TSTP Solution File: SET902+1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET902+1 : TPTP v8.1.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n027.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 : Tue Sep 20 05:08:33 EDT 2022
% Result : Theorem 0.12s 0.39s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET902+1 : TPTP v8.1.0. Released v3.2.0.
% 0.07/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.34 % Computer : n027.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 : Sat Sep 3 08:45:05 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.12/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.35 Usage: tptp [options] [-file:]file
% 0.12/0.35 -h, -? prints this message.
% 0.12/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.35 -m, -model generate model.
% 0.12/0.35 -p, -proof generate proof.
% 0.12/0.35 -c, -core generate unsat core of named formulas.
% 0.12/0.35 -st, -statistics display statistics.
% 0.12/0.35 -t:timeout set timeout (in second).
% 0.12/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.35 -<param>:<value> configuration parameter and value.
% 0.12/0.35 -o:<output-file> file to place output in.
% 0.12/0.39 % SZS status Theorem
% 0.12/0.39 % SZS output start Proof
% 0.12/0.39 tff(empty_set_type, type, (
% 0.12/0.39 empty_set: $i)).
% 0.12/0.39 tff(singleton_type, type, (
% 0.12/0.39 singleton: $i > $i)).
% 0.12/0.39 tff(tptp_fun_A_4_type, type, (
% 0.12/0.39 tptp_fun_A_4: $i)).
% 0.12/0.39 tff(set_union2_type, type, (
% 0.12/0.39 set_union2: ( $i * $i ) > $i)).
% 0.12/0.39 tff(tptp_fun_C_2_type, type, (
% 0.12/0.39 tptp_fun_C_2: $i)).
% 0.12/0.39 tff(tptp_fun_B_3_type, type, (
% 0.12/0.39 tptp_fun_B_3: $i)).
% 0.12/0.39 tff(subset_type, type, (
% 0.12/0.39 subset: ( $i * $i ) > $o)).
% 0.12/0.39 tff(1,plain,
% 0.12/0.39 (^[A: $i] : refl((set_union2(A, A) = A) <=> (set_union2(A, A) = A))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(2,plain,
% 0.12/0.39 (![A: $i] : (set_union2(A, A) = A) <=> ![A: $i] : (set_union2(A, A) = A)),
% 0.12/0.39 inference(quant_intro,[status(thm)],[1])).
% 0.12/0.39 tff(3,plain,
% 0.12/0.39 (![A: $i] : (set_union2(A, A) = A) <=> ![A: $i] : (set_union2(A, A) = A)),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(4,plain,
% 0.12/0.39 (![A: $i, B: $i] : (set_union2(A, A) = A) <=> ![A: $i] : (set_union2(A, A) = A)),
% 0.12/0.39 inference(elim_unused_vars,[status(thm)],[])).
% 0.12/0.39 tff(5,axiom,(![A: $i, B: $i] : (set_union2(A, A) = A)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','idempotence_k2_xboole_0')).
% 0.12/0.39 tff(6,plain,
% 0.12/0.39 (![A: $i] : (set_union2(A, A) = A)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.12/0.39 tff(7,plain,
% 0.12/0.39 (![A: $i] : (set_union2(A, A) = A)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[6, 3])).
% 0.12/0.39 tff(8,plain,(
% 0.12/0.39 ![A: $i] : (set_union2(A, A) = A)),
% 0.12/0.39 inference(skolemize,[status(sab)],[7])).
% 0.12/0.39 tff(9,plain,
% 0.12/0.39 (![A: $i] : (set_union2(A, A) = A)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[8, 2])).
% 0.12/0.39 tff(10,plain,
% 0.12/0.39 ((~![A: $i] : (set_union2(A, A) = A)) | (set_union2(empty_set, empty_set) = empty_set)),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(11,plain,
% 0.12/0.39 (set_union2(empty_set, empty_set) = empty_set),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[10, 9])).
% 0.12/0.39 tff(12,assumption,(C!2 = singleton(A!4)), introduced(assumption)).
% 0.12/0.39 tff(13,plain,
% 0.12/0.39 ((~(~((~(B!3 = singleton(A!4))) | (~(C!2 = singleton(A!4)))))) <=> ((~(B!3 = singleton(A!4))) | (~(C!2 = singleton(A!4))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(14,plain,
% 0.12/0.39 (((B!3 = singleton(A!4)) & (C!2 = singleton(A!4))) <=> (~((~(B!3 = singleton(A!4))) | (~(C!2 = singleton(A!4)))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(15,plain,
% 0.12/0.39 ((~((B!3 = singleton(A!4)) & (C!2 = singleton(A!4)))) <=> (~(~((~(B!3 = singleton(A!4))) | (~(C!2 = singleton(A!4))))))),
% 0.12/0.39 inference(monotonicity,[status(thm)],[14])).
% 0.12/0.39 tff(16,plain,
% 0.12/0.39 ((~((B!3 = singleton(A!4)) & (C!2 = singleton(A!4)))) <=> ((~(B!3 = singleton(A!4))) | (~(C!2 = singleton(A!4))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[15, 13])).
% 0.12/0.39 tff(17,plain,
% 0.12/0.39 ((~(~((singleton(A!4) = set_union2(B!3, C!2)) & (~((B!3 = singleton(A!4)) & (C!2 = singleton(A!4)))) & (~((B!3 = empty_set) & (C!2 = singleton(A!4)))) & (~((B!3 = singleton(A!4)) & (C!2 = empty_set)))))) <=> ((singleton(A!4) = set_union2(B!3, C!2)) & (~((B!3 = singleton(A!4)) & (C!2 = singleton(A!4)))) & (~((B!3 = empty_set) & (C!2 = singleton(A!4)))) & (~((B!3 = singleton(A!4)) & (C!2 = empty_set))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(18,plain,
% 0.12/0.40 ((~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set)))))) <=> (~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set))))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(19,plain,
% 0.12/0.40 ((~![A: $i, B: $i, C: $i] : (~((((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A))))) & (~((B = empty_set) & (C = singleton(A))))) & (~((B = singleton(A)) & (C = empty_set)))))) <=> (~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set))))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(20,axiom,(~![A: $i, B: $i, C: $i] : (~((((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A))))) & (~((B = empty_set) & (C = singleton(A))))) & (~((B = singleton(A)) & (C = empty_set)))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t43_zfmisc_1')).
% 0.12/0.40 tff(21,plain,
% 0.12/0.40 (~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set)))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.12/0.40 tff(22,plain,
% 0.12/0.40 (~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set)))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[21, 18])).
% 0.12/0.40 tff(23,plain,
% 0.12/0.40 (~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set)))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[22, 18])).
% 0.12/0.40 tff(24,plain,
% 0.12/0.40 (~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set)))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[23, 18])).
% 0.12/0.40 tff(25,plain,
% 0.12/0.40 (~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set)))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[24, 18])).
% 0.12/0.40 tff(26,plain,
% 0.12/0.40 (~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set)))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[25, 18])).
% 0.12/0.40 tff(27,plain,
% 0.12/0.40 (~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set)))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[26, 18])).
% 0.12/0.40 tff(28,plain,(
% 0.12/0.40 ~(~((singleton(A!4) = set_union2(B!3, C!2)) & (~((B!3 = singleton(A!4)) & (C!2 = singleton(A!4)))) & (~((B!3 = empty_set) & (C!2 = singleton(A!4)))) & (~((B!3 = singleton(A!4)) & (C!2 = empty_set)))))),
% 0.12/0.40 inference(skolemize,[status(sab)],[27])).
% 0.12/0.40 tff(29,plain,
% 0.12/0.40 ((singleton(A!4) = set_union2(B!3, C!2)) & (~((B!3 = singleton(A!4)) & (C!2 = singleton(A!4)))) & (~((B!3 = empty_set) & (C!2 = singleton(A!4)))) & (~((B!3 = singleton(A!4)) & (C!2 = empty_set)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[28, 17])).
% 0.12/0.40 tff(30,plain,
% 0.12/0.40 (~((B!3 = singleton(A!4)) & (C!2 = singleton(A!4)))),
% 0.12/0.40 inference(and_elim,[status(thm)],[29])).
% 0.12/0.40 tff(31,plain,
% 0.12/0.40 ((~(B!3 = singleton(A!4))) | (~(C!2 = singleton(A!4)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[30, 16])).
% 0.12/0.40 tff(32,plain,
% 0.12/0.40 (~(B!3 = singleton(A!4))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[31, 12])).
% 0.12/0.40 tff(33,plain,
% 0.12/0.40 ((~(~((~(B!3 = empty_set)) | (~(C!2 = singleton(A!4)))))) <=> ((~(B!3 = empty_set)) | (~(C!2 = singleton(A!4))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(34,plain,
% 0.12/0.40 (((B!3 = empty_set) & (C!2 = singleton(A!4))) <=> (~((~(B!3 = empty_set)) | (~(C!2 = singleton(A!4)))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(35,plain,
% 0.12/0.40 ((~((B!3 = empty_set) & (C!2 = singleton(A!4)))) <=> (~(~((~(B!3 = empty_set)) | (~(C!2 = singleton(A!4))))))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[34])).
% 0.12/0.40 tff(36,plain,
% 0.12/0.40 ((~((B!3 = empty_set) & (C!2 = singleton(A!4)))) <=> ((~(B!3 = empty_set)) | (~(C!2 = singleton(A!4))))),
% 0.12/0.40 inference(transitivity,[status(thm)],[35, 33])).
% 0.12/0.40 tff(37,plain,
% 0.12/0.40 (~((B!3 = empty_set) & (C!2 = singleton(A!4)))),
% 0.12/0.40 inference(and_elim,[status(thm)],[29])).
% 0.12/0.40 tff(38,plain,
% 0.12/0.40 ((~(B!3 = empty_set)) | (~(C!2 = singleton(A!4)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[37, 36])).
% 0.12/0.40 tff(39,plain,
% 0.12/0.40 (~(B!3 = empty_set)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[38, 12])).
% 0.12/0.40 tff(40,plain,
% 0.12/0.40 (^[A: $i, B: $i] : refl((subset(A, singleton(B)) <=> ((A = empty_set) | (A = singleton(B)))) <=> (subset(A, singleton(B)) <=> ((A = empty_set) | (A = singleton(B)))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(41,plain,
% 0.12/0.40 (![A: $i, B: $i] : (subset(A, singleton(B)) <=> ((A = empty_set) | (A = singleton(B)))) <=> ![A: $i, B: $i] : (subset(A, singleton(B)) <=> ((A = empty_set) | (A = singleton(B))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[40])).
% 0.12/0.40 tff(42,plain,
% 0.12/0.40 (![A: $i, B: $i] : (subset(A, singleton(B)) <=> ((A = empty_set) | (A = singleton(B)))) <=> ![A: $i, B: $i] : (subset(A, singleton(B)) <=> ((A = empty_set) | (A = singleton(B))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(43,axiom,(![A: $i, B: $i] : (subset(A, singleton(B)) <=> ((A = empty_set) | (A = singleton(B))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','l4_zfmisc_1')).
% 0.12/0.40 tff(44,plain,
% 0.12/0.40 (![A: $i, B: $i] : (subset(A, singleton(B)) <=> ((A = empty_set) | (A = singleton(B))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.12/0.40 tff(45,plain,(
% 0.12/0.40 ![A: $i, B: $i] : (subset(A, singleton(B)) <=> ((A = empty_set) | (A = singleton(B))))),
% 0.12/0.40 inference(skolemize,[status(sab)],[44])).
% 0.12/0.40 tff(46,plain,
% 0.12/0.40 (![A: $i, B: $i] : (subset(A, singleton(B)) <=> ((A = empty_set) | (A = singleton(B))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[45, 41])).
% 0.12/0.40 tff(47,plain,
% 0.12/0.40 ((~![A: $i, B: $i] : (subset(A, singleton(B)) <=> ((A = empty_set) | (A = singleton(B))))) | (subset(B!3, singleton(A!4)) <=> ((B!3 = empty_set) | (B!3 = singleton(A!4))))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(48,plain,
% 0.12/0.40 (subset(B!3, singleton(A!4)) <=> ((B!3 = empty_set) | (B!3 = singleton(A!4)))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[47, 46])).
% 0.12/0.40 tff(49,plain,
% 0.12/0.40 (singleton(A!4) = set_union2(B!3, C!2)),
% 0.12/0.40 inference(and_elim,[status(thm)],[29])).
% 0.12/0.40 tff(50,plain,
% 0.12/0.40 (subset(B!3, singleton(A!4)) <=> subset(B!3, set_union2(B!3, C!2))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[49])).
% 0.12/0.40 tff(51,plain,
% 0.12/0.40 (subset(B!3, set_union2(B!3, C!2)) <=> subset(B!3, singleton(A!4))),
% 0.12/0.40 inference(symmetry,[status(thm)],[50])).
% 0.12/0.40 tff(52,plain,
% 0.12/0.40 (^[A: $i, B: $i] : refl(subset(A, set_union2(A, B)) <=> subset(A, set_union2(A, B)))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(53,plain,
% 0.12/0.40 (![A: $i, B: $i] : subset(A, set_union2(A, B)) <=> ![A: $i, B: $i] : subset(A, set_union2(A, B))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[52])).
% 0.12/0.40 tff(54,plain,
% 0.12/0.40 (![A: $i, B: $i] : subset(A, set_union2(A, B)) <=> ![A: $i, B: $i] : subset(A, set_union2(A, B))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(55,axiom,(![A: $i, B: $i] : subset(A, set_union2(A, B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t7_xboole_1')).
% 0.12/0.40 tff(56,plain,
% 0.12/0.40 (![A: $i, B: $i] : subset(A, set_union2(A, B))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[55, 54])).
% 0.12/0.40 tff(57,plain,(
% 0.12/0.40 ![A: $i, B: $i] : subset(A, set_union2(A, B))),
% 0.12/0.40 inference(skolemize,[status(sab)],[56])).
% 0.12/0.40 tff(58,plain,
% 0.12/0.40 (![A: $i, B: $i] : subset(A, set_union2(A, B))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[57, 53])).
% 0.12/0.40 tff(59,plain,
% 0.12/0.40 ((~![A: $i, B: $i] : subset(A, set_union2(A, B))) | subset(B!3, set_union2(B!3, C!2))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(60,plain,
% 0.12/0.40 (subset(B!3, set_union2(B!3, C!2))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[59, 58])).
% 0.12/0.40 tff(61,plain,
% 0.12/0.40 (subset(B!3, singleton(A!4))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[60, 51])).
% 0.12/0.40 tff(62,plain,
% 0.12/0.40 ((~(subset(B!3, singleton(A!4)) <=> ((B!3 = empty_set) | (B!3 = singleton(A!4))))) | (~subset(B!3, singleton(A!4))) | ((B!3 = empty_set) | (B!3 = singleton(A!4)))),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(63,plain,
% 0.12/0.40 ((B!3 = empty_set) | (B!3 = singleton(A!4))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[62, 61, 48])).
% 0.12/0.40 tff(64,plain,
% 0.12/0.40 ((~((B!3 = empty_set) | (B!3 = singleton(A!4)))) | (B!3 = empty_set) | (B!3 = singleton(A!4))),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(65,plain,
% 0.12/0.40 ((B!3 = empty_set) | (B!3 = singleton(A!4))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[64, 63])).
% 0.12/0.40 tff(66,plain,
% 0.12/0.40 ($false),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[65, 39, 32])).
% 0.20/0.40 tff(67,plain,(~(C!2 = singleton(A!4))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.40 tff(68,plain,
% 0.20/0.40 ((~![A: $i, B: $i] : (subset(A, singleton(B)) <=> ((A = empty_set) | (A = singleton(B))))) | (subset(C!2, singleton(A!4)) <=> ((C!2 = empty_set) | (C!2 = singleton(A!4))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(69,plain,
% 0.20/0.40 (subset(C!2, singleton(A!4)) <=> ((C!2 = empty_set) | (C!2 = singleton(A!4)))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[68, 46])).
% 0.20/0.40 tff(70,plain,
% 0.20/0.40 (^[A: $i, B: $i] : refl((set_union2(A, B) = set_union2(B, A)) <=> (set_union2(A, B) = set_union2(B, A)))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(71,plain,
% 0.20/0.40 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A)) <=> ![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[70])).
% 0.20/0.40 tff(72,plain,
% 0.20/0.40 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A)) <=> ![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(73,axiom,(![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','commutativity_k2_xboole_0')).
% 0.20/0.40 tff(74,plain,
% 0.20/0.40 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[73, 72])).
% 0.20/0.40 tff(75,plain,(
% 0.20/0.40 ![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.20/0.40 inference(skolemize,[status(sab)],[74])).
% 0.20/0.40 tff(76,plain,
% 0.20/0.40 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[75, 71])).
% 0.20/0.40 tff(77,plain,
% 0.20/0.40 ((~![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))) | (set_union2(B!3, C!2) = set_union2(C!2, B!3))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(78,plain,
% 0.20/0.40 (set_union2(B!3, C!2) = set_union2(C!2, B!3)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[77, 76])).
% 0.20/0.40 tff(79,plain,
% 0.20/0.40 (singleton(A!4) = set_union2(C!2, B!3)),
% 0.20/0.40 inference(transitivity,[status(thm)],[49, 78])).
% 0.20/0.40 tff(80,plain,
% 0.20/0.40 (subset(C!2, singleton(A!4)) <=> subset(C!2, set_union2(C!2, B!3))),
% 0.20/0.40 inference(monotonicity,[status(thm)],[79])).
% 0.20/0.40 tff(81,plain,
% 0.20/0.40 (subset(C!2, set_union2(C!2, B!3)) <=> subset(C!2, singleton(A!4))),
% 0.20/0.40 inference(symmetry,[status(thm)],[80])).
% 0.20/0.40 tff(82,plain,
% 0.20/0.40 ((~![A: $i, B: $i] : subset(A, set_union2(A, B))) | subset(C!2, set_union2(C!2, B!3))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(83,plain,
% 0.20/0.40 (subset(C!2, set_union2(C!2, B!3))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[82, 58])).
% 0.20/0.40 tff(84,plain,
% 0.20/0.40 (subset(C!2, singleton(A!4))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[83, 81])).
% 0.20/0.40 tff(85,plain,
% 0.20/0.40 ((~(subset(C!2, singleton(A!4)) <=> ((C!2 = empty_set) | (C!2 = singleton(A!4))))) | (~subset(C!2, singleton(A!4))) | ((C!2 = empty_set) | (C!2 = singleton(A!4)))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(86,plain,
% 0.20/0.40 ((C!2 = empty_set) | (C!2 = singleton(A!4))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[85, 84, 69])).
% 0.20/0.40 tff(87,plain,
% 0.20/0.40 ((~((C!2 = empty_set) | (C!2 = singleton(A!4)))) | (C!2 = empty_set) | (C!2 = singleton(A!4))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(88,plain,
% 0.20/0.40 ((C!2 = empty_set) | (C!2 = singleton(A!4))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[87, 86])).
% 0.20/0.40 tff(89,plain,
% 0.20/0.40 (C!2 = empty_set),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[88, 67])).
% 0.20/0.40 tff(90,plain,
% 0.20/0.40 ((~(~((~(B!3 = singleton(A!4))) | (~(C!2 = empty_set))))) <=> ((~(B!3 = singleton(A!4))) | (~(C!2 = empty_set)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(91,plain,
% 0.20/0.40 (((B!3 = singleton(A!4)) & (C!2 = empty_set)) <=> (~((~(B!3 = singleton(A!4))) | (~(C!2 = empty_set))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(92,plain,
% 0.20/0.40 ((~((B!3 = singleton(A!4)) & (C!2 = empty_set))) <=> (~(~((~(B!3 = singleton(A!4))) | (~(C!2 = empty_set)))))),
% 0.20/0.40 inference(monotonicity,[status(thm)],[91])).
% 0.20/0.40 tff(93,plain,
% 0.20/0.40 ((~((B!3 = singleton(A!4)) & (C!2 = empty_set))) <=> ((~(B!3 = singleton(A!4))) | (~(C!2 = empty_set)))),
% 0.20/0.40 inference(transitivity,[status(thm)],[92, 90])).
% 0.20/0.40 tff(94,plain,
% 0.20/0.40 (~((B!3 = singleton(A!4)) & (C!2 = empty_set))),
% 0.20/0.40 inference(and_elim,[status(thm)],[29])).
% 0.20/0.41 tff(95,plain,
% 0.20/0.41 ((~(B!3 = singleton(A!4))) | (~(C!2 = empty_set))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[94, 93])).
% 0.20/0.41 tff(96,plain,
% 0.20/0.41 (~(B!3 = singleton(A!4))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[95, 89])).
% 0.20/0.41 tff(97,plain,
% 0.20/0.41 (B!3 = empty_set),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[65, 96])).
% 0.20/0.41 tff(98,plain,
% 0.20/0.41 (set_union2(B!3, C!2) = set_union2(empty_set, empty_set)),
% 0.20/0.41 inference(monotonicity,[status(thm)],[97, 89])).
% 0.20/0.41 tff(99,plain,
% 0.20/0.41 (singleton(A!4) = empty_set),
% 0.20/0.41 inference(transitivity,[status(thm)],[49, 98, 11])).
% 0.20/0.41 tff(100,plain,
% 0.20/0.41 (^[A: $i] : refl((~(singleton(A) = empty_set)) <=> (~(singleton(A) = empty_set)))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(101,plain,
% 0.20/0.41 (![A: $i] : (~(singleton(A) = empty_set)) <=> ![A: $i] : (~(singleton(A) = empty_set))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[100])).
% 0.20/0.41 tff(102,plain,
% 0.20/0.41 (![A: $i] : (~(singleton(A) = empty_set)) <=> ![A: $i] : (~(singleton(A) = empty_set))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(103,axiom,(![A: $i] : (~(singleton(A) = empty_set))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','l1_zfmisc_1')).
% 0.20/0.41 tff(104,plain,
% 0.20/0.41 (![A: $i] : (~(singleton(A) = empty_set))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[103, 102])).
% 0.20/0.41 tff(105,plain,(
% 0.20/0.41 ![A: $i] : (~(singleton(A) = empty_set))),
% 0.20/0.41 inference(skolemize,[status(sab)],[104])).
% 0.20/0.41 tff(106,plain,
% 0.20/0.41 (![A: $i] : (~(singleton(A) = empty_set))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[105, 101])).
% 0.20/0.41 tff(107,plain,
% 0.20/0.41 ((~![A: $i] : (~(singleton(A) = empty_set))) | (~(singleton(A!4) = empty_set))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(108,plain,
% 0.20/0.41 (~(singleton(A!4) = empty_set)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[107, 106])).
% 0.20/0.41 tff(109,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[108, 99])).
% 0.20/0.41 % SZS output end Proof
%------------------------------------------------------------------------------