%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SWW240+1 : TPTP v9.0.0. Released v5.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.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 05:36:40 AM UTC 2025
% Result : Theorem 0.64s 0.83s
% Output : Proof 0.72s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 70
% Syntax : Number of formulae : 181 ( 57 unt; 0 typ; 0 def)
% Number of atoms : 565 ( 193 equ)
% Maximal formula atoms : 8 ( 3 avg)
% Number of connectives : 637 ( 273 ~; 269 |; 0 &)
% ( 65 <=>; 30 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of FOOLs : 20 ( 20 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 9 ( 7 usr; 1 prp; 0-2 aty)
% Number of functors : 14 ( 14 usr; 5 con; 0-3 aty)
% Number of variables : 398 ( 354 !; 0 ?; 398 :)
% Comments :
%------------------------------------------------------------------------------
tff(hAPP_type,type,
hAPP: ( $i * $i ) > $i ).
tff(v_x_type,type,
v_x: $i ).
tff(c_Polynomial_Opoly_type,type,
c_Polynomial_Opoly: ( $i * $i ) > $i ).
tff(v_p_type,type,
v_p: $i ).
tff(t_a_type,type,
t_a: $i ).
tff(v_n_type,type,
v_n: $i ).
tff(c_Power_Opower__class_Opower_type,type,
c_Power_Opower__class_Opower: $i > $i ).
tff(c_Groups_Otimes__class_Otimes_type,type,
c_Groups_Otimes__class_Otimes: $i > $i ).
tff(c_Polynomial_Omonom_type,type,
c_Polynomial_Omonom: ( $i * $i * $i ) > $i ).
tff(c_Groups_Oone__class_Oone_type,type,
c_Groups_Oone__class_Oone: $i > $i ).
tff(tc_Polynomial_Opoly_type,type,
tc_Polynomial_Opoly: $i > $i ).
tff(class_Rings_Ocomm__semiring__1_type,type,
class_Rings_Ocomm__semiring__1: $i > $o ).
tff(class_Rings_Ocomm__ring__1_type,type,
class_Rings_Ocomm__ring__1: $i > $o ).
tff(class_Groups_Omonoid__mult_type,type,
class_Groups_Omonoid__mult: $i > $o ).
tff(c_Groups_Ozero__class_Ozero_type,type,
c_Groups_Ozero__class_Ozero: $i > $i ).
tff(tc_Nat_Onat_type,type,
tc_Nat_Onat: $i ).
tff(c_Nat_OSuc_type,type,
c_Nat_OSuc: $i > $i ).
tff(class_Power_Opower_type,type,
class_Power_Opower: $i > $o ).
tff(class_Rings_Ocomm__semiring__0_type,type,
class_Rings_Ocomm__semiring__0: $i > $o ).
tff(1,plain,
( class_Rings_Ocomm__ring__1(t_a)
<=> class_Rings_Ocomm__ring__1(t_a) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
class_Rings_Ocomm__ring__1(t_a),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tfree_0) ).
tff(3,plain,
class_Rings_Ocomm__ring__1(t_a),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [T: $i] :
refl(( ( class_Rings_Ocomm__semiring__1(T)
| ~ class_Rings_Ocomm__ring__1(T) )
<=> ( class_Rings_Ocomm__semiring__1(T)
| ~ class_Rings_Ocomm__ring__1(T) ) )),
inference(bind,[status(th)],[]) ).
tff(5,plain,
( ! [T: $i] :
( class_Rings_Ocomm__semiring__1(T)
| ~ class_Rings_Ocomm__ring__1(T) )
<=> ! [T: $i] :
( class_Rings_Ocomm__semiring__1(T)
| ~ class_Rings_Ocomm__ring__1(T) ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
( ! [T: $i] :
( class_Rings_Ocomm__semiring__1(T)
| ~ class_Rings_Ocomm__ring__1(T) )
<=> ! [T: $i] :
( class_Rings_Ocomm__semiring__1(T)
| ~ class_Rings_Ocomm__ring__1(T) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(7,plain,
^ [T: $i] :
rewrite(( ( class_Rings_Ocomm__ring__1(T)
=> class_Rings_Ocomm__semiring__1(T) )
<=> ( class_Rings_Ocomm__semiring__1(T)
| ~ class_Rings_Ocomm__ring__1(T) ) )),
inference(bind,[status(th)],[]) ).
tff(8,plain,
( ! [T: $i] :
( class_Rings_Ocomm__ring__1(T)
=> class_Rings_Ocomm__semiring__1(T) )
<=> ! [T: $i] :
( class_Rings_Ocomm__semiring__1(T)
| ~ class_Rings_Ocomm__ring__1(T) ) ),
inference(quant_intro,[status(thm)],[7]) ).
tff(9,axiom,
! [T: $i] :
( class_Rings_Ocomm__ring__1(T)
=> class_Rings_Ocomm__semiring__1(T) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clrel_Rings_Ocomm__ring__1__Rings_Ocomm__semiring__1) ).
tff(10,plain,
! [T: $i] :
( class_Rings_Ocomm__semiring__1(T)
| ~ class_Rings_Ocomm__ring__1(T) ),
inference(modus_ponens,[status(thm)],[9,8]) ).
tff(11,plain,
! [T: $i] :
( class_Rings_Ocomm__semiring__1(T)
| ~ class_Rings_Ocomm__ring__1(T) ),
inference(modus_ponens,[status(thm)],[10,6]) ).
tff(12,plain,
! [T: $i] :
( class_Rings_Ocomm__semiring__1(T)
| ~ class_Rings_Ocomm__ring__1(T) ),
inference(skolemize,[status(sab)],[11]) ).
tff(13,plain,
! [T: $i] :
( class_Rings_Ocomm__semiring__1(T)
| ~ class_Rings_Ocomm__ring__1(T) ),
inference(modus_ponens,[status(thm)],[12,5]) ).
tff(14,plain,
( ( ~ ! [T: $i] :
( class_Rings_Ocomm__semiring__1(T)
| ~ class_Rings_Ocomm__ring__1(T) )
| class_Rings_Ocomm__semiring__1(t_a)
| ~ class_Rings_Ocomm__ring__1(t_a) )
<=> ( ~ ! [T: $i] :
( class_Rings_Ocomm__semiring__1(T)
| ~ class_Rings_Ocomm__ring__1(T) )
| class_Rings_Ocomm__semiring__1(t_a)
| ~ class_Rings_Ocomm__ring__1(t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(15,plain,
( ~ ! [T: $i] :
( class_Rings_Ocomm__semiring__1(T)
| ~ class_Rings_Ocomm__ring__1(T) )
| class_Rings_Ocomm__semiring__1(t_a)
| ~ class_Rings_Ocomm__ring__1(t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(16,plain,
( ~ ! [T: $i] :
( class_Rings_Ocomm__semiring__1(T)
| ~ class_Rings_Ocomm__ring__1(T) )
| class_Rings_Ocomm__semiring__1(t_a)
| ~ class_Rings_Ocomm__ring__1(t_a) ),
inference(modus_ponens,[status(thm)],[15,14]) ).
tff(17,plain,
class_Rings_Ocomm__semiring__1(t_a),
inference(unit_resolution,[status(thm)],[16,13,3]) ).
tff(18,plain,
^ [V_b: $i,V_a: $i,T_a: $i] :
refl(( ( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
<=> ( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) ) )),
inference(bind,[status(th)],[]) ).
tff(19,plain,
( ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
<=> ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) ) ),
inference(quant_intro,[status(thm)],[18]) ).
tff(20,plain,
( ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
<=> ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(21,plain,
^ [V_b: $i,V_a: $i,T_a: $i] :
rewrite(( ( class_Rings_Ocomm__semiring__1(T_a)
=> ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
<=> ( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) ) )),
inference(bind,[status(th)],[]) ).
tff(22,plain,
( ! [V_b: $i,V_a: $i,T_a: $i] :
( class_Rings_Ocomm__semiring__1(T_a)
=> ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
<=> ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) ) ),
inference(quant_intro,[status(thm)],[21]) ).
tff(23,axiom,
! [V_b: $i,V_a: $i,T_a: $i] :
( class_Rings_Ocomm__semiring__1(T_a)
=> ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) ).
tff(24,plain,
! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) ),
inference(modus_ponens,[status(thm)],[23,22]) ).
tff(25,plain,
! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) ),
inference(modus_ponens,[status(thm)],[24,20]) ).
tff(26,plain,
! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) ),
inference(skolemize,[status(sab)],[25]) ).
tff(27,plain,
! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) ),
inference(modus_ponens,[status(thm)],[26,19]) ).
tff(28,plain,
( ( ~ ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
| ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) ) )
<=> ( ~ ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
| ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(29,plain,
( ~ ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
| ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(30,plain,
( ~ ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
| ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) ) ),
inference(modus_ponens,[status(thm)],[29,28]) ).
tff(31,plain,
( ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) ) ),
inference(unit_resolution,[status(thm)],[30,27]) ).
tff(32,plain,
hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),
inference(unit_resolution,[status(thm)],[31,17]) ).
tff(33,plain,
^ [T: $i] :
refl(( ( class_Groups_Omonoid__mult(T)
| ~ class_Rings_Ocomm__ring__1(T) )
<=> ( class_Groups_Omonoid__mult(T)
| ~ class_Rings_Ocomm__ring__1(T) ) )),
inference(bind,[status(th)],[]) ).
tff(34,plain,
( ! [T: $i] :
( class_Groups_Omonoid__mult(T)
| ~ class_Rings_Ocomm__ring__1(T) )
<=> ! [T: $i] :
( class_Groups_Omonoid__mult(T)
| ~ class_Rings_Ocomm__ring__1(T) ) ),
inference(quant_intro,[status(thm)],[33]) ).
tff(35,plain,
( ! [T: $i] :
( class_Groups_Omonoid__mult(T)
| ~ class_Rings_Ocomm__ring__1(T) )
<=> ! [T: $i] :
( class_Groups_Omonoid__mult(T)
| ~ class_Rings_Ocomm__ring__1(T) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(36,plain,
^ [T: $i] :
rewrite(( ( class_Rings_Ocomm__ring__1(T)
=> class_Groups_Omonoid__mult(T) )
<=> ( class_Groups_Omonoid__mult(T)
| ~ class_Rings_Ocomm__ring__1(T) ) )),
inference(bind,[status(th)],[]) ).
tff(37,plain,
( ! [T: $i] :
( class_Rings_Ocomm__ring__1(T)
=> class_Groups_Omonoid__mult(T) )
<=> ! [T: $i] :
( class_Groups_Omonoid__mult(T)
| ~ class_Rings_Ocomm__ring__1(T) ) ),
inference(quant_intro,[status(thm)],[36]) ).
tff(38,axiom,
! [T: $i] :
( class_Rings_Ocomm__ring__1(T)
=> class_Groups_Omonoid__mult(T) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clrel_Rings_Ocomm__ring__1__Groups_Omonoid__mult) ).
tff(39,plain,
! [T: $i] :
( class_Groups_Omonoid__mult(T)
| ~ class_Rings_Ocomm__ring__1(T) ),
inference(modus_ponens,[status(thm)],[38,37]) ).
tff(40,plain,
! [T: $i] :
( class_Groups_Omonoid__mult(T)
| ~ class_Rings_Ocomm__ring__1(T) ),
inference(modus_ponens,[status(thm)],[39,35]) ).
tff(41,plain,
! [T: $i] :
( class_Groups_Omonoid__mult(T)
| ~ class_Rings_Ocomm__ring__1(T) ),
inference(skolemize,[status(sab)],[40]) ).
tff(42,plain,
! [T: $i] :
( class_Groups_Omonoid__mult(T)
| ~ class_Rings_Ocomm__ring__1(T) ),
inference(modus_ponens,[status(thm)],[41,34]) ).
tff(43,plain,
( ( ~ ! [T: $i] :
( class_Groups_Omonoid__mult(T)
| ~ class_Rings_Ocomm__ring__1(T) )
| class_Groups_Omonoid__mult(t_a)
| ~ class_Rings_Ocomm__ring__1(t_a) )
<=> ( ~ ! [T: $i] :
( class_Groups_Omonoid__mult(T)
| ~ class_Rings_Ocomm__ring__1(T) )
| class_Groups_Omonoid__mult(t_a)
| ~ class_Rings_Ocomm__ring__1(t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(44,plain,
( ~ ! [T: $i] :
( class_Groups_Omonoid__mult(T)
| ~ class_Rings_Ocomm__ring__1(T) )
| class_Groups_Omonoid__mult(t_a)
| ~ class_Rings_Ocomm__ring__1(t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(45,plain,
( ~ ! [T: $i] :
( class_Groups_Omonoid__mult(T)
| ~ class_Rings_Ocomm__ring__1(T) )
| class_Groups_Omonoid__mult(t_a)
| ~ class_Rings_Ocomm__ring__1(t_a) ),
inference(modus_ponens,[status(thm)],[44,43]) ).
tff(46,plain,
class_Groups_Omonoid__mult(t_a),
inference(unit_resolution,[status(thm)],[45,42,3]) ).
tff(47,plain,
^ [V_a: $i,T_a: $i] :
refl(( ( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )
<=> ( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) ) )),
inference(bind,[status(th)],[]) ).
tff(48,plain,
( ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )
<=> ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) ) ),
inference(quant_intro,[status(thm)],[47]) ).
tff(49,plain,
( ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )
<=> ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(50,plain,
^ [V_a: $i,T_a: $i] :
rewrite(( ( class_Groups_Omonoid__mult(T_a)
=> ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )
<=> ( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) ) )),
inference(bind,[status(th)],[]) ).
tff(51,plain,
( ! [V_a: $i,T_a: $i] :
( class_Groups_Omonoid__mult(T_a)
=> ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )
<=> ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) ) ),
inference(quant_intro,[status(thm)],[50]) ).
tff(52,axiom,
! [V_a: $i,T_a: $i] :
( class_Groups_Omonoid__mult(T_a)
=> ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__1__right) ).
tff(53,plain,
! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) ),
inference(modus_ponens,[status(thm)],[52,51]) ).
tff(54,plain,
! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) ),
inference(modus_ponens,[status(thm)],[53,49]) ).
tff(55,plain,
! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) ),
inference(skolemize,[status(sab)],[54]) ).
tff(56,plain,
! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) ),
inference(modus_ponens,[status(thm)],[55,48]) ).
tff(57,plain,
( ( ~ ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )
| ~ class_Groups_Omonoid__mult(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),c_Groups_Oone__class_Oone(t_a)) = hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n) ) )
<=> ( ~ ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )
| ~ class_Groups_Omonoid__mult(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),c_Groups_Oone__class_Oone(t_a)) = hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(58,plain,
( ~ ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )
| ~ class_Groups_Omonoid__mult(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),c_Groups_Oone__class_Oone(t_a)) = hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(59,plain,
( ~ ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )
| ~ class_Groups_Omonoid__mult(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),c_Groups_Oone__class_Oone(t_a)) = hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n) ) ),
inference(modus_ponens,[status(thm)],[58,57]) ).
tff(60,plain,
( ~ class_Groups_Omonoid__mult(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),c_Groups_Oone__class_Oone(t_a)) = hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n) ) ),
inference(unit_resolution,[status(thm)],[59,56]) ).
tff(61,plain,
hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),c_Groups_Oone__class_Oone(t_a)) = hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n),
inference(unit_resolution,[status(thm)],[60,46]) ).
tff(62,plain,
( ( ~ ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
| ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),c_Groups_Oone__class_Oone(t_a)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),c_Groups_Oone__class_Oone(t_a)) ) )
<=> ( ~ ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
| ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),c_Groups_Oone__class_Oone(t_a)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),c_Groups_Oone__class_Oone(t_a)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(63,plain,
( ~ ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
| ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),c_Groups_Oone__class_Oone(t_a)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),c_Groups_Oone__class_Oone(t_a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(64,plain,
( ~ ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
| ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),c_Groups_Oone__class_Oone(t_a)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),c_Groups_Oone__class_Oone(t_a)) ) ),
inference(modus_ponens,[status(thm)],[63,62]) ).
tff(65,plain,
hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),c_Groups_Oone__class_Oone(t_a)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),c_Groups_Oone__class_Oone(t_a)),
inference(unit_resolution,[status(thm)],[64,27,17]) ).
tff(66,plain,
hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),c_Groups_Oone__class_Oone(t_a)),
inference(symmetry,[status(thm)],[61]) ).
tff(67,plain,
hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),c_Groups_Oone__class_Oone(t_a)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),c_Groups_Oone__class_Oone(t_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),c_Groups_Oone__class_Oone(t_a))),
inference(monotonicity,[status(thm)],[66]) ).
tff(68,plain,
hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),c_Groups_Oone__class_Oone(t_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),c_Groups_Oone__class_Oone(t_a))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),c_Groups_Oone__class_Oone(t_a)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),
inference(symmetry,[status(thm)],[67]) ).
tff(69,plain,
^ [V_a: $i,T_a: $i] :
refl(( ( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) )
<=> ( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) ) )),
inference(bind,[status(th)],[]) ).
tff(70,plain,
( ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) )
<=> ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) ) ),
inference(quant_intro,[status(thm)],[69]) ).
tff(71,plain,
( ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) )
<=> ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(72,plain,
^ [V_a: $i,T_a: $i] :
rewrite(( ( class_Groups_Omonoid__mult(T_a)
=> ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) )
<=> ( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) ) )),
inference(bind,[status(th)],[]) ).
tff(73,plain,
( ! [V_a: $i,T_a: $i] :
( class_Groups_Omonoid__mult(T_a)
=> ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) )
<=> ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) ) ),
inference(quant_intro,[status(thm)],[72]) ).
tff(74,axiom,
! [V_a: $i,T_a: $i] :
( class_Groups_Omonoid__mult(T_a)
=> ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__one__right) ).
tff(75,plain,
! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) ),
inference(modus_ponens,[status(thm)],[74,73]) ).
tff(76,plain,
! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) ),
inference(modus_ponens,[status(thm)],[75,71]) ).
tff(77,plain,
! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) ),
inference(skolemize,[status(sab)],[76]) ).
tff(78,plain,
! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) ),
inference(modus_ponens,[status(thm)],[77,70]) ).
tff(79,plain,
( ( ~ ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) )
| ~ class_Groups_Omonoid__mult(t_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = v_x ) )
<=> ( ~ ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) )
| ~ class_Groups_Omonoid__mult(t_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = v_x ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(80,plain,
( ~ ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) )
| ~ class_Groups_Omonoid__mult(t_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = v_x ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(81,plain,
( ~ ! [V_a: $i,T_a: $i] :
( ~ class_Groups_Omonoid__mult(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) )
| ~ class_Groups_Omonoid__mult(t_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = v_x ) ),
inference(modus_ponens,[status(thm)],[80,79]) ).
tff(82,plain,
( ~ class_Groups_Omonoid__mult(t_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = v_x ) ),
inference(unit_resolution,[status(thm)],[81,78]) ).
tff(83,plain,
hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = v_x,
inference(unit_resolution,[status(thm)],[82,46]) ).
tff(84,plain,
( ( c_Groups_Oone__class_Oone(tc_Nat_Onat) = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) )
<=> ( c_Groups_Oone__class_Oone(tc_Nat_Onat) = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(85,axiom,
c_Groups_Oone__class_Oone(tc_Nat_Onat) = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_One__nat__def) ).
tff(86,plain,
c_Groups_Oone__class_Oone(tc_Nat_Onat) = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),
inference(modus_ponens,[status(thm)],[85,84]) ).
tff(87,plain,
hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),
inference(monotonicity,[status(thm)],[86]) ).
tff(88,plain,
hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)),
inference(symmetry,[status(thm)],[87]) ).
tff(89,plain,
^ [T: $i] :
refl(( ( class_Power_Opower(T)
| ~ class_Rings_Ocomm__ring__1(T) )
<=> ( class_Power_Opower(T)
| ~ class_Rings_Ocomm__ring__1(T) ) )),
inference(bind,[status(th)],[]) ).
tff(90,plain,
( ! [T: $i] :
( class_Power_Opower(T)
| ~ class_Rings_Ocomm__ring__1(T) )
<=> ! [T: $i] :
( class_Power_Opower(T)
| ~ class_Rings_Ocomm__ring__1(T) ) ),
inference(quant_intro,[status(thm)],[89]) ).
tff(91,plain,
( ! [T: $i] :
( class_Power_Opower(T)
| ~ class_Rings_Ocomm__ring__1(T) )
<=> ! [T: $i] :
( class_Power_Opower(T)
| ~ class_Rings_Ocomm__ring__1(T) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(92,plain,
^ [T: $i] :
rewrite(( ( class_Rings_Ocomm__ring__1(T)
=> class_Power_Opower(T) )
<=> ( class_Power_Opower(T)
| ~ class_Rings_Ocomm__ring__1(T) ) )),
inference(bind,[status(th)],[]) ).
tff(93,plain,
( ! [T: $i] :
( class_Rings_Ocomm__ring__1(T)
=> class_Power_Opower(T) )
<=> ! [T: $i] :
( class_Power_Opower(T)
| ~ class_Rings_Ocomm__ring__1(T) ) ),
inference(quant_intro,[status(thm)],[92]) ).
tff(94,axiom,
! [T: $i] :
( class_Rings_Ocomm__ring__1(T)
=> class_Power_Opower(T) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clrel_Rings_Ocomm__ring__1__Power_Opower) ).
tff(95,plain,
! [T: $i] :
( class_Power_Opower(T)
| ~ class_Rings_Ocomm__ring__1(T) ),
inference(modus_ponens,[status(thm)],[94,93]) ).
tff(96,plain,
! [T: $i] :
( class_Power_Opower(T)
| ~ class_Rings_Ocomm__ring__1(T) ),
inference(modus_ponens,[status(thm)],[95,91]) ).
tff(97,plain,
! [T: $i] :
( class_Power_Opower(T)
| ~ class_Rings_Ocomm__ring__1(T) ),
inference(skolemize,[status(sab)],[96]) ).
tff(98,plain,
! [T: $i] :
( class_Power_Opower(T)
| ~ class_Rings_Ocomm__ring__1(T) ),
inference(modus_ponens,[status(thm)],[97,90]) ).
tff(99,plain,
( ( ~ ! [T: $i] :
( class_Power_Opower(T)
| ~ class_Rings_Ocomm__ring__1(T) )
| class_Power_Opower(t_a)
| ~ class_Rings_Ocomm__ring__1(t_a) )
<=> ( ~ ! [T: $i] :
( class_Power_Opower(T)
| ~ class_Rings_Ocomm__ring__1(T) )
| class_Power_Opower(t_a)
| ~ class_Rings_Ocomm__ring__1(t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(100,plain,
( ~ ! [T: $i] :
( class_Power_Opower(T)
| ~ class_Rings_Ocomm__ring__1(T) )
| class_Power_Opower(t_a)
| ~ class_Rings_Ocomm__ring__1(t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(101,plain,
( ~ ! [T: $i] :
( class_Power_Opower(T)
| ~ class_Rings_Ocomm__ring__1(T) )
| class_Power_Opower(t_a)
| ~ class_Rings_Ocomm__ring__1(t_a) ),
inference(modus_ponens,[status(thm)],[100,99]) ).
tff(102,plain,
class_Power_Opower(t_a),
inference(unit_resolution,[status(thm)],[101,98,3]) ).
tff(103,plain,
^ [V_n: $i,V_a: $i,T_a: $i] :
refl(( ( ~ class_Power_Opower(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )
<=> ( ~ class_Power_Opower(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(104,plain,
( ! [V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Power_Opower(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )
<=> ! [V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Power_Opower(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ) ),
inference(quant_intro,[status(thm)],[103]) ).
tff(105,plain,
( ! [V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Power_Opower(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )
<=> ! [V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Power_Opower(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(106,plain,
^ [V_n: $i,V_a: $i,T_a: $i] :
rewrite(( ( class_Power_Opower(T_a)
=> ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )
<=> ( ~ class_Power_Opower(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(107,plain,
( ! [V_n: $i,V_a: $i,T_a: $i] :
( class_Power_Opower(T_a)
=> ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )
<=> ! [V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Power_Opower(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ) ),
inference(quant_intro,[status(thm)],[106]) ).
tff(108,axiom,
! [V_n: $i,V_a: $i,T_a: $i] :
( class_Power_Opower(T_a)
=> ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__Suc) ).
tff(109,plain,
! [V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Power_Opower(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ),
inference(modus_ponens,[status(thm)],[108,107]) ).
tff(110,plain,
! [V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Power_Opower(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ),
inference(modus_ponens,[status(thm)],[109,105]) ).
tff(111,plain,
! [V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Power_Opower(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ),
inference(skolemize,[status(sab)],[110]) ).
tff(112,plain,
! [V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Power_Opower(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ),
inference(modus_ponens,[status(thm)],[111,104]) ).
tff(113,plain,
( ( ~ ! [V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Power_Opower(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )
| ~ class_Power_Opower(t_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) ) )
<=> ( ~ ! [V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Power_Opower(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )
| ~ class_Power_Opower(t_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(114,plain,
( ~ ! [V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Power_Opower(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )
| ~ class_Power_Opower(t_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(115,plain,
( ~ ! [V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Power_Opower(T_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )
| ~ class_Power_Opower(t_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) ) ),
inference(modus_ponens,[status(thm)],[114,113]) ).
tff(116,plain,
( ~ class_Power_Opower(t_a)
| ( hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) ) ),
inference(unit_resolution,[status(thm)],[115,112]) ).
tff(117,plain,
hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),
inference(unit_resolution,[status(thm)],[116,102]) ).
tff(118,plain,
hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),
inference(symmetry,[status(thm)],[117]) ).
tff(119,plain,
hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = v_x,
inference(transitivity,[status(thm)],[118,88,83]) ).
tff(120,plain,
hAPP(c_Power_Opower__class_Opower(t_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) = hAPP(c_Power_Opower__class_Opower(t_a),v_x),
inference(monotonicity,[status(thm)],[119]) ).
tff(121,plain,
hAPP(hAPP(c_Power_Opower__class_Opower(t_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),v_n) = hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n),
inference(monotonicity,[status(thm)],[120]) ).
tff(122,plain,
hAPP(hAPP(c_Power_Opower__class_Opower(t_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),v_n) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),c_Groups_Oone__class_Oone(t_a)),
inference(transitivity,[status(thm)],[121,66]) ).
tff(123,plain,
hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),c_Groups_Oone__class_Oone(t_a)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),v_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),c_Groups_Oone__class_Oone(t_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),c_Groups_Oone__class_Oone(t_a))),
inference(monotonicity,[status(thm)],[122]) ).
tff(124,plain,
^ [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
refl(( ( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) )
<=> ( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(125,plain,
( ! [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) )
<=> ! [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) ) ),
inference(quant_intro,[status(thm)],[124]) ).
tff(126,plain,
( ! [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) )
<=> ! [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(127,plain,
^ [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
rewrite(( ( class_Rings_Ocomm__semiring__1(T_a)
=> ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) )
<=> ( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(128,plain,
( ! [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
( class_Rings_Ocomm__semiring__1(T_a)
=> ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) )
<=> ! [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) ) ),
inference(quant_intro,[status(thm)],[127]) ).
tff(129,axiom,
! [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
( class_Rings_Ocomm__semiring__1(T_a)
=> ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_poly__monom) ).
tff(130,plain,
! [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) ),
inference(modus_ponens,[status(thm)],[129,128]) ).
tff(131,plain,
! [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) ),
inference(modus_ponens,[status(thm)],[130,126]) ).
tff(132,plain,
! [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) ),
inference(skolemize,[status(sab)],[131]) ).
tff(133,plain,
! [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) ),
inference(modus_ponens,[status(thm)],[132,125]) ).
tff(134,plain,
( ( ~ ! [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) )
| ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),c_Groups_Oone__class_Oone(t_a)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),v_n)) ) )
<=> ( ~ ! [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) )
| ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),c_Groups_Oone__class_Oone(t_a)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),v_n)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(135,plain,
( ~ ! [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) )
| ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),c_Groups_Oone__class_Oone(t_a)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),v_n)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(136,plain,
( ~ ! [V_x: $i,V_n: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) )
| ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),c_Groups_Oone__class_Oone(t_a)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),v_n)) ) ),
inference(modus_ponens,[status(thm)],[135,134]) ).
tff(137,plain,
hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),c_Groups_Oone__class_Oone(t_a)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),v_n)),
inference(unit_resolution,[status(thm)],[136,133,17]) ).
tff(138,plain,
v_x = hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)),
inference(symmetry,[status(thm)],[83]) ).
tff(139,plain,
v_x = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),
inference(transitivity,[status(thm)],[138,87,117]) ).
tff(140,plain,
hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x) = hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),v_x),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),
inference(monotonicity,[status(thm)],[139]) ).
tff(141,plain,
hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x) = hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n),
inference(transitivity,[status(thm)],[140,137,123,68,65,61]) ).
tff(142,plain,
hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),
inference(monotonicity,[status(thm)],[141]) ).
tff(143,plain,
( ( ~ ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
| ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) ) )
<=> ( ~ ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
| ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(144,plain,
( ~ ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
| ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(145,plain,
( ~ ! [V_b: $i,V_a: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__1(T_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )
| ~ class_Rings_Ocomm__semiring__1(t_a)
| ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) ) ),
inference(modus_ponens,[status(thm)],[144,143]) ).
tff(146,plain,
hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),
inference(unit_resolution,[status(thm)],[145,27,17]) ).
tff(147,plain,
hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)),
inference(symmetry,[status(thm)],[146]) ).
tff(148,plain,
^ [T: $i] :
refl(( ( class_Rings_Ocomm__semiring__0(T)
| ~ class_Rings_Ocomm__ring__1(T) )
<=> ( class_Rings_Ocomm__semiring__0(T)
| ~ class_Rings_Ocomm__ring__1(T) ) )),
inference(bind,[status(th)],[]) ).
tff(149,plain,
( ! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
| ~ class_Rings_Ocomm__ring__1(T) )
<=> ! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
| ~ class_Rings_Ocomm__ring__1(T) ) ),
inference(quant_intro,[status(thm)],[148]) ).
tff(150,plain,
( ! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
| ~ class_Rings_Ocomm__ring__1(T) )
<=> ! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
| ~ class_Rings_Ocomm__ring__1(T) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(151,plain,
^ [T: $i] :
rewrite(( ( class_Rings_Ocomm__ring__1(T)
=> class_Rings_Ocomm__semiring__0(T) )
<=> ( class_Rings_Ocomm__semiring__0(T)
| ~ class_Rings_Ocomm__ring__1(T) ) )),
inference(bind,[status(th)],[]) ).
tff(152,plain,
( ! [T: $i] :
( class_Rings_Ocomm__ring__1(T)
=> class_Rings_Ocomm__semiring__0(T) )
<=> ! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
| ~ class_Rings_Ocomm__ring__1(T) ) ),
inference(quant_intro,[status(thm)],[151]) ).
tff(153,axiom,
! [T: $i] :
( class_Rings_Ocomm__ring__1(T)
=> class_Rings_Ocomm__semiring__0(T) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clrel_Rings_Ocomm__ring__1__Rings_Ocomm__semiring__0) ).
tff(154,plain,
! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
| ~ class_Rings_Ocomm__ring__1(T) ),
inference(modus_ponens,[status(thm)],[153,152]) ).
tff(155,plain,
! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
| ~ class_Rings_Ocomm__ring__1(T) ),
inference(modus_ponens,[status(thm)],[154,150]) ).
tff(156,plain,
! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
| ~ class_Rings_Ocomm__ring__1(T) ),
inference(skolemize,[status(sab)],[155]) ).
tff(157,plain,
! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
| ~ class_Rings_Ocomm__ring__1(T) ),
inference(modus_ponens,[status(thm)],[156,149]) ).
tff(158,plain,
( ( ~ ! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
| ~ class_Rings_Ocomm__ring__1(T) )
| class_Rings_Ocomm__semiring__0(t_a)
| ~ class_Rings_Ocomm__ring__1(t_a) )
<=> ( ~ ! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
| ~ class_Rings_Ocomm__ring__1(T) )
| class_Rings_Ocomm__semiring__0(t_a)
| ~ class_Rings_Ocomm__ring__1(t_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(159,plain,
( ~ ! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
| ~ class_Rings_Ocomm__ring__1(T) )
| class_Rings_Ocomm__semiring__0(t_a)
| ~ class_Rings_Ocomm__ring__1(t_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(160,plain,
( ~ ! [T: $i] :
( class_Rings_Ocomm__semiring__0(T)
| ~ class_Rings_Ocomm__ring__1(T) )
| class_Rings_Ocomm__semiring__0(t_a)
| ~ class_Rings_Ocomm__ring__1(t_a) ),
inference(modus_ponens,[status(thm)],[159,158]) ).
tff(161,plain,
class_Rings_Ocomm__semiring__0(t_a),
inference(unit_resolution,[status(thm)],[160,157,3]) ).
tff(162,plain,
^ [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
refl(( ( ~ class_Rings_Ocomm__semiring__0(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) )
<=> ( ~ class_Rings_Ocomm__semiring__0(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(163,plain,
( ! [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) )
<=> ! [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) ) ),
inference(quant_intro,[status(thm)],[162]) ).
tff(164,plain,
( ! [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) )
<=> ! [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(165,plain,
^ [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
rewrite(( ( class_Rings_Ocomm__semiring__0(T_a)
=> ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) )
<=> ( ~ class_Rings_Ocomm__semiring__0(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(166,plain,
( ! [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
( class_Rings_Ocomm__semiring__0(T_a)
=> ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) )
<=> ! [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) ) ),
inference(quant_intro,[status(thm)],[165]) ).
tff(167,axiom,
! [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
( class_Rings_Ocomm__semiring__0(T_a)
=> ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_poly__mult) ).
tff(168,plain,
! [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) ),
inference(modus_ponens,[status(thm)],[167,166]) ).
tff(169,plain,
! [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) ),
inference(modus_ponens,[status(thm)],[168,164]) ).
tff(170,plain,
! [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) ),
inference(skolemize,[status(sab)],[169]) ).
tff(171,plain,
! [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) ),
inference(modus_ponens,[status(thm)],[170,163]) ).
tff(172,plain,
( ( ~ ! [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) )
| ~ class_Rings_Ocomm__semiring__0(t_a)
| ( hAPP(c_Polynomial_Opoly(t_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(t_a)),c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_p)),v_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) ) )
<=> ( ~ ! [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) )
| ~ class_Rings_Ocomm__semiring__0(t_a)
| ( hAPP(c_Polynomial_Opoly(t_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(t_a)),c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_p)),v_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(173,plain,
( ~ ! [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) )
| ~ class_Rings_Ocomm__semiring__0(t_a)
| ( hAPP(c_Polynomial_Opoly(t_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(t_a)),c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_p)),v_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(174,plain,
( ~ ! [V_x: $i,V_q: $i,V_p: $i,T_a: $i] :
( ~ class_Rings_Ocomm__semiring__0(T_a)
| ( hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) ) )
| ~ class_Rings_Ocomm__semiring__0(t_a)
| ( hAPP(c_Polynomial_Opoly(t_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(t_a)),c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_p)),v_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) ) ),
inference(modus_ponens,[status(thm)],[173,172]) ).
tff(175,plain,
( ~ class_Rings_Ocomm__semiring__0(t_a)
| ( hAPP(c_Polynomial_Opoly(t_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(t_a)),c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_p)),v_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) ) ),
inference(unit_resolution,[status(thm)],[174,171]) ).
tff(176,plain,
hAPP(c_Polynomial_Opoly(t_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(t_a)),c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_p)),v_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_x)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),
inference(unit_resolution,[status(thm)],[175,161]) ).
tff(177,plain,
hAPP(c_Polynomial_Opoly(t_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(t_a)),c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_p)),v_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),
inference(transitivity,[status(thm)],[176,147,142,32]) ).
tff(178,plain,
( ( hAPP(c_Polynomial_Opoly(t_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(t_a)),c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_p)),v_x) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) )
<=> ( hAPP(c_Polynomial_Opoly(t_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(t_a)),c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_p)),v_x) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(179,axiom,
hAPP(c_Polynomial_Opoly(t_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(t_a)),c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_p)),v_x) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).
tff(180,plain,
hAPP(c_Polynomial_Opoly(t_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(t_a)),c_Polynomial_Omonom(t_a,c_Groups_Oone__class_Oone(t_a),v_n)),v_p)),v_x) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),v_x),v_n)),hAPP(c_Polynomial_Opoly(t_a,v_p),v_x)),
inference(modus_ponens,[status(thm)],[179,178]) ).
tff(181,plain,
$false,
inference(unit_resolution,[status(thm)],[180,177]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : SWW240+1 : TPTP v9.0.0. Released v5.2.0.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n003.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 11:20:24 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.64/0.83 % SZS status Theorem
% 0.64/0.83 % SZS output start Proof
% See solution above
% 0.75/0.91 % E exiting
%------------------------------------------------------------------------------