TSTP Solution File: KLE007+3 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : KLE007+3 : TPTP v8.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n014.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 Sep 17 17:23:45 EDT 2022
% Result : Theorem 0.20s 0.41s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : KLE007+3 : TPTP v8.1.0. Released v4.0.0.
% 0.03/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n014.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 Sep 1 07:44:17 EDT 2022
% 0.13/0.34 % CPUTime :
% 0.13/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.35 Usage: tptp [options] [-file:]file
% 0.13/0.35 -h, -? prints this message.
% 0.13/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.35 -m, -model generate model.
% 0.13/0.35 -p, -proof generate proof.
% 0.13/0.35 -c, -core generate unsat core of named formulas.
% 0.13/0.35 -st, -statistics display statistics.
% 0.13/0.35 -t:timeout set timeout (in second).
% 0.13/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.35 -<param>:<value> configuration parameter and value.
% 0.13/0.35 -o:<output-file> file to place output in.
% 0.20/0.41 % SZS status Theorem
% 0.20/0.41 % SZS output start Proof
% 0.20/0.41 tff(addition_type, type, (
% 0.20/0.41 addition: ( $i * $i ) > $i)).
% 0.20/0.41 tff(multiplication_type, type, (
% 0.20/0.41 multiplication: ( $i * $i ) > $i)).
% 0.20/0.41 tff(c_type, type, (
% 0.20/0.41 c: $i > $i)).
% 0.20/0.41 tff(tptp_fun_X1_1_type, type, (
% 0.20/0.41 tptp_fun_X1_1: $i)).
% 0.20/0.41 tff(tptp_fun_X0_2_type, type, (
% 0.20/0.41 tptp_fun_X0_2: $i)).
% 0.20/0.41 tff(one_type, type, (
% 0.20/0.41 one: $i)).
% 0.20/0.41 tff(zero_type, type, (
% 0.20/0.41 zero: $i)).
% 0.20/0.41 tff(complement_type, type, (
% 0.20/0.41 complement: ( $i * $i ) > $o)).
% 0.20/0.41 tff(test_type, type, (
% 0.20/0.41 test: $i > $o)).
% 0.20/0.41 tff(1,plain,
% 0.20/0.41 (^[A: $i, B: $i, C: $i] : refl((multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C))) <=> (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(2,plain,
% 0.20/0.41 (![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C))) <=> ![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.41 tff(3,plain,
% 0.20/0.41 (![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C))) <=> ![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(4,axiom,(![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))), file('/export/starexec/sandbox2/benchmark/Axioms/KLE001+0.ax','right_distributivity')).
% 0.20/0.41 tff(5,plain,
% 0.20/0.41 (![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.20/0.41 tff(6,plain,(
% 0.20/0.41 ![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))),
% 0.20/0.41 inference(skolemize,[status(sab)],[5])).
% 0.20/0.41 tff(7,plain,
% 0.20/0.41 (![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.20/0.41 tff(8,plain,
% 0.20/0.41 ((~![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))) | (multiplication(addition(X0!2, c(X0!2)), addition(X1!1, c(X1!1))) = addition(multiplication(addition(X0!2, c(X0!2)), X1!1), multiplication(addition(X0!2, c(X0!2)), c(X1!1))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(9,plain,
% 0.20/0.41 (multiplication(addition(X0!2, c(X0!2)), addition(X1!1, c(X1!1))) = addition(multiplication(addition(X0!2, c(X0!2)), X1!1), multiplication(addition(X0!2, c(X0!2)), c(X1!1)))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.20/0.41 tff(10,plain,
% 0.20/0.41 (^[A: $i, B: $i] : refl((addition(A, B) = addition(B, A)) <=> (addition(A, B) = addition(B, A)))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(11,plain,
% 0.20/0.41 (![A: $i, B: $i] : (addition(A, B) = addition(B, A)) <=> ![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[10])).
% 0.20/0.41 tff(12,plain,
% 0.20/0.41 (![A: $i, B: $i] : (addition(A, B) = addition(B, A)) <=> ![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(13,axiom,(![A: $i, B: $i] : (addition(A, B) = addition(B, A))), file('/export/starexec/sandbox2/benchmark/Axioms/KLE001+0.ax','additive_commutativity')).
% 0.20/0.41 tff(14,plain,
% 0.20/0.41 (![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.20/0.41 tff(15,plain,(
% 0.20/0.41 ![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.20/0.41 inference(skolemize,[status(sab)],[14])).
% 0.20/0.41 tff(16,plain,
% 0.20/0.41 (![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[15, 11])).
% 0.20/0.41 tff(17,plain,
% 0.20/0.41 ((~![A: $i, B: $i] : (addition(A, B) = addition(B, A))) | (addition(X1!1, c(X1!1)) = addition(c(X1!1), X1!1))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(18,plain,
% 0.20/0.41 (addition(X1!1, c(X1!1)) = addition(c(X1!1), X1!1)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.20/0.41 tff(19,plain,
% 0.20/0.41 (addition(c(X1!1), X1!1) = addition(X1!1, c(X1!1))),
% 0.20/0.41 inference(symmetry,[status(thm)],[18])).
% 0.20/0.41 tff(20,plain,
% 0.20/0.41 ((~![A: $i, B: $i] : (addition(A, B) = addition(B, A))) | (addition(X0!2, c(X0!2)) = addition(c(X0!2), X0!2))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(21,plain,
% 0.20/0.41 (addition(X0!2, c(X0!2)) = addition(c(X0!2), X0!2)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[20, 16])).
% 0.20/0.41 tff(22,plain,
% 0.20/0.41 (addition(c(X0!2), X0!2) = addition(X0!2, c(X0!2))),
% 0.20/0.41 inference(symmetry,[status(thm)],[21])).
% 0.20/0.41 tff(23,plain,
% 0.20/0.41 (multiplication(addition(c(X0!2), X0!2), addition(c(X1!1), X1!1)) = multiplication(addition(X0!2, c(X0!2)), addition(X1!1, c(X1!1)))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[22, 19])).
% 0.20/0.41 tff(24,plain,
% 0.20/0.41 (^[X0: $i, X1: $i] : refl((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))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(25,plain,
% 0.20/0.41 (![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)))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[24])).
% 0.20/0.41 tff(26,plain,
% 0.20/0.41 (^[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))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(27,plain,
% 0.20/0.41 (![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)))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[26])).
% 0.20/0.41 tff(28,plain,
% 0.20/0.41 (![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)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(29,plain,
% 0.20/0.41 (^[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))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(30,plain,
% 0.20/0.41 (![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)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[29])).
% 0.20/0.41 tff(31,axiom,(![X0: $i, X1: $i] : (complement(X1, X0) <=> (((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero)) & (addition(X0, X1) = one)))), file('/export/starexec/sandbox2/benchmark/Axioms/KLE001+1.ax','test_2')).
% 0.20/0.41 tff(32,plain,
% 0.20/0.41 (![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[31, 30])).
% 0.20/0.41 tff(33,plain,
% 0.20/0.41 (![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[32, 28])).
% 0.20/0.41 tff(34,plain,(
% 0.20/0.41 ![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one)))),
% 0.20/0.41 inference(skolemize,[status(sab)],[33])).
% 0.20/0.41 tff(35,plain,
% 0.20/0.41 (![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[34, 27])).
% 0.20/0.41 tff(36,plain,
% 0.20/0.41 (![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[35, 25])).
% 0.20/0.41 tff(37,plain,
% 0.20/0.41 (((~![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))) | (complement(X0!2, c(X0!2)) <=> (~((~(multiplication(X0!2, c(X0!2)) = zero)) | (~(multiplication(c(X0!2), X0!2) = zero)) | (~(addition(c(X0!2), X0!2) = one)))))) <=> ((~![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))) | (complement(X0!2, c(X0!2)) <=> (~((~(multiplication(X0!2, c(X0!2)) = zero)) | (~(multiplication(c(X0!2), X0!2) = zero)) | (~(addition(c(X0!2), X0!2) = one))))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(38,plain,
% 0.20/0.41 ((complement(X0!2, c(X0!2)) <=> (~((~(multiplication(c(X0!2), X0!2) = zero)) | (~(multiplication(X0!2, c(X0!2)) = zero)) | (~(addition(c(X0!2), X0!2) = one))))) <=> (complement(X0!2, c(X0!2)) <=> (~((~(multiplication(X0!2, c(X0!2)) = zero)) | (~(multiplication(c(X0!2), X0!2) = zero)) | (~(addition(c(X0!2), X0!2) = one)))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(39,plain,
% 0.20/0.41 (((~![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))) | (complement(X0!2, c(X0!2)) <=> (~((~(multiplication(c(X0!2), X0!2) = zero)) | (~(multiplication(X0!2, c(X0!2)) = zero)) | (~(addition(c(X0!2), X0!2) = one)))))) <=> ((~![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))) | (complement(X0!2, c(X0!2)) <=> (~((~(multiplication(X0!2, c(X0!2)) = zero)) | (~(multiplication(c(X0!2), X0!2) = zero)) | (~(addition(c(X0!2), X0!2) = one))))))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[38])).
% 0.20/0.41 tff(40,plain,
% 0.20/0.41 (((~![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))) | (complement(X0!2, c(X0!2)) <=> (~((~(multiplication(c(X0!2), X0!2) = zero)) | (~(multiplication(X0!2, c(X0!2)) = zero)) | (~(addition(c(X0!2), X0!2) = one)))))) <=> ((~![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))) | (complement(X0!2, c(X0!2)) <=> (~((~(multiplication(X0!2, c(X0!2)) = zero)) | (~(multiplication(c(X0!2), X0!2) = zero)) | (~(addition(c(X0!2), X0!2) = one))))))),
% 0.20/0.41 inference(transitivity,[status(thm)],[39, 37])).
% 0.20/0.41 tff(41,plain,
% 0.20/0.41 ((~![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))) | (complement(X0!2, c(X0!2)) <=> (~((~(multiplication(c(X0!2), X0!2) = zero)) | (~(multiplication(X0!2, c(X0!2)) = zero)) | (~(addition(c(X0!2), X0!2) = one)))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(42,plain,
% 0.20/0.41 ((~![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))) | (complement(X0!2, c(X0!2)) <=> (~((~(multiplication(X0!2, c(X0!2)) = zero)) | (~(multiplication(c(X0!2), X0!2) = zero)) | (~(addition(c(X0!2), X0!2) = one)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[41, 40])).
% 0.20/0.41 tff(43,plain,
% 0.20/0.41 (complement(X0!2, c(X0!2)) <=> (~((~(multiplication(X0!2, c(X0!2)) = zero)) | (~(multiplication(c(X0!2), X0!2) = zero)) | (~(addition(c(X0!2), X0!2) = one))))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[42, 36])).
% 0.20/0.41 tff(44,plain,
% 0.20/0.41 ((~![X0: $i, X1: $i] : ((~(test(X1) & test(X0))) | (one = addition(multiplication(addition(X0, c(X0)), X1), multiplication(addition(X0, c(X0)), c(X1)))))) <=> (~![X0: $i, X1: $i] : ((~(test(X1) & test(X0))) | (one = addition(multiplication(addition(X0, c(X0)), X1), multiplication(addition(X0, c(X0)), c(X1))))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(45,plain,
% 0.20/0.41 ((~![X0: $i, X1: $i] : ((test(X1) & test(X0)) => (one = addition(multiplication(addition(X0, c(X0)), X1), multiplication(addition(X0, c(X0)), c(X1)))))) <=> (~![X0: $i, X1: $i] : ((~(test(X1) & test(X0))) | (one = addition(multiplication(addition(X0, c(X0)), X1), multiplication(addition(X0, c(X0)), c(X1))))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(46,axiom,(~![X0: $i, X1: $i] : ((test(X1) & test(X0)) => (one = addition(multiplication(addition(X0, c(X0)), X1), multiplication(addition(X0, c(X0)), c(X1)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','goals')).
% 0.20/0.41 tff(47,plain,
% 0.20/0.41 (~![X0: $i, X1: $i] : ((~(test(X1) & test(X0))) | (one = addition(multiplication(addition(X0, c(X0)), X1), multiplication(addition(X0, c(X0)), c(X1)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[46, 45])).
% 0.20/0.41 tff(48,plain,
% 0.20/0.41 (~![X0: $i, X1: $i] : ((~(test(X1) & test(X0))) | (one = addition(multiplication(addition(X0, c(X0)), X1), multiplication(addition(X0, c(X0)), c(X1)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[47, 44])).
% 0.20/0.41 tff(49,plain,
% 0.20/0.41 (~![X0: $i, X1: $i] : ((~(test(X1) & test(X0))) | (one = addition(multiplication(addition(X0, c(X0)), X1), multiplication(addition(X0, c(X0)), c(X1)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[48, 44])).
% 0.20/0.41 tff(50,plain,
% 0.20/0.41 (~![X0: $i, X1: $i] : ((~(test(X1) & test(X0))) | (one = addition(multiplication(addition(X0, c(X0)), X1), multiplication(addition(X0, c(X0)), c(X1)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[49, 44])).
% 0.20/0.41 tff(51,plain,
% 0.20/0.41 (~![X0: $i, X1: $i] : ((~(test(X1) & test(X0))) | (one = addition(multiplication(addition(X0, c(X0)), X1), multiplication(addition(X0, c(X0)), c(X1)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[50, 44])).
% 0.20/0.41 tff(52,plain,
% 0.20/0.41 (~![X0: $i, X1: $i] : ((~(test(X1) & test(X0))) | (one = addition(multiplication(addition(X0, c(X0)), X1), multiplication(addition(X0, c(X0)), c(X1)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[51, 44])).
% 0.20/0.41 tff(53,plain,
% 0.20/0.41 (~![X0: $i, X1: $i] : ((~(test(X1) & test(X0))) | (one = addition(multiplication(addition(X0, c(X0)), X1), multiplication(addition(X0, c(X0)), c(X1)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[52, 44])).
% 0.20/0.41 tff(54,plain,(
% 0.20/0.41 ~((~(test(X1!1) & test(X0!2))) | (one = addition(multiplication(addition(X0!2, c(X0!2)), X1!1), multiplication(addition(X0!2, c(X0!2)), c(X1!1)))))),
% 0.20/0.41 inference(skolemize,[status(sab)],[53])).
% 0.20/0.41 tff(55,plain,
% 0.20/0.41 (test(X1!1) & test(X0!2)),
% 0.20/0.41 inference(or_elim,[status(thm)],[54])).
% 0.20/0.41 tff(56,plain,
% 0.20/0.41 (test(X0!2)),
% 0.20/0.41 inference(and_elim,[status(thm)],[55])).
% 0.20/0.41 tff(57,plain,
% 0.20/0.41 (^[X0: $i, X1: $i] : refl(((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1))) <=> ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(58,plain,
% 0.20/0.41 (![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1))) <=> ![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[57])).
% 0.20/0.41 tff(59,plain,
% 0.20/0.41 (![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1))) <=> ![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(60,plain,
% 0.20/0.41 (^[X0: $i, X1: $i] : rewrite((test(X0) => ((c(X0) = X1) <=> complement(X0, X1))) <=> ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(61,plain,
% 0.20/0.41 (![X0: $i, X1: $i] : (test(X0) => ((c(X0) = X1) <=> complement(X0, X1))) <=> ![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[60])).
% 0.20/0.41 tff(62,axiom,(![X0: $i, X1: $i] : (test(X0) => ((c(X0) = X1) <=> complement(X0, X1)))), file('/export/starexec/sandbox2/benchmark/Axioms/KLE001+1.ax','test_3')).
% 0.20/0.41 tff(63,plain,
% 0.20/0.41 (![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.20/0.42 tff(64,plain,
% 0.20/0.42 (![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[63, 59])).
% 0.20/0.42 tff(65,plain,(
% 0.20/0.42 ![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.20/0.42 inference(skolemize,[status(sab)],[64])).
% 0.20/0.42 tff(66,plain,
% 0.20/0.42 (![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[65, 58])).
% 0.20/0.42 tff(67,plain,
% 0.20/0.42 (((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(X0!2)) | complement(X0!2, c(X0!2)))) <=> ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | (~test(X0!2)) | complement(X0!2, c(X0!2)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(68,plain,
% 0.20/0.42 (((~test(X0!2)) | ((c(X0!2) = c(X0!2)) <=> complement(X0!2, c(X0!2)))) <=> ((~test(X0!2)) | complement(X0!2, c(X0!2)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(69,plain,
% 0.20/0.42 (((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(X0!2)) | ((c(X0!2) = c(X0!2)) <=> complement(X0!2, c(X0!2))))) <=> ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(X0!2)) | complement(X0!2, c(X0!2))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[68])).
% 0.20/0.42 tff(70,plain,
% 0.20/0.42 (((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(X0!2)) | ((c(X0!2) = c(X0!2)) <=> complement(X0!2, c(X0!2))))) <=> ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | (~test(X0!2)) | complement(X0!2, c(X0!2)))),
% 0.20/0.42 inference(transitivity,[status(thm)],[69, 67])).
% 0.20/0.42 tff(71,plain,
% 0.20/0.42 ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(X0!2)) | ((c(X0!2) = c(X0!2)) <=> complement(X0!2, c(X0!2))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(72,plain,
% 0.20/0.42 ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | (~test(X0!2)) | complement(X0!2, c(X0!2))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.20/0.42 tff(73,plain,
% 0.20/0.42 (complement(X0!2, c(X0!2))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[72, 66, 56])).
% 0.20/0.42 tff(74,plain,
% 0.20/0.42 ((~(complement(X0!2, c(X0!2)) <=> (~((~(multiplication(X0!2, c(X0!2)) = zero)) | (~(multiplication(c(X0!2), X0!2) = zero)) | (~(addition(c(X0!2), X0!2) = one)))))) | (~complement(X0!2, c(X0!2))) | (~((~(multiplication(X0!2, c(X0!2)) = zero)) | (~(multiplication(c(X0!2), X0!2) = zero)) | (~(addition(c(X0!2), X0!2) = one))))),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(75,plain,
% 0.20/0.42 (~((~(multiplication(X0!2, c(X0!2)) = zero)) | (~(multiplication(c(X0!2), X0!2) = zero)) | (~(addition(c(X0!2), X0!2) = one)))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[74, 73, 43])).
% 0.20/0.42 tff(76,plain,
% 0.20/0.42 (((~(multiplication(X0!2, c(X0!2)) = zero)) | (~(multiplication(c(X0!2), X0!2) = zero)) | (~(addition(c(X0!2), X0!2) = one))) | (addition(c(X0!2), X0!2) = one)),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(77,plain,
% 0.20/0.42 (addition(c(X0!2), X0!2) = one),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[76, 75])).
% 0.20/0.42 tff(78,plain,
% 0.20/0.42 (one = addition(c(X0!2), X0!2)),
% 0.20/0.42 inference(symmetry,[status(thm)],[77])).
% 0.20/0.42 tff(79,plain,
% 0.20/0.42 (multiplication(one, addition(c(X1!1), X1!1)) = multiplication(addition(c(X0!2), X0!2), addition(c(X1!1), X1!1))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[78])).
% 0.20/0.42 tff(80,plain,
% 0.20/0.42 (^[A: $i] : refl((multiplication(one, A) = A) <=> (multiplication(one, A) = A))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(81,plain,
% 0.20/0.42 (![A: $i] : (multiplication(one, A) = A) <=> ![A: $i] : (multiplication(one, A) = A)),
% 0.20/0.42 inference(quant_intro,[status(thm)],[80])).
% 0.20/0.42 tff(82,plain,
% 0.20/0.42 (![A: $i] : (multiplication(one, A) = A) <=> ![A: $i] : (multiplication(one, A) = A)),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(83,axiom,(![A: $i] : (multiplication(one, A) = A)), file('/export/starexec/sandbox2/benchmark/Axioms/KLE001+0.ax','multiplicative_left_identity')).
% 0.20/0.42 tff(84,plain,
% 0.20/0.42 (![A: $i] : (multiplication(one, A) = A)),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[83, 82])).
% 0.20/0.42 tff(85,plain,(
% 0.20/0.42 ![A: $i] : (multiplication(one, A) = A)),
% 0.20/0.42 inference(skolemize,[status(sab)],[84])).
% 0.20/0.42 tff(86,plain,
% 0.20/0.42 (![A: $i] : (multiplication(one, A) = A)),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[85, 81])).
% 0.20/0.42 tff(87,plain,
% 0.20/0.42 ((~![A: $i] : (multiplication(one, A) = A)) | (multiplication(one, addition(c(X1!1), X1!1)) = addition(c(X1!1), X1!1))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(88,plain,
% 0.20/0.42 (multiplication(one, addition(c(X1!1), X1!1)) = addition(c(X1!1), X1!1)),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[87, 86])).
% 0.20/0.42 tff(89,plain,
% 0.20/0.42 (addition(c(X1!1), X1!1) = multiplication(one, addition(c(X1!1), X1!1))),
% 0.20/0.42 inference(symmetry,[status(thm)],[88])).
% 0.20/0.42 tff(90,plain,
% 0.20/0.42 (((~![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))) | (complement(X1!1, c(X1!1)) <=> (~((~(multiplication(X1!1, c(X1!1)) = zero)) | (~(multiplication(c(X1!1), X1!1) = zero)) | (~(addition(c(X1!1), X1!1) = one)))))) <=> ((~![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))) | (complement(X1!1, c(X1!1)) <=> (~((~(multiplication(X1!1, c(X1!1)) = zero)) | (~(multiplication(c(X1!1), X1!1) = zero)) | (~(addition(c(X1!1), X1!1) = one))))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(91,plain,
% 0.20/0.42 ((complement(X1!1, c(X1!1)) <=> (~((~(multiplication(c(X1!1), X1!1) = zero)) | (~(multiplication(X1!1, c(X1!1)) = zero)) | (~(addition(c(X1!1), X1!1) = one))))) <=> (complement(X1!1, c(X1!1)) <=> (~((~(multiplication(X1!1, c(X1!1)) = zero)) | (~(multiplication(c(X1!1), X1!1) = zero)) | (~(addition(c(X1!1), X1!1) = one)))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(92,plain,
% 0.20/0.42 (((~![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))) | (complement(X1!1, c(X1!1)) <=> (~((~(multiplication(c(X1!1), X1!1) = zero)) | (~(multiplication(X1!1, c(X1!1)) = zero)) | (~(addition(c(X1!1), X1!1) = one)))))) <=> ((~![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))) | (complement(X1!1, c(X1!1)) <=> (~((~(multiplication(X1!1, c(X1!1)) = zero)) | (~(multiplication(c(X1!1), X1!1) = zero)) | (~(addition(c(X1!1), X1!1) = one))))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[91])).
% 0.20/0.42 tff(93,plain,
% 0.20/0.42 (((~![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))) | (complement(X1!1, c(X1!1)) <=> (~((~(multiplication(c(X1!1), X1!1) = zero)) | (~(multiplication(X1!1, c(X1!1)) = zero)) | (~(addition(c(X1!1), X1!1) = one)))))) <=> ((~![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))) | (complement(X1!1, c(X1!1)) <=> (~((~(multiplication(X1!1, c(X1!1)) = zero)) | (~(multiplication(c(X1!1), X1!1) = zero)) | (~(addition(c(X1!1), X1!1) = one))))))),
% 0.20/0.42 inference(transitivity,[status(thm)],[92, 90])).
% 0.20/0.42 tff(94,plain,
% 0.20/0.42 ((~![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))) | (complement(X1!1, c(X1!1)) <=> (~((~(multiplication(c(X1!1), X1!1) = zero)) | (~(multiplication(X1!1, c(X1!1)) = zero)) | (~(addition(c(X1!1), X1!1) = one)))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(95,plain,
% 0.20/0.42 ((~![X0: $i, X1: $i] : (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one)))))) | (complement(X1!1, c(X1!1)) <=> (~((~(multiplication(X1!1, c(X1!1)) = zero)) | (~(multiplication(c(X1!1), X1!1) = zero)) | (~(addition(c(X1!1), X1!1) = one)))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[94, 93])).
% 0.20/0.42 tff(96,plain,
% 0.20/0.42 (complement(X1!1, c(X1!1)) <=> (~((~(multiplication(X1!1, c(X1!1)) = zero)) | (~(multiplication(c(X1!1), X1!1) = zero)) | (~(addition(c(X1!1), X1!1) = one))))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[95, 36])).
% 0.20/0.43 tff(97,plain,
% 0.20/0.43 (test(X1!1)),
% 0.20/0.43 inference(and_elim,[status(thm)],[55])).
% 0.20/0.43 tff(98,plain,
% 0.20/0.43 (((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(X1!1)) | complement(X1!1, c(X1!1)))) <=> ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | (~test(X1!1)) | complement(X1!1, c(X1!1)))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(99,plain,
% 0.20/0.43 (((~test(X1!1)) | ((c(X1!1) = c(X1!1)) <=> complement(X1!1, c(X1!1)))) <=> ((~test(X1!1)) | complement(X1!1, c(X1!1)))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(100,plain,
% 0.20/0.43 (((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(X1!1)) | ((c(X1!1) = c(X1!1)) <=> complement(X1!1, c(X1!1))))) <=> ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(X1!1)) | complement(X1!1, c(X1!1))))),
% 0.20/0.43 inference(monotonicity,[status(thm)],[99])).
% 0.20/0.43 tff(101,plain,
% 0.20/0.43 (((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(X1!1)) | ((c(X1!1) = c(X1!1)) <=> complement(X1!1, c(X1!1))))) <=> ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | (~test(X1!1)) | complement(X1!1, c(X1!1)))),
% 0.20/0.43 inference(transitivity,[status(thm)],[100, 98])).
% 0.20/0.43 tff(102,plain,
% 0.20/0.43 ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(X1!1)) | ((c(X1!1) = c(X1!1)) <=> complement(X1!1, c(X1!1))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(103,plain,
% 0.20/0.43 ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | (~test(X1!1)) | complement(X1!1, c(X1!1))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.20/0.43 tff(104,plain,
% 0.20/0.43 (complement(X1!1, c(X1!1))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[103, 66, 97])).
% 0.20/0.43 tff(105,plain,
% 0.20/0.43 ((~(complement(X1!1, c(X1!1)) <=> (~((~(multiplication(X1!1, c(X1!1)) = zero)) | (~(multiplication(c(X1!1), X1!1) = zero)) | (~(addition(c(X1!1), X1!1) = one)))))) | (~complement(X1!1, c(X1!1))) | (~((~(multiplication(X1!1, c(X1!1)) = zero)) | (~(multiplication(c(X1!1), X1!1) = zero)) | (~(addition(c(X1!1), X1!1) = one))))),
% 0.20/0.43 inference(tautology,[status(thm)],[])).
% 0.20/0.43 tff(106,plain,
% 0.20/0.43 (~((~(multiplication(X1!1, c(X1!1)) = zero)) | (~(multiplication(c(X1!1), X1!1) = zero)) | (~(addition(c(X1!1), X1!1) = one)))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[105, 104, 96])).
% 0.20/0.43 tff(107,plain,
% 0.20/0.43 (((~(multiplication(X1!1, c(X1!1)) = zero)) | (~(multiplication(c(X1!1), X1!1) = zero)) | (~(addition(c(X1!1), X1!1) = one))) | (addition(c(X1!1), X1!1) = one)),
% 0.20/0.43 inference(tautology,[status(thm)],[])).
% 0.20/0.43 tff(108,plain,
% 0.20/0.43 (addition(c(X1!1), X1!1) = one),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[107, 106])).
% 0.20/0.43 tff(109,plain,
% 0.20/0.43 (one = addition(c(X1!1), X1!1)),
% 0.20/0.43 inference(symmetry,[status(thm)],[108])).
% 0.20/0.43 tff(110,plain,
% 0.20/0.43 (one = addition(multiplication(addition(X0!2, c(X0!2)), X1!1), multiplication(addition(X0!2, c(X0!2)), c(X1!1)))),
% 0.20/0.43 inference(transitivity,[status(thm)],[109, 89, 79, 23, 9])).
% 0.20/0.43 tff(111,plain,
% 0.20/0.43 (~(one = addition(multiplication(addition(X0!2, c(X0!2)), X1!1), multiplication(addition(X0!2, c(X0!2)), c(X1!1))))),
% 0.20/0.43 inference(or_elim,[status(thm)],[54])).
% 0.20/0.43 tff(112,plain,
% 0.20/0.43 ($false),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[111, 110])).
% 0.20/0.43 % SZS output end Proof
%------------------------------------------------------------------------------