%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : HAL002+1 : TPTP v9.3.1. Released v2.6.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% Computer : n010.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 07:31:30 AM UTC 2026
% Result : Theorem 8.10s 3.47s
% Output : Refutation 8.59s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 17
% Syntax : Number of formulae : 36 ( 3 unt; 0 typ; 0 def)
% Number of atoms : 171 ( 44 equ; 0 cnn)
% Maximal formula atoms : 11 ( 4 avg)
% Number of connectives : 583 ( 2 ~; 0 |; 75 &; 446 @)
% ( 4 <=>; 56 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 11 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 15 ( 13 usr; 4 con; 0-4 aty)
% Number of variables : 156 ( 0 ^; 146 !; 10 ?; 156 :)
% Comments :
%------------------------------------------------------------------------------
thf(injection_decl,type,
injection: $i > $o ).
thf(x_decl,type,
x: $i ).
thf(injection_2_decl,type,
injection_2: $i > $o ).
thf(morphism_decl,type,
morphism: $i > $i > $i > $o ).
thf(element_decl,type,
element: $i > $i > $o ).
thf(apply_decl,type,
apply: $i > $i > $i ).
thf(commute_decl,type,
commute: $i > $i > $i > $i > $o ).
thf(surjection_decl,type,
surjection: $i > $o ).
thf(subtract_decl,type,
subtract: $i > $i > $i > $i ).
thf(zero_decl,type,
zero: $i > $i ).
thf(exact_decl,type,
exact: $i > $i > $o ).
thf(any1_decl,type,
any1: $i ).
thf(any2_decl,type,
any2: $i ).
thf(14,axiom,
any2 @ ( any1 @ ( x @ morphism ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',x_morphism) ).
thf(88,plain,
any2 @ ( any1 @ ( x @ morphism ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(9,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ! [F: $i] :
( ( ( ( F @ ( B @ apply ) )
= ( E @ zero ) )
& ( D @ ( F @ element ) ) )
<=> ? [G: $i] :
( ( ( G @ ( A @ apply ) )
= F )
& ( C @ ( G @ element ) ) ) )
& ( E @ ( D @ ( B @ morphism ) ) )
& ( D @ ( C @ ( A @ morphism ) ) ) )
=> ( B @ ( A @ exact ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',properties_for_exact) ).
thf(52,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ! [F: $i] :
( ( ? [G: $i] :
( ( ( G @ ( A @ apply ) )
= F )
& ( C @ ( G @ element ) ) )
=> ( ( ( F @ ( B @ apply ) )
= ( E @ zero ) )
& ( D @ ( F @ element ) ) ) )
& ( ( ( ( F @ ( B @ apply ) )
= ( E @ zero ) )
& ( D @ ( F @ element ) ) )
=> ? [G: $i] :
( ( ( G @ ( A @ apply ) )
= F )
& ( C @ ( G @ element ) ) ) ) )
& ( E @ ( D @ ( B @ morphism ) ) )
& ( D @ ( C @ ( A @ morphism ) ) ) )
=> ( B @ ( A @ exact ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(15,axiom,
! [A: $i,B: $i,C: $i] :
( ( ! [D: $i] :
( ( ( ( D @ ( A @ apply ) )
= ( C @ zero ) )
& ( B @ ( D @ element ) ) )
=> ( D
= ( B @ zero ) ) )
& ( C @ ( B @ ( A @ morphism ) ) ) )
=> ( A @ injection_2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',properties_for_injection_2) ).
thf(89,plain,
! [A: $i,B: $i,C: $i] :
( ( ! [D: $i] :
( ( ( ( D @ ( A @ apply ) )
= ( C @ zero ) )
& ( B @ ( D @ element ) ) )
=> ( D
= ( B @ zero ) ) )
& ( C @ ( B @ ( A @ morphism ) ) ) )
=> ( A @ injection_2 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(11,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ( E @ ( D @ ( B @ morphism ) ) )
& ( D @ ( C @ ( A @ morphism ) ) )
& ( B @ ( A @ exact ) ) )
=> ! [F: $i] :
( ( ( ( F @ ( B @ apply ) )
= ( E @ zero ) )
& ( D @ ( F @ element ) ) )
<=> ? [G: $i] :
( ( ( G @ ( A @ apply ) )
= F )
& ( C @ ( G @ element ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',exact_properties) ).
thf(70,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ( E @ ( D @ ( B @ morphism ) ) )
& ( D @ ( C @ ( A @ morphism ) ) )
& ( B @ ( A @ exact ) ) )
=> ! [F: $i] :
( ( ? [G: $i] :
( ( ( G @ ( A @ apply ) )
= F )
& ( C @ ( G @ element ) ) )
=> ( ( ( F @ ( B @ apply ) )
= ( E @ zero ) )
& ( D @ ( F @ element ) ) ) )
& ( ( ( ( F @ ( B @ apply ) )
= ( E @ zero ) )
& ( D @ ( F @ element ) ) )
=> ? [G: $i] :
( ( ( G @ ( A @ apply ) )
= F )
& ( C @ ( G @ element ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(18,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ ( A @ morphism ) ) )
& ( A @ injection ) )
=> ! [D: $i,E: $i] :
( ( ( ( D @ ( A @ apply ) )
= ( E @ ( A @ apply ) ) )
& ( B @ ( E @ element ) )
& ( B @ ( D @ element ) ) )
=> ( D = E ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',injection_properties) ).
thf(109,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ ( A @ morphism ) ) )
& ( A @ injection ) )
=> ! [D: $i,E: $i] :
( ( ( ( D @ ( A @ apply ) )
= ( E @ ( A @ apply ) ) )
& ( B @ ( E @ element ) )
& ( B @ ( D @ element ) ) )
=> ( D = E ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(7,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ morphism ) ) )
=> ( ( ( B @ zero @ ( A @ apply ) )
= ( C @ zero ) )
& ! [D: $i] :
( ( B @ ( D @ element ) )
=> ( C @ ( D @ ( A @ apply ) @ element ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',morphism) ).
thf(44,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ morphism ) ) )
=> ( ( ( B @ zero @ ( A @ apply ) )
= ( C @ zero ) )
& ! [D: $i] :
( ( B @ ( D @ element ) )
=> ( C @ ( D @ ( A @ apply ) @ element ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(4,axiom,
! [A: $i,B: $i,C: $i] :
( ( ! [D: $i] :
( ( C @ ( D @ element ) )
=> ? [E: $i] :
( ( ( E @ ( A @ apply ) )
= D )
& ( B @ ( E @ element ) ) ) )
& ( C @ ( B @ ( A @ morphism ) ) ) )
=> ( A @ surjection ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',properties_for_surjection) ).
thf(31,plain,
! [A: $i,B: $i,C: $i] :
( ( ! [D: $i] :
( ( C @ ( D @ element ) )
=> ? [E: $i] :
( ( ( E @ ( A @ apply ) )
= D )
& ( B @ ( E @ element ) ) ) )
& ( C @ ( B @ ( A @ morphism ) ) ) )
=> ( A @ surjection ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(6,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( ( H @ ( G @ ( D @ morphism ) ) )
& ( G @ ( E @ ( C @ morphism ) ) )
& ( H @ ( F @ ( B @ morphism ) ) )
& ( F @ ( E @ ( A @ morphism ) ) )
& ( D @ ( C @ ( B @ ( A @ commute ) ) ) ) )
=> ! [I: $i] :
( ( E @ ( I @ element ) )
=> ( ( I @ ( A @ apply ) @ ( B @ apply ) )
= ( I @ ( C @ apply ) @ ( D @ apply ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commute_properties) ).
thf(40,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( ( H @ ( G @ ( D @ morphism ) ) )
& ( G @ ( E @ ( C @ morphism ) ) )
& ( H @ ( F @ ( B @ morphism ) ) )
& ( F @ ( E @ ( A @ morphism ) ) )
& ( D @ ( C @ ( B @ ( A @ commute ) ) ) ) )
=> ! [I: $i] :
( ( E @ ( I @ element ) )
=> ( ( I @ ( A @ apply ) @ ( B @ apply ) )
= ( I @ ( C @ apply ) @ ( D @ apply ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(1,conjecture,
( ( x @ injection )
<=> ( x @ injection_2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',my) ).
thf(2,negated_conjecture,
~ ( ( x @ injection )
<=> ( x @ injection_2 ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(19,plain,
~ ( ( ( x @ injection_2 )
=> ( x @ injection ) )
& ( ( x @ injection )
=> ( x @ injection_2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(5,axiom,
! [A: $i,B: $i] :
( ( A @ ( B @ element ) )
=> ( ( B @ ( B @ ( A @ subtract ) ) )
= ( A @ zero ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subtract_to_0) ).
thf(37,plain,
! [A: $i,B: $i] :
( ( A @ ( B @ element ) )
=> ( ( B @ ( B @ ( A @ subtract ) ) )
= ( A @ zero ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(3,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( ! [I: $i] :
( ( E @ ( I @ element ) )
=> ( ( I @ ( A @ apply ) @ ( B @ apply ) )
= ( I @ ( C @ apply ) @ ( D @ apply ) ) ) )
& ( H @ ( G @ ( D @ morphism ) ) )
& ( G @ ( E @ ( C @ morphism ) ) )
& ( H @ ( F @ ( B @ morphism ) ) )
& ( F @ ( E @ ( A @ morphism ) ) ) )
=> ( D @ ( C @ ( B @ ( A @ commute ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',properties_for_commute) ).
thf(25,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( ! [I: $i] :
( ( E @ ( I @ element ) )
=> ( ( I @ ( A @ apply ) @ ( B @ apply ) )
= ( I @ ( C @ apply ) @ ( D @ apply ) ) ) )
& ( H @ ( G @ ( D @ morphism ) ) )
& ( G @ ( E @ ( C @ morphism ) ) )
& ( H @ ( F @ ( B @ morphism ) ) )
& ( F @ ( E @ ( A @ morphism ) ) ) )
=> ( D @ ( C @ ( B @ ( A @ commute ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(10,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ ( A @ morphism ) ) )
& ( A @ surjection ) )
=> ! [D: $i] :
( ( C @ ( D @ element ) )
=> ? [E: $i] :
( ( ( E @ ( A @ apply ) )
= D )
& ( B @ ( E @ element ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',surjection_properties) ).
thf(65,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ ( A @ morphism ) ) )
& ( A @ surjection ) )
=> ! [D: $i] :
( ( C @ ( D @ element ) )
=> ? [E: $i] :
( ( ( E @ ( A @ apply ) )
= D )
& ( B @ ( E @ element ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(16,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ ( A @ morphism ) ) )
& ( A @ injection_2 ) )
=> ! [D: $i] :
( ( ( ( D @ ( A @ apply ) )
= ( C @ zero ) )
& ( B @ ( D @ element ) ) )
=> ( D
= ( B @ zero ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',injection_properties_2) ).
thf(97,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ ( A @ morphism ) ) )
& ( A @ injection_2 ) )
=> ! [D: $i] :
( ( ( ( D @ ( A @ apply ) )
= ( C @ zero ) )
& ( B @ ( D @ element ) ) )
=> ( D
= ( B @ zero ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(8,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ morphism ) ) )
=> ! [D: $i,E: $i] :
( ( ( B @ ( E @ element ) )
& ( B @ ( D @ element ) ) )
=> ( ( E @ ( D @ ( B @ subtract ) ) @ ( A @ apply ) )
= ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ( C @ subtract ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subtract_distribution) ).
thf(49,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ morphism ) ) )
=> ! [D: $i,E: $i] :
( ( ( B @ ( E @ element ) )
& ( B @ ( D @ element ) ) )
=> ( ( E @ ( D @ ( B @ subtract ) ) @ ( A @ apply ) )
= ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ( C @ subtract ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(13,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( A @ ( C @ element ) )
& ( A @ ( B @ element ) ) )
=> ( A @ ( C @ ( B @ ( A @ subtract ) ) @ element ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subtract_in_domain) ).
thf(86,plain,
! [A: $i,B: $i,C: $i] :
( ( ( A @ ( C @ element ) )
& ( A @ ( B @ element ) ) )
=> ( A @ ( C @ ( B @ ( A @ subtract ) ) @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(12,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( A @ ( C @ element ) )
& ( A @ ( B @ element ) ) )
=> ( ( C @ ( B @ ( A @ subtract ) ) @ ( B @ ( A @ subtract ) ) )
= C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subtract_cancellation) ).
thf(83,plain,
! [A: $i,B: $i,C: $i] :
( ( ( A @ ( C @ element ) )
& ( A @ ( B @ element ) ) )
=> ( ( C @ ( B @ ( A @ subtract ) ) @ ( B @ ( A @ subtract ) ) )
= C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(17,axiom,
! [A: $i,B: $i,C: $i] :
( ( ! [D: $i,E: $i] :
( ( ( ( D @ ( A @ apply ) )
= ( E @ ( A @ apply ) ) )
& ( B @ ( E @ element ) )
& ( B @ ( D @ element ) ) )
=> ( D = E ) )
& ( C @ ( B @ ( A @ morphism ) ) ) )
=> ( A @ injection ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',properties_for_injection) ).
thf(100,plain,
! [A: $i,B: $i,C: $i] :
( ( ! [D: $i,E: $i] :
( ( ( ( D @ ( A @ apply ) )
= ( E @ ( A @ apply ) ) )
& ( B @ ( E @ element ) )
& ( B @ ( D @ element ) ) )
=> ( D = E ) )
& ( C @ ( B @ ( A @ morphism ) ) ) )
=> ( A @ injection ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(381,plain,
$false,
inference(e,[status(thm)],[88,52,89,70,109,44,31,40,19,37,25,65,97,49,86,83,100]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : HAL002+1 : TPTP v9.3.1. Released v2.6.0.
% 0.00/0.07 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.12/0.40 % Computer : n010.cluster.edu
% 0.12/0.40 % Model : x86_64 x86_64
% 0.12/0.40 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.40 % Memory : 8046.5625MB
% 0.12/0.40 % OS : Linux 6.8.0-71-generic
% 0.12/0.40 % CPULimit : 300
% 0.12/0.40 % WCLimit : 300
% 0.12/0.40 % DateTime : Fri Sep 25 15:54:14 UTC 2026
% 0.12/0.40 % CPUTime :
% 0.12/0.40 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.87/0.98 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 1.46/1.21 % [INFO] Parsing done (222ms).
% 1.46/1.22 % [INFO] Running in sequential loop mode.
% 2.16/1.63 % [INFO] eprover registered as external prover.
% 2.16/1.64 % [INFO] Scanning for conjecture ...
% 2.37/1.74 % [INFO] Found a conjecture (or negated_conjecture) and 16 axioms. Running axiom selection ...
% 2.53/1.80 % [INFO] Axiom selection finished. Selected 16 axioms (removed 0 axioms).
% 2.53/1.89 % [INFO] Problem is first-order (TPTP FOF).
% 2.78/1.90 % [INFO] Type checking passed.
% 2.78/1.90 % [CONFIG] Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>. Searching for refutation ...
% 8.10/3.47 % External prover 'e' found a proof!
% 8.10/3.47 % [INFO] Killing All external provers ...
% 8.10/3.47 % Time passed: 2926ms (effective reasoning time: 2237ms)
% 8.10/3.47 % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 8.10/3.47 % Axioms used in derivation (16): surjection_properties, subtract_cancellation, x_morphism, properties_for_injection, properties_for_commute, exact_properties, properties_for_surjection, morphism, properties_for_injection_2, subtract_in_domain, subtract_to_0, injection_properties, properties_for_exact, subtract_distribution, injection_properties_2, commute_properties
% 8.10/3.47 % No. of inferences in proof: 36
% 8.10/3.47 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 2926 ms resp. 2237 ms w/o parsing
% 8.59/3.52 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 8.59/3.52 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------