a_b_c,B: carrie722926983_a_b_c] :
( ( module_a_b_c_d @ A @ r )
=> ( ( module_a_b_c_d @ B @ r )
=> ( ( misomo1282343797_c_a_c @ r @ A @ m )
=> ( ( misomo1282343797_c_a_c @ r @ m @ B )
=> ( misomo1282343797_c_a_c @ r @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_91_misom__trans) ).
thf(1136,plain,
! [A: carrie722926983_a_b_c,B: carrie722926983_a_b_c] :
( ( module_a_b_c_d @ A @ r )
=> ( ( module_a_b_c_d @ B @ r )
=> ( ( misomo1282343797_c_a_c @ r @ A @ m )
=> ( ( misomo1282343797_c_a_c @ r @ m @ B )
=> ( misomo1282343797_c_a_c @ r @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).
thf(111,axiom,
! [A: nat,B: nat > $o] :
( ( ! [C: nat] :
( ( ord_less_nat @ C @ ( suc @ A ) )
=> ( B @ C ) ) )
= ( ( B @ zero_zero_nat )
& ! [C: nat] :
( ( ord_less_nat @ C @ A )
=> ( B @ ( suc @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_248_All__less__Suc2) ).
thf(692,plain,
! [A: nat,B: nat > $o] :
( ( ! [C: nat] :
( ( ord_less_nat @ C @ ( suc @ A ) )
=> ( B @ C ) ) )
= ( ( B @ zero_zero_nat )
& ! [C: nat] :
( ( ord_less_nat @ C @ A )
=> ( B @ ( suc @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(41,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_208_less__zeroE) ).
thf(453,plain,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(238,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a,D: set_a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( submod903911234t_unit @ B @ A @ C )
=> ( ( submod903911234t_unit @ B @ A @ D )
=> ( ideal_b_d @ B @ ( quotie1196826005t_unit @ B @ A @ C @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_153_Module_Oquotient__of__submodules__is__ideal) ).
thf(1119,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a,D: set_a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( submod903911234t_unit @ B @ A @ C )
=> ( ( submod903911234t_unit @ B @ A @ D )
=> ( ideal_b_d @ B @ ( quotie1196826005t_unit @ B @ A @ C @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).
thf(192,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a,D: set_a,E: set_a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( free_g1087686480_d_a_c @ B @ A @ C )
=> ( ( ord_less_eq_set_a @ D @ E )
=> ( ( ord_less_eq_set_a @ E @ C )
=> ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ B @ A @ D ) @ ( algebr1152250919_d_a_c @ B @ A @ E ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_24_Module_Ofgs__mono) ).
thf(988,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a,D: set_a,E: set_a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( free_g1087686480_d_a_c @ B @ A @ C )
=> ( ( ord_less_eq_set_a @ D @ E )
=> ( ( ord_less_eq_set_a @ E @ C )
=> ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ B @ A @ D ) @ ( algebr1152250919_d_a_c @ B @ A @ E ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(37,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_76_less__nat__zero__code) ).
thf(428,plain,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( suc @ A ) @ B )
=> ~ ! [C: nat] :
( ( ord_less_nat @ A @ C )
=> ( B
!= ( suc @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_202_Suc__lessE) ).
thf(312,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( suc @ A ) @ B )
=> ~ ! [C: nat] :
( ( ord_less_nat @ A @ C )
=> ( B
!= ( suc @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(222,axiom,
( member_nat_b @ t
@ ( pi_nat_b
@ ( collect_nat
@ ^ [A: nat] : ( ord_less_eq_nat @ A @ m2 ) )
@ ^ [A: nat] : ( carrie2079586589xt_b_d @ r ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_2) ).
thf(1074,plain,
( member_nat_b @ t
@ ( pi_nat_b
@ ( collect_nat
@ ^ [A: nat] : ( ord_less_eq_nat @ A @ m2 ) )
@ ^ [A: nat] : ( carrie2079586589xt_b_d @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).
thf(262,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [B: a] : ( member_a @ B @ A ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_72_Collect__mem__eq) ).
thf(1206,plain,
! [A: set_a] :
( ( collect_a
@ ^ [B: a] : ( member_a @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).
thf(174,axiom,
! [A: set_nat_b,B: set_nat_b] :
( ! [C: nat > b] :
( ( member_nat_b @ C @ A )
=> ( member_nat_b @ C @ B ) )
=> ( ord_le1296932838_nat_b @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_281_subsetI) ).
thf(920,plain,
! [A: set_nat_b,B: set_nat_b] :
( ! [C: nat > b] :
( ( member_nat_b @ C @ A )
=> ( member_nat_b @ C @ B ) )
=> ( ord_le1296932838_nat_b @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(23,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( minus_minus_nat @ C @ B ) @ ( minus_minus_nat @ C @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_223_diff__less__mono2) ).
thf(380,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( minus_minus_nat @ C @ B ) @ ( minus_minus_nat @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(146,axiom,
! [A: set_b,B: nat > b,C: nat,D: nat > a] :
( ( ideal_b_d @ r @ A )
=> ( ( ( member_nat_b @ B
@ ( pi_nat_b
@ ( collect_nat
@ ^ [E: nat] : ( ord_less_eq_nat @ E @ C ) )
@ ^ [E: nat] : A ) )
& ( member_nat_a @ D
@ ( pi_nat_a
@ ( collect_nat
@ ^ [E: nat] : ( ord_less_eq_nat @ E @ C ) )
@ ^ [E: nat] : ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) ) )
=> ( ( l_comb_b_d_a_c @ r @ m @ C @ B @ D )
= ( zero_a1261444626_a_b_c @ m ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_277_linear__comb0__2Tr) ).
thf(810,plain,
! [A: set_b,B: nat > b,C: nat,D: nat > a] :
( ( ideal_b_d @ r @ A )
=> ( ( ( member_nat_b @ B
@ ( pi_nat_b
@ ( collect_nat
@ ^ [E: nat] : ( ord_less_eq_nat @ E @ C ) )
@ ^ [E: nat] : A ) )
& ( member_nat_a @ D
@ ( pi_nat_a
@ ( collect_nat
@ ^ [E: nat] : ( ord_less_eq_nat @ E @ C ) )
@ ^ [E: nat] : ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) ) )
=> ( ( l_comb_b_d_a_c @ r @ m @ C @ B @ D )
= ( zero_a1261444626_a_b_c @ m ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(305,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_337_singletonI) ).
thf(1338,plain,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).
thf(219,axiom,
ideal_b_d @ r @ ( annihilator_b_d_a_c @ r @ m ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_13_Ann__is__ideal) ).
thf(1067,plain,
ideal_b_d @ r @ ( annihilator_b_d_a_c @ r @ m ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).
thf(212,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( minus_minus_nat @ B @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_256_diff__less) ).
thf(1047,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( minus_minus_nat @ B @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(60,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d] :
( ( module_a_b_c_d @ A @ B )
=> ( misomo1282343797_c_a_c @ B @ A @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_115_Module_Omisom__self) ).
thf(512,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d] :
( ( module_a_b_c_d @ A @ B )
=> ( misomo1282343797_c_a_c @ B @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(182,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ C @ A ) )
= ( ord_less_nat @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_253_less__diff__iff) ).
thf(942,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ C @ A ) )
= ( ord_less_nat @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(67,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( suc @ A ) @ B )
= ( ord_less_nat @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_243_Suc__le__eq) ).
thf(546,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( suc @ A ) @ B )
= ( ord_less_nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(142,axiom,
! [A: nat > $o,B: nat] :
( ( A @ zero_zero_nat )
=> ( ! [C: nat] :
( ( A @ C )
=> ( A @ ( suc @ C ) ) )
=> ( A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_177_nat__induct) ).
thf(793,plain,
! [A: nat > $o,B: nat] :
( ( A @ zero_zero_nat )
=> ( ! [C: nat] :
( ( A @ C )
=> ( A @ ( suc @ C ) ) )
=> ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(247,axiom,
! [A: nat > nat > $o,B: nat,C: nat] :
( ! [D: nat] : ( A @ D @ zero_zero_nat )
=> ( ! [D: nat] : ( A @ zero_zero_nat @ ( suc @ D ) )
=> ( ! [D: nat,E: nat] :
( ( A @ D @ E )
=> ( A @ ( suc @ D ) @ ( suc @ E ) ) )
=> ( A @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_176_diff__induct) ).
thf(1153,plain,
! [A: nat > nat > $o,B: nat,C: nat] :
( ! [D: nat] : ( A @ D @ zero_zero_nat )
=> ( ! [D: nat] : ( A @ zero_zero_nat @ ( suc @ D ) )
=> ( ! [D: nat,E: nat] :
( ( A @ D @ E )
=> ( A @ ( suc @ D ) @ ( suc @ E ) ) )
=> ( A @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).
thf(129,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_283_subset__antisym) ).
thf(761,plain,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(171,axiom,
! [A: nat] :
( A
!= ( suc @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_138_n__not__Suc__n) ).
thf(910,plain,
! [A: nat] :
( A
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(228,axiom,
! [A: set_a,B: set_a] :
( ( submodule_b_d_a_c @ r @ m @ A )
=> ( ( submodule_b_d_a_c @ r @ m @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( submod903911234t_unit @ r @ ( mdl_a_b_c @ m @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_313_submodule__of__mdl) ).
thf(1091,plain,
! [A: set_a,B: set_a] :
( ( submodule_b_d_a_c @ r @ m @ A )
=> ( ( submodule_b_d_a_c @ r @ m @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( submod903911234t_unit @ r @ ( mdl_a_b_c @ m @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).
thf(307,axiom,
! [A: set_b,B: set_a,C: nat > b,D: nat,E: nat > a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_nat_b @ C
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : A ) )
=> ( ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : ( linear_span_b_d_a_c @ r @ m @ A @ B ) ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ D @ C @ E ) @ ( linear_span_b_d_a_c @ r @ m @ A @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_347_l__span__closed) ).
thf(1343,plain,
! [A: set_b,B: set_a,C: nat > b,D: nat,E: nat > a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_nat_b @ C
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : A ) )
=> ( ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : ( linear_span_b_d_a_c @ r @ m @ A @ B ) ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ D @ C @ E ) @ ( linear_span_b_d_a_c @ r @ m @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).
thf(12,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: carrie722926983_a_b_c] :
( ( module1821517916unit_d @ A @ B )
=> ( ( module_a_b_c_d @ C @ B )
=> ( ( misomo1935876354it_a_c @ B @ A @ C )
=> ( misomo1857159554t_unit @ B @ C @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_122_Module_Omisom__sym) ).
thf(349,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: carrie722926983_a_b_c] :
( ( module1821517916unit_d @ A @ B )
=> ( ( module_a_b_c_d @ C @ B )
=> ( ( misomo1935876354it_a_c @ B @ A @ C )
=> ( misomo1857159554t_unit @ B @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(250,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( suc @ B ) )
=> ( ~ ( ord_less_eq_nat @ A @ B )
=> ( A
= ( suc @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_163_le__SucE) ).
thf(1174,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( suc @ B ) )
=> ( ~ ( ord_less_eq_nat @ A @ B )
=> ( A
= ( suc @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).
thf(31,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_77_neq0__conv) ).
thf(407,plain,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(272,axiom,
! [A: a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ A @ B ) @ C )
= ( ( member_a @ A @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_333_insert__subset) ).
thf(1238,plain,
! [A: a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ A @ B ) @ C )
= ( ( member_a @ A @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).
thf(203,axiom,
module_a_b_c_d @ m @ r,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_15_Module__axioms) ).
thf(1021,plain,
module_a_b_c_d @ m @ r,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(136,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_b] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ideal_b_d @ B @ C )
=> ( submod903911234t_unit @ B @ A @ ( smodul132175289t_unit @ B @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_150_Module_Osmodule__ideal__coeff__is__Submodule) ).
thf(778,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_b] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ideal_b_d @ B @ C )
=> ( submod903911234t_unit @ B @ A @ ( smodul132175289t_unit @ B @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(45,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( suc @ B ) )
=> ( ~ ( ord_less_nat @ A @ B )
=> ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_200_less__SucE) ).
thf(467,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( suc @ B ) )
=> ( ~ ( ord_less_nat @ A @ B )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(16,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_209_not__less0) ).
thf(361,plain,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(152,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( free_g1087686480_d_a_c @ B @ A @ C )
=> ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_269_Module_Ofree__generator__sub) ).
thf(830,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( free_g1087686480_d_a_c @ B @ A @ C )
=> ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(128,axiom,
! [A: nat] :
( zero_zero_nat
!= ( suc @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_179_old_Onat_Odistinct_I1_J) ).
thf(757,plain,
! [A: nat] :
( zero_zero_nat
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(258,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d] :
( ( module1821517916unit_d @ A @ B )
=> ( ideal_b_d @ B @ ( annihi259882159t_unit @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_132_Module_OAnn__is__ideal) ).
thf(1195,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d] :
( ( module1821517916unit_d @ A @ B )
=> ( ideal_b_d @ B @ ( annihi259882159t_unit @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).
thf(240,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( suc @ A ) @ B )
=> ( ord_less_eq_nat @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_164_Suc__leD) ).
thf(1128,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( suc @ A ) @ B )
=> ( ord_less_eq_nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).
thf(285,axiom,
! [A: set_a,B: set_a] :
( ! [C: a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_282_subsetI) ).
thf(1281,plain,
! [A: set_a,B: set_a] :
( ! [C: a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).
thf(42,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: carrie1963041556t_unit] :
( ( module_a_b_c_d @ A @ B )
=> ( ( module1821517916unit_d @ C @ B )
=> ( ( misomo1857159554t_unit @ B @ A @ C )
=> ( misomo1935876354it_a_c @ B @ C @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_120_Module_Omisom__sym) ).
thf(456,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: carrie1963041556t_unit] :
( ( module_a_b_c_d @ A @ B )
=> ( ( module1821517916unit_d @ C @ B )
=> ( ( misomo1857159554t_unit @ B @ A @ C )
=> ( misomo1935876354it_a_c @ B @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(200,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ ( suc @ A ) )
= ( B = A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_239_le__less__Suc__eq) ).
thf(1012,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ ( suc @ A ) )
= ( B = A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(234,axiom,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ? [B: nat] :
( A
= ( suc @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_247_gr0__implies__Suc) ).
thf(1106,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ? [B: nat] :
( A
= ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).
thf(6,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ A @ ( suc @ B ) )
=> ( B = A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_193_less__antisym) ).
thf(329,plain,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ A @ ( suc @ B ) )
=> ( B = A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(252,axiom,
! [A: a,B: set_a] :
( ( insert_a @ A @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_309_insert__absorb2) ).
thf(1178,plain,
! [A: a,B: set_a] :
( ( insert_a @ A @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).
thf(289,axiom,
( ( g @ m2 )
= h2 ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_3) ).
thf(1296,plain,
( ( g @ m2 )
= h2 ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).
thf(107,axiom,
! [A: nat > a,B: ( nat > a ) > $o] :
( ( member_nat_a @ A @ ( collect_nat_a @ B ) )
= ( B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_66_mem__Collect__eq) ).
thf(665,plain,
! [A: nat > a,B: ( nat > a ) > $o] :
( ( member_nat_a @ A @ ( collect_nat_a @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(310,axiom,
! [A: set_a,B: a] :
( ( A != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_a @ B @ ( algebr1152250919_d_a_c @ r @ m @ A ) )
=> ? [C: nat,D: nat > a] :
( ( member_nat_a @ D
@ ( pi_nat_a
@ ( collect_nat
@ ^ [E: nat] : ( ord_less_eq_nat @ E @ C ) )
@ ^ [E: nat] : A ) )
& ? [E: nat > b] :
( ( member_nat_b @ E
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : ( carrie2079586589xt_b_d @ r ) ) )
& ( B
= ( l_comb_b_d_a_c @ r @ m @ C @ E @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_14_mem__fgs__l__comb) ).
thf(1350,plain,
! [A: set_a,B: a] :
( ( A != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_a @ B @ ( algebr1152250919_d_a_c @ r @ m @ A ) )
=> ? [C: nat,D: nat > a] :
( ( member_nat_a @ D
@ ( pi_nat_a
@ ( collect_nat
@ ^ [E: nat] : ( ord_less_eq_nat @ E @ C ) )
@ ^ [E: nat] : A ) )
& ? [E: nat > b] :
( ( member_nat_b @ E
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : ( carrie2079586589xt_b_d @ r ) ) )
& ( B
= ( l_comb_b_d_a_c @ r @ m @ C @ E @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).
thf(113,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( free_g637607517t_unit @ B @ A @ C )
=> ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_270_Module_Ofree__generator__sub) ).
thf(697,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( free_g637607517t_unit @ B @ A @ C )
=> ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(246,axiom,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ ( suc @ D ) )
=> ( A @ D ) )
=> ( A @ ( minus_minus_nat @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_219_zero__induct__lemma) ).
thf(1143,plain,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ ( suc @ D ) )
=> ( A @ D ) )
=> ( A @ ( minus_minus_nat @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).
thf(102,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_284_Diff__empty) ).
thf(650,plain,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(268,axiom,
( x
= ( l_comb_b_d_a_c @ r @ m @ m2 @ t @ g ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_13) ).
thf(1227,plain,
( x
= ( l_comb_b_d_a_c @ r @ m @ m2 @ t @ g ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).
thf(30,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( B != A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_145_less__not__refl2) ).
thf(403,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( B != A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(230,axiom,
! [A: carrie1963041556t_unit] :
( ( module1821517916unit_d @ A @ r )
=> ( module589927589unit_d @ ( hOM_b_717364638t_unit @ r @ m @ A ) @ r ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_37_HOM__is__module) ).
thf(1096,plain,
! [A: carrie1963041556t_unit] :
( ( module1821517916unit_d @ A @ r )
=> ( module589927589unit_d @ ( hOM_b_717364638t_unit @ r @ m @ A ) @ r ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).
thf(198,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( submodule_b_d_a_c @ B @ A @ C )
=> ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_266_Module_Osubmodule__subset) ).
thf(1006,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( submodule_b_d_a_c @ B @ A @ C )
=> ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(163,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ~ ! [B: nat] :
( A
!= ( suc @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_171_old_Onat_Oexhaust) ).
thf(869,plain,
! [A: nat] :
( ( A != zero_zero_nat )
=> ~ ! [B: nat] :
( A
!= ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(34,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_143_less__irrefl__nat) ).
thf(420,plain,
! [A: nat] :
~ ( ord_less_nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(56,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( free_g1087686480_d_a_c @ B @ A @ C )
=> ( generator_b_d_a_c @ B @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_127_Module_Ofree__generator__generator) ).
thf(499,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( free_g1087686480_d_a_c @ B @ A @ C )
=> ( generator_b_d_a_c @ B @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(62,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ C @ A ) )
= ( ord_less_eq_nat @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_217_le__diff__iff) ).
thf(519,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ C @ A ) )
= ( ord_less_eq_nat @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(122,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a,D: set_a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( submodule_b_d_a_c @ B @ A @ C )
=> ( ( submodule_b_d_a_c @ B @ A @ D )
=> ( ideal_b_d @ B @ ( quotie1153752712_d_a_c @ B @ A @ C @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_154_Module_Oquotient__of__submodules__is__ideal) ).
thf(742,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a,D: set_a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( submodule_b_d_a_c @ B @ A @ C )
=> ( ( submodule_b_d_a_c @ B @ A @ D )
=> ( ideal_b_d @ B @ ( quotie1153752712_d_a_c @ B @ A @ C @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(167,axiom,
! [A: nat,B: nat,C: nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( C @ A )
=> ( ! [D: nat] :
( ( ord_less_eq_nat @ A @ D )
=> ( ( C @ D )
=> ( C @ ( suc @ D ) ) ) )
=> ( C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_156_nat__induct__at__least) ).
thf(884,plain,
! [A: nat,B: nat,C: nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( C @ A )
=> ( ! [D: nat] :
( ( ord_less_eq_nat @ A @ D )
=> ( ( C @ D )
=> ( C @ ( suc @ D ) ) ) )
=> ( C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(53,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_183_le__neq__implies__less) ).
thf(488,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(287,axiom,
member_a @ ( l_comb_b_d_a_c @ r @ m @ m2 @ t @ g ) @ ( algebr1152250919_d_a_c @ r @ m @ ( insert_a @ h2 @ h1 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_12) ).
thf(1292,plain,
member_a @ ( l_comb_b_d_a_c @ r @ m @ m2 @ t @ g ) @ ( algebr1152250919_d_a_c @ r @ m @ ( insert_a @ h2 @ h1 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).
thf(99,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ A ) )
= ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_316_l__span__l__span) ).
thf(642,plain,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ A ) )
= ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(64,axiom,
! [A: nat > $o,B: nat] :
( ( A @ B )
=> ( ~ ( A @ zero_zero_nat )
=> ? [C: nat] :
( ( ord_less_nat @ C @ B )
& ! [D: nat] :
( ( ord_less_eq_nat @ D @ C )
=> ~ ( A @ D ) )
& ( A @ ( suc @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_263_ex__least__nat__less) ).
thf(526,plain,
! [A: nat > $o,B: nat] :
( ( A @ B )
=> ( ~ ( A @ zero_zero_nat )
=> ? [C: nat] :
( ( ord_less_nat @ C @ B )
& ! [D: nat] :
( ( ord_less_eq_nat @ D @ C )
=> ~ ( A @ D ) )
& ( A @ ( suc @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(83,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
| ( ord_less_eq_nat @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_135_nat__le__linear) ).
thf(590,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
| ( ord_less_eq_nat @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(209,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( genera1692266857t_unit @ r @ ( mdl_a_b_c @ m @ ( algebr1152250919_d_a_c @ r @ m @ A ) ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_320_fgs__generator) ).
thf(1039,plain,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( genera1692266857t_unit @ r @ ( mdl_a_b_c @ m @ ( algebr1152250919_d_a_c @ r @ m @ A ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(169,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_323_le__zero__eq) ).
thf(901,plain,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(223,axiom,
! [A: set_nat_a] :
( ( collect_nat_a
@ ^ [B: nat > a] : ( member_nat_a @ B @ A ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_70_Collect__mem__eq) ).
thf(1075,plain,
! [A: set_nat_a] :
( ( collect_nat_a
@ ^ [B: nat > a] : ( member_nat_a @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).
thf(213,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: carrie1963041556t_unit] :
( ( module1821517916unit_d @ A @ B )
=> ( ( module1821517916unit_d @ C @ B )
=> ( ( misomo493403663t_unit @ B @ A @ C )
=> ( misomo493403663t_unit @ B @ C @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_123_Module_Omisom__sym) ).
thf(1050,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: carrie1963041556t_unit] :
( ( module1821517916unit_d @ A @ B )
=> ( ( module1821517916unit_d @ C @ B )
=> ( ( misomo493403663t_unit @ B @ A @ C )
=> ( misomo493403663t_unit @ B @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(120,axiom,
! [A: nat,B: nat,C: nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( C @ A )
=> ( ! [D: nat] :
( ( ord_less_eq_nat @ A @ D )
=> ( ( ord_less_nat @ D @ B )
=> ( ( C @ D )
=> ( C @ ( suc @ D ) ) ) ) )
=> ( C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_242_dec__induct) ).
thf(721,plain,
! [A: nat,B: nat,C: nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( C @ A )
=> ( ! [D: nat] :
( ( ord_less_eq_nat @ A @ D )
=> ( ( ord_less_nat @ D @ B )
=> ( ( C @ D )
=> ( C @ ( suc @ D ) ) ) ) )
=> ( C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(181,axiom,
! [A: nat] :
( zero_zero_nat
!= ( suc @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_181_nat_Odistinct_I1_J) ).
thf(938,plain,
! [A: nat] :
( zero_zero_nat
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(273,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_b,D: a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ideal_b_d @ B @ C )
=> ( ( member_a @ D @ ( smodul818989740_d_a_c @ B @ A @ C ) )
=> ? [E: nat,F: nat > b] :
( ( member_nat_b @ F
@ ( pi_nat_b
@ ( collect_nat
@ ^ [G: nat] : ( ord_less_eq_nat @ G @ E ) )
@ ^ [G: nat] : C ) )
& ? [G: nat > a] :
( ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : ( carrie2021454486_a_b_c @ A ) ) )
& ( D
= ( l_comb_b_d_a_c @ B @ A @ E @ F @ G ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_229_Module_Omem__smodule__ideal__coeff) ).
thf(1241,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_b,D: a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ideal_b_d @ B @ C )
=> ( ( member_a @ D @ ( smodul818989740_d_a_c @ B @ A @ C ) )
=> ? [E: nat,F: nat > b] :
( ( member_nat_b @ F
@ ( pi_nat_b
@ ( collect_nat
@ ^ [G: nat] : ( ord_less_eq_nat @ G @ E ) )
@ ^ [G: nat] : C ) )
& ? [G: nat > a] :
( ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : ( carrie2021454486_a_b_c @ A ) ) )
& ( D
= ( l_comb_b_d_a_c @ B @ A @ E @ F @ G ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).
thf(95,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_168_less__eq__nat_Osimps_I1_J) ).
thf(633,plain,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(59,axiom,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat,E: nat] :
( ( ord_less_nat @ D @ E )
=> ( ord_less_nat @ ( A @ D ) @ ( A @ E ) ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( A @ B ) @ ( A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_182_less__mono__imp__le__mono) ).
thf(508,plain,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat,E: nat] :
( ( ord_less_nat @ D @ E )
=> ( ord_less_nat @ ( A @ D ) @ ( A @ E ) ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( A @ B ) @ ( A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(1,conjecture,
member_a @ ( g @ xa ) @ h1,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_21) ).
thf(2,negated_conjecture,
~ ( member_a @ ( g @ xa ) @ h1 ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(311,plain,
~ ( member_a @ ( g @ xa ) @ h1 ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(221,axiom,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
= ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_57_nat_Oinject) ).
thf(1071,plain,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
= ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).
thf(110,axiom,
! [A: nat,B: nat,C: nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( C @ B )
=> ( ! [D: nat] :
( ( ord_less_eq_nat @ A @ D )
=> ( ( ord_less_nat @ D @ B )
=> ( ( C @ ( suc @ D ) )
=> ( C @ D ) ) ) )
=> ( C @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_241_inc__induct) ).
thf(674,plain,
! [A: nat,B: nat,C: nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( C @ B )
=> ( ! [D: nat] :
( ( ord_less_eq_nat @ A @ D )
=> ( ( ord_less_nat @ D @ B )
=> ( ( C @ ( suc @ D ) )
=> ( C @ D ) ) ) )
=> ( C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(255,axiom,
! [A: set_a,B: set_a] :
( ( generator_b_d_a_c @ r @ m @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ord_less_eq_set_a @ A @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ B ) )
=> ( generator_b_d_a_c @ r @ m @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_342_generator__generator) ).
thf(1185,plain,
! [A: set_a,B: set_a] :
( ( generator_b_d_a_c @ r @ m @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ord_less_eq_set_a @ A @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ B ) )
=> ( generator_b_d_a_c @ r @ m @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).
thf(134,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ ( suc @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_236_le__imp__less__Suc) ).
thf(773,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(176,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( suc @ B ) )
= ( ( ord_less_eq_nat @ A @ B )
| ( A
= ( suc @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_160_le__Suc__eq) ).
thf(926,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( suc @ B ) )
= ( ( ord_less_eq_nat @ A @ B )
| ( A
= ( suc @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(28,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_211_gr0I) ).
thf(398,plain,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(207,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( suc @ B ) )
= ( ( A = zero_zero_nat )
| ? [C: nat] :
( ( A
= ( suc @ C ) )
& ( ord_less_nat @ C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_246_less__Suc__eq__0__disj) ).
thf(1035,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( suc @ B ) )
= ( ( A = zero_zero_nat )
| ? [C: nat] :
( ( A
= ( suc @ C ) )
& ( ord_less_nat @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(106,axiom,
! [A: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ A ) )
= ( ! [B: nat] :
~ ( A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_295_empty__Collect__eq) ).
thf(662,plain,
! [A: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ A ) )
= ( ! [B: nat] :
~ ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(119,axiom,
! [A: set_b,B: set_a,C: set_a] :
( ( ideal_b_d @ r @ A )
=> ( ( submodule_b_d_a_c @ r @ m @ B )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_319_l__span__sub__submodule) ).
thf(718,plain,
! [A: set_b,B: set_a,C: set_a] :
( ( ideal_b_d @ r @ A )
=> ( ( submodule_b_d_a_c @ r @ m @ B )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(93,axiom,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ! [D: nat] :
( ( ord_less_eq_nat @ ( suc @ D ) @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_157_full__nat__induct) ).
thf(618,plain,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ! [D: nat] :
( ( ord_less_eq_nat @ ( suc @ D ) @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(36,axiom,
! [A: nat,B: nat] :
( ( A != B )
= ( ( ord_less_nat @ A @ B )
| ( ord_less_nat @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_147_nat__neq__iff) ).
thf(425,plain,
! [A: nat,B: nat] :
( ( A != B )
= ( ( ord_less_nat @ A @ B )
| ( ord_less_nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(233,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) )
=> ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ B @ A @ C ) @ ( carrie2021454486_a_b_c @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_33_Module_Ofgs__sub__carrier) ).
thf(1103,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) )
=> ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ B @ A @ C ) @ ( carrie2021454486_a_b_c @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).
thf(150,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ A ) @ B ) @ ( suc @ C ) )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_79_Suc__diff__diff) ).
thf(824,plain,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ A ) @ B ) @ ( suc @ C ) )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(282,axiom,
! [A: set_a,B: a] :
( ( submodule_b_d_a_c @ r @ m @ A )
=> ( ( member_a @ B @ A )
=> ( member_a @ B @ ( carrie2021454486_a_b_c @ m ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1_submodule__subset1) ).
thf(1273,plain,
! [A: set_a,B: a] :
( ( submodule_b_d_a_c @ r @ m @ A )
=> ( ( member_a @ B @ A )
=> ( member_a @ B @ ( carrie2021454486_a_b_c @ m ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).
thf(123,axiom,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( suc @ ( minus_minus_nat @ A @ ( suc @ zero_zero_nat ) ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_38_Suc__pred) ).
thf(745,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( suc @ ( minus_minus_nat @ A @ ( suc @ zero_zero_nat ) ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(21,axiom,
! [A: set_a] :
( ( free_g1087686480_d_a_c @ r @ m @ A )
=> ( generator_b_d_a_c @ r @ m @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_40_free__generator__generator) ).
thf(375,plain,
! [A: set_a] :
( ( free_g1087686480_d_a_c @ r @ m @ A )
=> ( generator_b_d_a_c @ r @ m @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(86,axiom,
! [A: set_nat_b] :
( ( ! [B: nat > b] :
~ ( member_nat_b @ B @ A ) )
= ( A = bot_bot_set_nat_b ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_291_all__not__in__conv) ).
thf(601,plain,
! [A: set_nat_b] :
( ( ! [B: nat > b] :
~ ( member_nat_b @ B @ A ) )
= ( A = bot_bot_set_nat_b ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(81,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_220_minus__nat_Odiff__0) ).
thf(584,plain,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(35,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
| ( A = B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_185_le__eq__less__or__eq) ).
thf(423,plain,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
| ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(79,axiom,
! [A: set_b] :
( ( ideal_b_d @ r @ A )
=> ( submodule_b_d_a_c @ r @ m @ ( smodul818989740_d_a_c @ r @ m @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_17_smodule__ideal__coeff__is__Submodule) ).
thf(579,plain,
! [A: set_b] :
( ( ideal_b_d @ r @ A )
=> ( submodule_b_d_a_c @ r @ m @ ( smodul818989740_d_a_c @ r @ m @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(276,axiom,
( member_nat_a @ g
@ ( pi_nat_a
@ ( collect_nat
@ ^ [A: nat] : ( ord_less_eq_nat @ A @ m2 ) )
@ ^ [A: nat] : ( insert_a @ h2 @ h1 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).
thf(1256,plain,
( member_nat_a @ g
@ ( pi_nat_a
@ ( collect_nat
@ ^ [A: nat] : ( ord_less_eq_nat @ A @ m2 ) )
@ ^ [A: nat] : ( insert_a @ h2 @ h1 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).
thf(33,axiom,
! [A: nat,B: nat,C: nat > nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ! [D: nat] : ( C @ D @ ( suc @ D ) )
=> ( ! [D: nat,E: nat,F: nat] :
( ( ord_less_nat @ D @ E )
=> ( ( ord_less_nat @ E @ F )
=> ( ( C @ D @ E )
=> ( ( C @ E @ F )
=> ( C @ D @ F ) ) ) ) )
=> ( C @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_190_less__Suc__induct) ).
thf(413,plain,
! [A: nat,B: nat,C: nat > nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ! [D: nat] : ( C @ D @ ( suc @ D ) )
=> ( ! [D: nat,E: nat,F: nat] :
( ( ord_less_nat @ D @ E )
=> ( ( ord_less_nat @ E @ F )
=> ( ( C @ D @ E )
=> ( ( C @ E @ F )
=> ( C @ D @ F ) ) ) ) )
=> ( C @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(170,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a,D: nat > b,E: nat,F: nat > a,G: nat > a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) )
=> ( ( ( member_nat_b @ D
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : ( carrie2079586589xt_b_d @ B ) ) )
& ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
& ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
& ! [H: nat] :
( ( member_nat @ H
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ E ) ) )
=> ( ( F @ H )
= ( G @ H ) ) ) )
=> ( ( l_comb1138968323t_unit @ B @ A @ E @ D @ F )
= ( l_comb1138968323t_unit @ B @ A @ E @ D @ G ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_53_Module_Olinear__comb__eqTr) ).
thf(904,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a,D: nat > b,E: nat,F: nat > a,G: nat > a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) )
=> ( ( ( member_nat_b @ D
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : ( carrie2079586589xt_b_d @ B ) ) )
& ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
& ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
& ! [H: nat] :
( ( member_nat @ H
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ E ) ) )
=> ( ( F @ H )
= ( G @ H ) ) ) )
=> ( ( l_comb1138968323t_unit @ B @ A @ E @ D @ F )
= ( l_comb1138968323t_unit @ B @ A @ E @ D @ G ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(191,axiom,
! [A: nat > $o,B: nat] :
( ( A @ B )
=> ( ~ ( A @ zero_zero_nat )
=> ? [C: nat] :
( ( ord_less_eq_nat @ C @ B )
& ! [D: nat] :
( ( ord_less_nat @ D @ C )
=> ~ ( A @ D ) )
& ( A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_245_ex__least__nat__le) ).
thf(975,plain,
! [A: nat > $o,B: nat] :
( ( A @ B )
=> ( ~ ( A @ zero_zero_nat )
=> ? [C: nat] :
( ( ord_less_eq_nat @ C @ B )
& ! [D: nat] :
( ( ord_less_nat @ D @ C )
=> ~ ( A @ D ) )
& ( A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(8,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ A @ ( suc @ B ) )
= ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_188_not__less__less__Suc__eq) ).
thf(335,plain,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ A @ ( suc @ B ) )
= ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(267,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_b,D: set_a,E: nat > b,F: nat,G: nat > a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ideal_b_d @ B @ C )
=> ( ( ord_less_eq_set_a @ D @ ( carrie2021454486_a_b_c @ A ) )
=> ( ( member_nat_b @ E
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ F ) )
@ ^ [H: nat] : C ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ F ) )
@ ^ [H: nat] : D ) )
=> ( member_a @ ( l_comb_b_d_a_c @ B @ A @ F @ E @ G ) @ ( carrie2021454486_a_b_c @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_42_Module_Ol__comb__mem) ).
thf(1224,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_b,D: set_a,E: nat > b,F: nat,G: nat > a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ideal_b_d @ B @ C )
=> ( ( ord_less_eq_set_a @ D @ ( carrie2021454486_a_b_c @ A ) )
=> ( ( member_nat_b @ E
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ F ) )
@ ^ [H: nat] : C ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ F ) )
@ ^ [H: nat] : D ) )
=> ( member_a @ ( l_comb_b_d_a_c @ B @ A @ F @ E @ G ) @ ( carrie2021454486_a_b_c @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).
thf(26,axiom,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat] : ( ord_less_nat @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less_nat @ ( A @ B ) @ ( A @ C ) )
= ( ord_less_nat @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_234_lift__Suc__mono__less__iff) ).
thf(391,plain,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat] : ( ord_less_nat @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less_nat @ ( A @ B ) @ ( A @ C ) )
= ( ord_less_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(244,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: carrie1963041556t_unit] :
( ( module_a_b_c_d @ A @ B )
=> ( ( module1821517916unit_d @ C @ B )
=> ( module589927589unit_d @ ( hOM_b_717364638t_unit @ B @ A @ C ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_99_Module_OHOM__is__module) ).
thf(1139,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: carrie1963041556t_unit] :
( ( module_a_b_c_d @ A @ B )
=> ( ( module1821517916unit_d @ C @ B )
=> ( module589927589unit_d @ ( hOM_b_717364638t_unit @ B @ A @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).
thf(90,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_327_cancel__comm__monoid__add__class_Odiff__cancel) ).
thf(611,plain,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(139,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d] :
( ( module1821517916unit_d @ A @ B )
=> ( misomo493403663t_unit @ B @ A @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_114_Module_Omisom__self) ).
thf(787,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d] :
( ( module1821517916unit_d @ A @ B )
=> ( misomo493403663t_unit @ B @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(263,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a,D: a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( submod903911234t_unit @ B @ A @ C )
=> ( ( member_a @ D @ C )
=> ( member_a @ D @ ( carrie1074654371t_unit @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_259_Module_Osubmodule__subset1) ).
thf(1209,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a,D: a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( submod903911234t_unit @ B @ A @ C )
=> ( ( member_a @ D @ C )
=> ( member_a @ D @ ( carrie1074654371t_unit @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).
thf(138,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_166_bot__nat__0_Oextremum__unique) ).
thf(784,plain,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(309,axiom,
member_a @ h2 @ ( carrie2021454486_a_b_c @ m ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_17) ).
thf(1349,plain,
member_a @ h2 @ ( carrie2021454486_a_b_c @ m ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).
thf(22,axiom,
! [A: carrie1963041556t_unit,B: carrie1963041556t_unit] :
( ( module1821517916unit_d @ A @ r )
=> ( ( module1821517916unit_d @ B @ r )
=> ( ( misomo1935876354it_a_c @ r @ A @ m )
=> ( ( misomo1857159554t_unit @ r @ m @ B )
=> ( misomo493403663t_unit @ r @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_86_misom__trans) ).
thf(377,plain,
! [A: carrie1963041556t_unit,B: carrie1963041556t_unit] :
( ( module1821517916unit_d @ A @ r )
=> ( ( module1821517916unit_d @ B @ r )
=> ( ( misomo1935876354it_a_c @ r @ A @ m )
=> ( ( misomo1857159554t_unit @ r @ m @ B )
=> ( misomo493403663t_unit @ r @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(5,axiom,
! [A: nat,B: nat,C: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ! [D: nat] :
( ( B
= ( suc @ D ) )
=> ( C @ D ) )
=> ( ! [D: nat] :
( ( ord_less_nat @ D @ B )
=> ( ( C @ ( suc @ D ) )
=> ( C @ D ) ) )
=> ( C @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_189_strict__inc__induct) ).
thf(318,plain,
! [A: nat,B: nat,C: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ! [D: nat] :
( ( B
= ( suc @ D ) )
=> ( C @ D ) )
=> ( ! [D: nat] :
( ( ord_less_nat @ D @ B )
=> ( ( C @ ( suc @ D ) )
=> ( C @ D ) ) )
=> ( C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(294,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a,D: a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( submodule_b_d_a_c @ B @ A @ C )
=> ( ( member_a @ D @ C )
=> ( member_a @ D @ ( carrie2021454486_a_b_c @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_258_Module_Osubmodule__subset1) ).
thf(1307,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a,D: a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( submodule_b_d_a_c @ B @ A @ C )
=> ( ( member_a @ D @ C )
=> ( member_a @ D @ ( carrie2021454486_a_b_c @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).
thf(130,axiom,
! [A: a,B: set_a,C: a] :
( ( ( insert_a @ A @ B )
= ( insert_a @ C @ bot_bot_set_a ) )
= ( ( A = C )
& ( ord_less_eq_set_a @ B @ ( insert_a @ C @ bot_bot_set_a ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_273_singleton__insert__inj__eq_H) ).
thf(764,plain,
! [A: a,B: set_a,C: a] :
( ( ( insert_a @ A @ B )
= ( insert_a @ C @ bot_bot_set_a ) )
= ( ( A = C )
& ( ord_less_eq_set_a @ B @ ( insert_a @ C @ bot_bot_set_a ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(87,axiom,
! [A: set_b,B: set_a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( submodule_b_d_a_c @ r @ m @ ( linear_span_b_d_a_c @ r @ m @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_338_linear__span__subModule) ).
thf(604,plain,
! [A: set_b,B: set_a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( submodule_b_d_a_c @ r @ m @ ( linear_span_b_d_a_c @ r @ m @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(166,axiom,
! [A: set_a,B: set_a,C: nat > a,D: nat,E: nat > b] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( ( member_nat_a @ C
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : B ) )
& ( member_nat_b @ E
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : ( carrie2079586589xt_b_d @ r ) ) ) )
=> ( ( l_comb1138968323t_unit @ r @ ( mdl_a_b_c @ m @ ( algebr1152250919_d_a_c @ r @ m @ B ) ) @ D @ E @ C )
= ( l_comb_b_d_a_c @ r @ m @ D @ E @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_271_eSum__in__SubmoduleTr) ).
thf(880,plain,
! [A: set_a,B: set_a,C: nat > a,D: nat,E: nat > b] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( ( member_nat_a @ C
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : B ) )
& ( member_nat_b @ E
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : ( carrie2079586589xt_b_d @ r ) ) ) )
=> ( ( l_comb1138968323t_unit @ r @ ( mdl_a_b_c @ m @ ( algebr1152250919_d_a_c @ r @ m @ B ) ) @ D @ E @ C )
= ( l_comb_b_d_a_c @ r @ m @ D @ E @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(39,axiom,
! [A: nat,B: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ B ) )
= ( B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_69_mem__Collect__eq) ).
thf(434,plain,
! [A: nat,B: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(92,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ord_less_eq_set_a @ A @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_315_l__span__cont__H) ).
thf(616,plain,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ord_less_eq_set_a @ A @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(239,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a,D: nat > b,E: nat,F: nat > a,G: nat > a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) )
=> ( ( ( member_nat_b @ D
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : ( carrie2079586589xt_b_d @ B ) ) )
& ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
& ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
& ! [H: nat] :
( ( member_nat @ H
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ E ) ) )
=> ( ( F @ H )
= ( G @ H ) ) ) )
=> ( ( l_comb_b_d_a_c @ B @ A @ E @ D @ F )
= ( l_comb_b_d_a_c @ B @ A @ E @ D @ G ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_52_Module_Olinear__comb__eqTr) ).
thf(1122,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a,D: nat > b,E: nat,F: nat > a,G: nat > a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) )
=> ( ( ( member_nat_b @ D
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : ( carrie2079586589xt_b_d @ B ) ) )
& ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
& ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
& ! [H: nat] :
( ( member_nat @ H
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ E ) ) )
=> ( ( F @ H )
= ( G @ H ) ) ) )
=> ( ( l_comb_b_d_a_c @ B @ A @ E @ D @ F )
= ( l_comb_b_d_a_c @ B @ A @ E @ D @ G ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).
thf(25,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_144_less__not__refl3) ).
thf(387,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(55,axiom,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less_nat @ D @ C )
& ~ ( A @ D ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_141_infinite__descent) ).
thf(494,plain,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less_nat @ D @ C )
& ~ ( A @ D ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(89,axiom,
! [A: carrie722926983_a_b_c] :
( ( module_a_b_c_d @ A @ r )
=> ( ( misomo1282343797_c_a_c @ r @ m @ A )
=> ( misomo1282343797_c_a_c @ r @ A @ m ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_60_misom__sym) ).
thf(609,plain,
! [A: carrie722926983_a_b_c] :
( ( module_a_b_c_d @ A @ r )
=> ( ( misomo1282343797_c_a_c @ r @ m @ A )
=> ( misomo1282343797_c_a_c @ r @ A @ m ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(51,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( suc @ A ) @ ( suc @ B ) )
=> ( ord_less_nat @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_192_Suc__less__SucD) ).
thf(483,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( suc @ A ) @ ( suc @ B ) )
=> ( ord_less_nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(132,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d] :
( ( module1821517916unit_d @ A @ B )
=> ( ring_b_d @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_226_Module_Osc__Ring) ).
thf(769,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d] :
( ( module1821517916unit_d @ A @ B )
=> ( ring_b_d @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(204,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_82_diff__0__eq__0) ).
thf(1022,plain,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(251,axiom,
ring_b_d @ r,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_0_sc__Ring) ).
thf(1177,plain,
ring_b_d @ r,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).
thf(236,axiom,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
= ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_56_old_Onat_Oinject) ).
thf(1112,plain,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
= ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).
thf(159,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( suc @ A ) @ B )
=> ? [C: nat] :
( B
= ( suc @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_161_Suc__le__D) ).
thf(858,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( suc @ A ) @ B )
=> ? [C: nat] :
( B
= ( suc @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(18,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_186_less__imp__le__nat) ).
thf(367,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(69,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_137_le__trans) ).
thf(552,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(158,axiom,
! [A: nat > $o,B: nat] :
( ( A @ zero_zero_nat )
=> ( ! [C: nat] :
( ( A @ C )
=> ( A @ ( suc @ C ) ) )
=> ( A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_170_old_Onat_Oinducts) ).
thf(848,plain,
! [A: nat > $o,B: nat] :
( ( A @ zero_zero_nat )
=> ( ! [C: nat] :
( ( A @ C )
=> ( A @ ( suc @ C ) ) )
=> ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(114,axiom,
! [A: set_b,B: set_a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A @ B ) @ ( carrie2021454486_a_b_c @ m ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_318_linear__span__sub) ).
thf(700,plain,
! [A: set_b,B: set_a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A @ B ) @ ( carrie2021454486_a_b_c @ m ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(288,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a,D: a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) )
=> ( ( member_a @ D @ C )
=> ( member_a @ D @ ( algebr1152250919_d_a_c @ B @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_21_Module_Oelem__fgs) ).
thf(1293,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a,D: a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) )
=> ( ( member_a @ D @ C )
=> ( member_a @ D @ ( algebr1152250919_d_a_c @ B @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).
thf(108,axiom,
! [A: set_b] :
( ( ideal_b_d @ r @ A )
=> ( ( linear_span_b_d_a_c @ r @ m @ A @ ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_343_l__span__zero) ).
thf(668,plain,
! [A: set_b] :
( ( ideal_b_d @ r @ A )
=> ( ( linear_span_b_d_a_c @ r @ m @ A @ ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(44,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_nat @ A @ B )
| ( A = B ) )
=> ( ord_less_eq_nat @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_184_less__or__eq__imp__le) ).
thf(462,plain,
! [A: nat,B: nat] :
( ( ( ord_less_nat @ A @ B )
| ( A = B ) )
=> ( ord_less_eq_nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(178,axiom,
! [A: set_a] :
( ( free_g1087686480_d_a_c @ r @ m @ A )
=> ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_4_free__generator__sub) ).
thf(932,plain,
! [A: set_a] :
( ( free_g1087686480_d_a_c @ r @ m @ A )
=> ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(249,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_325_zero__diff) ).
thf(1171,plain,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(186,axiom,
submodule_b_d_a_c @ r @ m @ ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_321_submodule__0) ).
thf(959,plain,
submodule_b_d_a_c @ r @ m @ ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(151,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_215_diff__le__mono) ).
thf(827,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(201,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ B @ ( minus_minus_nat @ B @ A ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_78_diff__diff__cancel) ).
thf(1015,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ B @ ( minus_minus_nat @ B @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).
thf(73,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ ( suc @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_162_le__SucI) ).
thf(563,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(58,axiom,
! [A: nat,B: nat > $o] :
( ( ? [C: nat] :
( ( ord_less_nat @ C @ ( suc @ A ) )
& ( B @ C ) ) )
= ( ( B @ A )
| ? [C: nat] :
( ( ord_less_nat @ C @ A )
& ( B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_198_Ex__less__Suc) ).
thf(505,plain,
! [A: nat,B: nat > $o] :
( ( ? [C: nat] :
( ( ord_less_nat @ C @ ( suc @ A ) )
& ( B @ C ) ) )
= ( ( B @ A )
| ? [C: nat] :
( ( ord_less_nat @ C @ A )
& ( B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(38,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [B: nat] : ( member_nat @ B @ A ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_73_Collect__mem__eq) ).
thf(431,plain,
! [A: set_nat] :
( ( collect_nat
@ ^ [B: nat] : ( member_nat @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(103,axiom,
! [A: set_nat_b] :
( ( collect_nat_b
@ ^ [B: nat > b] : ( member_nat_b @ B @ A ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_71_Collect__mem__eq) ).
thf(653,plain,
! [A: set_nat_b] :
( ( collect_nat_b
@ ^ [B: nat > b] : ( member_nat_b @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(211,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_285_empty__Diff) ).
thf(1044,plain,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(112,axiom,
( ( algebr1152250919_d_a_c @ r @ m @ bot_bot_set_a )
= ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_322_empty__fgs) ).
thf(695,plain,
( ( algebr1152250919_d_a_c @ r @ m @ bot_bot_set_a )
= ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(293,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_b,D: set_a,E: nat] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ideal_b_d @ B @ C )
=> ( ( ord_less_eq_set_a @ D @ ( carrie1074654371t_unit @ A ) )
=> ! [F: nat > b,G: nat > a] :
( ( ( member_nat_b @ F
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
& ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : D ) ) )
=> ( member_a @ ( l_comb1138968323t_unit @ B @ A @ E @ F @ G ) @ ( carrie1074654371t_unit @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_46_Module_Oliear__comb__memTr) ).
thf(1304,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_b,D: set_a,E: nat] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ideal_b_d @ B @ C )
=> ( ( ord_less_eq_set_a @ D @ ( carrie1074654371t_unit @ A ) )
=> ! [F: nat > b,G: nat > a] :
( ( ( member_nat_b @ F
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
& ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : D ) ) )
=> ( member_a @ ( l_comb1138968323t_unit @ B @ A @ E @ F @ G ) @ ( carrie1074654371t_unit @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).
thf(261,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ m2 )
=> ! [B: nat] :
( ( ord_less_eq_nat @ B @ m2 )
=> ( ( ( g @ A )
= ( g @ B ) )
=> ( A = B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_1) ).
thf(1203,plain,
! [A: nat] :
( ( ord_less_eq_nat @ A @ m2 )
=> ! [B: nat] :
( ( ord_less_eq_nat @ B @ m2 )
=> ( ( ( g @ A )
= ( g @ B ) )
=> ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).
thf(101,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_63_le0) ).
thf(648,plain,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(225,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ ( suc @ A ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_244_Suc__leI) ).
thf(1081,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ ( suc @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(306,axiom,
! [A: set_b,B: set_a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( member_a @ ( zero_a1261444626_a_b_c @ m ) @ ( linear_span_b_d_a_c @ r @ m @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_341_linear__span__inc__0) ).
thf(1340,plain,
! [A: set_b,B: set_a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( member_a @ ( zero_a1261444626_a_b_c @ m ) @ ( linear_span_b_d_a_c @ r @ m @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).
thf(202,axiom,
! [A: set_b,B: set_a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A @ B ) @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_340_lin__span__coeff__mono) ).
thf(1018,plain,
! [A: set_b,B: set_a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A @ B ) @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).
thf(14,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: carrie722926983_a_b_c] :
( ( module_a_b_c_d @ A @ B )
=> ( ( module_a_b_c_d @ C @ B )
=> ( ( misomo1282343797_c_a_c @ B @ A @ C )
=> ( misomo1282343797_c_a_c @ B @ C @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_124_Module_Omisom__sym) ).
thf(355,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: carrie722926983_a_b_c] :
( ( module_a_b_c_d @ A @ B )
=> ( ( module_a_b_c_d @ C @ B )
=> ( ( misomo1282343797_c_a_c @ B @ A @ C )
=> ( misomo1282343797_c_a_c @ B @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(126,axiom,
ord_less_nat @ zero_zero_nat @ m2,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_16) ).
thf(754,plain,
ord_less_nat @ zero_zero_nat @ m2,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(253,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( suc @ A ) @ ( suc @ B ) )
= ( ord_less_eq_nat @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_61_Suc__le__mono) ).
thf(1181,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( suc @ A ) @ ( suc @ B ) )
= ( ord_less_eq_nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(154,axiom,
! [A: carrie722926983_a_b_c,B: carrie1963041556t_unit] :
( ( module_a_b_c_d @ A @ r )
=> ( ( module1821517916unit_d @ B @ r )
=> ( ( misomo1282343797_c_a_c @ r @ A @ m )
=> ( ( misomo1857159554t_unit @ r @ m @ B )
=> ( misomo1857159554t_unit @ r @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_90_misom__trans) ).
thf(836,plain,
! [A: carrie722926983_a_b_c,B: carrie1963041556t_unit] :
( ( module_a_b_c_d @ A @ r )
=> ( ( module1821517916unit_d @ B @ r )
=> ( ( misomo1282343797_c_a_c @ r @ A @ m )
=> ( ( misomo1857159554t_unit @ r @ m @ B )
=> ( misomo1857159554t_unit @ r @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(98,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d] :
( ( module_a_b_c_d @ A @ B )
=> ( submodule_b_d_a_c @ B @ A @ ( carrie2021454486_a_b_c @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_261_Module_Osubmodule__whole) ).
thf(640,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d] :
( ( module_a_b_c_d @ A @ B )
=> ( submodule_b_d_a_c @ B @ A @ ( carrie2021454486_a_b_c @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(205,axiom,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ D )
=> ( ord_less_eq_nat @ D @ C ) )
=> ? [D: nat] :
( ( A @ D )
& ! [E: nat] :
( ( A @ E )
=> ( ord_less_eq_nat @ E @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_134_Nat_Oex__has__greatest__nat) ).
thf(1025,plain,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ D )
=> ( ord_less_eq_nat @ D @ C ) )
=> ? [D: nat] :
( ( A @ D )
& ! [E: nat] :
( ( A @ E )
=> ( ord_less_eq_nat @ E @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(121,axiom,
! [A: nat > b] :
~ ( member_nat_b @ A @ bot_bot_set_nat_b ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_288_empty__iff) ).
thf(739,plain,
! [A: nat > b] :
~ ( member_nat_b @ A @ bot_bot_set_nat_b ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(231,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_330_empty__subsetI) ).
thf(1098,plain,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).
thf(194,axiom,
! [A: nat,B: nat] : ( ord_less_nat @ ( minus_minus_nat @ A @ B ) @ ( suc @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_254_diff__less__Suc) ).
thf(993,plain,
! [A: nat,B: nat] : ( ord_less_nat @ ( minus_minus_nat @ A @ B ) @ ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(117,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) )
=> ( submod903911234t_unit @ B @ A @ ( algebr1039611956t_unit @ B @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_26_Module_Ofgs__submodule) ).
thf(712,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) )
=> ( submod903911234t_unit @ B @ A @ ( algebr1039611956t_unit @ B @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(71,axiom,
! [A: set_a,B: set_a] :
( ( submodule_b_d_a_c @ r @ m @ A )
=> ( ( submodule_b_d_a_c @ r @ m @ B )
=> ( ideal_b_d @ r @ ( quotie1153752712_d_a_c @ r @ m @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_16_quotient__of__submodules__is__ideal) ).
thf(557,plain,
! [A: set_a,B: set_a] :
( ( submodule_b_d_a_c @ r @ m @ A )
=> ( ( submodule_b_d_a_c @ r @ m @ B )
=> ( ideal_b_d @ r @ ( quotie1153752712_d_a_c @ r @ m @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(197,axiom,
! [A: nat > $o,B: nat > $o] :
( ! [C: nat] :
( ( A @ C )
= ( B @ C ) )
=> ( ( collect_nat @ A )
= ( collect_nat @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_74_Collect__cong) ).
thf(1003,plain,
! [A: nat > $o,B: nat > $o] :
( ! [C: nat] :
( ( A @ C )
= ( B @ C ) )
=> ( ( collect_nat @ A )
= ( collect_nat @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).
thf(214,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_167_le__0__eq) ).
thf(1053,plain,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(291,axiom,
! [A: set_b,B: set_a,C: nat > b,D: nat,E: nat > a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_nat_b @ C
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : A ) )
=> ( ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : B ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ D @ C @ E ) @ ( carrie2021454486_a_b_c @ m ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_11_l__comb__mem) ).
thf(1300,plain,
! [A: set_b,B: set_a,C: nat > b,D: nat,E: nat > a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_nat_b @ C
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : A ) )
=> ( ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : B ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ D @ C @ E ) @ ( carrie2021454486_a_b_c @ m ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).
thf(141,axiom,
! [A: set_a] :
( ( submodule_b_d_a_c @ r @ m @ A )
=> ( module1821517916unit_d @ ( mdl_a_b_c @ m @ A ) @ r ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_312_mdl__is__module) ).
thf(791,plain,
! [A: set_a] :
( ( submodule_b_d_a_c @ r @ m @ A )
=> ( module1821517916unit_d @ ( mdl_a_b_c @ m @ A ) @ r ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(256,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( suc @ B ) @ A )
= ( suc @ ( minus_minus_nat @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_251_Suc__diff__le) ).
thf(1188,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( suc @ B ) @ A )
= ( suc @ ( minus_minus_nat @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).
thf(208,axiom,
submodule_b_d_a_c @ r @ m @ ( carrie2021454486_a_b_c @ m ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_2_submodule__whole) ).
thf(1038,plain,
submodule_b_d_a_c @ r @ m @ ( carrie2021454486_a_b_c @ m ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(57,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( suc @ B ) )
= ( ( ord_less_nat @ A @ B )
| ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_197_less__Suc__eq) ).
thf(502,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( suc @ B ) )
= ( ( ord_less_nat @ A @ B )
| ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(271,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_b,D: set_a,E: nat] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ideal_b_d @ B @ C )
=> ( ( ord_less_eq_set_a @ D @ ( carrie2021454486_a_b_c @ A ) )
=> ! [F: nat > b,G: nat > a] :
( ( ( member_nat_b @ F
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
& ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : D ) ) )
=> ( member_a @ ( l_comb_b_d_a_c @ B @ A @ E @ F @ G ) @ ( carrie2021454486_a_b_c @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_45_Module_Oliear__comb__memTr) ).
thf(1235,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_b,D: set_a,E: nat] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ideal_b_d @ B @ C )
=> ( ( ord_less_eq_set_a @ D @ ( carrie2021454486_a_b_c @ A ) )
=> ! [F: nat > b,G: nat > a] :
( ( ( member_nat_b @ F
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
& ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : D ) ) )
=> ( member_a @ ( l_comb_b_d_a_c @ B @ A @ E @ F @ G ) @ ( carrie2021454486_a_b_c @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).
thf(298,axiom,
! [A: a,B: set_a,C: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = C ) )
=> ( member_a @ A @ ( insert_a @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_305_insertCI) ).
thf(1317,plain,
! [A: a,B: set_a,C: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = C ) )
=> ( member_a @ A @ ( insert_a @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).
thf(9,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( suc @ A ) @ ( suc @ B ) )
= ( ord_less_nat @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_64_Suc__less__eq) ).
thf(338,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( suc @ A ) @ ( suc @ B ) )
= ( ord_less_nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(145,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ A @ B ) @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_214_diff__le__self) ).
thf(808,plain,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ A @ B ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(241,axiom,
! [A: a,B: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ B @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_334_insert__Diff__single) ).
thf(1130,plain,
! [A: a,B: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ B @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).
thf(265,axiom,
! [A: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_a @ B @ A )
=> ( member_a @ B @ ( algebr1152250919_d_a_c @ r @ m @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_5_elem__fgs) ).
thf(1218,plain,
! [A: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_a @ B @ A )
=> ( member_a @ B @ ( algebr1152250919_d_a_c @ r @ m @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).
thf(46,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( suc @ A ) @ B )
=> ( ord_less_nat @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_203_Suc__lessD) ).
thf(470,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( suc @ A ) @ B )
=> ( ord_less_nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(78,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: carrie722926983_a_b_c] :
( ( module1821517916unit_d @ A @ B )
=> ( ( module_a_b_c_d @ C @ B )
=> ( module589927589unit_d @ ( hOM_b_796081438it_a_c @ B @ A @ C ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_101_Module_OHOM__is__module) ).
thf(576,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: carrie722926983_a_b_c] :
( ( module1821517916unit_d @ A @ B )
=> ( ( module_a_b_c_d @ C @ B )
=> ( module589927589unit_d @ ( hOM_b_796081438it_a_c @ B @ A @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(277,axiom,
! [A: set_b,B: set_a,C: nat] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ! [D: nat > b,E: nat > a] :
( ( ( member_nat_b @ D
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : A ) )
& ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : B ) ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ C @ D @ E ) @ ( carrie2021454486_a_b_c @ m ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_12_liear__comb__memTr) ).
thf(1257,plain,
! [A: set_b,B: set_a,C: nat] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ! [D: nat > b,E: nat > a] :
( ( ( member_nat_b @ D
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : A ) )
& ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : B ) ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ C @ D @ E ) @ ( carrie2021454486_a_b_c @ m ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).
thf(115,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a,D: nat > b,E: nat,F: nat > a,G: nat > a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) )
=> ( ( member_nat_b @ D
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : ( carrie2079586589xt_b_d @ B ) ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
=> ( ! [H: nat] :
( ( member_nat @ H
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ E ) ) )
=> ( ( F @ H )
= ( G @ H ) ) )
=> ( ( l_comb_b_d_a_c @ B @ A @ E @ D @ F )
= ( l_comb_b_d_a_c @ B @ A @ E @ D @ G ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_49_Module_Olinear__comb__eq) ).
thf(703,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a,D: nat > b,E: nat,F: nat > a,G: nat > a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) )
=> ( ( member_nat_b @ D
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : ( carrie2079586589xt_b_d @ B ) ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
=> ( ! [H: nat] :
( ( member_nat @ H
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ E ) ) )
=> ( ( F @ H )
= ( G @ H ) ) )
=> ( ( l_comb_b_d_a_c @ B @ A @ E @ D @ F )
= ( l_comb_b_d_a_c @ B @ A @ E @ D @ G ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(104,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) )
=> ( submodule_b_d_a_c @ B @ A @ ( algebr1152250919_d_a_c @ B @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_27_Module_Ofgs__submodule) ).
thf(656,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) )
=> ( submodule_b_d_a_c @ B @ A @ ( algebr1152250919_d_a_c @ B @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(161,axiom,
ord_less_eq_set_a @ h @ ( carrie2021454486_a_b_c @ m ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_9) ).
thf(864,plain,
ord_less_eq_set_a @ h @ ( carrie2021454486_a_b_c @ m ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(229,axiom,
! [A: carrie722926983_a_b_c] :
( ( module_a_b_c_d @ A @ r )
=> ( module589927589unit_d @ ( hOM_b_d_a_c_a_c @ r @ m @ A ) @ r ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_36_HOM__is__module) ).
thf(1094,plain,
! [A: carrie722926983_a_b_c] :
( ( module_a_b_c_d @ A @ r )
=> ( module589927589unit_d @ ( hOM_b_d_a_c_a_c @ r @ m @ A ) @ r ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).
thf(76,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( suc @ B ) )
= ( ord_less_eq_nat @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_238_less__Suc__eq__le) ).
thf(571,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( suc @ B ) )
= ( ord_less_eq_nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(85,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a,D: nat > b,E: nat,F: nat > a,G: nat > a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) )
=> ( ( member_nat_b @ D
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : ( carrie2079586589xt_b_d @ B ) ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
=> ( ! [H: nat] :
( ( member_nat @ H
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ E ) ) )
=> ( ( F @ H )
= ( G @ H ) ) )
=> ( ( l_comb1138968323t_unit @ B @ A @ E @ D @ F )
= ( l_comb1138968323t_unit @ B @ A @ E @ D @ G ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_50_Module_Olinear__comb__eq) ).
thf(595,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a,D: nat > b,E: nat,F: nat > a,G: nat > a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) )
=> ( ( member_nat_b @ D
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : ( carrie2079586589xt_b_d @ B ) ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : C ) )
=> ( ! [H: nat] :
( ( member_nat @ H
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ E ) ) )
=> ( ( F @ H )
= ( G @ H ) ) )
=> ( ( l_comb1138968323t_unit @ B @ A @ E @ D @ F )
= ( l_comb1138968323t_unit @ B @ A @ E @ D @ G ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(281,axiom,
ord_less_eq_set_a @ h1 @ h,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_6) ).
thf(1272,plain,
ord_less_eq_set_a @ h1 @ h,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).
thf(165,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ( ( minus_minus_nat @ B @ A )
= ( minus_minus_nat @ C @ A ) )
= ( B = C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_218_eq__diff__iff) ).
thf(876,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ( ( minus_minus_nat @ B @ A )
= ( minus_minus_nat @ C @ A ) )
= ( B = C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(193,axiom,
( ord_less_nat
= ( ^ [A: nat] : ( ord_less_eq_nat @ ( suc @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_237_less__eq__Suc__le) ).
thf(991,plain,
( ord_less_nat
= ( ^ [A: nat] : ( ord_less_eq_nat @ ( suc @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).
thf(266,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a,D: a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) )
=> ( ( member_a @ D @ C )
=> ( member_a @ D @ ( algebr1039611956t_unit @ B @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_20_Module_Oelem__fgs) ).
thf(1221,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a,D: a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) )
=> ( ( member_a @ D @ C )
=> ( member_a @ D @ ( algebr1039611956t_unit @ B @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).
thf(286,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a,D: a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( C != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) )
=> ( ( member_a @ D @ ( algebr1152250919_d_a_c @ B @ A @ C ) )
=> ? [E: nat,F: nat > a] :
( ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [G: nat] : ( ord_less_eq_nat @ G @ E ) )
@ ^ [G: nat] : C ) )
& ? [G: nat > b] :
( ( member_nat_b @ G
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : ( carrie2079586589xt_b_d @ B ) ) )
& ( D
= ( l_comb_b_d_a_c @ B @ A @ E @ G @ F ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_30_Module_Omem__fgs__l__comb) ).
thf(1284,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_a,D: a] :
( ( module_a_b_c_d @ A @ B )
=> ( ( C != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ C @ ( carrie2021454486_a_b_c @ A ) )
=> ( ( member_a @ D @ ( algebr1152250919_d_a_c @ B @ A @ C ) )
=> ? [E: nat,F: nat > a] :
( ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [G: nat] : ( ord_less_eq_nat @ G @ E ) )
@ ^ [G: nat] : C ) )
& ? [G: nat > b] :
( ( member_nat_b @ G
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : ( carrie2079586589xt_b_d @ B ) ) )
& ( D
= ( l_comb_b_d_a_c @ B @ A @ E @ G @ F ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).
thf(65,axiom,
! [A: nat] :
( ( suc @ A )
!= zero_zero_nat ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_174_Suc__neq__Zero) ).
thf(539,plain,
! [A: nat] :
( ( suc @ A )
!= zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(109,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ord_less_eq_nat @ ( suc @ B ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_158_not__less__eq__eq) ).
thf(671,plain,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ord_less_eq_nat @ ( suc @ B ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(248,axiom,
! [A: nat,B: nat,C: nat > nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ! [D: nat] : ( C @ D @ D )
=> ( ! [D: nat,E: nat,F: nat] :
( ( C @ D @ E )
=> ( ( C @ E @ F )
=> ( C @ D @ F ) ) )
=> ( ! [D: nat] : ( C @ D @ ( suc @ D ) )
=> ( C @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_155_transitive__stepwise__le) ).
thf(1157,plain,
! [A: nat,B: nat,C: nat > nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ! [D: nat] : ( C @ D @ D )
=> ( ! [D: nat,E: nat,F: nat] :
( ( C @ D @ E )
=> ( ( C @ E @ F )
=> ( C @ D @ F ) ) )
=> ( ! [D: nat] : ( C @ D @ ( suc @ D ) )
=> ( C @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).
thf(80,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_252_diff__less__mono) ).
thf(581,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(157,axiom,
! [A: set_b,B: set_a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( linear_span_b_d_a_c @ r @ m @ A @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ B ) )
= ( linear_span_b_d_a_c @ r @ m @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_339_l__spanA__l__span) ).
thf(844,plain,
! [A: set_b,B: set_a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( linear_span_b_d_a_c @ r @ m @ A @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ B ) )
= ( linear_span_b_d_a_c @ r @ m @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(133,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_62_bot__nat__0_Oextremum) ).
thf(771,plain,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(283,axiom,
! [A: set_a] :
( ( ! [B: a] :
~ ( member_a @ B @ A ) )
= ( A = bot_bot_set_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_292_all__not__in__conv) ).
thf(1276,plain,
! [A: set_a] :
( ( ! [B: a] :
~ ( member_a @ B @ A ) )
= ( A = bot_bot_set_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).
thf(301,axiom,
ord_less_eq_nat @ xa @ ( minus_minus_nat @ m2 @ ( suc @ zero_zero_nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_19) ).
thf(1326,plain,
ord_less_eq_nat @ xa @ ( minus_minus_nat @ m2 @ ( suc @ zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).
thf(275,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_289_empty__iff) ).
thf(1253,plain,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).
thf(188,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( submod903911234t_unit @ B @ A @ C )
=> ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_267_Module_Osubmodule__subset) ).
thf(963,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( submod903911234t_unit @ B @ A @ C )
=> ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(124,axiom,
! [A: set_a,B: set_a,C: set_b] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ideal_b_d @ r @ C )
=> ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ C @ A ) @ ( linear_span_b_d_a_c @ r @ m @ C @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_317_l__span__gen__mono) ).
thf(748,plain,
! [A: set_a,B: set_a,C: set_b] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ideal_b_d @ r @ C )
=> ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ C @ A ) @ ( linear_span_b_d_a_c @ r @ m @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(217,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_286_Diff__cancel) ).
thf(1062,plain,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).
thf(183,axiom,
! [A: nat > $o] :
( ( ( collect_nat @ A )
= bot_bot_set_nat )
= ( ! [B: nat] :
~ ( A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_293_Collect__empty__eq) ).
thf(946,plain,
! [A: nat > $o] :
( ( ( collect_nat @ A )
= bot_bot_set_nat )
= ( ! [B: nat] :
~ ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(29,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ A @ ( suc @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_199_less__SucI) ).
thf(401,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ A @ ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(137,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: carrie1963041556t_unit] :
( ( module1821517916unit_d @ A @ B )
=> ( ( module1821517916unit_d @ C @ B )
=> ( module589927589unit_d @ ( hOM_b_1103679019t_unit @ B @ A @ C ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_102_Module_OHOM__is__module) ).
thf(781,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: carrie1963041556t_unit] :
( ( module1821517916unit_d @ A @ B )
=> ( ( module1821517916unit_d @ C @ B )
=> ( module589927589unit_d @ ( hOM_b_1103679019t_unit @ B @ A @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(195,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ C @ A ) )
= ( minus_minus_nat @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_216_Nat_Odiff__diff__eq) ).
thf(995,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ C @ A ) )
= ( minus_minus_nat @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(75,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_148_diff__commute) ).
thf(568,plain,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(172,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ ( suc @ B ) ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_264_diff__Suc__less) ).
thf(914,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ ( suc @ B ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(177,axiom,
! [A: nat,B: nat] :
( ( ( minus_minus_nat @ A @ B )
= zero_zero_nat )
= ( ord_less_eq_nat @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_54_diff__is__0__eq) ).
thf(929,plain,
! [A: nat,B: nat] :
( ( ( minus_minus_nat @ A @ B )
= zero_zero_nat )
= ( ord_less_eq_nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(49,axiom,
! [A: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ A ) )
= ( A = zero_zero_nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_210_not__gr0) ).
thf(478,plain,
! [A: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ A ) )
= ( A = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(264,axiom,
! [A: set_b,B: a] :
( ( ideal_b_d @ r @ A )
=> ( ( member_a @ B @ ( smodul818989740_d_a_c @ r @ m @ A ) )
=> ? [C: nat,D: nat > b] :
( ( member_nat_b @ D
@ ( pi_nat_b
@ ( collect_nat
@ ^ [E: nat] : ( ord_less_eq_nat @ E @ C ) )
@ ^ [E: nat] : A ) )
& ? [E: nat > a] :
( ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : ( carrie2021454486_a_b_c @ m ) ) )
& ( B
= ( l_comb_b_d_a_c @ r @ m @ C @ D @ E ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_18_mem__smodule__ideal__coeff) ).
thf(1212,plain,
! [A: set_b,B: a] :
( ( ideal_b_d @ r @ A )
=> ( ( member_a @ B @ ( smodul818989740_d_a_c @ r @ m @ A ) )
=> ? [C: nat,D: nat > b] :
( ( member_nat_b @ D
@ ( pi_nat_b
@ ( collect_nat
@ ^ [E: nat] : ( ord_less_eq_nat @ E @ C ) )
@ ^ [E: nat] : A ) )
& ? [E: nat > a] :
( ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ C ) )
@ ^ [F: nat] : ( carrie2021454486_a_b_c @ m ) ) )
& ( B
= ( l_comb_b_d_a_c @ r @ m @ C @ D @ E ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).
thf(24,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( B != zero_zero_nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_207_gr__implies__not0) ).
thf(383,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( B != zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(11,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_75_bot__nat__0_Onot__eq__extremum) ).
thf(346,plain,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(68,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ? [B: nat] :
( A
= ( suc @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_169_not0__implies__Suc) ).
thf(549,plain,
! [A: nat] :
( ( A != zero_zero_nat )
=> ? [B: nat] :
( A
= ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(242,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_b] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ideal_b_d @ B @ C )
=> ( submodule_b_d_a_c @ B @ A @ ( smodul818989740_d_a_c @ B @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_151_Module_Osmodule__ideal__coeff__is__Submodule) ).
thf(1133,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: set_b] :
( ( module_a_b_c_d @ A @ B )
=> ( ( ideal_b_d @ B @ C )
=> ( submodule_b_d_a_c @ B @ A @ ( smodul818989740_d_a_c @ B @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).
thf(196,axiom,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_139_Suc__inject) ).
thf(999,plain,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(269,axiom,
! [A: a,B: a > $o] :
( ( member_a @ A @ ( collect_a @ B ) )
= ( B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_68_mem__Collect__eq) ).
thf(1229,plain,
! [A: a,B: a > $o] :
( ( member_a @ A @ ( collect_a @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).
thf(292,axiom,
ord_less_eq_set_a @ h1 @ ( carrie2021454486_a_b_c @ m ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_10) ).
thf(1303,plain,
ord_less_eq_set_a @ h1 @ ( carrie2021454486_a_b_c @ m ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).
thf(162,axiom,
! [A: set_a,B: set_a,C: nat > a,D: nat,E: nat > b] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( member_nat_a @ C
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : B ) )
=> ( ( member_nat_b @ E
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : ( carrie2079586589xt_b_d @ r ) ) )
=> ( ( l_comb1138968323t_unit @ r @ ( mdl_a_b_c @ m @ ( algebr1152250919_d_a_c @ r @ m @ B ) ) @ D @ E @ C )
= ( l_comb_b_d_a_c @ r @ m @ D @ E @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_272_eSum__in__Submodule) ).
thf(865,plain,
! [A: set_a,B: set_a,C: nat > a,D: nat,E: nat > b] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( member_nat_a @ C
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : B ) )
=> ( ( member_nat_b @ E
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : ( carrie2079586589xt_b_d @ r ) ) )
=> ( ( l_comb1138968323t_unit @ r @ ( mdl_a_b_c @ m @ ( algebr1152250919_d_a_c @ r @ m @ B ) ) @ D @ E @ C )
= ( l_comb_b_d_a_c @ r @ m @ D @ E @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(153,axiom,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
= ( ? [B: nat] :
( A
= ( suc @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_249_gr0__conv__Suc) ).
thf(833,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
= ( ? [B: nat] :
( A
= ( suc @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(32,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( suc @ A ) @ B )
= ( ? [C: nat] :
( ( B
= ( suc @ C ) )
& ( ord_less_nat @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_194_Suc__less__eq2) ).
thf(410,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( suc @ A ) @ B )
= ( ? [C: nat] :
( ( B
= ( suc @ C ) )
& ( ord_less_nat @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(297,axiom,
! [A: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_a @ B @ A )
=> ( member_a @ B @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_314_h__in__linear__span) ).
thf(1314,plain,
! [A: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_a @ B @ A )
=> ( member_a @ B @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).
thf(10,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B
!= ( suc @ A ) )
=> ~ ! [C: nat] :
( ( ord_less_nat @ A @ C )
=> ( B
!= ( suc @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_204_Nat_OlessE) ).
thf(341,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B
!= ( suc @ A ) )
=> ~ ! [C: nat] :
( ( ord_less_nat @ A @ C )
=> ( B
!= ( suc @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(15,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( free_g637607517t_unit @ B @ A @ C )
=> ( genera1692266857t_unit @ B @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_126_Module_Ofree__generator__generator) ).
thf(358,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( free_g637607517t_unit @ B @ A @ C )
=> ( genera1692266857t_unit @ B @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(40,axiom,
! [A: nat > $o,B: nat] :
( ( A @ zero_zero_nat )
=> ( ! [C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less_nat @ D @ C )
& ~ ( A @ D ) ) ) )
=> ( A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_206_infinite__descent0) ).
thf(437,plain,
! [A: nat > $o,B: nat] :
( ( A @ zero_zero_nat )
=> ( ! [C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less_nat @ D @ C )
& ~ ( A @ D ) ) ) )
=> ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(131,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( suc @ A ) @ B )
=> ( ord_less_nat @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_240_Suc__le__lessD) ).
thf(767,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( suc @ A ) @ B )
=> ( ord_less_nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(215,axiom,
! [A: a > $o] :
( ( ( collect_a @ A )
= bot_bot_set_a )
= ( ! [B: a] :
~ ( A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_294_Collect__empty__eq) ).
thf(1056,plain,
! [A: a > $o] :
( ( ( collect_a @ A )
= bot_bot_set_a )
= ( ! [B: a] :
~ ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(259,axiom,
! [A: carrie1963041556t_unit,B: carrie722926983_a_b_c] :
( ( module1821517916unit_d @ A @ r )
=> ( ( module_a_b_c_d @ B @ r )
=> ( ( misomo1935876354it_a_c @ r @ A @ m )
=> ( ( misomo1282343797_c_a_c @ r @ m @ B )
=> ( misomo1935876354it_a_c @ r @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_88_misom__trans) ).
thf(1197,plain,
! [A: carrie1963041556t_unit,B: carrie722926983_a_b_c] :
( ( module1821517916unit_d @ A @ r )
=> ( ( module_a_b_c_d @ B @ r )
=> ( ( misomo1935876354it_a_c @ r @ A @ m )
=> ( ( misomo1282343797_c_a_c @ r @ m @ B )
=> ( misomo1935876354it_a_c @ r @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).
thf(218,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d] :
( ( module_a_b_c_d @ A @ B )
=> ( ideal_b_d @ B @ ( annihilator_b_d_a_c @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_133_Module_OAnn__is__ideal) ).
thf(1065,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d] :
( ( module_a_b_c_d @ A @ B )
=> ( ideal_b_d @ B @ ( annihilator_b_d_a_c @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(185,axiom,
! [A: nat] :
( zero_zero_nat
!= ( suc @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_173_Zero__neq__Suc) ).
thf(955,plain,
! [A: nat] :
( zero_zero_nat
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(308,axiom,
! [A: set_b,B: set_a,C: nat > b,D: nat,E: nat > a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_nat_b @ C
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : A ) )
=> ( ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ B ) ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ D @ C @ E ) @ ( linear_span_b_d_a_c @ r @ m @ A @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_276_l__span__closed2) ).
thf(1346,plain,
! [A: set_b,B: set_a,C: nat > b,D: nat,E: nat > a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_nat_b @ C
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : A ) )
=> ( ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ B ) ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ D @ C @ E ) @ ( linear_span_b_d_a_c @ r @ m @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).
thf(187,axiom,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat] : ( ord_less_eq_nat @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( A @ B ) @ ( A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_232_lift__Suc__mono__le) ).
thf(960,plain,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat] : ( ord_less_eq_nat @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( A @ B ) @ ( A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(96,axiom,
! [A: set_b,B: nat > b,C: nat,D: nat > a] :
( ( ideal_b_d @ r @ A )
=> ( ( member_nat_b @ B
@ ( pi_nat_b
@ ( collect_nat
@ ^ [E: nat] : ( ord_less_eq_nat @ E @ C ) )
@ ^ [E: nat] : A ) )
=> ( ( member_nat_a @ D
@ ( pi_nat_a
@ ( collect_nat
@ ^ [E: nat] : ( ord_less_eq_nat @ E @ C ) )
@ ^ [E: nat] : ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) )
=> ( ( l_comb_b_d_a_c @ r @ m @ C @ B @ D )
= ( zero_a1261444626_a_b_c @ m ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_348_linear__comb0__2) ).
thf(635,plain,
! [A: set_b,B: nat > b,C: nat,D: nat > a] :
( ( ideal_b_d @ r @ A )
=> ( ( member_nat_b @ B
@ ( pi_nat_b
@ ( collect_nat
@ ^ [E: nat] : ( ord_less_eq_nat @ E @ C ) )
@ ^ [E: nat] : A ) )
=> ( ( member_nat_a @ D
@ ( pi_nat_a
@ ( collect_nat
@ ^ [E: nat] : ( ord_less_eq_nat @ E @ C ) )
@ ^ [E: nat] : ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) )
=> ( ( l_comb_b_d_a_c @ r @ m @ C @ B @ D )
= ( zero_a1261444626_a_b_c @ m ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(7,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ord_less_nat @ B @ ( suc @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_196_not__less__eq) ).
thf(332,plain,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ord_less_nat @ B @ ( suc @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(279,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a,D: a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( C != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) )
=> ( ( member_a @ D @ ( algebr1039611956t_unit @ B @ A @ C ) )
=> ? [E: nat,F: nat > a] :
( ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [G: nat] : ( ord_less_eq_nat @ G @ E ) )
@ ^ [G: nat] : C ) )
& ? [G: nat > b] :
( ( member_nat_b @ G
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : ( carrie2079586589xt_b_d @ B ) ) )
& ( D
= ( l_comb1138968323t_unit @ B @ A @ E @ G @ F ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_29_Module_Omem__fgs__l__comb) ).
thf(1261,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a,D: a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( C != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) )
=> ( ( member_a @ D @ ( algebr1039611956t_unit @ B @ A @ C ) )
=> ? [E: nat,F: nat > a] :
( ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [G: nat] : ( ord_less_eq_nat @ G @ E ) )
@ ^ [G: nat] : C ) )
& ? [G: nat > b] :
( ( member_nat_b @ G
@ ( pi_nat_b
@ ( collect_nat
@ ^ [H: nat] : ( ord_less_eq_nat @ H @ E ) )
@ ^ [H: nat] : ( carrie2079586589xt_b_d @ B ) ) )
& ( D
= ( l_comb1138968323t_unit @ B @ A @ E @ G @ F ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).
thf(43,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: carrie722926983_a_b_c] :
( ( module_a_b_c_d @ A @ B )
=> ( ( module_a_b_c_d @ C @ B )
=> ( module589927589unit_d @ ( hOM_b_d_a_c_a_c @ B @ A @ C ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_98_Module_OHOM__is__module) ).
thf(459,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d,C: carrie722926983_a_b_c] :
( ( module_a_b_c_d @ A @ B )
=> ( ( module_a_b_c_d @ C @ B )
=> ( module589927589unit_d @ ( hOM_b_d_a_c_a_c @ B @ A @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(27,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_146_less__not__refl) ).
thf(395,plain,
! [A: nat] :
~ ( ord_less_nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(155,axiom,
! [A: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_92_zero__less__Suc) ).
thf(839,plain,
! [A: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(19,axiom,
! [A: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ A ) )
= ( A = zero_zero_nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_324_not__gr__zero) ).
thf(369,plain,
! [A: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ A ) )
= ( A = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(88,axiom,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d] :
( ( module_a_b_c_d @ A @ B )
=> ( ring_b_d @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_225_Module_Osc__Ring) ).
thf(607,plain,
! [A: carrie722926983_a_b_c,B: carrie1950868226xt_b_d] :
( ( module_a_b_c_d @ A @ B )
=> ( ring_b_d @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(303,axiom,
! [A: set_b,B: set_a,C: nat > b,D: nat,E: nat > a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_nat_b @ C
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : A ) )
=> ( ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : B ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ D @ C @ E ) @ ( linear_span_b_d_a_c @ r @ m @ A @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_346_l__comb__mem__linear__span) ).
thf(1331,plain,
! [A: set_b,B: set_a,C: nat > b,D: nat,E: nat > a] :
( ( ideal_b_d @ r @ A )
=> ( ( ord_less_eq_set_a @ B @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_nat_b @ C
@ ( pi_nat_b
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : A ) )
=> ( ( member_nat_a @ E
@ ( pi_nat_a
@ ( collect_nat
@ ^ [F: nat] : ( ord_less_eq_nat @ F @ D ) )
@ ^ [F: nat] : B ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ D @ C @ E ) @ ( linear_span_b_d_a_c @ r @ m @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).
thf(143,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( suc @ A ) @ ( suc @ B ) )
= ( minus_minus_nat @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_80_diff__Suc__Suc) ).
thf(803,plain,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( suc @ A ) @ ( suc @ B ) )
= ( minus_minus_nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(175,axiom,
! [A: a] :
( ( collect_a
@ ^ [B: a] : ( B = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_345_singleton__conv) ).
thf(923,plain,
! [A: a] :
( ( collect_a
@ ^ [B: a] : ( B = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(254,axiom,
misomo1282343797_c_a_c @ r @ m @ m,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_34_misom__self) ).
thf(1184,plain,
misomo1282343797_c_a_c @ r @ m @ m,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).
thf(105,axiom,
! [A: set_a] :
( ( submodule_b_d_a_c @ r @ m @ A )
=> ( ( carrie1074654371t_unit @ ( mdl_a_b_c @ m @ A ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_311_mdl__carrier) ).
thf(659,plain,
! [A: set_a] :
( ( submodule_b_d_a_c @ r @ m @ A )
=> ( ( carrie1074654371t_unit @ ( mdl_a_b_c @ m @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(66,axiom,
! [A: nat] :
~ ( ord_less_eq_nat @ ( suc @ A ) @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_159_Suc__n__not__le__n) ).
thf(543,plain,
! [A: nat] :
~ ( ord_less_eq_nat @ ( suc @ A ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(97,axiom,
submodule_b_d_a_c @ r @ m @ ( algebr1152250919_d_a_c @ r @ m @ ( insert_a @ h2 @ bot_bot_set_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_18) ).
thf(639,plain,
submodule_b_d_a_c @ r @ m @ ( algebr1152250919_d_a_c @ r @ m @ ( insert_a @ h2 @ bot_bot_set_a ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(127,axiom,
! [A: set_a] :
( ( submodule_b_d_a_c @ r @ m @ A )
=> ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_3_submodule__subset) ).
thf(755,plain,
! [A: set_a] :
( ( submodule_b_d_a_c @ r @ m @ A )
=> ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(84,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( free_g1087686480_d_a_c @ r @ m @ A )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ r @ m @ B ) @ ( algebr1152250919_d_a_c @ r @ m @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_7_fgs__mono) ).
thf(592,plain,
! [A: set_a,B: set_a,C: set_a] :
( ( free_g1087686480_d_a_c @ r @ m @ A )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ r @ m @ B ) @ ( algebr1152250919_d_a_c @ r @ m @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(17,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ~ ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_140_linorder__neqE__nat) ).
thf(364,plain,
! [A: nat,B: nat] :
( ( A != B )
=> ( ~ ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(180,axiom,
! [A: set_nat_a,B: set_nat_a] :
( ! [C: nat > a] :
( ( member_nat_a @ C @ A )
=> ( member_nat_a @ C @ B ) )
=> ( ord_le157368549_nat_a @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_280_subsetI) ).
thf(935,plain,
! [A: set_nat_a,B: set_nat_a] :
( ! [C: nat > a] :
( ( member_nat_a @ C @ A )
=> ( member_nat_a @ C @ B ) )
=> ( ord_le157368549_nat_a @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(199,axiom,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) )
=> ( ord_less_eq_set_a @ ( algebr1039611956t_unit @ B @ A @ C ) @ ( carrie1074654371t_unit @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_32_Module_Ofgs__sub__carrier) ).
thf(1009,plain,
! [A: carrie1963041556t_unit,B: carrie1950868226xt_b_d,C: set_a] :
( ( module1821517916unit_d @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ ( carrie1074654371t_unit @ A ) )
=> ( ord_less_eq_set_a @ ( algebr1039611956t_unit @ B @ A @ C ) @ ( carrie1074654371t_unit @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(135,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_165_bot__nat__0_Oextremum__uniqueI) ).
thf(775,plain,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(72,axiom,
! [A: nat] :
( ( ord_less_nat @ A @ ( suc @ zero_zero_nat ) )
= ( A = zero_zero_nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_93_less__Suc0) ).
thf(560,plain,
! [A: nat] :
( ( ord_less_nat @ A @ ( suc @ zero_zero_nat ) )
= ( A = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(54,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_205_bot__nat__0_Oextremum__strict) ).
thf(491,plain,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(227,axiom,
! [A: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ A ) )
= ( ! [B: a] :
~ ( A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_296_empty__Collect__eq) ).
thf(1088,plain,
! [A: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ A ) )
= ( ! [B: a] :
~ ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).
thf(1401,plain,
$false,
inference(e,[status(thm)],[645,1322,1031,1083,555,481,352,1200,523,1068,873,614,1142,1269,1041,861,1078,1310,715,472,1260,814,1327,898,709,841,565,1298,372,1059,949,587,1323,917,751,1313,1279,934,1191,817,1247,969,1115,316,1100,1232,475,789,1109,623,1334,514,574,806,966,821,485,1136,692,453,1119,988,428,312,1074,1206,920,380,810,1338,1067,1047,512,942,546,793,1153,761,910,1091,1343,349,1174,407,1238,1021,778,467,361,830,757,1195,1128,1281,456,1012,1106,329,1178,1296,665,1350,697,1143,650,1227,403,1096,1006,869,420,499,519,742,884,488,1292,642,526,590,1039,901,1075,1050,721,938,1241,633,508,311,1071,674,1185,773,926,398,1035,662,718,618,425,1103,824,1273,745,375,601,584,423,579,1256,413,904,975,335,1224,391,1139,611,787,1209,784,1349,377,318,1307,764,604,880,434,616,1122,387,494,609,483,769,1022,1177,1112,858,367,552,848,700,1293,668,462,932,1171,959,827,1015,563,505,431,653,1044,695,1304,1203,648,1081,1340,1018,355,754,1181,836,640,1025,739,1098,993,712,557,1003,1053,1300,791,1188,1038,502,1235,1317,338,808,1130,1218,470,576,1257,703,656,864,1094,571,595,1272,876,991,1221,1284,539,671,1157,581,844,771,1276,1326,1253,963,748,1062,946,401,781,995,568,914,929,478,1212,383,346,549,1133,999,1229,1303,865,833,410,1314,341,358,437,767,1056,1197,1065,955,1346,960,635,332,1261,459,395,839,369,607,1331,803,923,1184,659,543,639,755,592,364,935,1009,775,560,491,1088]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : ITP026^1 : TPTP v8.1.2. Released v7.5.0.
% 0.06/0.15 % Command : run_Leo-III %s %d
% 0.15/0.36 % Computer : n012.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 300
% 0.15/0.36 % DateTime : Mon May 6 14:27:54 EDT 2024
% 0.15/0.36 % CPUTime :
% 0.92/0.84 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 1.89/1.09 % [INFO] Parsing done (241ms).
% 1.89/1.09 % [INFO] Running in sequential loop mode.
% 2.53/1.29 % [INFO] eprover registered as external prover.
% 2.53/1.30 % [INFO] cvc4 registered as external prover.
% 2.53/1.30 % [INFO] Scanning for conjecture ...
% 2.84/1.39 % [INFO] Found a conjecture and 370 axioms. Running axiom selection ...
% 3.32/1.49 % [INFO] Axiom selection finished. Selected 308 axioms (removed 62 axioms).
% 3.83/1.67 % [INFO] Problem is higher-order (TPTP THF).
% 4.00/1.69 % [INFO] Type checking passed.
% 4.00/1.69 % [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 ...
% 12.29/3.55 % External prover 'e' found a proof!
% 12.29/3.55 % [INFO] Killing All external provers ...
% 12.29/3.55 % Time passed: 3032ms (effective reasoning time: 2454ms)
% 12.29/3.55 % 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)>
% 12.29/3.55 % Axioms used in derivation (308): fact_328_Diff__eq__empty__iff, fact_296_empty__Collect__eq, fact_59_misom__sym, fact_4_free__generator__sub, fact_288_empty__iff, fact_292_all__not__in__conv, fact_43_Module_Ol__comb__mem, fact_254_diff__less__Suc, fact_201_Suc__lessI, fact_145_less__not__refl2, fact_333_insert__subset, fact_82_diff__0__eq__0, fact_220_minus__nat_Odiff__0, fact_211_gr0I, fact_71_Collect__mem__eq, fact_199_less__SucI, fact_9_linear__comb__eq, fact_102_Module_OHOM__is__module, fact_311_mdl__carrier, fact_217_le__diff__iff, fact_198_Ex__less__Suc, fact_343_l__span__zero, fact_170_old_Onat_Oinducts, fact_286_Diff__cancel, fact_57_nat_Oinject, fact_337_singletonI, fact_289_empty__iff, fact_293_Collect__empty__eq, fact_11_l__comb__mem, fact_261_Module_Osubmodule__whole, fact_200_less__SucE, fact_196_not__less__eq, fact_64_Suc__less__eq, fact_346_l__comb__mem__linear__span, fact_325_zero__diff, fact_189_strict__inc__induct, fact_168_less__eq__nat_Osimps_I1_J, conj_11, fact_150_Module_Osmodule__ideal__coeff__is__Submodule, fact_160_le__Suc__eq, fact_156_nat__induct__at__least, conj_9, fact_225_Module_Osc__Ring, fact_45_Module_Oliear__comb__memTr, fact_295_empty__Collect__eq, fact_240_Suc__le__lessD, fact_324_not__gr__zero, conj_3, fact_75_bot__nat__0_Onot__eq__extremum, fact_163_le__SucE, fact_281_subsetI, fact_266_Module_Osubmodule__subset, fact_54_diff__is__0__eq, fact_144_less__not__refl3, fact_173_Zero__neq__Suc, fact_193_less__antisym, fact_315_l__span__cont__H, conj_14, fact_93_less__Suc0, fact_222_less__imp__diff__less, fact_147_nat__neq__iff, fact_155_transitive__stepwise__le, fact_81_diff__self__eq__0, fact_191_less__trans__Suc, fact_320_fgs__generator, fact_36_HOM__is__module, fact_73_Collect__mem__eq, fact_326_diff__zero, fact_153_Module_Oquotient__of__submodules__is__ideal, fact_2_submodule__whole, conj_10, fact_276_l__span__closed2, fact_177_nat__induct, fact_12_liear__comb__memTr, fact_56_old_Onat_Oinject, fact_7_fgs__mono, fact_80_diff__Suc__Suc, fact_162_le__SucI, fact_74_Collect__cong, fact_158_not__less__eq__eq, fact_120_Module_Omisom__sym, fact_283_subset__antisym, fact_237_less__eq__Suc__le, fact_101_Module_OHOM__is__module, fact_341_linear__span__inc__0, fact_26_Module_Ofgs__submodule, fact_49_Module_Olinear__comb__eq, fact_294_Collect__empty__eq, fact_135_nat__le__linear, fact_18_mem__smodule__ideal__coeff, fact_129_Module_Ogenerator__sub__carrier, fact_165_bot__nat__0_Oextremum__uniqueI, fact_195_All__less__Suc, fact_127_Module_Ofree__generator__generator, fact_66_mem__Collect__eq, fact_130_Module_Ogenerator__sub__carrier, fact_91_misom__trans, fact_88_misom__trans, fact_67_mem__Collect__eq, fact_330_empty__subsetI, fact_139_Suc__inject, fact_238_less__Suc__eq__le, fact_47_zero__less__diff, fact_52_Module_Olinear__comb__eqTr, fact_187_nat__less__le, fact_347_l__span__closed, fact_16_quotient__of__submodules__is__ideal, fact_339_l__spanA__l__span, fact_70_Collect__mem__eq, fact_14_mem__fgs__l__comb, conj_18, fact_345_singleton__conv, fact_319_l__span__sub__submodule, fact_141_infinite__descent, fact_126_Module_Ofree__generator__generator, fact_185_le__eq__less__or__eq, fact_273_singleton__insert__inj__eq_H, fact_182_less__mono__imp__le__mono, conj_2, fact_20_Module_Oelem__fgs, fact_282_subsetI, fact_27_Module_Ofgs__submodule, fact_236_le__imp__less__Suc, fact_244_Suc__leI, fact_239_le__less__Suc__eq, fact_290_all__not__in__conv, fact_3_submodule__subset, fact_342_generator__generator, fact_327_cancel__comm__monoid__add__class_Odiff__cancel, fact_334_insert__Diff__single, fact_60_misom__sym, fact_234_lift__Suc__mono__less__iff, fact_137_le__trans, fact_340_lin__span__coeff__mono, conj_15, fact_148_diff__commute, fact_246_less__Suc__eq__0__disj, fact_21_Module_Oelem__fgs, fact_10_linear__comb__eqTr, fact_8_fgs__submodule, fact_250_Ex__less__Suc2, fact_62_bot__nat__0_Oextremum, fact_344_l__span__closed3, fact_212_diff__le__mono2, fact_143_less__irrefl__nat, fact_213_le__diff__iff_H, fact_249_gr0__conv__Suc, fact_53_Module_Olinear__comb__eqTr, fact_65_Suc__mono, fact_166_bot__nat__0_Oextremum__unique, fact_329_subset__empty, fact_317_l__span__gen__mono, fact_63_le0, fact_40_free__generator__generator, fact_270_Module_Ofree__generator__sub, fact_256_diff__less, fact_232_lift__Suc__mono__le, fact_204_Nat_OlessE, fact_142_nat__less__induct, fact_86_misom__trans, fact_72_Collect__mem__eq, fact_248_All__less__Suc2, fact_242_dec__induct, fact_90_misom__trans, fact_151_Module_Osmodule__ideal__coeff__is__Submodule, fact_207_gr__implies__not0, fact_262_Module_Osubmodule__whole, fact_223_diff__less__mono2, fact_210_not__gr0, fact_136_le__antisym, fact_34_misom__self, fact_279_psubsetI, fact_231_lift__Suc__antimono__le, fact_68_mem__Collect__eq, fact_50_Module_Olinear__comb__eq, fact_275_l__span__closed2Tr, fact_284_Diff__empty, fact_258_Module_Osubmodule__subset1, fact_274_singleton__insert__inj__eq, fact_203_Suc__lessD, fact_215_diff__le__mono, fact_264_diff__Suc__less, fact_267_Module_Osubmodule__subset, fact_1_submodule__subset1, fact_24_Module_Ofgs__mono, fact_194_Suc__less__eq2, fact_98_Module_OHOM__is__module, conj_16, fact_38_Suc__pred, fact_171_old_Onat_Oexhaust, fact_69_mem__Collect__eq, fact_272_eSum__in__Submodule, fact_218_eq__diff__iff, fact_178_nat_OdiscI, fact_219_zero__induct__lemma, fact_208_less__zeroE, fact_312_mdl__is__module, fact_186_less__imp__le__nat, fact_291_all__not__in__conv, fact_138_n__not__Suc__n, fact_205_bot__nat__0_Oextremum__strict, fact_321_submodule__0, fact_209_not__less0, fact_29_Module_Omem__fgs__l__comb, fact_115_Module_Omisom__self, fact_226_Module_Osc__Ring, fact_192_Suc__less__SucD, fact_313_submodule__of__mdl, conj_6, fact_348_linear__comb0__2, fact_23_Module_Ofgs__mono, fact_123_Module_Omisom__sym, conj_20, conj_1, fact_180_old_Onat_Odistinct_I2_J, fact_247_gr0__implies__Suc, conj_19, fact_251_Suc__diff__le, fact_269_Module_Ofree__generator__sub, fact_132_Module_OAnn__is__ideal, fact_174_Suc__neq__Zero, fact_78_diff__diff__cancel, fact_252_diff__less__mono, fact_175_zero__induct, fact_17_smodule__ideal__coeff__is__Submodule, fact_13_Ann__is__ideal, fact_15_Module__axioms, fact_280_subsetI, fact_255_Suc__diff__Suc, fact_6_fgs__sub__carrier, fact_228_Module_Omem__smodule__ideal__coeff, fact_172_Zero__not__Suc, fact_314_h__in__linear__span, conj_8, fact_169_not0__implies__Suc, fact_42_Module_Ol__comb__mem, fact_122_Module_Omisom__sym, fact_338_linear__span__subModule, fact_285_empty__Diff, fact_114_Module_Omisom__self, fact_39_generator__sub__carrier, fact_302_insert__Diff1, fact_322_empty__fgs, fact_310_submodule__inc__0, fact_164_Suc__leD, fact_133_Module_OAnn__is__ideal, fact_230_lift__Suc__antimono__le, conj_12, fact_243_Suc__le__eq, fact_214_diff__le__self, fact_92_zero__less__Suc, fact_221_diffs0__imp__equal, conj_17, fact_229_Module_Omem__smodule__ideal__coeff, fact_305_insertCI, fact_263_ex__least__nat__less, fact_55_diff__is__0__eq_H, fact_299_Diff__insert0, fact_253_less__diff__iff, fact_157_full__nat__induct, fact_190_less__Suc__induct, fact_184_less__or__eq__imp__le, fact_271_eSum__in__SubmoduleTr, fact_308_insert__iff, fact_259_Module_Osubmodule__subset1, fact_176_diff__induct, fact_5_elem__fgs, conj_13, fact_235_lift__Suc__mono__less, fact_124_Module_Omisom__sym, fact_46_Module_Oliear__comb__memTr, fact_277_linear__comb0__2Tr, fact_206_infinite__descent0, fact_161_Suc__le__D, fact_76_less__nat__zero__code, fact_33_Module_Ofgs__sub__carrier, fact_316_l__span__l__span, conj_5, fact_167_le__0__eq, fact_202_Suc__lessE, fact_309_insert__absorb2, conj_0, fact_241_inc__induct, fact_183_le__neq__implies__less, fact_159_Suc__n__not__le__n, fact_140_linorder__neqE__nat, fact_79_Suc__diff__diff, fact_154_Module_Oquotient__of__submodules__is__ideal, fact_197_less__Suc__eq, fact_146_less__not__refl, fact_30_Module_Omem__fgs__l__comb, fact_233_lift__Suc__mono__le, fact_323_le__zero__eq, fact_318_linear__span__sub, fact_37_HOM__is__module, fact_179_old_Onat_Odistinct_I1_J, fact_99_Module_OHOM__is__module, fact_0_sc__Ring, fact_77_neq0__conv, fact_134_Nat_Oex__has__greatest__nat, fact_245_ex__least__nat__le, fact_61_Suc__le__mono, fact_181_nat_Odistinct_I1_J, fact_32_Module_Ofgs__sub__carrier, fact_278_module__inc__zero, fact_287_empty__iff, fact_216_Nat_Odiff__diff__eq, fact_188_not__less__less__Suc__eq
% 12.29/3.55 % No. of inferences in proof: 620
% 12.29/3.55 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 3032 ms resp. 2454 ms w/o parsing
% 13.18/3.74 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 13.18/3.74 % [INFO] Killing All external provers ...
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