%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET813+4 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n002.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 05:26:24 AM UTC 2025
% Result : Theorem 0.12s 0.39s
% Output : Proof 0.12s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.13 % Problem : SET813+4 : TPTP v9.0.0. Released v3.2.0.
% 0.11/0.13 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n002.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 05:45:01 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.12/0.39 % SZS status Theorem
% 0.12/0.39 % SZS output start Proof
% 0.12/0.39 tff(member_type, type, (
% 0.12/0.39 member: ( $i * $i ) > $o)).
% 0.12/0.39 tff(singleton_type, type, (
% 0.12/0.39 singleton: $i > $i)).
% 0.12/0.39 tff(tptp_fun_A_13_type, type, (
% 0.12/0.39 tptp_fun_A_13: $i)).
% 0.12/0.39 tff(union_type, type, (
% 0.12/0.39 union: ( $i * $i ) > $i)).
% 0.12/0.39 tff(suc_type, type, (
% 0.12/0.39 suc: $i > $i)).
% 0.12/0.39 tff(on_type, type, (
% 0.12/0.39 on: $i)).
% 0.12/0.39 tff(1,plain,
% 0.12/0.39 (^[X: $i, A: $i, B: $i] : refl((member(X, union(A, B)) <=> (member(X, B) | member(X, A))) <=> (member(X, union(A, B)) <=> (member(X, B) | member(X, A))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(2,plain,
% 0.12/0.39 (![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A))) <=> ![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[1])).
% 0.12/0.39 tff(3,plain,
% 0.12/0.39 (![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A))) <=> ![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(4,plain,
% 0.12/0.39 (^[X: $i, A: $i, B: $i] : rewrite((member(X, union(A, B)) <=> (member(X, A) | member(X, B))) <=> (member(X, union(A, B)) <=> (member(X, B) | member(X, A))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(5,plain,
% 0.12/0.39 (![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, A) | member(X, B))) <=> ![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[4])).
% 0.12/0.39 tff(6,axiom,(![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, A) | member(X, B)))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+0.ax','union')).
% 0.12/0.39 tff(7,plain,
% 0.12/0.39 (![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[6, 5])).
% 0.12/0.39 tff(8,plain,
% 0.12/0.39 (![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.12/0.39 tff(9,plain,(
% 0.12/0.39 ![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))),
% 0.12/0.39 inference(skolemize,[status(sab)],[8])).
% 0.12/0.39 tff(10,plain,
% 0.12/0.39 (![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[9, 2])).
% 0.12/0.39 tff(11,plain,
% 0.12/0.39 ((~![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))) | (member(A!13, union(A!13, singleton(A!13))) <=> (member(A!13, singleton(A!13)) | member(A!13, A!13)))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(12,plain,
% 0.12/0.39 (member(A!13, union(A!13, singleton(A!13))) <=> (member(A!13, singleton(A!13)) | member(A!13, A!13))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[11, 10])).
% 0.12/0.39 tff(13,plain,
% 0.12/0.39 (^[A: $i, X: $i] : refl((member(X, suc(A)) <=> member(X, union(A, singleton(A)))) <=> (member(X, suc(A)) <=> member(X, union(A, singleton(A)))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(14,plain,
% 0.12/0.39 (![A: $i, X: $i] : (member(X, suc(A)) <=> member(X, union(A, singleton(A)))) <=> ![A: $i, X: $i] : (member(X, suc(A)) <=> member(X, union(A, singleton(A))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[13])).
% 0.12/0.39 tff(15,plain,
% 0.12/0.39 (![A: $i, X: $i] : (member(X, suc(A)) <=> member(X, union(A, singleton(A)))) <=> ![A: $i, X: $i] : (member(X, suc(A)) <=> member(X, union(A, singleton(A))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(16,axiom,(![A: $i, X: $i] : (member(X, suc(A)) <=> member(X, union(A, singleton(A))))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+4.ax','successor')).
% 0.12/0.39 tff(17,plain,
% 0.12/0.39 (![A: $i, X: $i] : (member(X, suc(A)) <=> member(X, union(A, singleton(A))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.12/0.39 tff(18,plain,(
% 0.12/0.39 ![A: $i, X: $i] : (member(X, suc(A)) <=> member(X, union(A, singleton(A))))),
% 0.12/0.39 inference(skolemize,[status(sab)],[17])).
% 0.12/0.39 tff(19,plain,
% 0.12/0.39 (![A: $i, X: $i] : (member(X, suc(A)) <=> member(X, union(A, singleton(A))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[18, 14])).
% 0.12/0.39 tff(20,plain,
% 0.12/0.39 ((~![A: $i, X: $i] : (member(X, suc(A)) <=> member(X, union(A, singleton(A))))) | (member(A!13, suc(A!13)) <=> member(A!13, union(A!13, singleton(A!13))))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(21,plain,
% 0.12/0.39 (member(A!13, suc(A!13)) <=> member(A!13, union(A!13, singleton(A!13)))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[20, 19])).
% 0.12/0.39 tff(22,plain,
% 0.12/0.39 ((~![A: $i] : ((~member(A, on)) | member(A, suc(A)))) <=> (~![A: $i] : ((~member(A, on)) | member(A, suc(A))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(23,plain,
% 0.12/0.39 ((~![A: $i] : (member(A, on) => member(A, suc(A)))) <=> (~![A: $i] : ((~member(A, on)) | member(A, suc(A))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(24,axiom,(~![A: $i] : (member(A, on) => member(A, suc(A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','thV12')).
% 0.12/0.39 tff(25,plain,
% 0.12/0.39 (~![A: $i] : ((~member(A, on)) | member(A, suc(A)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[24, 23])).
% 0.12/0.39 tff(26,plain,
% 0.12/0.39 (~![A: $i] : ((~member(A, on)) | member(A, suc(A)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[25, 22])).
% 0.12/0.39 tff(27,plain,
% 0.12/0.39 (~![A: $i] : ((~member(A, on)) | member(A, suc(A)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[26, 22])).
% 0.12/0.39 tff(28,plain,
% 0.12/0.39 (~![A: $i] : ((~member(A, on)) | member(A, suc(A)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[27, 22])).
% 0.12/0.39 tff(29,plain,(
% 0.12/0.39 ~((~member(A!13, on)) | member(A!13, suc(A!13)))),
% 0.12/0.39 inference(skolemize,[status(sab)],[28])).
% 0.12/0.39 tff(30,plain,
% 0.12/0.39 (~member(A!13, suc(A!13))),
% 0.12/0.39 inference(or_elim,[status(thm)],[29])).
% 0.12/0.39 tff(31,plain,
% 0.12/0.39 ((~(member(A!13, suc(A!13)) <=> member(A!13, union(A!13, singleton(A!13))))) | member(A!13, suc(A!13)) | (~member(A!13, union(A!13, singleton(A!13))))),
% 0.12/0.39 inference(tautology,[status(thm)],[])).
% 0.12/0.39 tff(32,plain,
% 0.12/0.39 ((~(member(A!13, suc(A!13)) <=> member(A!13, union(A!13, singleton(A!13))))) | (~member(A!13, union(A!13, singleton(A!13))))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[31, 30])).
% 0.12/0.39 tff(33,plain,
% 0.12/0.39 (~member(A!13, union(A!13, singleton(A!13)))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[32, 21])).
% 0.12/0.39 tff(34,plain,
% 0.12/0.39 ((~(member(A!13, union(A!13, singleton(A!13))) <=> (member(A!13, singleton(A!13)) | member(A!13, A!13)))) | member(A!13, union(A!13, singleton(A!13))) | (~(member(A!13, singleton(A!13)) | member(A!13, A!13)))),
% 0.12/0.39 inference(tautology,[status(thm)],[])).
% 0.12/0.39 tff(35,plain,
% 0.12/0.39 ((~(member(A!13, union(A!13, singleton(A!13))) <=> (member(A!13, singleton(A!13)) | member(A!13, A!13)))) | (~(member(A!13, singleton(A!13)) | member(A!13, A!13)))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[34, 33])).
% 0.12/0.39 tff(36,plain,
% 0.12/0.39 (~(member(A!13, singleton(A!13)) | member(A!13, A!13))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[35, 12])).
% 0.12/0.39 tff(37,plain,
% 0.12/0.39 ((member(A!13, singleton(A!13)) | member(A!13, A!13)) | (~member(A!13, singleton(A!13)))),
% 0.12/0.39 inference(tautology,[status(thm)],[])).
% 0.12/0.39 tff(38,plain,
% 0.12/0.39 (~member(A!13, singleton(A!13))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[37, 36])).
% 0.12/0.39 tff(39,plain,
% 0.12/0.39 (^[X: $i, A: $i] : refl((member(X, singleton(A)) <=> (X = A)) <=> (member(X, singleton(A)) <=> (X = A)))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(40,plain,
% 0.12/0.39 (![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A)) <=> ![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[39])).
% 0.12/0.39 tff(41,plain,
% 0.12/0.39 (![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A)) <=> ![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(42,axiom,(![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+0.ax','singleton')).
% 0.12/0.39 tff(43,plain,
% 0.12/0.39 (![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.12/0.39 tff(44,plain,(
% 0.12/0.39 ![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))),
% 0.12/0.39 inference(skolemize,[status(sab)],[43])).
% 0.12/0.39 tff(45,plain,
% 0.12/0.39 (![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[44, 40])).
% 0.12/0.39 tff(46,plain,
% 0.12/0.39 (((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | member(A!13, singleton(A!13))) <=> ((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | member(A!13, singleton(A!13)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(47,plain,
% 0.12/0.39 ((member(A!13, singleton(A!13)) <=> $true) <=> member(A!13, singleton(A!13))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(48,plain,
% 0.12/0.39 ((A!13 = A!13) <=> $true),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(49,plain,
% 0.12/0.39 ((member(A!13, singleton(A!13)) <=> (A!13 = A!13)) <=> (member(A!13, singleton(A!13)) <=> $true)),
% 0.12/0.39 inference(monotonicity,[status(thm)],[48])).
% 0.12/0.39 tff(50,plain,
% 0.12/0.39 ((member(A!13, singleton(A!13)) <=> (A!13 = A!13)) <=> member(A!13, singleton(A!13))),
% 0.12/0.39 inference(transitivity,[status(thm)],[49, 47])).
% 0.12/0.39 tff(51,plain,
% 0.12/0.39 (((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | (member(A!13, singleton(A!13)) <=> (A!13 = A!13))) <=> ((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | member(A!13, singleton(A!13)))),
% 0.12/0.39 inference(monotonicity,[status(thm)],[50])).
% 0.12/0.39 tff(52,plain,
% 0.12/0.39 (((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | (member(A!13, singleton(A!13)) <=> (A!13 = A!13))) <=> ((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | member(A!13, singleton(A!13)))),
% 0.12/0.39 inference(transitivity,[status(thm)],[51, 46])).
% 0.12/0.39 tff(53,plain,
% 0.12/0.39 ((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | (member(A!13, singleton(A!13)) <=> (A!13 = A!13))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(54,plain,
% 0.12/0.39 ((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | member(A!13, singleton(A!13))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.12/0.39 tff(55,plain,
% 0.12/0.39 ($false),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[54, 45, 38])).
% 0.12/0.39 % SZS output end Proof
% 0.12/0.39 % E exiting
%------------------------------------------------------------------------------