%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SWW188+1 : TPTP v9.0.0. Released v5.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n012.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:36:35 AM UTC 2025
% Result : Theorem 1.21s 1.38s
% Output : Proof 1.29s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 53
% Syntax : Number of formulae : 138 ( 38 unt; 0 typ; 0 def)
% Number of atoms : 566 ( 202 equ)
% Maximal formula atoms : 12 ( 4 avg)
% Number of connectives : 662 ( 260 ~; 273 |; 0 &)
% ( 92 <=>; 37 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of FOOLs : 26 ( 26 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 11 ( 9 usr; 1 prp; 0-3 aty)
% Number of functors : 13 ( 13 usr; 5 con; 0-3 aty)
% Number of variables : 296 ( 264 !; 0 ?; 296 :)
% Comments :
%------------------------------------------------------------------------------
tff(c_Orderings_Oord__class_Oless_type,type,
c_Orderings_Oord__class_Oless: ( $i * $i * $i ) > $o ).
tff(c_Nat_OSuc_type,type,
c_Nat_OSuc: $i > $i ).
tff(c_Polynomial_Odegree_type,type,
c_Polynomial_Odegree: ( $i * $i ) > $i ).
tff(v_p_type,type,
v_p: $i ).
tff(t_a_type,type,
t_a: $i ).
tff(c_Polynomial_Osmult_type,type,
c_Polynomial_Osmult: ( $i * $i * $i ) > $i ).
tff(c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly_type,type,
c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly: ( $i * $i * $i ) > $i ).
tff(v_h_type,type,
v_h: $i ).
tff(tc_Nat_Onat_type,type,
tc_Nat_Onat: $i ).
tff(c_Polynomial_OpCons_type,type,
c_Polynomial_OpCons: ( $i * $i * $i ) > $i ).
tff(v_a_type,type,
v_a: $i ).
tff(class_Groups_Ozero_type,type,
class_Groups_Ozero: $i > $o ).
tff(class_Rings_Ocomm__semiring__0_type,type,
class_Rings_Ocomm__semiring__0: $i > $o ).
tff(c_Groups_Ozero__class_Ozero_type,type,
c_Groups_Ozero__class_Ozero: $i > $i ).
tff(tc_Polynomial_Opoly_type,type,
tc_Polynomial_Opoly: $i > $i ).
tff(class_Groups_Ocomm__monoid__add_type,type,
class_Groups_Ocomm__monoid__add: $i > $o ).
tff(c_Groups_Oplus__class_Oplus_type,type,
c_Groups_Oplus__class_Oplus: ( $i * $i * $i ) > $i ).
tff(c_Orderings_Oord__class_Oless__eq_type,type,
c_Orderings_Oord__class_Oless__eq: ( $i * $i * $i ) > $o ).
tff(1,plain,
( class_Rings_Ocomm__semiring__0(t_a)
<=> class_Rings_Ocomm__semiring__0(t_a) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
class_Rings_Ocomm__semiring__0(t_a),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tfree_0) ).
tff(3,plain,
class_Rings_Ocomm__semiring__0(t_a),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [T: $i] :
refl(( ( class_Groups_Ozero(T)
| ~ class_Rings_Ocomm__semiring__0(T) )
<=> ( class_Groups_Ozero(T)
| ~ class_Rings_Ocomm__semiring__0(T) ) )),
inference(bind,[status(th)],[]) ).
tff(5,plain,
( ! [T: $i] :
( class_Groups_Ozero(T)
| ~ class_Rings_Ocomm__semiring__0(T) )
<=> ! [T: $i] :
( class_Groups_Ozero(T)
| ~ class_Rings_Ocomm__semiring__0(T) ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
( ! [T: $i] :
( class_Groups_Ozero(T)
| ~ class_Rings_Ocomm__semiring__0(T) )
<=> ! [T: $i] :
( class_Groups_Ozero(T)
| ~ class_Rings_Ocomm__semiring__0(T) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(7,plain,
^ [T: $i] :
rewrite(( ( class_Rings_Ocomm__semiring__0(T)
=> class_Groups_Ozero(T) )
<=> ( class_Groups_Ozero(T)
| ~ class_Rings_Ocomm__semiring__0(T) ) )),
inference(bind,[status(th)],[]) ).
tff(8,plain,
( ! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
=> class_Groups_Ozero(T) )
<=> ! [T: $i] :
( class_Groups_Ozero(T)
| ~ class_Rings_Ocomm__semiring__0(T) ) ),
inference(quant_intro,[status(thm)],[7]) ).
tff(9,axiom,
! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
=> class_Groups_Ozero(T) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clrel_Rings_Ocomm__semiring__0__Groups_Ozero) ).
tff(10,plain,
! [T: $i] :
( class_Groups_Ozero(T)
| ~ class_Rings_Ocomm__semiring__0(T) ),
inference(modus_ponens,[status(thm)],[9,8]) ).
tff(11,plain,
! [T: $i] :
( class_Groups_Ozero(T)
| ~ class_Rings_Ocomm__semiring__0(T) ),
inference(modus_ponens,[status(thm)],[10,6]) ).
tff(12,plain,
! [T: $i] :
( class_Groups_Ozero(T)
| ~ class_Rings_Ocomm__semiring__0(T) ),
inference(skolemize,[status(sab)],[11]) ).
tff(13,plain,
! [T: $i] :
( class_Groups_Ozero(T)
| ~ class_Rings_Ocomm__semiring__0(T) ),
inference(modus_ponens,[status(thm)],[12,5]) ).
tff(14,plain,
( ( ~ ! [T: $i] :
( class_Groups_Ozero(T)
| ~ class_Rings_Ocomm__semiring__0(T) )
| class_Groups_Ozero(t_a)
| ~ class_Rings_Ocomm__semiring__0(t_a) )
<=> ( ~ ! [T: $i] :
( class_Groups_Ozero(T)
| ~ class_Rings_Ocomm__semiring__0(T) )
| class_Groups_Ozero(t_a)
| ~ class_Rings_Ocomm__semiring__0(t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(15,plain,
( ~ ! [T: $i] :
( class_Groups_Ozero(T)
| ~ class_Rings_Ocomm__semiring__0(T) )
| class_Groups_Ozero(t_a)
| ~ class_Rings_Ocomm__semiring__0(t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(16,plain,
( ~ ! [T: $i] :
( class_Groups_Ozero(T)
| ~ class_Rings_Ocomm__semiring__0(T) )
| class_Groups_Ozero(t_a)
| ~ class_Rings_Ocomm__semiring__0(t_a) ),
inference(modus_ponens,[status(thm)],[15,14]) ).
tff(17,plain,
class_Groups_Ozero(t_a),
inference(unit_resolution,[status(thm)],[16,13,3]) ).
tff(18,plain,
^ [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
refl(( ( ~ class_Rings_Ocomm__semiring__0(T_b)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) )
<=> ( ~ class_Rings_Ocomm__semiring__0(T_b)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(19,plain,
( ! [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_b)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) )
<=> ! [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_b)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) ) ),
inference(quant_intro,[status(thm)],[18]) ).
tff(20,plain,
( ! [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_b)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) )
<=> ! [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_b)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(21,plain,
^ [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
rewrite(( ( class_Rings_Ocomm__semiring__0(T_b)
=> ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) )
<=> ( ~ class_Rings_Ocomm__semiring__0(T_b)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(22,plain,
( ! [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
( class_Rings_Ocomm__semiring__0(T_b)
=> ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) )
<=> ! [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_b)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) ) ),
inference(quant_intro,[status(thm)],[21]) ).
tff(23,axiom,
! [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
( class_Rings_Ocomm__semiring__0(T_b)
=> ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_offset__poly__eq__0__iff) ).
tff(24,plain,
! [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_b)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) ),
inference(modus_ponens,[status(thm)],[23,22]) ).
tff(25,plain,
! [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_b)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) ),
inference(modus_ponens,[status(thm)],[24,20]) ).
tff(26,plain,
! [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_b)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) ),
inference(skolemize,[status(sab)],[25]) ).
tff(27,plain,
! [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_b)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) ),
inference(modus_ponens,[status(thm)],[26,19]) ).
tff(28,plain,
( ( ~ ! [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_b)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) )
| ~ class_Rings_Ocomm__semiring__0(t_a)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
<=> ( v_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) ) ) )
<=> ( ~ ! [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_b)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) )
| ~ class_Rings_Ocomm__semiring__0(t_a)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
<=> ( v_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) ) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(29,plain,
( ~ ! [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_b)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) )
| ~ class_Rings_Ocomm__semiring__0(t_a)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
<=> ( v_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) ) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(30,plain,
( ~ ! [V_ha_2: $i,V_pa_2: $i,T_b: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_b)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(T_b,V_pa_2,V_ha_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) )
<=> ( V_pa_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_b)) ) ) )
| ~ class_Rings_Ocomm__semiring__0(t_a)
| ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
<=> ( v_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) ) ) ),
inference(modus_ponens,[status(thm)],[29,28]) ).
tff(31,plain,
( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
<=> ( v_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) ) ),
inference(unit_resolution,[status(thm)],[30,27,3]) ).
tff(32,plain,
( ( v_p != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
<=> ( v_p != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(33,axiom,
v_p != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_1) ).
tff(34,plain,
v_p != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)),
inference(modus_ponens,[status(thm)],[33,32]) ).
tff(35,plain,
( ~ ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
<=> ( v_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) ) )
| ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( v_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) ) ),
inference(tautology,[status(thm)],[]) ).
tff(36,plain,
( ~ ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
<=> ( v_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) ) )
| ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) ) ),
inference(unit_resolution,[status(thm)],[35,34]) ).
tff(37,plain,
c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)),
inference(unit_resolution,[status(thm)],[36,31]) ).
tff(38,plain,
^ [V_a: $i,V_p: $i,T_a: $i] :
refl(( ( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) )
<=> ( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) ) )),
inference(bind,[status(th)],[]) ).
tff(39,plain,
( ! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) )
<=> ! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) ) ),
inference(quant_intro,[status(thm)],[38]) ).
tff(40,plain,
( ! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) )
<=> ! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(41,plain,
^ [V_a: $i,V_p: $i,T_a: $i] :
trans(monotonicity(rewrite(( ( ( V_p != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
=> ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) ) )
<=> ( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) ) ) )),
( ( class_Groups_Ozero(T_a)
=> ( ( V_p != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
=> ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) ) ) )
<=> ( class_Groups_Ozero(T_a)
=> ( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) ) ) ) )),
rewrite(( ( class_Groups_Ozero(T_a)
=> ( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) ) ) )
<=> ( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) ) )),
( ( class_Groups_Ozero(T_a)
=> ( ( V_p != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
=> ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) ) ) )
<=> ( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) ) )),
inference(bind,[status(th)],[]) ).
tff(42,plain,
( ! [V_a: $i,V_p: $i,T_a: $i] :
( class_Groups_Ozero(T_a)
=> ( ( V_p != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
=> ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) ) ) )
<=> ! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) ) ),
inference(quant_intro,[status(thm)],[41]) ).
tff(43,axiom,
! [V_a: $i,V_p: $i,T_a: $i] :
( class_Groups_Ozero(T_a)
=> ( ( V_p != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
=> ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_degree__pCons__eq) ).
tff(44,plain,
! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) ),
inference(modus_ponens,[status(thm)],[43,42]) ).
tff(45,plain,
! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) ),
inference(modus_ponens,[status(thm)],[44,40]) ).
tff(46,plain,
! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) ),
inference(skolemize,[status(sab)],[45]) ).
tff(47,plain,
! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) ),
inference(modus_ponens,[status(thm)],[46,39]) ).
tff(48,plain,
( ( ~ ! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) )
| ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ~ class_Groups_Ozero(t_a)
| ( c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) ) )
<=> ( ~ ! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) )
| ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ~ class_Groups_Ozero(t_a)
| ( c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(49,plain,
( ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
| ~ class_Groups_Ozero(t_a) )
<=> ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ~ class_Groups_Ozero(t_a)
| ( c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(50,plain,
( ( ~ ! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) )
| ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
| ~ class_Groups_Ozero(t_a) )
<=> ( ~ ! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) )
| ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ~ class_Groups_Ozero(t_a)
| ( c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) ) ) ),
inference(monotonicity,[status(thm)],[49]) ).
tff(51,plain,
( ( ~ ! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) )
| ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
| ~ class_Groups_Ozero(t_a) )
<=> ( ~ ! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) )
| ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ~ class_Groups_Ozero(t_a)
| ( c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) ) ) ),
inference(transitivity,[status(thm)],[50,48]) ).
tff(52,plain,
( ~ ! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) )
| ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
| ~ class_Groups_Ozero(t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(53,plain,
( ~ ! [V_a: $i,V_p: $i,T_a: $i] :
( ( V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) )
| ( c_Polynomial_Odegree(T_a,c_Polynomial_OpCons(T_a,V_a,V_p)) = c_Nat_OSuc(c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Groups_Ozero(T_a) )
| ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ~ class_Groups_Ozero(t_a)
| ( c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) ) ),
inference(modus_ponens,[status(thm)],[52,51]) ).
tff(54,plain,
( ( c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ~ class_Groups_Ozero(t_a)
| ( c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) ) ),
inference(unit_resolution,[status(thm)],[53,47]) ).
tff(55,plain,
c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),
inference(unit_resolution,[status(thm)],[54,37,17]) ).
tff(56,plain,
c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) = c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),
inference(symmetry,[status(thm)],[55]) ).
tff(57,plain,
( ( c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)) = c_Polynomial_Odegree(t_a,v_p) )
<=> ( c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)) = c_Polynomial_Odegree(t_a,v_p) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(58,axiom,
c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)) = c_Polynomial_Odegree(t_a,v_p),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).
tff(59,plain,
c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)) = c_Polynomial_Odegree(t_a,v_p),
inference(modus_ponens,[status(thm)],[58,57]) ).
tff(60,plain,
c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p)),
inference(monotonicity,[status(thm)],[59]) ).
tff(61,plain,
c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),
inference(symmetry,[status(thm)],[60]) ).
tff(62,plain,
c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p)) = c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),
inference(transitivity,[status(thm)],[61,56]) ).
tff(63,plain,
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p)))
<=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) ),
inference(monotonicity,[status(thm)],[62]) ).
tff(64,plain,
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))))
<=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p))) ),
inference(symmetry,[status(thm)],[63]) ).
tff(65,plain,
( ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))))
<=> ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p))) ),
inference(monotonicity,[status(thm)],[64]) ).
tff(66,plain,
^ [T: $i] :
refl(( ( class_Groups_Ocomm__monoid__add(T)
| ~ class_Rings_Ocomm__semiring__0(T) )
<=> ( class_Groups_Ocomm__monoid__add(T)
| ~ class_Rings_Ocomm__semiring__0(T) ) )),
inference(bind,[status(th)],[]) ).
tff(67,plain,
( ! [T: $i] :
( class_Groups_Ocomm__monoid__add(T)
| ~ class_Rings_Ocomm__semiring__0(T) )
<=> ! [T: $i] :
( class_Groups_Ocomm__monoid__add(T)
| ~ class_Rings_Ocomm__semiring__0(T) ) ),
inference(quant_intro,[status(thm)],[66]) ).
tff(68,plain,
( ! [T: $i] :
( class_Groups_Ocomm__monoid__add(T)
| ~ class_Rings_Ocomm__semiring__0(T) )
<=> ! [T: $i] :
( class_Groups_Ocomm__monoid__add(T)
| ~ class_Rings_Ocomm__semiring__0(T) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(69,plain,
^ [T: $i] :
rewrite(( ( class_Rings_Ocomm__semiring__0(T)
=> class_Groups_Ocomm__monoid__add(T) )
<=> ( class_Groups_Ocomm__monoid__add(T)
| ~ class_Rings_Ocomm__semiring__0(T) ) )),
inference(bind,[status(th)],[]) ).
tff(70,plain,
( ! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
=> class_Groups_Ocomm__monoid__add(T) )
<=> ! [T: $i] :
( class_Groups_Ocomm__monoid__add(T)
| ~ class_Rings_Ocomm__semiring__0(T) ) ),
inference(quant_intro,[status(thm)],[69]) ).
tff(71,axiom,
! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
=> class_Groups_Ocomm__monoid__add(T) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clrel_Rings_Ocomm__semiring__0__Groups_Ocomm__monoid__add) ).
tff(72,plain,
! [T: $i] :
( class_Groups_Ocomm__monoid__add(T)
| ~ class_Rings_Ocomm__semiring__0(T) ),
inference(modus_ponens,[status(thm)],[71,70]) ).
tff(73,plain,
! [T: $i] :
( class_Groups_Ocomm__monoid__add(T)
| ~ class_Rings_Ocomm__semiring__0(T) ),
inference(modus_ponens,[status(thm)],[72,68]) ).
tff(74,plain,
! [T: $i] :
( class_Groups_Ocomm__monoid__add(T)
| ~ class_Rings_Ocomm__semiring__0(T) ),
inference(skolemize,[status(sab)],[73]) ).
tff(75,plain,
! [T: $i] :
( class_Groups_Ocomm__monoid__add(T)
| ~ class_Rings_Ocomm__semiring__0(T) ),
inference(modus_ponens,[status(thm)],[74,67]) ).
tff(76,plain,
( ( ~ ! [T: $i] :
( class_Groups_Ocomm__monoid__add(T)
| ~ class_Rings_Ocomm__semiring__0(T) )
| class_Groups_Ocomm__monoid__add(t_a)
| ~ class_Rings_Ocomm__semiring__0(t_a) )
<=> ( ~ ! [T: $i] :
( class_Groups_Ocomm__monoid__add(T)
| ~ class_Rings_Ocomm__semiring__0(T) )
| class_Groups_Ocomm__monoid__add(t_a)
| ~ class_Rings_Ocomm__semiring__0(t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(77,plain,
( ~ ! [T: $i] :
( class_Groups_Ocomm__monoid__add(T)
| ~ class_Rings_Ocomm__semiring__0(T) )
| class_Groups_Ocomm__monoid__add(t_a)
| ~ class_Rings_Ocomm__semiring__0(t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(78,plain,
( ~ ! [T: $i] :
( class_Groups_Ocomm__monoid__add(T)
| ~ class_Rings_Ocomm__semiring__0(T) )
| class_Groups_Ocomm__monoid__add(t_a)
| ~ class_Rings_Ocomm__semiring__0(t_a) ),
inference(modus_ponens,[status(thm)],[77,76]) ).
tff(79,plain,
class_Groups_Ocomm__monoid__add(t_a),
inference(unit_resolution,[status(thm)],[78,75,3]) ).
tff(80,plain,
c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p)),
inference(transitivity,[status(thm)],[55,60]) ).
tff(81,plain,
( ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) = c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
<=> ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p)) ) ),
inference(monotonicity,[status(thm)],[80]) ).
tff(82,plain,
( ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p)) )
<=> ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) = c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) ) ),
inference(symmetry,[status(thm)],[81]) ).
tff(83,plain,
( ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) != c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p)) )
<=> ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) != c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) ) ),
inference(monotonicity,[status(thm)],[82]) ).
tff(84,plain,
( ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) != c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p)) )
<=> ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) != c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(85,axiom,
c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) != c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_2) ).
tff(86,plain,
c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) != c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p)),
inference(modus_ponens,[status(thm)],[85,84]) ).
tff(87,plain,
c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) != c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),
inference(modus_ponens,[status(thm)],[86,83]) ).
tff(88,plain,
^ [V_q: $i,V_p: $i,T_a: $i] :
refl(( ( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) )
<=> ( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) ) )),
inference(bind,[status(th)],[]) ).
tff(89,plain,
( ! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) )
<=> ! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) ) ),
inference(quant_intro,[status(thm)],[88]) ).
tff(90,plain,
( ! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) )
<=> ! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(91,plain,
^ [V_q: $i,V_p: $i,T_a: $i] :
trans(monotonicity(rewrite(( ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q))
=> ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) ) )
<=> ( ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q))
| ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) ) ) )),
( ( class_Groups_Ocomm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q))
=> ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) ) ) )
<=> ( class_Groups_Ocomm__monoid__add(T_a)
=> ( ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q))
| ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) ) ) ) )),
rewrite(( ( class_Groups_Ocomm__monoid__add(T_a)
=> ( ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q))
| ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) ) ) )
<=> ( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) ) )),
( ( class_Groups_Ocomm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q))
=> ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) ) ) )
<=> ( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) ) )),
inference(bind,[status(th)],[]) ).
tff(92,plain,
( ! [V_q: $i,V_p: $i,T_a: $i] :
( class_Groups_Ocomm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q))
=> ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) ) ) )
<=> ! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) ) ),
inference(quant_intro,[status(thm)],[91]) ).
tff(93,axiom,
! [V_q: $i,V_p: $i,T_a: $i] :
( class_Groups_Ocomm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q))
=> ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_degree__add__eq__right) ).
tff(94,plain,
! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) ),
inference(modus_ponens,[status(thm)],[93,92]) ).
tff(95,plain,
! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) ),
inference(modus_ponens,[status(thm)],[94,90]) ).
tff(96,plain,
! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) ),
inference(skolemize,[status(sab)],[95]) ).
tff(97,plain,
! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) ),
inference(modus_ponens,[status(thm)],[96,89]) ).
tff(98,plain,
( ( ~ ! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) )
| ~ class_Groups_Ocomm__monoid__add(t_a)
| ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) = c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) )
<=> ( ~ ! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) )
| ~ class_Groups_Ocomm__monoid__add(t_a)
| ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) = c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(99,plain,
( ( ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) = c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
| ~ class_Groups_Ocomm__monoid__add(t_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) )
<=> ( ~ class_Groups_Ocomm__monoid__add(t_a)
| ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) = c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(100,plain,
( ( ~ ! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) )
| ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) = c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
| ~ class_Groups_Ocomm__monoid__add(t_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) )
<=> ( ~ ! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) )
| ~ class_Groups_Ocomm__monoid__add(t_a)
| ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) = c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) ) ),
inference(monotonicity,[status(thm)],[99]) ).
tff(101,plain,
( ( ~ ! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) )
| ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) = c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
| ~ class_Groups_Ocomm__monoid__add(t_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) )
<=> ( ~ ! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) )
| ~ class_Groups_Ocomm__monoid__add(t_a)
| ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) = c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) ) ),
inference(transitivity,[status(thm)],[100,98]) ).
tff(102,plain,
( ~ ! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) )
| ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) = c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
| ~ class_Groups_Ocomm__monoid__add(t_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(103,plain,
( ~ ! [V_q: $i,V_p: $i,T_a: $i] :
( ( c_Polynomial_Odegree(T_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a),V_p,V_q)) = c_Polynomial_Odegree(T_a,V_q) )
| ~ class_Groups_Ocomm__monoid__add(T_a)
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p),c_Polynomial_Odegree(T_a,V_q)) )
| ~ class_Groups_Ocomm__monoid__add(t_a)
| ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) = c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) ),
inference(modus_ponens,[status(thm)],[102,101]) ).
tff(104,plain,
( ~ class_Groups_Ocomm__monoid__add(t_a)
| ( c_Polynomial_Odegree(t_a,c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(t_a),c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) = c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
| ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))) ),
inference(unit_resolution,[status(thm)],[103,97]) ).
tff(105,plain,
~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))),
inference(unit_resolution,[status(thm)],[104,87,79]) ).
tff(106,plain,
~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p))),
inference(modus_ponens,[status(thm)],[105,65]) ).
tff(107,plain,
c_Polynomial_Odegree(t_a,v_p) = c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)),
inference(symmetry,[status(thm)],[59]) ).
tff(108,plain,
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,v_p))
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) ),
inference(monotonicity,[status(thm)],[107]) ).
tff(109,plain,
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h)))
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,v_p)) ),
inference(symmetry,[status(thm)],[108]) ).
tff(110,plain,
^ [V_p: $i,V_a: $i,T_a: $i] :
refl(( ( ~ class_Rings_Ocomm__semiring__0(T_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) )
<=> ( ~ class_Rings_Ocomm__semiring__0(T_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) ) )),
inference(bind,[status(th)],[]) ).
tff(111,plain,
( ! [V_p: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) )
<=> ! [V_p: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) ) ),
inference(quant_intro,[status(thm)],[110]) ).
tff(112,plain,
( ! [V_p: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) )
<=> ! [V_p: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(113,plain,
^ [V_p: $i,V_a: $i,T_a: $i] :
rewrite(( ( class_Rings_Ocomm__semiring__0(T_a)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) )
<=> ( ~ class_Rings_Ocomm__semiring__0(T_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) ) )),
inference(bind,[status(th)],[]) ).
tff(114,plain,
( ! [V_p: $i,V_a: $i,T_a: $i] :
( class_Rings_Ocomm__semiring__0(T_a)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) )
<=> ! [V_p: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) ) ),
inference(quant_intro,[status(thm)],[113]) ).
tff(115,axiom,
! [V_p: $i,V_a: $i,T_a: $i] :
( class_Rings_Ocomm__semiring__0(T_a)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_degree__smult__le) ).
tff(116,plain,
! [V_p: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) ),
inference(modus_ponens,[status(thm)],[115,114]) ).
tff(117,plain,
! [V_p: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) ),
inference(modus_ponens,[status(thm)],[116,112]) ).
tff(118,plain,
! [V_p: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) ),
inference(skolemize,[status(sab)],[117]) ).
tff(119,plain,
! [V_p: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) ),
inference(modus_ponens,[status(thm)],[118,111]) ).
tff(120,plain,
( ( ~ ! [V_p: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Rings_Ocomm__semiring__0(t_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) )
<=> ( ~ ! [V_p: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Rings_Ocomm__semiring__0(t_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(121,plain,
( ~ ! [V_p: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Rings_Ocomm__semiring__0(t_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(122,plain,
( ~ ! [V_p: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)) )
| ~ class_Rings_Ocomm__semiring__0(t_a)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))) ),
inference(modus_ponens,[status(thm)],[121,120]) ).
tff(123,plain,
c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),
inference(unit_resolution,[status(thm)],[122,119,3]) ).
tff(124,plain,
c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,v_p)),
inference(modus_ponens,[status(thm)],[123,109]) ).
tff(125,plain,
^ [V_n: $i,V_m: $i] :
refl(( ( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) )
<=> ( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) ) )),
inference(bind,[status(th)],[]) ).
tff(126,plain,
( ! [V_n: $i,V_m: $i] :
( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) )
<=> ! [V_n: $i,V_m: $i] :
( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) ) ),
inference(quant_intro,[status(thm)],[125]) ).
tff(127,plain,
( ! [V_n: $i,V_m: $i] :
( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) )
<=> ! [V_n: $i,V_m: $i] :
( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(128,plain,
^ [V_n: $i,V_m: $i] :
rewrite(( ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) )
<=> ( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) ) )),
inference(bind,[status(th)],[]) ).
tff(129,plain,
( ! [V_n: $i,V_m: $i] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) )
<=> ! [V_n: $i,V_m: $i] :
( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) ) ),
inference(quant_intro,[status(thm)],[128]) ).
tff(130,axiom,
! [V_n: $i,V_m: $i] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__imp__less__Suc) ).
tff(131,plain,
! [V_n: $i,V_m: $i] :
( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) ),
inference(modus_ponens,[status(thm)],[130,129]) ).
tff(132,plain,
! [V_n: $i,V_m: $i] :
( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) ),
inference(modus_ponens,[status(thm)],[131,127]) ).
tff(133,plain,
! [V_n: $i,V_m: $i] :
( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) ),
inference(skolemize,[status(sab)],[132]) ).
tff(134,plain,
! [V_n: $i,V_m: $i] :
( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) ),
inference(modus_ponens,[status(thm)],[133,126]) ).
tff(135,plain,
( ( ~ ! [V_n: $i,V_m: $i] :
( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) )
| ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,v_p))
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p))) )
<=> ( ~ ! [V_n: $i,V_m: $i] :
( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) )
| ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,v_p))
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(136,plain,
( ~ ! [V_n: $i,V_m: $i] :
( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) )
| ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,v_p))
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(137,plain,
( ~ ! [V_n: $i,V_m: $i] :
( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) )
| ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Polynomial_Odegree(t_a,v_p))
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Polynomial_Odegree(t_a,c_Polynomial_Osmult(t_a,v_h,c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly(t_a,v_p,v_h))),c_Nat_OSuc(c_Polynomial_Odegree(t_a,v_p))) ),
inference(modus_ponens,[status(thm)],[136,135]) ).
tff(138,plain,
$false,
inference(unit_resolution,[status(thm)],[137,134,124,106]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SWW188+1 : TPTP v9.0.0. Released v5.2.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n012.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Jun 20 11:14:10 EDT 2025
% 0.13/0.34 % CPUTime :
% 1.21/1.38 % SZS status Theorem
% 1.21/1.38 % SZS output start Proof
% See solution above
% 1.29/1.45 % E exiting
%------------------------------------------------------------------------------