%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SWC040+1 : TPTP v9.0.0. Released v2.4.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n001.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 05:30:00 AM UTC 2025
% Result : Theorem 0.22s 0.40s
% Output : Proof 0.22s
% Verified :
% SZS Type : Refutation
% Derivation depth : 5
% Number of leaves : 4
% Syntax : Number of formulae : 9 ( 2 unt; 0 typ; 0 def)
% Number of atoms : 1148 ( 653 equ)
% Maximal formula atoms : 16 ( 127 avg)
% Number of connectives : 1722 ( 699 ~; 678 |; 178 &)
% ( 63 <=>; 104 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 13 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 116 ( 116 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 14 ( 11 usr; 2 prp; 0-3 aty)
% Number of functors : 3 ( 3 usr; 1 con; 0-2 aty)
% Number of variables : 246 ( 242 !; 0 ?; 246 :)
% Comments :
%------------------------------------------------------------------------------
tff(neq_type,type,
neq: ( $i * $i ) > $o ).
tff(nil_type,type,
nil: $i ).
tff(app_type,type,
app: ( $i * $i ) > $i ).
tff(cons_type,type,
cons: ( $i * $i ) > $i ).
tff(ssList_type,type,
ssList: $i > $o ).
tff(ssItem_type,type,
ssItem: $i > $o ).
tff(1,plain,
( ~ $true
<=> $false ),
inference(rewrite,[status(thm)],[]) ).
tff(2,plain,
( ! [U: $i] : $true
<=> $true ),
inference(elim_unused_vars,[status(thm)],[]) ).
tff(3,plain,
^ [U: $i] :
trans(monotonicity(trans(quant_intro(proof_bind(^ [V: $i] :
trans(monotonicity(trans(quant_intro(proof_bind(^ [W: $i] :
trans(monotonicity(trans(quant_intro(proof_bind(^ [X: $i] :
trans(monotonicity(trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite(( ( ( nil != V )
| ( V != X )
| ( U != W ) )
<=> ( ( U != W )
| ( V != X )
| ( nil != V ) ) )),
( ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U ) )
<=> ( ( U != W )
| ( V != X )
| ( nil != V )
| ( nil = U ) ) )),
rewrite(( ( ( U != W )
| ( V != X )
| ( nil != V )
| ( nil = U ) )
<=> ( ( nil = U )
| ( U != W )
| ( V != X )
| ( nil != V ) ) )),
( ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U ) )
<=> ( ( nil = U )
| ( U != W )
| ( V != X )
| ( nil != V ) ) )),
( ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) ) )
<=> ( ( nil = U )
| ( U != W )
| ( V != X )
| ( nil != V )
| ( ( nil != W )
& ( nil = X ) ) ) )),
rewrite(( ( ( nil = U )
| ( U != W )
| ( V != X )
| ( nil != V )
| ( ( nil != W )
& ( nil = X ) ) )
<=> ( ( nil = U )
| ( U != W )
| ( V != X )
| ( ( nil != W )
& ( nil = X ) )
| ( nil != V ) ) )),
( ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) ) )
<=> ( ( nil = U )
| ( U != W )
| ( V != X )
| ( ( nil != W )
& ( nil = X ) )
| ( nil != V ) ) )),
rewrite(( ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) )
<=> ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X )
| ~ ssList(Z) ) )
& neq(X,nil) ) )),
( ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) )
<=> ( ( nil = U )
| ( U != W )
| ( V != X )
| ( ( nil != W )
& ( nil = X ) )
| ( nil != V )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X )
| ~ ssList(Z) ) )
& neq(X,nil) ) ) )),
rewrite(( ( ( nil = U )
| ( U != W )
| ( V != X )
| ( ( nil != W )
& ( nil = X ) )
| ( nil != V )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X )
| ~ ssList(Z) ) )
& neq(X,nil) ) )
<=> ( ( nil = U )
| ( U != W )
| ( V != X )
| ( ( nil != W )
& ( nil = X ) )
| ( nil != V )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X )
| ~ ssList(Z) ) )
& neq(X,nil) ) ) )),
( ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) )
<=> ( ( nil = U )
| ( U != W )
| ( V != X )
| ( ( nil != W )
& ( nil = X ) )
| ( nil != V )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X )
| ~ ssList(Z) ) )
& neq(X,nil) ) ) )),
( ( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) )
<=> ( ssList(X)
=> ( ( nil = U )
| ( U != W )
| ( V != X )
| ( ( nil != W )
& ( nil = X ) )
| ( nil != V )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X )
| ~ ssList(Z) ) )
& neq(X,nil) ) ) ) )),
rewrite(( ( ssList(X)
=> ( ( nil = U )
| ( U != W )
| ( V != X )
| ( ( nil != W )
& ( nil = X ) )
| ( nil != V )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X )
| ~ ssList(Z) ) )
& neq(X,nil) ) ) )
<=> ( ( nil = U )
| ( U != W )
| ( V != X )
| ( ( nil != W )
& ( nil = X ) )
| ( nil != V )
| ~ ssList(X)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X )
| ~ ssList(Z) ) )
& neq(X,nil) ) ) )),
( ( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) )
<=> ( ( nil = U )
| ( U != W )
| ( V != X )
| ( ( nil != W )
& ( nil = X ) )
| ( nil != V )
| ~ ssList(X)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X )
| ~ ssList(Z) ) )
& neq(X,nil) ) ) ))),
( ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) )
<=> ! [X: $i] :
( ( nil = U )
| ( U != W )
| ( V != X )
| ( ( nil != W )
& ( nil = X ) )
| ( nil != V )
| ~ ssList(X)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X )
| ~ ssList(Z) ) )
& neq(X,nil) ) ) )),
trans(trans(der(( ! [X: $i] :
( ( nil = U )
| ( U != W )
| ( V != X )
| ( ( nil != W )
& ( nil = X ) )
| ( nil != V )
| ~ ssList(X)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X )
| ~ ssList(Z) ) )
& neq(X,nil) ) )
<=> ! [X: $i] :
( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V )
| ~ ssList(Z) ) )
& neq(V,nil) )
| ( nil = U )
| ( U != W )
| ( ( nil != W )
& ( nil = V ) )
| ( nil != V )
| ~ ssList(V) ) )),
elim_unused(( ! [X: $i] :
( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V )
| ~ ssList(Z) ) )
& neq(V,nil) )
| ( nil = U )
| ( U != W )
| ( ( nil != W )
& ( nil = V ) )
| ( nil != V )
| ~ ssList(V) )
<=> ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V )
| ~ ssList(Z) ) )
& neq(V,nil) )
| ( nil = U )
| ( U != W )
| ( ( nil != W )
& ( nil = V ) )
| ( nil != V )
| ~ ssList(V) ) )),
( ! [X: $i] :
( ( nil = U )
| ( U != W )
| ( V != X )
| ( ( nil != W )
& ( nil = X ) )
| ( nil != V )
| ~ ssList(X)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X )
| ~ ssList(Z) ) )
& neq(X,nil) ) )
<=> ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V )
| ~ ssList(Z) ) )
& neq(V,nil) )
| ( nil = U )
| ( U != W )
| ( ( nil != W )
& ( nil = V ) )
| ( nil != V )
| ~ ssList(V) ) )),
trans(monotonicity(rewrite(( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V )
| ~ ssList(Z) ) )
& neq(V,nil) )
<=> ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) )),
( ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V )
| ~ ssList(Z) ) )
& neq(V,nil) )
| ( nil = U )
| ( U != W )
| ( ( nil != W )
& ( nil = V ) )
| ( nil != V )
| ~ ssList(V) )
<=> ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) )
| ( nil = U )
| ( U != W )
| ( ( nil != W )
& ( nil = V ) )
| ( nil != V )
| ~ ssList(V) ) )),
rewrite(( ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) )
| ( nil = U )
| ( U != W )
| ( ( nil != W )
& ( nil = V ) )
| ( nil != V )
| ~ ssList(V) )
<=> ( ( nil != V )
| ( U != W )
| ( nil = U )
| ~ ssList(V)
| ( ( nil != W )
& ( nil = V ) )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) )),
( ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V )
| ~ ssList(Z) ) )
& neq(V,nil) )
| ( nil = U )
| ( U != W )
| ( ( nil != W )
& ( nil = V ) )
| ( nil != V )
| ~ ssList(V) )
<=> ( ( nil != V )
| ( U != W )
| ( nil = U )
| ~ ssList(V)
| ( ( nil != W )
& ( nil = V ) )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) )),
( ! [X: $i] :
( ( nil = U )
| ( U != W )
| ( V != X )
| ( ( nil != W )
& ( nil = X ) )
| ( nil != V )
| ~ ssList(X)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X )
| ~ ssList(Z) ) )
& neq(X,nil) ) )
<=> ( ( nil != V )
| ( U != W )
| ( nil = U )
| ~ ssList(V)
| ( ( nil != W )
& ( nil = V ) )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) )),
( ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) )
<=> ( ( nil != V )
| ( U != W )
| ( nil = U )
| ~ ssList(V)
| ( ( nil != W )
& ( nil = V ) )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) )),
( ( ssList(W)
=> ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) ) )
<=> ( ssList(W)
=> ( ( nil != V )
| ( U != W )
| ( nil = U )
| ~ ssList(V)
| ( ( nil != W )
& ( nil = V ) )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) ) )),
rewrite(( ( ssList(W)
=> ( ( nil != V )
| ( U != W )
| ( nil = U )
| ~ ssList(V)
| ( ( nil != W )
& ( nil = V ) )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) )
<=> ( ( nil != V )
| ( U != W )
| ( nil = U )
| ~ ssList(W)
| ~ ssList(V)
| ( ( nil != W )
& ( nil = V ) )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) )),
( ( ssList(W)
=> ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) ) )
<=> ( ( nil != V )
| ( U != W )
| ( nil = U )
| ~ ssList(W)
| ~ ssList(V)
| ( ( nil != W )
& ( nil = V ) )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) ))),
( ! [W: $i] :
( ssList(W)
=> ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) ) )
<=> ! [W: $i] :
( ( nil != V )
| ( U != W )
| ( nil = U )
| ~ ssList(W)
| ~ ssList(V)
| ( ( nil != W )
& ( nil = V ) )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) )),
trans(trans(der(( ! [W: $i] :
( ( nil != V )
| ( U != W )
| ( nil = U )
| ~ ssList(W)
| ~ ssList(V)
| ( ( nil != W )
& ( nil = V ) )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) )
<=> ! [W: $i] :
( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) )
| ( nil != V )
| ( nil = U )
| ~ ssList(U)
| ~ ssList(V)
| ( ( nil != U )
& ( nil = V ) ) ) )),
elim_unused(( ! [W: $i] :
( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) )
| ( nil != V )
| ( nil = U )
| ~ ssList(U)
| ~ ssList(V)
| ( ( nil != U )
& ( nil = V ) ) )
<=> ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) )
| ( nil != V )
| ( nil = U )
| ~ ssList(U)
| ~ ssList(V)
| ( ( nil != U )
& ( nil = V ) ) ) )),
( ! [W: $i] :
( ( nil != V )
| ( U != W )
| ( nil = U )
| ~ ssList(W)
| ~ ssList(V)
| ( ( nil != W )
& ( nil = V ) )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) )
<=> ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) )
| ( nil != V )
| ( nil = U )
| ~ ssList(U)
| ~ ssList(V)
| ( ( nil != U )
& ( nil = V ) ) ) )),
trans(monotonicity(rewrite(( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) )
<=> ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) )),
( ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) )
| ( nil != V )
| ( nil = U )
| ~ ssList(U)
| ~ ssList(V)
| ( ( nil != U )
& ( nil = V ) ) )
<=> ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) )
| ( nil != V )
| ( nil = U )
| ~ ssList(U)
| ~ ssList(V)
| ( ( nil != U )
& ( nil = V ) ) ) )),
rewrite(( ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) )
| ( nil != V )
| ( nil = U )
| ~ ssList(U)
| ~ ssList(V)
| ( ( nil != U )
& ( nil = V ) ) )
<=> ( ( nil != V )
| ( nil = U )
| ( ( nil != U )
& ( nil = V ) )
| ~ ssList(V)
| ~ ssList(U)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) )),
( ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) )
| ( nil != V )
| ( nil = U )
| ~ ssList(U)
| ~ ssList(V)
| ( ( nil != U )
& ( nil = V ) ) )
<=> ( ( nil != V )
| ( nil = U )
| ( ( nil != U )
& ( nil = V ) )
| ~ ssList(V)
| ~ ssList(U)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) )),
( ! [W: $i] :
( ( nil != V )
| ( U != W )
| ( nil = U )
| ~ ssList(W)
| ~ ssList(V)
| ( ( nil != W )
& ( nil = V ) )
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) )
<=> ( ( nil != V )
| ( nil = U )
| ( ( nil != U )
& ( nil = V ) )
| ~ ssList(V)
| ~ ssList(U)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) )),
( ! [W: $i] :
( ssList(W)
=> ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) ) )
<=> ( ( nil != V )
| ( nil = U )
| ( ( nil != U )
& ( nil = V ) )
| ~ ssList(V)
| ~ ssList(U)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) )),
( ( ssList(V)
=> ! [W: $i] :
( ssList(W)
=> ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) ) ) )
<=> ( ssList(V)
=> ( ( nil != V )
| ( nil = U )
| ( ( nil != U )
& ( nil = V ) )
| ~ ssList(V)
| ~ ssList(U)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) ) )),
rewrite(( ( ssList(V)
=> ( ( nil != V )
| ( nil = U )
| ( ( nil != U )
& ( nil = V ) )
| ~ ssList(V)
| ~ ssList(U)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) )
<=> ( ( nil != V )
| ( nil = U )
| ( ( nil != U )
& ( nil = V ) )
| ~ ssList(V)
| ~ ssList(U)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) )),
( ( ssList(V)
=> ! [W: $i] :
( ssList(W)
=> ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) ) ) )
<=> ( ( nil != V )
| ( nil = U )
| ( ( nil != U )
& ( nil = V ) )
| ~ ssList(V)
| ~ ssList(U)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) ))),
( ! [V: $i] :
( ssList(V)
=> ! [W: $i] :
( ssList(W)
=> ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) ) ) )
<=> ! [V: $i] :
( ( nil != V )
| ( nil = U )
| ( ( nil != U )
& ( nil = V ) )
| ~ ssList(V)
| ~ ssList(U)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) ) )),
trans(trans(der(( ! [V: $i] :
( ( nil != V )
| ( nil = U )
| ( ( nil != U )
& ( nil = V ) )
| ~ ssList(V)
| ~ ssList(U)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) )
<=> ! [V: $i] :
( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != nil ) ) )
& neq(nil,nil) )
| ( nil = U )
| ( ( nil != U )
& ( nil = nil ) )
| ~ ssList(nil)
| ~ ssList(U) ) )),
elim_unused(( ! [V: $i] :
( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != nil ) ) )
& neq(nil,nil) )
| ( nil = U )
| ( ( nil != U )
& ( nil = nil ) )
| ~ ssList(nil)
| ~ ssList(U) )
<=> ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != nil ) ) )
& neq(nil,nil) )
| ( nil = U )
| ( ( nil != U )
& ( nil = nil ) )
| ~ ssList(nil)
| ~ ssList(U) ) )),
( ! [V: $i] :
( ( nil != V )
| ( nil = U )
| ( ( nil != U )
& ( nil = V ) )
| ~ ssList(V)
| ~ ssList(U)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) )
<=> ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != nil ) ) )
& neq(nil,nil) )
| ( nil = U )
| ( ( nil != U )
& ( nil = nil ) )
| ~ ssList(nil)
| ~ ssList(U) ) )),
trans(monotonicity(rewrite(( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != nil ) ) )
& neq(nil,nil) )
<=> ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != nil ) ) )
& neq(nil,nil) ) )),
trans(monotonicity(rewrite(( ( nil = nil )
<=> $true )),
( ( ( nil != U )
& ( nil = nil ) )
<=> ( ( nil != U )
& $true ) )),
rewrite(( ( ( nil != U )
& $true )
<=> ( nil != U ) )),
( ( ( nil != U )
& ( nil = nil ) )
<=> ( nil != U ) )),
( ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != nil ) ) )
& neq(nil,nil) )
| ( nil = U )
| ( ( nil != U )
& ( nil = nil ) )
| ~ ssList(nil)
| ~ ssList(U) )
<=> ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != nil ) ) )
& neq(nil,nil) )
| ( nil = U )
| ( nil != U )
| ~ ssList(nil)
| ~ ssList(U) ) )),
rewrite(( ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != nil ) ) )
& neq(nil,nil) )
| ( nil = U )
| ( nil != U )
| ~ ssList(nil)
| ~ ssList(U) )
<=> $true )),
( ( ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != nil ) ) )
& neq(nil,nil) )
| ( nil = U )
| ( ( nil != U )
& ( nil = nil ) )
| ~ ssList(nil)
| ~ ssList(U) )
<=> $true )),
( ! [V: $i] :
( ( nil != V )
| ( nil = U )
| ( ( nil != U )
& ( nil = V ) )
| ~ ssList(V)
| ~ ssList(U)
| ( ! [Y: $i] :
( ~ ssItem(Y)
| ! [Z: $i] :
( ~ ssList(Z)
| ( app(cons(Y,nil),Z) != U )
| ( app(Z,cons(Y,nil)) != V ) ) )
& neq(V,nil) ) )
<=> $true )),
( ! [V: $i] :
( ssList(V)
=> ! [W: $i] :
( ssList(W)
=> ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) ) ) )
<=> $true )),
( ( ssList(U)
=> ! [V: $i] :
( ssList(V)
=> ! [W: $i] :
( ssList(W)
=> ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) ) ) ) )
<=> ( ssList(U)
=> $true ) )),
rewrite(( ( ssList(U)
=> $true )
<=> $true )),
( ( ssList(U)
=> ! [V: $i] :
( ssList(V)
=> ! [W: $i] :
( ssList(W)
=> ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) ) ) ) )
<=> $true )),
inference(bind,[status(th)],[]) ).
tff(4,plain,
( ! [U: $i] :
( ssList(U)
=> ! [V: $i] :
( ssList(V)
=> ! [W: $i] :
( ssList(W)
=> ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) ) ) ) )
<=> ! [U: $i] : $true ),
inference(quant_intro,[status(thm)],[3]) ).
tff(5,plain,
( ! [U: $i] :
( ssList(U)
=> ! [V: $i] :
( ssList(V)
=> ! [W: $i] :
( ssList(W)
=> ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) ) ) ) )
<=> $true ),
inference(transitivity,[status(thm)],[4,2]) ).
tff(6,plain,
( ~ ! [U: $i] :
( ssList(U)
=> ! [V: $i] :
( ssList(V)
=> ! [W: $i] :
( ssList(W)
=> ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) ) ) ) )
<=> ~ $true ),
inference(monotonicity,[status(thm)],[5]) ).
tff(7,plain,
( ~ ! [U: $i] :
( ssList(U)
=> ! [V: $i] :
( ssList(V)
=> ! [W: $i] :
( ssList(W)
=> ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) ) ) ) )
<=> $false ),
inference(transitivity,[status(thm)],[6,1]) ).
tff(8,axiom,
~ ! [U: $i] :
( ssList(U)
=> ! [V: $i] :
( ssList(V)
=> ! [W: $i] :
( ssList(W)
=> ! [X: $i] :
( ssList(X)
=> ( ( nil != V )
| ( V != X )
| ( U != W )
| ( nil = U )
| ( ( nil != W )
& ( nil = X ) )
| ( ! [Y: $i] :
( ssItem(Y)
=> ! [Z: $i] :
( ssList(Z)
=> ( ( app(cons(Y,nil),Z) != W )
| ( app(Z,cons(Y,nil)) != X ) ) ) )
& neq(X,nil) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',co1) ).
tff(9,plain,
$false,
inference(modus_ponens,[status(thm)],[8,7]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SWC040+1 : TPTP v9.0.0. Released v2.4.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n001.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Jun 20 08:09:50 EDT 2025
% 0.12/0.35 % CPUTime :
% 0.22/0.40 % SZS status Theorem
% 0.22/0.40 % SZS output start Proof
% See solution above
% 0.22/0.42 % E exiting
%------------------------------------------------------------------------------