TSTP Solution File: ITP026^2 by Leo-III---1.7.12
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%------------------------------------------------------------------------------
% File : Leo-III---1.7.12
% Problem : ITP026^2 : TPTP v8.2.0. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n002.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Mon May 20 22:22:11 EDT 2024
% Result : Theorem 62.60s 45.49s
% Output : Refutation 63.66s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 338
% Syntax : Number of formulae : 624 ( 204 unt; 54 typ; 0 def)
% Number of atoms : 2118 ( 378 equ; 2 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 11285 ( 134 ~; 22 |; 80 &;10289 @)
% ( 0 <=>; 760 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 341 ( 341 >; 0 *; 0 +; 0 <<)
% Number of symbols : 54 ( 51 usr; 16 con; 0-9 aty)
% Number of variables : 1869 ( 206 ^1524 !; 58 ?;1869 :)
% ( 81 !>; 0 ?*; 0 @-; 0 @+)
% Comments :
%------------------------------------------------------------------------------
thf(product_unit_type,type,
product_unit: $tType ).
thf(nat_type,type,
nat: $tType ).
thf(f_type,type,
f: $tType ).
thf(e_type,type,
e: $tType ).
thf(d_type,type,
d: $tType ).
thf(c_type,type,
c: $tType ).
thf(b_type,type,
b: $tType ).
thf(a_type,type,
a: $tType ).
thf(zero_type,type,
zero:
!>[TA: $tType] : $o ).
thf(order_type,type,
order:
!>[TA: $tType] : $o ).
thf(group_add_type,type,
group_add:
!>[TA: $tType] : $o ).
thf(wellorder_type,type,
wellorder:
!>[TA: $tType] : $o ).
thf(comm_monoid_diff_type,type,
comm_monoid_diff:
!>[TA: $tType] : $o ).
thf(ordered_ab_group_add_type,type,
ordered_ab_group_add:
!>[TA: $tType] : $o ).
thf(cancel1352612707id_add_type,type,
cancel1352612707id_add:
!>[TA: $tType] : $o ).
thf(canoni770627133id_add_type,type,
canoni770627133id_add:
!>[TA: $tType] : $o ).
thf(carrier_type,type,
carrier:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ TA ) > ( set @ TB ) ) ).
thf(ring_type,type,
ring:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > $o ) ).
thf(zero2_type,type,
zero2:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) > TB ) ).
thf(ideal_type,type,
ideal:
!>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > $o ) ).
thf(annihilator_type,type,
annihilator:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > ( set @ TD ) ) ).
thf(hom_type,type,
hom:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType] : ( ( carrier_ext @ TF @ ( aGroup_ext @ TF @ ( ring_ext @ TF @ TE ) ) ) > ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TF @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TF @ TA ) ) ) > ( carrier_ext @ ( TD > TB ) @ ( aGroup_ext @ ( TD > TB ) @ ( module_ext @ ( TD > TB ) @ TF @ product_unit ) ) ) ) ).
thf(module_type,type,
module:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TC @ TB ) ) ) > ( carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TA ) ) ) > $o ) ).
thf(free_generator_type,type,
free_generator:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > ( set @ TB ) > $o ) ).
thf(generator_type,type,
generator:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > ( set @ TB ) > $o ) ).
thf(l_comb_type,type,
l_comb:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > nat > ( nat > TD ) > ( nat > TB ) > TB ) ).
thf(linear_span_type,type,
linear_span:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > ( set @ TD ) > ( set @ TB ) > ( set @ TB ) ) ).
thf(mdl_type,type,
mdl:
!>[TA: $tType,TB: $tType,TC: $tType] : ( ( carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TB @ TA ) ) ) > ( set @ TC ) > ( carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TB @ product_unit ) ) ) ) ).
thf(misomorphic_type,type,
misomorphic:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType] : ( ( carrier_ext @ TF @ ( aGroup_ext @ TF @ ( ring_ext @ TF @ TE ) ) ) > ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TF @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TF @ TA ) ) ) > $o ) ).
thf(quotie135653060odules_type,type,
quotie135653060odules:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > ( set @ TB ) > ( set @ TB ) > ( set @ TD ) ) ).
thf(smodule_ideal_coeff_type,type,
smodule_ideal_coeff:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > ( set @ TD ) > ( set @ TB ) ) ).
thf(submodule_type,type,
submodule:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > ( set @ TB ) > $o ) ).
thf(algebr833503410le_fgs_type,type,
algebr833503410le_fgs:
!>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > ( set @ TB ) > ( set @ TB ) ) ).
thf(pi_type,type,
pi:
!>[TA: $tType,TB: $tType] : ( ( set @ TB ) > ( TB > ( set @ TA ) ) > ( set @ ( TB > TA ) ) ) ).
thf(minus_minus_type,type,
minus_minus:
!>[TA: $tType] : ( TA > TA > TA ) ).
thf(zero_zero_type,type,
zero_zero:
!>[TA: $tType] : TA ).
thf(suc_type,type,
suc: nat > nat ).
thf(bot_bot_type,type,
bot_bot:
!>[TA: $tType] : TA ).
thf(ord_less_type,type,
ord_less:
!>[TA: $tType] : ( TA > TA > $o ) ).
thf(ord_less_eq_type,type,
ord_less_eq:
!>[TA: $tType] : ( TA > TA > $o ) ).
thf(collect_type,type,
collect:
!>[TA: $tType] : ( ( TA > $o ) > ( set @ TA ) ) ).
thf(insert_type,type,
insert:
!>[TA: $tType] : ( TA > ( set @ TA ) > ( set @ TA ) ) ).
thf(member_type,type,
member:
!>[TA: $tType] : ( TA > ( set @ TA ) > $o ) ).
thf(h_type,type,
h: set @ a ).
thf(h1_type,type,
h1: set @ a ).
thf(m_type,type,
m: carrier_ext @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) ).
thf(n_type,type,
n: carrier_ext @ e @ ( aGroup_ext @ e @ ( module_ext @ e @ b @ f ) ) ).
thf(r_type,type,
r: carrier_ext @ b @ ( aGroup_ext @ b @ ( ring_ext @ 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(111,axiom,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
=> ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_Suc__inject) ).
thf(645,plain,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(218,axiom,
! [A: set @ b,B: set @ a] :
( ( ideal @ b @ d @ r @ A )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( linear_span @ b @ d @ a @ c @ r @ m @ A @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ B ) )
= ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_240_l__spanA__l__span) ).
thf(1024,plain,
! [A: set @ b,B: set @ a] :
( ( ideal @ b @ d @ r @ A )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( linear_span @ b @ d @ a @ c @ r @ m @ A @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ B ) )
= ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(17,axiom,
order @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oorder_1) ).
thf(333,plain,
order @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(276,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 @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) )
=> ( ( l_comb @ b @ d @ a @ c @ r @ m @ C @ B @ D )
= ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_250_linear__comb0__2) ).
thf(1179,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 @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) )
=> ( ( l_comb @ b @ d @ a @ c @ r @ m @ C @ B @ D )
= ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).
thf(155,axiom,
comm_monoid_diff @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__diff) ).
thf(777,plain,
comm_monoid_diff @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(248,axiom,
! [A: set @ a] :
( ( free_generator @ b @ d @ a @ c @ r @ m @ A )
=> ( generator @ b @ d @ a @ c @ r @ m @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_free__generator__generator) ).
thf(1110,plain,
! [A: set @ a] :
( ( free_generator @ b @ d @ a @ c @ r @ m @ A )
=> ( generator @ b @ d @ a @ c @ r @ m @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).
thf(209,axiom,
ideal @ b @ d @ r @ ( annihilator @ b @ d @ a @ c @ r @ m ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_Ann__is__ideal) ).
thf(1005,plain,
ideal @ b @ d @ r @ ( annihilator @ b @ d @ a @ c @ r @ m ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(121,axiom,
! [A: nat,B: nat > $o] :
( ( ! [C: nat] :
( ( ord_less @ nat @ C @ ( suc @ A ) )
=> ( B @ C ) ) )
= ( ( B @ ( zero_zero @ nat ) )
& ! [C: nat] :
( ( ord_less @ nat @ C @ A )
=> ( B @ ( suc @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_All__less__Suc2) ).
thf(683,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)],[121]) ).
thf(119,axiom,
! [TA: $tType,A: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ ( bot_bot @ ( set @ TA ) ) )
= ( A
= ( bot_bot @ ( set @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_234_subset__empty) ).
thf(677,plain,
! [TA: $tType,A: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ ( bot_bot @ ( set @ TA ) ) )
= ( A
= ( bot_bot @ ( set @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(96,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less @ nat @ A @ B )
=> ( ( ord_less @ nat @ A @ ( suc @ B ) )
= ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_not__less__less__Suc__eq) ).
thf(597,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)],[96]) ).
thf(175,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ ( suc @ A ) @ B )
= ( ord_less @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_168_Suc__le__eq) ).
thf(861,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)],[175]) ).
thf(273,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ~ ( member @ TA @ A @ B )
=> ( ( minus_minus @ ( set @ TA ) @ B @ ( insert @ TA @ A @ C ) )
= ( minus_minus @ ( set @ TA ) @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_Diff__insert0) ).
thf(1173,plain,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ~ ( member @ TA @ A @ B )
=> ( ( minus_minus @ ( set @ TA ) @ B @ ( insert @ TA @ A @ C ) )
= ( minus_minus @ ( set @ TA ) @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).
thf(263,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 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_248_l__comb__mem__linear__span) ).
thf(1142,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 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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)],[263]) ).
thf(35,axiom,
order @ product_unit,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Ounit___Orderings_Oorder_5) ).
thf(379,plain,
order @ product_unit,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(237,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: TA] :
( ( module @ TA @ TD @ TC @ TB @ A @ B )
=> ( ( C
!= ( bot_bot @ ( set @ TA ) ) )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
=> ( ( member @ TA @ D @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ C ) )
=> ? [E: nat,F: nat > TA] :
( ( member @ ( nat > TA ) @ F
@ ( pi @ nat @ TA
@ ( collect @ nat
@ ^ [G: nat] : ( ord_less_eq @ nat @ G @ E ) )
@ ^ [G: nat] : C ) )
& ? [G: nat > TD] :
( ( member @ ( nat > TD ) @ G
@ ( pi @ nat @ TD
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
@ ^ [H: nat] : ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ) @ B ) ) )
& ( D
= ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ G @ F ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_Module_Omem__fgs__l__comb) ).
thf(1078,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: TA] :
( ( module @ TA @ TD @ TC @ TB @ A @ B )
=> ( ( C
!= ( bot_bot @ ( set @ TA ) ) )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
=> ( ( member @ TA @ D @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ C ) )
=> ? [E: nat,F: nat > TA] :
( ( member @ ( nat > TA ) @ F
@ ( pi @ nat @ TA
@ ( collect @ nat
@ ^ [G: nat] : ( ord_less_eq @ nat @ G @ E ) )
@ ^ [G: nat] : C ) )
& ? [G: nat > TD] :
( ( member @ ( nat > TD ) @ G
@ ( pi @ nat @ TD
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
@ ^ [H: nat] : ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ) @ B ) ) )
& ( D
= ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ G @ F ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).
thf(38,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ( ord_less @ nat @ B @ C )
=> ( ord_less @ nat @ ( suc @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_less__trans__Suc) ).
thf(385,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)],[38]) ).
thf(77,axiom,
! [A: nat > $o,B: nat] :
( ( A @ B )
=> ( ! [C: nat] :
( ( A @ ( suc @ C ) )
=> ( A @ C ) )
=> ( A @ ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_105_zero__induct) ).
thf(500,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)],[77]) ).
thf(132,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_48_less__nat__zero__code) ).
thf(715,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(167,axiom,
! [A: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ A )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_diff__0__eq__0) ).
thf(814,plain,
! [A: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ A )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(144,axiom,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
= ( ord_less_eq @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_diff__ge__0__iff__ge) ).
thf(747,plain,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
= ( ord_less_eq @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(83,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_less__imp__le__nat) ).
thf(538,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(271,axiom,
! [A: set @ a,B: a] :
( ( A
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( member @ a @ B @ ( algebr833503410le_fgs @ b @ 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] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
& ( B
= ( l_comb @ b @ d @ a @ c @ r @ m @ C @ E @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_mem__fgs__l__comb) ).
thf(1164,plain,
! [A: set @ a,B: a] :
( ( A
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( member @ a @ B @ ( algebr833503410le_fgs @ b @ 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] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
& ( B
= ( l_comb @ b @ d @ a @ c @ r @ m @ C @ E @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).
thf(162,axiom,
! [A: set @ a,B: set @ a,C: set @ b] :
( ( ord_less_eq @ ( set @ a ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_220_l__span__gen__mono) ).
thf(797,plain,
! [A: set @ a,B: set @ a,C: set @ b] :
( ( ord_less_eq @ ( set @ a ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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)],[162]) ).
thf(130,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ? [B: nat] :
( A
= ( suc @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_not0__implies__Suc) ).
thf(709,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ? [B: nat] :
( A
= ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(7,axiom,
! [TA: $tType] : ( order @ ( set @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oorder_2) ).
thf(301,plain,
! [TA: $tType] : ( order @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(54,axiom,
! [A: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_zero__less__Suc) ).
thf(436,plain,
! [A: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(113,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) )
= ( ord_less @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_Suc__less__eq) ).
thf(651,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)],[113]) ).
thf(27,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/sandbox/benchmark/theBenchmark.p',fact_87_full__nat__induct) ).
thf(357,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)],[27]) ).
thf(230,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: nat > TD,E: nat,F: nat > TA,G: nat > TA] :
( ( module @ TA @ TD @ TC @ TB @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
=> ( ( member @ ( nat > TD ) @ D
@ ( pi @ nat @ TD
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
@ ^ [H: nat] : ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ) @ B ) ) )
=> ( ( member @ ( nat > TA ) @ F
@ ( pi @ nat @ TA
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
@ ^ [H: nat] : C ) )
=> ( ( member @ ( nat > TA ) @ G
@ ( pi @ nat @ TA
@ ( 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 @ TD @ TB @ TA @ TC @ B @ A @ E @ D @ F )
= ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ D @ G ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_Module_Olinear__comb__eq) ).
thf(1059,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: nat > TD,E: nat,F: nat > TA,G: nat > TA] :
( ( module @ TA @ TD @ TC @ TB @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
=> ( ( member @ ( nat > TD ) @ D
@ ( pi @ nat @ TD
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
@ ^ [H: nat] : ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ) @ B ) ) )
=> ( ( member @ ( nat > TA ) @ F
@ ( pi @ nat @ TA
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
@ ^ [H: nat] : C ) )
=> ( ( member @ ( nat > TA ) @ G
@ ( pi @ nat @ TA
@ ( 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 @ TD @ TB @ TA @ TC @ B @ A @ E @ D @ F )
= ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ D @ G ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).
thf(85,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) )
=> ( ord_less @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_Suc__less__SucD) ).
thf(542,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)],[85]) ).
thf(105,axiom,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_Zero__not__Suc) ).
thf(619,plain,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(173,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/sandbox/benchmark/theBenchmark.p',fact_150_zero__induct__lemma) ).
thf(838,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)],[173]) ).
thf(168,axiom,
! [A: nat > $o,B: nat] :
( ( A @ ( zero_zero @ nat ) )
=> ( ! [C: nat] :
( ( A @ C )
=> ( A @ ( suc @ C ) ) )
=> ( A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_old_Onat_Oinducts) ).
thf(817,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)],[168]) ).
thf(198,axiom,
! [A: nat,B: nat > $o] :
( ( ? [C: nat] :
( ( ord_less @ nat @ C @ ( suc @ A ) )
& ( B @ C ) ) )
= ( ( B @ ( zero_zero @ nat ) )
| ? [C: nat] :
( ( ord_less @ nat @ C @ A )
& ( B @ ( suc @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_Ex__less__Suc2) ).
thf(969,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)],[198]) ).
thf(213,axiom,
( ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( bot_bot @ ( set @ a ) ) )
= ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_225_empty__fgs) ).
thf(1013,plain,
( ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( bot_bot @ ( set @ a ) ) )
= ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(252,axiom,
ord_less_eq @ nat @ xa @ ( minus_minus @ nat @ m2 @ ( suc @ ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_19) ).
thf(1115,plain,
ord_less_eq @ nat @ xa @ ( minus_minus @ nat @ m2 @ ( suc @ ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).
thf(247,axiom,
member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_module__inc__zero) ).
thf(1109,plain,
member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).
thf(34,axiom,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_97_le__0__eq) ).
thf(376,plain,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(106,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ~ ! [B: nat] :
( A
!= ( suc @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_101_old_Onat_Oexhaust) ).
thf(623,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ~ ! [B: nat] :
( A
!= ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(260,axiom,
! [A: set @ a,B: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( member @ a @ B @ A )
=> ( member @ a @ B @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_217_h__in__linear__span) ).
thf(1136,plain,
! [A: set @ a,B: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( member @ a @ B @ A )
=> ( member @ a @ B @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).
thf(266,axiom,
! [A: set @ b,B: set @ a] :
( ( ideal @ b @ d @ r @ A )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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 @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_lin__span__coeff__mono) ).
thf(1151,plain,
! [A: set @ b,B: set @ a] :
( ( ideal @ b @ d @ r @ A )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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 @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).
thf(25,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_Suc__mono) ).
thf(353,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)],[25]) ).
thf(277,axiom,
! [A: set @ a,B: nat > b,C: nat,D: nat > a,E: nat > a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( ( member @ ( nat > b ) @ B
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
@ ^ [F: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_10_linear__comb__eqTr) ).
thf(1183,plain,
! [A: set @ a,B: nat > b,C: nat,D: nat > a,E: nat > a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( ( member @ ( nat > b ) @ B
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
@ ^ [F: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ 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)],[277]) ).
thf(226,axiom,
! [A: set @ a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ A )
=> ( member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_213_submodule__inc__0) ).
thf(1051,plain,
! [A: set @ a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ A )
=> ( member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).
thf(70,axiom,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
= ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_nat_Oinject) ).
thf(480,plain,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
= ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(151,axiom,
! [A: set @ b,B: set @ a] :
( ( ideal @ b @ d @ r @ A )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_221_linear__span__sub) ).
thf(766,plain,
! [A: set @ b,B: set @ a] :
( ( ideal @ b @ d @ r @ A )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(139,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TA @ TD ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),C: carrier_ext @ TF @ ( aGroup_ext @ TF @ ( module_ext @ TF @ TA @ TE ) )] :
( ( module @ TC @ TA @ TD @ TB @ A @ B )
=> ( ( module @ TF @ TA @ TE @ TB @ C @ B )
=> ( module @ ( TC > TF ) @ TA @ product_unit @ TB @ ( hom @ TA @ TB @ TC @ TD @ TF @ TE @ B @ A @ C ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_Module_OHOM__is__module) ).
thf(734,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TA @ TD ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),C: carrier_ext @ TF @ ( aGroup_ext @ TF @ ( module_ext @ TF @ TA @ TE ) )] :
( ( module @ TC @ TA @ TD @ TB @ A @ B )
=> ( ( module @ TF @ TA @ TE @ TB @ C @ B )
=> ( module @ ( TC > TF ) @ TA @ product_unit @ TB @ ( hom @ TA @ TB @ TC @ TD @ TF @ TE @ B @ A @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(5,axiom,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ! [D: nat] :
( ( ord_less @ nat @ D @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_nat__less__induct) ).
thf(293,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)],[5]) ).
thf(177,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( suc @ A ) @ B )
= ( ? [C: nat] :
( ( B
= ( suc @ C ) )
& ( ord_less @ nat @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_Suc__less__eq2) ).
thf(866,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)],[177]) ).
thf(235,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 )
=> ( submodule @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_216_submodule__of__mdl) ).
thf(1072,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 )
=> ( submodule @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).
thf(154,axiom,
! [A: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_141_not__gr0) ).
thf(774,plain,
! [A: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(115,axiom,
! [A: nat,B: nat,C: nat > nat > $o] :
( ( ord_less @ nat @ A @ B )
=> ( ! [D: nat] : ( C @ D @ ( suc @ D ) )
=> ( ! [D: nat,E: nat,F: nat] :
( ( ord_less @ nat @ D @ E )
=> ( ( ord_less @ nat @ E @ F )
=> ( ( C @ D @ E )
=> ( ( C @ E @ F )
=> ( C @ D @ F ) ) ) ) )
=> ( C @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_less__Suc__induct) ).
thf(660,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)],[115]) ).
thf(207,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TD,D: TA] :
( ( module @ TA @ TD @ TC @ TB @ A @ B )
=> ( ( ideal @ TD @ TB @ B @ C )
=> ( ( member @ TA @ D @ ( smodule_ideal_coeff @ TD @ TB @ TA @ TC @ B @ A @ C ) )
=> ? [E: nat,F: nat > TD] :
( ( member @ ( nat > TD ) @ F
@ ( pi @ nat @ TD
@ ( collect @ nat
@ ^ [G: nat] : ( ord_less_eq @ nat @ G @ E ) )
@ ^ [G: nat] : C ) )
& ? [G: nat > TA] :
( ( member @ ( nat > TA ) @ G
@ ( pi @ nat @ TA
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
@ ^ [H: nat] : ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) ) )
& ( D
= ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ F @ G ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_156_Module_Omem__smodule__ideal__coeff) ).
thf(998,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TD,D: TA] :
( ( module @ TA @ TD @ TC @ TB @ A @ B )
=> ( ( ideal @ TD @ TB @ B @ C )
=> ( ( member @ TA @ D @ ( smodule_ideal_coeff @ TD @ TB @ TA @ TC @ B @ A @ C ) )
=> ? [E: nat,F: nat > TD] :
( ( member @ ( nat > TD ) @ F
@ ( pi @ nat @ TD
@ ( collect @ nat
@ ^ [G: nat] : ( ord_less_eq @ nat @ G @ E ) )
@ ^ [G: nat] : C ) )
& ? [G: nat > TA] :
( ( member @ ( nat > TA ) @ G
@ ( pi @ nat @ TA
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
@ ^ [H: nat] : ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) ) )
& ( D
= ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ F @ G ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(204,axiom,
! [A: set @ a,B: set @ a,C: nat > a,D: nat,E: nat > b] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
=> ( ( l_comb @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ B ) ) @ D @ E @ C )
= ( l_comb @ b @ d @ a @ c @ r @ m @ D @ E @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_eSum__in__Submodule) ).
thf(988,plain,
! [A: set @ a,B: set @ a,C: nat > a,D: nat,E: nat > b] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
=> ( ( l_comb @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ 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)],[204]) ).
thf(284,axiom,
! [A: set @ a,B: set @ a,C: set @ a] :
( ( free_generator @ b @ 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 ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ B ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_fgs__mono) ).
thf(1201,plain,
! [A: set @ a,B: set @ a,C: set @ a] :
( ( free_generator @ b @ 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 ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ B ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).
thf(245,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 @ ( quotie135653060odules @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_quotient__of__submodules__is__ideal) ).
thf(1104,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 @ ( quotie135653060odules @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(181,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/sandbox/benchmark/theBenchmark.p',fact_86_nat__induct__at__least) ).
thf(877,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)],[181]) ).
thf(124,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( A != B )
=> ( ord_less @ nat @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_le__neq__implies__less) ).
thf(692,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)],[124]) ).
thf(36,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( submodule @ TB @ TD @ TA @ TC @ B @ A @ C )
=> ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_Module_Osubmodule__subset) ).
thf(380,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( submodule @ TB @ TD @ TA @ TC @ B @ A @ C )
=> ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(87,axiom,
! [A: nat] :
( ( minus_minus @ nat @ A @ A )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_diff__self__eq__0) ).
thf(546,plain,
! [A: nat] :
( ( minus_minus @ nat @ A @ A )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(283,axiom,
! [A: set @ a] :
( ( free_generator @ b @ d @ a @ c @ r @ m @ A )
=> ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4_free__generator__sub) ).
thf(1199,plain,
! [A: set @ a] :
( ( free_generator @ b @ d @ a @ c @ r @ m @ A )
=> ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).
thf(233,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA,D: TA] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( submodule @ TB @ TD @ TA @ TC @ B @ A @ C )
=> ( ( member @ TA @ D @ C )
=> ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_182_Module_Osubmodule__subset1) ).
thf(1067,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA,D: TA] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( submodule @ TB @ TD @ TA @ TC @ B @ A @ C )
=> ( ( member @ TA @ D @ C )
=> ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).
thf(229,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ b @ TA ) )] :
( ( module @ TB @ b @ TA @ d @ A @ r )
=> ( ( misomorphic @ b @ d @ a @ c @ TB @ TA @ r @ m @ A )
=> ( misomorphic @ b @ d @ TB @ TA @ a @ c @ r @ A @ m ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_misom__sym) ).
thf(1057,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ b @ TA ) )] :
( ( module @ TB @ b @ TA @ d @ A @ r )
=> ( ( misomorphic @ b @ d @ a @ c @ TB @ TA @ r @ m @ A )
=> ( misomorphic @ b @ d @ TB @ TA @ a @ c @ r @ A @ m ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).
thf(18,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: nat > TA,B: nat,C: nat] :
( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less @ nat @ B @ C )
=> ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_160_lift__Suc__mono__less) ).
thf(334,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: nat > TA,B: nat,C: nat] :
( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less @ nat @ B @ C )
=> ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(185,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: nat > TA,B: nat,C: nat] :
( ! [D: nat] : ( ord_less_eq @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ TA @ ( A @ B ) @ ( A @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_158_lift__Suc__mono__le) ).
thf(910,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: nat > TA,B: nat,C: nat] :
( ! [D: nat] : ( ord_less_eq @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ TA @ ( A @ B ) @ ( A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(240,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/sandbox/benchmark/theBenchmark.p',fact_222_l__span__sub__submodule) ).
thf(1091,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)],[240]) ).
thf(156,axiom,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ A @ A )
= ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_232_cancel__comm__monoid__add__class_Odiff__cancel) ).
thf(778,plain,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ A @ A )
= ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(223,axiom,
! [A: set @ a,B: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( member @ a @ B @ A )
=> ( member @ a @ B @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5_elem__fgs) ).
thf(1042,plain,
! [A: set @ a,B: a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( member @ a @ B @ A )
=> ( member @ a @ B @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).
thf(40,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( minus_minus @ nat @ A @ B )
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_diff__is__0__eq_H) ).
thf(392,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)],[40]) ).
thf(159,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ ( suc @ B ) )
= ( ( ord_less_eq @ nat @ A @ B )
| ( A
= ( suc @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_le__Suc__eq) ).
thf(788,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)],[159]) ).
thf(137,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A @ C ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_less__imp__diff__less) ).
thf(729,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)],[137]) ).
thf(16,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_less__not__refl3) ).
thf(329,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(170,axiom,
! [A: set @ a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ A )
=> ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_submodule__subset) ).
thf(830,plain,
! [A: set @ a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ A )
=> ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(269,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: TA] :
( ( module @ TA @ TD @ TC @ TB @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
=> ( ( member @ TA @ D @ C )
=> ( member @ TA @ D @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_Module_Oelem__fgs) ).
thf(1159,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: TA] :
( ( module @ TA @ TD @ TC @ TB @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
=> ( ( member @ TA @ D @ C )
=> ( member @ TA @ D @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).
thf(280,axiom,
! [A: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ord_less_eq @ ( set @ a ) @ A @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_218_l__span__cont__H) ).
thf(1195,plain,
! [A: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ord_less_eq @ ( set @ a ) @ A @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).
thf(257,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: TA] :
( ( ~ ( member @ TA @ A @ B )
=> ( A = C ) )
=> ( member @ TA @ A @ ( insert @ TA @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_insertCI) ).
thf(1128,plain,
! [TA: $tType,A: TA,B: set @ TA,C: TA] :
( ( ~ ( member @ TA @ A @ B )
=> ( A = C ) )
=> ( member @ TA @ A @ ( insert @ TA @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).
thf(62,axiom,
! [A: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_fgs__sub__carrier) ).
thf(456,plain,
! [A: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(200,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: nat > TA,B: nat,C: nat] :
( ! [D: nat] : ( ord_less_eq @ TA @ ( A @ ( suc @ D ) ) @ ( A @ D ) )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ TA @ ( A @ C ) @ ( A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_157_lift__Suc__antimono__le) ).
thf(974,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: nat > TA,B: nat,C: nat] :
( ! [D: nat] : ( ord_less_eq @ TA @ ( A @ ( suc @ D ) ) @ ( A @ D ) )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ TA @ ( A @ C ) @ ( A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(142,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less @ nat @ B @ ( suc @ A ) )
= ( B = A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_164_le__less__Suc__eq) ).
thf(742,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)],[142]) ).
thf(254,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( member @ TA @ A @ B )
=> ( ( minus_minus @ ( set @ TA ) @ ( insert @ TA @ A @ C ) @ B )
= ( minus_minus @ ( set @ TA ) @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_insert__Diff1) ).
thf(1121,plain,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( member @ TA @ A @ B )
=> ( ( minus_minus @ ( set @ TA ) @ ( insert @ TA @ A @ C ) @ B )
= ( minus_minus @ ( set @ TA ) @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).
thf(231,axiom,
! [TA: $tType,A: TA] : ( member @ TA @ A @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_238_singletonI) ).
thf(1064,plain,
! [TA: $tType,A: TA] : ( member @ TA @ A @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).
thf(210,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/sandbox/benchmark/theBenchmark.p',conj_1) ).
thf(1006,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)],[210]) ).
thf(39,axiom,
! [A: nat] :
( ( suc @ A )
!= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_104_Suc__neq__Zero) ).
thf(388,plain,
! [A: nat] :
( ( suc @ A )
!= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(178,axiom,
! [TA: $tType,A: TA,B: set @ TA] :
( ( insert @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) )
= ( insert @ TA @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_237_insert__Diff__single) ).
thf(869,plain,
! [TA: $tType,A: TA,B: set @ TA] :
( ( insert @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) )
= ( insert @ TA @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(126,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/sandbox/benchmark/theBenchmark.p',fact_178_less__diff__iff) ).
thf(697,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)],[126]) ).
thf(239,axiom,
! [A: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( submodule @ b @ d @ a @ c @ r @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8_fgs__submodule) ).
thf(1089,plain,
! [A: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( submodule @ b @ d @ a @ c @ r @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).
thf(219,axiom,
! [A: set @ a,B: nat > b,C: nat,D: nat > a,E: nat > a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( member @ ( nat > b ) @ B
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
@ ^ [F: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_9_linear__comb__eq) ).
thf(1028,plain,
! [A: set @ a,B: nat > b,C: nat,D: nat > a,E: nat > a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( member @ ( nat > b ) @ B
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
@ ^ [F: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ 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)],[219]) ).
thf(73,axiom,
! [A: nat,B: nat > $o] :
( ( ? [C: nat] :
( ( ord_less @ nat @ C @ ( suc @ A ) )
& ( B @ C ) ) )
= ( ( B @ A )
| ? [C: nat] :
( ( ord_less @ nat @ C @ A )
& ( B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_Ex__less__Suc) ).
thf(488,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)],[73]) ).
thf(20,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
| ( ord_less_eq @ nat @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_68_nat__le__linear) ).
thf(339,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)],[20]) ).
thf(108,axiom,
! [TA: $tType,A: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ ( bot_bot @ ( set @ TA ) ) @ A )
= ( bot_bot @ ( set @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_empty__Diff) ).
thf(637,plain,
! [TA: $tType,A: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ ( bot_bot @ ( set @ TA ) ) @ A )
= ( bot_bot @ ( set @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(215,axiom,
! [TA: $tType,A: TA] :
~ ( member @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_empty__iff) ).
thf(1016,plain,
! [TA: $tType,A: TA] :
~ ( member @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(122,axiom,
! [TA: $tType] :
( ( comm_monoid_diff @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ ( zero_zero @ TA ) @ A )
= ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_230_zero__diff) ).
thf(686,plain,
! [TA: $tType] :
( ( comm_monoid_diff @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ ( zero_zero @ TA ) @ A )
= ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(236,axiom,
! [A: set @ b] :
( ( ideal @ b @ d @ r @ A )
=> ( ( linear_span @ b @ d @ a @ c @ r @ m @ A @ ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) )
= ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_244_l__span__zero) ).
thf(1075,plain,
! [A: set @ b] :
( ( ideal @ b @ d @ r @ A )
=> ( ( linear_span @ b @ d @ a @ c @ r @ m @ A @ ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) )
= ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).
thf(251,axiom,
ord_less_eq @ ( set @ a ) @ h1 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_10) ).
thf(1114,plain,
ord_less_eq @ ( set @ a ) @ h1 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).
thf(243,axiom,
( ( ( g @ xa )
= h2 )
| ( member @ a @ ( g @ xa ) @ h1 ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_20) ).
thf(1099,plain,
( ( ( g @ xa )
= h2 )
| ( member @ a @ ( g @ xa ) @ h1 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).
thf(127,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ ( suc @ B ) )
= ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_163_less__Suc__eq__le) ).
thf(701,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)],[127]) ).
thf(80,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: set @ TA,E: set @ TA] :
( ( module @ TA @ TD @ TC @ TB @ A @ B )
=> ( ( free_generator @ TD @ TB @ TA @ TC @ B @ A @ C )
=> ( ( ord_less_eq @ ( set @ TA ) @ D @ E )
=> ( ( ord_less_eq @ ( set @ TA ) @ E @ C )
=> ( ord_less_eq @ ( set @ TA ) @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ D ) @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ E ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_Module_Ofgs__mono) ).
thf(515,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: set @ TA,E: set @ TA] :
( ( module @ TA @ TD @ TC @ TB @ A @ B )
=> ( ( free_generator @ TD @ TB @ TA @ TC @ B @ A @ C )
=> ( ( ord_less_eq @ ( set @ TA ) @ D @ E )
=> ( ( ord_less_eq @ ( set @ TA ) @ E @ C )
=> ( ord_less_eq @ ( set @ TA ) @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ D ) @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ E ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(114,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/sandbox/benchmark/theBenchmark.p',fact_67_Nat_Oex__has__greatest__nat) ).
thf(654,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)],[114]) ).
thf(279,axiom,
! [A: set @ a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ A )
=> ( ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ product_unit ) ) @ ( mdl @ a @ b @ c @ m @ A ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_214_mdl__carrier) ).
thf(1192,plain,
! [A: set @ a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ A )
=> ( ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ product_unit ) ) @ ( mdl @ a @ b @ c @ m @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).
thf(22,axiom,
! [A: nat,B: nat > $o] :
( ( ! [C: nat] :
( ( ord_less @ nat @ C @ ( suc @ A ) )
=> ( B @ C ) ) )
= ( ( B @ A )
& ! [C: nat] :
( ( ord_less @ nat @ C @ A )
=> ( B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_All__less__Suc) ).
thf(343,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)],[22]) ).
thf(165,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) )] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ideal @ TB @ TD @ B @ ( annihilator @ TB @ TD @ TA @ TC @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_Module_OAnn__is__ideal) ).
thf(805,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) )] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ideal @ TB @ TD @ B @ ( annihilator @ TB @ TD @ TA @ TC @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(42,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/sandbox/benchmark/theBenchmark.p',fact_152_diffs0__imp__equal) ).
thf(398,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)],[42]) ).
thf(118,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ( minus_minus @ ( set @ TA ) @ A @ B )
= ( bot_bot @ ( set @ TA ) ) )
= ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_233_Diff__eq__empty__iff) ).
thf(674,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ( minus_minus @ ( set @ TA ) @ A @ B )
= ( bot_bot @ ( set @ TA ) ) )
= ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(161,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ( ( suc @ A )
!= B )
=> ( ord_less @ nat @ ( suc @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_Suc__lessI) ).
thf(794,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)],[161]) ).
thf(133,axiom,
! [A: nat,B: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ A @ B ) @ ( suc @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_diff__less__Suc) ).
thf(718,plain,
! [A: nat,B: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ A @ B ) @ ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(6,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_less__not__refl) ).
thf(298,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(50,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA,D: set @ TA] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( submodule @ TB @ TD @ TA @ TC @ B @ A @ C )
=> ( ( submodule @ TB @ TD @ TA @ TC @ B @ A @ D )
=> ( ideal @ TB @ TD @ B @ ( quotie135653060odules @ TB @ TD @ TA @ TC @ B @ A @ C @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_84_Module_Oquotient__of__submodules__is__ideal) ).
thf(425,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA,D: set @ TA] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( submodule @ TB @ TD @ TA @ TC @ B @ A @ C )
=> ( ( submodule @ TB @ TD @ TA @ TC @ B @ A @ D )
=> ( ideal @ TB @ TD @ B @ ( quotie135653060odules @ TB @ TD @ TA @ TC @ B @ A @ C @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(143,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) )] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( submodule @ TB @ TD @ TA @ TC @ B @ A @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_Module_Osubmodule__whole) ).
thf(745,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) )] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( submodule @ TB @ TD @ TA @ TC @ B @ A @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(123,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/sandbox/benchmark/theBenchmark.p',fact_154_diff__less__mono2) ).
thf(689,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)],[123]) ).
thf(57,axiom,
zero @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ozero) ).
thf(444,plain,
zero @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(98,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( generator @ TB @ TD @ TA @ TC @ B @ A @ C )
=> ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_Module_Ogenerator__sub__carrier) ).
thf(601,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( generator @ TB @ TD @ TA @ TC @ B @ A @ C )
=> ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(196,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less @ nat @ A @ B )
=> ( ( ord_less @ nat @ A @ ( suc @ B ) )
=> ( B = A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_less__antisym) ).
thf(965,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)],[196]) ).
thf(145,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ ( suc @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_le__SucI) ).
thf(750,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)],[145]) ).
thf(79,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) )
=> ( ord_less_eq @ ( set @ TA ) @ ( algebr833503410le_fgs @ TB @ TD @ TA @ TC @ B @ A @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_Module_Ofgs__sub__carrier) ).
thf(513,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) )
=> ( ord_less_eq @ ( set @ TA ) @ ( algebr833503410le_fgs @ TB @ TD @ TA @ TC @ B @ A @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(52,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ TA] :
( ( ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) )
= ( insert @ TA @ B @ C ) )
= ( ( B = A )
& ( ord_less_eq @ ( set @ TA ) @ C @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_singleton__insert__inj__eq) ).
thf(430,plain,
! [TA: $tType,A: TA,B: TA,C: set @ TA] :
( ( ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) )
= ( insert @ TA @ B @ C ) )
= ( ( B = A )
& ( ord_less_eq @ ( set @ TA ) @ C @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(267,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ TA] :
( ( member @ TA @ A @ ( insert @ TA @ B @ C ) )
= ( ( A = B )
| ( member @ TA @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_211_insert__iff) ).
thf(1154,plain,
! [TA: $tType,A: TA,B: TA,C: set @ TA] :
( ( member @ TA @ A @ ( insert @ TA @ B @ C ) )
= ( ( A = B )
| ( member @ TA @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).
thf(183,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/sandbox/benchmark/theBenchmark.p',fact_177_diff__less__mono) ).
thf(904,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)],[183]) ).
thf(138,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( suc @ A ) @ B )
=> ( ord_less @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_Suc__lessD) ).
thf(732,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( suc @ A ) @ B )
=> ( ord_less @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(256,axiom,
! [TA: $tType,A: set @ TA] :
( ( ! [B: TA] :
~ ( member @ TA @ B @ A ) )
= ( A
= ( bot_bot @ ( set @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_all__not__in__conv) ).
thf(1125,plain,
! [TA: $tType,A: set @ TA] :
( ( ! [B: TA] :
~ ( member @ TA @ B @ A ) )
= ( A
= ( bot_bot @ ( set @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).
thf(153,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ B ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_252_Diff__subset) ).
thf(772,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ B ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(1,conjecture,
member @ a @ ( g @ xa ) @ h1,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_21) ).
thf(2,negated_conjecture,
~ ( member @ a @ ( g @ xa ) @ h1 ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(286,plain,
~ ( member @ a @ ( g @ xa ) @ h1 ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(194,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq @ nat @ A @ B ) )
= ( ord_less_eq @ nat @ ( suc @ B ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_not__less__eq__eq) ).
thf(958,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)],[194]) ).
thf(255,axiom,
ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ h1 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_11) ).
thf(1124,plain,
ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ h1 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).
thf(46,axiom,
! [A: nat,B: nat] :
( ( ( ord_less @ nat @ A @ B )
| ( A = B ) )
=> ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_114_less__or__eq__imp__le) ).
thf(413,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)],[46]) ).
thf(180,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less @ nat @ A @ ( suc @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_161_le__imp__less__Suc) ).
thf(875,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)],[180]) ).
thf(93,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_neq0__conv) ).
thf(577,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(261,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TB,D: set @ TA,E: nat] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( ideal @ TB @ TD @ B @ C )
=> ( ( ord_less_eq @ ( set @ TA ) @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) )
=> ! [F: nat > TB,G: nat > TA] :
( ( ( member @ ( nat > TB ) @ F
@ ( pi @ nat @ TB
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
@ ^ [H: nat] : C ) )
& ( member @ ( nat > TA ) @ G
@ ( pi @ nat @ TA
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
@ ^ [H: nat] : D ) ) )
=> ( member @ TA @ ( l_comb @ TB @ TD @ TA @ TC @ B @ A @ E @ F @ G ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_Module_Oliear__comb__memTr) ).
thf(1139,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TB,D: set @ TA,E: nat] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( ideal @ TB @ TD @ B @ C )
=> ( ( ord_less_eq @ ( set @ TA ) @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) )
=> ! [F: nat > TB,G: nat > TA] :
( ( ( member @ ( nat > TB ) @ F
@ ( pi @ nat @ TB
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
@ ^ [H: nat] : C ) )
& ( member @ ( nat > TA ) @ G
@ ( pi @ nat @ TA
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
@ ^ [H: nat] : D ) ) )
=> ( member @ TA @ ( l_comb @ TB @ TD @ TA @ TC @ B @ A @ E @ F @ G ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).
thf(37,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_subset__antisym) ).
thf(382,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(91,axiom,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
=> ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_bot__nat__0_Oextremum__uniqueI) ).
thf(572,plain,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
=> ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(11,axiom,
! [TA: $tType,TB: $tType] :
( ( order @ TA )
=> ( order @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oorder) ).
thf(313,plain,
! [TA: $tType,TB: $tType] :
( ( order @ TA )
=> ( order @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(102,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
| ( A = B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_le__eq__less__or__eq) ).
thf(611,plain,
( ( ord_less_eq @ nat )
= ( ^ [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
| ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(246,axiom,
! [A: set @ b] :
( ( ideal @ b @ d @ r @ A )
=> ( submodule @ b @ d @ a @ c @ r @ m @ ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_smodule__ideal__coeff__is__Submodule) ).
thf(1107,plain,
! [A: set @ b] :
( ( ideal @ b @ d @ r @ A )
=> ( submodule @ b @ d @ a @ c @ r @ m @ ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).
thf(66,axiom,
! [A: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
=> ? [B: nat] :
( A
= ( suc @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_gr0__implies__Suc) ).
thf(466,plain,
! [A: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
=> ? [B: nat] :
( A
= ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(128,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ A @ B ) )
= ( ord_less @ nat @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_zero__less__diff) ).
thf(704,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)],[128]) ).
thf(190,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_226_le__zero__eq) ).
thf(947,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(24,axiom,
! [A: nat] :
( ( minus_minus @ nat @ A @ ( zero_zero @ nat ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_151_minus__nat_Odiff__0) ).
thf(350,plain,
! [A: nat] :
( ( minus_minus @ nat @ A @ ( zero_zero @ nat ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(84,axiom,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_less__eq__nat_Osimps_I1_J) ).
thf(540,plain,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(117,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,TH: $tType,A: carrier_ext @ TG @ ( aGroup_ext @ TG @ ( module_ext @ TG @ TB @ TH ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: carrier_ext @ TF @ ( aGroup_ext @ TF @ ( module_ext @ TF @ TB @ TE ) ),D: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) )] :
( ( module @ TG @ TB @ TH @ TD @ A @ B )
=> ( ( module @ TF @ TB @ TE @ TD @ C @ B )
=> ( ( module @ TA @ TB @ TC @ TD @ D @ B )
=> ( ( misomorphic @ TB @ TD @ TF @ TE @ TG @ TH @ B @ C @ A )
=> ( ( misomorphic @ TB @ TD @ TG @ TH @ TA @ TC @ B @ A @ D )
=> ( misomorphic @ TB @ TD @ TF @ TE @ TA @ TC @ B @ C @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_Module_Omisom__trans) ).
thf(672,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,TH: $tType,A: carrier_ext @ TG @ ( aGroup_ext @ TG @ ( module_ext @ TG @ TB @ TH ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: carrier_ext @ TF @ ( aGroup_ext @ TF @ ( module_ext @ TF @ TB @ TE ) ),D: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) )] :
( ( module @ TG @ TB @ TH @ TD @ A @ B )
=> ( ( module @ TF @ TB @ TE @ TD @ C @ B )
=> ( ( module @ TA @ TB @ TC @ TD @ D @ B )
=> ( ( misomorphic @ TB @ TD @ TF @ TE @ TG @ TH @ B @ C @ A )
=> ( ( misomorphic @ TB @ TD @ TG @ TH @ TA @ TC @ B @ A @ D )
=> ( misomorphic @ TB @ TD @ TF @ TE @ TA @ TC @ B @ C @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(58,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TA @ TC ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),C: set @ TA] :
( ( module @ TD @ TA @ TC @ TB @ A @ B )
=> ( ( ideal @ TA @ TB @ B @ C )
=> ( submodule @ TA @ TB @ TD @ TC @ B @ A @ ( smodule_ideal_coeff @ TA @ TB @ TD @ TC @ B @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_83_Module_Osmodule__ideal__coeff__is__Submodule) ).
thf(445,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TA @ TC ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),C: set @ TA] :
( ( module @ TD @ TA @ TC @ TB @ A @ B )
=> ( ( ideal @ TA @ TB @ B @ C )
=> ( submodule @ TA @ TB @ TD @ TC @ B @ A @ ( smodule_ideal_coeff @ TA @ TB @ TD @ TC @ B @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(189,axiom,
! [A: nat,B: nat,C: nat > $o] :
( ( ord_less @ nat @ A @ B )
=> ( ! [D: nat] :
( ( B
= ( suc @ D ) )
=> ( C @ D ) )
=> ( ! [D: nat] :
( ( ord_less @ nat @ D @ B )
=> ( ( C @ ( suc @ D ) )
=> ( C @ D ) ) )
=> ( C @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_strict__inc__induct) ).
thf(936,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)],[189]) ).
thf(202,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 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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 @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_194_l__span__closed2Tr) ).
thf(979,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 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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 @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ 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)],[202]) ).
thf(158,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/sandbox/benchmark/theBenchmark.p',fact_147_Nat_Odiff__diff__eq) ).
thf(784,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)],[158]) ).
thf(28,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ m2,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_16) ).
thf(362,plain,
ord_less @ nat @ ( zero_zero @ nat ) @ m2,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(13,axiom,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less @ nat @ D @ C )
& ~ ( A @ D ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_infinite__descent) ).
thf(318,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)],[13]) ).
thf(112,axiom,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_bot__nat__0_Oextremum) ).
thf(649,plain,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(212,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ b @ TC ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ b @ TB ) )] :
( ( module @ TD @ b @ TC @ d @ A @ r )
=> ( ( module @ TA @ b @ TB @ d @ B @ r )
=> ( ( misomorphic @ b @ d @ TD @ TC @ a @ c @ r @ A @ m )
=> ( ( misomorphic @ b @ d @ a @ c @ TA @ TB @ r @ m @ B )
=> ( misomorphic @ b @ d @ TD @ TC @ TA @ TB @ r @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_misom__trans) ).
thf(1011,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ b @ TC ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ b @ TB ) )] :
( ( module @ TD @ b @ TC @ d @ A @ r )
=> ( ( module @ TA @ b @ TB @ d @ B @ r )
=> ( ( misomorphic @ b @ d @ TD @ TC @ a @ c @ r @ A @ m )
=> ( ( misomorphic @ b @ d @ a @ c @ TA @ TB @ r @ m @ B )
=> ( misomorphic @ b @ d @ TD @ TC @ TA @ TB @ r @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(48,axiom,
! [A: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
=> ( ( suc @ ( minus_minus @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26_Suc__pred) ).
thf(419,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)],[48]) ).
thf(225,axiom,
! [A: set @ a,B: set @ a] :
( ( generator @ b @ d @ a @ c @ r @ m @ A )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( ord_less_eq @ ( set @ a ) @ A @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ B ) )
=> ( generator @ b @ d @ a @ c @ r @ m @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_243_generator__generator) ).
thf(1048,plain,
! [A: set @ a,B: set @ a] :
( ( generator @ b @ d @ a @ c @ r @ m @ A )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( ord_less_eq @ ( set @ a ) @ A @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ B ) )
=> ( generator @ b @ d @ a @ c @ r @ m @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(140,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/sandbox/benchmark/theBenchmark.p',fact_185_diff__Suc__less) ).
thf(736,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)],[140]) ).
thf(104,axiom,
! [A: nat] :
~ ( ord_less_eq @ nat @ ( suc @ A ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_Suc__n__not__le__n) ).
thf(616,plain,
! [A: nat] :
~ ( ord_less_eq @ nat @ ( suc @ A ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(8,axiom,
! [TA: $tType,TB: $tType] :
( ( wellorder @ TB )
=> ! [A: TA > TB,B: TA > $o,C: TA] :
( ! [D: TA] :
( ! [E: TA] :
( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
=> ( B @ E ) )
=> ( B @ D ) )
=> ( B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_66_measure__induct) ).
thf(303,plain,
! [TA: $tType,TB: $tType] :
( ( wellorder @ TB )
=> ! [A: TA > TB,B: TA > $o,C: TA] :
( ! [D: TA] :
( ! [E: TA] :
( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
=> ( B @ E ) )
=> ( B @ D ) )
=> ( B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(69,axiom,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
= ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_old_Onat_Oinject) ).
thf(477,plain,
! [A: nat,B: nat] :
( ( ( suc @ A )
= ( suc @ B ) )
= ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(75,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( minus_minus @ nat @ B @ ( minus_minus @ nat @ B @ A ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50_diff__diff__cancel) ).
thf(494,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)],[75]) ).
thf(148,axiom,
wellorder @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Owellorder) ).
thf(758,plain,
wellorder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(101,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA] :
( ( module @ TA @ TD @ TC @ TB @ A @ B )
=> ( ( free_generator @ TD @ TB @ TA @ TC @ B @ A @ C )
=> ( generator @ TD @ TB @ TA @ TC @ B @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_Module_Ofree__generator__generator) ).
thf(609,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA] :
( ( module @ TA @ TD @ TC @ TB @ A @ B )
=> ( ( free_generator @ TD @ TB @ TA @ TC @ B @ A @ C )
=> ( generator @ TD @ TB @ TA @ TC @ B @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(14,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ~ ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_74_linorder__neqE__nat) ).
thf(323,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)],[14]) ).
thf(95,axiom,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
= ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_diff__gt__0__iff__gt) ).
thf(594,plain,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
= ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(136,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/sandbox/benchmark/theBenchmark.p',fact_70_le__trans) ).
thf(726,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)],[136]) ).
thf(71,axiom,
! [TA: $tType,A: TA > $o] :
( ( ( bot_bot @ ( set @ TA ) )
= ( collect @ TA @ A ) )
= ( ! [B: TA] :
~ ( A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_empty__Collect__eq) ).
thf(483,plain,
! [TA: $tType,A: TA > $o] :
( ( ( bot_bot @ ( set @ TA ) )
= ( collect @ TA @ A ) )
= ( ! [B: TA] :
~ ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(187,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less @ nat @ A @ B ) )
= ( ord_less @ nat @ B @ ( suc @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_126_not__less__eq) ).
thf(915,plain,
! [A: nat,B: nat] :
( ( ~ ( ord_less @ nat @ A @ B ) )
= ( ord_less @ nat @ B @ ( suc @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(152,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/sandbox/benchmark/theBenchmark.p',fact_143_diff__le__mono2) ).
thf(769,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)],[152]) ).
thf(217,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TB,D: set @ TA,E: nat > TB,F: nat,G: nat > TA] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( ideal @ TB @ TD @ B @ C )
=> ( ( ord_less_eq @ ( set @ TA ) @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) )
=> ( ( member @ ( nat > TB ) @ E
@ ( pi @ nat @ TB
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ F ) )
@ ^ [H: nat] : C ) )
=> ( ( member @ ( nat > TA ) @ G
@ ( pi @ nat @ TA
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ F ) )
@ ^ [H: nat] : D ) )
=> ( member @ TA @ ( l_comb @ TB @ TD @ TA @ TC @ B @ A @ F @ E @ G ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29_Module_Ol__comb__mem) ).
thf(1022,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TB,D: set @ TA,E: nat > TB,F: nat,G: nat > TA] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( ideal @ TB @ TD @ B @ C )
=> ( ( ord_less_eq @ ( set @ TA ) @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) )
=> ( ( member @ ( nat > TB ) @ E
@ ( pi @ nat @ TB
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ F ) )
@ ^ [H: nat] : C ) )
=> ( ( member @ ( nat > TA ) @ G
@ ( pi @ nat @ TA
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ F ) )
@ ^ [H: nat] : D ) )
=> ( member @ TA @ ( l_comb @ TB @ TD @ TA @ TC @ B @ A @ F @ E @ G ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).
thf(275,axiom,
! [A: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( generator @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_223_fgs__generator) ).
thf(1177,plain,
! [A: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( generator @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).
thf(249,axiom,
module @ a @ b @ c @ d @ m @ r,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_Module__axioms) ).
thf(1112,plain,
module @ a @ b @ c @ d @ m @ r,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(30,axiom,
module @ e @ b @ f @ d @ n @ r,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_4) ).
thf(367,plain,
module @ e @ b @ f @ d @ n @ r,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(89,axiom,
! [A: nat] :
( A
!= ( suc @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_n__not__Suc__n) ).
thf(552,plain,
! [A: nat] :
( A
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(174,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/sandbox/benchmark/theBenchmark.p',fact_170_ex__least__nat__le) ).
thf(848,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)],[174]) ).
thf(82,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/sandbox/benchmark/theBenchmark.p',fact_167_dec__induct) ).
thf(520,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)],[82]) ).
thf(186,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ ( suc @ B ) )
=> ( ~ ( ord_less @ nat @ A @ B )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_less__SucE) ).
thf(912,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)],[186]) ).
thf(282,axiom,
member @ a @ h2 @ h,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_14) ).
thf(1198,plain,
member @ a @ h2 @ h,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).
thf(169,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ B @ A )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_le__antisym) ).
thf(827,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)],[169]) ).
thf(125,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: carrier_ext @ TF @ ( aGroup_ext @ TF @ ( module_ext @ TF @ TB @ TE ) )] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( module @ TF @ TB @ TE @ TD @ C @ B )
=> ( ( misomorphic @ TB @ TD @ TA @ TC @ TF @ TE @ B @ A @ C )
=> ( misomorphic @ TB @ TD @ TF @ TE @ TA @ TC @ B @ C @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_Module_Omisom__sym) ).
thf(695,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: carrier_ext @ TF @ ( aGroup_ext @ TF @ ( module_ext @ TF @ TB @ TE ) )] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( module @ TF @ TB @ TE @ TD @ C @ B )
=> ( ( misomorphic @ TB @ TD @ TA @ TC @ TF @ TE @ B @ A @ C )
=> ( misomorphic @ TB @ TD @ TF @ TE @ TA @ TC @ B @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(232,axiom,
member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ m2 @ t @ g ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( insert @ a @ h2 @ h1 ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_12) ).
thf(1066,plain,
member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ m2 @ t @ g ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( insert @ a @ h2 @ h1 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(214,axiom,
ord_less_eq @ ( set @ a ) @ h @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_9) ).
thf(1015,plain,
ord_less_eq @ ( set @ a ) @ h @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(274,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/sandbox/benchmark/theBenchmark.p',conj_0) ).
thf(1176,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)],[274]) ).
thf(9,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( B != A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_less__not__refl2) ).
thf(306,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( B != A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(250,axiom,
submodule @ b @ d @ a @ c @ r @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( insert @ a @ h2 @ ( bot_bot @ ( set @ a ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_18) ).
thf(1113,plain,
submodule @ b @ d @ a @ c @ r @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( insert @ a @ h2 @ ( bot_bot @ ( set @ a ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).
thf(244,axiom,
( ( g @ m2 )
= h2 ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_3) ).
thf(1102,plain,
( ( g @ m2 )
= h2 ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).
thf(55,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_not__less0) ).
thf(438,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(68,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/sandbox/benchmark/theBenchmark.p',fact_112_less__mono__imp__le__mono) ).
thf(473,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)],[68]) ).
thf(220,axiom,
! [A: set @ a,B: a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ A )
=> ( ( member @ a @ B @ A )
=> ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1_submodule__subset1) ).
thf(1034,plain,
! [A: set @ a,B: a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ A )
=> ( ( member @ a @ B @ A )
=> ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(120,axiom,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_bot__nat__0_Oextremum__unique) ).
thf(680,plain,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(259,axiom,
( x
= ( l_comb @ b @ d @ a @ c @ r @ m @ m2 @ t @ g ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_13) ).
thf(1134,plain,
( x
= ( l_comb @ b @ d @ a @ c @ r @ m @ m2 @ t @ g ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).
thf(26,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( free_generator @ TB @ TD @ TA @ TC @ B @ A @ C )
=> ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_Module_Ofree__generator__sub) ).
thf(355,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( ( free_generator @ TB @ TD @ TA @ TC @ B @ A @ C )
=> ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(264,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 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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 @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_195_l__span__closed2) ).
thf(1145,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 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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 @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ 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)],[264]) ).
thf(216,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 @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ) )
=> ( ( l_comb @ b @ d @ a @ c @ r @ m @ C @ B @ D )
= ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_linear__comb0__2Tr) ).
thf(1018,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 @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ) )
=> ( ( l_comb @ b @ d @ a @ c @ r @ m @ C @ B @ D )
= ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(197,axiom,
wellorder @ product_unit,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Ounit___Orderings_Owellorder_4) ).
thf(968,plain,
wellorder @ product_unit,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).
thf(63,axiom,
ring @ b @ d @ r,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0_sc__Ring) ).
thf(458,plain,
ring @ b @ d @ r,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(94,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/sandbox/benchmark/theBenchmark.p',fact_85_transitive__stepwise__le) ).
thf(580,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)],[94]) ).
thf(109,axiom,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_229_diff__0__right) ).
thf(640,plain,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(179,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ ( suc @ B ) )
= ( ( ord_less @ nat @ A @ B )
| ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_127_less__Suc__eq) ).
thf(872,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)],[179]) ).
thf(129,axiom,
! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ ( bot_bot @ ( set @ TA ) ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_empty__subsetI) ).
thf(707,plain,
! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ ( bot_bot @ ( set @ TA ) ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(193,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/sandbox/benchmark/theBenchmark.p',fact_144_le__diff__iff_H) ).
thf(954,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)],[193]) ).
thf(141,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_less__zeroE) ).
thf(739,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(10,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: nat > TA,B: nat,C: nat] :
( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) )
= ( ord_less @ nat @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_159_lift__Suc__mono__less__iff) ).
thf(310,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: nat > TA,B: nat,C: nat] :
( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
=> ( ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) )
= ( ord_less @ nat @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(131,axiom,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_231_diff__zero) ).
thf(712,plain,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(208,axiom,
! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ b @ TA ) )] :
( ( module @ TB @ b @ TA @ d @ A @ r )
=> ( module @ ( a > TB ) @ b @ product_unit @ d @ ( hom @ b @ d @ a @ c @ TB @ TA @ r @ m @ A ) @ r ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_HOM__is__module) ).
thf(1003,plain,
! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ b @ TA ) )] :
( ( module @ TB @ b @ TA @ d @ A @ r )
=> ( module @ ( a > TB ) @ b @ product_unit @ d @ ( hom @ b @ d @ a @ c @ TB @ TA @ r @ m @ A ) @ r ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(227,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_subsetI) ).
thf(1053,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).
thf(160,axiom,
! [A: nat] :
( ( ord_less @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_less__Suc0) ).
thf(791,plain,
! [A: nat] :
( ( ord_less @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(56,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_142_gr0I) ).
thf(441,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(65,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ ( suc @ A ) @ B )
=> ? [C: nat] :
( B
= ( suc @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_Suc__le__D) ).
thf(463,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)],[65]) ).
thf(238,axiom,
! [A: set @ b,B: set @ a] :
( ( ideal @ b @ d @ r @ A )
=> ( ( generator @ b @ d @ a @ c @ r @ m @ B )
=> ( ( ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A )
= ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B )
= ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_245_l__span__closed3) ).
thf(1085,plain,
! [A: set @ b,B: set @ a] :
( ( ideal @ b @ d @ r @ A )
=> ( ( generator @ b @ d @ a @ c @ r @ m @ B )
=> ( ( ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A )
= ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B )
= ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).
thf(192,axiom,
! [TA: $tType,A: TA,B: set @ TA] :
( ( insert @ TA @ A @ ( insert @ TA @ A @ B ) )
= ( insert @ TA @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_212_insert__absorb2) ).
thf(951,plain,
! [TA: $tType,A: TA,B: set @ TA] :
( ( insert @ TA @ A @ ( insert @ TA @ A @ B ) )
= ( insert @ TA @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(110,axiom,
! [A: nat,B: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ B ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_diff__le__self) ).
thf(643,plain,
! [A: nat,B: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ B ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(222,axiom,
! [A: set @ a,B: set @ a,C: nat > a,D: nat,E: nat > b] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) ) )
=> ( ( l_comb @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ B ) ) @ D @ E @ C )
= ( l_comb @ b @ d @ a @ c @ r @ m @ D @ E @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_eSum__in__SubmoduleTr) ).
thf(1038,plain,
! [A: set @ a,B: set @ a,C: nat > a,D: nat,E: nat > b] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) ) )
=> ( ( l_comb @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ 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)],[222]) ).
thf(32,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
= ( B @ C ) )
=> ( ( collect @ TA @ A )
= ( collect @ TA @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_Collect__cong) ).
thf(370,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
= ( B @ C ) )
=> ( ( collect @ TA @ A )
= ( collect @ TA @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(86,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA] :
( ( module @ TA @ TD @ TC @ TB @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
=> ( submodule @ TD @ TB @ TA @ TC @ B @ A @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_Module_Ofgs__submodule) ).
thf(544,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA] :
( ( module @ TA @ TD @ TC @ TB @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
=> ( submodule @ TD @ TB @ TA @ TC @ B @ A @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(184,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ ( suc @ B ) )
=> ( ~ ( ord_less_eq @ nat @ A @ B )
=> ( A
= ( suc @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_le__SucE) ).
thf(907,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)],[184]) ).
thf(19,axiom,
! [A: nat,B: nat] :
( ( A != B )
= ( ( ord_less @ nat @ A @ B )
| ( ord_less @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_nat__neq__iff) ).
thf(336,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)],[19]) ).
thf(262,axiom,
misomorphic @ b @ d @ a @ c @ a @ c @ r @ m @ m,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_misom__self) ).
thf(1141,plain,
misomorphic @ b @ d @ a @ c @ a @ c @ r @ m @ m,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).
thf(103,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ ( suc @ B ) )
= ( ( A
= ( zero_zero @ nat ) )
| ? [C: nat] :
( ( A
= ( suc @ C ) )
& ( ord_less @ nat @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_171_less__Suc__eq__0__disj) ).
thf(613,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)],[103]) ).
thf(97,axiom,
cancel1352612707id_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add) ).
thf(600,plain,
cancel1352612707id_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(176,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ ( suc @ A ) @ B )
=> ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_Suc__leD) ).
thf(864,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)],[176]) ).
thf(241,axiom,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_mem__Collect__eq) ).
thf(1094,plain,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).
thf(76,axiom,
! [A: nat,B: nat] :
( ( minus_minus @ nat @ ( suc @ A ) @ ( suc @ B ) )
= ( minus_minus @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_diff__Suc__Suc) ).
thf(497,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)],[76]) ).
thf(107,axiom,
! [A: nat > $o,B: nat] :
( ( A @ ( zero_zero @ nat ) )
=> ( ! [C: nat] :
( ( A @ C )
=> ( A @ ( suc @ C ) ) )
=> ( A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_nat__induct) ).
thf(627,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)],[107]) ).
thf(60,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/sandbox/benchmark/theBenchmark.p',fact_148_le__diff__iff) ).
thf(450,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)],[60]) ).
thf(268,axiom,
! [A: set @ a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ A )
=> ( module @ a @ b @ product_unit @ d @ ( mdl @ a @ b @ c @ m @ A ) @ r ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_215_mdl__is__module) ).
thf(1157,plain,
! [A: set @ a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ A )
=> ( module @ a @ b @ product_unit @ d @ ( mdl @ a @ b @ c @ m @ A ) @ r ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).
thf(201,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/sandbox/benchmark/theBenchmark.p',fact_146_diff__le__mono) ).
thf(976,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)],[201]) ).
thf(149,axiom,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_old_Onat_Odistinct_I1_J) ).
thf(759,plain,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(182,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/sandbox/benchmark/theBenchmark.p',fact_184_ex__least__nat__less) ).
thf(891,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)],[182]) ).
thf(278,axiom,
! [A: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ A ) )
= ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_l__span__l__span) ).
thf(1189,plain,
! [A: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ A ) )
= ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).
thf(31,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) )] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( misomorphic @ TB @ TD @ TA @ TC @ TA @ TC @ B @ A @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_Module_Omisom__self) ).
thf(368,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) )] :
( ( module @ TA @ TB @ TC @ TD @ A @ B )
=> ( misomorphic @ TB @ TD @ TA @ TC @ TA @ TC @ B @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(29,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/sandbox/benchmark/theBenchmark.p',fact_149_eq__diff__iff) ).
thf(363,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)],[29]) ).
thf(90,axiom,
! [A: nat > $o,B: nat] :
( ( A @ ( zero_zero @ nat ) )
=> ( ! [C: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
=> ( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less @ nat @ D @ C )
& ~ ( A @ D ) ) ) )
=> ( A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_infinite__descent0) ).
thf(556,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)],[90]) ).
thf(195,axiom,
! [A: nat] :
( ( suc @ A )
!= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_old_Onat_Odistinct_I2_J) ).
thf(961,plain,
! [A: nat] :
( ( suc @ A )
!= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(81,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/sandbox/benchmark/theBenchmark.p',fact_181_diff__less) ).
thf(517,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)],[81]) ).
thf(43,axiom,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_103_Zero__neq__Suc) ).
thf(401,plain,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(47,axiom,
canoni770627133id_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add) ).
thf(418,plain,
canoni770627133id_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(157,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_47_bot__nat__0_Onot__eq__extremum) ).
thf(781,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(206,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 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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/sandbox/benchmark/theBenchmark.p',fact_249_l__span__closed) ).
thf(995,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 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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)],[206]) ).
thf(4,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_227_not__gr__zero) ).
thf(290,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(116,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ( B
!= ( suc @ A ) )
=> ~ ! [C: nat] :
( ( ord_less @ nat @ A @ C )
=> ( B
!= ( suc @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_Nat_OlessE) ).
thf(667,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)],[116]) ).
thf(150,axiom,
! [A: nat,B: nat] :
( ( ( minus_minus @ nat @ A @ B )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_diff__is__0__eq) ).
thf(763,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)],[150]) ).
thf(171,axiom,
! [A: set @ b,B: set @ a] :
( ( ideal @ b @ d @ r @ A )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( submodule @ b @ d @ a @ c @ r @ m @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_239_linear__span__subModule) ).
thf(832,plain,
! [A: set @ b,B: set @ a] :
( ( ideal @ b @ d @ r @ A )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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)],[171]) ).
thf(23,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( suc @ A ) @ B )
=> ~ ! [C: nat] :
( ( ord_less @ nat @ A @ C )
=> ( B
!= ( suc @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_Suc__lessE) ).
thf(346,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)],[23]) ).
thf(88,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/sandbox/benchmark/theBenchmark.p',fact_180_Suc__diff__Suc) ).
thf(549,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)],[88]) ).
thf(224,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 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_l__comb__mem) ).
thf(1045,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 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).
thf(53,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: TA] :
( ( ( insert @ TA @ A @ B )
= ( insert @ TA @ C @ ( bot_bot @ ( set @ TA ) ) ) )
= ( ( A = C )
& ( ord_less_eq @ ( set @ TA ) @ B @ ( insert @ TA @ C @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_singleton__insert__inj__eq_H) ).
thf(433,plain,
! [TA: $tType,A: TA,B: set @ TA,C: TA] :
( ( ( insert @ TA @ A @ B )
= ( insert @ TA @ C @ ( bot_bot @ ( set @ TA ) ) ) )
= ( ( A = C )
& ( ord_less_eq @ ( set @ TA ) @ B @ ( insert @ TA @ C @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(78,axiom,
! [TA: $tType,A: TA] :
( ( collect @ TA @ ( (=) @ TA @ A ) )
= ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_singleton__conv2) ).
thf(510,plain,
! [TA: $tType,A: TA] :
( ( collect @ TA @ ( (=) @ TA @ A ) )
= ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(270,axiom,
! [TA: $tType,A: set @ TA] :
( ( collect @ TA
@ ^ [B: TA] : ( member @ TA @ B @ A ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_Collect__mem__eq) ).
thf(1161,plain,
! [TA: $tType,A: set @ TA] :
( ( collect @ TA
@ ^ [B: TA] : ( member @ TA @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).
thf(163,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/sandbox/benchmark/theBenchmark.p',fact_51_Suc__diff__diff) ).
thf(800,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)],[163]) ).
thf(3,axiom,
! [TA: $tType,TB: $tType] :
( ( wellorder @ TB )
=> ! [A: TA > TB,B: TA > $o,C: TA] :
( ! [D: TA] :
( ! [E: TA] :
( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
=> ( B @ E ) )
=> ( B @ D ) )
=> ( B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_measure__induct__rule) ).
thf(287,plain,
! [TA: $tType,TB: $tType] :
( ( wellorder @ TB )
=> ! [A: TA > TB,B: TA > $o,C: TA] :
( ! [D: TA] :
( ! [E: TA] :
( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
=> ( B @ E ) )
=> ( B @ D ) )
=> ( B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(253,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: nat > TD,E: nat,F: nat > TA,G: nat > TA] :
( ( module @ TA @ TD @ TC @ TB @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
=> ( ( ( member @ ( nat > TD ) @ D
@ ( pi @ nat @ TD
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
@ ^ [H: nat] : ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ) @ B ) ) )
& ( member @ ( nat > TA ) @ F
@ ( pi @ nat @ TA
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
@ ^ [H: nat] : C ) )
& ( member @ ( nat > TA ) @ G
@ ( pi @ nat @ TA
@ ( 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 @ TD @ TB @ TA @ TC @ B @ A @ E @ D @ F )
= ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ D @ G ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_Module_Olinear__comb__eqTr) ).
thf(1116,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: nat > TD,E: nat,F: nat > TA,G: nat > TA] :
( ( module @ TA @ TD @ TC @ TB @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
=> ( ( ( member @ ( nat > TD ) @ D
@ ( pi @ nat @ TD
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
@ ^ [H: nat] : ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ) @ B ) ) )
& ( member @ ( nat > TA ) @ F
@ ( pi @ nat @ TA
@ ( collect @ nat
@ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
@ ^ [H: nat] : C ) )
& ( member @ ( nat > TA ) @ G
@ ( pi @ nat @ TA
@ ( 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 @ TD @ TB @ TA @ TC @ B @ A @ E @ D @ F )
= ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ D @ G ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(258,axiom,
submodule @ b @ d @ a @ c @ r @ m @ ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_224_submodule__0) ).
thf(1133,plain,
submodule @ b @ d @ a @ c @ r @ m @ ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).
thf(146,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_bot__nat__0_Oextremum__strict) ).
thf(752,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(59,axiom,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ A @ A )
= ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_228_diff__self) ).
thf(447,plain,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ A @ A )
= ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(12,axiom,
! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
( ! [D: TA] :
( ~ ( A @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
& ~ ( A @ E ) ) )
=> ( A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_infinite__descent__measure) ).
thf(315,plain,
! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
( ! [D: TA] :
( ~ ( A @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
& ~ ( A @ E ) ) )
=> ( A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(45,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/sandbox/benchmark/theBenchmark.p',fact_82_diff__commute) ).
thf(410,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)],[45]) ).
thf(44,axiom,
! [A: nat,B: nat] :
( ( A
= ( suc @ B ) )
=> ( A
!= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_nat_OdiscI) ).
thf(405,plain,
! [A: nat,B: nat] :
( ( A
= ( suc @ B ) )
=> ( A
!= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(265,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ ( insert @ TA @ A @ B ) @ C )
= ( ( member @ TA @ A @ C )
& ( ord_less_eq @ ( set @ TA ) @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_236_insert__subset) ).
thf(1148,plain,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ ( insert @ TA @ A @ B ) @ C )
= ( ( member @ TA @ A @ C )
& ( ord_less_eq @ ( set @ TA ) @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).
thf(49,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/sandbox/benchmark/theBenchmark.p',fact_176_Suc__diff__le) ).
thf(422,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)],[49]) ).
thf(191,axiom,
ideal @ b @ d @ r @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_8) ).
thf(950,plain,
ideal @ b @ d @ r @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(242,axiom,
~ ( member @ a @ h2 @ h1 ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_15) ).
thf(1097,plain,
~ ( member @ a @ h2 @ h1 ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).
thf(172,axiom,
! [TA: $tType,A: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ A @ ( bot_bot @ ( set @ TA ) ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_Diff__empty) ).
thf(835,plain,
! [TA: $tType,A: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ A @ ( bot_bot @ ( set @ TA ) ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(33,axiom,
! [A: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
= ( ? [B: nat] :
( A
= ( suc @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_gr0__conv__Suc) ).
thf(373,plain,
! [A: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
= ( ? [B: nat] :
( A
= ( suc @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(99,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( B
!= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_gr__implies__not0) ).
thf(603,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( B
!= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(15,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_less__irrefl__nat) ).
thf(326,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(188,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/sandbox/benchmark/theBenchmark.p',fact_166_inc__induct) ).
thf(918,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)],[188]) ).
thf(21,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TA @ TD ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( module @ TC @ TA @ TD @ TB @ A @ B )
=> ( ring @ TA @ TB @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_155_Module_Osc__Ring) ).
thf(341,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TA @ TD ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
( ( module @ TC @ TA @ TD @ TB @ A @ B )
=> ( ring @ TA @ TB @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(166,axiom,
! [TA: $tType,A: TA > nat,B: TA > $o,C: TA] :
( ! [D: TA] :
( ( ( A @ D )
= ( zero_zero @ nat ) )
=> ( B @ D ) )
=> ( ! [D: TA] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( A @ D ) )
=> ( ~ ( B @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( A @ E ) @ ( A @ D ) )
& ~ ( B @ E ) ) ) )
=> ( B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_infinite__descent0__measure) ).
thf(807,plain,
! [TA: $tType,A: TA > nat,B: TA > $o,C: TA] :
( ! [D: TA] :
( ( ( A @ D )
= ( zero_zero @ nat ) )
=> ( B @ D ) )
=> ( ! [D: TA] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( A @ D ) )
=> ( ~ ( B @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( A @ E ) @ ( A @ D ) )
& ~ ( B @ E ) ) ) )
=> ( B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(272,axiom,
free_generator @ b @ d @ a @ c @ r @ m @ h,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_5) ).
thf(1172,plain,
free_generator @ b @ d @ a @ c @ r @ m @ h,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).
thf(228,axiom,
ord_less_eq @ ( set @ a ) @ h1 @ h,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_6) ).
thf(1056,plain,
ord_less_eq @ ( set @ a ) @ h1 @ h,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).
thf(281,axiom,
submodule @ b @ d @ a @ c @ r @ m @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_submodule__whole) ).
thf(1197,plain,
submodule @ b @ d @ a @ c @ r @ m @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).
thf(134,axiom,
! [TA: $tType,A: TA] :
( ( collect @ TA
@ ^ [B: TA] : ( B = A ) )
= ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_singleton__conv) ).
thf(720,plain,
! [TA: $tType,A: TA] :
( ( collect @ TA
@ ^ [B: TA] : ( B = A ) )
= ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(41,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ ( suc @ A ) @ ( suc @ B ) )
= ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_Suc__le__mono) ).
thf(395,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)],[41]) ).
thf(92,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less_eq @ nat @ ( suc @ A ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_Suc__leI) ).
thf(575,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)],[92]) ).
thf(51,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ C )
=> ( ( minus_minus @ ( set @ TA ) @ B @ ( minus_minus @ ( set @ TA ) @ C @ A ) )
= A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_251_double__diff) ).
thf(427,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ C )
=> ( ( minus_minus @ ( set @ TA ) @ B @ ( minus_minus @ ( set @ TA ) @ C @ A ) )
= A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(64,axiom,
! [A: nat > nat > $o,B: nat,C: nat] :
( ! [D: nat] : ( A @ D @ ( zero_zero @ nat ) )
=> ( ! [D: nat] : ( A @ ( zero_zero @ nat ) @ ( suc @ D ) )
=> ( ! [D: nat,E: nat] :
( ( A @ D @ E )
=> ( A @ ( suc @ D ) @ ( suc @ E ) ) )
=> ( A @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_106_diff__induct) ).
thf(459,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)],[64]) ).
thf(205,axiom,
! [A: set @ b,B: set @ a] :
( ( ideal @ b @ d @ r @ A )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_242_linear__span__inc__0) ).
thf(992,plain,
! [A: set @ b,B: set @ a] :
( ( ideal @ b @ d @ r @ A )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(135,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( A != B )
=> ( ord_less @ ( set @ TA ) @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_psubsetI) ).
thf(723,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( A != B )
=> ( ord_less @ ( set @ TA ) @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(203,axiom,
! [A: set @ b,B: a] :
( ( ideal @ b @ d @ r @ A )
=> ( ( member @ a @ B @ ( smodule_ideal_coeff @ b @ 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] : ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) )
& ( B
= ( l_comb @ b @ d @ a @ c @ r @ m @ C @ D @ E ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_mem__smodule__ideal__coeff) ).
thf(982,plain,
! [A: set @ b,B: a] :
( ( ideal @ b @ d @ r @ A )
=> ( ( member @ a @ B @ ( smodule_ideal_coeff @ b @ 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] : ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) )
& ( B
= ( l_comb @ b @ d @ a @ c @ r @ m @ C @ D @ E ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(100,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ A @ ( suc @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_less__SucI) ).
thf(607,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ A @ ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(72,axiom,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_le0) ).
thf(486,plain,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(164,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ ( suc @ A ) @ B )
=> ( ord_less @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_165_Suc__le__lessD) ).
thf(803,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)],[164]) ).
thf(67,axiom,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_111_nat_Odistinct_I1_J) ).
thf(469,plain,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(211,axiom,
! [A: set @ a] :
( ( generator @ b @ d @ a @ c @ r @ m @ A )
=> ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_generator__sub__carrier) ).
thf(1009,plain,
! [A: set @ a] :
( ( generator @ b @ d @ a @ c @ r @ m @ A )
=> ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(221,axiom,
( member @ ( nat > b ) @ t
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [A: nat] : ( ord_less_eq @ nat @ A @ m2 ) )
@ ^ [A: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_2) ).
thf(1037,plain,
( member @ ( nat > b ) @ t
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [A: nat] : ( ord_less_eq @ nat @ A @ m2 ) )
@ ^ [A: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).
thf(147,axiom,
! [TA: $tType,A: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ A @ A )
= ( bot_bot @ ( set @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_Diff__cancel) ).
thf(755,plain,
! [TA: $tType,A: set @ TA] :
( ( minus_minus @ ( set @ TA ) @ A @ A )
= ( bot_bot @ ( set @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(74,axiom,
! [TA: $tType,A: TA > $o] :
( ( ( collect @ TA @ A )
= ( bot_bot @ ( set @ TA ) ) )
= ( ! [B: TA] :
~ ( A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_Collect__empty__eq) ).
thf(491,plain,
! [TA: $tType,A: TA > $o] :
( ( ( collect @ TA @ A )
= ( bot_bot @ ( set @ TA ) ) )
= ( ! [B: TA] :
~ ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(285,axiom,
member @ a @ h2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_17) ).
thf(1204,plain,
member @ a @ h2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).
thf(199,axiom,
( ( ord_less @ nat )
= ( ^ [A: nat] : ( ord_less_eq @ nat @ ( suc @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_162_less__eq__Suc__le) ).
thf(972,plain,
( ( ord_less @ nat )
= ( ^ [A: nat] : ( ord_less_eq @ nat @ ( suc @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(61,axiom,
( ( ord_less @ nat )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
& ( A != B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_nat__less__le) ).
thf(454,plain,
( ( ord_less @ nat )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
& ( A != B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(234,axiom,
! [A: set @ b,B: set @ a,C: nat] :
( ( ideal @ b @ d @ r @ A )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_liear__comb__memTr) ).
thf(1069,plain,
! [A: set @ b,B: set @ a,C: nat] :
( ( ideal @ b @ d @ r @ A )
=> ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ 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 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).
thf(1205,plain,
$false,
inference(e,[status(thm)],[645,1024,333,1179,777,1110,1005,683,677,597,861,1173,1142,379,1078,385,500,715,814,747,538,1164,797,709,301,436,651,357,1059,542,619,838,817,969,1013,1115,1109,376,623,1136,1151,353,1183,1051,480,766,734,293,866,1072,774,660,998,988,1201,1104,877,692,380,546,1199,1067,1057,334,910,1091,778,1042,392,788,729,329,830,1159,1195,1128,456,974,742,1121,1064,1006,388,869,697,1089,1028,488,339,637,1016,686,1075,1114,1099,701,515,654,1192,343,805,398,674,794,718,298,425,745,689,444,601,965,750,513,430,1154,904,732,1125,772,286,958,1124,413,875,577,1139,382,572,313,611,1107,466,704,947,350,540,672,445,936,979,784,362,318,649,1011,419,1048,736,616,303,477,494,758,609,323,594,726,483,915,769,1022,1177,1112,367,552,848,520,912,1198,827,695,1066,1015,1176,306,1113,1102,438,473,1034,680,1134,355,1145,1018,968,458,580,640,872,707,954,739,310,712,1003,1053,791,441,463,1085,951,643,1038,370,544,907,336,1141,613,600,864,1094,497,627,450,1157,976,759,891,1189,368,363,556,961,517,401,418,781,995,290,667,763,832,346,549,1045,433,510,1161,800,287,1116,1133,752,447,315,410,405,1148,422,950,1097,835,373,603,326,918,341,807,1172,1056,1197,720,395,575,427,459,992,723,982,607,486,803,469,1009,1037,755,491,1204,972,454,1069]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : ITP026^2 : TPTP v8.2.0. Released v7.5.0.
% 0.14/0.15 % Command : run_Leo-III %s %d
% 0.16/0.36 % Computer : n002.cluster.edu
% 0.16/0.36 % Model : x86_64 x86_64
% 0.16/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.36 % Memory : 8042.1875MB
% 0.16/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.36 % CPULimit : 300
% 0.16/0.36 % WCLimit : 300
% 0.16/0.36 % DateTime : Sat May 18 18:34:54 EDT 2024
% 0.16/0.36 % CPUTime :
% 0.86/0.83 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.85/1.10 % [INFO] Parsing done (265ms).
% 1.85/1.11 % [INFO] Running in sequential loop mode.
% 2.72/1.32 % [INFO] eprover registered as external prover.
% 2.72/1.32 % [INFO] cvc4 registered as external prover.
% 2.72/1.33 % [INFO] Scanning for conjecture ...
% 2.87/1.43 % [INFO] Found a conjecture (or negated_conjecture) and 285 axioms. Running axiom selection ...
% 3.45/1.53 % [INFO] Axiom selection finished. Selected 283 axioms (removed 2 axioms).
% 4.10/1.76 % [INFO] Problem is higher-order (TPTP THF).
% 4.48/1.78 % [INFO] Type checking passed.
% 4.48/1.79 % [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 ...
% 62.60/45.48 % External prover 'e' found a proof!
% 62.60/45.48 % [INFO] Killing All external provers ...
% 62.60/45.48 % Time passed: 45026ms (effective reasoning time: 44370ms)
% 62.60/45.48 % 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)>
% 62.60/45.49 % Axioms used in derivation (283): fact_159_lift__Suc__mono__less__iff, fact_51_Suc__diff__diff, fact_103_Zero__neq__Suc, fact_129_less__SucI, fact_99_not0__implies__Suc, fact_4_free__generator__sub, fact_37_nat_Oinject, fact_249_l__span__closed, fact_111_nat_Odistinct_I1_J, fact_67_Nat_Oex__has__greatest__nat, fact_204_empty__iff, fact_241_lin__span__coeff__mono, fact_9_linear__comb__eq, fact_41_le0, fact_144_le__diff__iff_H, fact_168_Suc__le__eq, fact_165_Suc__le__lessD, fact_57_less__Suc0, fact_238_singletonI, fact_185_diff__Suc__less, fact_22_Module_Omem__fgs__l__comb, fact_85_transitive__stepwise__le, fact_25_HOM__is__module, fact_187_Module_Ofree__generator__sub, tcon_Nat_Onat___Groups_Ocomm__monoid__diff, fact_11_l__comb__mem, conj_9, conj_11, fact_245_l__span__closed3, fact_147_Nat_Odiff__diff__eq, fact_184_ex__least__nat__less, fact_69_le__antisym, fact_142_gr0I, fact_218_l__span__cont__H, conj_3, tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, fact_220_l__span__gen__mono, fact_60_Module_Omisom__self, fact_149_eq__diff__iff, fact_93_le__SucE, conj_14, fact_29_Module_Ol__comb__mem, fact_171_less__Suc__eq__0__disj, fact_216_submodule__of__mdl, fact_23_Module_Ofgs__sub__carrier, fact_235_empty__subsetI, fact_2_submodule__whole, conj_10, fact_181_diff__less, fact_12_liear__comb__memTr, fact_7_fgs__mono, fact_152_diffs0__imp__equal, fact_160_lift__Suc__mono__less, fact_101_old_Onat_Oexhaust, fact_153_less__imp__diff__less, fact_72_Suc__inject, fact_182_Module_Osubmodule__subset1, fact_169_Suc__leI, fact_163_less__Suc__eq__le, fact_200_subset__antisym, fact_202_empty__Diff, tcon_Product__Type_Ounit___Orderings_Owellorder_4, fact_180_Suc__diff__Suc, fact_91_Suc__le__D, fact_92_le__SucI, fact_94_Suc__leD, fact_248_l__comb__mem__linear__span, fact_18_mem__smodule__ideal__coeff, fact_199_subsetI, fact_164_le__less__Suc__eq, fact_161_le__imp__less__Suc, fact_219_l__span__l__span, fact_122_Suc__less__SucD, fact_224_submodule__0, fact_114_less__or__eq__imp__le, fact_98_less__eq__nat_Osimps_I1_J, fact_190_singleton__insert__inj__eq_H, fact_188_eSum__in__SubmoduleTr, fact_112_less__mono__imp__le__mono, fact_73_infinite__descent__measure, fact_186_Module_Osubmodule__subset, fact_206_Collect__empty__eq, fact_77_less__irrefl__nat, fact_16_quotient__of__submodules__is__ideal, fact_80_less__not__refl, tcon_Product__Type_Ounit___Orderings_Oorder_5, fact_26_Suc__pred, fact_14_mem__fgs__l__comb, conj_18, fact_232_cancel__comm__monoid__add__class_Odiff__cancel, fact_88_not__less__eq__eq, fact_3_submodule__subset, fact_113_le__neq__implies__less, fact_139_less__zeroE, fact_177_diff__less__mono, fact_31_zero__less__diff, conj_2, fact_36_old_Onat_Oinject, fact_28_free__generator__generator, fact_138_gr__implies__not0, fact_21_Module_Ofgs__submodule, conj_15, fact_146_diff__le__mono, fact_134_Nat_OlessE, fact_240_l__spanA__l__span, fact_84_Module_Oquotient__of__submodules__is__ideal, fact_203_Diff__cancel, fact_140_not__less0, fact_189_eSum__in__Submodule, fact_68_nat__le__linear, fact_10_linear__comb__eqTr, fact_59_Module_Omisom__trans, fact_194_l__span__closed2Tr, fact_8_fgs__submodule, fact_137_infinite__descent0, fact_81_nat__neq__iff, tcon_Nat_Onat___Groups_Ozero, fact_52_diff__Suc__Suc, fact_131_Suc__lessI, fact_74_linorder__neqE__nat, fact_205_all__not__in__conv, tcon_Nat_Onat___Orderings_Owellorder, fact_24_misom__self, fact_225_empty__fgs, fact_53_diff__self__eq__0, fact_243_generator__generator, fact_167_dec__induct, fact_237_insert__Diff__single, fact_65_measure__induct__rule, fact_106_diff__induct, fact_35_diff__is__0__eq_H, fact_229_diff__0__right, fact_208_Diff__insert0, fact_110_old_Onat_Odistinct_I2_J, fact_47_bot__nat__0_Onot__eq__extremum, fact_252_Diff__subset, fact_135_infinite__descent0__measure, fact_148_le__diff__iff, fact_63_Module_Ogenerator__sub__carrier, fact_192_diff__gt__0__iff__gt, fact_230_zero__diff, fact_87_full__nat__induct, fact_158_lift__Suc__mono__le, fact_1_submodule__subset1, fact_150_zero__induct__lemma, fact_251_double__diff, fact_166_inc__induct, fact_78_less__not__refl3, fact_127_less__Suc__eq, fact_62_Module_Ofree__generator__generator, fact_212_insert__absorb2, fact_217_h__in__linear__span, conj_16, fact_54_diff__0__eq__0, fact_75_infinite__descent, fact_236_insert__subset, fact_191_singleton__insert__inj__eq, fact_64_Module_OAnn__is__ideal, fact_71_n__not__Suc__n, fact_132_Suc__lessE, fact_197_module__inc__zero, fact_125_All__less__Suc, fact_231_diff__zero, fact_221_linear__span__sub, fact_228_diff__self, fact_100_old_Onat_Oinducts, fact_82_diff__commute, fact_210_insertCI, fact_242_linear__span__inc__0, fact_30_Module_Oliear__comb__memTr, fact_117_nat__less__le, fact_176_Suc__diff__le, fact_130_less__SucE, fact_234_subset__empty, conj_6, fact_214_mdl__carrier, fact_179_diff__less__Suc, fact_124_Suc__less__eq2, fact_223_fgs__generator, fact_120_less__Suc__induct, fact_173_All__less__Suc2, conj_20, fact_157_lift__Suc__antimono__le, conj_1, fact_34_diff__is__0__eq, fact_95_bot__nat__0_Oextremum__uniqueI, fact_193_diff__ge__0__iff__ge, fact_155_Module_Osc__Ring, tcon_Set_Oset___Orderings_Oorder_2, fact_247_singleton__conv2, fact_97_le__0__eq, fact_45_Collect__mem__eq, conj_19, fact_49_neq0__conv, fact_70_le__trans, fact_40_bot__nat__0_Oextremum, fact_213_submodule__inc__0, fact_76_nat__less__induct, fact_56_zero__less__Suc, fact_61_Module_Omisom__sym, fact_244_l__span__zero, fact_201_Diff__empty, fact_108_nat_OdiscI, fact_162_less__eq__Suc__le, fact_105_zero__induct, fact_17_smodule__ideal__coeff__is__Submodule, fact_50_diff__diff__cancel, fact_13_Ann__is__ideal, fact_15_Module__axioms, fact_33_Module_Olinear__comb__eqTr, fact_96_bot__nat__0_Oextremum__unique, fact_6_fgs__sub__carrier, fact_246_singleton__conv, fact_115_le__eq__less__or__eq, fact_83_Module_Osmodule__ideal__coeff__is__Submodule, conj_8, fact_90_le__Suc__eq, fact_172_gr0__implies__Suc, fact_175_Ex__less__Suc2, fact_239_linear__span__subModule, conj_17, fact_107_nat__induct, fact_178_less__diff__iff, fact_143_diff__le__mono2, fact_170_ex__least__nat__le, fact_174_gr0__conv__Suc, fact_102_Zero__not__Suc, fact_116_less__imp__le__nat, fact_133_Suc__lessD, conj_12, fact_145_diff__le__self, tcon_fun___Orderings_Oorder, fact_38_misom__sym, tcon_Nat_Onat___Orderings_Oorder_1, fact_104_Suc__neq__Zero, fact_79_less__not__refl2, fact_250_linear__comb0__2, fact_156_Module_Omem__smodule__ideal__coeff, fact_233_Diff__eq__empty__iff, fact_55_misom__trans, conj_4, fact_128_Ex__less__Suc, fact_226_le__zero__eq, fact_109_old_Onat_Odistinct_I1_J, fact_196_linear__comb0__2Tr, fact_126_not__less__eq, fact_151_minus__nat_Odiff__0, fact_5_elem__fgs, conj_13, fact_207_empty__Collect__eq, fact_42_Suc__less__eq, tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, fact_209_insert__Diff1, conj_5, fact_195_l__span__closed2, fact_227_not__gr__zero, fact_19_Module_Oelem__fgs, conj_0, fact_183_Module_Osubmodule__whole, fact_119_strict__inc__induct, fact_141_not__gr0, fact_32_Module_Olinear__comb__eq, fact_43_Suc__mono, fact_89_Suc__n__not__le__n, fact_20_Module_Ofgs__mono, fact_0_sc__Ring, fact_211_insert__iff, fact_123_less__antisym, fact_86_nat__induct__at__least, fact_39_Suc__le__mono, fact_27_generator__sub__carrier, fact_121_less__trans__Suc, fact_222_l__span__sub__submodule, fact_118_not__less__less__Suc__eq, fact_48_less__nat__zero__code, fact_215_mdl__is__module, fact_66_measure__induct, fact_198_psubsetI, fact_154_diff__less__mono2, fact_46_Collect__cong, fact_136_bot__nat__0_Oextremum__strict, fact_58_Module_OHOM__is__module, fact_44_mem__Collect__eq
% 62.60/45.49 % No. of inferences in proof: 570
% 62.60/45.49 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 45026 ms resp. 44370 ms w/o parsing
% 63.66/45.76 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 63.66/45.76 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------