%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : ALG211+1 : TPTP v9.3.1. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n017.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 06:49:47 AM UTC 2026
% Result : Theorem 0.09s 0.39s
% Output : CNFRefutation 0.09s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 6
% Syntax : Number of formulae : 35 ( 9 unt; 0 def)
% Number of atoms : 88 ( 0 equ)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 91 ( 38 ~; 30 |; 15 &)
% ( 1 <=>; 7 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 6 ( 5 usr; 1 prp; 0-2 aty)
% Number of functors : 6 ( 6 usr; 2 con; 0-3 aty)
% Number of variables : 66 ( 4 sgn 33 !; 10 ?)
% Comments :
%------------------------------------------------------------------------------
fof(basis_of,axiom,
! [B,V] :
( basis_of(B,V)
=> ( a_subset_of(B,vec_to_class(V))
& lin_ind_subset(B,V) ) ),
file('theBenchmark.p',basis_of) ).
fof(bg_2_2_5,axiom,
! [S,T,V] :
( ( basis_of(T,V)
& lin_ind_subset(S,V) )
=> ? [U] :
( basis_of(union(S,U),V)
& a_subset_of(U,T) ) ),
file('theBenchmark.p',bg_2_2_5) ).
fof(bg_remark_63_a,axiom,
! [A] :
( a_vector_space(A)
=> ? [B] : basis_of(B,A) ),
file('theBenchmark.p',bg_remark_63_a) ).
fof(bg_2_4_a,axiom,
! [A,B] :
( a_vector_subspace_of(A,B)
=> a_vector_space(A) ),
file('theBenchmark.p',bg_2_4_a) ).
fof(bg_2_4_2,axiom,
! [W,V,E] :
( ( a_subset_of(E,vec_to_class(W))
& a_vector_subspace_of(W,V) )
=> ( lin_ind_subset(E,W)
<=> lin_ind_subset(E,V) ) ),
file('theBenchmark.p',bg_2_4_2) ).
fof(bg_2_4_3,conjecture,
! [W,V] :
( ( a_vector_space(V)
& a_vector_subspace_of(W,V) )
=> ? [E,F] :
( basis_of(E,W)
& basis_of(union(E,F),V) ) ),
file('theBenchmark.p',bg_2_4_3) ).
fof(bg_2_4_3_neg,negated_conjecture,
~ ! [W,V] :
( ( a_vector_space(V)
& a_vector_subspace_of(W,V) )
=> ? [E,F] :
( basis_of(E,W)
& basis_of(union(E,F),V) ) ),
inference(assume_negation,[status(cth)],[bg_2_4_3]) ).
cnf(c_7,plain,
( lin_ind_subset(C,B)
| ~ lin_ind_subset(C,A)
| ~ a_subset_of(C,vec_to_class(A))
| ~ a_vector_subspace_of(A,B) ),
inference(clausify,[status(thm)],[bg_2_4_2]) ).
cnf(c_9,plain,
( a_subset_of(A,vec_to_class(B))
| ~ basis_of(A,B) ),
inference(clausify,[status(thm)],[basis_of]) ).
fof(nnf_3,plain,
! [A] :
( ? [B] : basis_of(B,A)
| ~ a_vector_space(A) ),
inference(fof_nnf,[status(thm)],[bg_remark_63_a]) ).
fof(sk_bg_remark_63_a,plain,
! [A] :
( basis_of(sK0(A),A)
| ~ a_vector_space(A) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(B,sK0(A))],[nnf_3]) ).
cnf(c_11,plain,
( basis_of(sK0(A),A)
| ~ a_vector_space(A) ),
inference(split_conjunct,[status(thm)],[sk_bg_remark_63_a]) ).
cnf(c_12,plain,
( a_vector_space(A)
| ~ a_vector_subspace_of(A,B) ),
inference(clausify,[status(thm)],[bg_2_4_a]) ).
fof(nnf_6,plain,
? [W,V] :
( ! [E,F] :
( ~ basis_of(E,W)
| ~ basis_of(union(E,F),V) )
& a_vector_space(V)
& a_vector_subspace_of(W,V) ),
inference(fof_nnf,[status(thm)],[bg_2_4_3_neg]) ).
fof(sk_bg_2_4_3_neg,plain,
( ! [E,F] :
( ~ basis_of(E,sK2)
| ~ basis_of(union(E,F),sK1) )
& a_vector_space(sK1)
& a_vector_subspace_of(sK2,sK1) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK2,sK1]),skolemize(W,sK2),skolemize(V,sK1)],[nnf_6]) ).
cnf(c_13,negated_conjecture,
a_vector_space(sK1),
inference(split_conjunct,[status(thm)],[sk_bg_2_4_3_neg]) ).
cnf(c_14,negated_conjecture,
a_vector_subspace_of(sK2,sK1),
inference(split_conjunct,[status(thm)],[sk_bg_2_4_3_neg]) ).
cnf(c_15,plain,
( ~ a_vector_subspace_of(A,B)
| basis_of(sK0(A),A) ),
inference(resolve,[status(thm)],[c_11,c_12]) ).
cnf(c_17,plain,
( ~ basis_of(C,A)
| lin_ind_subset(C,B)
| ~ lin_ind_subset(C,A)
| ~ a_vector_subspace_of(A,B) ),
inference(resolve,[status(thm)],[c_7,c_9]) ).
cnf(c_22,negated_conjecture,
( ~ basis_of(A,sK2)
| lin_ind_subset(A,sK1)
| ~ lin_ind_subset(A,sK2) ),
inference(resolve,[status(thm)],[c_17,c_14]) ).
cnf(c_24,negated_conjecture,
basis_of(sK0(sK2),sK2),
inference(resolve,[status(thm)],[c_14,c_15]) ).
cnf(c_25,negated_conjecture,
( ~ basis_of(A,sK2)
| ~ basis_of(union(A,B),sK1) ),
inference(split_conjunct,[status(thm)],[sk_bg_2_4_3_neg]) ).
fof(nnf_2,plain,
! [S,T,V] :
( ? [U] :
( basis_of(union(S,U),V)
& a_subset_of(U,T) )
| ~ basis_of(T,V)
| ~ lin_ind_subset(S,V) ),
inference(fof_nnf,[status(thm)],[bg_2_2_5]) ).
fof(sk_bg_2_2_5_sk,plain,
! [S,T,V] :
( ( basis_of(union(S,sK3(S,T,V)),V)
& a_subset_of(sK3(S,T,V),T) )
| ~ basis_of(T,V)
| ~ lin_ind_subset(S,V) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK3]),skolemize(U,sK3(S,T,V))],[nnf_2]) ).
fof(sk_bg_2_2_5,plain,
( ! [VAR_0,VAR_1,VAR_2] :
( basis_of(union(VAR_0,sK3(VAR_0,VAR_2,VAR_1)),VAR_1)
| ~ basis_of(VAR_2,VAR_1)
| ~ lin_ind_subset(VAR_0,VAR_1) )
& ! [VAR_0,VAR_1,VAR_2] :
( a_subset_of(sK3(VAR_0,VAR_2,VAR_1),VAR_2)
| ~ basis_of(VAR_2,VAR_1)
| ~ lin_ind_subset(VAR_0,VAR_1) ) ),
inference(cnf_transformation,[status(thm)],[sk_bg_2_2_5_sk]) ).
cnf(c_26,plain,
( basis_of(union(A,sK3(A,C,B)),B)
| ~ basis_of(C,B)
| ~ lin_ind_subset(A,B) ),
inference(split_conjunct,[status(thm)],[sk_bg_2_2_5]) ).
cnf(c_27,plain,
( lin_ind_subset(A,B)
| ~ basis_of(A,B) ),
inference(clausify,[status(thm)],[basis_of]) ).
cnf(c_28,negated_conjecture,
basis_of(sK0(sK1),sK1),
inference(resolve,[status(thm)],[c_11,c_13]) ).
cnf(c_29,plain,
~ basis_of(union(sK0(sK2),A),sK1),
inference(resolve,[status(thm)],[c_25,c_24]) ).
cnf(c_32,negated_conjecture,
lin_ind_subset(sK0(sK2),sK2),
inference(resolve,[status(thm)],[c_27,c_24]) ).
cnf(c_38,negated_conjecture,
( basis_of(union(A,sK3(A,sK0(sK1),sK1)),sK1)
| ~ lin_ind_subset(A,sK1) ),
inference(resolve,[status(thm)],[c_26,c_28]) ).
cnf(c_35,negated_conjecture,
~ lin_ind_subset(sK0(sK2),sK1),
inference(resolve,[status(thm)],[c_38,c_29]) ).
cnf(c_39,negated_conjecture,
( ~ basis_of(sK0(sK2),sK2)
| ~ lin_ind_subset(sK0(sK2),sK2) ),
inference(resolve,[status(thm)],[c_35,c_22]) ).
cnf(c_40,negated_conjecture,
~ basis_of(sK0(sK2),sK2),
inference(resolve,[status(thm)],[c_32,c_39]) ).
cnf(c_37,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_24,c_40]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : ALG211+1 : TPTP v9.3.1. Released v3.1.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.36 % Computer : n017.cluster.edu
% 0.09/0.36 % Model : x86_64 x86_64
% 0.09/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.36 % Memory : 8046.5625MB
% 0.09/0.36 % OS : Linux 6.8.0-71-generic
% 0.09/0.36 % CPULimit : 300
% 0.09/0.36 % WCLimit : 300
% 0.09/0.36 % DateTime : Sat Sep 26 21:21:04 UTC 2026
% 0.09/0.37 % CPUTime :
% 0.09/0.37 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.09/0.37 % Process 1662342 was started by sandbox2 on n017,
% 0.09/0.37 % Sat Sep 26 21:21:04 2026
% 0.09/0.37 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.09/0.37
% 0.09/0.37 % From the command line: assign(max_seconds, 300).
% 0.09/0.39
% 0.09/0.39 % SZS status Theorem for theBenchmark
% 0.09/0.39
% 0.09/0.39 % Proof 1 at 0.00 (+ 0.00) seconds.
% 0.09/0.39 % Length of proof is 24.
% 0.09/0.39 % Level of proof is 6.
% 0.09/0.39 % Maximum clause weight is 14.000.
% 0.09/0.39 % Given clauses 10.
% 0.09/0.39
% 0.09/0.39 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------