TSTP Solution File: SWW490_5 by Leo-III---1.7.7
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%------------------------------------------------------------------------------
% File : Leo-III---1.7.7
% Problem : SWW490_5 : TPTP v8.1.2. Released v6.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n014.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri May 19 12:24:53 EDT 2023
% Result : Theorem 13.03s 5.04s
% Output : Refutation 13.29s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 176
% Syntax : Number of formulae : 313 ( 60 unt; 41 typ; 0 def)
% Number of atoms : 851 ( 352 equ; 0 cnn)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 3137 ( 53 ~; 12 |; 41 &;2719 @)
% ( 20 <=>; 292 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 37 ( 37 >; 0 *; 0 +; 0 <<)
% Number of symbols : 42 ( 40 usr; 6 con; 0-5 aty)
% Number of variables : 662 ( 0 ^; 625 !; 2 ?; 662 :)
% ( 35 !>; 0 ?*; 0 @-; 0 @+)
% Comments :
%------------------------------------------------------------------------------
thf(a_type,type,
a: $tType ).
thf(bool_type,type,
bool: $tType ).
thf(nat_type,type,
nat: $tType ).
thf(cancel1352612707id_add_type,type,
cancel1352612707id_add:
!>[TA: $tType] : $o ).
thf(linordered_idom_type,type,
linordered_idom:
!>[TA: $tType] : $o ).
thf(idom_type,type,
idom:
!>[TA: $tType] : $o ).
thf(zero_type,type,
zero:
!>[TA: $tType] : $o ).
thf(ring_char_0_type,type,
ring_char_0:
!>[TA: $tType] : $o ).
thf(semiring_1_type,type,
semiring_1:
!>[TA: $tType] : $o ).
thf(monoid_add_type,type,
monoid_add:
!>[TA: $tType] : $o ).
thf(comm_semiring_0_type,type,
comm_semiring_0:
!>[TA: $tType] : $o ).
thf(comm_semiring_1_type,type,
comm_semiring_1:
!>[TA: $tType] : $o ).
thf(comm_monoid_add_type,type,
comm_monoid_add:
!>[TA: $tType] : $o ).
thf(ab_semigroup_add_type,type,
ab_semigroup_add:
!>[TA: $tType] : $o ).
thf(cancel_semigroup_add_type,type,
cancel_semigroup_add:
!>[TA: $tType] : $o ).
thf(cancel146912293up_add_type,type,
cancel146912293up_add:
!>[TA: $tType] : $o ).
thf(linord219039673up_add_type,type,
linord219039673up_add:
!>[TA: $tType] : $o ).
thf(fundam296178794t_poly_type,type,
fundam296178794t_poly:
!>[TA: $tType] : ( ( poly @ TA ) > TA > ( poly @ TA ) ) ).
thf(plus_plus_type,type,
plus_plus:
!>[TA: $tType] : ( TA > TA > TA ) ).
thf(zero_zero_type,type,
zero_zero:
!>[TA: $tType] : TA ).
thf(if_type,type,
if:
!>[TA: $tType] : ( bool > TA > TA > TA ) ).
thf(suc_type,type,
suc: nat > nat ).
thf(nat_case_type,type,
nat_case:
!>[TA: $tType] : ( TA > ( fun @ nat @ TA ) > ( fun @ nat @ TA ) ) ).
thf(semiri532925092at_aux_type,type,
semiri532925092at_aux:
!>[TA: $tType] : ( ( fun @ TA @ TA ) > nat > TA > TA ) ).
thf(abs_poly_type,type,
abs_poly:
!>[TA: $tType] : ( ( fun @ nat @ TA ) > ( poly @ TA ) ) ).
thf(coeff_type,type,
coeff:
!>[TA: $tType] : ( ( poly @ TA ) > ( fun @ nat @ TA ) ) ).
thf(degree_type,type,
degree:
!>[TA: $tType] : ( ( poly @ TA ) > nat ) ).
thf(monom_type,type,
monom:
!>[TA: $tType] : ( TA > nat > ( poly @ TA ) ) ).
thf(order_type,type,
order:
!>[TA: $tType] : ( TA > ( poly @ TA ) > nat ) ).
thf(pCons_type,type,
pCons:
!>[TA: $tType] : ( TA > ( poly @ TA ) > ( poly @ TA ) ) ).
thf(pcompose_type,type,
pcompose:
!>[TA: $tType] : ( ( poly @ TA ) > ( poly @ TA ) > ( poly @ TA ) ) ).
thf(poly1_type,type,
poly1:
!>[TA: $tType] : ( ( poly @ TA ) > ( fun @ TA @ TA ) ) ).
thf(poly_rec_type,type,
poly_rec:
!>[TA: $tType,TB: $tType] : ( TB > ( fun @ TA @ ( fun @ ( poly @ TA ) @ ( fun @ TB @ TB ) ) ) > ( poly @ TA ) > TB ) ).
thf(smult_type,type,
smult:
!>[TA: $tType] : ( TA > ( poly @ TA ) > ( poly @ TA ) ) ).
thf(synthetic_div_type,type,
synthetic_div:
!>[TA: $tType] : ( ( poly @ TA ) > TA > ( poly @ TA ) ) ).
thf(aa_type,type,
aa:
!>[TA: $tType,TB: $tType] : ( ( fun @ TB @ TA ) > TB > TA ) ).
thf(fFalse_type,type,
fFalse: bool ).
thf(fTrue_type,type,
fTrue: bool ).
thf(fequal_type,type,
fequal:
!>[TA: $tType] : ( TA > TA > bool ) ).
thf(pp_type,type,
pp: bool > $o ).
thf(h_type,type,
h: a ).
thf(112,axiom,
! [A: nat] :
( ( suc @ A )
!= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_55_nat_Osimps_I3_J) ).
thf(504,plain,
! [A: nat] :
( ( suc @ A )
!= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(74,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( ( aa @ nat @ TA @ ( coeff @ TA @ A ) @ ( degree @ TA @ A ) )
= ( zero_zero @ TA ) )
<=> ( A
= ( zero_zero @ ( poly @ TA ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_3_leading__coeff__0__iff) ).
thf(369,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( ( ( aa @ nat @ TA @ ( coeff @ TA @ A ) @ ( degree @ TA @ A ) )
= ( zero_zero @ TA ) )
=> ( A
= ( zero_zero @ ( poly @ TA ) ) ) )
& ( ( A
= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( aa @ nat @ TA @ ( coeff @ TA @ A ) @ ( degree @ TA @ A ) )
= ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(103,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ( ab_semigroup_add @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial_Opoly___Groups_Oab__semigroup__add) ).
thf(479,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ( ab_semigroup_add @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(64,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA] :
( ( degree @ TA @ ( pCons @ TA @ A @ ( zero_zero @ ( poly @ TA ) ) ) )
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_27_degree__pCons__0) ).
thf(333,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA] :
( ( degree @ TA @ ( pCons @ TA @ A @ ( zero_zero @ ( poly @ TA ) ) ) )
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(127,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA] :
( ( aa @ TA @ TA @ ( poly1 @ TA @ ( zero_zero @ ( poly @ TA ) ) ) @ A )
= ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_10_poly__0) ).
thf(555,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA] :
( ( aa @ TA @ TA @ ( poly1 @ TA @ ( zero_zero @ ( poly @ TA ) ) ) @ A )
= ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(104,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA,C: poly @ TA] :
( ( aa @ TA @ TA @ ( poly1 @ TA @ ( plus_plus @ ( poly @ TA ) @ C @ B ) ) @ A )
= ( plus_plus @ TA @ ( aa @ TA @ TA @ ( poly1 @ TA @ C ) @ A ) @ ( aa @ TA @ TA @ ( poly1 @ TA @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_70_poly__add) ).
thf(481,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA,C: poly @ TA] :
( ( aa @ TA @ TA @ ( poly1 @ TA @ ( plus_plus @ ( poly @ TA ) @ C @ B ) ) @ A )
= ( plus_plus @ TA @ ( aa @ TA @ TA @ ( poly1 @ TA @ C ) @ A ) @ ( aa @ TA @ TA @ ( poly1 @ TA @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(94,axiom,
! [TA: $tType,TB: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: TA,C: fun @ TA @ ( fun @ ( poly @ TA ) @ ( fun @ TB @ TB ) ),D: TB] :
( ( poly_rec @ TB @ TA @ D @ C @ ( pCons @ TA @ B @ A ) )
= ( aa @ TB @ TB @ ( aa @ ( poly @ TA ) @ ( fun @ TB @ TB ) @ ( aa @ TA @ ( fun @ ( poly @ TA ) @ ( fun @ TB @ TB ) ) @ C @ B ) @ A ) @ ( if @ TB @ ( fequal @ ( poly @ TA ) @ A @ ( zero_zero @ ( poly @ TA ) ) ) @ D @ ( poly_rec @ TB @ TA @ D @ C @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_25_poly__rec_Osimps) ).
thf(449,plain,
! [TA: $tType,TB: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: TA,C: fun @ TA @ ( fun @ ( poly @ TA ) @ ( fun @ TB @ TB ) ),D: TB] :
( ( poly_rec @ TB @ TA @ D @ C @ ( pCons @ TA @ B @ A ) )
= ( aa @ TB @ TB @ ( aa @ ( poly @ TA ) @ ( fun @ TB @ TB ) @ ( aa @ TA @ ( fun @ ( poly @ TA ) @ ( fun @ TB @ TB ) ) @ C @ B ) @ A ) @ ( if @ TB @ ( fequal @ ( poly @ TA ) @ A @ ( zero_zero @ ( poly @ TA ) ) ) @ D @ ( poly_rec @ TB @ TA @ D @ C @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(114,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: TA] :
( ( fundam296178794t_poly @ TA @ ( pCons @ TA @ B @ ( zero_zero @ ( poly @ TA ) ) ) @ A )
= ( pCons @ TA @ B @ ( zero_zero @ ( poly @ TA ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_4_offset__poly__single) ).
thf(511,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: TA] :
( ( fundam296178794t_poly @ TA @ ( pCons @ TA @ B @ ( zero_zero @ ( poly @ TA ) ) ) @ A )
= ( pCons @ TA @ B @ ( zero_zero @ ( poly @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(117,axiom,
! [TA: $tType] :
( ( monoid_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_72_add__0__left) ).
thf(523,plain,
! [TA: $tType] :
( ( monoid_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(135,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA,B: poly @ TA] :
( ( ( B
= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( degree @ TA @ ( pCons @ TA @ A @ B ) )
= ( zero_zero @ nat ) ) )
& ( ( B
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( degree @ TA @ ( pCons @ TA @ A @ B ) )
= ( suc @ ( degree @ TA @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_33_degree__pCons__eq__if) ).
thf(582,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA,B: poly @ TA] :
( ( ( B
= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( degree @ TA @ ( pCons @ TA @ A @ B ) )
= ( zero_zero @ nat ) ) )
& ( ( B
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( degree @ TA @ ( pCons @ TA @ A @ B ) )
= ( suc @ ( degree @ TA @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(78,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA] :
( ( ( zero_zero @ TA )
= A )
<=> ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_2_zero__reorient) ).
thf(385,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA] :
( ( ( ( zero_zero @ TA )
= A )
=> ( A
= ( zero_zero @ TA ) ) )
& ( ( A
= ( zero_zero @ TA ) )
=> ( ( zero_zero @ TA )
= A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(4,axiom,
! [TA: $tType,A: TA,B: TA] :
( ( if @ TA @ fFalse @ B @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_If_2_1_T) ).
thf(142,plain,
! [TA: $tType,A: TA,B: TA] :
( ( if @ TA @ fFalse @ B @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(121,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( aa @ nat @ TA @ ( coeff @ TA @ ( pCons @ TA @ B @ A ) ) @ ( zero_zero @ nat ) )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_23_coeff__pCons__0) ).
thf(538,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( aa @ nat @ TA @ ( coeff @ TA @ ( pCons @ TA @ B @ A ) ) @ ( zero_zero @ nat ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(8,axiom,
semiring_1 @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Osemiring__1) ).
thf(153,plain,
semiring_1 @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(14,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat,B: poly @ TA,C: TA] :
( ( aa @ nat @ TA @ ( coeff @ TA @ ( pCons @ TA @ C @ B ) ) @ ( suc @ A ) )
= ( aa @ nat @ TA @ ( coeff @ TA @ B ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_42_coeff__pCons__Suc) ).
thf(174,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat,B: poly @ TA,C: TA] :
( ( aa @ nat @ TA @ ( coeff @ TA @ ( pCons @ TA @ C @ B ) ) @ ( suc @ A ) )
= ( aa @ nat @ TA @ ( coeff @ TA @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(18,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( abs_poly @ TA @ ( coeff @ TA @ A ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_85_coeff__inverse) ).
thf(185,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( abs_poly @ TA @ ( coeff @ TA @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(87,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: poly @ TA,B: TA,C: TA] :
( ( smult @ TA @ ( plus_plus @ TA @ C @ B ) @ A )
= ( plus_plus @ ( poly @ TA ) @ ( smult @ TA @ C @ A ) @ ( smult @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_71_smult__add__left) ).
thf(421,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: poly @ TA,B: TA,C: TA] :
( ( smult @ TA @ ( plus_plus @ TA @ C @ B ) @ A )
= ( plus_plus @ ( poly @ TA ) @ ( smult @ TA @ C @ A ) @ ( smult @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(7,axiom,
zero @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ozero) ).
thf(152,plain,
zero @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(50,axiom,
cancel146912293up_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add) ).
thf(289,plain,
cancel146912293up_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(59,axiom,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ( cancel146912293up_add @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial_Opoly___Groups_Ocancel__ab__semigroup__add) ).
thf(316,plain,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ( cancel146912293up_add @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(11,axiom,
comm_semiring_1 @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Ocomm__semiring__1) ).
thf(164,plain,
comm_semiring_1 @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(61,axiom,
! [TA: $tType,TB: $tType] :
( ( zero @ TB )
=> ! [A: TA,B: fun @ TB @ ( fun @ ( poly @ TB ) @ ( fun @ TA @ TA ) )] :
( ( ( aa @ TA @ TA @ ( aa @ ( poly @ TB ) @ ( fun @ TA @ TA ) @ ( aa @ TB @ ( fun @ ( poly @ TB ) @ ( fun @ TA @ TA ) ) @ B @ ( zero_zero @ TB ) ) @ ( zero_zero @ ( poly @ TB ) ) ) @ A )
= A )
=> ( ( poly_rec @ TA @ TB @ A @ B @ ( zero_zero @ ( poly @ TB ) ) )
= A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_5_poly__rec__0) ).
thf(321,plain,
! [TA: $tType,TB: $tType] :
( ( zero @ TB )
=> ! [A: TA,B: fun @ TB @ ( fun @ ( poly @ TB ) @ ( fun @ TA @ TA ) )] :
( ( ( aa @ TA @ TA @ ( aa @ ( poly @ TB ) @ ( fun @ TA @ TA ) @ ( aa @ TB @ ( fun @ ( poly @ TB ) @ ( fun @ TA @ TA ) ) @ B @ ( zero_zero @ TB ) ) @ ( zero_zero @ ( poly @ TB ) ) ) @ A )
= A )
=> ( ( poly_rec @ TA @ TB @ A @ B @ ( zero_zero @ ( poly @ TB ) ) )
= A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(115,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat,B: TA] :
( ( ( monom @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) )
<=> ( B
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_8_monom__eq__0__iff) ).
thf(514,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat,B: TA] :
( ( ( ( monom @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) )
=> ( B
= ( zero_zero @ TA ) ) )
& ( ( B
= ( zero_zero @ TA ) )
=> ( ( monom @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(69,axiom,
! [TA: $tType] :
( ( semiring_1 @ TA )
=> ! [A: TA,B: fun @ TA @ TA] :
( ( semiri532925092at_aux @ TA @ B @ ( zero_zero @ nat ) @ A )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_87_of__nat__aux_Osimps_I1_J) ).
thf(348,plain,
! [TA: $tType] :
( ( semiring_1 @ TA )
=> ! [A: TA,B: fun @ TA @ TA] :
( ( semiri532925092at_aux @ TA @ B @ ( zero_zero @ nat ) @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(89,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( linordered_idom @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial_Opoly___Rings_Olinordered__idom) ).
thf(428,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( linordered_idom @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(57,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ( comm_semiring_0 @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial_Opoly___Rings_Ocomm__semiring__0) ).
thf(312,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ( comm_semiring_0 @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(24,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: poly @ TA] :
( ! [C: nat] :
( ( aa @ nat @ TA @ ( coeff @ TA @ B ) @ C )
= ( aa @ nat @ TA @ ( coeff @ TA @ A ) @ C ) )
=> ( B = A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_34_poly__ext) ).
thf(206,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: poly @ TA] :
( ! [C: nat] :
( ( aa @ nat @ TA @ ( coeff @ TA @ B ) @ C )
= ( aa @ nat @ TA @ ( coeff @ TA @ A ) @ C ) )
=> ( B = A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(42,axiom,
! [TA: $tType,A: TA,B: TA] :
( ( if @ TA @ fTrue @ B @ A )
= B ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_If_1_1_T) ).
thf(270,plain,
! [TA: $tType,A: TA,B: TA] :
( ( if @ TA @ fTrue @ B @ A )
= B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(124,axiom,
! [TA: $tType,TB: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: TA,C: TB,D: fun @ TA @ ( fun @ ( poly @ TA ) @ ( fun @ TB @ TB ) )] :
( ( ( aa @ TB @ TB @ ( aa @ ( poly @ TA ) @ ( fun @ TB @ TB ) @ ( aa @ TA @ ( fun @ ( poly @ TA ) @ ( fun @ TB @ TB ) ) @ D @ ( zero_zero @ TA ) ) @ ( zero_zero @ ( poly @ TA ) ) ) @ C )
= C )
=> ( ( poly_rec @ TB @ TA @ C @ D @ ( pCons @ TA @ B @ A ) )
= ( aa @ TB @ TB @ ( aa @ ( poly @ TA ) @ ( fun @ TB @ TB ) @ ( aa @ TA @ ( fun @ ( poly @ TA ) @ ( fun @ TB @ TB ) ) @ D @ B ) @ A ) @ ( poly_rec @ TB @ TA @ C @ D @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_28_poly__rec__pCons) ).
thf(546,plain,
! [TA: $tType,TB: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: TA,C: TB,D: fun @ TA @ ( fun @ ( poly @ TA ) @ ( fun @ TB @ TB ) )] :
( ( ( aa @ TB @ TB @ ( aa @ ( poly @ TA ) @ ( fun @ TB @ TB ) @ ( aa @ TA @ ( fun @ ( poly @ TA ) @ ( fun @ TB @ TB ) ) @ D @ ( zero_zero @ TA ) ) @ ( zero_zero @ ( poly @ TA ) ) ) @ C )
= C )
=> ( ( poly_rec @ TB @ TA @ C @ D @ ( pCons @ TA @ B @ A ) )
= ( aa @ TB @ TB @ ( aa @ ( poly @ TA ) @ ( fun @ TB @ TB ) @ ( aa @ TA @ ( fun @ ( poly @ TA ) @ ( fun @ TB @ TB ) ) @ D @ B ) @ A ) @ ( poly_rec @ TB @ TA @ C @ D @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(30,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ( zero @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clar_Rings_Ocomm__semiring__0__Groups_Ozero) ).
thf(233,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ( zero @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(95,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_93_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) ).
thf(452,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(39,axiom,
! [A: nat,B: nat] :
( ( plus_plus @ nat @ B @ ( suc @ A ) )
= ( suc @ ( plus_plus @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_95_add__Suc__right) ).
thf(260,plain,
! [A: nat,B: nat] :
( ( plus_plus @ nat @ B @ ( suc @ A ) )
= ( suc @ ( plus_plus @ nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(129,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ( comm_monoid_add @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial_Opoly___Groups_Ocomm__monoid__add) ).
thf(561,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ( comm_monoid_add @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(79,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( pCons @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) )
<=> ( ( B
= ( zero_zero @ TA ) )
& ( A
= ( zero_zero @ ( poly @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_15_pCons__eq__0__iff) ).
thf(392,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( ( pCons @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( B
= ( zero_zero @ TA ) )
& ( A
= ( zero_zero @ ( poly @ TA ) ) ) ) )
& ( ( ( B
= ( zero_zero @ TA ) )
& ( A
= ( zero_zero @ ( poly @ TA ) ) ) )
=> ( ( pCons @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(54,axiom,
! [TA: $tType] :
( ( idom @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( smult @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) )
<=> ( ( B
= ( zero_zero @ TA ) )
| ( A
= ( zero_zero @ ( poly @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_41_smult__eq__0__iff) ).
thf(297,plain,
! [TA: $tType] :
( ( idom @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( ( smult @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( B
= ( zero_zero @ TA ) )
| ( A
= ( zero_zero @ ( poly @ TA ) ) ) ) )
& ( ( ( B
= ( zero_zero @ TA ) )
| ( A
= ( zero_zero @ ( poly @ TA ) ) ) )
=> ( ( smult @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(88,axiom,
! [A: nat] :
( ( suc @ A )
!= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_53_Suc__neq__Zero) ).
thf(424,plain,
! [A: nat] :
( ( suc @ A )
!= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(12,axiom,
! [TA: $tType,TB: $tType,A: fun @ TA @ TB,B: fun @ TA @ TB] :
( ! [C: TA] :
( ( aa @ TA @ TB @ B @ C )
= ( aa @ TA @ TB @ A @ C ) )
=> ( B = A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_73_ext) ).
thf(165,plain,
! [TA: $tType,TB: $tType,A: fun @ TA @ TB,B: fun @ TA @ TB] :
( ! [C: TA] :
( ( aa @ TA @ TB @ B @ C )
= ( aa @ TA @ TB @ A @ C ) )
=> ( B = A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(45,axiom,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ C @ B )
= ( plus_plus @ TA @ C @ A ) )
=> ( B = A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_79_add__left__imp__eq) ).
thf(275,plain,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ C @ B )
= ( plus_plus @ TA @ C @ A ) )
=> ( B = A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(1,conjecture,
( ( degree @ a @ ( fundam296178794t_poly @ a @ ( zero_zero @ ( poly @ a ) ) @ h ) )
= ( degree @ a @ ( zero_zero @ ( poly @ a ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).
thf(2,negated_conjecture,
( ( degree @ a @ ( fundam296178794t_poly @ a @ ( zero_zero @ ( poly @ a ) ) @ h ) )
!= ( degree @ a @ ( zero_zero @ ( poly @ a ) ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(137,plain,
( ( degree @ a @ ( fundam296178794t_poly @ a @ ( zero_zero @ ( poly @ a ) ) @ h ) )
!= ( degree @ a @ ( zero_zero @ ( poly @ a ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(101,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA,C: TA] :
( ( ( plus_plus @ ( poly @ TA ) @ ( smult @ TA @ C @ B ) @ ( pCons @ TA @ A @ B ) )
= ( zero_zero @ ( poly @ TA ) ) )
=> ( B
= ( zero_zero @ ( poly @ TA ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_88_offset__poly__eq__0__lemma) ).
thf(467,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA,C: TA] :
( ( ( plus_plus @ ( poly @ TA ) @ ( smult @ TA @ C @ B ) @ ( pCons @ TA @ A @ B ) )
= ( zero_zero @ ( poly @ TA ) ) )
=> ( B
= ( zero_zero @ ( poly @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(105,axiom,
! [A: nat] :
( ( suc @ A )
!= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_56_Suc__not__Zero) ).
thf(484,plain,
! [A: nat] :
( ( suc @ A )
!= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(22,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: poly @ TA] :
( ( ( coeff @ TA @ B )
= ( coeff @ TA @ A ) )
<=> ( B = A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_22_coeff__inject) ).
thf(197,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: poly @ TA] :
( ( ( ( coeff @ TA @ B )
= ( coeff @ TA @ A ) )
=> ( B = A ) )
& ( ( B = A )
=> ( ( coeff @ TA @ B )
= ( coeff @ TA @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(92,axiom,
! [TA: $tType] :
( ( ( ring_char_0 @ TA )
& ( idom @ TA ) )
=> ! [A: poly @ TA] :
( ( ( poly1 @ TA @ A )
= ( poly1 @ TA @ ( zero_zero @ ( poly @ TA ) ) ) )
<=> ( A
= ( zero_zero @ ( poly @ TA ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_26_poly__zero) ).
thf(435,plain,
! [TA: $tType] :
( ( ( ring_char_0 @ TA )
& ( idom @ TA ) )
=> ! [A: poly @ TA] :
( ( ( ( poly1 @ TA @ A )
= ( poly1 @ TA @ ( zero_zero @ ( poly @ TA ) ) ) )
=> ( A
= ( zero_zero @ ( poly @ TA ) ) ) )
& ( ( A
= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( poly1 @ TA @ A )
= ( poly1 @ TA @ ( zero_zero @ ( poly @ TA ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(98,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ring_char_0 @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial_Opoly___Int_Oring__char__0) ).
thf(461,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ring_char_0 @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(108,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA] :
( ( synthetic_div @ TA @ ( zero_zero @ ( poly @ TA ) ) @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_39_synthetic__div__0) ).
thf(493,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA] :
( ( synthetic_div @ TA @ ( zero_zero @ ( poly @ TA ) ) @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(62,axiom,
! [TA: $tType] :
( ( idom @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( B
= ( zero_zero @ TA ) )
=> ( ( degree @ TA @ ( smult @ TA @ B @ A ) )
= ( zero_zero @ nat ) ) )
& ( ( B
!= ( zero_zero @ TA ) )
=> ( ( degree @ TA @ ( smult @ TA @ B @ A ) )
= ( degree @ TA @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_37_degree__smult__eq) ).
thf(324,plain,
! [TA: $tType] :
( ( idom @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( ( B
= ( zero_zero @ TA ) )
=> ( ( degree @ TA @ ( smult @ TA @ B @ A ) )
= ( zero_zero @ nat ) ) )
& ( ( B
!= ( zero_zero @ TA ) )
=> ( ( degree @ TA @ ( smult @ TA @ B @ A ) )
= ( degree @ TA @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(10,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: TA,C: poly @ TA,D: TA] :
( ( ( pCons @ TA @ D @ C )
= ( pCons @ TA @ B @ A ) )
<=> ( ( D = B )
& ( C = A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_13_pCons__eq__iff) ).
thf(156,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: TA,C: poly @ TA,D: TA] :
( ( ( ( pCons @ TA @ D @ C )
= ( pCons @ TA @ B @ A ) )
=> ( ( D = B )
& ( C = A ) ) )
& ( ( ( D = B )
& ( C = A ) )
=> ( ( pCons @ TA @ D @ C )
= ( pCons @ TA @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(81,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: TA,C: poly @ TA] :
( ( aa @ TA @ TA @ ( poly1 @ TA @ ( fundam296178794t_poly @ TA @ C @ B ) ) @ A )
= ( aa @ TA @ TA @ ( poly1 @ TA @ C ) @ ( plus_plus @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_86_poly__offset__poly) ).
thf(403,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: TA,C: poly @ TA] :
( ( aa @ TA @ TA @ ( poly1 @ TA @ ( fundam296178794t_poly @ TA @ C @ B ) ) @ A )
= ( aa @ TA @ TA @ ( poly1 @ TA @ C ) @ ( plus_plus @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(19,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( coeff @ TA @ ( pCons @ TA @ B @ A ) )
= ( nat_case @ TA @ B @ ( coeff @ TA @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_36_coeff__pCons) ).
thf(188,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( coeff @ TA @ ( pCons @ TA @ B @ A ) )
= ( nat_case @ TA @ B @ ( coeff @ TA @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(110,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ~ ! [B: nat] :
( A
!= ( suc @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_58_nat_Oexhaust) ).
thf(499,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ~ ! [B: nat] :
( A
!= ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(106,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat] :
( ( monom @ TA @ ( zero_zero @ TA ) @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_7_monom__eq__0) ).
thf(488,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat] :
( ( monom @ TA @ ( zero_zero @ TA ) @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(21,axiom,
! [A: nat,B: nat] :
( ( ( suc @ B )
= ( suc @ A ) )
=> ( B = A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_52_Suc__inject) ).
thf(193,plain,
! [A: nat,B: nat] :
( ( ( suc @ B )
= ( suc @ A ) )
=> ( B = A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(118,axiom,
! [TA: $tType] :
( ( linord219039673up_add @ TA )
=> ! [A: TA] :
( ( ( plus_plus @ TA @ A @ A )
= ( zero_zero @ TA ) )
<=> ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_91_double__eq__0__iff) ).
thf(526,plain,
! [TA: $tType] :
( ( linord219039673up_add @ TA )
=> ! [A: TA] :
( ( ( ( plus_plus @ TA @ A @ A )
= ( zero_zero @ TA ) )
=> ( A
= ( zero_zero @ TA ) ) )
& ( ( A
= ( zero_zero @ TA ) )
=> ( ( plus_plus @ TA @ A @ A )
= ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(131,axiom,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_57_nat_Osimps_I2_J) ).
thf(569,plain,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(111,axiom,
comm_semiring_0 @ a,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tfree_0) ).
thf(503,plain,
comm_semiring_0 @ a,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(33,axiom,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ C @ B )
= ( plus_plus @ TA @ A @ B ) )
=> ( C = A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_77_add__right__imp__eq) ).
thf(237,plain,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ C @ B )
= ( plus_plus @ TA @ A @ B ) )
=> ( C = A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(113,axiom,
! [A: nat,B: fun @ nat @ bool] :
( ( pp @ ( aa @ nat @ bool @ B @ ( zero_zero @ nat ) ) )
=> ( ! [C: nat] :
( ( pp @ ( aa @ nat @ bool @ B @ C ) )
=> ( pp @ ( aa @ nat @ bool @ B @ ( suc @ C ) ) ) )
=> ( pp @ ( aa @ nat @ bool @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_60_nat__induct) ).
thf(508,plain,
! [A: nat,B: fun @ nat @ bool] :
( ( pp @ ( aa @ nat @ bool @ B @ ( zero_zero @ nat ) ) )
=> ( ! [C: nat] :
( ( pp @ ( aa @ nat @ bool @ B @ C ) )
=> ( pp @ ( aa @ nat @ bool @ B @ ( suc @ C ) ) ) )
=> ( pp @ ( aa @ nat @ bool @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(122,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( comm_semiring_1 @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial_Opoly___Rings_Ocomm__semiring__1) ).
thf(541,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( comm_semiring_1 @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(128,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: poly @ TA] :
( ( plus_plus @ ( poly @ TA ) @ ( zero_zero @ ( poly @ TA ) ) @ A )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_83_add__poly__code_I1_J) ).
thf(558,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: poly @ TA] :
( ( plus_plus @ ( poly @ TA ) @ ( zero_zero @ ( poly @ TA ) ) @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(84,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_92_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) ).
thf(412,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(90,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA,B: poly @ TA] :
( ( B
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( degree @ TA @ ( pCons @ TA @ A @ B ) )
= ( suc @ ( degree @ TA @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_44_degree__pCons__eq) ).
thf(430,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA,B: poly @ TA] :
( ( B
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( degree @ TA @ ( pCons @ TA @ A @ B ) )
= ( suc @ ( degree @ TA @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(6,axiom,
! [A: nat] :
( A
!= ( suc @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_50_n__not__Suc__n) ).
thf(148,plain,
! [A: nat] :
( A
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(41,axiom,
! [TA: $tType] :
( ( ( ring_char_0 @ TA )
& ( idom @ TA ) )
=> ! [A: poly @ TA,B: poly @ TA] :
( ( ( poly1 @ TA @ B )
= ( poly1 @ TA @ A ) )
<=> ( B = A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_21_poly__eq__iff) ).
thf(264,plain,
! [TA: $tType] :
( ( ( ring_char_0 @ TA )
& ( idom @ TA ) )
=> ! [A: poly @ TA,B: poly @ TA] :
( ( ( ( poly1 @ TA @ B )
= ( poly1 @ TA @ A ) )
=> ( B = A ) )
& ( ( B = A )
=> ( ( poly1 @ TA @ B )
= ( poly1 @ TA @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(132,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA] :
( ( monom @ TA @ A @ ( zero_zero @ nat ) )
= ( pCons @ TA @ A @ ( zero_zero @ ( poly @ TA ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_19_monom__0) ).
thf(573,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA] :
( ( monom @ TA @ A @ ( zero_zero @ nat ) )
= ( pCons @ TA @ A @ ( zero_zero @ ( poly @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(75,axiom,
! [TA: $tType] :
( ( idom @ TA )
=> ( idom @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial_Opoly___Rings_Oidom) ).
thf(375,plain,
! [TA: $tType] :
( ( idom @ TA )
=> ( idom @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(20,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ( monoid_add @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clar_Rings_Ocomm__semiring__0__Groups_Omonoid__add) ).
thf(191,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ( monoid_add @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(49,axiom,
! [TA: $tType,A: nat,B: fun @ nat @ TA,C: TA] :
( ( aa @ nat @ TA @ ( nat_case @ TA @ C @ B ) @ ( suc @ A ) )
= ( aa @ nat @ TA @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_47_nat__case__Suc) ).
thf(286,plain,
! [TA: $tType,A: nat,B: fun @ nat @ TA,C: TA] :
( ( aa @ nat @ TA @ ( nat_case @ TA @ C @ B ) @ ( suc @ A ) )
= ( aa @ nat @ TA @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(47,axiom,
~ ( pp @ fFalse ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_pp_1_1_U) ).
thf(281,plain,
~ ( pp @ fFalse ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(134,axiom,
! [A: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_97_plus__nat_Oadd__0) ).
thf(579,plain,
! [A: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(52,axiom,
! [TA: $tType,A: fun @ nat @ TA,B: TA] :
( ( aa @ nat @ TA @ ( nat_case @ TA @ B @ A ) @ ( zero_zero @ nat ) )
= B ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_48_nat__case__0) ).
thf(291,plain,
! [TA: $tType,A: fun @ nat @ TA,B: TA] :
( ( aa @ nat @ TA @ ( nat_case @ TA @ B @ A ) @ ( zero_zero @ nat ) )
= B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(77,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: poly @ TA] :
( ( plus_plus @ ( poly @ TA ) @ A @ ( zero_zero @ ( poly @ TA ) ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_84_add__poly__code_I2_J) ).
thf(382,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: poly @ TA] :
( ( plus_plus @ ( poly @ TA ) @ A @ ( zero_zero @ ( poly @ TA ) ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(32,axiom,
pp @ fTrue,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_pp_2_1_U) ).
thf(236,plain,
pp @ fTrue,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(16,axiom,
ab_semigroup_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oab__semigroup__add) ).
thf(181,plain,
ab_semigroup_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(82,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat,B: TA] :
( ( B
!= ( zero_zero @ TA ) )
=> ( ( degree @ TA @ ( monom @ TA @ B @ A ) )
= A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_11_degree__monom__eq) ).
thf(406,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat,B: TA] :
( ( B
!= ( zero_zero @ TA ) )
=> ( ( degree @ TA @ ( monom @ TA @ B @ A ) )
= A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(44,axiom,
cancel_semigroup_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ocancel__semigroup__add) ).
thf(274,plain,
cancel_semigroup_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(76,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA] :
( ( ( synthetic_div @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) )
<=> ( ( degree @ TA @ B )
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_32_synthetic__div__eq__0__iff) ).
thf(377,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA] :
( ( ( ( synthetic_div @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( degree @ TA @ B )
= ( zero_zero @ nat ) ) )
& ( ( ( degree @ TA @ B )
= ( zero_zero @ nat ) )
=> ( ( synthetic_div @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(29,axiom,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ ( plus_plus @ TA @ C @ B ) @ A )
= ( plus_plus @ TA @ C @ ( plus_plus @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_81_ab__semigroup__add__class_Oadd__ac_I1_J) ).
thf(230,plain,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ ( plus_plus @ TA @ C @ B ) @ A )
= ( plus_plus @ TA @ C @ ( plus_plus @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(36,axiom,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ C @ B )
= ( plus_plus @ TA @ C @ A ) )
<=> ( B = A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_66_add__left__cancel) ).
thf(245,plain,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( ( plus_plus @ TA @ C @ B )
= ( plus_plus @ TA @ C @ A ) )
=> ( B = A ) )
& ( ( B = A )
=> ( ( plus_plus @ TA @ C @ B )
= ( plus_plus @ TA @ C @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(60,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA] :
( ( fundam296178794t_poly @ TA @ ( zero_zero @ ( poly @ TA ) ) @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_0_offset__poly__0) ).
thf(318,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA] :
( ( fundam296178794t_poly @ TA @ ( zero_zero @ ( poly @ TA ) ) @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(120,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_74_add__0) ).
thf(535,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(116,axiom,
! [TA: $tType] :
( ( monoid_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_75_add__0__right) ).
thf(520,plain,
! [TA: $tType] :
( ( monoid_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(63,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat] :
( ( aa @ nat @ TA @ ( coeff @ TA @ ( zero_zero @ ( poly @ TA ) ) ) @ A )
= ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_17_coeff__0) ).
thf(330,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat] :
( ( aa @ nat @ TA @ ( coeff @ TA @ ( zero_zero @ ( poly @ TA ) ) ) @ A )
= ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(68,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_76_add_Ocomm__neutral) ).
thf(345,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(126,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA] :
( ( smult @ TA @ A @ ( zero_zero @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_38_smult__0__right) ).
thf(552,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA] :
( ( smult @ TA @ A @ ( zero_zero @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(34,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ( ab_semigroup_add @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clar_Rings_Ocomm__semiring__0__Groups_Oab__semigroup__add) ).
thf(240,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ( ab_semigroup_add @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(37,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( pCons @ TA @ B @ A )
= ( abs_poly @ TA @ ( nat_case @ TA @ B @ ( coeff @ TA @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_63_pCons__def) ).
thf(251,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: TA] :
( ( pCons @ TA @ B @ A )
= ( abs_poly @ TA @ ( nat_case @ TA @ B @ ( coeff @ TA @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(46,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA,C: poly @ TA] :
( ( aa @ TA @ TA @ ( poly1 @ TA @ ( pcompose @ TA @ C @ B ) ) @ A )
= ( aa @ TA @ TA @ ( poly1 @ TA @ C ) @ ( aa @ TA @ TA @ ( poly1 @ TA @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_24_poly__pcompose) ).
thf(278,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA,C: poly @ TA] :
( ( aa @ TA @ TA @ ( poly1 @ TA @ ( pcompose @ TA @ C @ B ) ) @ A )
= ( aa @ TA @ TA @ ( poly1 @ TA @ C ) @ ( aa @ TA @ TA @ ( poly1 @ TA @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(96,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA,C: TA] :
( ( fundam296178794t_poly @ TA @ ( pCons @ TA @ C @ B ) @ A )
= ( plus_plus @ ( poly @ TA ) @ ( smult @ TA @ A @ ( fundam296178794t_poly @ TA @ B @ A ) ) @ ( pCons @ TA @ C @ ( fundam296178794t_poly @ TA @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_89_offset__poly__pCons) ).
thf(455,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA,C: TA] :
( ( fundam296178794t_poly @ TA @ ( pCons @ TA @ C @ B ) @ A )
= ( plus_plus @ ( poly @ TA ) @ ( smult @ TA @ A @ ( fundam296178794t_poly @ TA @ B @ A ) ) @ ( pCons @ TA @ C @ ( fundam296178794t_poly @ TA @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(27,axiom,
! [TA: $tType,A: TA,B: TA] :
( ( B != A )
| ( pp @ ( fequal @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_2_1_T) ).
thf(223,plain,
! [TA: $tType,A: TA,B: TA] :
( ( B != A )
| ( pp @ ( fequal @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(55,axiom,
! [A: nat,B: fun @ nat @ bool] :
( ( pp @ ( aa @ nat @ bool @ B @ A ) )
=> ( ! [C: nat] :
( ( pp @ ( aa @ nat @ bool @ B @ ( suc @ C ) ) )
=> ( pp @ ( aa @ nat @ bool @ B @ C ) ) )
=> ( pp @ ( aa @ nat @ bool @ B @ ( zero_zero @ nat ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_59_zero__induct) ).
thf(306,plain,
! [A: nat,B: fun @ nat @ bool] :
( ( pp @ ( aa @ nat @ bool @ B @ A ) )
=> ( ! [C: nat] :
( ( pp @ ( aa @ nat @ bool @ B @ ( suc @ C ) ) )
=> ( pp @ ( aa @ nat @ bool @ B @ C ) ) )
=> ( pp @ ( aa @ nat @ bool @ B @ ( zero_zero @ nat ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(130,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA,B: nat,C: nat] :
( ( ( C = B )
=> ( ( aa @ nat @ TA @ ( coeff @ TA @ ( monom @ TA @ A @ C ) ) @ B )
= A ) )
& ( ( C != B )
=> ( ( aa @ nat @ TA @ ( coeff @ TA @ ( monom @ TA @ A @ C ) ) @ B )
= ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_18_coeff__monom) ).
thf(563,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA,B: nat,C: nat] :
( ( ( C = B )
=> ( ( aa @ nat @ TA @ ( coeff @ TA @ ( monom @ TA @ A @ C ) ) @ B )
= A ) )
& ( ( C != B )
=> ( ( aa @ nat @ TA @ ( coeff @ TA @ ( monom @ TA @ A @ C ) ) @ B )
= ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(25,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA,B: nat,C: TA] :
( ( ( monom @ TA @ C @ B )
= ( monom @ TA @ A @ B ) )
<=> ( C = A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_14_monom__eq__iff) ).
thf(209,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA,B: nat,C: TA] :
( ( ( ( monom @ TA @ C @ B )
= ( monom @ TA @ A @ B ) )
=> ( C = A ) )
& ( ( C = A )
=> ( ( monom @ TA @ C @ B )
= ( monom @ TA @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(67,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: poly @ TA,B: TA,C: poly @ TA,D: TA] :
( ( plus_plus @ ( poly @ TA ) @ ( pCons @ TA @ D @ C ) @ ( pCons @ TA @ B @ A ) )
= ( pCons @ TA @ ( plus_plus @ TA @ D @ B ) @ ( plus_plus @ ( poly @ TA ) @ C @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_68_add__pCons) ).
thf(342,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: poly @ TA,B: TA,C: poly @ TA,D: TA] :
( ( plus_plus @ ( poly @ TA ) @ ( pCons @ TA @ D @ C ) @ ( pCons @ TA @ B @ A ) )
= ( pCons @ TA @ ( plus_plus @ TA @ D @ B ) @ ( plus_plus @ ( poly @ TA ) @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(97,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( degree @ TA @ ( zero_zero @ ( poly @ TA ) ) )
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_9_degree__0) ).
thf(458,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( degree @ TA @ ( zero_zero @ ( poly @ TA ) ) )
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(5,axiom,
! [TA: $tType,A: TA,B: TA] :
( ~ ( pp @ ( fequal @ TA @ B @ A ) )
| ( B = A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_1_1_T) ).
thf(145,plain,
! [TA: $tType,A: TA,B: TA] :
( ~ ( pp @ ( fequal @ TA @ B @ A ) )
| ( B = A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(83,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA] :
( ( plus_plus @ ( poly @ TA ) @ B @ ( smult @ TA @ A @ ( synthetic_div @ TA @ B @ A ) ) )
= ( pCons @ TA @ ( aa @ TA @ TA @ ( poly1 @ TA @ B ) @ A ) @ ( synthetic_div @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_90_synthetic__div__correct) ).
thf(409,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA] :
( ( plus_plus @ ( poly @ TA ) @ B @ ( smult @ TA @ A @ ( synthetic_div @ TA @ B @ A ) ) )
= ( pCons @ TA @ ( aa @ TA @ TA @ ( poly1 @ TA @ B ) @ A ) @ ( synthetic_div @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(40,axiom,
comm_monoid_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ocomm__monoid__add) ).
thf(263,plain,
comm_monoid_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(93,axiom,
! [TA: $tType] :
( ( idom @ TA )
=> ! [A: TA,B: poly @ TA] :
( ( ( aa @ TA @ TA @ ( poly1 @ TA @ B ) @ A )
= ( zero_zero @ TA ) )
<=> ( ( B
= ( zero_zero @ ( poly @ TA ) ) )
| ( ( order @ TA @ A @ B )
!= ( zero_zero @ nat ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_30_order__root) ).
thf(441,plain,
! [TA: $tType] :
( ( idom @ TA )
=> ! [A: TA,B: poly @ TA] :
( ( ( ( aa @ TA @ TA @ ( poly1 @ TA @ B ) @ A )
= ( zero_zero @ TA ) )
=> ( ( B
= ( zero_zero @ ( poly @ TA ) ) )
| ( ( order @ TA @ A @ B )
!= ( zero_zero @ nat ) ) ) )
& ( ( ( B
= ( zero_zero @ ( poly @ TA ) ) )
| ( ( order @ TA @ A @ B )
!= ( zero_zero @ nat ) ) )
=> ( ( aa @ TA @ TA @ ( poly1 @ TA @ B ) @ A )
= ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(99,axiom,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ( cancel1352612707id_add @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial_Opoly___Groups_Ocancel__comm__monoid__add) ).
thf(463,plain,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ( cancel1352612707id_add @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(66,axiom,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_54_Zero__neq__Suc) ).
thf(338,plain,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(72,axiom,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_49_Zero__not__Suc) ).
thf(359,plain,
! [A: nat] :
( ( zero_zero @ nat )
!= ( suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(15,axiom,
! [A: nat] :
( ( suc @ A )
!= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_51_Suc__n__not__n) ).
thf(177,plain,
! [A: nat] :
( ( suc @ A )
!= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(17,axiom,
! [A: nat,B: nat] :
( ( plus_plus @ nat @ ( suc @ B ) @ A )
= ( suc @ ( plus_plus @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_96_add__Suc) ).
thf(182,plain,
! [A: nat,B: nat] :
( ( plus_plus @ nat @ ( suc @ B ) @ A )
= ( suc @ ( plus_plus @ nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(102,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus @ nat @ B @ A )
= ( zero_zero @ nat ) )
<=> ( ( B
= ( zero_zero @ nat ) )
& ( A
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_94_add__is__0) ).
thf(470,plain,
! [A: nat,B: nat] :
( ( ( ( plus_plus @ nat @ B @ A )
= ( zero_zero @ nat ) )
=> ( ( B
= ( zero_zero @ nat ) )
& ( A
= ( zero_zero @ nat ) ) ) )
& ( ( ( B
= ( zero_zero @ nat ) )
& ( A
= ( zero_zero @ nat ) ) )
=> ( ( plus_plus @ nat @ B @ A )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(133,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: poly @ TA] :
( ( pcompose @ TA @ ( zero_zero @ ( poly @ TA ) ) @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_12_pcompose__0) ).
thf(576,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: poly @ TA] :
( ( pcompose @ TA @ ( zero_zero @ ( poly @ TA ) ) @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(65,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( linord219039673up_add @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial_Opoly___Groups_Olinordered__ab__group__add) ).
thf(336,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( linord219039673up_add @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(9,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ( comm_monoid_add @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clar_Rings_Ocomm__semiring__0__Groups_Ocomm__monoid__add) ).
thf(154,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ( comm_monoid_add @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(58,axiom,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ( cancel_semigroup_add @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial_Opoly___Groups_Ocancel__semigroup__add) ).
thf(314,plain,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ( cancel_semigroup_add @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(100,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ( zero @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial_Opoly___Groups_Ozero) ).
thf(465,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ( zero @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(73,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA] :
( ( ( fundam296178794t_poly @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) )
<=> ( B
= ( zero_zero @ ( poly @ TA ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1_offset__poly__eq__0__iff) ).
thf(363,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA] :
( ( ( ( fundam296178794t_poly @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) )
=> ( B
= ( zero_zero @ ( poly @ TA ) ) ) )
& ( ( B
= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( fundam296178794t_poly @ TA @ B @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(23,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA,C: TA] :
( ( synthetic_div @ TA @ ( pCons @ TA @ C @ B ) @ A )
= ( pCons @ TA @ ( aa @ TA @ TA @ ( poly1 @ TA @ B ) @ A ) @ ( synthetic_div @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_31_synthetic__div__pCons) ).
thf(203,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA,C: TA] :
( ( synthetic_div @ TA @ ( pCons @ TA @ C @ B ) @ A )
= ( pCons @ TA @ ( aa @ TA @ TA @ ( poly1 @ TA @ B ) @ A ) @ ( synthetic_div @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(136,axiom,
! [TA: $tType] :
( ( linord219039673up_add @ TA )
=> ! [A: TA] :
( ( ( zero_zero @ TA )
= ( plus_plus @ TA @ A @ A ) )
<=> ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_67_double__zero__sym) ).
thf(588,plain,
! [TA: $tType] :
( ( linord219039673up_add @ TA )
=> ! [A: TA] :
( ( ( ( zero_zero @ TA )
= ( plus_plus @ TA @ A @ A ) )
=> ( A
= ( zero_zero @ TA ) ) )
& ( ( A
= ( zero_zero @ TA ) )
=> ( ( zero_zero @ TA )
= ( plus_plus @ TA @ A @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(70,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: poly @ TA] :
( ( smult @ TA @ ( zero_zero @ TA ) @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_40_smult__0__left) ).
thf(351,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: poly @ TA] :
( ( smult @ TA @ ( zero_zero @ TA ) @ A )
= ( zero_zero @ ( poly @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(3,axiom,
! [A: bool] :
( ( A = fTrue )
| ( A = fFalse ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_If_3_1_T) ).
thf(139,plain,
! [A: bool] :
( ( A = fTrue )
| ( A = fFalse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(86,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( pCons @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_16_pCons__0__0) ).
thf(418,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ( ( pCons @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ ( poly @ TA ) ) )
= ( zero_zero @ ( poly @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(51,axiom,
cancel1352612707id_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ocancel__comm__monoid__add) ).
thf(290,plain,
cancel1352612707id_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(31,axiom,
comm_semiring_0 @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Ocomm__semiring__0) ).
thf(235,plain,
comm_semiring_0 @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(91,axiom,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( semiring_1 @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial_Opoly___Rings_Osemiring__1) ).
thf(433,plain,
! [TA: $tType] :
( ( comm_semiring_1 @ TA )
=> ( semiring_1 @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(125,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat,B: TA] :
( ( monom @ TA @ B @ ( suc @ A ) )
= ( pCons @ TA @ ( zero_zero @ TA ) @ ( monom @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_45_monom__Suc) ).
thf(549,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: nat,B: TA] :
( ( monom @ TA @ B @ ( suc @ A ) )
= ( pCons @ TA @ ( zero_zero @ TA ) @ ( monom @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(13,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: poly @ TA] :
( ( B = A )
<=> ! [C: nat] :
( ( aa @ nat @ TA @ ( coeff @ TA @ B ) @ C )
= ( aa @ nat @ TA @ ( coeff @ TA @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_20_expand__poly__eq) ).
thf(168,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: poly @ TA] :
( ( ( B = A )
=> ! [C: nat] :
( ( aa @ nat @ TA @ ( coeff @ TA @ B ) @ C )
= ( aa @ nat @ TA @ ( coeff @ TA @ A ) @ C ) ) )
& ( ! [C: nat] :
( ( aa @ nat @ TA @ ( coeff @ TA @ B ) @ C )
= ( aa @ nat @ TA @ ( coeff @ TA @ A ) @ C ) )
=> ( B = A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(43,axiom,
monoid_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Omonoid__add) ).
thf(273,plain,
monoid_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(35,axiom,
! [TA: $tType] :
( ( semiring_1 @ TA )
=> ! [A: TA,B: nat,C: fun @ TA @ TA] :
( ( semiri532925092at_aux @ TA @ C @ ( suc @ B ) @ A )
= ( semiri532925092at_aux @ TA @ C @ B @ ( aa @ TA @ TA @ C @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_62_of__nat__aux_Osimps_I2_J) ).
thf(242,plain,
! [TA: $tType] :
( ( semiring_1 @ TA )
=> ! [A: TA,B: nat,C: fun @ TA @ TA] :
( ( semiri532925092at_aux @ TA @ C @ ( suc @ B ) @ A )
= ( semiri532925092at_aux @ TA @ C @ B @ ( aa @ TA @ TA @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(53,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ? [B: nat] :
( A
= ( suc @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_61_not0__implies__Suc) ).
thf(294,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ? [B: nat] :
( A
= ( suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(80,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA,C: TA] :
( ( ( smult @ TA @ C @ B )
= ( pCons @ TA @ A @ B ) )
=> ( B
= ( zero_zero @ ( poly @ TA ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_43_synthetic__div__unique__lemma) ).
thf(400,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA,C: TA] :
( ( ( smult @ TA @ C @ B )
= ( pCons @ TA @ A @ B ) )
=> ( B
= ( zero_zero @ ( poly @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(119,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: fun @ ( poly @ TA ) @ bool] :
( ( pp @ ( aa @ ( poly @ TA ) @ bool @ B @ ( zero_zero @ ( poly @ TA ) ) ) )
=> ( ! [C: TA,D: poly @ TA] :
( ( pp @ ( aa @ ( poly @ TA ) @ bool @ B @ D ) )
=> ( pp @ ( aa @ ( poly @ TA ) @ bool @ B @ ( pCons @ TA @ C @ D ) ) ) )
=> ( pp @ ( aa @ ( poly @ TA ) @ bool @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_29_pCons__induct) ).
thf(532,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA,B: fun @ ( poly @ TA ) @ bool] :
( ( pp @ ( aa @ ( poly @ TA ) @ bool @ B @ ( zero_zero @ ( poly @ TA ) ) ) )
=> ( ! [C: TA,D: poly @ TA] :
( ( pp @ ( aa @ ( poly @ TA ) @ bool @ B @ D ) )
=> ( pp @ ( aa @ ( poly @ TA ) @ bool @ B @ ( pCons @ TA @ C @ D ) ) ) )
=> ( pp @ ( aa @ ( poly @ TA ) @ bool @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(48,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
~ ! [B: TA,C: poly @ TA] :
( A
!= ( pCons @ TA @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_35_pCons__cases) ).
thf(283,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
~ ! [B: TA,C: poly @ TA] :
( A
!= ( pCons @ TA @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(85,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: nat,B: poly @ TA,C: poly @ TA] :
( ( aa @ nat @ TA @ ( coeff @ TA @ ( plus_plus @ ( poly @ TA ) @ C @ B ) ) @ A )
= ( plus_plus @ TA @ ( aa @ nat @ TA @ ( coeff @ TA @ C ) @ A ) @ ( aa @ nat @ TA @ ( coeff @ TA @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_69_coeff__add) ).
thf(415,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: nat,B: poly @ TA,C: poly @ TA] :
( ( aa @ nat @ TA @ ( coeff @ TA @ ( plus_plus @ ( poly @ TA ) @ C @ B ) ) @ A )
= ( plus_plus @ TA @ ( aa @ nat @ TA @ ( coeff @ TA @ C ) @ A ) @ ( aa @ nat @ TA @ ( coeff @ TA @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(71,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA,C: TA,D: poly @ TA] :
( ( ( plus_plus @ ( poly @ TA ) @ D @ ( smult @ TA @ C @ B ) )
= ( pCons @ TA @ A @ B ) )
=> ( ( A
= ( aa @ TA @ TA @ ( poly1 @ TA @ D ) @ C ) )
& ( B
= ( synthetic_div @ TA @ D @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_64_synthetic__div__unique) ).
thf(354,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: TA,B: poly @ TA,C: TA,D: poly @ TA] :
( ( ( plus_plus @ ( poly @ TA ) @ D @ ( smult @ TA @ C @ B ) )
= ( pCons @ TA @ A @ B ) )
=> ( ( A
= ( aa @ TA @ TA @ ( poly1 @ TA @ D ) @ C ) )
& ( B
= ( synthetic_div @ TA @ D @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(56,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA,B: nat,C: TA] :
( ( plus_plus @ ( poly @ TA ) @ ( monom @ TA @ C @ B ) @ ( monom @ TA @ A @ B ) )
= ( monom @ TA @ ( plus_plus @ TA @ C @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_80_add__monom) ).
thf(309,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA,B: nat,C: TA] :
( ( plus_plus @ ( poly @ TA ) @ ( monom @ TA @ C @ B ) @ ( monom @ TA @ A @ B ) )
= ( monom @ TA @ ( plus_plus @ TA @ C @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(123,axiom,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: poly @ TA,B: poly @ TA,C: TA] :
( ( smult @ TA @ C @ ( plus_plus @ ( poly @ TA ) @ B @ A ) )
= ( plus_plus @ ( poly @ TA ) @ ( smult @ TA @ C @ B ) @ ( smult @ TA @ C @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_82_smult__add__right) ).
thf(543,plain,
! [TA: $tType] :
( ( comm_semiring_0 @ TA )
=> ! [A: poly @ TA,B: poly @ TA,C: TA] :
( ( smult @ TA @ C @ ( plus_plus @ ( poly @ TA ) @ B @ A ) )
= ( plus_plus @ ( poly @ TA ) @ ( smult @ TA @ C @ B ) @ ( smult @ TA @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(26,axiom,
! [A: nat,B: nat] :
( ( ( suc @ B )
= ( suc @ A ) )
<=> ( B = A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_46_nat_Oinject) ).
thf(215,plain,
! [A: nat,B: nat] :
( ( ( ( suc @ B )
= ( suc @ A ) )
=> ( B = A ) )
& ( ( B = A )
=> ( ( suc @ B )
= ( suc @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(107,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ( monoid_add @ ( poly @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial_Opoly___Groups_Omonoid__add) ).
thf(491,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ( monoid_add @ ( poly @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(109,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( A
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( aa @ nat @ TA @ ( coeff @ TA @ A ) @ ( degree @ TA @ A ) )
!= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_6_leading__coeff__neq__0) ).
thf(496,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: poly @ TA] :
( ( A
!= ( zero_zero @ ( poly @ TA ) ) )
=> ( ( aa @ nat @ TA @ ( coeff @ TA @ A ) @ ( degree @ TA @ A ) )
!= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(38,axiom,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ C @ B )
= ( plus_plus @ TA @ A @ B ) )
<=> ( C = A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_65_add__right__cancel) ).
thf(254,plain,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( ( plus_plus @ TA @ C @ B )
= ( plus_plus @ TA @ A @ B ) )
=> ( C = A ) )
& ( ( C = A )
=> ( ( plus_plus @ TA @ C @ B )
= ( plus_plus @ TA @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(28,axiom,
! [TA: $tType] :
( ( cancel146912293up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ C @ B )
= ( plus_plus @ TA @ C @ A ) )
=> ( B = A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_78_add__imp__eq) ).
thf(227,plain,
! [TA: $tType] :
( ( cancel146912293up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ C @ B )
= ( plus_plus @ TA @ C @ A ) )
=> ( B = A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(594,plain,
$false,
inference(e,[status(thm)],[504,369,479,333,555,481,449,511,523,582,385,142,538,153,174,185,421,152,289,316,164,321,514,348,428,312,206,270,546,233,452,260,561,392,297,424,165,275,137,467,484,197,435,461,493,324,156,403,188,499,488,193,526,569,503,237,508,541,558,412,430,148,264,573,375,191,286,281,579,291,382,236,181,406,274,377,230,245,318,535,520,330,345,552,240,251,278,455,223,306,563,209,342,458,145,409,263,441,463,338,359,177,182,470,576,336,154,314,465,363,203,588,351,139,418,290,235,433,549,168,273,242,294,400,532,283,415,354,309,543,215,491,496,254,227]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SWW490_5 : TPTP v8.1.2. Released v6.0.0.
% 0.15/0.15 % Command : run_Leo-III %s %d
% 0.15/0.36 % Computer : n014.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 300
% 0.15/0.36 % DateTime : Thu May 18 22:18:43 EDT 2023
% 0.15/0.36 % CPUTime :
% 0.94/0.85 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 1.62/1.04 % [INFO] Parsing done (193ms).
% 1.62/1.05 % [INFO] Running in sequential loop mode.
% 2.02/1.24 % [INFO] eprover registered as external prover.
% 2.02/1.25 % [INFO] cvc4 registered as external prover.
% 2.07/1.25 % [INFO] Scanning for conjecture ...
% 2.23/1.34 % [INFO] Found a conjecture and 134 axioms. Running axiom selection ...
% 2.39/1.42 % [INFO] Axiom selection finished. Selected 134 axioms (removed 0 axioms).
% 3.01/1.55 % [INFO] Problem is typed first-order (TPTP TFF).
% 3.01/1.57 % [INFO] Type checking passed.
% 3.01/1.57 % [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 ...
% 3.39/1.72 % [INFO] [Domain constraints] Detected constraint on bool
% 3.39/1.72 % [INFO] [Domain constraints] dom(bool) ⊆ {fTrue,fFalse}
% 12.65/5.03 % External prover 'e' found a proof!
% 12.65/5.03 % [INFO] Killing All external provers ...
% 13.03/5.04 % Time passed: 4511ms (effective reasoning time: 3983ms)
% 13.03/5.04 % 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)>
% 13.03/5.04 % Axioms used in derivation (134): fact_37_degree__smult__eq, fact_71_smult__add__left, help_fequal_1_1_T, tfree_0, fact_45_monom__Suc, fact_17_coeff__0, fact_69_coeff__add, fact_39_synthetic__div__0, arity_Nat_Onat___Rings_Osemiring__1, fact_49_Zero__not__Suc, fact_2_zero__reorient, fact_35_pCons__cases, fact_78_add__imp__eq, fact_56_Suc__not__Zero, fact_43_synthetic__div__unique__lemma, fact_90_synthetic__div__correct, fact_6_leading__coeff__neq__0, fact_0_offset__poly__0, fact_88_offset__poly__eq__0__lemma, arity_Polynomial_Opoly___Groups_Olinordered__ab__group__add, fact_93_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J, fact_89_offset__poly__pCons, fact_31_synthetic__div__pCons, arity_Polynomial_Opoly___Groups_Omonoid__add, fact_97_plus__nat_Oadd__0, fact_86_poly__offset__poly, fact_29_pCons__induct, fact_79_add__left__imp__eq, fact_40_smult__0__left, fact_52_Suc__inject, arity_Nat_Onat___Groups_Ocomm__monoid__add, fact_83_add__poly__code_I1_J, fact_10_poly__0, arity_Nat_Onat___Groups_Ocancel__comm__monoid__add, fact_84_add__poly__code_I2_J, arity_Polynomial_Opoly___Groups_Ocancel__semigroup__add, arity_Polynomial_Opoly___Rings_Oidom, fact_53_Suc__neq__Zero, fact_41_smult__eq__0__iff, fact_34_poly__ext, fact_5_poly__rec__0, arity_Polynomial_Opoly___Groups_Ozero, clar_Rings_Ocomm__semiring__0__Groups_Ocomm__monoid__add, help_If_1_1_T, arity_Polynomial_Opoly___Int_Oring__char__0, fact_46_nat_Oinject, arity_Polynomial_Opoly___Groups_Ocancel__ab__semigroup__add, fact_92_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J, arity_Polynomial_Opoly___Rings_Ocomm__semiring__1, clar_Rings_Ocomm__semiring__0__Groups_Oab__semigroup__add, fact_21_poly__eq__iff, fact_23_coeff__pCons__0, clar_Rings_Ocomm__semiring__0__Groups_Omonoid__add, fact_94_add__is__0, fact_73_ext, fact_54_Zero__neq__Suc, fact_27_degree__pCons__0, fact_77_add__right__imp__eq, arity_Polynomial_Opoly___Rings_Olinordered__idom, fact_61_not0__implies__Suc, arity_Polynomial_Opoly___Groups_Ocomm__monoid__add, fact_33_degree__pCons__eq__if, clar_Rings_Ocomm__semiring__0__Groups_Ozero, fact_80_add__monom, fact_19_monom__0, arity_Nat_Onat___Rings_Ocomm__semiring__1, fact_9_degree__0, fact_62_of__nat__aux_Osimps_I2_J, arity_Nat_Onat___Groups_Ozero, fact_13_pCons__eq__iff, fact_15_pCons__eq__0__iff, fact_22_coeff__inject, fact_3_leading__coeff__0__iff, fact_26_poly__zero, help_If_3_1_T, fact_81_ab__semigroup__add__class_Oadd__ac_I1_J, help_If_2_1_T, fact_24_poly__pcompose, fact_4_offset__poly__single, fact_66_add__left__cancel, fact_64_synthetic__div__unique, fact_47_nat__case__Suc, arity_Nat_Onat___Rings_Ocomm__semiring__0, fact_96_add__Suc, fact_70_poly__add, fact_82_smult__add__right, fact_16_pCons__0__0, arity_Polynomial_Opoly___Groups_Ocancel__comm__monoid__add, arity_Polynomial_Opoly___Rings_Osemiring__1, fact_63_pCons__def, fact_59_zero__induct, fact_58_nat_Oexhaust, fact_44_degree__pCons__eq, fact_76_add_Ocomm__neutral, fact_11_degree__monom__eq, help_fequal_2_1_T, fact_38_smult__0__right, fact_7_monom__eq__0, fact_75_add__0__right, fact_25_poly__rec_Osimps, fact_72_add__0__left, fact_20_expand__poly__eq, fact_85_coeff__inverse, fact_12_pcompose__0, fact_91_double__eq__0__iff, fact_32_synthetic__div__eq__0__iff, fact_65_add__right__cancel, fact_36_coeff__pCons, fact_57_nat_Osimps_I2_J, fact_95_add__Suc__right, help_pp_1_1_U, fact_30_order__root, fact_8_monom__eq__0__iff, fact_74_add__0, fact_1_offset__poly__eq__0__iff, fact_60_nat__induct, fact_18_coeff__monom, arity_Polynomial_Opoly___Groups_Oab__semigroup__add, arity_Nat_Onat___Groups_Oab__semigroup__add, arity_Nat_Onat___Groups_Omonoid__add, fact_55_nat_Osimps_I3_J, fact_48_nat__case__0, arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add, fact_67_double__zero__sym, fact_87_of__nat__aux_Osimps_I1_J, help_pp_2_1_U, fact_51_Suc__n__not__n, fact_68_add__pCons, arity_Polynomial_Opoly___Rings_Ocomm__semiring__0, fact_50_n__not__Suc__n, fact_28_poly__rec__pCons, arity_Nat_Onat___Groups_Ocancel__semigroup__add, fact_42_coeff__pCons__Suc, fact_14_monom__eq__iff
% 13.03/5.04 % No. of inferences in proof: 272
% 13.03/5.04 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 4511 ms resp. 3983 ms w/o parsing
% 13.29/5.12 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 13.29/5.12 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------