%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : GRP194+1 : TPTP v9.3.1. Released v2.0.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sun Sep 27 07:28:20 AM UTC 2026
% Result : Theorem 0.18s 5.49s
% Output : CNFRefutation 0.18s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 6
% Syntax : Number of formulae : 39 ( 16 unt; 0 def)
% Number of atoms : 96 ( 26 equ)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 94 ( 37 ~; 37 |; 15 &)
% ( 1 <=>; 4 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 3 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 4 ( 2 usr; 1 prp; 0-2 aty)
% Number of functors : 7 ( 7 usr; 3 con; 0-3 aty)
% Number of variables : 47 ( 0 sgn 26 !; 3 ?)
% Comments :
%------------------------------------------------------------------------------
fof(homomorphism1,axiom,
! [X] :
( group_member(X,f)
=> group_member(phi(X),h) ),
file('theBenchmark.p',homomorphism1) ).
fof(homomorphism2,axiom,
! [X,Y] :
( ( group_member(Y,f)
& group_member(X,f) )
=> multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ),
file('theBenchmark.p',homomorphism2) ).
fof(surjective,axiom,
! [X] :
( group_member(X,h)
=> ? [Y] :
( phi(Y) = X
& group_member(Y,f) ) ),
file('theBenchmark.p',surjective) ).
fof(left_zero,axiom,
! [G,X] :
( left_zero(G,X)
<=> ( ! [Y] :
( group_member(Y,G)
=> multiply(G,X,Y) = X )
& group_member(X,G) ) ),
file('theBenchmark.p',left_zero) ).
fof(prove_left_zero_h,conjecture,
left_zero(h,phi(f_left_zero)),
file('theBenchmark.p',prove_left_zero_h) ).
fof(prove_left_zero_h_neg,negated_conjecture,
~ left_zero(h,phi(f_left_zero)),
inference(assume_negation,[status(cth)],[prove_left_zero_h]) ).
cnf(c_10,plain,
( group_member(phi(A),h)
| ~ group_member(A,f) ),
inference(clausify,[status(thm)],[homomorphism1]) ).
cnf(c_11,plain,
( phi(multiply(f,A,B)) = multiply(h,phi(A),phi(B))
| ~ group_member(B,f)
| ~ group_member(A,f) ),
inference(clausify,[status(thm)],[homomorphism2]) ).
cnf(c_12,plain,
( multiply(h,phi(A),phi(B)) = phi(multiply(f,A,B))
| ~ group_member(B,f)
| ~ group_member(A,f) ),
inference(copy,[status(thm)],[c_11]) ).
fof(nnf_5,plain,
! [X] :
( ? [Y] :
( phi(Y) = X
& group_member(Y,f) )
| ~ group_member(X,h) ),
inference(fof_nnf,[status(thm)],[surjective]) ).
fof(sk_surjective_sk,plain,
! [X] :
( ( phi(sK0(X)) = X
& group_member(sK0(X),f) )
| ~ group_member(X,h) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(Y,sK0(X))],[nnf_5]) ).
fof(sk_surjective,plain,
( ! [VAR_0] :
( phi(sK0(VAR_0)) = VAR_0
| ~ group_member(VAR_0,h) )
& ! [VAR_0] :
( group_member(sK0(VAR_0),f)
| ~ group_member(VAR_0,h) ) ),
inference(cnf_transformation,[status(thm)],[sk_surjective_sk]) ).
cnf(c_13,plain,
( group_member(sK0(A),f)
| ~ group_member(A,h) ),
inference(split_conjunct,[status(thm)],[sk_surjective]) ).
cnf(c_14,plain,
( phi(sK0(A)) = A
| ~ group_member(A,h) ),
inference(split_conjunct,[status(thm)],[sk_surjective]) ).
fof(nnf_6,plain,
! [G,X] :
( ( ? [Y] :
( multiply(G,X,Y) != X
& group_member(Y,G) )
| ~ group_member(X,G)
| left_zero(G,X) )
& ( ( ! [X0] :
( multiply(G,X,X0) = X
| ~ group_member(X0,G) )
& group_member(X,G) )
| ~ left_zero(G,X) ) ),
inference(fof_nnf,[status(thm)],[left_zero]) ).
fof(sk_left_zero_sk,plain,
! [G,X] :
( ( ( multiply(G,X,sK1(G,X)) != X
& group_member(sK1(G,X),G) )
| ~ group_member(X,G)
| left_zero(G,X) )
& ( ( ! [X0] :
( multiply(G,X,X0) = X
| ~ group_member(X0,G) )
& group_member(X,G) )
| ~ left_zero(G,X) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1]),skolemize(Y,sK1(G,X))],[nnf_6]) ).
fof(sk_left_zero,plain,
( ! [VAR_0,VAR_1] :
( multiply(VAR_0,VAR_1,sK1(VAR_0,VAR_1)) != VAR_1
| ~ group_member(VAR_1,VAR_0)
| left_zero(VAR_0,VAR_1) )
& ! [VAR_0,VAR_1] :
( group_member(sK1(VAR_0,VAR_1),VAR_0)
| ~ group_member(VAR_1,VAR_0)
| left_zero(VAR_0,VAR_1) )
& ! [VAR_0,VAR_1,VAR_2] :
( multiply(VAR_0,VAR_1,VAR_2) = VAR_1
| ~ group_member(VAR_2,VAR_0)
| ~ left_zero(VAR_0,VAR_1) )
& ! [VAR_0,VAR_1] :
( group_member(VAR_1,VAR_0)
| ~ left_zero(VAR_0,VAR_1) ) ),
inference(cnf_transformation,[status(thm)],[sk_left_zero_sk]) ).
cnf(c_15,plain,
( group_member(B,A)
| ~ left_zero(A,B) ),
inference(split_conjunct,[status(thm)],[sk_left_zero]) ).
cnf(c_16,plain,
( multiply(A,B,C) = B
| ~ group_member(C,A)
| ~ left_zero(A,B) ),
inference(split_conjunct,[status(thm)],[sk_left_zero]) ).
cnf(c_17,plain,
( group_member(sK1(A,B),A)
| ~ group_member(B,A)
| left_zero(A,B) ),
inference(split_conjunct,[status(thm)],[sk_left_zero]) ).
cnf(c_18,plain,
( multiply(A,B,sK1(A,B)) != B
| ~ group_member(B,A)
| left_zero(A,B) ),
inference(split_conjunct,[status(thm)],[sk_left_zero]) ).
fof(left_zero_for_f,axiom,
left_zero(f,f_left_zero),
file('theBenchmark.p',left_zero_for_f) ).
cnf(c_19,plain,
left_zero(f,f_left_zero),
inference(clausify,[status(thm)],[left_zero_for_f]) ).
cnf(c_20,negated_conjecture,
~ left_zero(h,phi(f_left_zero)),
inference(clausify,[status(thm)],[prove_left_zero_h_neg]) ).
cnf(c_27,plain,
( multiply(f,f_left_zero,A) = f_left_zero
| ~ group_member(A,f) ),
inference(resolve,[status(thm)],[c_19,c_16]) ).
cnf(c_28,plain,
group_member(f_left_zero,f),
inference(resolve,[status(thm)],[c_19,c_15]) ).
cnf(c_39,plain,
( multiply(h,phi(f_left_zero),phi(A)) = phi(multiply(f,f_left_zero,A))
| ~ group_member(A,f) ),
inference(resolve,[status(thm)],[c_28,c_12]) ).
cnf(c_40,plain,
group_member(phi(f_left_zero),h),
inference(resolve,[status(thm)],[c_28,c_10]) ).
cnf(c_175,plain,
( multiply(h,phi(f_left_zero),sK1(h,phi(f_left_zero))) != phi(f_left_zero)
| left_zero(h,phi(f_left_zero)) ),
inference(resolve,[status(thm)],[c_40,c_18]) ).
cnf(c_52,negated_conjecture,
multiply(h,phi(f_left_zero),sK1(h,phi(f_left_zero))) != phi(f_left_zero),
inference(resolve,[status(thm)],[c_20,c_175]) ).
cnf(c_176,plain,
( group_member(sK1(h,phi(f_left_zero)),h)
| left_zero(h,phi(f_left_zero)) ),
inference(resolve,[status(thm)],[c_40,c_17]) ).
cnf(c_53,negated_conjecture,
group_member(sK1(h,phi(f_left_zero)),h),
inference(resolve,[status(thm)],[c_20,c_176]) ).
cnf(c_95,plain,
phi(sK0(sK1(h,phi(f_left_zero)))) = sK1(h,phi(f_left_zero)),
inference(resolve,[status(thm)],[c_53,c_14]) ).
cnf(c_96,plain,
group_member(sK0(sK1(h,phi(f_left_zero))),f),
inference(resolve,[status(thm)],[c_53,c_13]) ).
cnf(c_110,plain,
multiply(f,f_left_zero,sK0(sK1(h,phi(f_left_zero)))) = f_left_zero,
inference(resolve,[status(thm)],[c_96,c_27]) ).
cnf(c_177,plain,
multiply(h,phi(f_left_zero),phi(sK0(sK1(h,phi(f_left_zero))))) = phi(multiply(f,f_left_zero,sK0(sK1(h,phi(f_left_zero))))),
inference(resolve,[status(thm)],[c_39,c_96]) ).
cnf(c_178,plain,
multiply(h,phi(f_left_zero),sK1(h,phi(f_left_zero))) = phi(multiply(f,f_left_zero,sK0(sK1(h,phi(f_left_zero))))),
inference(paramod,[status(thm)],[c_95,c_177]) ).
cnf(c_179,plain,
multiply(h,phi(f_left_zero),sK1(h,phi(f_left_zero))) = phi(f_left_zero),
inference(paramod,[status(thm)],[c_110,c_178]) ).
cnf(c_174,plain,
$false,
inference(resolve,[status(thm)],[c_52,c_179]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : GRP194+1 : TPTP v9.3.1. Released v2.0.0.
% 0.00/0.06 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.18/5.44 % Computer : n005.cluster.edu
% 0.18/5.44 % Model : x86_64 x86_64
% 0.18/5.44 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/5.44 % Memory : 8046.5625MB
% 0.18/5.44 % OS : Linux 6.8.0-71-generic
% 0.18/5.44 % CPULimit : 300
% 0.18/5.44 % WCLimit : 300
% 0.18/5.44 % DateTime : Fri Sep 25 14:37:39 UTC 2026
% 0.18/5.45 % CPUTime :
% 0.18/5.45 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.18/5.45 % Process 3710035 was started by sandbox2 on n005,
% 0.18/5.45 % Fri Sep 25 14:37:39 2026
% 0.18/5.45 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.18/5.46
% 0.18/5.46 % From the command line: assign(max_seconds, 300).
% 0.18/5.49
% 0.18/5.49 % SZS status Theorem for theBenchmark
% 0.18/5.49
% 0.18/5.49 % Proof 1 at 0.02 (+ 0.01) seconds.
% 0.18/5.49 % Length of proof is 26.
% 0.18/5.49 % Level of proof is 7.
% 0.18/5.49 % Maximum clause weight is 36.000.
% 0.18/5.49 % Given clauses 44.
% 0.18/5.49
% 0.18/5.49 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------