%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : KLE006+1 : TPTP v9.0.0. Released v4.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n012.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 05:12:50 AM UTC 2025
% Result : Theorem 0.21s 0.38s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.13 % Problem : KLE006+1 : TPTP v9.0.0. Released v4.0.0.
% 0.03/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n012.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Jun 19 18:49:41 EDT 2025
% 0.13/0.35 % CPUTime :
% 0.21/0.38 % SZS status Theorem
% 0.21/0.38 % SZS output start Proof
% 0.21/0.38 tff(complement_type, type, (
% 0.21/0.38 complement: ( $i * $i ) > $o)).
% 0.21/0.38 tff(c_type, type, (
% 0.21/0.38 c: $i > $i)).
% 0.21/0.38 tff(tptp_fun_X0_1_type, type, (
% 0.21/0.38 tptp_fun_X0_1: $i)).
% 0.21/0.38 tff(one_type, type, (
% 0.21/0.38 one: $i)).
% 0.21/0.38 tff(addition_type, type, (
% 0.21/0.38 addition: ( $i * $i ) > $i)).
% 0.21/0.38 tff(zero_type, type, (
% 0.21/0.38 zero: $i)).
% 0.21/0.38 tff(multiplication_type, type, (
% 0.21/0.38 multiplication: ( $i * $i ) > $i)).
% 0.21/0.38 tff(test_type, type, (
% 0.21/0.38 test: $i > $o)).
% 0.21/0.38 tff(1,plain,
% 0.21/0.38 (^[X0: $i, X1: $i] : refl((~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0))) <=> (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0))))),
% 0.21/0.38 inference(bind,[status(th)],[])).
% 0.21/0.38 tff(2,plain,
% 0.21/0.38 (![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0))) <=> ![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))),
% 0.21/0.39 inference(quant_intro,[status(thm)],[1])).
% 0.21/0.39 tff(3,plain,
% 0.21/0.39 (^[X0: $i, X1: $i] : trans(trans(monotonicity(rewrite(((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one)) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))))), ((complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one))) <=> (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))))))), rewrite((complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))))) <=> (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))), ((complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one))) <=> (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0))))), rewrite((~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0))) <=> (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))), ((complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one))) <=> (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))))),
% 0.21/0.39 inference(bind,[status(th)],[])).
% 0.21/0.39 tff(4,plain,
% 0.21/0.39 (![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one))) <=> ![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))),
% 0.21/0.39 inference(quant_intro,[status(thm)],[3])).
% 0.21/0.39 tff(5,plain,
% 0.21/0.39 (![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one))) <=> ![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one)))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(6,plain,
% 0.21/0.39 (^[X0: $i, X1: $i] : rewrite((complement(X1, X0) <=> (((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero)) & (addition(X0, X1) = one))) <=> (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one))))),
% 0.21/0.39 inference(bind,[status(th)],[])).
% 0.21/0.39 tff(7,plain,
% 0.21/0.39 (![X0: $i, X1: $i] : (complement(X1, X0) <=> (((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero)) & (addition(X0, X1) = one))) <=> ![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one)))),
% 0.21/0.39 inference(quant_intro,[status(thm)],[6])).
% 0.21/0.39 tff(8,axiom,(![X0: $i, X1: $i] : (complement(X1, X0) <=> (((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero)) & (addition(X0, X1) = one)))), file('/export/starexec/sandbox/benchmark/Axioms/KLE001+1.ax','test_2')).
% 0.21/0.39 tff(9,plain,
% 0.21/0.39 (![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one)))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[8, 7])).
% 0.21/0.39 tff(10,plain,
% 0.21/0.39 (![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one)))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[9, 5])).
% 0.21/0.39 tff(11,plain,(
% 0.21/0.39 ![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one)))),
% 0.21/0.39 inference(skolemize,[status(sab)],[10])).
% 0.21/0.39 tff(12,plain,
% 0.21/0.39 (![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[11, 4])).
% 0.21/0.39 tff(13,plain,
% 0.21/0.39 (![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[12, 2])).
% 0.21/0.39 tff(14,plain,
% 0.21/0.39 (((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(X0!1, c(X0!1)) = zero)) | (~(multiplication(c(X0!1), X0!1) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1))))) <=> ((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(X0!1, c(X0!1)) = zero)) | (~(multiplication(c(X0!1), X0!1) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1)))))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(15,plain,
% 0.21/0.39 ((~(((~(multiplication(X0!1, c(X0!1)) = zero)) | (~(multiplication(c(X0!1), X0!1) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1)))) <=> (~(((~(multiplication(X0!1, c(X0!1)) = zero)) | (~(multiplication(c(X0!1), X0!1) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1))))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(16,plain,
% 0.21/0.39 ((((~(multiplication(c(X0!1), X0!1) = zero)) | (~(multiplication(X0!1, c(X0!1)) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1))) <=> (((~(multiplication(X0!1, c(X0!1)) = zero)) | (~(multiplication(c(X0!1), X0!1) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1)))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(17,plain,
% 0.21/0.39 ((~(((~(multiplication(c(X0!1), X0!1) = zero)) | (~(multiplication(X0!1, c(X0!1)) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1)))) <=> (~(((~(multiplication(X0!1, c(X0!1)) = zero)) | (~(multiplication(c(X0!1), X0!1) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1))))),
% 0.21/0.39 inference(monotonicity,[status(thm)],[16])).
% 0.21/0.39 tff(18,plain,
% 0.21/0.39 ((~(((~(multiplication(c(X0!1), X0!1) = zero)) | (~(multiplication(X0!1, c(X0!1)) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1)))) <=> (~(((~(multiplication(X0!1, c(X0!1)) = zero)) | (~(multiplication(c(X0!1), X0!1) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1))))),
% 0.21/0.39 inference(transitivity,[status(thm)],[17, 15])).
% 0.21/0.39 tff(19,plain,
% 0.21/0.39 (((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(c(X0!1), X0!1) = zero)) | (~(multiplication(X0!1, c(X0!1)) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1))))) <=> ((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(X0!1, c(X0!1)) = zero)) | (~(multiplication(c(X0!1), X0!1) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1)))))),
% 0.21/0.39 inference(monotonicity,[status(thm)],[18])).
% 0.21/0.39 tff(20,plain,
% 0.21/0.39 (((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(c(X0!1), X0!1) = zero)) | (~(multiplication(X0!1, c(X0!1)) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1))))) <=> ((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(X0!1, c(X0!1)) = zero)) | (~(multiplication(c(X0!1), X0!1) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1)))))),
% 0.21/0.39 inference(transitivity,[status(thm)],[19, 14])).
% 0.21/0.39 tff(21,plain,
% 0.21/0.39 ((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(c(X0!1), X0!1) = zero)) | (~(multiplication(X0!1, c(X0!1)) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1))))),
% 0.21/0.39 inference(quant_inst,[status(thm)],[])).
% 0.21/0.39 tff(22,plain,
% 0.21/0.39 ((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(X0!1, c(X0!1)) = zero)) | (~(multiplication(c(X0!1), X0!1) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1))))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.21/0.39 tff(23,plain,
% 0.21/0.39 (~(((~(multiplication(X0!1, c(X0!1)) = zero)) | (~(multiplication(c(X0!1), X0!1) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1)))),
% 0.21/0.39 inference(unit_resolution,[status(thm)],[22, 13])).
% 0.21/0.39 tff(24,plain,
% 0.21/0.39 ((~![X0: $i] : ((~test(X0)) | (one = addition(X0, c(X0))))) <=> (~![X0: $i] : ((~test(X0)) | (one = addition(X0, c(X0)))))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(25,plain,
% 0.21/0.39 ((~![X0: $i] : (test(X0) => (one = addition(X0, c(X0))))) <=> (~![X0: $i] : ((~test(X0)) | (one = addition(X0, c(X0)))))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(26,axiom,(~![X0: $i] : (test(X0) => (one = addition(X0, c(X0))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','goals')).
% 0.21/0.39 tff(27,plain,
% 0.21/0.39 (~![X0: $i] : ((~test(X0)) | (one = addition(X0, c(X0))))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.21/0.39 tff(28,plain,
% 0.21/0.39 (~![X0: $i] : ((~test(X0)) | (one = addition(X0, c(X0))))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[27, 24])).
% 0.21/0.39 tff(29,plain,
% 0.21/0.39 (~![X0: $i] : ((~test(X0)) | (one = addition(X0, c(X0))))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[28, 24])).
% 0.21/0.39 tff(30,plain,
% 0.21/0.39 (~![X0: $i] : ((~test(X0)) | (one = addition(X0, c(X0))))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[29, 24])).
% 0.21/0.39 tff(31,plain,(
% 0.21/0.39 ~((~test(X0!1)) | (one = addition(X0!1, c(X0!1))))),
% 0.21/0.39 inference(skolemize,[status(sab)],[30])).
% 0.21/0.39 tff(32,plain,
% 0.21/0.39 (test(X0!1)),
% 0.21/0.39 inference(or_elim,[status(thm)],[31])).
% 0.21/0.39 tff(33,plain,
% 0.21/0.39 (^[X0: $i, X1: $i] : refl(((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1))) <=> ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1))))),
% 0.21/0.39 inference(bind,[status(th)],[])).
% 0.21/0.39 tff(34,plain,
% 0.21/0.39 (![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1))) <=> ![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.21/0.39 inference(quant_intro,[status(thm)],[33])).
% 0.21/0.39 tff(35,plain,
% 0.21/0.39 (![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1))) <=> ![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(36,plain,
% 0.21/0.39 (^[X0: $i, X1: $i] : rewrite((test(X0) => ((c(X0) = X1) <=> complement(X0, X1))) <=> ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1))))),
% 0.21/0.39 inference(bind,[status(th)],[])).
% 0.21/0.39 tff(37,plain,
% 0.21/0.39 (![X0: $i, X1: $i] : (test(X0) => ((c(X0) = X1) <=> complement(X0, X1))) <=> ![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.21/0.39 inference(quant_intro,[status(thm)],[36])).
% 0.21/0.39 tff(38,axiom,(![X0: $i, X1: $i] : (test(X0) => ((c(X0) = X1) <=> complement(X0, X1)))), file('/export/starexec/sandbox/benchmark/Axioms/KLE001+1.ax','test_3')).
% 0.21/0.39 tff(39,plain,
% 0.21/0.39 (![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.21/0.39 tff(40,plain,
% 0.21/0.39 (![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[39, 35])).
% 0.21/0.39 tff(41,plain,(
% 0.21/0.39 ![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.21/0.39 inference(skolemize,[status(sab)],[40])).
% 0.21/0.39 tff(42,plain,
% 0.21/0.39 (![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[41, 34])).
% 0.21/0.39 tff(43,plain,
% 0.21/0.39 (((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(X0!1)) | complement(X0!1, c(X0!1)))) <=> ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | (~test(X0!1)) | complement(X0!1, c(X0!1)))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(44,plain,
% 0.21/0.39 (((~test(X0!1)) | ((c(X0!1) = c(X0!1)) <=> complement(X0!1, c(X0!1)))) <=> ((~test(X0!1)) | complement(X0!1, c(X0!1)))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(45,plain,
% 0.21/0.39 (((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(X0!1)) | ((c(X0!1) = c(X0!1)) <=> complement(X0!1, c(X0!1))))) <=> ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(X0!1)) | complement(X0!1, c(X0!1))))),
% 0.21/0.39 inference(monotonicity,[status(thm)],[44])).
% 0.21/0.39 tff(46,plain,
% 0.21/0.39 (((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(X0!1)) | ((c(X0!1) = c(X0!1)) <=> complement(X0!1, c(X0!1))))) <=> ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | (~test(X0!1)) | complement(X0!1, c(X0!1)))),
% 0.21/0.39 inference(transitivity,[status(thm)],[45, 43])).
% 0.21/0.39 tff(47,plain,
% 0.21/0.39 ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(X0!1)) | ((c(X0!1) = c(X0!1)) <=> complement(X0!1, c(X0!1))))),
% 0.21/0.39 inference(quant_inst,[status(thm)],[])).
% 0.21/0.39 tff(48,plain,
% 0.21/0.39 ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | (~test(X0!1)) | complement(X0!1, c(X0!1))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.21/0.39 tff(49,plain,
% 0.21/0.39 (complement(X0!1, c(X0!1))),
% 0.21/0.39 inference(unit_resolution,[status(thm)],[48, 42, 32])).
% 0.21/0.39 tff(50,plain,
% 0.21/0.39 ((addition(X0!1, c(X0!1)) = one) <=> (one = addition(X0!1, c(X0!1)))),
% 0.21/0.39 inference(commutativity,[status(thm)],[])).
% 0.21/0.39 tff(51,plain,
% 0.21/0.39 (^[A: $i, B: $i] : refl((addition(A, B) = addition(B, A)) <=> (addition(A, B) = addition(B, A)))),
% 0.21/0.39 inference(bind,[status(th)],[])).
% 0.21/0.39 tff(52,plain,
% 0.21/0.39 (![A: $i, B: $i] : (addition(A, B) = addition(B, A)) <=> ![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.21/0.39 inference(quant_intro,[status(thm)],[51])).
% 0.21/0.39 tff(53,plain,
% 0.21/0.39 (![A: $i, B: $i] : (addition(A, B) = addition(B, A)) <=> ![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(54,axiom,(![A: $i, B: $i] : (addition(A, B) = addition(B, A))), file('/export/starexec/sandbox/benchmark/Axioms/KLE001+0.ax','additive_commutativity')).
% 0.21/0.39 tff(55,plain,
% 0.21/0.39 (![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[54, 53])).
% 0.21/0.39 tff(56,plain,(
% 0.21/0.39 ![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.21/0.39 inference(skolemize,[status(sab)],[55])).
% 0.21/0.39 tff(57,plain,
% 0.21/0.39 (![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[56, 52])).
% 0.21/0.39 tff(58,plain,
% 0.21/0.39 ((~![A: $i, B: $i] : (addition(A, B) = addition(B, A))) | (addition(X0!1, c(X0!1)) = addition(c(X0!1), X0!1))),
% 0.21/0.39 inference(quant_inst,[status(thm)],[])).
% 0.21/0.39 tff(59,plain,
% 0.21/0.39 (addition(X0!1, c(X0!1)) = addition(c(X0!1), X0!1)),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[58, 57])).
% 0.21/0.40 tff(60,plain,
% 0.21/0.40 (addition(c(X0!1), X0!1) = addition(X0!1, c(X0!1))),
% 0.21/0.40 inference(symmetry,[status(thm)],[59])).
% 0.21/0.40 tff(61,plain,
% 0.21/0.40 ((addition(c(X0!1), X0!1) = one) <=> (addition(X0!1, c(X0!1)) = one)),
% 0.21/0.40 inference(monotonicity,[status(thm)],[60])).
% 0.21/0.40 tff(62,plain,
% 0.21/0.40 ((addition(c(X0!1), X0!1) = one) <=> (one = addition(X0!1, c(X0!1)))),
% 0.21/0.40 inference(transitivity,[status(thm)],[61, 50])).
% 0.21/0.40 tff(63,plain,
% 0.21/0.40 ((one = addition(X0!1, c(X0!1))) <=> (addition(c(X0!1), X0!1) = one)),
% 0.21/0.40 inference(symmetry,[status(thm)],[62])).
% 0.21/0.40 tff(64,plain,
% 0.21/0.40 ((~(one = addition(X0!1, c(X0!1)))) <=> (~(addition(c(X0!1), X0!1) = one))),
% 0.21/0.40 inference(monotonicity,[status(thm)],[63])).
% 0.21/0.40 tff(65,plain,
% 0.21/0.40 (~(one = addition(X0!1, c(X0!1)))),
% 0.21/0.40 inference(or_elim,[status(thm)],[31])).
% 0.21/0.40 tff(66,plain,
% 0.21/0.40 (~(addition(c(X0!1), X0!1) = one)),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.21/0.40 tff(67,plain,
% 0.21/0.40 (((~(multiplication(X0!1, c(X0!1)) = zero)) | (~(multiplication(c(X0!1), X0!1) = zero)) | (~(addition(c(X0!1), X0!1) = one))) | (addition(c(X0!1), X0!1) = one)),
% 0.21/0.40 inference(tautology,[status(thm)],[])).
% 0.21/0.40 tff(68,plain,
% 0.21/0.40 ((~(multiplication(X0!1, c(X0!1)) = zero)) | (~(multiplication(c(X0!1), X0!1) = zero)) | (~(addition(c(X0!1), X0!1) = one))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[67, 66])).
% 0.21/0.40 tff(69,plain,
% 0.21/0.40 ((((~(multiplication(X0!1, c(X0!1)) = zero)) | (~(multiplication(c(X0!1), X0!1) = zero)) | (~(addition(c(X0!1), X0!1) = one))) <=> complement(X0!1, c(X0!1))) | (~((~(multiplication(X0!1, c(X0!1)) = zero)) | (~(multiplication(c(X0!1), X0!1) = zero)) | (~(addition(c(X0!1), X0!1) = one)))) | (~complement(X0!1, c(X0!1)))),
% 0.21/0.40 inference(tautology,[status(thm)],[])).
% 0.21/0.40 tff(70,plain,
% 0.21/0.40 ($false),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[69, 68, 49, 23])).
% 0.21/0.40 % SZS output end Proof
% 0.21/0.40 % E exiting
%------------------------------------------------------------------------------