TSTP Solution File: SWW245+1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SWW245+1 : TPTP v8.1.0. Released v5.2.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n015.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 : Thu Sep 29 20:58:02 EDT 2022
% Result : Theorem 0.46s 0.62s
% Output : Proof 0.58s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 33
% Syntax : Number of formulae : 59 ( 4 unt; 17 typ; 0 def)
% Number of atoms : 395 ( 344 equ)
% Maximal formula atoms : 16 ( 9 avg)
% Number of connectives : 503 ( 151 ~; 120 |; 150 &)
% ( 30 <=>; 52 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 9 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of FOOLs : 1 ( 1 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 24 ( 13 >; 11 *; 0 +; 0 <<)
% Number of predicates : 5 ( 2 usr; 2 prp; 0-2 aty)
% Number of functors : 16 ( 16 usr; 4 con; 0-4 aty)
% Number of variables : 198 ( 50 !; 146 ?; 198 :)
% Comments :
%------------------------------------------------------------------------------
tff(hAPP_type,type,
hAPP: ( $i * $i ) > $i ).
tff(c_Polynomial_Opoly_type,type,
c_Polynomial_Opoly: ( $i * $i ) > $i ).
tff(c_Polynomial_OpCons_type,type,
c_Polynomial_OpCons: ( $i * $i * $i ) > $i ).
tff(t_a_type,type,
t_a: $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(v_cs_____type,type,
v_cs____: $i ).
tff(v_c_____type,type,
v_c____: $i ).
tff(c_Nat_OSuc_type,type,
c_Nat_OSuc: $i > $i ).
tff(c_Polynomial_Odegree_type,type,
c_Polynomial_Odegree: ( $i * $i ) > $i ).
tff(c_Groups_Oplus__class_Oplus_type,type,
c_Groups_Oplus__class_Oplus: ( $i * $i * $i ) > $i ).
tff(c_If_type,type,
c_If: ( $i * $i * $i * $i ) > $i ).
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_fequal_type,type,
c_fequal: ( $i * $i ) > $i ).
tff(tc_Polynomial_Opoly_type,type,
tc_Polynomial_Opoly: $i > $i ).
tff(class_Rings_Oidom_type,type,
class_Rings_Oidom: $i > $o ).
tff(1,plain,
( ( $false
| $false
| $false )
<=> $false ),
inference(rewrite,[status(thm)],]) ).
tff(2,plain,
( ~ ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) )
<=> ~ ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(3,plain,
( ~ ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) )
<=> ~ ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(4,axiom,
~ ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).
tff(5,plain,
~ ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
~ ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ),
inference(modus_ponens,[status(thm)],[5,2]) ).
tff(7,plain,
( ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) )
<=> $false ),
inference(iff_false,[status(thm)],[6]) ).
tff(8,plain,
( ~ $true
<=> $false ),
inference(rewrite,[status(thm)],]) ).
tff(9,axiom,
class_Rings_Oidom(t_a),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tfree_0) ).
tff(10,plain,
( class_Rings_Oidom(t_a)
<=> $true ),
inference(iff_true,[status(thm)],[9]) ).
tff(11,plain,
( ~ class_Rings_Oidom(t_a)
<=> ~ $true ),
inference(monotonicity,[status(thm)],[10]) ).
tff(12,plain,
( ~ class_Rings_Oidom(t_a)
<=> $false ),
inference(transitivity,[status(thm)],[11,8]) ).
tff(13,plain,
( ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
| $false
| $false )
<=> ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(14,plain,
( ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
| ~ class_Rings_Oidom(t_a)
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) )
<=> ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
| $false
| $false ) ),
inference(monotonicity,[status(thm)],[12,7]) ).
tff(15,plain,
( ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
| ~ class_Rings_Oidom(t_a)
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) )
<=> ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) ) ),
inference(transitivity,[status(thm)],[14,13]) ).
tff(16,plain,
( ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
| ~ class_Rings_Oidom(t_a)
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) )
<=> ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
| ~ class_Rings_Oidom(t_a)
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(17,plain,
( ( class_Rings_Oidom(t_a)
=> ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) )
<=> ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
| ~ class_Rings_Oidom(t_a)
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(18,plain,
( ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
=> ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) )
<=> ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(19,plain,
^ [B_k: $i,B_a: $i] :
rewrite(
( ( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) )
<=> ( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) )),
inference(bind,[status(th)],]) ).
tff(20,plain,
( ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) )
<=> ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ),
inference(quant_intro,[status(thm)],[19]) ).
tff(21,plain,
( ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
=> ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) )
<=> ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
=> ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) ),
inference(monotonicity,[status(thm)],[20]) ).
tff(22,plain,
( ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
=> ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) )
<=> ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) ),
inference(transitivity,[status(thm)],[21,18]) ).
tff(23,plain,
( ( class_Rings_Oidom(t_a)
=> ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
=> ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) )
<=> ( class_Rings_Oidom(t_a)
=> ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) ) ),
inference(monotonicity,[status(thm)],[22]) ).
tff(24,plain,
( ( class_Rings_Oidom(t_a)
=> ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
=> ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) )
<=> ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
| ~ class_Rings_Oidom(t_a)
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) ),
inference(transitivity,[status(thm)],[23,17]) ).
tff(25,axiom,
( class_Rings_Oidom(t_a)
=> ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
=> ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact__096c_A_061_A_I0_058_058_Ha_J_061_061_062_AEX_Ak_Aa_Aq_O_Aa_A_126_061_A_I0_058_058_Ha_J_A_G_ASuc_A_I_Iif_Aq_A_061_A0_Athen_A0_Aelse_ASuc_A_Idegree_Aq_J_J_A_L_Ak_J_A_061_A_Iif_ApCons_Ac_Acs_A_061_A0_Athen_A0_Aelse_ASuc_A_Idegree_A_IpCons_Ac_Acs_J_J_J_A_G_A_IALL_Az_O_Apoly_A_IpCons_Ac_Acs_J_Az_A_061_Az_A_094_Ak_A_K_Apoly_A_IpCons_Aa_Aq_J_Az_J_096) ).
tff(26,plain,
( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
| ~ class_Rings_Oidom(t_a)
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ),
inference(modus_ponens,[status(thm)],[25,24]) ).
tff(27,plain,
( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
| ~ class_Rings_Oidom(t_a)
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ),
inference(modus_ponens,[status(thm)],[26,16]) ).
tff(28,plain,
v_c____ != c_Groups_Ozero__class_Ozero(t_a),
inference(modus_ponens,[status(thm)],[27,15]) ).
tff(29,plain,
( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
<=> $false ),
inference(iff_false,[status(thm)],[28]) ).
tff(30,plain,
( ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
| ~ class_Rings_Oidom(t_a)
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) )
<=> ( $false
| $false
| $false ) ),
inference(monotonicity,[status(thm)],[29,12,7]) ).
tff(31,plain,
( ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
| ~ class_Rings_Oidom(t_a)
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) )
<=> $false ),
inference(transitivity,[status(thm)],[30,1]) ).
tff(32,plain,
( ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
| ~ class_Rings_Oidom(t_a)
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) )
<=> ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
| ~ class_Rings_Oidom(t_a)
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(33,plain,
( ( class_Rings_Oidom(t_a)
=> ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) )
<=> ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
| ~ class_Rings_Oidom(t_a)
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(34,plain,
( ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
=> ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) )
<=> ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(35,plain,
( ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
=> ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) )
<=> ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
=> ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) ),
inference(monotonicity,[status(thm)],[20]) ).
tff(36,plain,
( ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
=> ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) )
<=> ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) ),
inference(transitivity,[status(thm)],[35,34]) ).
tff(37,plain,
( ( class_Rings_Oidom(t_a)
=> ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
=> ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) )
<=> ( class_Rings_Oidom(t_a)
=> ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) ) ),
inference(monotonicity,[status(thm)],[36]) ).
tff(38,plain,
( ( class_Rings_Oidom(t_a)
=> ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
=> ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) )
<=> ( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
| ~ class_Rings_Oidom(t_a)
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) ),
inference(transitivity,[status(thm)],[37,33]) ).
tff(39,axiom,
( class_Rings_Oidom(t_a)
=> ( ( v_c____ != c_Groups_Ozero__class_Ozero(t_a) )
=> ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
=> ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact__096c_A_126_061_A_I0_058_058_Ha_J_061_061_062_AEX_Ak_Aa_Aq_O_Aa_A_126_061_A_I0_058_058_Ha_J_A_G_ASuc_A_I_Iif_Aq_A_061_A0_Athen_A0_Aelse_ASuc_A_Idegree_Aq_J_J_A_L_Ak_J_A_061_A_Iif_ApCons_Ac_Acs_A_061_A0_Athen_A0_Aelse_ASuc_A_Idegree_A_IpCons_Ac_Acs_J_J_J_A_G_A_IALL_Az_O_Apoly_A_IpCons_Ac_Acs_J_Az_A_061_Az_A_094_Ak_A_K_Apoly_A_IpCons_Aa_Aq_J_Az_J_096) ).
tff(40,plain,
( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
| ~ class_Rings_Oidom(t_a)
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ),
inference(modus_ponens,[status(thm)],[39,38]) ).
tff(41,plain,
( ( v_c____ = c_Groups_Ozero__class_Ozero(t_a) )
| ~ class_Rings_Oidom(t_a)
| ? [B_k: $i,B_a: $i] :
( ( B_a != c_Groups_Ozero__class_Ozero(t_a) )
& ? [B_q: $i] :
( ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )
& ( ( c_Polynomial_OpCons(t_a,v_c____,v_cs____) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)) )
| ( c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_If(tc_Nat_Onat,c_fequal(B_q,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(c_Polynomial_Odegree(t_a,B_q))),B_k)) = c_Nat_OSuc(c_Polynomial_Odegree(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____))) ) )
& ! [B_z: $i] : ( hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,v_c____,v_cs____)),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(t_a),hAPP(hAPP(c_Power_Opower__class_Opower(t_a),B_z),B_k)),hAPP(c_Polynomial_Opoly(t_a,c_Polynomial_OpCons(t_a,B_a,B_q)),B_z)) ) ) ) ),
inference(modus_ponens,[status(thm)],[40,32]) ).
tff(42,plain,
$false,
inference(modus_ponens,[status(thm)],[41,31]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.10 % Problem : SWW245+1 : TPTP v8.1.0. Released v5.2.0.
% 0.00/0.10 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.10/0.30 % Computer : n015.cluster.edu
% 0.10/0.30 % Model : x86_64 x86_64
% 0.10/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.30 % Memory : 8042.1875MB
% 0.10/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.30 % CPULimit : 300
% 0.10/0.30 % WCLimit : 300
% 0.10/0.30 % DateTime : Sun Sep 4 13:31:11 EDT 2022
% 0.10/0.31 % CPUTime :
% 0.10/0.31 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.10/0.31 Usage: tptp [options] [-file:]file
% 0.10/0.31 -h, -? prints this message.
% 0.10/0.31 -smt2 print SMT-LIB2 benchmark.
% 0.10/0.31 -m, -model generate model.
% 0.10/0.31 -p, -proof generate proof.
% 0.10/0.31 -c, -core generate unsat core of named formulas.
% 0.10/0.31 -st, -statistics display statistics.
% 0.10/0.31 -t:timeout set timeout (in second).
% 0.10/0.31 -smt2status display status in smt2 format instead of SZS.
% 0.10/0.31 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.10/0.31 -<param>:<value> configuration parameter and value.
% 0.10/0.31 -o:<output-file> file to place output in.
% 0.46/0.62 % SZS status Theorem
% 0.46/0.62 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------