%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : LCL414+1 : TPTP v9.3.1. Released v2.5.0.
% Transfm : NO INFORMATION
% Format : NO INFORMATION
% Command : run_E %s %d THM
% Computer : n027.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 : Mon Sep 7 02:59:58 PM UTC 2026
% Result : Theorem 0.13s 0.41s
% Output : Proof 0.13s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : LCL414+1 : TPTP v9.3.1. Released v2.5.0.
% 0.00/0.03 % Command : run_E %s %d THM
% 0.09/0.35 % Computer : n027.cluster.edu
% 0.09/0.35 % Model : x86_64 x86_64
% 0.09/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.35 % Memory : 8046.5625MB
% 0.09/0.35 % OS : Linux 6.8.0-71-generic
% 0.09/0.35 % CPULimit : 300
% 0.09/0.35 % WCLimit : 300
% 0.09/0.35 % DateTime : Sat Sep 5 13:03:30 UTC 2026
% 0.09/0.35 % CPUTime :
% 0.13/0.41 % SZS status Theorem
% 0.13/0.41 % SZS output start Proof
% 0.13/0.41 tff(a_truth_type, type, (
% 0.13/0.41 a_truth: $i > $o)).
% 0.13/0.41 tff(implies_type, type, (
% 0.13/0.41 implies: ( $i * $i ) > $i)).
% 0.13/0.41 tff(tptp_fun_A_0_type, type, (
% 0.13/0.41 tptp_fun_A_0: $i)).
% 0.13/0.41 tff(not_type, type, (
% 0.13/0.41 not: $i > $i)).
% 0.13/0.41 tff(1,plain,
% 0.13/0.41 (^[P: $i, Q: $i] : refl(a_truth(implies(P, implies(Q, P))) <=> a_truth(implies(P, implies(Q, P))))),
% 0.13/0.41 inference(bind,[status(th)],[])).
% 0.13/0.41 tff(2,plain,
% 0.13/0.41 (![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P))) <=> ![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P)))),
% 0.13/0.41 inference(quant_intro,[status(thm)],[1])).
% 0.13/0.41 tff(3,plain,
% 0.13/0.41 (![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P))) <=> ![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P)))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(4,plain,
% 0.13/0.41 ((~(~(![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q)))) & ![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P))) & ![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R)))) & ![P: $i, Q: $i] : a_truth(implies(implies(not(P), not(Q)), implies(Q, P))) & ?[A: $i] : (~a_truth(implies(A, A)))))) <=> (![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q)))) & ![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P))) & ![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R)))) & ![P: $i, Q: $i] : a_truth(implies(implies(not(P), not(Q)), implies(Q, P))) & ?[A: $i] : (~a_truth(implies(A, A))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(5,plain,
% 0.13/0.41 ((~((((![P: $i, Q: $i] : (((~a_truth(implies(P, Q))) | (~a_truth(P))) | a_truth(Q)) & ![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P)))) & ![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R))))) & ![P: $i, Q: $i] : a_truth(implies(implies(not(P), not(Q)), implies(Q, P)))) & ?[A: $i] : (~a_truth(implies(A, A))))) <=> (~(![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q)))) & ![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P))) & ![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R)))) & ![P: $i, Q: $i] : a_truth(implies(implies(not(P), not(Q)), implies(Q, P))) & ?[A: $i] : (~a_truth(implies(A, A)))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(6,plain,
% 0.13/0.41 ((~(~((((![P: $i, Q: $i] : (((~a_truth(implies(P, Q))) | (~a_truth(P))) | a_truth(Q)) & ![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P)))) & ![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R))))) & ![P: $i, Q: $i] : a_truth(implies(implies(not(P), not(Q)), implies(Q, P)))) & ?[A: $i] : (~a_truth(implies(A, A)))))) <=> (~(~(![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q)))) & ![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P))) & ![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R)))) & ![P: $i, Q: $i] : a_truth(implies(implies(not(P), not(Q)), implies(Q, P))) & ?[A: $i] : (~a_truth(implies(A, A))))))),
% 0.13/0.41 inference(monotonicity,[status(thm)],[5])).
% 0.13/0.41 tff(7,plain,
% 0.13/0.41 ((~(~((((![P: $i, Q: $i] : (((~a_truth(implies(P, Q))) | (~a_truth(P))) | a_truth(Q)) & ![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P)))) & ![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R))))) & ![P: $i, Q: $i] : a_truth(implies(implies(not(P), not(Q)), implies(Q, P)))) & ?[A: $i] : (~a_truth(implies(A, A)))))) <=> (![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q)))) & ![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P))) & ![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R)))) & ![P: $i, Q: $i] : a_truth(implies(implies(not(P), not(Q)), implies(Q, P))) & ?[A: $i] : (~a_truth(implies(A, A))))),
% 0.13/0.41 inference(transitivity,[status(thm)],[6, 4])).
% 0.13/0.41 tff(8,axiom,(~(~((((![P: $i, Q: $i] : (((~a_truth(implies(P, Q))) | (~a_truth(P))) | a_truth(Q)) & ![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P)))) & ![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R))))) & ![P: $i, Q: $i] : a_truth(implies(implies(not(P), not(Q)), implies(Q, P)))) & ?[A: $i] : (~a_truth(implies(A, A)))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','thm147')).
% 0.13/0.41 tff(9,plain,
% 0.13/0.41 (![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q)))) & ![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P))) & ![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R)))) & ![P: $i, Q: $i] : a_truth(implies(implies(not(P), not(Q)), implies(Q, P))) & ?[A: $i] : (~a_truth(implies(A, A)))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[8, 7])).
% 0.13/0.41 tff(10,plain,
% 0.13/0.41 (![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P)))),
% 0.13/0.41 inference(and_elim,[status(thm)],[9])).
% 0.13/0.41 tff(11,plain,
% 0.13/0.41 (![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P)))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[10, 3])).
% 0.13/0.41 tff(12,plain,(
% 0.13/0.41 ![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P)))),
% 0.13/0.41 inference(skolemize,[status(sab)],[11])).
% 0.13/0.41 tff(13,plain,
% 0.13/0.41 (![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P)))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[12, 2])).
% 0.13/0.41 tff(14,plain,
% 0.13/0.41 ((~![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P)))) | a_truth(implies(A!0, implies(A!0, A!0)))),
% 0.13/0.41 inference(quant_inst,[status(thm)],[])).
% 0.13/0.41 tff(15,plain,
% 0.13/0.41 (a_truth(implies(A!0, implies(A!0, A!0)))),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[14, 13])).
% 0.13/0.41 tff(16,plain,
% 0.13/0.41 (?[A: $i] : (~a_truth(implies(A, A))) <=> ?[A: $i] : (~a_truth(implies(A, A)))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(17,plain,
% 0.13/0.41 (?[A: $i] : (~a_truth(implies(A, A)))),
% 0.13/0.41 inference(and_elim,[status(thm)],[9])).
% 0.13/0.41 tff(18,plain,
% 0.13/0.41 (?[A: $i] : (~a_truth(implies(A, A)))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[17, 16])).
% 0.13/0.41 tff(19,plain,(
% 0.13/0.41 ~a_truth(implies(A!0, A!0))),
% 0.13/0.41 inference(skolemize,[status(sab)],[18])).
% 0.13/0.41 tff(20,plain,
% 0.13/0.41 (^[P: $i, Q: $i] : refl((a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q)))) <=> (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q)))))),
% 0.13/0.41 inference(bind,[status(th)],[])).
% 0.13/0.41 tff(21,plain,
% 0.13/0.41 (![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q)))) <=> ![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q))))),
% 0.13/0.41 inference(quant_intro,[status(thm)],[20])).
% 0.13/0.41 tff(22,plain,
% 0.13/0.41 (![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q)))) <=> ![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(23,plain,
% 0.13/0.41 (![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q))))),
% 0.13/0.41 inference(and_elim,[status(thm)],[9])).
% 0.13/0.41 tff(24,plain,
% 0.13/0.41 (![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.13/0.41 tff(25,plain,(
% 0.13/0.41 ![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q))))),
% 0.13/0.41 inference(skolemize,[status(sab)],[24])).
% 0.13/0.41 tff(26,plain,
% 0.13/0.41 (![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[25, 21])).
% 0.13/0.41 tff(27,plain,
% 0.13/0.41 (((~![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q))))) | (a_truth(implies(A!0, A!0)) | (~a_truth(implies(A!0, implies(A!0, A!0)))) | (~a_truth(implies(implies(A!0, implies(A!0, A!0)), implies(A!0, A!0)))))) <=> ((~![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q))))) | a_truth(implies(A!0, A!0)) | (~a_truth(implies(A!0, implies(A!0, A!0)))) | (~a_truth(implies(implies(A!0, implies(A!0, A!0)), implies(A!0, A!0)))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(28,plain,
% 0.13/0.41 ((~![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q))))) | (a_truth(implies(A!0, A!0)) | (~a_truth(implies(A!0, implies(A!0, A!0)))) | (~a_truth(implies(implies(A!0, implies(A!0, A!0)), implies(A!0, A!0)))))),
% 0.13/0.41 inference(quant_inst,[status(thm)],[])).
% 0.13/0.41 tff(29,plain,
% 0.13/0.41 ((~![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q))))) | a_truth(implies(A!0, A!0)) | (~a_truth(implies(A!0, implies(A!0, A!0)))) | (~a_truth(implies(implies(A!0, implies(A!0, A!0)), implies(A!0, A!0))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[28, 27])).
% 0.13/0.41 tff(30,plain,
% 0.13/0.41 (~a_truth(implies(implies(A!0, implies(A!0, A!0)), implies(A!0, A!0)))),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[29, 26, 19, 15])).
% 0.13/0.41 tff(31,plain,
% 0.13/0.41 ((~![P: $i, Q: $i] : a_truth(implies(P, implies(Q, P)))) | a_truth(implies(A!0, implies(implies(A!0, A!0), A!0)))),
% 0.13/0.41 inference(quant_inst,[status(thm)],[])).
% 0.13/0.41 tff(32,plain,
% 0.13/0.41 (a_truth(implies(A!0, implies(implies(A!0, A!0), A!0)))),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[31, 13])).
% 0.13/0.41 tff(33,plain,
% 0.13/0.41 (^[P: $i, Q: $i, R: $i] : refl(a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R)))) <=> a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R)))))),
% 0.13/0.41 inference(bind,[status(th)],[])).
% 0.13/0.41 tff(34,plain,
% 0.13/0.41 (![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R)))) <=> ![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R))))),
% 0.13/0.41 inference(quant_intro,[status(thm)],[33])).
% 0.13/0.41 tff(35,plain,
% 0.13/0.41 (![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R)))) <=> ![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(36,plain,
% 0.13/0.41 (![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R))))),
% 0.13/0.41 inference(and_elim,[status(thm)],[9])).
% 0.13/0.41 tff(37,plain,
% 0.13/0.41 (![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.13/0.41 tff(38,plain,(
% 0.13/0.41 ![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R))))),
% 0.13/0.41 inference(skolemize,[status(sab)],[37])).
% 0.13/0.41 tff(39,plain,
% 0.13/0.41 (![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[38, 34])).
% 0.13/0.41 tff(40,plain,
% 0.13/0.41 ((~![P: $i, Q: $i, R: $i] : a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R))))) | a_truth(implies(implies(A!0, implies(implies(A!0, A!0), A!0)), implies(implies(A!0, implies(A!0, A!0)), implies(A!0, A!0))))),
% 0.13/0.41 inference(quant_inst,[status(thm)],[])).
% 0.13/0.41 tff(41,plain,
% 0.13/0.41 (a_truth(implies(implies(A!0, implies(implies(A!0, A!0), A!0)), implies(implies(A!0, implies(A!0, A!0)), implies(A!0, A!0))))),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[40, 39])).
% 0.13/0.41 tff(42,plain,
% 0.13/0.41 (((~![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q))))) | (a_truth(implies(implies(A!0, implies(A!0, A!0)), implies(A!0, A!0))) | (~a_truth(implies(A!0, implies(implies(A!0, A!0), A!0)))) | (~a_truth(implies(implies(A!0, implies(implies(A!0, A!0), A!0)), implies(implies(A!0, implies(A!0, A!0)), implies(A!0, A!0))))))) <=> ((~![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q))))) | a_truth(implies(implies(A!0, implies(A!0, A!0)), implies(A!0, A!0))) | (~a_truth(implies(A!0, implies(implies(A!0, A!0), A!0)))) | (~a_truth(implies(implies(A!0, implies(implies(A!0, A!0), A!0)), implies(implies(A!0, implies(A!0, A!0)), implies(A!0, A!0))))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(43,plain,
% 0.13/0.41 ((~![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q))))) | (a_truth(implies(implies(A!0, implies(A!0, A!0)), implies(A!0, A!0))) | (~a_truth(implies(A!0, implies(implies(A!0, A!0), A!0)))) | (~a_truth(implies(implies(A!0, implies(implies(A!0, A!0), A!0)), implies(implies(A!0, implies(A!0, A!0)), implies(A!0, A!0))))))),
% 0.13/0.41 inference(quant_inst,[status(thm)],[])).
% 0.13/0.41 tff(44,plain,
% 0.13/0.41 ((~![P: $i, Q: $i] : (a_truth(Q) | (~a_truth(P)) | (~a_truth(implies(P, Q))))) | a_truth(implies(implies(A!0, implies(A!0, A!0)), implies(A!0, A!0))) | (~a_truth(implies(A!0, implies(implies(A!0, A!0), A!0)))) | (~a_truth(implies(implies(A!0, implies(implies(A!0, A!0), A!0)), implies(implies(A!0, implies(A!0, A!0)), implies(A!0, A!0)))))),
% 0.13/0.42 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.13/0.42 tff(45,plain,
% 0.13/0.42 (a_truth(implies(implies(A!0, implies(A!0, A!0)), implies(A!0, A!0))) | (~a_truth(implies(A!0, implies(implies(A!0, A!0), A!0))))),
% 0.13/0.42 inference(unit_resolution,[status(thm)],[44, 26, 41])).
% 0.13/0.42 tff(46,plain,
% 0.13/0.42 ($false),
% 0.13/0.42 inference(unit_resolution,[status(thm)],[45, 32, 30])).
% 0.13/0.42 % SZS output end Proof
% 0.13/0.42 % E exiting
%------------------------------------------------------------------------------