%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : ITP131^1 : TPTP v9.3.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% Computer : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Sep 30 08:06:53 AM UTC 2026
% Result : Theorem 155.12s 70.47s
% Output : Refutation 156.17s
% Verified :
% SZS Type : Refutation
% Derivation depth : 35
% Number of leaves : 218
% Syntax : Number of formulae : 1408 ( 617 unt; 0 typ; 5 def)
% Number of atoms : 3347 (2154 equ; 82 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 10843 (1199 ~; 759 |; 66 &;8502 @)
% ( 0 <=>; 317 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 53 ( 53 >; 0 *; 0 +; 0 <<)
% Number of symbols : 22 ( 18 usr; 10 con; 0-2 aty)
% Number of variables : 2656 ( 93 ^;2530 !; 28 ?;2656 :)
% ( 0 !>; 0 ?*; 0 @-; 5 @+)
% Comments :
%------------------------------------------------------------------------------
thf(nat_type,type,
nat: $tType ).
thf(minus_minus_nat_decl,type,
minus_minus_nat: nat > nat > nat ).
thf(plus_plus_nat_decl,type,
plus_plus_nat: nat > nat > nat ).
thf(times_times_nat_decl,type,
times_times_nat: nat > nat > nat ).
thf(zero_zero_nat_decl,type,
zero_zero_nat: nat ).
thf(number1551313001itions_decl,type,
number1551313001itions: ( nat > nat ) > nat > $o ).
thf(ord_less_nat_decl,type,
ord_less_nat: nat > nat > $o ).
thf(ord_less_eq_nat_decl,type,
ord_less_eq_nat: nat > nat > $o ).
thf(i_decl,type,
i: nat ).
thf(k_decl,type,
k: nat ).
thf(n_decl,type,
n: nat ).
thf(p_decl,type,
p: nat > nat ).
thf(sk1_decl,type,
sk1: nat > nat > nat ).
thf(sk2_decl,type,
sk2: nat > nat > nat ).
thf(sk3_decl,type,
sk3: ( nat > $o ) > nat ).
thf(sk8_decl,type,
sk8: ( nat > $o ) > nat ).
thf(sk11_decl,type,
sk11: nat ).
thf(sk12_decl,type,
sk12: nat ).
thf(sk17_decl,type,
sk17: nat ).
thf(sk1_def,definition,
( sk1
= ( @+[A: nat > nat > nat] :
! [B: nat,C: nat] :
( ( A @ C @ ( plus_plus_nat @ B @ C ) )
= B ) ) ) ).
thf(sk2_def,definition,
( sk2
= ( @+[A: nat > nat > nat] :
! [B: nat,C: nat] :
( ( A @ B @ ( plus_plus_nat @ B @ C ) )
= C ) ) ) ).
thf(sk3_def,definition,
( sk3
= ( ^ [A: nat > $o] :
@+[B: nat] :
~ ( ~ ( A @ B )
=> ? [C: nat] :
( ( ord_less_nat @ C @ B )
& ~ ( A @ C ) ) ) ) ) ).
thf(sk8_def,definition,
( sk8
= ( ^ [A: nat > $o] :
@+[B: nat] :
~ ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ~ ( A @ B )
=> ? [C: nat] :
( ( ord_less_nat @ C @ B )
& ~ ( A @ C ) ) ) ) ) ) ).
thf(sk17_def,definition,
( sk17
= ( @+[A: nat] :
~ ( ( n
= ( plus_plus_nat @ k @ A ) )
=> ( zero_zero_nat = A ) ) ) ) ).
thf(139,axiom,
( times_times_nat
= ( ^ [A: nat,B: nat] : ( times_times_nat @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_mult_Ocommute) ).
thf(645,plain,
( times_times_nat
= ( ^ [A: nat,B: nat] : ( times_times_nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(134,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( A
= ( plus_plus_nat @ B @ C ) )
=> ( ( plus_plus_nat @ D @ A )
= ( plus_plus_nat @ B @ ( plus_plus_nat @ D @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_group__cancel_Oadd2) ).
thf(625,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( A
= ( plus_plus_nat @ B @ C ) )
=> ( ( plus_plus_nat @ D @ A )
= ( plus_plus_nat @ B @ ( plus_plus_nat @ D @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(626,plain,
! [A: nat,B: nat,C: nat] :
( ( A
= ( plus_plus_nat @ B @ C ) )
=> ! [D: nat] :
( ( plus_plus_nat @ D @ A )
= ( plus_plus_nat @ B @ ( plus_plus_nat @ D @ C ) ) ) ),
inference(miniscope,[status(thm)],[625]) ).
thf(627,plain,
! [D: nat,C: nat,B: nat,A: nat] :
( ( A
!= ( plus_plus_nat @ B @ C ) )
| ( ( plus_plus_nat @ D @ A )
= ( plus_plus_nat @ B @ ( plus_plus_nat @ D @ C ) ) ) ),
inference(cnf,[status(esa)],[626]) ).
thf(628,plain,
! [D: nat,C: nat,B: nat,A: nat] :
( ( A
!= ( plus_plus_nat @ B @ C ) )
| ( ( plus_plus_nat @ D @ A )
= ( plus_plus_nat @ B @ ( plus_plus_nat @ D @ C ) ) ) ),
inference(lifteq,[status(thm)],[627]) ).
thf(629,plain,
! [C: nat,B: nat,A: nat] :
( ( plus_plus_nat @ C @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ C @ B ) ) ),
inference(simp,[status(thm)],[628]) ).
thf(7,axiom,
ord_less_eq_nat @ k @ n,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4__092_060open_062k_A_092_060le_062_An_092_060close_062) ).
thf(220,plain,
ord_less_eq_nat @ k @ n,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(70,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_67_diff__is__0__eq_H) ).
thf(401,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)],[70]) ).
thf(402,plain,
! [B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ( minus_minus_nat @ A @ B )
= zero_zero_nat ) ),
inference(cnf,[status(esa)],[401]) ).
thf(403,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
= zero_zero_nat )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[402]) ).
thf(9086,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
= zero_zero_nat )
| ~ $true
| ( ( ord_less_eq_nat @ k @ n )
!= ( ord_less_eq_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[220,403]) ).
thf(9087,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
= zero_zero_nat )
| ( ( ord_less_eq_nat @ k @ n )
!= ( ord_less_eq_nat @ A @ B ) ) ),
inference(simp,[status(thm)],[9086]) ).
thf(9088,plain,
( ( minus_minus_nat @ k @ n )
= zero_zero_nat ),
inference(pattern_uni,[status(thm)],[9087:[bind(A,$thf( k )),bind(B,$thf( n ))]]) ).
thf(97,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse) ).
thf(491,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(492,plain,
! [B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ),
inference(cnf,[status(esa)],[491]) ).
thf(493,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[492]) ).
thf(71,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_68_diff__is__0__eq) ).
thf(404,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)],[71]) ).
thf(405,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
= zero_zero_nat )
= ( ord_less_eq_nat @ A @ B ) ),
inference(cnf,[status(esa)],[404]) ).
thf(406,plain,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ A @ B )
= ( ( minus_minus_nat @ A @ B )
= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[405]) ).
thf(21135,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[493,406]) ).
thf(21136,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ),
inference(lifteq,[status(thm)],[21135]) ).
thf(21190,plain,
! [B: nat,A: nat] :
( ( zero_zero_nat != zero_zero_nat )
| ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B )
| ( ( minus_minus_nat @ k @ n )
!= ( minus_minus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[9088,21136]) ).
thf(21191,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B )
| ( ( minus_minus_nat @ k @ n )
!= ( minus_minus_nat @ A @ B ) ) ),
inference(simp,[status(thm)],[21190]) ).
thf(21192,plain,
( ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) )
= n ),
inference(pattern_uni,[status(thm)],[21191:[bind(A,$thf( k )),bind(B,$thf( n ))]]) ).
thf(34,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_add__cancel__left__right) ).
thf(295,plain,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(296,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ),
inference(cnf,[status(esa)],[295]) ).
thf(297,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ),
inference(lifteq,[status(thm)],[296]) ).
thf(21638,plain,
! [B: nat,A: nat] :
( ( ( n = A )
= ( B = zero_zero_nat ) )
| ( ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[21192,297]) ).
thf(21639,plain,
( ( ( minus_minus_nat @ n @ k )
= zero_zero_nat )
= ( n = k ) ),
inference(pattern_uni,[status(thm)],[21638:[bind(A,$thf( k )),bind(B,$thf( minus_minus_nat @ n @ k ))]]) ).
thf(128,axiom,
! [A: nat] :
( ( zero_zero_nat = A )
= ( A = zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_zero__reorient) ).
thf(604,plain,
! [A: nat] :
( ( zero_zero_nat = A )
= ( A = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(605,plain,
! [A: nat] :
( ( zero_zero_nat = A )
= ( A = zero_zero_nat ) ),
inference(cnf,[status(esa)],[604]) ).
thf(606,plain,
! [A: nat] :
( ( zero_zero_nat = A )
= ( A = zero_zero_nat ) ),
inference(lifteq,[status(thm)],[605]) ).
thf(22833,plain,
! [A: nat] :
( ( ( n = k )
= ( zero_zero_nat = A ) )
| ( ( ( minus_minus_nat @ n @ k )
= zero_zero_nat )
!= ( A = zero_zero_nat ) ) ),
inference(paramod_ordered,[status(thm)],[21639,606]) ).
thf(22834,plain,
( ( zero_zero_nat
= ( minus_minus_nat @ n @ k ) )
= ( n = k ) ),
inference(pattern_uni,[status(thm)],[22833:[bind(A,$thf( minus_minus_nat @ n @ k ))]]) ).
thf(118,axiom,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ ( minus_minus_nat @ B @ C ) )
= ( ( ( ord_less_nat @ B @ C )
=> ( A @ zero_zero_nat ) )
& ! [D: nat] :
( ( B
= ( plus_plus_nat @ C @ D ) )
=> ( A @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_nat__diff__split) ).
thf(565,plain,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ ( minus_minus_nat @ B @ C ) )
= ( ( ( ord_less_nat @ B @ C )
=> ( A @ zero_zero_nat ) )
& ! [D: nat] :
( ( B
= ( plus_plus_nat @ C @ D ) )
=> ( A @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(566,plain,
! [C: nat,B: nat,A: nat > $o] :
( ( A @ ( minus_minus_nat @ B @ C ) )
= ( ( ( ord_less_nat @ B @ C )
=> ( A @ zero_zero_nat ) )
& ! [D: nat] :
( ( B
= ( plus_plus_nat @ C @ D ) )
=> ( A @ D ) ) ) ),
inference(cnf,[status(esa)],[565]) ).
thf(567,plain,
! [C: nat,B: nat,A: nat > $o] :
( ( A @ ( minus_minus_nat @ B @ C ) )
= ( ( ( ord_less_nat @ B @ C )
=> ( A @ zero_zero_nat ) )
& ! [D: nat] :
( ( B
= ( plus_plus_nat @ C @ D ) )
=> ( A @ D ) ) ) ),
inference(lifteq,[status(thm)],[566]) ).
thf(69,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_66_zero__less__diff) ).
thf(398,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)],[69]) ).
thf(399,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ A @ B ) )
= ( ord_less_nat @ B @ A ) ),
inference(cnf,[status(esa)],[398]) ).
thf(400,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ A @ B ) )
= ( ord_less_nat @ B @ A ) ),
inference(lifteq,[status(thm)],[399]) ).
thf(47,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_bot__nat__0_Onot__eq__extremum) ).
thf(334,plain,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(335,plain,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(cnf,[status(esa)],[334]) ).
thf(336,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
= ( A != zero_zero_nat ) ),
inference(lifteq,[status(thm)],[335]) ).
thf(8881,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
= ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[400,336]) ).
thf(34567,plain,
! [C: nat,B: nat,A: nat > $o] :
( ( A @ ( minus_minus_nat @ B @ C ) )
= ( ( ( ( minus_minus_nat @ C @ B )
!= zero_zero_nat )
=> ( A @ zero_zero_nat ) )
& ! [D: nat] :
( ( B
= ( plus_plus_nat @ C @ D ) )
=> ( A @ D ) ) ) ),
inference(rewrite,[status(thm)],[567,8881]) ).
thf(34630,plain,
! [C: nat,B: nat,A: nat > $o] :
( ( ( ( ( ( minus_minus_nat @ C @ B )
!= zero_zero_nat )
=> ( A @ zero_zero_nat ) )
& ! [D: nat] :
( ( B
= ( plus_plus_nat @ C @ D ) )
=> ( A @ D ) ) )
= ( n = k ) )
| ( ( zero_zero_nat
= ( minus_minus_nat @ n @ k ) )
!= ( A @ ( minus_minus_nat @ B @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[22834,34567]) ).
thf(34929,plain,
( ( ( ( ( minus_minus_nat @ k @ n )
!= zero_zero_nat )
=> ( zero_zero_nat = zero_zero_nat ) )
& ! [A: nat] :
( ( n
= ( plus_plus_nat @ k @ A ) )
=> ( zero_zero_nat = A ) ) )
= ( n = k ) ),
inference(pre_uni,[status(thm)],[34630:[bind(A,$thf( (=) @ nat @ zero_zero_nat )),bind(B,$thf( n )),bind(C,$thf( k ))]]) ).
thf(35280,plain,
( ( ! [A: nat] :
( ( n
= ( plus_plus_nat @ k @ A ) )
=> ( zero_zero_nat = A ) ) )
= ( n = k ) ),
inference(simp,[status(thm)],[34929]) ).
thf(48276,plain,
( ~ ! [A: nat] :
( ( n
= ( plus_plus_nat @ k @ A ) )
=> ( zero_zero_nat = A ) )
| ( n = k ) ),
inference(bool_ext,[status(thm)],[35280]) ).
thf(48286,plain,
( ( n = k )
| ~ ! [A: nat] :
( ( n
= ( plus_plus_nat @ k @ A ) )
=> ( zero_zero_nat = A ) ) ),
inference(lifteq,[status(thm)],[48276]) ).
thf(48289,plain,
( ( ( n = k )
| ( n
= ( plus_plus_nat @ k @ sk17 ) ) )
& ( ( n = k )
| ( zero_zero_nat != sk17 ) ) ),
inference(cnf,[status(esa)],[48286]) ).
thf(48290,plain,
( ( n = k )
| ( n
= ( plus_plus_nat @ k @ sk17 ) ) ),
inference(cnfConj,[status(thm)],[48289]) ).
thf(48292,plain,
( ( ( plus_plus_nat @ k @ sk17 )
= n )
| ( n = k ) ),
inference(lifteq,[status(thm)],[48290]) ).
thf(52,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_Nat_Oadd__0__right) ).
thf(347,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(104,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_101_le__add__diff) ).
thf(518,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(90,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_87_diff__add__inverse2) ).
thf(468,plain,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(182,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_179_infinite__descent0) ).
thf(797,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)],[182]) ).
thf(798,plain,
! [A: nat > $o] :
( ( A @ zero_zero_nat )
=> ( ! [B: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ~ ( A @ B )
=> ? [C: nat] :
( ( ord_less_nat @ C @ B )
& ~ ( A @ C ) ) ) )
=> ! [B: nat] : ( A @ B ) ) ),
inference(miniscope,[status(thm)],[797]) ).
thf(799,plain,
! [C: nat,B: nat,A: nat > $o] :
( ( ~ ( A @ zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ ( sk8 @ A ) )
| ( A @ C ) )
& ( ~ ( A @ zero_zero_nat )
| ~ ( A @ ( sk8 @ A ) )
| ( A @ C ) )
& ( ~ ( A @ zero_zero_nat )
| ~ ( ord_less_nat @ B @ ( sk8 @ A ) )
| ( A @ B )
| ( A @ C ) ) ),
inference(cnf,[status(esa)],[798]) ).
thf(800,plain,
! [B: nat,A: nat > $o] :
( ~ ( A @ zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ ( sk8 @ A ) )
| ( A @ B ) ),
inference(cnfConj,[status(thm)],[799]) ).
thf(803,plain,
! [A: nat] :
( ( zero_zero_nat != zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( zero_zero_nat = A ) ),
inference(replace_leibeq,[status(thm)],[800:[bind(A,$thf( (=) @ nat @ zero_zero_nat ))]]) ).
thf(812,plain,
! [A: nat] :
( ( zero_zero_nat != zero_zero_nat )
| ( zero_zero_nat = A )
| ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) ) ),
inference(lifteq,[status(thm)],[803]) ).
thf(813,plain,
! [A: nat] :
( ( zero_zero_nat = A )
| ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) ) ),
inference(simp,[status(thm)],[812]) ).
thf(1283,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_eq_nat @ k @ zero_zero_nat )
| ( A != n ) ),
inference(paramod_ordered,[status(thm)],[813,220]) ).
thf(1284,plain,
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_eq_nat @ k @ zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[1283:[bind(A,$thf( n ))]]) ).
thf(9905,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ k @ zero_zero_nat )
= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[1284,8881,406]) ).
thf(9906,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ k @ zero_zero_nat )
= zero_zero_nat ) ),
inference(simp,[status(thm)],[9905]) ).
thf(9907,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ k @ zero_zero_nat )
= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[9906]) ).
thf(22,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_less__add__same__cancel1) ).
thf(259,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(260,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ),
inference(cnf,[status(esa)],[259]) ).
thf(261,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ),
inference(lifteq,[status(thm)],[260]) ).
thf(160,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_157_gr__zeroI) ).
thf(733,plain,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(734,plain,
! [A: nat] :
( ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(cnf,[status(esa)],[733]) ).
thf(735,plain,
! [A: nat] :
( ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(lifteq,[status(thm)],[734]) ).
thf(8,axiom,
( ( p @ i )
!= zero_zero_nat ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5__092_060open_062p_Ai_A_092_060noteq_062_A0_092_060close_062) ).
thf(221,plain,
( ( p @ i )
!= zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(222,plain,
( ( p @ i )
!= zero_zero_nat ),
inference(polarity_switch,[status(thm)],[221]) ).
thf(223,plain,
( ( p @ i )
!= zero_zero_nat ),
inference(lifteq,[status(thm)],[222]) ).
thf(2189,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( zero_zero_nat != zero_zero_nat )
| ( A
!= ( p @ i ) ) ),
inference(paramod_ordered,[status(thm)],[735,223]) ).
thf(2190,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( A
!= ( p @ i ) ) ),
inference(simp,[status(thm)],[2189]) ).
thf(2191,plain,
ord_less_nat @ zero_zero_nat @ ( p @ i ),
inference(pattern_uni,[status(thm)],[2190:[bind(A,$thf( p @ i ))]]) ).
thf(3125,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
| ( ( ord_less_nat @ zero_zero_nat @ B )
!= ( ord_less_nat @ zero_zero_nat @ ( p @ i ) ) ) ),
inference(paramod_ordered,[status(thm)],[261,2191]) ).
thf(3126,plain,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ ( p @ i ) ) ),
inference(pattern_uni,[status(thm)],[3125:[bind(A,$thf( A )),bind(B,$thf( p @ i ))]]) ).
thf(145,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_142_less__irrefl__nat) ).
thf(674,plain,
! [A: nat] :
~ ( ord_less_nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(675,plain,
~ ? [A: nat] : ( ord_less_nat @ A @ A ),
inference(miniscope,[status(thm)],[674]) ).
thf(676,plain,
! [A: nat] :
~ ( ord_less_nat @ A @ A ),
inference(cnf,[status(esa)],[675]) ).
thf(6063,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ ( p @ i ) ) )
!= ( ord_less_nat @ B @ B ) ) ),
inference(paramod_ordered,[status(thm)],[3126,676]) ).
thf(6064,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ ( p @ i ) ) )
!= ( ord_less_nat @ B @ B ) ),
inference(simp,[status(thm)],[6063]) ).
thf(6072,plain,
! [B: nat,A: nat] :
( ( A != B )
| ( ( plus_plus_nat @ A @ ( p @ i ) )
!= B ) ),
inference(simp,[status(thm)],[6064]) ).
thf(6110,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ ( p @ i ) )
!= A ),
inference(simp,[status(thm)],[6072]) ).
thf(21630,plain,
! [A: nat] :
( ( n != A )
| ( ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) )
!= ( plus_plus_nat @ A @ ( p @ i ) ) ) ),
inference(paramod_ordered,[status(thm)],[21192,6110]) ).
thf(21818,plain,
( ( plus_plus_nat @ n @ ( p @ i ) )
!= ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) ) ),
inference(simp,[status(thm)],[21630]) ).
thf(51671,plain,
( ( plus_plus_nat @ n @ ( p @ i ) )
!= n ),
inference(rewrite,[status(thm)],[21818,21192]) ).
thf(58,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C ) )
= ( ( B = C )
| ( A = zero_zero_nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_mult__cancel1) ).
thf(365,plain,
! [A: nat,B: nat,C: nat] :
( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C ) )
= ( ( B = C )
| ( A = zero_zero_nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(74,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_Nat_Oadd__diff__assoc) ).
thf(415,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(416,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat] :
( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ),
inference(miniscope,[status(thm)],[415]) ).
thf(417,plain,
! [C: nat,B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ),
inference(cnf,[status(esa)],[416]) ).
thf(418,plain,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[417]) ).
thf(10258,plain,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[418,406]) ).
thf(10259,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ),
inference(lifteq,[status(thm)],[10258]) ).
thf(46,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
= ( minus_minus_nat @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_add__diff__cancel__left) ).
thf(331,plain,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
= ( minus_minus_nat @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(332,plain,
! [C: nat,B: nat,A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
= ( minus_minus_nat @ B @ C ) ),
inference(cnf,[status(esa)],[331]) ).
thf(333,plain,
! [C: nat,B: nat,A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
= ( minus_minus_nat @ B @ C ) ),
inference(lifteq,[status(thm)],[332]) ).
thf(9,axiom,
ord_less_nat @ zero_zero_nat @ ( p @ k ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_assms_I2_J) ).
thf(224,plain,
ord_less_nat @ zero_zero_nat @ ( p @ k ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(48,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_less__nat__zero__code) ).
thf(337,plain,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(338,plain,
~ ? [A: nat] : ( ord_less_nat @ A @ zero_zero_nat ),
inference(miniscope,[status(thm)],[337]) ).
thf(339,plain,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
inference(cnf,[status(esa)],[338]) ).
thf(915,plain,
! [A: nat] :
( ~ $true
| ( ( ord_less_nat @ zero_zero_nat @ ( p @ k ) )
!= ( ord_less_nat @ A @ zero_zero_nat ) ) ),
inference(paramod_ordered,[status(thm)],[224,339]) ).
thf(916,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( p @ k ) )
!= ( ord_less_nat @ A @ zero_zero_nat ) ),
inference(simp,[status(thm)],[915]) ).
thf(917,plain,
! [A: nat] :
( ( zero_zero_nat != A )
| ( ( p @ k )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[916]) ).
thf(918,plain,
( ( p @ k )
!= zero_zero_nat ),
inference(simp,[status(thm)],[917]) ).
thf(2039,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( ( p @ zero_zero_nat )
!= zero_zero_nat )
| ( A != k ) ),
inference(paramod_ordered,[status(thm)],[735,918]) ).
thf(2040,plain,
( ( ord_less_nat @ zero_zero_nat @ k )
| ( ( p @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[2039:[bind(A,$thf( k ))]]) ).
thf(20,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
= ( ord_less_nat @ B @ zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_add__less__same__cancel1) ).
thf(253,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
= ( ord_less_nat @ B @ zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(254,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
= ( ord_less_nat @ B @ zero_zero_nat ) ),
inference(cnf,[status(esa)],[253]) ).
thf(255,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
= ( ord_less_nat @ B @ zero_zero_nat ) ),
inference(lifteq,[status(thm)],[254]) ).
thf(2618,plain,
! [B: nat,A: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A ),
inference(rewrite,[status(thm)],[255,339]) ).
thf(3932,plain,
! [B: nat,A: nat] :
( ( ( p @ zero_zero_nat )
!= zero_zero_nat )
| ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ zero_zero_nat @ k ) ) ),
inference(paramod_ordered,[status(thm)],[2040,2618]) ).
thf(3933,plain,
! [B: nat,A: nat] :
( ( ( p @ zero_zero_nat )
!= zero_zero_nat )
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ zero_zero_nat @ k ) ) ),
inference(simp,[status(thm)],[3932]) ).
thf(3972,plain,
! [B: nat,A: nat] :
( ( ( p @ zero_zero_nat )
!= zero_zero_nat )
| ( ( plus_plus_nat @ A @ B )
!= zero_zero_nat )
| ( A != k ) ),
inference(simp,[status(thm)],[3933]) ).
thf(3979,plain,
! [A: nat] :
( ( ( p @ zero_zero_nat )
!= zero_zero_nat )
| ( ( plus_plus_nat @ k @ A )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[3972]) ).
thf(43,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_add__diff__cancel__right_H) ).
thf(322,plain,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(323,plain,
! [B: nat,A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ),
inference(cnf,[status(esa)],[322]) ).
thf(324,plain,
! [B: nat,A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ),
inference(lifteq,[status(thm)],[323]) ).
thf(63,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_diff__diff__left) ).
thf(380,plain,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(381,plain,
! [C: nat,B: nat,A: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
inference(cnf,[status(esa)],[380]) ).
thf(382,plain,
! [C: nat,B: nat,A: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
inference(lifteq,[status(thm)],[381]) ).
thf(75,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_cancel__ab__semigroup__add__class_Odiff__right__commute) ).
thf(419,plain,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(420,plain,
! [C: nat,B: nat,A: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B ) ),
inference(cnf,[status(esa)],[419]) ).
thf(421,plain,
! [C: nat,B: nat,A: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B ) ),
inference(lifteq,[status(thm)],[420]) ).
thf(85,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ A @ B ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_diff__le__self) ).
thf(451,plain,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ A @ B ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(452,plain,
! [B: nat,A: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ A @ B ) @ A ),
inference(cnf,[status(esa)],[451]) ).
thf(15702,plain,
! [B: nat,A: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ A )
= zero_zero_nat ),
inference(rewrite,[status(thm)],[452,406]) ).
thf(15703,plain,
! [B: nat,A: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ A )
= zero_zero_nat ),
inference(lifteq,[status(thm)],[15702]) ).
thf(3,axiom,
ord_less_eq_nat @ i @ n,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0__092_060open_062i_A_092_060le_062_An_092_060close_062) ).
thf(216,plain,
ord_less_eq_nat @ i @ n,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(62,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_59_diff__diff__cancel) ).
thf(377,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)],[62]) ).
thf(378,plain,
! [B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ( minus_minus_nat @ B @ ( minus_minus_nat @ B @ A ) )
= A ) ),
inference(cnf,[status(esa)],[377]) ).
thf(379,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ ( minus_minus_nat @ B @ A ) )
= A )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[378]) ).
thf(7750,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ ( minus_minus_nat @ B @ A ) )
= A )
| ~ $true
| ( ( ord_less_eq_nat @ i @ n )
!= ( ord_less_eq_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[216,379]) ).
thf(7751,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ ( minus_minus_nat @ B @ A ) )
= A )
| ( ( ord_less_eq_nat @ i @ n )
!= ( ord_less_eq_nat @ A @ B ) ) ),
inference(simp,[status(thm)],[7750]) ).
thf(7752,plain,
( ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ i ) )
= i ),
inference(pattern_uni,[status(thm)],[7751:[bind(A,$thf( i )),bind(B,$thf( n ))]]) ).
thf(11,axiom,
ord_less_nat @ ( minus_minus_nat @ n @ k ) @ i,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8__092_060open_062n_A_N_Ak_A_060_Ai_092_060close_062) ).
thf(228,plain,
ord_less_nat @ ( minus_minus_nat @ n @ k ) @ i,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(8888,plain,
( ( minus_minus_nat @ i @ ( minus_minus_nat @ n @ k ) )
!= zero_zero_nat ),
inference(rewrite,[status(thm)],[228,8881]) ).
thf(8889,plain,
( ( minus_minus_nat @ i @ ( minus_minus_nat @ n @ k ) )
!= zero_zero_nat ),
inference(simp,[status(thm)],[8888]) ).
thf(8942,plain,
( ( minus_minus_nat @ i @ ( minus_minus_nat @ n @ k ) )
!= zero_zero_nat ),
inference(lifteq,[status(thm)],[8889]) ).
thf(15781,plain,
! [B: nat,A: nat] :
( ( zero_zero_nat != zero_zero_nat )
| ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ A )
!= ( minus_minus_nat @ i @ ( minus_minus_nat @ n @ k ) ) ) ),
inference(paramod_ordered,[status(thm)],[15703,8942]) ).
thf(15782,plain,
! [B: nat,A: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ A )
!= ( minus_minus_nat @ i @ ( minus_minus_nat @ n @ k ) ) ),
inference(simp,[status(thm)],[15781]) ).
thf(15971,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= i )
| ( A
!= ( minus_minus_nat @ n @ k ) ) ),
inference(simp,[status(thm)],[15782]) ).
thf(16031,plain,
! [A: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ n @ k ) @ A )
!= i ),
inference(simp,[status(thm)],[15971]) ).
thf(18820,plain,
! [D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) )
!= i )
| ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
!= ( minus_minus_nat @ ( minus_minus_nat @ n @ k ) @ D ) ) ),
inference(paramod_ordered,[status(thm)],[382,16031]) ).
thf(18821,plain,
! [A: nat] :
( ( minus_minus_nat @ n @ ( plus_plus_nat @ k @ A ) )
!= i ),
inference(pattern_uni,[status(thm)],[18820:[bind(A,$thf( n )),bind(B,$thf( k )),bind(C,$thf( C )),bind(D,$thf( C ))]]) ).
thf(18885,plain,
! [A: nat] :
( ( minus_minus_nat @ n @ ( plus_plus_nat @ k @ A ) )
!= i ),
inference(simp,[status(thm)],[18821]) ).
thf(19104,plain,
! [A: nat] :
( ( i != i )
| ( ( minus_minus_nat @ n @ ( plus_plus_nat @ k @ A ) )
!= ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ i ) ) ) ),
inference(paramod_ordered,[status(thm)],[7752,18885]) ).
thf(19105,plain,
! [A: nat] :
( ( minus_minus_nat @ n @ ( plus_plus_nat @ k @ A ) )
!= ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ i ) ) ),
inference(simp,[status(thm)],[19104]) ).
thf(19123,plain,
! [A: nat] :
( ( n != n )
| ( ( plus_plus_nat @ k @ A )
!= ( minus_minus_nat @ n @ i ) ) ),
inference(simp,[status(thm)],[19105]) ).
thf(19151,plain,
! [A: nat] :
( ( plus_plus_nat @ k @ A )
!= ( minus_minus_nat @ n @ i ) ),
inference(simp,[status(thm)],[19123]) ).
thf(21358,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( B
!= ( minus_minus_nat @ n @ i ) )
| ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
!= ( plus_plus_nat @ k @ C ) ) ),
inference(paramod_ordered,[status(thm)],[21136,19151]) ).
thf(21359,plain,
! [A: nat] :
( ( ( minus_minus_nat @ k @ A )
!= zero_zero_nat )
| ( A
!= ( minus_minus_nat @ n @ i ) ) ),
inference(pattern_uni,[status(thm)],[21358:[bind(A,$thf( k )),bind(B,$thf( D )),bind(C,$thf( minus_minus_nat @ D @ k ))]]) ).
thf(21483,plain,
( ( minus_minus_nat @ k @ ( minus_minus_nat @ n @ i ) )
!= zero_zero_nat ),
inference(simp,[status(thm)],[21359]) ).
thf(22204,plain,
! [B: nat,A: nat] :
( ( zero_zero_nat != zero_zero_nat )
| ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ A )
!= ( minus_minus_nat @ k @ ( minus_minus_nat @ n @ i ) ) ) ),
inference(paramod_ordered,[status(thm)],[15703,21483]) ).
thf(22205,plain,
! [B: nat,A: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ A )
!= ( minus_minus_nat @ k @ ( minus_minus_nat @ n @ i ) ) ),
inference(simp,[status(thm)],[22204]) ).
thf(22294,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= k )
| ( A
!= ( minus_minus_nat @ n @ i ) ) ),
inference(simp,[status(thm)],[22205]) ).
thf(22317,plain,
! [A: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ n @ i ) @ A )
!= k ),
inference(simp,[status(thm)],[22294]) ).
thf(35704,plain,
! [D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
!= k )
| ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
!= ( minus_minus_nat @ ( minus_minus_nat @ n @ i ) @ D ) ) ),
inference(paramod_ordered,[status(thm)],[421,22317]) ).
thf(35705,plain,
! [A: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ n @ A ) @ i )
!= k ),
inference(pattern_uni,[status(thm)],[35704:[bind(A,$thf( n )),bind(B,$thf( i )),bind(C,$thf( C )),bind(D,$thf( C ))]]) ).
thf(35845,plain,
! [A: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ n @ A ) @ i )
!= k ),
inference(simp,[status(thm)],[35705]) ).
thf(42664,plain,
! [D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) )
!= k )
| ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
!= ( minus_minus_nat @ ( minus_minus_nat @ n @ D ) @ i ) ) ),
inference(paramod_ordered,[status(thm)],[382,35845]) ).
thf(42665,plain,
! [A: nat] :
( ( minus_minus_nat @ n @ ( plus_plus_nat @ A @ i ) )
!= k ),
inference(pattern_uni,[status(thm)],[42664:[bind(A,$thf( n )),bind(B,$thf( B )),bind(C,$thf( i )),bind(D,$thf( B ))]]) ).
thf(42760,plain,
! [A: nat] :
( ( minus_minus_nat @ n @ ( plus_plus_nat @ A @ i ) )
!= k ),
inference(simp,[status(thm)],[42665]) ).
thf(44871,plain,
! [C: nat,B: nat,A: nat] :
( ( A != k )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( minus_minus_nat @ n @ ( plus_plus_nat @ C @ i ) ) ) ),
inference(paramod_ordered,[status(thm)],[324,42760]) ).
thf(44925,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ k @ A )
!= n )
| ( A
!= ( plus_plus_nat @ B @ i ) ) ),
inference(simp,[status(thm)],[44871]) ).
thf(44967,plain,
! [A: nat] :
( ( plus_plus_nat @ k @ ( plus_plus_nat @ A @ i ) )
!= n ),
inference(simp,[status(thm)],[44925]) ).
thf(120,axiom,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( minus_minus_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_nat__diff__add__eq2) ).
thf(572,plain,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( minus_minus_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(573,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat,D: nat,E: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( minus_minus_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) ) ),
inference(miniscope,[status(thm)],[572]) ).
thf(574,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( minus_minus_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) ) ),
inference(cnf,[status(esa)],[573]) ).
thf(575,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( minus_minus_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[574]) ).
thf(95,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_diff__mult__distrib) ).
thf(483,plain,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(484,plain,
! [C: nat,B: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ),
inference(cnf,[status(esa)],[483]) ).
thf(485,plain,
! [C: nat,B: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ),
inference(lifteq,[status(thm)],[484]) ).
thf(36140,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( minus_minus_nat @ D @ ( plus_plus_nat @ ( minus_minus_nat @ ( times_times_nat @ B @ C ) @ ( times_times_nat @ A @ C ) ) @ E ) ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[575,485,406]) ).
thf(36141,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( minus_minus_nat @ D @ ( plus_plus_nat @ ( minus_minus_nat @ ( times_times_nat @ B @ C ) @ ( times_times_nat @ A @ C ) ) @ E ) ) ) ),
inference(lifteq,[status(thm)],[36140]) ).
thf(115,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_112_Nat_Ole__diff__conv2) ).
thf(555,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(6,axiom,
ord_less_nat @ n @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_lower__bound) ).
thf(219,plain,
ord_less_nat @ n @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(933,plain,
! [A: nat] :
( ~ $true
| ( ( ord_less_nat @ n @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) )
!= ( ord_less_nat @ A @ zero_zero_nat ) ) ),
inference(paramod_ordered,[status(thm)],[219,339]) ).
thf(934,plain,
! [A: nat] :
( ( ord_less_nat @ n @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) )
!= ( ord_less_nat @ A @ zero_zero_nat ) ),
inference(simp,[status(thm)],[933]) ).
thf(935,plain,
! [A: nat] :
( ( n != A )
| ( ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[934]) ).
thf(936,plain,
( ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) )
!= zero_zero_nat ),
inference(simp,[status(thm)],[935]) ).
thf(1231,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( zero_zero_nat != zero_zero_nat )
| ( A
!= ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) ) ),
inference(paramod_ordered,[status(thm)],[813,936]) ).
thf(1232,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( A
!= ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) ) ),
inference(simp,[status(thm)],[1231]) ).
thf(1233,plain,
ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[1232:[bind(A,$thf( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ))]]) ).
thf(3042,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
| ( ( ord_less_nat @ zero_zero_nat @ B )
!= ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[261,1233]) ).
thf(3043,plain,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) ) ),
inference(pattern_uni,[status(thm)],[3042:[bind(A,$thf( A )),bind(B,$thf( sk8 @ ( (=) @ nat @ zero_zero_nat ) ))]]) ).
thf(7702,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) ) )
!= ( ord_less_nat @ B @ B ) ) ),
inference(paramod_ordered,[status(thm)],[3043,676]) ).
thf(7703,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) ) )
!= ( ord_less_nat @ B @ B ) ),
inference(simp,[status(thm)],[7702]) ).
thf(7712,plain,
! [B: nat,A: nat] :
( ( A != B )
| ( ( plus_plus_nat @ A @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
!= B ) ),
inference(simp,[status(thm)],[7703]) ).
thf(7713,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
!= A ),
inference(simp,[status(thm)],[7712]) ).
thf(21642,plain,
! [A: nat] :
( ( n != A )
| ( ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) )
!= ( plus_plus_nat @ A @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[21192,7713]) ).
thf(21791,plain,
( ( plus_plus_nat @ n @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
!= ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) ) ),
inference(simp,[status(thm)],[21642]) ).
thf(54916,plain,
( ( plus_plus_nat @ n @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
!= n ),
inference(rewrite,[status(thm)],[21791,21192]) ).
thf(23,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_less__add__same__cancel2) ).
thf(262,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(263,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ),
inference(cnf,[status(esa)],[262]) ).
thf(264,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ),
inference(lifteq,[status(thm)],[263]) ).
thf(929,plain,
! [A: nat] :
( ~ $true
| ( ( ord_less_nat @ ( minus_minus_nat @ n @ k ) @ i )
!= ( ord_less_nat @ A @ zero_zero_nat ) ) ),
inference(paramod_ordered,[status(thm)],[228,339]) ).
thf(930,plain,
! [A: nat] :
( ( ord_less_nat @ ( minus_minus_nat @ n @ k ) @ i )
!= ( ord_less_nat @ A @ zero_zero_nat ) ),
inference(simp,[status(thm)],[929]) ).
thf(931,plain,
! [A: nat] :
( ( ( minus_minus_nat @ n @ k )
!= A )
| ( i != zero_zero_nat ) ),
inference(simp,[status(thm)],[930]) ).
thf(932,plain,
i != zero_zero_nat,
inference(simp,[status(thm)],[931]) ).
thf(2145,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( zero_zero_nat != zero_zero_nat )
| ( A != i ) ),
inference(paramod_ordered,[status(thm)],[735,932]) ).
thf(2146,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( A != i ) ),
inference(simp,[status(thm)],[2145]) ).
thf(2147,plain,
ord_less_nat @ zero_zero_nat @ i,
inference(pattern_uni,[status(thm)],[2146:[bind(A,$thf( i ))]]) ).
thf(3525,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
| ( ( ord_less_nat @ zero_zero_nat @ B )
!= ( ord_less_nat @ zero_zero_nat @ i ) ) ),
inference(paramod_ordered,[status(thm)],[264,2147]) ).
thf(3526,plain,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ i @ A ) ),
inference(pattern_uni,[status(thm)],[3525:[bind(A,$thf( A )),bind(B,$thf( i ))]]) ).
thf(5055,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ A @ ( plus_plus_nat @ i @ A ) )
!= ( ord_less_nat @ B @ B ) ) ),
inference(paramod_ordered,[status(thm)],[3526,676]) ).
thf(5056,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ i @ A ) )
!= ( ord_less_nat @ B @ B ) ),
inference(simp,[status(thm)],[5055]) ).
thf(5058,plain,
! [B: nat,A: nat] :
( ( A != B )
| ( ( plus_plus_nat @ i @ A )
!= B ) ),
inference(simp,[status(thm)],[5056]) ).
thf(5069,plain,
! [A: nat] :
( ( plus_plus_nat @ i @ A )
!= A ),
inference(simp,[status(thm)],[5058]) ).
thf(39,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_diff__zero) ).
thf(310,plain,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(311,plain,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ),
inference(cnf,[status(esa)],[310]) ).
thf(312,plain,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ),
inference(lifteq,[status(thm)],[311]) ).
thf(9071,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
= zero_zero_nat )
| ~ $true
| ( ( ord_less_eq_nat @ i @ n )
!= ( ord_less_eq_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[216,403]) ).
thf(9072,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
= zero_zero_nat )
| ( ( ord_less_eq_nat @ i @ n )
!= ( ord_less_eq_nat @ A @ B ) ) ),
inference(simp,[status(thm)],[9071]) ).
thf(9073,plain,
( ( minus_minus_nat @ i @ n )
= zero_zero_nat ),
inference(pattern_uni,[status(thm)],[9072:[bind(A,$thf( i )),bind(B,$thf( n ))]]) ).
thf(9212,plain,
( ( zero_zero_nat != zero_zero_nat )
| ( ( minus_minus_nat @ i @ ( minus_minus_nat @ n @ k ) )
!= ( minus_minus_nat @ i @ n ) ) ),
inference(paramod_ordered,[status(thm)],[9073,8942]) ).
thf(9213,plain,
( ( minus_minus_nat @ i @ ( minus_minus_nat @ n @ k ) )
!= ( minus_minus_nat @ i @ n ) ),
inference(simp,[status(thm)],[9212]) ).
thf(9272,plain,
( ( i != i )
| ( ( minus_minus_nat @ n @ k )
!= n ) ),
inference(simp,[status(thm)],[9213]) ).
thf(9298,plain,
( ( minus_minus_nat @ n @ k )
!= n ),
inference(simp,[status(thm)],[9272]) ).
thf(9419,plain,
! [A: nat] :
( ( A != n )
| ( ( minus_minus_nat @ A @ zero_zero_nat )
!= ( minus_minus_nat @ n @ k ) ) ),
inference(paramod_ordered,[status(thm)],[312,9298]) ).
thf(9447,plain,
( ( n != n )
| ( k != zero_zero_nat ) ),
inference(simp,[status(thm)],[9419]) ).
thf(9456,plain,
k != zero_zero_nat,
inference(simp,[status(thm)],[9447]) ).
thf(18,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
= ( ( ord_less_nat @ zero_zero_nat @ A )
& ( ord_less_nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_nat__mult__less__cancel__disj) ).
thf(247,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
= ( ( ord_less_nat @ zero_zero_nat @ A )
& ( ord_less_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(197,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
| ( A = B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194_le__eq__less__or__eq) ).
thf(861,plain,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
| ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).
thf(159,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_156_not__less__zero) ).
thf(730,plain,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(4,axiom,
number1551313001itions @ p @ n,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1_assms_I1_J) ).
thf(217,plain,
number1551313001itions @ p @ n,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(130,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_127_add__left__imp__eq) ).
thf(612,plain,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(613,plain,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
!= ( plus_plus_nat @ A @ C ) )
| ( B = C ) ),
inference(cnf,[status(esa)],[612]) ).
thf(614,plain,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
!= ( plus_plus_nat @ A @ C ) )
| ( B = C ) ),
inference(lifteq,[status(thm)],[613]) ).
thf(2093,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( ord_less_eq_nat @ i @ zero_zero_nat )
| ( A != n ) ),
inference(paramod_ordered,[status(thm)],[735,216]) ).
thf(2094,plain,
( ( ord_less_nat @ zero_zero_nat @ n )
| ( ord_less_eq_nat @ i @ zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[2093:[bind(A,$thf( n ))]]) ).
thf(26,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_le__zero__eq) ).
thf(271,plain,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(272,plain,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ),
inference(cnf,[status(esa)],[271]) ).
thf(273,plain,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ),
inference(lifteq,[status(thm)],[272]) ).
thf(3263,plain,
( ( ord_less_nat @ zero_zero_nat @ n )
| ( i = zero_zero_nat ) ),
inference(rewrite,[status(thm)],[2094,273]) ).
thf(3264,plain,
( ( i = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ n ) ),
inference(lifteq,[status(thm)],[3263]) ).
thf(3322,plain,
( ( ord_less_nat @ zero_zero_nat @ n )
| ( zero_zero_nat != zero_zero_nat )
| ( i != i ) ),
inference(paramod_ordered,[status(thm)],[3264,932]) ).
thf(3323,plain,
( ( ord_less_nat @ zero_zero_nat @ n )
| ( i != i ) ),
inference(simp,[status(thm)],[3322]) ).
thf(3324,plain,
ord_less_nat @ zero_zero_nat @ n,
inference(pattern_uni,[status(thm)],[3323:[]]) ).
thf(3565,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
| ( ( ord_less_nat @ zero_zero_nat @ B )
!= ( ord_less_nat @ zero_zero_nat @ n ) ) ),
inference(paramod_ordered,[status(thm)],[264,3324]) ).
thf(3566,plain,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ n @ A ) ),
inference(pattern_uni,[status(thm)],[3565:[bind(A,$thf( A )),bind(B,$thf( n ))]]) ).
thf(8882,plain,
! [A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ n @ A ) @ A )
!= zero_zero_nat ),
inference(rewrite,[status(thm)],[3566,8881]) ).
thf(8883,plain,
! [A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ n @ A ) @ A )
!= zero_zero_nat ),
inference(simp,[status(thm)],[8882]) ).
thf(8928,plain,
! [A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ n @ A ) @ A )
!= zero_zero_nat ),
inference(lifteq,[status(thm)],[8883]) ).
thf(146,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_143_less__not__refl3) ).
thf(677,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(18795,plain,
! [D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
!= i )
| ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
!= ( minus_minus_nat @ ( minus_minus_nat @ n @ k ) @ D ) ) ),
inference(paramod_ordered,[status(thm)],[421,16031]) ).
thf(18796,plain,
! [A: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ n @ A ) @ k )
!= i ),
inference(pattern_uni,[status(thm)],[18795:[bind(A,$thf( n )),bind(B,$thf( k )),bind(C,$thf( C )),bind(D,$thf( C ))]]) ).
thf(18920,plain,
! [A: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ n @ A ) @ k )
!= i ),
inference(simp,[status(thm)],[18796]) ).
thf(19190,plain,
! [D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) )
!= i )
| ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
!= ( minus_minus_nat @ ( minus_minus_nat @ n @ D ) @ k ) ) ),
inference(paramod_ordered,[status(thm)],[382,18920]) ).
thf(19191,plain,
! [A: nat] :
( ( minus_minus_nat @ n @ ( plus_plus_nat @ A @ k ) )
!= i ),
inference(pattern_uni,[status(thm)],[19190:[bind(A,$thf( n )),bind(B,$thf( B )),bind(C,$thf( k )),bind(D,$thf( B ))]]) ).
thf(19291,plain,
! [A: nat] :
( ( minus_minus_nat @ n @ ( plus_plus_nat @ A @ k ) )
!= i ),
inference(simp,[status(thm)],[19191]) ).
thf(109,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_106_less__diff__iff) ).
thf(533,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)],[109]) ).
thf(21,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_add__less__same__cancel2) ).
thf(256,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(257,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ),
inference(cnf,[status(esa)],[256]) ).
thf(258,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ),
inference(lifteq,[status(thm)],[257]) ).
thf(2849,plain,
! [B: nat,A: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B ),
inference(rewrite,[status(thm)],[258,339]) ).
thf(2880,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ zero_zero_nat @ ( p @ i ) ) ) ),
inference(paramod_ordered,[status(thm)],[2191,2849]) ).
thf(2881,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ zero_zero_nat @ ( p @ i ) ) ),
inference(simp,[status(thm)],[2880]) ).
thf(2896,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
!= zero_zero_nat )
| ( B
!= ( p @ i ) ) ),
inference(simp,[status(thm)],[2881]) ).
thf(2918,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ ( p @ i ) )
!= zero_zero_nat ),
inference(simp,[status(thm)],[2896]) ).
thf(183,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( B != zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_180_gr__implies__not0) ).
thf(814,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( B != zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(646,plain,
( times_times_nat
= ( ^ [A: nat,B: nat] : ( times_times_nat @ B @ A ) ) ),
inference(lifteq,[status(thm)],[645]) ).
thf(913,plain,
! [A: nat] :
( ( times_times_nat @ A )
= ( ^ [B: nat] : ( times_times_nat @ B @ A ) ) ),
inference(func_ext,[status(esa)],[646]) ).
thf(3412,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ zero_zero_nat @ n ) ) ),
inference(paramod_ordered,[status(thm)],[3324,2618]) ).
thf(3413,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ zero_zero_nat @ n ) ),
inference(simp,[status(thm)],[3412]) ).
thf(3431,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
!= zero_zero_nat )
| ( A != n ) ),
inference(simp,[status(thm)],[3413]) ).
thf(3446,plain,
! [A: nat] :
( ( plus_plus_nat @ n @ A )
!= zero_zero_nat ),
inference(simp,[status(thm)],[3431]) ).
thf(49,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_neq0__conv) ).
thf(340,plain,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(192,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
=> ( B = zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_add__eq__self__zero) ).
thf(841,plain,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
=> ( B = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(13,axiom,
! [A: nat > nat,B: nat,C: nat] :
( ( number1551313001itions @ A @ B )
=> ( ord_less_eq_nat @ ( A @ C ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_partitions__bounds) ).
thf(232,plain,
! [A: nat > nat,B: nat,C: nat] :
( ( number1551313001itions @ A @ B )
=> ( ord_less_eq_nat @ ( A @ C ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(233,plain,
! [A: nat > nat,B: nat] :
( ( number1551313001itions @ A @ B )
=> ! [C: nat] : ( ord_less_eq_nat @ ( A @ C ) @ B ) ),
inference(miniscope,[status(thm)],[232]) ).
thf(234,plain,
! [C: nat,B: nat,A: nat > nat] :
( ~ ( number1551313001itions @ A @ B )
| ( ord_less_eq_nat @ ( A @ C ) @ B ) ),
inference(cnf,[status(esa)],[233]) ).
thf(66,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_add__le__same__cancel2) ).
thf(389,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(390,plain,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ),
inference(cnf,[status(esa)],[389]) ).
thf(391,plain,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ),
inference(lifteq,[status(thm)],[390]) ).
thf(8490,plain,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( A = zero_zero_nat ) ),
inference(rewrite,[status(thm)],[391,273]) ).
thf(8494,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat > nat] :
( ~ ( number1551313001itions @ A @ B )
| ( D = zero_zero_nat )
| ( ( ord_less_eq_nat @ ( A @ C ) @ B )
!= ( ord_less_eq_nat @ ( plus_plus_nat @ D @ E ) @ E ) ) ),
inference(paramod_ordered,[status(thm)],[234,8490]) ).
thf(8537,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat > nat] :
( ( D = zero_zero_nat )
| ~ ( number1551313001itions @ A @ B )
| ( ( ord_less_eq_nat @ ( A @ C ) @ B )
!= ( ord_less_eq_nat @ ( plus_plus_nat @ D @ E ) @ E ) ) ),
inference(lifteq,[status(thm)],[8494]) ).
thf(8549,plain,
! [B: nat,A: nat] :
( ~ ( number1551313001itions
@ ^ [C: nat] : C
@ B )
| ( A = zero_zero_nat ) ),
inference(pre_uni,[status(thm)],[8537:[bind(A,$thf( ^ [F: nat] : F )),bind(B,$thf( E )),bind(C,$thf( plus_plus_nat @ D @ E ))]]) ).
thf(8562,plain,
! [B: nat,A: nat] :
( ~ ( number1551313001itions
@ ^ [C: nat] : C
@ B )
| ( A = zero_zero_nat ) ),
inference(simp,[status(thm)],[8549]) ).
thf(14379,plain,
! [C: nat,B: nat,A: nat] :
( ~ ( number1551313001itions
@ ^ [D: nat] : D
@ B )
| ( zero_zero_nat != C )
| ( A
!= ( plus_plus_nat @ i @ C ) ) ),
inference(paramod_ordered,[status(thm)],[8562,5069]) ).
thf(14380,plain,
! [B: nat,A: nat] :
( ~ ( number1551313001itions
@ ^ [C: nat] : C
@ A )
| ( zero_zero_nat != B ) ),
inference(pattern_uni,[status(thm)],[14379:[bind(A,$thf( plus_plus_nat @ i @ E )),bind(B,$thf( B )),bind(C,$thf( E ))]]) ).
thf(15318,plain,
! [A: nat] :
~ ( number1551313001itions
@ ^ [B: nat] : B
@ A ),
inference(simp,[status(thm)],[14380]) ).
thf(31,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_add__eq__0__iff__both__eq__0) ).
thf(286,plain,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(287,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ),
inference(cnf,[status(esa)],[286]) ).
thf(288,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ),
inference(lifteq,[status(thm)],[287]) ).
thf(3489,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
| ( ( ord_less_nat @ zero_zero_nat @ B )
!= ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[264,1233]) ).
thf(3490,plain,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ A ) ),
inference(pattern_uni,[status(thm)],[3489:[bind(A,$thf( A )),bind(B,$thf( sk8 @ ( (=) @ nat @ zero_zero_nat ) ))]]) ).
thf(8458,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ A @ ( plus_plus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ A ) )
!= ( ord_less_nat @ B @ B ) ) ),
inference(paramod_ordered,[status(thm)],[3490,676]) ).
thf(8459,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ A ) )
!= ( ord_less_nat @ B @ B ) ),
inference(simp,[status(thm)],[8458]) ).
thf(8465,plain,
! [B: nat,A: nat] :
( ( A != B )
| ( ( plus_plus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ A )
!= B ) ),
inference(simp,[status(thm)],[8459]) ).
thf(8472,plain,
! [A: nat] :
( ( plus_plus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ A )
!= A ),
inference(simp,[status(thm)],[8465]) ).
thf(36,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_add_Oright__neutral) ).
thf(301,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(8921,plain,
! [A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ i @ A ) @ A )
!= zero_zero_nat ),
inference(rewrite,[status(thm)],[3526,8881]) ).
thf(8922,plain,
! [A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ i @ A ) @ A )
!= zero_zero_nat ),
inference(simp,[status(thm)],[8921]) ).
thf(8933,plain,
! [A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ i @ A ) @ A )
!= zero_zero_nat ),
inference(lifteq,[status(thm)],[8922]) ).
thf(12611,plain,
! [A: nat] :
( ( zero_zero_nat != zero_zero_nat )
| ( ( minus_minus_nat @ ( plus_plus_nat @ i @ A ) @ A )
!= ( minus_minus_nat @ k @ n ) ) ),
inference(paramod_ordered,[status(thm)],[9088,8933]) ).
thf(12612,plain,
! [A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ i @ A ) @ A )
!= ( minus_minus_nat @ k @ n ) ),
inference(simp,[status(thm)],[12611]) ).
thf(12647,plain,
! [A: nat] :
( ( ( plus_plus_nat @ i @ A )
!= k )
| ( A != n ) ),
inference(simp,[status(thm)],[12612]) ).
thf(12661,plain,
( ( plus_plus_nat @ i @ n )
!= k ),
inference(simp,[status(thm)],[12647]) ).
thf(152,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_149_le__antisym) ).
thf(709,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)],[152]) ).
thf(32,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29_add__cancel__right__right) ).
thf(289,plain,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(290,plain,
! [B: nat,A: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ),
inference(cnf,[status(esa)],[289]) ).
thf(291,plain,
! [B: nat,A: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ),
inference(lifteq,[status(thm)],[290]) ).
thf(21665,plain,
! [B: nat,A: nat] :
( ( ( A = n )
= ( B = zero_zero_nat ) )
| ( ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[21192,291]) ).
thf(21666,plain,
( ( ( minus_minus_nat @ n @ k )
= zero_zero_nat )
= ( k = n ) ),
inference(pattern_uni,[status(thm)],[21665:[bind(A,$thf( k )),bind(B,$thf( minus_minus_nat @ n @ k ))]]) ).
thf(23019,plain,
( ( n = k )
= ( k = n ) ),
inference(rewrite,[status(thm)],[21666,21639]) ).
thf(110,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_add__diff__inverse__nat) ).
thf(537,plain,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(538,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ B )
| ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ),
inference(cnf,[status(esa)],[537]) ).
thf(539,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A )
| ( ord_less_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[538]) ).
thf(30241,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A )
| ( ( minus_minus_nat @ B @ A )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[539,8881]) ).
thf(30242,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A )
| ( ( minus_minus_nat @ B @ A )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[30241]) ).
thf(30243,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ A )
!= zero_zero_nat )
| ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ),
inference(lifteq,[status(thm)],[30242]) ).
thf(19179,plain,
! [C: nat,B: nat,A: nat] :
( ( A != i )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( minus_minus_nat @ ( minus_minus_nat @ n @ C ) @ k ) ) ),
inference(paramod_ordered,[status(thm)],[324,18920]) ).
thf(19226,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ i @ A )
!= ( minus_minus_nat @ n @ B ) )
| ( A != k ) ),
inference(simp,[status(thm)],[19179]) ).
thf(19264,plain,
! [A: nat] :
( ( plus_plus_nat @ i @ k )
!= ( minus_minus_nat @ n @ A ) ),
inference(simp,[status(thm)],[19226]) ).
thf(20795,plain,
! [C: nat,B: nat,A: nat] :
( ( A
!= ( plus_plus_nat @ i @ k ) )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( minus_minus_nat @ n @ C ) ) ),
inference(paramod_ordered,[status(thm)],[324,19264]) ).
thf(20830,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ ( plus_plus_nat @ i @ k ) @ A )
!= n )
| ( A != B ) ),
inference(simp,[status(thm)],[20795]) ).
thf(20859,plain,
! [A: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ i @ k ) @ A )
!= n ),
inference(simp,[status(thm)],[20830]) ).
thf(31884,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ A )
!= zero_zero_nat )
| ( A != n )
| ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
!= ( plus_plus_nat @ ( plus_plus_nat @ i @ k ) @ C ) ) ),
inference(paramod_ordered,[status(thm)],[30243,20859]) ).
thf(31885,plain,
! [A: nat] :
( ( ( minus_minus_nat @ ( plus_plus_nat @ i @ k ) @ A )
!= zero_zero_nat )
| ( A != n ) ),
inference(pattern_uni,[status(thm)],[31884:[bind(A,$thf( F )),bind(B,$thf( plus_plus_nat @ i @ k )),bind(C,$thf( minus_minus_nat @ F @ ( plus_plus_nat @ i @ k ) ))]]) ).
thf(31907,plain,
( ( minus_minus_nat @ ( plus_plus_nat @ i @ k ) @ n )
!= zero_zero_nat ),
inference(simp,[status(thm)],[31885]) ).
thf(45,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_add__diff__cancel__left_H) ).
thf(328,plain,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(329,plain,
! [B: nat,A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ),
inference(cnf,[status(esa)],[328]) ).
thf(330,plain,
! [B: nat,A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ),
inference(lifteq,[status(thm)],[329]) ).
thf(19196,plain,
! [C: nat,B: nat,A: nat] :
( ( B != i )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( minus_minus_nat @ ( minus_minus_nat @ n @ C ) @ k ) ) ),
inference(paramod_ordered,[status(thm)],[330,18920]) ).
thf(19230,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ i )
!= ( minus_minus_nat @ n @ B ) )
| ( A != k ) ),
inference(simp,[status(thm)],[19196]) ).
thf(19268,plain,
! [A: nat] :
( ( plus_plus_nat @ k @ i )
!= ( minus_minus_nat @ n @ A ) ),
inference(simp,[status(thm)],[19230]) ).
thf(29845,plain,
! [C: nat,B: nat,A: nat] :
( ( B
!= ( plus_plus_nat @ k @ i ) )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( minus_minus_nat @ n @ C ) ) ),
inference(paramod_ordered,[status(thm)],[330,19268]) ).
thf(29870,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ ( plus_plus_nat @ k @ i ) )
!= n )
| ( A != B ) ),
inference(simp,[status(thm)],[29845]) ).
thf(29916,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ k @ i ) )
!= n ),
inference(simp,[status(thm)],[29870]) ).
thf(48621,plain,
! [B: nat,A: nat] :
( ( n = k )
| ( ( minus_minus_nat @ n @ A )
= B )
| ( ( plus_plus_nat @ k @ sk17 )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[48292,330]) ).
thf(48622,plain,
( ( n = k )
| ( ( minus_minus_nat @ n @ k )
= sk17 ) ),
inference(pattern_uni,[status(thm)],[48621:[bind(A,$thf( k )),bind(B,$thf( sk17 ))]]) ).
thf(51741,plain,
! [B: nat,A: nat] :
( ( n = k )
| ( ( minus_minus_nat @ sk17 @ A )
= zero_zero_nat )
| ( ( minus_minus_nat @ n @ k )
!= ( minus_minus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[48622,15703]) ).
thf(51742,plain,
( ( n = k )
| ( ( minus_minus_nat @ sk17 @ n )
= zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[51741:[bind(A,$thf( n )),bind(B,$thf( k ))]]) ).
thf(21230,plain,
! [B: nat,A: nat] :
( ( zero_zero_nat != zero_zero_nat )
| ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B )
| ( ( minus_minus_nat @ i @ n )
!= ( minus_minus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[9073,21136]) ).
thf(21231,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B )
| ( ( minus_minus_nat @ i @ n )
!= ( minus_minus_nat @ A @ B ) ) ),
inference(simp,[status(thm)],[21230]) ).
thf(21232,plain,
( ( plus_plus_nat @ i @ ( minus_minus_nat @ n @ i ) )
= n ),
inference(pattern_uni,[status(thm)],[21231:[bind(A,$thf( i )),bind(B,$thf( n ))]]) ).
thf(21967,plain,
! [B: nat,A: nat] :
( ( ( A = n )
= ( B = zero_zero_nat ) )
| ( ( plus_plus_nat @ i @ ( minus_minus_nat @ n @ i ) )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[21232,291]) ).
thf(21968,plain,
( ( ( minus_minus_nat @ n @ i )
= zero_zero_nat )
= ( i = n ) ),
inference(pattern_uni,[status(thm)],[21967:[bind(A,$thf( i )),bind(B,$thf( minus_minus_nat @ n @ i ))]]) ).
thf(21942,plain,
! [B: nat,A: nat] :
( ( ( n = A )
= ( B = zero_zero_nat ) )
| ( ( plus_plus_nat @ i @ ( minus_minus_nat @ n @ i ) )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[21232,297]) ).
thf(21943,plain,
( ( ( minus_minus_nat @ n @ i )
= zero_zero_nat )
= ( n = i ) ),
inference(pattern_uni,[status(thm)],[21942:[bind(A,$thf( i )),bind(B,$thf( minus_minus_nat @ n @ i ))]]) ).
thf(23936,plain,
( ( n = i )
= ( i = n ) ),
inference(rewrite,[status(thm)],[21968,21943]) ).
thf(11456,plain,
! [A: nat] :
( ( zero_zero_nat != zero_zero_nat )
| ( ( minus_minus_nat @ ( plus_plus_nat @ n @ A ) @ A )
!= ( minus_minus_nat @ i @ n ) ) ),
inference(paramod_ordered,[status(thm)],[9073,8928]) ).
thf(11457,plain,
! [A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ n @ A ) @ A )
!= ( minus_minus_nat @ i @ n ) ),
inference(simp,[status(thm)],[11456]) ).
thf(11488,plain,
! [A: nat] :
( ( ( plus_plus_nat @ n @ A )
!= i )
| ( A != n ) ),
inference(simp,[status(thm)],[11457]) ).
thf(11491,plain,
( ( plus_plus_nat @ n @ n )
!= i ),
inference(simp,[status(thm)],[11488]) ).
thf(83,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_80_diff__le__mono2) ).
thf(444,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)],[83]) ).
thf(445,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ C @ B ) @ ( minus_minus_nat @ C @ A ) ) ),
inference(miniscope,[status(thm)],[444]) ).
thf(446,plain,
! [C: nat,B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ord_less_eq_nat @ ( minus_minus_nat @ C @ B ) @ ( minus_minus_nat @ C @ A ) ) ),
inference(cnf,[status(esa)],[445]) ).
thf(12883,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( minus_minus_nat @ C @ B ) @ ( minus_minus_nat @ C @ A ) )
= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[446,406]) ).
thf(12884,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( minus_minus_nat @ C @ B ) @ ( minus_minus_nat @ C @ A ) )
= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[12883]) ).
thf(2852,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[1233,2849]) ).
thf(2853,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) ) ),
inference(simp,[status(thm)],[2852]) ).
thf(2894,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
!= zero_zero_nat )
| ( B
!= ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) ) ),
inference(simp,[status(thm)],[2853]) ).
thf(2916,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
!= zero_zero_nat ),
inference(simp,[status(thm)],[2894]) ).
thf(862,plain,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
| ( A = B ) ) ) ),
inference(lifteq,[status(thm)],[861]) ).
thf(8896,plain,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
( ( ( minus_minus_nat @ B @ A )
!= zero_zero_nat )
| ( A = B ) ) ) ),
inference(rewrite,[status(thm)],[862,8881]) ).
thf(9529,plain,
( ( ^ [A: nat,B: nat] :
( ( minus_minus_nat @ A @ B )
= zero_zero_nat ) )
= ( ^ [A: nat,B: nat] :
( ( ( minus_minus_nat @ B @ A )
!= zero_zero_nat )
| ( A = B ) ) ) ),
inference(rewrite,[status(thm)],[8896,406]) ).
thf(210,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
? [C: nat] :
( B
= ( plus_plus_nat @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_nat__le__iff__add) ).
thf(897,plain,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
? [C: nat] :
( B
= ( plus_plus_nat @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).
thf(898,plain,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
? [C: nat] :
( B
= ( plus_plus_nat @ A @ C ) ) ) ),
inference(lifteq,[status(thm)],[897]) ).
thf(924,plain,
( ( ^ [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
| ( A = B ) ) )
= ( ^ [A: nat,B: nat] :
? [C: nat] :
( B
= ( plus_plus_nat @ A @ C ) ) ) ),
inference(rewrite,[status(thm)],[898,862]) ).
thf(8890,plain,
( ( ^ [A: nat,B: nat] :
( ( ( minus_minus_nat @ B @ A )
!= zero_zero_nat )
| ( A = B ) ) )
= ( ^ [A: nat,B: nat] :
? [C: nat] :
( B
= ( plus_plus_nat @ A @ C ) ) ) ),
inference(rewrite,[status(thm)],[924,8881]) ).
thf(11298,plain,
( ( ( ^ [A: nat,B: nat] :
( ( minus_minus_nat @ A @ B )
= zero_zero_nat ) )
= ( ^ [A: nat,B: nat] :
? [C: nat] :
( B
= ( plus_plus_nat @ A @ C ) ) ) )
| ( ( ^ [A: nat,B: nat] :
( ( ( minus_minus_nat @ B @ A )
!= zero_zero_nat )
| ( A = B ) ) )
!= ( ^ [A: nat,B: nat] :
( ( ( minus_minus_nat @ B @ A )
!= zero_zero_nat )
| ( A = B ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[9529,8890]) ).
thf(11299,plain,
( ( ^ [A: nat,B: nat] :
( ( minus_minus_nat @ A @ B )
= zero_zero_nat ) )
= ( ^ [A: nat,B: nat] :
? [C: nat] :
( B
= ( plus_plus_nat @ A @ C ) ) ) ),
inference(pattern_uni,[status(thm)],[11298:[]]) ).
thf(111,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_less__diff__conv) ).
thf(540,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(541,plain,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ B ) ),
inference(cnf,[status(esa)],[540]) ).
thf(542,plain,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ B )
= ( ord_less_nat @ A @ ( minus_minus_nat @ B @ C ) ) ),
inference(lifteq,[status(thm)],[541]) ).
thf(30794,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ ( plus_plus_nat @ A @ C ) )
!= zero_zero_nat )
= ( ( minus_minus_nat @ ( minus_minus_nat @ B @ C ) @ A )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[542,8881]) ).
thf(30795,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= zero_zero_nat )
= ( ( minus_minus_nat @ ( minus_minus_nat @ B @ C ) @ A )
= zero_zero_nat ) ),
inference(simp,[status(thm)],[30794]) ).
thf(131,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_add_Oleft__commute) ).
thf(617,plain,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(618,plain,
! [C: nat,B: nat,A: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) ) ),
inference(cnf,[status(esa)],[617]) ).
thf(619,plain,
! [C: nat,B: nat,A: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) ) ),
inference(lifteq,[status(thm)],[618]) ).
thf(1505,plain,
! [A: nat] :
( ~ $true
| ( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
!= ( ord_less_nat @ A @ zero_zero_nat ) ) ),
inference(paramod_ordered,[status(thm)],[1233,339]) ).
thf(1506,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
!= ( ord_less_nat @ A @ zero_zero_nat ) ),
inference(simp,[status(thm)],[1505]) ).
thf(1509,plain,
! [A: nat] :
( ( zero_zero_nat != A )
| ( ( sk8 @ ( (=) @ nat @ zero_zero_nat ) )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[1506]) ).
thf(1511,plain,
( ( sk8 @ ( (=) @ nat @ zero_zero_nat ) )
!= zero_zero_nat ),
inference(simp,[status(thm)],[1509]) ).
thf(2856,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ ( minus_minus_nat @ n @ k ) @ i ) ) ),
inference(paramod_ordered,[status(thm)],[228,2849]) ).
thf(2857,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ ( minus_minus_nat @ n @ k ) @ i ) ),
inference(simp,[status(thm)],[2856]) ).
thf(2882,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
!= ( minus_minus_nat @ n @ k ) )
| ( B != i ) ),
inference(simp,[status(thm)],[2857]) ).
thf(2897,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ i )
!= ( minus_minus_nat @ n @ k ) ),
inference(simp,[status(thm)],[2882]) ).
thf(302,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ),
inference(cnf,[status(esa)],[301]) ).
thf(303,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ),
inference(lifteq,[status(thm)],[302]) ).
thf(1126,plain,
! [A: nat] :
( ( A != zero_zero_nat )
| ( ( plus_plus_nat @ A @ zero_zero_nat )
!= ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) ) ),
inference(paramod_ordered,[status(thm)],[303,936]) ).
thf(1132,plain,
( ( ( times_times_nat @ ( p @ i ) @ i )
!= zero_zero_nat )
| ( ( times_times_nat @ ( p @ k ) @ k )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[1126]) ).
thf(88,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_85_le__diff__iff) ).
thf(460,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)],[88]) ).
thf(73,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_Nat_Oadd__diff__assoc2) ).
thf(411,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(412,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ),
inference(miniscope,[status(thm)],[411]) ).
thf(413,plain,
! [C: nat,B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ),
inference(cnf,[status(esa)],[412]) ).
thf(414,plain,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[413]) ).
thf(9959,plain,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[414,406]) ).
thf(9960,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ),
inference(lifteq,[status(thm)],[9959]) ).
thf(38,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_cancel__comm__monoid__add__class_Odiff__cancel) ).
thf(307,plain,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(308,plain,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ),
inference(cnf,[status(esa)],[307]) ).
thf(309,plain,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ),
inference(lifteq,[status(thm)],[308]) ).
thf(7876,plain,
! [A: nat] :
( ( i = zero_zero_nat )
| ( ( minus_minus_nat @ A @ A )
!= ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ i ) ) ) ),
inference(paramod_ordered,[status(thm)],[309,7752]) ).
thf(7893,plain,
! [A: nat] :
( ( i = zero_zero_nat )
| ( A != n )
| ( A
!= ( minus_minus_nat @ n @ i ) ) ),
inference(simp,[status(thm)],[7876]) ).
thf(7918,plain,
( ( i = zero_zero_nat )
| ( ( minus_minus_nat @ n @ i )
!= n ) ),
inference(simp,[status(thm)],[7893]) ).
thf(8837,plain,
( ( minus_minus_nat @ n @ i )
!= n ),
inference(simplifyReflect,[status(thm)],[7918,932]) ).
thf(8842,plain,
! [B: nat,A: nat] :
( ( A != n )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( minus_minus_nat @ n @ i ) ) ),
inference(paramod_ordered,[status(thm)],[324,8837]) ).
thf(8856,plain,
! [A: nat] :
( ( ( plus_plus_nat @ n @ A )
!= n )
| ( A != i ) ),
inference(simp,[status(thm)],[8842]) ).
thf(8875,plain,
( ( plus_plus_nat @ n @ i )
!= n ),
inference(simp,[status(thm)],[8856]) ).
thf(155,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_152_le__refl) ).
thf(719,plain,
! [A: nat] : ( ord_less_eq_nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(116,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( ord_less_eq_nat @ A @ ( plus_plus_nat @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_le__diff__conv) ).
thf(559,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( ord_less_eq_nat @ A @ ( plus_plus_nat @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(199,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( A != B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_nat__less__le) ).
thf(865,plain,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( A != B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(866,plain,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( A != B ) ) ) ),
inference(lifteq,[status(thm)],[865]) ).
thf(8895,plain,
( ( ^ [A: nat,B: nat] :
( ( minus_minus_nat @ B @ A )
!= zero_zero_nat ) )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( A != B ) ) ) ),
inference(rewrite,[status(thm)],[866,8881]) ).
thf(11285,plain,
( ( ^ [A: nat,B: nat] :
( ( minus_minus_nat @ B @ A )
!= zero_zero_nat ) )
= ( ^ [A: nat,B: nat] :
( ( ( minus_minus_nat @ A @ B )
= zero_zero_nat )
& ( A != B ) ) ) ),
inference(rewrite,[status(thm)],[8895,406]) ).
thf(1004,plain,
! [A: nat] :
( ~ $true
| ( ( ord_less_nat @ ( minus_minus_nat @ n @ k ) @ i )
!= ( ord_less_nat @ A @ A ) ) ),
inference(paramod_ordered,[status(thm)],[228,676]) ).
thf(1005,plain,
! [A: nat] :
( ( ord_less_nat @ ( minus_minus_nat @ n @ k ) @ i )
!= ( ord_less_nat @ A @ A ) ),
inference(simp,[status(thm)],[1004]) ).
thf(1010,plain,
! [A: nat] :
( ( ( minus_minus_nat @ n @ k )
!= A )
| ( i != A ) ),
inference(simp,[status(thm)],[1005]) ).
thf(1013,plain,
( ( minus_minus_nat @ n @ k )
!= i ),
inference(simp,[status(thm)],[1010]) ).
thf(41,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_add__less__cancel__right) ).
thf(316,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(615,plain,
? [A: nat > nat > nat] :
! [B: nat,C: nat] :
( ( A @ B @ ( plus_plus_nat @ B @ C ) )
= C ),
inference(detectInverse,[status(thm)],[614]) ).
thf(616,plain,
! [B: nat,A: nat] :
( ( sk2 @ A @ ( plus_plus_nat @ A @ B ) )
= B ),
inference(cnf,[status(esa)],[615]) ).
thf(48937,plain,
! [B: nat,A: nat] :
( ( sk2 @ A @ ( plus_plus_nat @ A @ B ) )
= B ),
inference(lifteq,[status(thm)],[616]) ).
thf(48948,plain,
! [B: nat,A: nat] :
( ( ( sk2 @ A @ n )
= B )
| ( ( plus_plus_nat @ i @ ( minus_minus_nat @ n @ i ) )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[21232,48937]) ).
thf(48949,plain,
( ( sk2 @ i @ n )
= ( minus_minus_nat @ n @ i ) ),
inference(pattern_uni,[status(thm)],[48948:[bind(A,$thf( i )),bind(B,$thf( minus_minus_nat @ n @ i ))]]) ).
thf(212,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_trans__le__add1) ).
thf(902,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(44,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_add__diff__cancel__right) ).
thf(325,plain,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(519,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat] : ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ),
inference(miniscope,[status(thm)],[518]) ).
thf(520,plain,
! [C: nat,B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ),
inference(cnf,[status(esa)],[519]) ).
thf(25652,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) )
= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[520,406]) ).
thf(25653,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) )
= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[25652]) ).
thf(2645,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ zero_zero_nat @ ( p @ i ) ) ) ),
inference(paramod_ordered,[status(thm)],[2191,2618]) ).
thf(2646,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ zero_zero_nat @ ( p @ i ) ) ),
inference(simp,[status(thm)],[2645]) ).
thf(2649,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
!= zero_zero_nat )
| ( A
!= ( p @ i ) ) ),
inference(simp,[status(thm)],[2646]) ).
thf(2673,plain,
! [A: nat] :
( ( plus_plus_nat @ ( p @ i ) @ A )
!= zero_zero_nat ),
inference(simp,[status(thm)],[2649]) ).
thf(2698,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( zero_zero_nat != zero_zero_nat )
| ( A
!= ( plus_plus_nat @ ( p @ i ) @ B ) ) ),
inference(paramod_ordered,[status(thm)],[735,2673]) ).
thf(2699,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( A
!= ( plus_plus_nat @ ( p @ i ) @ B ) ) ),
inference(simp,[status(thm)],[2698]) ).
thf(2700,plain,
! [A: nat] : ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ ( p @ i ) @ A ) ),
inference(pattern_uni,[status(thm)],[2699:[bind(A,$thf( plus_plus_nat @ ( p @ i ) @ D )),bind(B,$thf( D ))]]) ).
thf(2710,plain,
! [A: nat] : ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ ( p @ i ) @ A ) ),
inference(simp,[status(thm)],[2700]) ).
thf(19,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) )
= ( ( ord_less_nat @ zero_zero_nat @ A )
| ( ord_less_nat @ zero_zero_nat @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_add__gr__0) ).
thf(250,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) )
= ( ( ord_less_nat @ zero_zero_nat @ A )
| ( ord_less_nat @ zero_zero_nat @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(251,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) )
= ( ( ord_less_nat @ zero_zero_nat @ A )
| ( ord_less_nat @ zero_zero_nat @ B ) ) ),
inference(cnf,[status(esa)],[250]) ).
thf(252,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) )
= ( ( ord_less_nat @ zero_zero_nat @ A )
| ( ord_less_nat @ zero_zero_nat @ B ) ) ),
inference(lifteq,[status(thm)],[251]) ).
thf(3261,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( p @ i ) )
| ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(rewrite,[status(thm)],[2710,252]) ).
thf(3262,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( p @ i ) )
| ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(cnf,[status(esa)],[3261]) ).
thf(44972,plain,
! [A: nat] :
( ( ( minus_minus_nat @ ( p @ i ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[3262,8881]) ).
thf(44973,plain,
! [A: nat] :
( ( ( minus_minus_nat @ ( p @ i ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[44972]) ).
thf(44974,plain,
! [A: nat] :
( ( ( minus_minus_nat @ ( p @ i ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[44973]) ).
thf(2087,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( ord_less_nat @ ( minus_minus_nat @ zero_zero_nat @ k ) @ i )
| ( A != n ) ),
inference(paramod_ordered,[status(thm)],[735,228]) ).
thf(2088,plain,
( ( ord_less_nat @ zero_zero_nat @ n )
| ( ord_less_nat @ ( minus_minus_nat @ zero_zero_nat @ k ) @ i ) ),
inference(pattern_uni,[status(thm)],[2087:[bind(A,$thf( n ))]]) ).
thf(13310,plain,
( ( ( minus_minus_nat @ n @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ ( minus_minus_nat @ zero_zero_nat @ k ) )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[2088,8881]) ).
thf(13311,plain,
( ( ( minus_minus_nat @ n @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ ( minus_minus_nat @ zero_zero_nat @ k ) )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[13310]) ).
thf(13312,plain,
( ( ( minus_minus_nat @ n @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ ( minus_minus_nat @ zero_zero_nat @ k ) )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[13311]) ).
thf(3418,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ zero_zero_nat @ n ) ) ),
inference(paramod_ordered,[status(thm)],[3324,2849]) ).
thf(3419,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ zero_zero_nat @ n ) ),
inference(simp,[status(thm)],[3418]) ).
thf(3435,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
!= zero_zero_nat )
| ( B != n ) ),
inference(simp,[status(thm)],[3419]) ).
thf(3453,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ n )
!= zero_zero_nat ),
inference(simp,[status(thm)],[3435]) ).
thf(7856,plain,
( ( i != i )
| ( ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ i ) )
!= ( minus_minus_nat @ n @ k ) ) ),
inference(paramod_ordered,[status(thm)],[7752,1013]) ).
thf(7857,plain,
( ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ i ) )
!= ( minus_minus_nat @ n @ k ) ),
inference(simp,[status(thm)],[7856]) ).
thf(7902,plain,
( ( n != n )
| ( ( minus_minus_nat @ n @ i )
!= k ) ),
inference(simp,[status(thm)],[7857]) ).
thf(7926,plain,
( ( minus_minus_nat @ n @ i )
!= k ),
inference(simp,[status(thm)],[7902]) ).
thf(15,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
= ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_nat__mult__le__cancel__disj) ).
thf(238,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
= ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(103,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2) ).
thf(514,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(54,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
= ( ord_less_nat @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_nat__add__left__cancel__less) ).
thf(353,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
= ( ord_less_nat @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(185,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_182_not__less0) ).
thf(821,plain,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(3407,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
| ( ( ord_less_nat @ zero_zero_nat @ n )
!= ( ord_less_nat @ zero_zero_nat @ B ) ) ),
inference(paramod_ordered,[status(thm)],[3324,261]) ).
thf(3408,plain,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ n ) ),
inference(pattern_uni,[status(thm)],[3407:[bind(A,$thf( A )),bind(B,$thf( n ))]]) ).
thf(8897,plain,
! [A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ n ) @ A )
!= zero_zero_nat ),
inference(rewrite,[status(thm)],[3408,8881]) ).
thf(8898,plain,
! [A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ n ) @ A )
!= zero_zero_nat ),
inference(simp,[status(thm)],[8897]) ).
thf(8929,plain,
! [A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ n ) @ A )
!= zero_zero_nat ),
inference(lifteq,[status(thm)],[8898]) ).
thf(213,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_add__le__mono1) ).
thf(905,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(94,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_diff__mult__distrib2) ).
thf(480,plain,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(68,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_diff__add__zero) ).
thf(395,plain,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(396,plain,
! [B: nat,A: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ),
inference(cnf,[status(esa)],[395]) ).
thf(397,plain,
! [B: nat,A: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ),
inference(lifteq,[status(thm)],[396]) ).
thf(7757,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ ( minus_minus_nat @ B @ A ) )
= A )
| ~ $true
| ( ( ord_less_eq_nat @ k @ n )
!= ( ord_less_eq_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[220,379]) ).
thf(7758,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ ( minus_minus_nat @ B @ A ) )
= A )
| ( ( ord_less_eq_nat @ k @ n )
!= ( ord_less_eq_nat @ A @ B ) ) ),
inference(simp,[status(thm)],[7757]) ).
thf(7759,plain,
( ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ k ) )
= k ),
inference(pattern_uni,[status(thm)],[7758:[bind(A,$thf( k )),bind(B,$thf( n ))]]) ).
thf(9430,plain,
( ( n != k )
| ( ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ k ) )
!= ( minus_minus_nat @ n @ k ) ) ),
inference(paramod_ordered,[status(thm)],[7759,9298]) ).
thf(9438,plain,
( ( n != k )
| ( n != n )
| ( ( minus_minus_nat @ n @ k )
!= k ) ),
inference(simp,[status(thm)],[9430]) ).
thf(9463,plain,
( ( n != k )
| ( ( minus_minus_nat @ n @ k )
!= k ) ),
inference(simp,[status(thm)],[9438]) ).
thf(17386,plain,
! [B: nat,A: nat] :
( ( n != k )
| ( B != k )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( minus_minus_nat @ n @ k ) ) ),
inference(paramod_ordered,[status(thm)],[330,9463]) ).
thf(17413,plain,
! [A: nat] :
( ( n != k )
| ( ( plus_plus_nat @ A @ k )
!= n )
| ( A != k ) ),
inference(simp,[status(thm)],[17386]) ).
thf(17425,plain,
( ( n != k )
| ( ( plus_plus_nat @ k @ k )
!= n ) ),
inference(simp,[status(thm)],[17413]) ).
thf(29,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
= ( ord_less_eq_nat @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26_add__le__cancel__left) ).
thf(280,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
= ( ord_less_eq_nat @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(143,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_140_infinite__descent) ).
thf(656,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)],[143]) ).
thf(657,plain,
! [A: nat > $o] :
( ! [B: nat] :
( ~ ( A @ B )
=> ? [C: nat] :
( ( ord_less_nat @ C @ B )
& ~ ( A @ C ) ) )
=> ! [B: nat] : ( A @ B ) ),
inference(miniscope,[status(thm)],[656]) ).
thf(658,plain,
! [C: nat,B: nat,A: nat > $o] :
( ( ~ ( A @ ( sk3 @ A ) )
| ( A @ C ) )
& ( ~ ( ord_less_nat @ B @ ( sk3 @ A ) )
| ( A @ B )
| ( A @ C ) ) ),
inference(cnf,[status(esa)],[657]) ).
thf(660,plain,
! [C: nat,B: nat,A: nat > $o] :
( ~ ( ord_less_nat @ B @ ( sk3 @ A ) )
| ( A @ B )
| ( A @ C ) ),
inference(cnfConj,[status(thm)],[658]) ).
thf(55579,plain,
! [C: nat,B: nat,A: nat > $o] :
( ~ ( ( ( minus_minus_nat @ ( sk3 @ A ) @ B )
!= zero_zero_nat ) )
| ( A @ B )
| ( A @ C ) ),
inference(rewrite,[status(thm)],[660,8881]) ).
thf(55580,plain,
! [C: nat,B: nat,A: nat > $o] :
( ( ( minus_minus_nat @ ( sk3 @ A ) @ B )
= zero_zero_nat )
| ( A @ B )
| ( A @ C ) ),
inference(simp,[status(thm)],[55579]) ).
thf(55581,plain,
! [C: nat,B: nat,A: nat > $o] :
( ( ( minus_minus_nat @ ( sk3 @ A ) @ B )
= zero_zero_nat )
| ( A @ B )
| ( A @ C ) ),
inference(lifteq,[status(thm)],[55580]) ).
thf(56454,plain,
! [A: nat] :
( ( ( minus_minus_nat
@ ( sk3
@ ^ [B: nat] : $false )
@ A )
= zero_zero_nat )
| $false
| $false ),
inference(prim_subst,[status(thm)],[55581:[bind(A,$thf( ^ [D: nat] : $false ))]]) ).
thf(57756,plain,
! [A: nat] :
( ( minus_minus_nat
@ ( sk3
@ ^ [B: nat] : $false )
@ A )
= zero_zero_nat ),
inference(simp,[status(thm)],[56454]) ).
thf(203,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_add__less__mono1) ).
thf(877,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(1008,plain,
! [A: nat] :
( ~ $true
| ( ( ord_less_nat @ n @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) )
!= ( ord_less_nat @ A @ A ) ) ),
inference(paramod_ordered,[status(thm)],[219,676]) ).
thf(1009,plain,
! [A: nat] :
( ( ord_less_nat @ n @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) )
!= ( ord_less_nat @ A @ A ) ),
inference(simp,[status(thm)],[1008]) ).
thf(1012,plain,
! [A: nat] :
( ( n != A )
| ( ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) )
!= A ) ),
inference(simp,[status(thm)],[1009]) ).
thf(1015,plain,
( ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) )
!= n ),
inference(simp,[status(thm)],[1012]) ).
thf(1882,plain,
! [A: nat] :
( ( A != n )
| ( ( plus_plus_nat @ A @ zero_zero_nat )
!= ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) ) ),
inference(paramod_ordered,[status(thm)],[303,1015]) ).
thf(1886,plain,
( ( ( times_times_nat @ ( p @ i ) @ i )
!= n )
| ( ( times_times_nat @ ( p @ k ) @ k )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[1882]) ).
thf(86,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_83_diff__le__mono) ).
thf(453,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)],[86]) ).
thf(4658,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ n ) )
!= ( ord_less_nat @ B @ B ) ) ),
inference(paramod_ordered,[status(thm)],[3408,676]) ).
thf(4659,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ n ) )
!= ( ord_less_nat @ B @ B ) ),
inference(simp,[status(thm)],[4658]) ).
thf(4663,plain,
! [B: nat,A: nat] :
( ( A != B )
| ( ( plus_plus_nat @ A @ n )
!= B ) ),
inference(simp,[status(thm)],[4659]) ).
thf(4688,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ n )
!= A ),
inference(simp,[status(thm)],[4663]) ).
thf(21700,plain,
! [A: nat] :
( ( n != A )
| ( ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) )
!= ( plus_plus_nat @ A @ n ) ) ),
inference(paramod_ordered,[status(thm)],[21192,4688]) ).
thf(21768,plain,
( ( plus_plus_nat @ n @ n )
!= ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) ) ),
inference(simp,[status(thm)],[21700]) ).
thf(23909,plain,
( ( plus_plus_nat @ n @ n )
!= n ),
inference(rewrite,[status(thm)],[21768,21192]) ).
thf(24,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ C @ B ) )
= ( A = C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_add__right__cancel) ).
thf(265,plain,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ C @ B ) )
= ( A = C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(107,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_104_diff__add__0) ).
thf(527,plain,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(105,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_ordered__cancel__comm__monoid__diff__class_Odiff__add) ).
thf(521,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(522,plain,
! [B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ),
inference(cnf,[status(esa)],[521]) ).
thf(523,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[522]) ).
thf(26414,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[523,406]) ).
thf(26415,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ),
inference(lifteq,[status(thm)],[26414]) ).
thf(26479,plain,
! [B: nat,A: nat] :
( ( zero_zero_nat != zero_zero_nat )
| ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B )
| ( ( minus_minus_nat @ k @ n )
!= ( minus_minus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[9088,26415]) ).
thf(26480,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B )
| ( ( minus_minus_nat @ k @ n )
!= ( minus_minus_nat @ A @ B ) ) ),
inference(simp,[status(thm)],[26479]) ).
thf(26481,plain,
( ( plus_plus_nat @ ( minus_minus_nat @ n @ k ) @ k )
= n ),
inference(pattern_uni,[status(thm)],[26480:[bind(A,$thf( k )),bind(B,$thf( n ))]]) ).
thf(129,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ C @ B ) )
=> ( A = C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_126_add__right__imp__eq) ).
thf(607,plain,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ C @ B ) )
=> ( A = C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(608,plain,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
!= ( plus_plus_nat @ C @ B ) )
| ( A = C ) ),
inference(cnf,[status(esa)],[607]) ).
thf(609,plain,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
!= ( plus_plus_nat @ C @ B ) )
| ( A = C ) ),
inference(lifteq,[status(thm)],[608]) ).
thf(610,plain,
? [A: nat > nat > nat] :
! [B: nat,C: nat] :
( ( A @ C @ ( plus_plus_nat @ B @ C ) )
= B ),
inference(detectInverse,[status(thm)],[609]) ).
thf(611,plain,
! [B: nat,A: nat] :
( ( sk1 @ B @ ( plus_plus_nat @ A @ B ) )
= A ),
inference(cnf,[status(esa)],[610]) ).
thf(48022,plain,
! [B: nat,A: nat] :
( ( sk1 @ B @ ( plus_plus_nat @ A @ B ) )
= A ),
inference(lifteq,[status(thm)],[611]) ).
thf(48039,plain,
! [B: nat,A: nat] :
( ( ( sk1 @ B @ n )
= A )
| ( ( plus_plus_nat @ ( minus_minus_nat @ n @ k ) @ k )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[26481,48022]) ).
thf(48040,plain,
( ( sk1 @ k @ n )
= ( minus_minus_nat @ n @ k ) ),
inference(pattern_uni,[status(thm)],[48039:[bind(A,$thf( minus_minus_nat @ n @ k )),bind(B,$thf( k ))]]) ).
thf(25,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_add__left__cancel) ).
thf(268,plain,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(269,plain,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ),
inference(cnf,[status(esa)],[268]) ).
thf(270,plain,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ),
inference(lifteq,[status(thm)],[269]) ).
thf(117,axiom,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ ( minus_minus_nat @ B @ C ) )
= ( ~ ( ( ( ord_less_nat @ B @ C )
& ~ ( A @ zero_zero_nat ) )
| ? [D: nat] :
( ( B
= ( plus_plus_nat @ C @ D ) )
& ~ ( A @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_114_nat__diff__split__asm) ).
thf(562,plain,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ ( minus_minus_nat @ B @ C ) )
= ( ~ ( ( ( ord_less_nat @ B @ C )
& ~ ( A @ zero_zero_nat ) )
| ? [D: nat] :
( ( B
= ( plus_plus_nat @ C @ D ) )
& ~ ( A @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(563,plain,
! [C: nat,B: nat,A: nat > $o] :
( ( A @ ( minus_minus_nat @ B @ C ) )
= ( ~ ( ( ( ord_less_nat @ B @ C )
& ~ ( A @ zero_zero_nat ) )
| ? [D: nat] :
( ( B
= ( plus_plus_nat @ C @ D ) )
& ~ ( A @ D ) ) ) ) ),
inference(cnf,[status(esa)],[562]) ).
thf(564,plain,
! [C: nat,B: nat,A: nat > $o] :
( ( A @ ( minus_minus_nat @ B @ C ) )
= ( ~ ( ( ( ord_less_nat @ B @ C )
& ~ ( A @ zero_zero_nat ) )
| ? [D: nat] :
( ( B
= ( plus_plus_nat @ C @ D ) )
& ~ ( A @ D ) ) ) ) ),
inference(lifteq,[status(thm)],[563]) ).
thf(33295,plain,
! [C: nat,B: nat,A: nat > $o] :
( ( A @ ( minus_minus_nat @ B @ C ) )
= ( ~ ( ( ( ( minus_minus_nat @ C @ B )
!= zero_zero_nat )
& ~ ( A @ zero_zero_nat ) )
| ? [D: nat] :
( ( B
= ( plus_plus_nat @ C @ D ) )
& ~ ( A @ D ) ) ) ) ),
inference(rewrite,[status(thm)],[564,8881]) ).
thf(170,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( ord_less_eq_nat @ A @ B )
& ( C = D ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_167_add__mono__thms__linordered__semiring_I3_J) ).
thf(761,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( ord_less_eq_nat @ A @ B )
& ( C = D ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(26526,plain,
! [B: nat,A: nat] :
( ( zero_zero_nat != zero_zero_nat )
| ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B )
| ( ( minus_minus_nat @ i @ n )
!= ( minus_minus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[9073,26415]) ).
thf(26527,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B )
| ( ( minus_minus_nat @ i @ n )
!= ( minus_minus_nat @ A @ B ) ) ),
inference(simp,[status(thm)],[26526]) ).
thf(26528,plain,
( ( plus_plus_nat @ ( minus_minus_nat @ n @ i ) @ i )
= n ),
inference(pattern_uni,[status(thm)],[26527:[bind(A,$thf( i )),bind(B,$thf( n ))]]) ).
thf(31871,plain,
! [A: nat] :
( ( n != n )
| ( ( plus_plus_nat @ ( plus_plus_nat @ i @ k ) @ A )
!= ( plus_plus_nat @ ( minus_minus_nat @ n @ i ) @ i ) ) ),
inference(paramod_ordered,[status(thm)],[26528,20859]) ).
thf(31872,plain,
! [A: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ i @ k ) @ A )
!= ( plus_plus_nat @ ( minus_minus_nat @ n @ i ) @ i ) ),
inference(simp,[status(thm)],[31871]) ).
thf(31897,plain,
! [A: nat] :
( ( ( plus_plus_nat @ i @ k )
!= ( minus_minus_nat @ n @ i ) )
| ( A != i ) ),
inference(simp,[status(thm)],[31872]) ).
thf(31918,plain,
( ( plus_plus_nat @ i @ k )
!= ( minus_minus_nat @ n @ i ) ),
inference(simp,[status(thm)],[31897]) ).
thf(31539,plain,
! [A: nat] :
( ( i != i )
| ( ( minus_minus_nat @ n @ ( plus_plus_nat @ A @ k ) )
!= ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ i ) ) ) ),
inference(paramod_ordered,[status(thm)],[7752,19291]) ).
thf(31540,plain,
! [A: nat] :
( ( minus_minus_nat @ n @ ( plus_plus_nat @ A @ k ) )
!= ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ i ) ) ),
inference(simp,[status(thm)],[31539]) ).
thf(31573,plain,
! [A: nat] :
( ( n != n )
| ( ( plus_plus_nat @ A @ k )
!= ( minus_minus_nat @ n @ i ) ) ),
inference(simp,[status(thm)],[31540]) ).
thf(31596,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ k )
!= ( minus_minus_nat @ n @ i ) ),
inference(simp,[status(thm)],[31573]) ).
thf(3076,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
| ( ( ord_less_nat @ zero_zero_nat @ B )
!= ( ord_less_nat @ zero_zero_nat @ ( p @ k ) ) ) ),
inference(paramod_ordered,[status(thm)],[261,224]) ).
thf(3077,plain,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ ( p @ k ) ) ),
inference(pattern_uni,[status(thm)],[3076:[bind(A,$thf( A )),bind(B,$thf( p @ k ))]]) ).
thf(5675,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ ( p @ k ) ) )
!= ( ord_less_nat @ B @ B ) ) ),
inference(paramod_ordered,[status(thm)],[3077,676]) ).
thf(5676,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ ( p @ k ) ) )
!= ( ord_less_nat @ B @ B ) ),
inference(simp,[status(thm)],[5675]) ).
thf(5681,plain,
! [B: nat,A: nat] :
( ( A != B )
| ( ( plus_plus_nat @ A @ ( p @ k ) )
!= B ) ),
inference(simp,[status(thm)],[5676]) ).
thf(5705,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ ( p @ k ) )
!= A ),
inference(simp,[status(thm)],[5681]) ).
thf(21657,plain,
! [A: nat] :
( ( n != A )
| ( ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) )
!= ( plus_plus_nat @ A @ ( p @ k ) ) ) ),
inference(paramod_ordered,[status(thm)],[21192,5705]) ).
thf(21795,plain,
( ( plus_plus_nat @ n @ ( p @ k ) )
!= ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) ) ),
inference(simp,[status(thm)],[21657]) ).
thf(51618,plain,
( ( plus_plus_nat @ n @ ( p @ k ) )
!= n ),
inference(rewrite,[status(thm)],[21795,21192]) ).
thf(33,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_add__cancel__right__left) ).
thf(292,plain,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(20805,plain,
! [C: nat,B: nat,A: nat] :
( ( B
!= ( plus_plus_nat @ i @ k ) )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( minus_minus_nat @ n @ C ) ) ),
inference(paramod_ordered,[status(thm)],[330,19264]) ).
thf(20842,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ ( plus_plus_nat @ i @ k ) )
!= n )
| ( A != B ) ),
inference(simp,[status(thm)],[20805]) ).
thf(20871,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ i @ k ) )
!= n ),
inference(simp,[status(thm)],[20842]) ).
thf(48027,plain,
! [B: nat,A: nat] :
( ( ( sk1 @ B @ n )
= A )
| ( ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[21192,48022]) ).
thf(48028,plain,
( ( sk1 @ ( minus_minus_nat @ n @ k ) @ n )
= k ),
inference(pattern_uni,[status(thm)],[48027:[bind(A,$thf( k )),bind(B,$thf( minus_minus_nat @ n @ k ))]]) ).
thf(2862,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ zero_zero_nat @ ( p @ k ) ) ) ),
inference(paramod_ordered,[status(thm)],[224,2849]) ).
thf(2863,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ zero_zero_nat @ ( p @ k ) ) ),
inference(simp,[status(thm)],[2862]) ).
thf(2889,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
!= zero_zero_nat )
| ( B
!= ( p @ k ) ) ),
inference(simp,[status(thm)],[2863]) ).
thf(2910,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ ( p @ k ) )
!= zero_zero_nat ),
inference(simp,[status(thm)],[2889]) ).
thf(209,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_mult__0) ).
thf(894,plain,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(176,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_add__strict__left__mono) ).
thf(778,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(48956,plain,
! [B: nat,A: nat] :
( ( ( sk2 @ A @ n )
= B )
| ( ( plus_plus_nat @ ( minus_minus_nat @ n @ k ) @ k )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[26481,48937]) ).
thf(48957,plain,
( ( sk2 @ ( minus_minus_nat @ n @ k ) @ n )
= k ),
inference(pattern_uni,[status(thm)],[48956:[bind(A,$thf( minus_minus_nat @ n @ k )),bind(B,$thf( k ))]]) ).
thf(67,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ A )
= ( ord_less_eq_nat @ B @ zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_add__le__same__cancel1) ).
thf(392,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ A )
= ( ord_less_eq_nat @ B @ zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(78,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_diff__diff__add) ).
thf(429,plain,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(179,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A = B )
& ( ord_less_nat @ C @ D ) )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_add__mono__thms__linordered__field_I2_J) ).
thf(788,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A = B )
& ( ord_less_nat @ C @ D ) )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(72,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_Nat_Odiff__diff__right) ).
thf(407,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(135,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( A
= ( plus_plus_nat @ B @ C ) )
=> ( ( plus_plus_nat @ A @ D )
= ( plus_plus_nat @ B @ ( plus_plus_nat @ C @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_group__cancel_Oadd1) ).
thf(630,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( A
= ( plus_plus_nat @ B @ C ) )
=> ( ( plus_plus_nat @ A @ D )
= ( plus_plus_nat @ B @ ( plus_plus_nat @ C @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(35,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= B )
= ( A = zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_add__cancel__left__left) ).
thf(298,plain,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= B )
= ( A = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(299,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
= B )
= ( A = zero_zero_nat ) ),
inference(cnf,[status(esa)],[298]) ).
thf(300,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
= B )
= ( A = zero_zero_nat ) ),
inference(lifteq,[status(thm)],[299]) ).
thf(22028,plain,
! [B: nat,A: nat] :
( ( ( n = B )
= ( A = zero_zero_nat ) )
| ( ( plus_plus_nat @ i @ ( minus_minus_nat @ n @ i ) )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[21232,300]) ).
thf(22029,plain,
( ( n
= ( minus_minus_nat @ n @ i ) )
= ( i = zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[22028:[bind(A,$thf( i )),bind(B,$thf( minus_minus_nat @ n @ i ))]]) ).
thf(9426,plain,
! [B: nat,A: nat] :
( ( B != n )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( minus_minus_nat @ n @ k ) ) ),
inference(paramod_ordered,[status(thm)],[330,9298]) ).
thf(9445,plain,
! [A: nat] :
( ( ( plus_plus_nat @ A @ n )
!= n )
| ( A != k ) ),
inference(simp,[status(thm)],[9426]) ).
thf(9455,plain,
( ( plus_plus_nat @ k @ n )
!= n ),
inference(simp,[status(thm)],[9445]) ).
thf(461,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat] :
( ( 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(miniscope,[status(thm)],[460]) ).
thf(462,plain,
! [C: nat,B: nat,A: 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(cnf,[status(esa)],[461]) ).
thf(463,plain,
! [C: nat,B: nat,A: nat] :
( ( ( ord_less_eq_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ C @ A ) )
= ( ord_less_eq_nat @ B @ C ) )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ A @ C ) ),
inference(lifteq,[status(thm)],[462]) ).
thf(17658,plain,
! [C: nat,B: nat,A: nat] :
( ( ( ( minus_minus_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ C @ A ) )
= zero_zero_nat )
= ( ( minus_minus_nat @ B @ C )
= zero_zero_nat ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ C )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[463,406]) ).
thf(17659,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ C )
!= zero_zero_nat )
| ( ( ( minus_minus_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ C @ A ) )
= zero_zero_nat )
= ( ( minus_minus_nat @ B @ C )
= zero_zero_nat ) ) ),
inference(lifteq,[status(thm)],[17658]) ).
thf(35735,plain,
! [D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) )
!= k )
| ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
!= ( minus_minus_nat @ ( minus_minus_nat @ n @ i ) @ D ) ) ),
inference(paramod_ordered,[status(thm)],[382,22317]) ).
thf(35736,plain,
! [A: nat] :
( ( minus_minus_nat @ n @ ( plus_plus_nat @ i @ A ) )
!= k ),
inference(pattern_uni,[status(thm)],[35735:[bind(A,$thf( n )),bind(B,$thf( i )),bind(C,$thf( C )),bind(D,$thf( C ))]]) ).
thf(35807,plain,
! [A: nat] :
( ( minus_minus_nat @ n @ ( plus_plus_nat @ i @ A ) )
!= k ),
inference(simp,[status(thm)],[35736]) ).
thf(188,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_bot__nat__0_Oextremum__uniqueI) ).
thf(830,plain,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(207,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ C )
=> ( ord_less_nat @ A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_add__lessD1) ).
thf(889,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ C )
=> ( ord_less_nat @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(169,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A = B )
& ( ord_less_eq_nat @ C @ D ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_166_add__mono__thms__linordered__semiring_I2_J) ).
thf(757,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A = B )
& ( ord_less_eq_nat @ C @ D ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(114,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_111_Nat_Odiff__add__assoc) ).
thf(551,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(87,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_84_Nat_Odiff__diff__eq) ).
thf(456,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)],[87]) ).
thf(12,axiom,
! [A: nat > nat] :
( ( number1551313001itions @ A @ zero_zero_nat )
= ( A
= ( ^ [B: nat] : zero_zero_nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_partitions__zero) ).
thf(229,plain,
! [A: nat > nat] :
( ( number1551313001itions @ A @ zero_zero_nat )
= ( A
= ( ^ [B: nat] : zero_zero_nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(194,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_191_less__mono__imp__le__mono) ).
thf(847,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)],[194]) ).
thf(195,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_192_le__neq__implies__less) ).
thf(852,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)],[195]) ).
thf(80,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_77_diffs0__imp__equal) ).
thf(435,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)],[80]) ).
thf(37,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_add_Oleft__neutral) ).
thf(304,plain,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(305,plain,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ),
inference(cnf,[status(esa)],[304]) ).
thf(306,plain,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ),
inference(lifteq,[status(thm)],[305]) ).
thf(1883,plain,
! [A: nat] :
( ( A != n )
| ( ( plus_plus_nat @ zero_zero_nat @ A )
!= ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) ) ),
inference(paramod_ordered,[status(thm)],[306,1015]) ).
thf(1888,plain,
( ( ( times_times_nat @ ( p @ i ) @ i )
!= zero_zero_nat )
| ( ( times_times_nat @ ( p @ k ) @ k )
!= n ) ),
inference(simp,[status(thm)],[1883]) ).
thf(144,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_141_nat__less__induct) ).
thf(665,plain,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ! [D: nat] :
( ( ord_less_nat @ D @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(1271,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ n @ k ) @ zero_zero_nat )
| ( A != i ) ),
inference(paramod_ordered,[status(thm)],[813,228]) ).
thf(1272,plain,
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ n @ k ) @ zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[1271:[bind(A,$thf( i ))]]) ).
thf(29996,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ zero_zero_nat @ ( minus_minus_nat @ n @ k ) )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[1272,8881]) ).
thf(29997,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ zero_zero_nat @ ( minus_minus_nat @ n @ k ) )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[29996]) ).
thf(29998,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ zero_zero_nat @ ( minus_minus_nat @ n @ k ) )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[29997]) ).
thf(29825,plain,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ k @ i )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
!= ( minus_minus_nat @ n @ C ) ) ),
inference(paramod_ordered,[status(thm)],[397,19268]) ).
thf(29826,plain,
( ( plus_plus_nat @ k @ i )
!= zero_zero_nat ),
inference(pattern_uni,[status(thm)],[29825:[bind(A,$thf( n )),bind(B,$thf( E )),bind(C,$thf( plus_plus_nat @ n @ E ))]]) ).
thf(30,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_zero__eq__add__iff__both__eq__0) ).
thf(283,plain,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(284,plain,
! [B: nat,A: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ),
inference(cnf,[status(esa)],[283]) ).
thf(285,plain,
! [B: nat,A: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ),
inference(lifteq,[status(thm)],[284]) ).
thf(164,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_161_add__right__mono) ).
thf(742,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(1019,plain,
! [C: nat,B: nat,A: nat > nat] :
( ~ $true
| ( ord_less_eq_nat @ ( A @ C ) @ B )
| ( ( number1551313001itions @ p @ n )
!= ( number1551313001itions @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[217,234]) ).
thf(1020,plain,
! [C: nat,B: nat,A: nat > nat] :
( ( ord_less_eq_nat @ ( A @ C ) @ B )
| ( ( number1551313001itions @ p @ n )
!= ( number1551313001itions @ A @ B ) ) ),
inference(simp,[status(thm)],[1019]) ).
thf(1021,plain,
! [A: nat] : ( ord_less_eq_nat @ ( p @ A ) @ n ),
inference(pattern_uni,[status(thm)],[1020:[bind(A,$thf( p )),bind(B,$thf( n )),bind(C,$thf( C ))]]) ).
thf(1047,plain,
! [A: nat] : ( ord_less_eq_nat @ ( p @ A ) @ n ),
inference(simp,[status(thm)],[1021]) ).
thf(9089,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ C )
= zero_zero_nat )
| ~ $true
| ( ( ord_less_eq_nat @ ( p @ A ) @ n )
!= ( ord_less_eq_nat @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[1047,403]) ).
thf(9090,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ C )
= zero_zero_nat )
| ( ( ord_less_eq_nat @ ( p @ A ) @ n )
!= ( ord_less_eq_nat @ B @ C ) ) ),
inference(simp,[status(thm)],[9089]) ).
thf(9091,plain,
! [A: nat] :
( ( minus_minus_nat @ ( p @ A ) @ n )
= zero_zero_nat ),
inference(pattern_uni,[status(thm)],[9090:[bind(A,$thf( D )),bind(B,$thf( p @ D )),bind(C,$thf( n ))]]) ).
thf(9171,plain,
! [A: nat] :
( ( minus_minus_nat @ ( p @ A ) @ n )
= zero_zero_nat ),
inference(simp,[status(thm)],[9091]) ).
thf(31955,plain,
! [A: nat] :
( ( zero_zero_nat != zero_zero_nat )
| ( ( minus_minus_nat @ ( p @ A ) @ n )
!= ( minus_minus_nat @ ( plus_plus_nat @ i @ k ) @ n ) ) ),
inference(paramod_ordered,[status(thm)],[9171,31907]) ).
thf(31956,plain,
! [A: nat] :
( ( minus_minus_nat @ ( p @ A ) @ n )
!= ( minus_minus_nat @ ( plus_plus_nat @ i @ k ) @ n ) ),
inference(simp,[status(thm)],[31955]) ).
thf(31993,plain,
! [A: nat] :
( ( ( p @ A )
!= ( plus_plus_nat @ i @ k ) )
| ( n != n ) ),
inference(simp,[status(thm)],[31956]) ).
thf(32042,plain,
! [A: nat] :
( ( p @ A )
!= ( plus_plus_nat @ i @ k ) ),
inference(simp,[status(thm)],[31993]) ).
thf(202,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_trans__less__add1) ).
thf(874,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).
thf(51813,plain,
( ( n = k )
| ( ( sk2 @ sk17 @ n )
= k )
| ( ( minus_minus_nat @ n @ k )
!= ( minus_minus_nat @ n @ k ) ) ),
inference(paramod_ordered,[status(thm)],[48622,48957]) ).
thf(51814,plain,
( ( n = k )
| ( ( sk2 @ sk17 @ n )
= k ) ),
inference(pattern_uni,[status(thm)],[51813:[]]) ).
thf(408,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat] :
( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ),
inference(miniscope,[status(thm)],[407]) ).
thf(409,plain,
! [C: nat,B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ),
inference(cnf,[status(esa)],[408]) ).
thf(410,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[409]) ).
thf(9575,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[410,406]) ).
thf(9576,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ),
inference(lifteq,[status(thm)],[9575]) ).
thf(2089,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( ord_less_nat @ zero_zero_nat @ i )
| ( A
!= ( minus_minus_nat @ n @ k ) ) ),
inference(paramod_ordered,[status(thm)],[735,228]) ).
thf(2090,plain,
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ n @ k ) )
| ( ord_less_nat @ zero_zero_nat @ i ) ),
inference(pattern_uni,[status(thm)],[2089:[bind(A,$thf( minus_minus_nat @ n @ k ))]]) ).
thf(14119,plain,
( ( ( minus_minus_nat @ ( minus_minus_nat @ n @ k ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[2090,8881]) ).
thf(14120,plain,
( ( ( minus_minus_nat @ ( minus_minus_nat @ n @ k ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[14119]) ).
thf(14121,plain,
( ( ( minus_minus_nat @ ( minus_minus_nat @ n @ k ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[14120]) ).
thf(55,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_diff__self__eq__0) ).
thf(356,plain,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(12624,plain,
! [B: nat,A: nat] :
( ( zero_zero_nat != zero_zero_nat )
| ( ( minus_minus_nat @ ( p @ A ) @ n )
!= ( minus_minus_nat @ ( plus_plus_nat @ i @ B ) @ B ) ) ),
inference(paramod_ordered,[status(thm)],[9171,8933]) ).
thf(12625,plain,
! [B: nat,A: nat] :
( ( minus_minus_nat @ ( p @ A ) @ n )
!= ( minus_minus_nat @ ( plus_plus_nat @ i @ B ) @ B ) ),
inference(simp,[status(thm)],[12624]) ).
thf(12653,plain,
! [B: nat,A: nat] :
( ( ( p @ A )
!= ( plus_plus_nat @ i @ B ) )
| ( n != B ) ),
inference(simp,[status(thm)],[12625]) ).
thf(12667,plain,
! [A: nat] :
( ( p @ A )
!= ( plus_plus_nat @ i @ n ) ),
inference(simp,[status(thm)],[12653]) ).
thf(2997,plain,
( ( ( times_times_nat @ ( p @ i ) @ i )
!= zero_zero_nat )
| ( ( times_times_nat @ ( p @ k ) @ k )
!= ( times_times_nat @ ( p @ i ) @ i ) )
| ( zero_zero_nat != zero_zero_nat ) ),
inference(eqfactor_ordered,[status(thm)],[1132]) ).
thf(3001,plain,
( ( ( times_times_nat @ ( p @ i ) @ i )
!= zero_zero_nat )
| ( ( times_times_nat @ ( p @ k ) @ k )
!= ( times_times_nat @ ( p @ i ) @ i ) ) ),
inference(simp,[status(thm)],[2997]) ).
thf(51940,plain,
( ( n = k )
| ( ( plus_plus_nat @ sk17 @ k )
= n )
| ( ( minus_minus_nat @ n @ k )
!= ( minus_minus_nat @ n @ k ) ) ),
inference(paramod_ordered,[status(thm)],[48622,26481]) ).
thf(51941,plain,
( ( n = k )
| ( ( plus_plus_nat @ sk17 @ k )
= n ) ),
inference(pattern_uni,[status(thm)],[51940:[]]) ).
thf(48952,plain,
! [C: nat,B: nat,A: nat] :
( ( ( sk2 @ B @ A )
= C )
| ( ( plus_plus_nat @ A @ zero_zero_nat )
!= ( plus_plus_nat @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[303,48937]) ).
thf(48953,plain,
! [A: nat] :
( ( sk2 @ A @ A )
= zero_zero_nat ),
inference(pattern_uni,[status(thm)],[48952:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( zero_zero_nat ))]]) ).
thf(3594,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
| ( ( ord_less_nat @ zero_zero_nat @ B )
!= ( ord_less_nat @ zero_zero_nat @ ( p @ i ) ) ) ),
inference(paramod_ordered,[status(thm)],[264,2191]) ).
thf(3595,plain,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ ( p @ i ) @ A ) ),
inference(pattern_uni,[status(thm)],[3594:[bind(A,$thf( A )),bind(B,$thf( p @ i ))]]) ).
thf(6768,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ A @ ( plus_plus_nat @ ( p @ i ) @ A ) )
!= ( ord_less_nat @ B @ B ) ) ),
inference(paramod_ordered,[status(thm)],[3595,676]) ).
thf(6769,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ ( p @ i ) @ A ) )
!= ( ord_less_nat @ B @ B ) ),
inference(simp,[status(thm)],[6768]) ).
thf(6775,plain,
! [B: nat,A: nat] :
( ( A != B )
| ( ( plus_plus_nat @ ( p @ i ) @ A )
!= B ) ),
inference(simp,[status(thm)],[6769]) ).
thf(6801,plain,
! [A: nat] :
( ( plus_plus_nat @ ( p @ i ) @ A )
!= A ),
inference(simp,[status(thm)],[6775]) ).
thf(141,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_ab__semigroup__mult__class_Omult__ac_I1_J) ).
thf(650,plain,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(10,axiom,
i != k,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7__092_060open_062i_A_092_060noteq_062_Ak_092_060close_062) ).
thf(225,plain,
i != k,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(1966,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( ord_less_nat @ zero_zero_nat @ ( p @ zero_zero_nat ) )
| ( A != k ) ),
inference(paramod_ordered,[status(thm)],[735,224]) ).
thf(1967,plain,
( ( ord_less_nat @ zero_zero_nat @ k )
| ( ord_less_nat @ zero_zero_nat @ ( p @ zero_zero_nat ) ) ),
inference(pattern_uni,[status(thm)],[1966:[bind(A,$thf( k ))]]) ).
thf(6171,plain,
( ( ord_less_nat @ zero_zero_nat @ k )
| ( ( ord_less_nat @ zero_zero_nat @ ( p @ zero_zero_nat ) )
!= ( ord_less_nat @ zero_zero_nat @ k ) )
| ~ $true ),
inference(eqfactor_ordered,[status(thm)],[1967]) ).
thf(6185,plain,
( ( ord_less_nat @ zero_zero_nat @ k )
| ( zero_zero_nat != zero_zero_nat )
| ( ( p @ zero_zero_nat )
!= k ) ),
inference(simp,[status(thm)],[6171]) ).
thf(6212,plain,
( ( ord_less_nat @ zero_zero_nat @ k )
| ( ( p @ zero_zero_nat )
!= k ) ),
inference(simp,[status(thm)],[6185]) ).
thf(6243,plain,
! [B: nat,A: nat] :
( ( ( p @ zero_zero_nat )
!= k )
| ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ zero_zero_nat @ k ) ) ),
inference(paramod_ordered,[status(thm)],[6212,2618]) ).
thf(6244,plain,
! [B: nat,A: nat] :
( ( ( p @ zero_zero_nat )
!= k )
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ zero_zero_nat @ k ) ) ),
inference(simp,[status(thm)],[6243]) ).
thf(6284,plain,
! [B: nat,A: nat] :
( ( ( p @ zero_zero_nat )
!= k )
| ( ( plus_plus_nat @ A @ B )
!= zero_zero_nat )
| ( A != k ) ),
inference(simp,[status(thm)],[6244]) ).
thf(6304,plain,
! [A: nat] :
( ( ( p @ zero_zero_nat )
!= k )
| ( ( plus_plus_nat @ k @ A )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[6284]) ).
thf(138,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( times_times_nat @ B @ C ) )
= ( times_times_nat @ B @ ( times_times_nat @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_mult_Oleft__commute) ).
thf(642,plain,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( times_times_nat @ B @ C ) )
= ( times_times_nat @ B @ ( times_times_nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(560,plain,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( ord_less_eq_nat @ A @ ( plus_plus_nat @ C @ B ) ) ),
inference(cnf,[status(esa)],[559]) ).
thf(561,plain,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( ord_less_eq_nat @ A @ ( plus_plus_nat @ C @ B ) ) ),
inference(lifteq,[status(thm)],[560]) ).
thf(32564,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= zero_zero_nat )
= ( ( minus_minus_nat @ A @ ( plus_plus_nat @ C @ B ) )
= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[561,406]) ).
thf(48291,plain,
( ( n = k )
| ( zero_zero_nat != sk17 ) ),
inference(cnfConj,[status(thm)],[48289]) ).
thf(48293,plain,
( ( sk17 != zero_zero_nat )
| ( n = k ) ),
inference(lifteq,[status(thm)],[48291]) ).
thf(19204,plain,
! [A: nat] :
( ( ( minus_minus_nat @ i @ k )
!= i )
| ( ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ i ) )
!= ( minus_minus_nat @ n @ A ) ) ),
inference(paramod_ordered,[status(thm)],[7752,18920]) ).
thf(19205,plain,
( ( minus_minus_nat @ i @ k )
!= i ),
inference(pattern_uni,[status(thm)],[19204:[bind(A,$thf( minus_minus_nat @ n @ i ))]]) ).
thf(19302,plain,
! [B: nat,A: nat] :
( ( A != i )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( minus_minus_nat @ i @ k ) ) ),
inference(paramod_ordered,[status(thm)],[324,19205]) ).
thf(19324,plain,
! [A: nat] :
( ( ( plus_plus_nat @ i @ A )
!= i )
| ( A != k ) ),
inference(simp,[status(thm)],[19302]) ).
thf(19358,plain,
( ( plus_plus_nat @ i @ k )
!= i ),
inference(simp,[status(thm)],[19324]) ).
thf(91,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_diff__add__inverse) ).
thf(471,plain,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(41411,plain,
! [C: nat,B: nat,A: nat] :
( ( A != k )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( minus_minus_nat @ n @ ( plus_plus_nat @ i @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[324,35807]) ).
thf(41463,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ k @ A )
!= n )
| ( A
!= ( plus_plus_nat @ i @ B ) ) ),
inference(simp,[status(thm)],[41411]) ).
thf(41506,plain,
! [A: nat] :
( ( plus_plus_nat @ k @ ( plus_plus_nat @ i @ A ) )
!= n ),
inference(simp,[status(thm)],[41463]) ).
thf(113,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_Nat_Odiff__add__assoc2) ).
thf(547,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(133,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_add_Oassoc) ).
thf(622,plain,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(140,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_mult_Oassoc) ).
thf(647,plain,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(162,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
=> ( ord_less_eq_nat @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_159_add__le__imp__le__left) ).
thf(738,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
=> ( ord_less_eq_nat @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(132,axiom,
( plus_plus_nat
= ( ^ [A: nat,B: nat] : ( plus_plus_nat @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_add_Ocommute) ).
thf(620,plain,
( plus_plus_nat
= ( ^ [A: nat,B: nat] : ( plus_plus_nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(621,plain,
( plus_plus_nat
= ( ^ [A: nat,B: nat] : ( plus_plus_nat @ B @ A ) ) ),
inference(lifteq,[status(thm)],[620]) ).
thf(911,plain,
! [A: nat] :
( ( plus_plus_nat @ A )
= ( ^ [B: nat] : ( plus_plus_nat @ B @ A ) ) ),
inference(func_ext,[status(esa)],[621]) ).
thf(156,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_zero__le) ).
thf(721,plain,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(21648,plain,
! [A: nat] :
( ( n != A )
| ( ( plus_plus_nat @ ( p @ i ) @ A )
!= ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) ) ) ),
inference(paramod_ordered,[status(thm)],[21192,6801]) ).
thf(21746,plain,
( ( plus_plus_nat @ ( p @ i ) @ n )
!= ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) ) ),
inference(simp,[status(thm)],[21648]) ).
thf(47977,plain,
( ( plus_plus_nat @ ( p @ i ) @ n )
!= n ),
inference(rewrite,[status(thm)],[21746,21192]) ).
thf(6159,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( ord_less_nat @ zero_zero_nat @ k )
| ( ord_less_nat @ zero_zero_nat @ ( p @ zero_zero_nat ) )
| ( A != zero_zero_nat ) ),
inference(paramod_ordered,[status(thm)],[735,1967]) ).
thf(6160,plain,
( ( ord_less_nat @ zero_zero_nat @ zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ k )
| ( ord_less_nat @ zero_zero_nat @ ( p @ zero_zero_nat ) ) ),
inference(pattern_uni,[status(thm)],[6159:[bind(A,$thf( zero_zero_nat ))]]) ).
thf(34303,plain,
( ( ( minus_minus_nat @ zero_zero_nat @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ k @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( p @ zero_zero_nat ) @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[6160,8881]) ).
thf(34304,plain,
( ( ( minus_minus_nat @ zero_zero_nat @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ k @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( p @ zero_zero_nat ) @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[34303]) ).
thf(34305,plain,
( ( ( minus_minus_nat @ zero_zero_nat @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ k @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( p @ zero_zero_nat ) @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[34304]) ).
thf(60,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_mult__is__0) ).
thf(371,plain,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(50,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_47_bot__nat__0_Oextremum) ).
thf(343,plain,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(17,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
= ( ( ord_less_nat @ zero_zero_nat @ A )
& ( ord_less_nat @ zero_zero_nat @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_nat__0__less__mult__iff) ).
thf(244,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
= ( ( ord_less_nat @ zero_zero_nat @ A )
& ( ord_less_nat @ zero_zero_nat @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(180,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( ord_less_nat @ A @ B )
& ( ord_less_nat @ C @ D ) )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_add__mono__thms__linordered__field_I5_J) ).
thf(792,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( ord_less_nat @ A @ B )
& ( ord_less_nat @ C @ D ) )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(48035,plain,
! [C: nat,B: nat,A: nat] :
( ( ( sk1 @ C @ A )
= B )
| ( ( plus_plus_nat @ A @ zero_zero_nat )
!= ( plus_plus_nat @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[303,48022]) ).
thf(48036,plain,
! [A: nat] :
( ( sk1 @ zero_zero_nat @ A )
= A ),
inference(pattern_uni,[status(thm)],[48035:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( zero_zero_nat ))]]) ).
thf(106,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_103_diff__less) ).
thf(524,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)],[106]) ).
thf(525,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ! [B: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( minus_minus_nat @ B @ A ) @ B ) ) ),
inference(miniscope,[status(thm)],[524]) ).
thf(526,plain,
! [B: nat,A: 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(cnf,[status(esa)],[525]) ).
thf(27247,plain,
! [B: nat,A: nat] :
( ~ ( ( ( minus_minus_nat @ A @ zero_zero_nat )
!= zero_zero_nat ) )
| ~ ( ( ( minus_minus_nat @ B @ zero_zero_nat )
!= zero_zero_nat ) )
| ( ( minus_minus_nat @ B @ ( minus_minus_nat @ B @ A ) )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[526,8881]) ).
thf(27248,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ zero_zero_nat )
= zero_zero_nat )
| ( ( minus_minus_nat @ B @ zero_zero_nat )
= zero_zero_nat )
| ( ( minus_minus_nat @ B @ ( minus_minus_nat @ B @ A ) )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[27247]) ).
thf(27249,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ zero_zero_nat )
= zero_zero_nat )
| ( ( minus_minus_nat @ B @ zero_zero_nat )
= zero_zero_nat )
| ( ( minus_minus_nat @ B @ ( minus_minus_nat @ B @ A ) )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[27248]) ).
thf(171,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_168_add_Ocomm__neutral) ).
thf(765,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(108,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_105_diff__less__mono) ).
thf(530,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)],[108]) ).
thf(31518,plain,
! [C: nat,B: nat,A: nat] :
( ( A != i )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( minus_minus_nat @ n @ ( plus_plus_nat @ C @ k ) ) ) ),
inference(paramod_ordered,[status(thm)],[324,19291]) ).
thf(31554,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ i @ A )
!= n )
| ( A
!= ( plus_plus_nat @ B @ k ) ) ),
inference(simp,[status(thm)],[31518]) ).
thf(31583,plain,
! [A: nat] :
( ( plus_plus_nat @ i @ ( plus_plus_nat @ A @ k ) )
!= n ),
inference(simp,[status(thm)],[31554]) ).
thf(2859,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ zero_zero_nat @ i ) ) ),
inference(paramod_ordered,[status(thm)],[2147,2849]) ).
thf(2860,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ zero_zero_nat @ i ) ),
inference(simp,[status(thm)],[2859]) ).
thf(2888,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
!= zero_zero_nat )
| ( B != i ) ),
inference(simp,[status(thm)],[2860]) ).
thf(2909,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ i )
!= zero_zero_nat ),
inference(simp,[status(thm)],[2888]) ).
thf(174,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
=> ( ord_less_nat @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_171_add__less__imp__less__left) ).
thf(773,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
=> ( ord_less_nat @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(181,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_bot__nat__0_Oextremum__strict) ).
thf(794,plain,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(21612,plain,
! [A: nat] :
( ( n != A )
| ( ( plus_plus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ A )
!= ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) ) ) ),
inference(paramod_ordered,[status(thm)],[21192,8472]) ).
thf(21799,plain,
( ( plus_plus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ n )
!= ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) ) ),
inference(simp,[status(thm)],[21612]) ).
thf(54979,plain,
( ( plus_plus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ n )
!= n ),
inference(rewrite,[status(thm)],[21799,21192]) ).
thf(77,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= C )
=> ( A
= ( minus_minus_nat @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_74_add__implies__diff) ).
thf(425,plain,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= C )
=> ( A
= ( minus_minus_nat @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(186,axiom,
! [A: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ A ) )
= ( A = zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_not__gr0) ).
thf(824,plain,
! [A: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ A ) )
= ( A = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(48944,plain,
! [B: nat,A: nat] :
( ( ( sk2 @ A @ n )
= B )
| ( ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[21192,48937]) ).
thf(48945,plain,
( ( sk2 @ k @ n )
= ( minus_minus_nat @ n @ k ) ),
inference(pattern_uni,[status(thm)],[48944:[bind(A,$thf( k )),bind(B,$thf( minus_minus_nat @ n @ k ))]]) ).
thf(6536,plain,
! [B: nat,A: nat] :
( ( B != i )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( minus_minus_nat @ n @ k ) ) ),
inference(paramod_ordered,[status(thm)],[330,1013]) ).
thf(6567,plain,
! [A: nat] :
( ( ( plus_plus_nat @ A @ i )
!= n )
| ( A != k ) ),
inference(simp,[status(thm)],[6536]) ).
thf(6578,plain,
( ( plus_plus_nat @ k @ i )
!= n ),
inference(simp,[status(thm)],[6567]) ).
thf(122,axiom,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( ord_less_eq_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_nat__le__add__iff2) ).
thf(580,plain,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( ord_less_eq_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(581,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( ord_less_eq_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) ) ),
inference(miniscope,[status(thm)],[580]) ).
thf(582,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( ord_less_eq_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) ) ),
inference(cnf,[status(esa)],[581]) ).
thf(583,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( ord_less_eq_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[582]) ).
thf(40119,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= zero_zero_nat )
= ( ( minus_minus_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) )
= zero_zero_nat ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[583,406]) ).
thf(40120,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= zero_zero_nat )
= ( ( minus_minus_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) )
= zero_zero_nat ) ) ),
inference(lifteq,[status(thm)],[40119]) ).
thf(16094,plain,
! [A: nat] :
( ~ $true
| ( ( number1551313001itions @ p @ n )
!= ( number1551313001itions
@ ^ [B: nat] : B
@ A ) ) ),
inference(paramod_ordered,[status(thm)],[217,15318]) ).
thf(16095,plain,
! [A: nat] :
( ( number1551313001itions @ p @ n )
!= ( number1551313001itions
@ ^ [B: nat] : B
@ A ) ),
inference(simp,[status(thm)],[16094]) ).
thf(16099,plain,
! [A: nat] :
( ( p
!= ( ^ [B: nat] : B ) )
| ( n != A ) ),
inference(simp,[status(thm)],[16095]) ).
thf(16100,plain,
( p
!= ( ^ [A: nat] : A ) ),
inference(simp,[status(thm)],[16099]) ).
thf(2631,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ zero_zero_nat @ ( p @ k ) ) ) ),
inference(paramod_ordered,[status(thm)],[224,2618]) ).
thf(2632,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ zero_zero_nat @ ( p @ k ) ) ),
inference(simp,[status(thm)],[2631]) ).
thf(2651,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
!= zero_zero_nat )
| ( A
!= ( p @ k ) ) ),
inference(simp,[status(thm)],[2632]) ).
thf(2675,plain,
! [A: nat] :
( ( plus_plus_nat @ ( p @ k ) @ A )
!= zero_zero_nat ),
inference(simp,[status(thm)],[2651]) ).
thf(2715,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( zero_zero_nat != zero_zero_nat )
| ( A
!= ( plus_plus_nat @ ( p @ k ) @ B ) ) ),
inference(paramod_ordered,[status(thm)],[735,2675]) ).
thf(2716,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( A
!= ( plus_plus_nat @ ( p @ k ) @ B ) ) ),
inference(simp,[status(thm)],[2715]) ).
thf(2717,plain,
! [A: nat] : ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ ( p @ k ) @ A ) ),
inference(pattern_uni,[status(thm)],[2716:[bind(A,$thf( plus_plus_nat @ ( p @ k ) @ D )),bind(B,$thf( D ))]]) ).
thf(2727,plain,
! [A: nat] : ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ ( p @ k ) @ A ) ),
inference(simp,[status(thm)],[2717]) ).
thf(3768,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( p @ k ) )
| ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(rewrite,[status(thm)],[2727,252]) ).
thf(3769,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( p @ k ) )
| ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(cnf,[status(esa)],[3768]) ).
thf(46355,plain,
! [A: nat] :
( ( ( minus_minus_nat @ ( p @ k ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[3769,8881]) ).
thf(46356,plain,
! [A: nat] :
( ( ( minus_minus_nat @ ( p @ k ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[46355]) ).
thf(46357,plain,
! [A: nat] :
( ( ( minus_minus_nat @ ( p @ k ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[46356]) ).
thf(165,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C: nat] :
( B
!= ( plus_plus_nat @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_162_less__eqE) ).
thf(745,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C: nat] :
( B
!= ( plus_plus_nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(2085,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( ord_less_nat @ ( minus_minus_nat @ n @ k ) @ zero_zero_nat )
| ( A != i ) ),
inference(paramod_ordered,[status(thm)],[735,228]) ).
thf(2086,plain,
( ( ord_less_nat @ zero_zero_nat @ i )
| ( ord_less_nat @ ( minus_minus_nat @ n @ k ) @ zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[2085:[bind(A,$thf( i ))]]) ).
thf(12702,plain,
( ( ( minus_minus_nat @ i @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ zero_zero_nat @ ( minus_minus_nat @ n @ k ) )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[2086,8881]) ).
thf(12703,plain,
( ( ( minus_minus_nat @ i @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ zero_zero_nat @ ( minus_minus_nat @ n @ k ) )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[12702]) ).
thf(12704,plain,
( ( ( minus_minus_nat @ i @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ zero_zero_nat @ ( minus_minus_nat @ n @ k ) )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[12703]) ).
thf(123,axiom,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_nat__le__add__iff1) ).
thf(584,plain,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(42690,plain,
! [A: nat] :
( ( ( minus_minus_nat @ k @ i )
!= k )
| ( ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ k ) )
!= ( minus_minus_nat @ n @ A ) ) ),
inference(paramod_ordered,[status(thm)],[7759,35845]) ).
thf(42691,plain,
( ( minus_minus_nat @ k @ i )
!= k ),
inference(pattern_uni,[status(thm)],[42690:[bind(A,$thf( minus_minus_nat @ n @ k ))]]) ).
thf(12619,plain,
! [A: nat] :
( ( zero_zero_nat != zero_zero_nat )
| ( ( minus_minus_nat @ ( plus_plus_nat @ i @ A ) @ A )
!= ( minus_minus_nat @ i @ n ) ) ),
inference(paramod_ordered,[status(thm)],[9073,8933]) ).
thf(12620,plain,
! [A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ i @ A ) @ A )
!= ( minus_minus_nat @ i @ n ) ),
inference(simp,[status(thm)],[12619]) ).
thf(12649,plain,
! [A: nat] :
( ( ( plus_plus_nat @ i @ A )
!= i )
| ( A != n ) ),
inference(simp,[status(thm)],[12620]) ).
thf(12663,plain,
( ( plus_plus_nat @ i @ n )
!= i ),
inference(simp,[status(thm)],[12649]) ).
thf(8848,plain,
( ( n != i )
| ( ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ i ) )
!= ( minus_minus_nat @ n @ i ) ) ),
inference(paramod_ordered,[status(thm)],[7752,8837]) ).
thf(8853,plain,
( ( n != i )
| ( n != n )
| ( ( minus_minus_nat @ n @ i )
!= i ) ),
inference(simp,[status(thm)],[8848]) ).
thf(8872,plain,
( ( n != i )
| ( ( minus_minus_nat @ n @ i )
!= i ) ),
inference(simp,[status(thm)],[8853]) ).
thf(531,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ! [C: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ),
inference(miniscope,[status(thm)],[530]) ).
thf(532,plain,
! [C: nat,B: nat,A: 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(cnf,[status(esa)],[531]) ).
thf(28583,plain,
! [C: nat,B: nat,A: nat] :
( ~ ( ( ( minus_minus_nat @ B @ A )
!= zero_zero_nat ) )
| ( ( minus_minus_nat @ C @ A )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( minus_minus_nat @ B @ C ) @ ( minus_minus_nat @ A @ C ) )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[532,8881,406]) ).
thf(28584,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ A )
= zero_zero_nat )
| ( ( minus_minus_nat @ C @ A )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( minus_minus_nat @ B @ C ) @ ( minus_minus_nat @ A @ C ) )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[28583]) ).
thf(28585,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ A )
= zero_zero_nat )
| ( ( minus_minus_nat @ C @ A )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( minus_minus_nat @ B @ C ) @ ( minus_minus_nat @ A @ C ) )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[28584]) ).
thf(293,plain,
! [B: nat,A: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ),
inference(cnf,[status(esa)],[292]) ).
thf(294,plain,
! [B: nat,A: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ),
inference(lifteq,[status(thm)],[293]) ).
thf(21734,plain,
! [B: nat,A: nat] :
( ( ( A = n )
= ( B = zero_zero_nat ) )
| ( ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) )
!= ( plus_plus_nat @ B @ A ) ) ),
inference(paramod_ordered,[status(thm)],[21192,294]) ).
thf(21735,plain,
( ( ( minus_minus_nat @ n @ k )
= n )
= ( k = zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[21734:[bind(A,$thf( minus_minus_nat @ n @ k )),bind(B,$thf( k ))]]) ).
thf(40,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_zero__diff) ).
thf(313,plain,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(2083,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( ord_less_nat @ ( minus_minus_nat @ n @ zero_zero_nat ) @ i )
| ( A != k ) ),
inference(paramod_ordered,[status(thm)],[735,228]) ).
thf(2084,plain,
( ( ord_less_nat @ zero_zero_nat @ k )
| ( ord_less_nat @ ( minus_minus_nat @ n @ zero_zero_nat ) @ i ) ),
inference(pattern_uni,[status(thm)],[2083:[bind(A,$thf( k ))]]) ).
thf(12065,plain,
( ( ( minus_minus_nat @ k @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ ( minus_minus_nat @ n @ zero_zero_nat ) )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[2084,8881]) ).
thf(12066,plain,
( ( ( minus_minus_nat @ k @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ ( minus_minus_nat @ n @ zero_zero_nat ) )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[12065]) ).
thf(12067,plain,
( ( ( minus_minus_nat @ k @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ ( minus_minus_nat @ n @ zero_zero_nat ) )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[12066]) ).
thf(99,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2) ).
thf(498,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(48041,plain,
! [C: nat,B: nat,A: nat] :
( ( ( sk1 @ C @ A )
= B )
| ( ( plus_plus_nat @ zero_zero_nat @ A )
!= ( plus_plus_nat @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[306,48022]) ).
thf(48042,plain,
! [A: nat] :
( ( sk1 @ A @ A )
= zero_zero_nat ),
inference(pattern_uni,[status(thm)],[48041:[bind(A,$thf( A )),bind(B,$thf( zero_zero_nat )),bind(C,$thf( A ))]]) ).
thf(53,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50_add__is__0) ).
thf(350,plain,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(35866,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( B != n )
| ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
!= ( plus_plus_nat @ C @ ( plus_plus_nat @ k @ i ) ) ) ),
inference(paramod_ordered,[status(thm)],[26415,29916]) ).
thf(35867,plain,
! [A: nat] :
( ( ( minus_minus_nat @ ( plus_plus_nat @ k @ i ) @ A )
!= zero_zero_nat )
| ( A != n ) ),
inference(pattern_uni,[status(thm)],[35866:[bind(A,$thf( plus_plus_nat @ k @ i )),bind(B,$thf( D )),bind(C,$thf( minus_minus_nat @ D @ ( plus_plus_nat @ k @ i ) ))]]) ).
thf(35900,plain,
( ( minus_minus_nat @ ( plus_plus_nat @ k @ i ) @ n )
!= zero_zero_nat ),
inference(simp,[status(thm)],[35867]) ).
thf(35956,plain,
! [A: nat] :
( ( zero_zero_nat != zero_zero_nat )
| ( ( minus_minus_nat @ ( p @ A ) @ n )
!= ( minus_minus_nat @ ( plus_plus_nat @ k @ i ) @ n ) ) ),
inference(paramod_ordered,[status(thm)],[9171,35900]) ).
thf(35957,plain,
! [A: nat] :
( ( minus_minus_nat @ ( p @ A ) @ n )
!= ( minus_minus_nat @ ( plus_plus_nat @ k @ i ) @ n ) ),
inference(simp,[status(thm)],[35956]) ).
thf(36004,plain,
! [A: nat] :
( ( ( p @ A )
!= ( plus_plus_nat @ k @ i ) )
| ( n != n ) ),
inference(simp,[status(thm)],[35957]) ).
thf(36036,plain,
! [A: nat] :
( ( p @ A )
!= ( plus_plus_nat @ k @ i ) ),
inference(simp,[status(thm)],[36004]) ).
thf(147,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( B != A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_144_less__not__refl2) ).
thf(681,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( B != A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(27,axiom,
! [A: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ A ) )
= ( A = zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_not__gr__zero) ).
thf(274,plain,
! [A: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ A ) )
= ( A = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(178,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( ord_less_nat @ A @ B )
& ( C = D ) )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_add__mono__thms__linordered__field_I1_J) ).
thf(784,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( ord_less_nat @ A @ B )
& ( C = D ) )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(161,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_158_add__le__imp__le__right) ).
thf(736,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(198,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_195_less__imp__le__nat) ).
thf(863,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(127,axiom,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_nat__less__add__iff1) ).
thf(600,plain,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(601,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat,D: nat,E: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
inference(miniscope,[status(thm)],[600]) ).
thf(602,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
inference(cnf,[status(esa)],[601]) ).
thf(603,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) ) )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[602]) ).
thf(46566,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( ( minus_minus_nat @ E @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) )
!= zero_zero_nat )
= ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) )
!= zero_zero_nat ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[603,8881,406]) ).
thf(46567,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( ( minus_minus_nat @ E @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) )
= zero_zero_nat )
= ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) )
= zero_zero_nat ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[46566]) ).
thf(46568,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( ( minus_minus_nat @ E @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) )
= zero_zero_nat )
= ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) )
= zero_zero_nat ) ) ),
inference(lifteq,[status(thm)],[46567]) ).
thf(2625,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ ( minus_minus_nat @ n @ k ) @ i ) ) ),
inference(paramod_ordered,[status(thm)],[228,2618]) ).
thf(2626,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ ( minus_minus_nat @ n @ k ) @ i ) ),
inference(simp,[status(thm)],[2625]) ).
thf(2654,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
!= ( minus_minus_nat @ n @ k ) )
| ( A != i ) ),
inference(simp,[status(thm)],[2626]) ).
thf(2679,plain,
! [A: nat] :
( ( plus_plus_nat @ i @ A )
!= ( minus_minus_nat @ n @ k ) ),
inference(simp,[status(thm)],[2654]) ).
thf(6257,plain,
! [B: nat,A: nat] :
( ( ( p @ zero_zero_nat )
!= k )
| ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ zero_zero_nat @ k ) ) ),
inference(paramod_ordered,[status(thm)],[6212,2849]) ).
thf(6258,plain,
! [B: nat,A: nat] :
( ( ( p @ zero_zero_nat )
!= k )
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ zero_zero_nat @ k ) ) ),
inference(simp,[status(thm)],[6257]) ).
thf(6286,plain,
! [B: nat,A: nat] :
( ( ( p @ zero_zero_nat )
!= k )
| ( ( plus_plus_nat @ A @ B )
!= zero_zero_nat )
| ( B != k ) ),
inference(simp,[status(thm)],[6258]) ).
thf(6308,plain,
! [A: nat] :
( ( ( p @ zero_zero_nat )
!= k )
| ( ( plus_plus_nat @ A @ k )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[6286]) ).
thf(22035,plain,
! [B: nat,A: nat] :
( ( ( A = n )
= ( B = zero_zero_nat ) )
| ( ( plus_plus_nat @ i @ ( minus_minus_nat @ n @ i ) )
!= ( plus_plus_nat @ B @ A ) ) ),
inference(paramod_ordered,[status(thm)],[21232,294]) ).
thf(22036,plain,
( ( ( minus_minus_nat @ n @ i )
= n )
= ( i = zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[22035:[bind(A,$thf( minus_minus_nat @ n @ i )),bind(B,$thf( i ))]]) ).
thf(57,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ C @ B ) )
= ( ( A = C )
| ( B = zero_zero_nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_mult__cancel2) ).
thf(362,plain,
! [A: nat,B: nat,C: nat] :
( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ C @ B ) )
= ( ( A = C )
| ( B = zero_zero_nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(204,axiom,
! [A: nat,B: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_not__add__less2) ).
thf(880,plain,
! [A: nat,B: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(648,plain,
! [C: nat,B: nat,A: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ),
inference(cnf,[status(esa)],[647]) ).
thf(649,plain,
! [C: nat,B: nat,A: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ),
inference(lifteq,[status(thm)],[648]) ).
thf(48656,plain,
! [B: nat,A: nat] :
( ( n = k )
| ( ( sk1 @ B @ n )
= A )
| ( ( plus_plus_nat @ k @ sk17 )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[48292,48022]) ).
thf(48657,plain,
( ( n = k )
| ( ( sk1 @ sk17 @ n )
= k ) ),
inference(pattern_uni,[status(thm)],[48656:[bind(A,$thf( k )),bind(B,$thf( sk17 ))]]) ).
thf(158,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( B != zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_155_gr__implies__not__zero) ).
thf(726,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( B != zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(230,plain,
! [A: nat > nat] :
( ( number1551313001itions @ A @ zero_zero_nat )
= ( A
= ( ^ [B: nat] : zero_zero_nat ) ) ),
inference(cnf,[status(esa)],[229]) ).
thf(231,plain,
! [A: nat > nat] :
( ( number1551313001itions @ A @ zero_zero_nat )
= ( A
= ( ^ [B: nat] : zero_zero_nat ) ) ),
inference(lifteq,[status(thm)],[230]) ).
thf(16092,plain,
! [B: nat,A: nat > nat] :
( ( A
!= ( ^ [C: nat] : zero_zero_nat ) )
| ( ( number1551313001itions @ A @ zero_zero_nat )
!= ( number1551313001itions
@ ^ [C: nat] : C
@ B ) ) ),
inference(paramod_ordered,[status(thm)],[231,15318]) ).
thf(16093,plain,
( ( ^ [A: nat] : A )
!= ( ^ [A: nat] : zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[16092:[bind(A,$thf( ^ [C: nat] : C )),bind(B,$thf( zero_zero_nat ))]]) ).
thf(16098,plain,
( ( ^ [A: nat] : A )
!= ( ^ [A: nat] : zero_zero_nat ) ),
inference(lifteq,[status(thm)],[16093]) ).
thf(16101,plain,
sk11 != zero_zero_nat,
inference(func_ext,[status(esa)],[16098]) ).
thf(93,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
= ( minus_minus_nat @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_Nat_Odiff__cancel) ).
thf(477,plain,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
= ( minus_minus_nat @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(534,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat] :
( ( 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(miniscope,[status(thm)],[533]) ).
thf(535,plain,
! [C: nat,B: nat,A: 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(cnf,[status(esa)],[534]) ).
thf(536,plain,
! [C: nat,B: nat,A: nat] :
( ( ( ord_less_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ C @ A ) )
= ( ord_less_nat @ B @ C ) )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ A @ C ) ),
inference(lifteq,[status(thm)],[535]) ).
thf(29172,plain,
! [C: nat,B: nat,A: nat] :
( ( ( ( minus_minus_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ B @ A ) )
!= zero_zero_nat )
= ( ( minus_minus_nat @ C @ B )
!= zero_zero_nat ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ C )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[536,8881,406]) ).
thf(29173,plain,
! [C: nat,B: nat,A: nat] :
( ( ( ( minus_minus_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ B @ A ) )
= zero_zero_nat )
= ( ( minus_minus_nat @ C @ B )
= zero_zero_nat ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ C )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[29172]) ).
thf(29174,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ C )
!= zero_zero_nat )
| ( ( ( minus_minus_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ B @ A ) )
= zero_zero_nat )
= ( ( minus_minus_nat @ C @ B )
= zero_zero_nat ) ) ),
inference(lifteq,[status(thm)],[29173]) ).
thf(98,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right) ).
thf(494,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(84,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_81_le__diff__iff_H) ).
thf(447,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)],[84]) ).
thf(448,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat] :
( ( 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(miniscope,[status(thm)],[447]) ).
thf(449,plain,
! [C: nat,B: nat,A: 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(cnf,[status(esa)],[448]) ).
thf(450,plain,
! [C: nat,B: nat,A: nat] :
( ( ( ord_less_eq_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ B @ C ) )
= ( ord_less_eq_nat @ C @ A ) )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ C @ B ) ),
inference(lifteq,[status(thm)],[449]) ).
thf(13498,plain,
! [C: nat,B: nat,A: nat] :
( ( ( ( minus_minus_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ B @ C ) )
= zero_zero_nat )
= ( ( minus_minus_nat @ C @ A )
= zero_zero_nat ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ C @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[450,406]) ).
thf(13499,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ C @ B )
!= zero_zero_nat )
| ( ( ( minus_minus_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ B @ C ) )
= zero_zero_nat )
= ( ( minus_minus_nat @ C @ A )
= zero_zero_nat ) ) ),
inference(lifteq,[status(thm)],[13498]) ).
thf(51,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_48_le0) ).
thf(345,plain,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(366,plain,
! [C: nat,B: nat,A: nat] :
( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C ) )
= ( ( B = C )
| ( A = zero_zero_nat ) ) ),
inference(cnf,[status(esa)],[365]) ).
thf(367,plain,
! [C: nat,B: nat,A: nat] :
( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C ) )
= ( ( B = C )
| ( A = zero_zero_nat ) ) ),
inference(lifteq,[status(thm)],[366]) ).
thf(5955,plain,
! [B: nat,A: nat] :
( ( A != i )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( minus_minus_nat @ n @ k ) ) ),
inference(paramod_ordered,[status(thm)],[324,1013]) ).
thf(5977,plain,
! [A: nat] :
( ( ( plus_plus_nat @ i @ A )
!= n )
| ( A != k ) ),
inference(simp,[status(thm)],[5955]) ).
thf(5994,plain,
( ( plus_plus_nat @ i @ k )
!= n ),
inference(simp,[status(thm)],[5977]) ).
thf(1269,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ n @ zero_zero_nat ) @ i )
| ( A != k ) ),
inference(paramod_ordered,[status(thm)],[813,228]) ).
thf(1270,plain,
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ n @ zero_zero_nat ) @ i ) ),
inference(pattern_uni,[status(thm)],[1269:[bind(A,$thf( k ))]]) ).
thf(28926,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ ( minus_minus_nat @ n @ zero_zero_nat ) )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[1270,8881]) ).
thf(28927,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ ( minus_minus_nat @ n @ zero_zero_nat ) )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[28926]) ).
thf(28928,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ ( minus_minus_nat @ n @ zero_zero_nat ) )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[28927]) ).
thf(7976,plain,
! [A: nat] :
( ( A = k )
| ( ( minus_minus_nat @ A @ zero_zero_nat )
!= ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ k ) ) ) ),
inference(paramod_ordered,[status(thm)],[312,7759]) ).
thf(8026,plain,
! [A: nat] :
( ( A = k )
| ( A != n )
| ( ( minus_minus_nat @ n @ k )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[7976]) ).
thf(8052,plain,
( ( n = k )
| ( ( minus_minus_nat @ n @ k )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[8026]) ).
thf(205,axiom,
! [A: nat,B: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_not__add__less1) ).
thf(883,plain,
! [A: nat,B: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(266,plain,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ C @ B ) )
= ( A = C ) ),
inference(cnf,[status(esa)],[265]) ).
thf(267,plain,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ C @ B ) )
= ( A = C ) ),
inference(lifteq,[status(thm)],[266]) ).
thf(81,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_78_less__imp__diff__less) ).
thf(438,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)],[81]) ).
thf(11461,plain,
! [B: nat,A: nat] :
( ( zero_zero_nat != zero_zero_nat )
| ( ( minus_minus_nat @ ( p @ A ) @ n )
!= ( minus_minus_nat @ ( plus_plus_nat @ n @ B ) @ B ) ) ),
inference(paramod_ordered,[status(thm)],[9171,8928]) ).
thf(11462,plain,
! [B: nat,A: nat] :
( ( minus_minus_nat @ ( p @ A ) @ n )
!= ( minus_minus_nat @ ( plus_plus_nat @ n @ B ) @ B ) ),
inference(simp,[status(thm)],[11461]) ).
thf(11490,plain,
! [B: nat,A: nat] :
( ( ( p @ A )
!= ( plus_plus_nat @ n @ B ) )
| ( n != B ) ),
inference(simp,[status(thm)],[11462]) ).
thf(11493,plain,
! [A: nat] :
( ( p @ A )
!= ( plus_plus_nat @ n @ n ) ),
inference(simp,[status(thm)],[11490]) ).
thf(314,plain,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ),
inference(cnf,[status(esa)],[313]) ).
thf(315,plain,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ),
inference(lifteq,[status(thm)],[314]) ).
thf(8142,plain,
! [D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= zero_zero_nat )
| ( ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) )
!= ( minus_minus_nat @ zero_zero_nat @ D ) ) ),
inference(paramod_ordered,[status(thm)],[382,315]) ).
thf(8143,plain,
! [B: nat,A: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ zero_zero_nat @ A ) @ B )
= zero_zero_nat ),
inference(pattern_uni,[status(thm)],[8142:[bind(A,$thf( zero_zero_nat )),bind(B,$thf( E )),bind(C,$thf( F )),bind(D,$thf( plus_plus_nat @ E @ F ))]]) ).
thf(8237,plain,
! [B: nat,A: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ zero_zero_nat @ A ) @ B )
= zero_zero_nat ),
inference(simp,[status(thm)],[8143]) ).
thf(8280,plain,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ),
inference(rewrite,[status(thm)],[8237,315]) ).
thf(148,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_less__not__refl) ).
thf(685,plain,
! [A: nat] :
~ ( ord_less_nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(48547,plain,
( ( n = k )
| ( n != n )
| ( ( plus_plus_nat @ k @ sk17 )
!= ( plus_plus_nat @ k @ n ) ) ),
inference(paramod_ordered,[status(thm)],[48292,9455]) ).
thf(48548,plain,
( ( n = k )
| ( ( plus_plus_nat @ k @ sk17 )
!= ( plus_plus_nat @ k @ n ) ) ),
inference(simp,[status(thm)],[48547]) ).
thf(48794,plain,
( ( n = k )
| ( k != k )
| ( sk17 != n ) ),
inference(simp,[status(thm)],[48548]) ).
thf(48904,plain,
( ( n = k )
| ( sk17 != n ) ),
inference(simp,[status(thm)],[48794]) ).
thf(187,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_184_gr0I) ).
thf(827,plain,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(1613,plain,
! [A: nat] :
( ( A != i )
| ( ( minus_minus_nat @ A @ zero_zero_nat )
!= ( minus_minus_nat @ n @ k ) ) ),
inference(paramod_ordered,[status(thm)],[312,1013]) ).
thf(1623,plain,
( ( n != i )
| ( k != zero_zero_nat ) ),
inference(simp,[status(thm)],[1613]) ).
thf(1959,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( n != i )
| ( zero_zero_nat != zero_zero_nat )
| ( A != k ) ),
inference(paramod_ordered,[status(thm)],[735,1623]) ).
thf(1960,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( n != i )
| ( A != k ) ),
inference(simp,[status(thm)],[1959]) ).
thf(1961,plain,
( ( ord_less_nat @ zero_zero_nat @ k )
| ( n != i ) ),
inference(pattern_uni,[status(thm)],[1960:[bind(A,$thf( k ))]]) ).
thf(2793,plain,
! [B: nat,A: nat] :
( ( n != i )
| ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ zero_zero_nat @ k ) ) ),
inference(paramod_ordered,[status(thm)],[1961,2618]) ).
thf(2794,plain,
! [B: nat,A: nat] :
( ( n != i )
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ zero_zero_nat @ k ) ) ),
inference(simp,[status(thm)],[2793]) ).
thf(2810,plain,
! [B: nat,A: nat] :
( ( n != i )
| ( ( plus_plus_nat @ A @ B )
!= zero_zero_nat )
| ( A != k ) ),
inference(simp,[status(thm)],[2794]) ).
thf(2835,plain,
! [A: nat] :
( ( n != i )
| ( ( plus_plus_nat @ k @ A )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[2810]) ).
thf(121,axiom,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_nat__diff__add__eq1) ).
thf(576,plain,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(577,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat,D: nat,E: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
inference(miniscope,[status(thm)],[576]) ).
thf(578,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
inference(cnf,[status(esa)],[577]) ).
thf(579,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E ) )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[578]) ).
thf(37697,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( times_times_nat @ B @ C ) @ ( times_times_nat @ A @ C ) ) @ D ) @ E ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[579,485,406]) ).
thf(37698,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( times_times_nat @ B @ C ) @ ( times_times_nat @ A @ C ) ) @ D ) @ E ) ) ),
inference(lifteq,[status(thm)],[37697]) ).
thf(172,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_comm__monoid__add__class_Oadd__0) ).
thf(768,plain,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(439,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ! [C: nat] : ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ B ) ),
inference(miniscope,[status(thm)],[438]) ).
thf(440,plain,
! [C: nat,B: nat,A: nat] :
( ~ ( ord_less_nat @ A @ B )
| ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ B ) ),
inference(cnf,[status(esa)],[439]) ).
thf(11872,plain,
! [C: nat,B: nat,A: nat] :
( ~ ( ( ( minus_minus_nat @ B @ A )
!= zero_zero_nat ) )
| ( ( minus_minus_nat @ B @ ( minus_minus_nat @ A @ C ) )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[440,8881]) ).
thf(11873,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ A )
= zero_zero_nat )
| ( ( minus_minus_nat @ B @ ( minus_minus_nat @ A @ C ) )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[11872]) ).
thf(11874,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ A )
= zero_zero_nat )
| ( ( minus_minus_nat @ B @ ( minus_minus_nat @ A @ C ) )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[11873]) ).
thf(16,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ C @ B ) )
= ( ( ord_less_nat @ zero_zero_nat @ B )
& ( ord_less_nat @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_mult__less__cancel2) ).
thf(241,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ C @ B ) )
= ( ( ord_less_nat @ zero_zero_nat @ B )
& ( ord_less_nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(142,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_139_linorder__neqE__nat) ).
thf(653,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)],[142]) ).
thf(3062,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
| ( ( ord_less_nat @ zero_zero_nat @ B )
!= ( ord_less_nat @ zero_zero_nat @ i ) ) ),
inference(paramod_ordered,[status(thm)],[261,2147]) ).
thf(3063,plain,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ i ) ),
inference(pattern_uni,[status(thm)],[3062:[bind(A,$thf( A )),bind(B,$thf( i ))]]) ).
thf(4307,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ i ) )
!= ( ord_less_nat @ B @ B ) ) ),
inference(paramod_ordered,[status(thm)],[3063,676]) ).
thf(4308,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ i ) )
!= ( ord_less_nat @ B @ B ) ),
inference(simp,[status(thm)],[4307]) ).
thf(4313,plain,
! [B: nat,A: nat] :
( ( A != B )
| ( ( plus_plus_nat @ A @ i )
!= B ) ),
inference(simp,[status(thm)],[4308]) ).
thf(4331,plain,
! [A: nat] :
( ( plus_plus_nat @ A @ i )
!= A ),
inference(simp,[status(thm)],[4313]) ).
thf(61,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
= ( ord_less_eq_nat @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_nat__add__left__cancel__le) ).
thf(374,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
= ( ord_less_eq_nat @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(166,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_163_add__left__mono) ).
thf(749,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(21727,plain,
! [B: nat,A: nat] :
( ( ( n = B )
= ( A = zero_zero_nat ) )
| ( ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[21192,300]) ).
thf(21728,plain,
( ( n
= ( minus_minus_nat @ n @ k ) )
= ( k = zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[21727:[bind(A,$thf( k )),bind(B,$thf( minus_minus_nat @ n @ k ))]]) ).
thf(190,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_le__0__eq) ).
thf(836,plain,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(29853,plain,
! [A: nat] :
( ( ( plus_plus_nat @ k @ i )
!= k )
| ( ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ k ) )
!= ( minus_minus_nat @ n @ A ) ) ),
inference(paramod_ordered,[status(thm)],[7759,19268]) ).
thf(29854,plain,
( ( plus_plus_nat @ k @ i )
!= k ),
inference(pattern_uni,[status(thm)],[29853:[bind(A,$thf( minus_minus_nat @ n @ k ))]]) ).
thf(957,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ i )
| ( ( minus_minus_nat @ zero_zero_nat @ A )
!= ( minus_minus_nat @ n @ k ) ) ),
inference(paramod_ordered,[status(thm)],[315,228]) ).
thf(958,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ i )
| ( n != zero_zero_nat )
| ( A != k ) ),
inference(simp,[status(thm)],[957]) ).
thf(959,plain,
( ( ord_less_nat @ zero_zero_nat @ i )
| ( n != zero_zero_nat ) ),
inference(simp,[status(thm)],[958]) ).
thf(1413,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_nat @ zero_zero_nat @ i )
| ( zero_zero_nat != zero_zero_nat )
| ( A != n ) ),
inference(paramod_ordered,[status(thm)],[813,959]) ).
thf(1414,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_nat @ zero_zero_nat @ i )
| ( A != n ) ),
inference(simp,[status(thm)],[1413]) ).
thf(1415,plain,
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_nat @ zero_zero_nat @ i ) ),
inference(pattern_uni,[status(thm)],[1414:[bind(A,$thf( n ))]]) ).
thf(11314,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[1415,8881]) ).
thf(11315,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[11314]) ).
thf(11316,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[11315]) ).
thf(2874,plain,
! [B: nat,A: nat] :
( ( n != i )
| ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ zero_zero_nat @ k ) ) ),
inference(paramod_ordered,[status(thm)],[1961,2849]) ).
thf(2875,plain,
! [B: nat,A: nat] :
( ( n != i )
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ zero_zero_nat @ k ) ) ),
inference(simp,[status(thm)],[2874]) ).
thf(2892,plain,
! [B: nat,A: nat] :
( ( n != i )
| ( ( plus_plus_nat @ A @ B )
!= zero_zero_nat )
| ( B != k ) ),
inference(simp,[status(thm)],[2875]) ).
thf(2914,plain,
! [A: nat] :
( ( n != i )
| ( ( plus_plus_nat @ A @ k )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[2892]) ).
thf(1273,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ zero_zero_nat @ k ) @ i )
| ( A != n ) ),
inference(paramod_ordered,[status(thm)],[813,228]) ).
thf(1274,plain,
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ zero_zero_nat @ k ) @ i ) ),
inference(pattern_uni,[status(thm)],[1273:[bind(A,$thf( n ))]]) ).
thf(31612,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ ( minus_minus_nat @ zero_zero_nat @ k ) )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[1274,8881]) ).
thf(31613,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ ( minus_minus_nat @ zero_zero_nat @ k ) )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[31612]) ).
thf(31614,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ i @ ( minus_minus_nat @ zero_zero_nat @ k ) )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[31613]) ).
thf(153,axiom,
! [A: nat,B: nat] :
( ( A = B )
=> ( ord_less_eq_nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_150_eq__imp__le) ).
thf(712,plain,
! [A: nat,B: nat] :
( ( A = B )
=> ( ord_less_eq_nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(151,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_148_nat__le__linear) ).
thf(707,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)],[151]) ).
thf(326,plain,
! [C: nat,B: nat,A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ C ) ),
inference(cnf,[status(esa)],[325]) ).
thf(327,plain,
! [C: nat,B: nat,A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ C ) ),
inference(lifteq,[status(thm)],[326]) ).
thf(1279,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_eq_nat @ i @ zero_zero_nat )
| ( A != n ) ),
inference(paramod_ordered,[status(thm)],[813,216]) ).
thf(1280,plain,
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_eq_nat @ i @ zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[1279:[bind(A,$thf( n ))]]) ).
thf(9465,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( i = zero_zero_nat ) ),
inference(rewrite,[status(thm)],[1280,273,8881]) ).
thf(9466,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( i = zero_zero_nat ) ),
inference(simp,[status(thm)],[9465]) ).
thf(9467,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( i = zero_zero_nat ) ),
inference(lifteq,[status(thm)],[9466]) ).
thf(196,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_nat @ A @ B )
| ( A = B ) )
=> ( ord_less_eq_nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_less__or__eq__imp__le) ).
thf(855,plain,
! [A: nat,B: nat] :
( ( ( ord_less_nat @ A @ B )
| ( A = B ) )
=> ( ord_less_eq_nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(48551,plain,
! [B: nat,A: nat] :
( ( n = k )
| ( ( minus_minus_nat @ n @ B )
= A )
| ( ( plus_plus_nat @ k @ sk17 )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[48292,324]) ).
thf(48552,plain,
( ( n = k )
| ( ( minus_minus_nat @ n @ sk17 )
= k ) ),
inference(pattern_uni,[status(thm)],[48551:[bind(A,$thf( k )),bind(B,$thf( sk17 ))]]) ).
thf(2628,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ zero_zero_nat @ i ) ) ),
inference(paramod_ordered,[status(thm)],[2147,2618]) ).
thf(2629,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ zero_zero_nat @ i ) ),
inference(simp,[status(thm)],[2628]) ).
thf(2653,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
!= zero_zero_nat )
| ( A != i ) ),
inference(simp,[status(thm)],[2629]) ).
thf(2678,plain,
! [A: nat] :
( ( plus_plus_nat @ i @ A )
!= zero_zero_nat ),
inference(simp,[status(thm)],[2653]) ).
thf(19090,plain,
! [C: nat,B: nat,A: nat] :
( ( A != i )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( minus_minus_nat @ n @ ( plus_plus_nat @ k @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[324,18885]) ).
thf(19118,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ i @ A )
!= n )
| ( A
!= ( plus_plus_nat @ k @ B ) ) ),
inference(simp,[status(thm)],[19090]) ).
thf(19147,plain,
! [A: nat] :
( ( plus_plus_nat @ i @ ( plus_plus_nat @ k @ A ) )
!= n ),
inference(simp,[status(thm)],[19118]) ).
thf(35858,plain,
! [A: nat] :
( ( n != n )
| ( ( plus_plus_nat @ i @ ( minus_minus_nat @ n @ i ) )
!= ( plus_plus_nat @ A @ ( plus_plus_nat @ k @ i ) ) ) ),
inference(paramod_ordered,[status(thm)],[21232,29916]) ).
thf(35859,plain,
! [A: nat] :
( ( plus_plus_nat @ i @ ( minus_minus_nat @ n @ i ) )
!= ( plus_plus_nat @ A @ ( plus_plus_nat @ k @ i ) ) ),
inference(simp,[status(thm)],[35858]) ).
thf(35894,plain,
! [A: nat] :
( ( i != A )
| ( ( plus_plus_nat @ k @ i )
!= ( minus_minus_nat @ n @ i ) ) ),
inference(simp,[status(thm)],[35859]) ).
thf(35916,plain,
( ( plus_plus_nat @ k @ i )
!= ( minus_minus_nat @ n @ i ) ),
inference(simp,[status(thm)],[35894]) ).
thf(126,axiom,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( ord_less_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_nat__less__add__iff2) ).
thf(596,plain,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( ord_less_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(597,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat,D: nat,E: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( ord_less_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) ) ),
inference(miniscope,[status(thm)],[596]) ).
thf(598,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( ord_less_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) ) ),
inference(cnf,[status(esa)],[597]) ).
thf(599,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( ord_less_nat @ D @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[598]) ).
thf(45183,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) )
!= zero_zero_nat )
= ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) @ D )
!= zero_zero_nat ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[599,8881,406]) ).
thf(45184,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) )
= zero_zero_nat )
= ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) @ D )
= zero_zero_nat ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[45183]) ).
thf(45185,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D ) )
= zero_zero_nat )
= ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) @ D )
= zero_zero_nat ) ) ),
inference(lifteq,[status(thm)],[45184]) ).
thf(59,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_mult__0__right) ).
thf(368,plain,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(369,plain,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ),
inference(cnf,[status(esa)],[368]) ).
thf(370,plain,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ),
inference(lifteq,[status(thm)],[369]) ).
thf(100,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_97_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2) ).
thf(502,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(89,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_86_eq__diff__iff) ).
thf(464,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)],[89]) ).
thf(465,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ( minus_minus_nat @ B @ A )
= ( minus_minus_nat @ C @ A ) )
= ( B = C ) ) ) ),
inference(miniscope,[status(thm)],[464]) ).
thf(466,plain,
! [C: nat,B: nat,A: 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(cnf,[status(esa)],[465]) ).
thf(467,plain,
! [C: nat,B: nat,A: nat] :
( ( ( ( minus_minus_nat @ B @ A )
= ( minus_minus_nat @ C @ A ) )
= ( B = C ) )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ A @ C ) ),
inference(lifteq,[status(thm)],[466]) ).
thf(18317,plain,
! [C: nat,B: nat,A: nat] :
( ( ( ( minus_minus_nat @ B @ A )
= ( minus_minus_nat @ C @ A ) )
= ( B = C ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ C )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[467,406]) ).
thf(18318,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ C )
!= zero_zero_nat )
| ( ( ( minus_minus_nat @ B @ A )
= ( minus_minus_nat @ C @ A ) )
= ( B = C ) ) ),
inference(lifteq,[status(thm)],[18317]) ).
thf(56,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_diff__0__eq__0) ).
thf(359,plain,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(19309,plain,
! [B: nat,A: nat] :
( ( B != i )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( minus_minus_nat @ i @ k ) ) ),
inference(paramod_ordered,[status(thm)],[330,19205]) ).
thf(19336,plain,
! [A: nat] :
( ( ( plus_plus_nat @ A @ i )
!= i )
| ( A != k ) ),
inference(simp,[status(thm)],[19309]) ).
thf(19348,plain,
( ( plus_plus_nat @ k @ i )
!= i ),
inference(simp,[status(thm)],[19336]) ).
thf(82,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_79_diff__less__mono2) ).
thf(441,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)],[82]) ).
thf(96,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add) ).
thf(486,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(487,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat] :
( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ),
inference(miniscope,[status(thm)],[486]) ).
thf(488,plain,
! [C: nat,B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ A @ B )
| ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ),
inference(cnf,[status(esa)],[487]) ).
thf(489,plain,
! [C: nat,B: nat,A: nat] :
( ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[488]) ).
thf(490,plain,
! [C: nat,B: nat,A: nat] :
( ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(simp,[status(thm)],[489]) ).
thf(20286,plain,
! [C: nat,B: nat,A: nat] :
( ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[490,406]) ).
thf(20287,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ),
inference(lifteq,[status(thm)],[20286]) ).
thf(16106,plain,
( ( p @ sk12 )
!= sk12 ),
inference(func_ext,[status(esa)],[16100]) ).
thf(48583,plain,
( ( n = k )
| ( n != n )
| ( ( plus_plus_nat @ k @ sk17 )
!= ( plus_plus_nat @ k @ i ) ) ),
inference(paramod_ordered,[status(thm)],[48292,6578]) ).
thf(48584,plain,
( ( n = k )
| ( ( plus_plus_nat @ k @ sk17 )
!= ( plus_plus_nat @ k @ i ) ) ),
inference(simp,[status(thm)],[48583]) ).
thf(48806,plain,
( ( n = k )
| ( k != k )
| ( sk17 != i ) ),
inference(simp,[status(thm)],[48584]) ).
thf(48912,plain,
( ( n = k )
| ( sk17 != i ) ),
inference(simp,[status(thm)],[48806]) ).
thf(125,axiom,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D )
= E ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_nat__eq__add__iff1) ).
thf(592,plain,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D )
= E ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(593,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat,D: nat,E: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D )
= E ) ) ),
inference(miniscope,[status(thm)],[592]) ).
thf(594,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D )
= E ) ) ),
inference(cnf,[status(esa)],[593]) ).
thf(595,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D )
= E ) )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[594]) ).
thf(43790,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( ( plus_plus_nat @ ( minus_minus_nat @ ( times_times_nat @ B @ C ) @ ( times_times_nat @ A @ C ) ) @ D )
= E ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[595,485,406]) ).
thf(43791,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( ( plus_plus_nat @ ( minus_minus_nat @ ( times_times_nat @ B @ C ) @ ( times_times_nat @ A @ C ) ) @ D )
= E ) ) ),
inference(lifteq,[status(thm)],[43790]) ).
thf(65,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_le__add__same__cancel1) ).
thf(386,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(442,plain,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ! [C: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( minus_minus_nat @ C @ B ) @ ( minus_minus_nat @ C @ A ) ) ) ),
inference(miniscope,[status(thm)],[441]) ).
thf(443,plain,
! [C: nat,B: nat,A: 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(cnf,[status(esa)],[442]) ).
thf(12239,plain,
! [C: nat,B: nat,A: nat] :
( ~ ( ( ( minus_minus_nat @ B @ A )
!= zero_zero_nat ) )
| ~ ( ( ( minus_minus_nat @ C @ A )
!= zero_zero_nat ) )
| ( ( minus_minus_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[443,8881]) ).
thf(12240,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ A )
= zero_zero_nat )
| ( ( minus_minus_nat @ C @ A )
= zero_zero_nat )
| ( ( minus_minus_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[12239]) ).
thf(12241,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ B @ A )
= zero_zero_nat )
| ( ( minus_minus_nat @ C @ A )
= zero_zero_nat )
| ( ( minus_minus_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[12240]) ).
thf(1201,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_nat @ zero_zero_nat @ ( p @ zero_zero_nat ) )
| ( A != k ) ),
inference(paramod_ordered,[status(thm)],[813,224]) ).
thf(1202,plain,
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_nat @ zero_zero_nat @ ( p @ zero_zero_nat ) ) ),
inference(pattern_uni,[status(thm)],[1201:[bind(A,$thf( k ))]]) ).
thf(19361,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( p @ zero_zero_nat ) @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[1202,8881]) ).
thf(19362,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( p @ zero_zero_nat ) @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[19361]) ).
thf(19363,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( p @ zero_zero_nat ) @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[19362]) ).
thf(48950,plain,
! [B: nat,A: nat] :
( ( ( sk2 @ A @ n )
= B )
| ( ( plus_plus_nat @ ( minus_minus_nat @ n @ i ) @ i )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[26528,48937]) ).
thf(48951,plain,
( ( sk2 @ ( minus_minus_nat @ n @ i ) @ n )
= i ),
inference(pattern_uni,[status(thm)],[48950:[bind(A,$thf( minus_minus_nat @ n @ i )),bind(B,$thf( i ))]]) ).
thf(20791,plain,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ i @ k )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
!= ( minus_minus_nat @ n @ C ) ) ),
inference(paramod_ordered,[status(thm)],[397,19264]) ).
thf(20792,plain,
( ( plus_plus_nat @ i @ k )
!= zero_zero_nat ),
inference(pattern_uni,[status(thm)],[20791:[bind(A,$thf( n )),bind(B,$thf( E )),bind(C,$thf( plus_plus_nat @ n @ E ))]]) ).
thf(9420,plain,
! [B: nat,A: nat] :
( ( A != n )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( minus_minus_nat @ n @ k ) ) ),
inference(paramod_ordered,[status(thm)],[324,9298]) ).
thf(9442,plain,
! [A: nat] :
( ( ( plus_plus_nat @ n @ A )
!= n )
| ( A != k ) ),
inference(simp,[status(thm)],[9420]) ).
thf(9453,plain,
( ( plus_plus_nat @ n @ k )
!= n ),
inference(simp,[status(thm)],[9442]) ).
thf(1197,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_nat @ zero_zero_nat @ ( p @ k ) )
| ( A != zero_zero_nat ) ),
inference(paramod_ordered,[status(thm)],[813,224]) ).
thf(1198,plain,
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_nat @ zero_zero_nat @ ( p @ k ) ) ),
inference(pattern_uni,[status(thm)],[1197:[bind(A,$thf( zero_zero_nat ))]]) ).
thf(18924,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( p @ k ) @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[1198,8881]) ).
thf(18925,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( p @ k ) @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[18924]) ).
thf(18926,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( p @ k ) @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[18925]) ).
thf(149,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_146_nat__neq__iff) ).
thf(688,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)],[149]) ).
thf(208,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C ) )
= ( ( A = zero_zero_nat )
| ( B = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_nat__mult__eq__cancel__disj) ).
thf(891,plain,
! [A: nat,B: nat,C: nat] :
( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C ) )
= ( ( A = zero_zero_nat )
| ( B = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(5,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) @ n,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_upper__bound) ).
thf(218,plain,
ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) @ n,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(173,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_add__less__imp__less__right) ).
thf(771,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(481,plain,
! [C: nat,B: nat,A: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ),
inference(cnf,[status(esa)],[480]) ).
thf(482,plain,
! [C: nat,B: nat,A: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ),
inference(lifteq,[status(thm)],[481]) ).
thf(20813,plain,
! [A: nat] :
( ( ( plus_plus_nat @ i @ k )
!= k )
| ( ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ k ) )
!= ( minus_minus_nat @ n @ A ) ) ),
inference(paramod_ordered,[status(thm)],[7759,19264]) ).
thf(20814,plain,
( ( plus_plus_nat @ i @ k )
!= k ),
inference(pattern_uni,[status(thm)],[20813:[bind(A,$thf( minus_minus_nat @ n @ k ))]]) ).
thf(3539,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
| ( ( ord_less_nat @ zero_zero_nat @ B )
!= ( ord_less_nat @ zero_zero_nat @ ( p @ k ) ) ) ),
inference(paramod_ordered,[status(thm)],[264,224]) ).
thf(3540,plain,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ ( p @ k ) @ A ) ),
inference(pattern_uni,[status(thm)],[3539:[bind(A,$thf( A )),bind(B,$thf( p @ k ))]]) ).
thf(6469,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ A @ ( plus_plus_nat @ ( p @ k ) @ A ) )
!= ( ord_less_nat @ B @ B ) ) ),
inference(paramod_ordered,[status(thm)],[3540,676]) ).
thf(6470,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ ( p @ k ) @ A ) )
!= ( ord_less_nat @ B @ B ) ),
inference(simp,[status(thm)],[6469]) ).
thf(6474,plain,
! [B: nat,A: nat] :
( ( A != B )
| ( ( plus_plus_nat @ ( p @ k ) @ A )
!= B ) ),
inference(simp,[status(thm)],[6470]) ).
thf(6494,plain,
! [A: nat] :
( ( plus_plus_nat @ ( p @ k ) @ A )
!= A ),
inference(simp,[status(thm)],[6474]) ).
thf(21723,plain,
! [A: nat] :
( ( n != A )
| ( ( plus_plus_nat @ ( p @ k ) @ A )
!= ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) ) ) ),
inference(paramod_ordered,[status(thm)],[21192,6494]) ).
thf(21760,plain,
( ( plus_plus_nat @ ( p @ k ) @ n )
!= ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) ) ),
inference(simp,[status(thm)],[21723]) ).
thf(48431,plain,
( ( plus_plus_nat @ ( p @ k ) @ n )
!= n ),
inference(rewrite,[status(thm)],[21760,21192]) ).
thf(193,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_plus__nat_Oadd__0) ).
thf(844,plain,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).
thf(3428,plain,
! [A: nat] :
( ~ $true
| ( ( ord_less_nat @ zero_zero_nat @ n )
!= ( ord_less_nat @ A @ A ) ) ),
inference(paramod_ordered,[status(thm)],[3324,676]) ).
thf(3429,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ n )
!= ( ord_less_nat @ A @ A ) ),
inference(simp,[status(thm)],[3428]) ).
thf(3439,plain,
! [A: nat] :
( ( zero_zero_nat != A )
| ( n != A ) ),
inference(simp,[status(thm)],[3429]) ).
thf(3465,plain,
n != zero_zero_nat,
inference(simp,[status(thm)],[3439]) ).
thf(124,axiom,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D )
= ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( D
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_nat__eq__add__iff2) ).
thf(588,plain,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D )
= ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( D
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(42,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
= ( ord_less_nat @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_add__less__cancel__left) ).
thf(319,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
= ( ord_less_nat @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(895,plain,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ),
inference(cnf,[status(esa)],[894]) ).
thf(896,plain,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ),
inference(lifteq,[status(thm)],[895]) ).
thf(5247,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ A @ ( plus_plus_nat @ n @ A ) )
!= ( ord_less_nat @ B @ B ) ) ),
inference(paramod_ordered,[status(thm)],[3566,676]) ).
thf(5248,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ n @ A ) )
!= ( ord_less_nat @ B @ B ) ),
inference(simp,[status(thm)],[5247]) ).
thf(5253,plain,
! [B: nat,A: nat] :
( ( A != B )
| ( ( plus_plus_nat @ n @ A )
!= B ) ),
inference(simp,[status(thm)],[5248]) ).
thf(5272,plain,
! [A: nat] :
( ( plus_plus_nat @ n @ A )
!= A ),
inference(simp,[status(thm)],[5253]) ).
thf(2684,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( zero_zero_nat != zero_zero_nat )
| ( A
!= ( plus_plus_nat @ i @ B ) ) ),
inference(paramod_ordered,[status(thm)],[735,2678]) ).
thf(2685,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( A
!= ( plus_plus_nat @ i @ B ) ) ),
inference(simp,[status(thm)],[2684]) ).
thf(2686,plain,
! [A: nat] : ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ i @ A ) ),
inference(pattern_uni,[status(thm)],[2685:[bind(A,$thf( plus_plus_nat @ i @ D )),bind(B,$thf( D ))]]) ).
thf(2694,plain,
! [A: nat] : ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ i @ A ) ),
inference(simp,[status(thm)],[2686]) ).
thf(2729,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ i )
| ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(rewrite,[status(thm)],[2694,252]) ).
thf(2730,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ i )
| ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(cnf,[status(esa)],[2729]) ).
thf(32137,plain,
! [A: nat] :
( ( ( minus_minus_nat @ i @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[2730,8881]) ).
thf(32138,plain,
! [A: nat] :
( ( ( minus_minus_nat @ i @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[32137]) ).
thf(32139,plain,
! [A: nat] :
( ( ( minus_minus_nat @ i @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[32138]) ).
thf(515,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat] :
( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ),
inference(miniscope,[status(thm)],[514]) ).
thf(516,plain,
! [C: nat,B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ),
inference(cnf,[status(esa)],[515]) ).
thf(517,plain,
! [C: nat,B: nat,A: nat] :
( ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B )
= ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) ) )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[516]) ).
thf(24241,plain,
! [C: nat,B: nat,A: nat] :
( ( ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B )
= zero_zero_nat )
= ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= zero_zero_nat ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[517,406]) ).
thf(24242,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B )
= zero_zero_nat )
= ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= zero_zero_nat ) ) ),
inference(lifteq,[status(thm)],[24241]) ).
thf(177,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_add__strict__mono) ).
thf(781,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(3703,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( zero_zero_nat != zero_zero_nat )
| ( A
!= ( plus_plus_nat @ n @ B ) ) ),
inference(paramod_ordered,[status(thm)],[735,3446]) ).
thf(3704,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( A
!= ( plus_plus_nat @ n @ B ) ) ),
inference(simp,[status(thm)],[3703]) ).
thf(3705,plain,
! [A: nat] : ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ n @ A ) ),
inference(pattern_uni,[status(thm)],[3704:[bind(A,$thf( plus_plus_nat @ n @ D )),bind(B,$thf( D ))]]) ).
thf(3710,plain,
! [A: nat] : ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ n @ A ) ),
inference(simp,[status(thm)],[3705]) ).
thf(3728,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ n )
| ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(rewrite,[status(thm)],[3710,252]) ).
thf(3729,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ n )
| ( ord_less_nat @ zero_zero_nat @ A ) ),
inference(cnf,[status(esa)],[3728]) ).
thf(32380,plain,
! [A: nat] :
( ( ( minus_minus_nat @ n @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[3729,8881]) ).
thf(32381,plain,
! [A: nat] :
( ( ( minus_minus_nat @ n @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[32380]) ).
thf(32382,plain,
! [A: nat] :
( ( ( minus_minus_nat @ n @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ zero_zero_nat )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[32381]) ).
thf(7859,plain,
! [A: nat] :
( ( i = A )
| ( ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ i ) )
!= ( minus_minus_nat @ A @ zero_zero_nat ) ) ),
inference(paramod_ordered,[status(thm)],[7752,312]) ).
thf(7883,plain,
! [A: nat] :
( ( i = A )
| ( n != A )
| ( ( minus_minus_nat @ n @ i )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[7859]) ).
thf(7912,plain,
( ( n = i )
| ( ( minus_minus_nat @ n @ i )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[7883]) ).
thf(589,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat,D: nat,E: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D )
= ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( D
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) ) ),
inference(miniscope,[status(thm)],[588]) ).
thf(590,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D )
= ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( D
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) ) ),
inference(cnf,[status(esa)],[589]) ).
thf(591,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D )
= ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( D
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ E ) ) )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[590]) ).
thf(42893,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D )
= ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( D
= ( plus_plus_nat @ ( minus_minus_nat @ ( times_times_nat @ B @ C ) @ ( times_times_nat @ A @ C ) ) @ E ) ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[591,485,406]) ).
thf(42894,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ D )
= ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ E ) )
= ( D
= ( plus_plus_nat @ ( minus_minus_nat @ ( times_times_nat @ B @ C ) @ ( times_times_nat @ A @ C ) ) @ E ) ) ) ),
inference(lifteq,[status(thm)],[42893]) ).
thf(11448,plain,
! [A: nat] :
( ( zero_zero_nat != zero_zero_nat )
| ( ( minus_minus_nat @ ( plus_plus_nat @ n @ A ) @ A )
!= ( minus_minus_nat @ k @ n ) ) ),
inference(paramod_ordered,[status(thm)],[9088,8928]) ).
thf(11449,plain,
! [A: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ n @ A ) @ A )
!= ( minus_minus_nat @ k @ n ) ),
inference(simp,[status(thm)],[11448]) ).
thf(11477,plain,
! [A: nat] :
( ( ( plus_plus_nat @ n @ A )
!= k )
| ( A != n ) ),
inference(simp,[status(thm)],[11449]) ).
thf(11497,plain,
( ( plus_plus_nat @ n @ n )
!= k ),
inference(simp,[status(thm)],[11477]) ).
thf(119,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( ord_less_nat @ B @ ( plus_plus_nat @ C @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_less__diff__conv2) ).
thf(568,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( ord_less_nat @ B @ ( plus_plus_nat @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(14,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ C @ B ) )
= ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_mult__le__cancel2) ).
thf(235,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ C @ B ) )
= ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(1295,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_nat @ n @ zero_zero_nat )
| ( A
!= ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) ) ),
inference(paramod_ordered,[status(thm)],[813,219]) ).
thf(1296,plain,
( ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) )
| ( ord_less_nat @ n @ zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[1295:[bind(A,$thf( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ))]]) ).
thf(10736,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ zero_zero_nat @ n )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[1296,8881]) ).
thf(10737,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ zero_zero_nat @ n )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[10736]) ).
thf(10738,plain,
( ( ( minus_minus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ zero_zero_nat @ n )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[10737]) ).
thf(2621,plain,
! [B: nat,A: nat] :
( ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[1233,2618]) ).
thf(2622,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( ord_less_nat @ zero_zero_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) ) ),
inference(simp,[status(thm)],[2621]) ).
thf(2655,plain,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ A @ B )
!= zero_zero_nat )
| ( A
!= ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) ) ),
inference(simp,[status(thm)],[2622]) ).
thf(2680,plain,
! [A: nat] :
( ( plus_plus_nat @ ( sk8 @ ( (=) @ nat @ zero_zero_nat ) ) @ A )
!= zero_zero_nat ),
inference(simp,[status(thm)],[2655]) ).
thf(64,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_le__add__same__cancel2) ).
thf(383,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(154,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_151_le__trans) ).
thf(716,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)],[154]) ).
thf(102,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc) ).
thf(510,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(457,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat] :
( ( 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(miniscope,[status(thm)],[456]) ).
thf(458,plain,
! [C: nat,B: nat,A: 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(cnf,[status(esa)],[457]) ).
thf(459,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ C @ A ) )
= ( minus_minus_nat @ B @ C ) )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ A @ C ) ),
inference(lifteq,[status(thm)],[458]) ).
thf(16777,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ C @ A ) )
= ( minus_minus_nat @ B @ C ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ C )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[459,406]) ).
thf(16778,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ A @ C )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( minus_minus_nat @ B @ A ) @ ( minus_minus_nat @ C @ A ) )
= ( minus_minus_nat @ B @ C ) ) ),
inference(lifteq,[status(thm)],[16777]) ).
thf(211,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ ( plus_plus_nat @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_trans__le__add2) ).
thf(899,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ ( plus_plus_nat @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(167,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_164_add__mono) ).
thf(752,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(79,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_minus__nat_Odiff__0) ).
thf(432,plain,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(189,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_bot__nat__0_Oextremum__unique) ).
thf(833,plain,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(48958,plain,
! [C: nat,B: nat,A: nat] :
( ( ( sk2 @ B @ A )
= C )
| ( ( plus_plus_nat @ zero_zero_nat @ A )
!= ( plus_plus_nat @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[306,48937]) ).
thf(48959,plain,
! [A: nat] :
( ( sk2 @ zero_zero_nat @ A )
= A ),
inference(pattern_uni,[status(thm)],[48958:[bind(A,$thf( A )),bind(B,$thf( zero_zero_nat )),bind(C,$thf( A ))]]) ).
thf(184,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_181_less__zeroE) ).
thf(818,plain,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(76,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_73_diff__commute) ).
thf(422,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)],[76]) ).
thf(200,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( plus_plus_nat @ C @ B )
= ( plus_plus_nat @ A @ D ) )
=> ( ord_less_nat @ C @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_less__add__eq__less) ).
thf(867,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( plus_plus_nat @ C @ B )
= ( plus_plus_nat @ A @ D ) )
=> ( ord_less_nat @ C @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(372,plain,
! [B: nat,A: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ),
inference(cnf,[status(esa)],[371]) ).
thf(373,plain,
! [B: nat,A: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ),
inference(lifteq,[status(thm)],[372]) ).
thf(48033,plain,
! [B: nat,A: nat] :
( ( ( sk1 @ B @ n )
= A )
| ( ( plus_plus_nat @ ( minus_minus_nat @ n @ i ) @ i )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[26528,48022]) ).
thf(48034,plain,
( ( sk1 @ i @ n )
= ( minus_minus_nat @ n @ i ) ),
inference(pattern_uni,[status(thm)],[48033:[bind(A,$thf( minus_minus_nat @ n @ i )),bind(B,$thf( i ))]]) ).
thf(48031,plain,
! [B: nat,A: nat] :
( ( ( sk1 @ B @ n )
= A )
| ( ( plus_plus_nat @ i @ ( minus_minus_nat @ n @ i ) )
!= ( plus_plus_nat @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[21232,48022]) ).
thf(48032,plain,
( ( sk1 @ ( minus_minus_nat @ n @ i ) @ n )
= i ),
inference(pattern_uni,[status(thm)],[48031:[bind(A,$thf( i )),bind(B,$thf( minus_minus_nat @ n @ i ))]]) ).
thf(454,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ),
inference(miniscope,[status(thm)],[453]) ).
thf(455,plain,
! [C: nat,B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ord_less_eq_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ),
inference(cnf,[status(esa)],[454]) ).
thf(16255,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) )
= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[455,406]) ).
thf(16256,plain,
! [C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) )
= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[16255]) ).
thf(3286,plain,
( ( ord_less_nat @ zero_zero_nat @ n )
| ( ord_less_nat @ ( minus_minus_nat @ n @ k ) @ zero_zero_nat )
| ( i != i ) ),
inference(paramod_ordered,[status(thm)],[3264,228]) ).
thf(3287,plain,
( ( ord_less_nat @ zero_zero_nat @ n )
| ( ord_less_nat @ ( minus_minus_nat @ n @ k ) @ zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[3286:[]]) ).
thf(17441,plain,
( ( ( minus_minus_nat @ n @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ zero_zero_nat @ ( minus_minus_nat @ n @ k ) )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[3287,8881]) ).
thf(17442,plain,
( ( ( minus_minus_nat @ n @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ zero_zero_nat @ ( minus_minus_nat @ n @ k ) )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[17441]) ).
thf(17443,plain,
( ( ( minus_minus_nat @ n @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ zero_zero_nat @ ( minus_minus_nat @ n @ k ) )
!= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[17442]) ).
thf(136,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A = B )
& ( C = D ) )
=> ( ( plus_plus_nat @ A @ C )
= ( plus_plus_nat @ B @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_add__mono__thms__linordered__semiring_I4_J) ).
thf(635,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A = B )
& ( C = D ) )
=> ( ( plus_plus_nat @ A @ C )
= ( plus_plus_nat @ B @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(21205,plain,
! [C: nat,B: nat,A: nat] :
( ( A != zero_zero_nat )
| ( ( plus_plus_nat @ B @ ( minus_minus_nat @ C @ B ) )
= C )
| ( ( minus_minus_nat @ A @ zero_zero_nat )
!= ( minus_minus_nat @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[312,21136]) ).
thf(21206,plain,
! [A: nat] :
( ( A != zero_zero_nat )
| ( ( plus_plus_nat @ A @ ( minus_minus_nat @ zero_zero_nat @ A ) )
= zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[21205:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( zero_zero_nat ))]]) ).
thf(21569,plain,
( ( plus_plus_nat @ zero_zero_nat @ ( minus_minus_nat @ zero_zero_nat @ zero_zero_nat ) )
= zero_zero_nat ),
inference(simp,[status(thm)],[21206]) ).
thf(22337,plain,
( ( minus_minus_nat @ zero_zero_nat @ zero_zero_nat )
= zero_zero_nat ),
inference(rewrite,[status(thm)],[21569,306]) ).
thf(101,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc) ).
thf(506,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(436,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ B @ A )
!= zero_zero_nat )
| ( A = B ) ),
inference(cnf,[status(esa)],[435]) ).
thf(437,plain,
! [B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( minus_minus_nat @ B @ A )
!= zero_zero_nat )
| ( A = B ) ),
inference(lifteq,[status(thm)],[436]) ).
thf(8845,plain,
! [B: nat,A: nat] :
( ( B != n )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( minus_minus_nat @ n @ i ) ) ),
inference(paramod_ordered,[status(thm)],[330,8837]) ).
thf(8866,plain,
! [A: nat] :
( ( ( plus_plus_nat @ A @ n )
!= n )
| ( A != i ) ),
inference(simp,[status(thm)],[8845]) ).
thf(8869,plain,
( ( plus_plus_nat @ i @ n )
!= n ),
inference(simp,[status(thm)],[8866]) ).
thf(92,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_diff__cancel2) ).
thf(474,plain,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(206,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_add__less__mono) ).
thf(886,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(163,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
? [C: nat] :
( B
= ( plus_plus_nat @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_160_le__iff__add) ).
thf(740,plain,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
? [C: nat] :
( B
= ( plus_plus_nat @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(643,plain,
! [C: nat,B: nat,A: nat] :
( ( times_times_nat @ A @ ( times_times_nat @ B @ C ) )
= ( times_times_nat @ B @ ( times_times_nat @ A @ C ) ) ),
inference(cnf,[status(esa)],[642]) ).
thf(644,plain,
! [C: nat,B: nat,A: nat] :
( ( times_times_nat @ A @ ( times_times_nat @ B @ C ) )
= ( times_times_nat @ B @ ( times_times_nat @ A @ C ) ) ),
inference(lifteq,[status(thm)],[643]) ).
thf(214,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_211_add__le__mono) ).
thf(908,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(157,axiom,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
= ( A != zero_zero_nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_154_zero__less__iff__neq__zero) ).
thf(723,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
= ( A != zero_zero_nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(201,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ A @ ( plus_plus_nat @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_trans__less__add2) ).
thf(871,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ A @ ( plus_plus_nat @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).
thf(623,plain,
! [C: nat,B: nat,A: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
inference(cnf,[status(esa)],[622]) ).
thf(624,plain,
! [C: nat,B: nat,A: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
inference(lifteq,[status(thm)],[623]) ).
thf(191,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_less__eq__nat_Osimps_I1_J) ).
thf(839,plain,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(585,plain,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ! [C: nat,D: nat,E: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
inference(miniscope,[status(thm)],[584]) ).
thf(586,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
inference(cnf,[status(esa)],[585]) ).
thf(587,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) ) )
| ~ ( ord_less_eq_nat @ A @ B ) ),
inference(lifteq,[status(thm)],[586]) ).
thf(41517,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E )
= zero_zero_nat )
= ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= zero_zero_nat ) )
| ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[587,406]) ).
thf(41518,plain,
! [E: nat,D: nat,C: nat,B: nat,A: nat] :
( ( ( minus_minus_nat @ A @ B )
!= zero_zero_nat )
| ( ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ B @ A ) @ C ) @ D ) @ E )
= zero_zero_nat )
= ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ D ) @ ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ E ) )
= zero_zero_nat ) ) ),
inference(lifteq,[status(thm)],[41517]) ).
thf(31879,plain,
! [A: nat] :
( ( n != n )
| ( ( plus_plus_nat @ ( plus_plus_nat @ i @ k ) @ A )
!= ( plus_plus_nat @ ( minus_minus_nat @ n @ k ) @ k ) ) ),
inference(paramod_ordered,[status(thm)],[26481,20859]) ).
thf(31880,plain,
! [A: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ i @ k ) @ A )
!= ( plus_plus_nat @ ( minus_minus_nat @ n @ k ) @ k ) ),
inference(simp,[status(thm)],[31879]) ).
thf(31895,plain,
! [A: nat] :
( ( ( plus_plus_nat @ i @ k )
!= ( minus_minus_nat @ n @ k ) )
| ( A != k ) ),
inference(simp,[status(thm)],[31880]) ).
thf(31916,plain,
( ( plus_plus_nat @ i @ k )
!= ( minus_minus_nat @ n @ k ) ),
inference(simp,[status(thm)],[31895]) ).
thf(3938,plain,
! [B: nat,A: nat] :
( ( ( p @ zero_zero_nat )
!= zero_zero_nat )
| ~ $true
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ zero_zero_nat @ k ) ) ),
inference(paramod_ordered,[status(thm)],[2040,2849]) ).
thf(3939,plain,
! [B: nat,A: nat] :
( ( ( p @ zero_zero_nat )
!= zero_zero_nat )
| ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
!= ( ord_less_nat @ zero_zero_nat @ k ) ) ),
inference(simp,[status(thm)],[3938]) ).
thf(3969,plain,
! [B: nat,A: nat] :
( ( ( p @ zero_zero_nat )
!= zero_zero_nat )
| ( ( plus_plus_nat @ A @ B )
!= zero_zero_nat )
| ( B != k ) ),
inference(simp,[status(thm)],[3939]) ).
thf(3974,plain,
! [A: nat] :
( ( ( p @ zero_zero_nat )
!= zero_zero_nat )
| ( ( plus_plus_nat @ A @ k )
!= zero_zero_nat ) ),
inference(simp,[status(thm)],[3969]) ).
thf(23705,plain,
! [A: nat] :
( ( ( n = i )
= ( zero_zero_nat = A ) )
| ( ( ( minus_minus_nat @ n @ i )
= zero_zero_nat )
!= ( A = zero_zero_nat ) ) ),
inference(paramod_ordered,[status(thm)],[21943,606]) ).
thf(23706,plain,
( ( zero_zero_nat
= ( minus_minus_nat @ n @ i ) )
= ( n = i ) ),
inference(pattern_uni,[status(thm)],[23705:[bind(A,$thf( minus_minus_nat @ n @ i ))]]) ).
thf(28,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_add__le__cancel__right) ).
thf(277,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(150,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_147_Nat_Oex__has__greatest__nat) ).
thf(691,plain,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ D )
=> ( ord_less_eq_nat @ D @ C ) )
=> ? [D: nat] :
( ( A @ D )
& ! [E: nat] :
( ( A @ E )
=> ( ord_less_eq_nat @ E @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(2113,plain,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
| ( ord_less_eq_nat @ ( p @ B ) @ zero_zero_nat )
| ( A != n ) ),
inference(paramod_ordered,[status(thm)],[735,1047]) ).
thf(2114,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ n )
| ( ord_less_eq_nat @ ( p @ A ) @ zero_zero_nat ) ),
inference(pattern_uni,[status(thm)],[2113:[bind(A,$thf( n ))]]) ).
thf(2254,plain,
! [A: nat] :
( ( ord_less_nat @ zero_zero_nat @ n )
| ( ord_less_eq_nat @ ( p @ A ) @ zero_zero_nat ) ),
inference(simp,[status(thm)],[2114]) ).
thf(39337,plain,
! [A: nat] :
( ( ( minus_minus_nat @ n @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( p @ A ) @ zero_zero_nat )
= zero_zero_nat ) ),
inference(rewrite,[status(thm)],[2254,8881,406]) ).
thf(39338,plain,
! [A: nat] :
( ( ( minus_minus_nat @ n @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( p @ A ) @ zero_zero_nat )
= zero_zero_nat ) ),
inference(simp,[status(thm)],[39337]) ).
thf(39339,plain,
! [A: nat] :
( ( ( minus_minus_nat @ n @ zero_zero_nat )
!= zero_zero_nat )
| ( ( minus_minus_nat @ ( p @ A ) @ zero_zero_nat )
= zero_zero_nat ) ),
inference(lifteq,[status(thm)],[39338]) ).
thf(226,plain,
i != k,
inference(polarity_switch,[status(thm)],[225]) ).
thf(227,plain,
k != i,
inference(lifteq,[status(thm)],[226]) ).
thf(112,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_Nat_Ole__imp__diff__is__add) ).
thf(543,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(35852,plain,
! [A: nat] :
( ( n != n )
| ( ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) )
!= ( plus_plus_nat @ A @ ( plus_plus_nat @ k @ i ) ) ) ),
inference(paramod_ordered,[status(thm)],[21192,29916]) ).
thf(35853,plain,
! [A: nat] :
( ( plus_plus_nat @ k @ ( minus_minus_nat @ n @ k ) )
!= ( plus_plus_nat @ A @ ( plus_plus_nat @ k @ i ) ) ),
inference(simp,[status(thm)],[35852]) ).
thf(35892,plain,
! [A: nat] :
( ( k != A )
| ( ( plus_plus_nat @ k @ i )
!= ( minus_minus_nat @ n @ k ) ) ),
inference(simp,[status(thm)],[35853]) ).
thf(35915,plain,
( ( plus_plus_nat @ k @ i )
!= ( minus_minus_nat @ n @ k ) ),
inference(simp,[status(thm)],[35892]) ).
thf(1,conjecture,
$false,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).
thf(2,negated_conjecture,
~ $false,
inference(neg_conjecture,[status(cth)],[1]) ).
thf(215,plain,
~ $false,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(175,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_add__strict__right__mono) ).
thf(775,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(137,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_ab__semigroup__add__class_Oadd__ac_I1_J) ).
thf(639,plain,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(168,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ C @ D ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_165_add__mono__thms__linordered__semiring_I1_J) ).
thf(755,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ C @ D ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(363,plain,
! [C: nat,B: nat,A: nat] :
( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ C @ B ) )
= ( ( A = C )
| ( B = zero_zero_nat ) ) ),
inference(cnf,[status(esa)],[362]) ).
thf(364,plain,
! [C: nat,B: nat,A: nat] :
( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ C @ B ) )
= ( ( A = C )
| ( B = zero_zero_nat ) ) ),
inference(lifteq,[status(thm)],[363]) ).
thf(10693,plain,
! [B: nat,A: nat] :
( ( n != i )
| ( B != i )
| ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
!= ( minus_minus_nat @ n @ i ) ) ),
inference(paramod_ordered,[status(thm)],[330,8872]) ).
thf(10704,plain,
! [A: nat] :
( ( n != i )
| ( ( plus_plus_nat @ A @ i )
!= n )
| ( A != i ) ),
inference(simp,[status(thm)],[10693]) ).
thf(10723,plain,
( ( n != i )
| ( ( plus_plus_nat @ i @ i )
!= n ) ),
inference(simp,[status(thm)],[10704]) ).
thf(61627,plain,
$false,
inference(e,[status(thm)],[645,629,48292,347,518,468,9907,51671,365,10259,333,3979,44967,36141,555,21136,54916,5069,9456,15703,247,861,730,217,614,8928,677,19291,533,21192,2918,814,913,797,3446,340,8881,404,35845,841,15318,288,8472,301,12661,709,23019,31907,565,29916,51742,23936,11491,12884,18920,2916,11299,30795,619,1511,2897,1132,22834,460,9960,389,8875,719,559,11285,421,606,1013,228,289,316,48949,902,9529,325,216,9298,25653,44974,9088,13312,3453,7926,238,514,353,821,646,253,8929,485,905,480,397,17425,280,57756,877,1886,453,411,23909,21943,265,21232,6110,221,527,312,307,380,48040,270,33295,48622,761,220,31918,31596,51618,292,20871,48028,2910,2673,334,894,778,48957,392,297,429,788,407,630,22029,9455,625,8942,17659,35807,830,889,7713,757,551,456,324,229,847,852,435,1888,665,29998,29826,285,224,742,32042,874,9073,51814,9576,14121,356,12667,16031,3001,51941,48953,6801,650,225,18885,5705,537,6304,642,32564,48293,19358,328,471,256,41506,547,622,647,738,911,721,47977,34305,371,343,244,792,48036,27249,765,398,9171,674,530,31583,2909,773,794,298,42760,54979,425,26528,824,48945,6578,40120,444,733,16100,46357,19268,745,12704,584,562,42691,286,291,19264,12663,8872,259,28585,21735,936,382,572,313,12067,498,48042,350,540,35900,604,36036,681,274,784,419,736,303,271,863,46568,406,2679,6308,219,377,22036,35280,362,451,880,649,26415,48657,726,621,483,16101,932,48937,477,29174,494,13499,345,367,5994,609,28928,26481,330,8052,883,223,306,267,438,11493,8280,685,521,55581,48904,827,2835,37698,768,11874,34567,1015,241,653,4331,21639,374,749,617,21728,4688,836,29854,11316,2914,31614,712,295,707,8890,327,30243,9467,310,580,855,48552,2678,19147,35916,48022,45185,331,370,502,18318,576,359,19348,441,20287,2675,16106,48912,656,600,43791,386,12241,19363,612,7752,48951,20792,9453,18926,688,524,891,218,771,250,231,482,21483,20814,48431,368,844,3465,588,319,896,5272,304,32139,24242,781,32382,7912,401,6494,22317,42894,11497,568,235,10738,2680,383,716,8837,510,16778,899,300,620,752,447,16098,7759,432,833,262,48959,19205,818,422,867,865,9463,373,8933,19151,48034,48032,16256,17443,635,283,294,918,22337,506,268,437,8869,897,474,886,415,740,644,908,723,871,607,395,624,839,41518,464,31916,3974,23706,596,277,691,39339,227,322,486,309,543,35915,215,775,639,755,491,592,364,232,337,10723]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : ITP131^1 : TPTP v9.3.1. Released v7.5.0.
% 0.00/0.08 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.10/0.21 % Computer : n003.cluster.edu
% 0.10/0.21 % Model : x86_64 x86_64
% 0.10/0.21 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.21 % Memory : 8046.5625MB
% 0.10/0.21 % OS : Linux 6.8.0-71-generic
% 0.10/0.21 % CPULimit : 300
% 0.10/0.21 % WCLimit : 300
% 0.10/0.21 % DateTime : Tue Sep 29 19:59:58 UTC 2026
% 0.10/0.21 % CPUTime :
% 0.10/0.21 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.86/0.77 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 2.09/1.13 % [INFO] Parsing done (361ms).
% 2.09/1.14 % [INFO] Running in sequential loop mode.
% 2.87/1.46 % [INFO] eprover registered as external prover.
% 2.87/1.47 % [INFO] Scanning for conjecture ...
% 3.20/1.58 % [INFO] Found a conjecture (or negated_conjecture) and 212 axioms. Running axiom selection ...
% 3.39/1.62 % [INFO] Axiom selection finished. Selected 212 axioms (removed 0 axioms).
% 3.96/1.84 % [INFO] Problem is higher-order (TPTP THF).
% 3.96/1.87 % [INFO] Type checking passed.
% 3.96/1.87 % [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 ...
% 155.12/70.47 % External prover 'e' found a proof!
% 155.12/70.47 % [INFO] Killing All external provers ...
% 155.12/70.47 % Time passed: 70132ms (effective reasoning time: 69317ms)
% 155.12/70.47 % 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)>
% 155.12/70.47 % Axioms used in derivation (212): fact_164_add__mono, fact_89_diff__cancel2, fact_169_comm__monoid__add__class_Oadd__0, fact_181_less__zeroE, fact_148_nat__le__linear, fact_103_diff__less, fact_32_add__cancel__left__left, fact_3_lower__bound, fact_180_gr__implies__not0, fact_95_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right, fact_10_partitions__bounds, fact_68_diff__is__0__eq, fact_207_nat__le__iff__add, fact_204_add__lessD1, fact_161_add__right__mono, fact_11_mult__le__cancel2, fact_74_add__implies__diff, fact_158_add__le__imp__le__right, fact_189_add__eq__self__zero, fact_90_Nat_Odiff__cancel, fact_15_nat__mult__less__cancel__disj, fact_200_add__less__mono1, fact_71_Nat_Oadd__diff__assoc, fact_159_add__le__imp__le__left, fact_201_not__add__less2, fact_35_cancel__comm__monoid__add__class_Odiff__cancel, fact_171_add__less__imp__less__left, fact_47_bot__nat__0_Oextremum, fact_4__092_060open_062k_A_092_060le_062_An_092_060close_062, fact_197_less__add__eq__less, fact_27_zero__eq__add__iff__both__eq__0, fact_205_nat__mult__eq__cancel__disj, fact_102_ordered__cancel__comm__monoid__diff__class_Odiff__add, fact_14_nat__0__less__mult__iff, fact_113_le__diff__conv, fact_24_not__gr__zero, fact_165_add__mono__thms__linordered__semiring_I1_J, fact_56_mult__0__right, fact_99_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc, fact_179_infinite__descent0, fact_57_mult__is__0, fact_45_less__nat__zero__code, fact_172_add__strict__right__mono, fact_174_add__strict__mono, fact_86_eq__diff__iff, fact_59_diff__diff__cancel, fact_80_diff__le__mono2, fact_131_group__cancel_Oadd2, fact_120_nat__le__add__iff1, fact_107_add__diff__inverse__nat, fact_117_nat__diff__add__eq2, fact_64_add__le__same__cancel1, fact_192_le__neq__implies__less, fact_188_less__eq__nat_Osimps_I1_J, fact_121_nat__eq__add__iff2, fact_157_gr__zeroI, fact_1_assms_I1_J, fact_141_nat__less__induct, fact_12_nat__mult__le__cancel__disj, fact_65_diff__add__zero, fact_46_neq0__conv, fact_168_add_Ocomm__neutral, fact_39_add__less__cancel__left, fact_151_le__trans, fact_144_less__not__refl2, fact_173_add__strict__left__mono, fact_123_nat__less__add__iff2, fact_60_diff__diff__left, fact_93_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add, fact_94_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse, fact_77_diffs0__imp__equal, fact_163_add__left__mono, fact_155_gr__implies__not__zero, fact_52_diff__self__eq__0, fact_76_minus__nat_Odiff__0, fact_166_add__mono__thms__linordered__semiring_I2_J, fact_28_add__eq__0__iff__both__eq__0, fact_127_add__left__imp__eq, fact_183_not__gr0, fact_8__092_060open_062n_A_N_Ak_A_060_Ai_092_060close_062, fact_111_Nat_Odiff__add__assoc, fact_114_nat__diff__split__asm, fact_43_add__diff__cancel__left, fact_138_ab__semigroup__mult__class_Omult__ac_I1_J, fact_70_Nat_Oadd__diff__assoc2, fact_154_zero__less__iff__neq__zero, fact_26_add__le__cancel__left, fact_36_diff__zero, fact_135_mult_Oleft__commute, fact_72_cancel__ab__semigroup__add__class_Odiff__right__commute, fact_119_nat__le__add__iff2, fact_150_eq__imp__le, fact_100_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2, fact_206_mult__0, fact_186_bot__nat__0_Oextremum__unique, fact_51_nat__add__left__cancel__less, fact_92_diff__mult__distrib, fact_91_diff__mult__distrib2, fact_19_less__add__same__cancel1, fact_101_le__add__diff, fact_128_add_Oleft__commute, fact_62_le__add__same__cancel1, fact_55_mult__cancel1, fact_124_nat__less__add__iff1, fact_140_infinite__descent, fact_88_diff__add__inverse, fact_139_linorder__neqE__nat, fact_191_less__mono__imp__le__mono, fact_61_le__add__same__cancel2, fact_63_add__le__same__cancel2, fact_152_le__refl, fact_182_not__less0, fact_79_diff__less__mono2, fact_211_add__le__mono, fact_203_add__less__mono, fact_18_add__less__same__cancel2, fact_116_less__diff__conv2, fact_34_add_Oleft__neutral, fact_136_mult_Ocommute, fact_134_ab__semigroup__add__class_Oadd__ac_I1_J, fact_44_bot__nat__0_Onot__eq__extremum, fact_16_add__gr__0, fact_153_zero__le, fact_162_less__eqE, fact_193_less__or__eq__imp__le, fact_54_mult__cancel2, fact_175_add__mono__thms__linordered__field_I1_J, fact_85_le__diff__iff, fact_38_add__less__cancel__right, fact_49_Nat_Oadd__0__right, fact_129_add_Ocommute, fact_58_nat__add__left__cancel__le, fact_50_add__is__0, fact_109_Nat_Ole__imp__diff__is__add, fact_53_diff__0__eq__0, fact_0__092_060open_062i_A_092_060le_062_An_092_060close_062, fact_143_less__not__refl3, fact_82_diff__le__self, fact_199_trans__less__add1, fact_184_gr0I, fact_137_mult_Oassoc, fact_105_diff__less__mono, fact_130_add_Oassoc, fact_29_add__cancel__right__right, fact_75_diff__diff__add, fact_167_add__mono__thms__linordered__semiring_I3_J, fact_20_less__add__same__cancel2, fact_115_nat__diff__split, fact_106_less__diff__iff, fact_145_less__not__refl, fact_142_less__irrefl__nat, fact_6_assms_I2_J, fact_196_nat__less__le, fact_110_Nat_Odiff__add__assoc2, fact_41_add__diff__cancel__right, fact_190_plus__nat_Oadd__0, fact_209_trans__le__add1, fact_104_diff__add__0, fact_69_Nat_Odiff__diff__right, fact_2_upper__bound, fact_96_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2, fact_198_trans__less__add2, fact_98_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc, fact_78_less__imp__diff__less, fact_176_add__mono__thms__linordered__field_I2_J, fact_9_partitions__zero, fact_108_less__diff__conv, fact_40_add__diff__cancel__right_H, fact_87_diff__add__inverse2, fact_17_add__less__same__cancel1, fact_42_add__diff__cancel__left_H, fact_202_not__add__less1, fact_81_le__diff__iff_H, fact_126_add__right__imp__eq, fact_178_bot__nat__0_Oextremum__strict, fact_195_less__imp__le__nat, fact_208_trans__le__add2, fact_84_Nat_Odiff__diff__eq, fact_146_nat__neq__iff, fact_83_diff__le__mono, fact_194_le__eq__less__or__eq, fact_37_zero__diff, fact_122_nat__eq__add__iff1, fact_156_not__less__zero, fact_147_Nat_Oex__has__greatest__nat, fact_23_le__zero__eq, fact_13_mult__less__cancel2, fact_22_add__left__cancel, fact_185_bot__nat__0_Oextremum__uniqueI, fact_30_add__cancel__right__left, fact_132_group__cancel_Oadd1, fact_97_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2, fact_125_zero__reorient, fact_112_Nat_Ole__diff__conv2, fact_118_nat__diff__add__eq1, fact_149_le__antisym, fact_133_add__mono__thms__linordered__semiring_I4_J, fact_66_zero__less__diff, fact_25_add__le__cancel__right, fact_170_add__less__imp__less__right, fact_210_add__le__mono1, fact_5__092_060open_062p_Ai_A_092_060noteq_062_A0_092_060close_062, fact_187_le__0__eq, fact_31_add__cancel__left__right, fact_48_le0, fact_73_diff__commute, fact_67_diff__is__0__eq_H, fact_7__092_060open_062i_A_092_060noteq_062_Ak_092_060close_062, fact_21_add__right__cancel, fact_160_le__iff__add, fact_33_add_Oright__neutral, fact_177_add__mono__thms__linordered__field_I5_J
% 155.12/70.47 % No. of inferences in proof: 1403
% 155.12/70.47 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 70132 ms resp. 69317 ms w/o parsing
% 156.17/70.78 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 156.17/70.79 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------