TSTP Solution File: SET985+1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET985+1 : TPTP v8.1.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n015.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 05:08:49 EDT 2022
% Result : Theorem 0.13s 0.40s
% Output : Proof 0.13s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET985+1 : TPTP v8.1.0. Released v3.2.0.
% 0.12/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.35 % Computer : n015.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Sat Sep 3 09:02:55 EDT 2022
% 0.13/0.35 % 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.
% 0.13/0.40 % SZS status Theorem
% 0.13/0.40 % SZS output start Proof
% 0.13/0.40 tff(subset_type, type, (
% 0.13/0.40 subset: ( $i * $i ) > $o)).
% 0.13/0.40 tff(tptp_fun_D_3_type, type, (
% 0.13/0.40 tptp_fun_D_3: $i)).
% 0.13/0.40 tff(tptp_fun_B_5_type, type, (
% 0.13/0.40 tptp_fun_B_5: $i)).
% 0.13/0.40 tff(tptp_fun_C_4_type, type, (
% 0.13/0.40 tptp_fun_C_4: $i)).
% 0.13/0.40 tff(tptp_fun_A_2_type, type, (
% 0.13/0.40 tptp_fun_A_2: $i)).
% 0.13/0.40 tff(cartesian_product2_type, type, (
% 0.13/0.40 cartesian_product2: ( $i * $i ) > $i)).
% 0.13/0.40 tff(empty_type, type, (
% 0.13/0.40 empty: $i > $o)).
% 0.13/0.40 tff(empty_set_type, type, (
% 0.13/0.40 empty_set: $i)).
% 0.13/0.40 tff(1,plain,
% 0.13/0.40 ((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) <=> (~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3))))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(2,plain,
% 0.13/0.40 (((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) | subset(B!5, D!3)) <=> ((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) | subset(B!5, D!3))),
% 0.13/0.40 inference(monotonicity,[status(thm)],[1])).
% 0.13/0.40 tff(3,plain,
% 0.13/0.40 ((~((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) | subset(B!5, D!3))) <=> (~((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) | subset(B!5, D!3)))),
% 0.13/0.40 inference(monotonicity,[status(thm)],[2])).
% 0.13/0.40 tff(4,plain,
% 0.13/0.40 (((~empty(A!2)) & (~((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) | subset(B!5, D!3)))) <=> ((~empty(A!2)) & (~((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) | subset(B!5, D!3))))),
% 0.13/0.40 inference(monotonicity,[status(thm)],[3])).
% 0.13/0.40 tff(5,plain,
% 0.13/0.40 ((~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D)))) <=> (~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D))))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(6,plain,
% 0.13/0.40 ((~![A: $i] : ((~empty(A)) => ![B: $i, C: $i, D: $i] : ((subset(cartesian_product2(A, B), cartesian_product2(C, D)) | subset(cartesian_product2(B, A), cartesian_product2(D, C))) => subset(B, D)))) <=> (~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D))))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(7,axiom,(~![A: $i] : ((~empty(A)) => ![B: $i, C: $i, D: $i] : ((subset(cartesian_product2(A, B), cartesian_product2(C, D)) | subset(cartesian_product2(B, A), cartesian_product2(D, C))) => subset(B, D)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t139_zfmisc_1')).
% 0.13/0.40 tff(8,plain,
% 0.13/0.40 (~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[7, 6])).
% 0.13/0.40 tff(9,plain,
% 0.13/0.40 (~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[8, 5])).
% 0.13/0.40 tff(10,plain,
% 0.13/0.40 (~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[9, 5])).
% 0.13/0.40 tff(11,plain,
% 0.13/0.40 (~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[10, 5])).
% 0.13/0.40 tff(12,plain,
% 0.13/0.40 (~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[11, 5])).
% 0.13/0.40 tff(13,plain,
% 0.13/0.40 (~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[12, 5])).
% 0.13/0.40 tff(14,plain,
% 0.13/0.40 (~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[13, 5])).
% 0.13/0.40 tff(15,plain,
% 0.13/0.40 ((~empty(A!2)) & (~((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) | subset(B!5, D!3)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[14, 4])).
% 0.13/0.40 tff(16,plain,
% 0.13/0.40 (~((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) | subset(B!5, D!3))),
% 0.13/0.40 inference(and_elim,[status(thm)],[15])).
% 0.13/0.40 tff(17,plain,
% 0.13/0.40 (~subset(B!5, D!3)),
% 0.13/0.40 inference(or_elim,[status(thm)],[16])).
% 0.13/0.40 tff(18,plain,
% 0.13/0.40 (((~subset(A!2, C!4)) | (~subset(B!5, D!3))) | subset(B!5, D!3)),
% 0.13/0.40 inference(tautology,[status(thm)],[])).
% 0.13/0.40 tff(19,plain,
% 0.13/0.40 ((~subset(A!2, C!4)) | (~subset(B!5, D!3))),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[18, 17])).
% 0.13/0.40 tff(20,assumption,(~((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))), introduced(assumption)).
% 0.13/0.40 tff(21,plain,
% 0.13/0.40 (^[A: $i, B: $i] : refl(((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set))) <=> ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set))))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(22,plain,
% 0.13/0.40 (![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set))) <=> ![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[21])).
% 0.13/0.40 tff(23,plain,
% 0.13/0.40 (![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set))) <=> ![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(24,plain,
% 0.13/0.40 (^[A: $i, B: $i] : rewrite(((cartesian_product2(A, B) = empty_set) <=> ((A = empty_set) | (B = empty_set))) <=> ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set))))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(25,plain,
% 0.13/0.40 (![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((A = empty_set) | (B = empty_set))) <=> ![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[24])).
% 0.13/0.40 tff(26,axiom,(![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((A = empty_set) | (B = empty_set)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t113_zfmisc_1')).
% 0.13/0.40 tff(27,plain,
% 0.13/0.40 (![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.13/0.40 tff(28,plain,
% 0.13/0.40 (![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[27, 23])).
% 0.13/0.40 tff(29,plain,(
% 0.13/0.40 ![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))),
% 0.13/0.40 inference(skolemize,[status(sab)],[28])).
% 0.13/0.40 tff(30,plain,
% 0.13/0.40 (![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[29, 22])).
% 0.13/0.40 tff(31,plain,
% 0.13/0.40 (((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))) <=> ((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set))))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(32,plain,
% 0.13/0.40 (((cartesian_product2(A!2, B!5) = empty_set) <=> ((B!5 = empty_set) | (A!2 = empty_set))) <=> ((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(33,plain,
% 0.13/0.40 (((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(A!2, B!5) = empty_set) <=> ((B!5 = empty_set) | (A!2 = empty_set)))) <=> ((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set))))),
% 0.13/0.40 inference(monotonicity,[status(thm)],[32])).
% 0.13/0.40 tff(34,plain,
% 0.13/0.40 (((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(A!2, B!5) = empty_set) <=> ((B!5 = empty_set) | (A!2 = empty_set)))) <=> ((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set))))),
% 0.13/0.40 inference(transitivity,[status(thm)],[33, 31])).
% 0.13/0.40 tff(35,plain,
% 0.13/0.40 ((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(A!2, B!5) = empty_set) <=> ((B!5 = empty_set) | (A!2 = empty_set)))),
% 0.13/0.40 inference(quant_inst,[status(thm)],[])).
% 0.13/0.40 tff(36,plain,
% 0.13/0.40 ((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.13/0.40 tff(37,plain,
% 0.13/0.40 ($false),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[36, 30, 20])).
% 0.13/0.40 tff(38,plain,((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set))), inference(lemma,lemma(discharge,[]))).
% 0.13/0.40 tff(39,assumption,(B!5 = empty_set), introduced(assumption)).
% 0.13/0.40 tff(40,plain,
% 0.13/0.40 (empty_set = B!5),
% 0.13/0.40 inference(symmetry,[status(thm)],[39])).
% 0.13/0.40 tff(41,plain,
% 0.13/0.40 (subset(empty_set, D!3) <=> subset(B!5, D!3)),
% 0.13/0.40 inference(monotonicity,[status(thm)],[40])).
% 0.13/0.40 tff(42,plain,
% 0.13/0.40 (subset(B!5, D!3) <=> subset(empty_set, D!3)),
% 0.13/0.40 inference(symmetry,[status(thm)],[41])).
% 0.13/0.40 tff(43,plain,
% 0.13/0.40 ((~subset(B!5, D!3)) <=> (~subset(empty_set, D!3))),
% 0.13/0.40 inference(monotonicity,[status(thm)],[42])).
% 0.13/0.40 tff(44,plain,
% 0.13/0.40 (~subset(empty_set, D!3)),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[17, 43])).
% 0.13/0.40 tff(45,plain,
% 0.13/0.40 (^[A: $i] : refl(subset(empty_set, A) <=> subset(empty_set, A))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(46,plain,
% 0.13/0.40 (![A: $i] : subset(empty_set, A) <=> ![A: $i] : subset(empty_set, A)),
% 0.13/0.40 inference(quant_intro,[status(thm)],[45])).
% 0.13/0.40 tff(47,plain,
% 0.13/0.40 (![A: $i] : subset(empty_set, A) <=> ![A: $i] : subset(empty_set, A)),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(48,axiom,(![A: $i] : subset(empty_set, A)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t2_xboole_1')).
% 0.13/0.40 tff(49,plain,
% 0.13/0.40 (![A: $i] : subset(empty_set, A)),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[48, 47])).
% 0.13/0.40 tff(50,plain,(
% 0.13/0.40 ![A: $i] : subset(empty_set, A)),
% 0.13/0.40 inference(skolemize,[status(sab)],[49])).
% 0.13/0.40 tff(51,plain,
% 0.13/0.40 (![A: $i] : subset(empty_set, A)),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[50, 46])).
% 0.13/0.40 tff(52,plain,
% 0.13/0.40 ((~![A: $i] : subset(empty_set, A)) | subset(empty_set, D!3)),
% 0.13/0.40 inference(quant_inst,[status(thm)],[])).
% 0.13/0.40 tff(53,plain,
% 0.13/0.40 (subset(empty_set, D!3)),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[52, 51])).
% 0.13/0.40 tff(54,plain,
% 0.13/0.41 ($false),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[53, 44])).
% 0.13/0.41 tff(55,plain,(~(B!5 = empty_set)), inference(lemma,lemma(discharge,[]))).
% 0.13/0.41 tff(56,assumption,(A!2 = empty_set), introduced(assumption)).
% 0.13/0.41 tff(57,plain,
% 0.13/0.41 (empty(A!2) <=> empty(empty_set)),
% 0.13/0.41 inference(monotonicity,[status(thm)],[56])).
% 0.13/0.41 tff(58,plain,
% 0.13/0.41 (empty(empty_set) <=> empty(A!2)),
% 0.13/0.41 inference(symmetry,[status(thm)],[57])).
% 0.13/0.41 tff(59,plain,
% 0.13/0.41 (empty(empty_set) <=> empty(empty_set)),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(60,axiom,(empty(empty_set)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','fc1_xboole_0')).
% 0.13/0.41 tff(61,plain,
% 0.13/0.41 (empty(empty_set)),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[60, 59])).
% 0.13/0.41 tff(62,plain,
% 0.13/0.41 (empty(A!2)),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[61, 58])).
% 0.13/0.41 tff(63,plain,
% 0.13/0.41 (~empty(A!2)),
% 0.13/0.41 inference(and_elim,[status(thm)],[15])).
% 0.13/0.41 tff(64,plain,
% 0.13/0.41 ($false),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[63, 62])).
% 0.13/0.41 tff(65,plain,(~(A!2 = empty_set)), inference(lemma,lemma(discharge,[]))).
% 0.13/0.41 tff(66,plain,
% 0.13/0.41 ((~((A!2 = empty_set) | (B!5 = empty_set))) | (A!2 = empty_set) | (B!5 = empty_set)),
% 0.13/0.41 inference(tautology,[status(thm)],[])).
% 0.13/0.41 tff(67,plain,
% 0.13/0.41 (~((A!2 = empty_set) | (B!5 = empty_set))),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[66, 65, 55])).
% 0.13/0.41 tff(68,plain,
% 0.13/0.41 ((~((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))) | (~(cartesian_product2(A!2, B!5) = empty_set)) | ((A!2 = empty_set) | (B!5 = empty_set))),
% 0.13/0.41 inference(tautology,[status(thm)],[])).
% 0.13/0.41 tff(69,plain,
% 0.13/0.41 ((~((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))) | (~(cartesian_product2(A!2, B!5) = empty_set))),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[68, 67])).
% 0.13/0.41 tff(70,plain,
% 0.13/0.41 (~(cartesian_product2(A!2, B!5) = empty_set)),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[69, 38])).
% 0.13/0.41 tff(71,plain,
% 0.13/0.41 (((~subset(B!5, D!3)) | (~subset(A!2, C!4))) | subset(B!5, D!3)),
% 0.13/0.41 inference(tautology,[status(thm)],[])).
% 0.13/0.41 tff(72,plain,
% 0.13/0.41 ((~subset(B!5, D!3)) | (~subset(A!2, C!4))),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[71, 17])).
% 0.13/0.41 tff(73,assumption,(~((cartesian_product2(B!5, A!2) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))), introduced(assumption)).
% 0.13/0.41 tff(74,plain,
% 0.13/0.41 ((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(B!5, A!2) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))),
% 0.13/0.41 inference(quant_inst,[status(thm)],[])).
% 0.13/0.41 tff(75,plain,
% 0.13/0.41 ($false),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[74, 30, 73])).
% 0.13/0.41 tff(76,plain,((cartesian_product2(B!5, A!2) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set))), inference(lemma,lemma(discharge,[]))).
% 0.13/0.41 tff(77,plain,
% 0.13/0.41 ((~((cartesian_product2(B!5, A!2) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))) | (~(cartesian_product2(B!5, A!2) = empty_set)) | ((A!2 = empty_set) | (B!5 = empty_set))),
% 0.13/0.41 inference(tautology,[status(thm)],[])).
% 0.13/0.41 tff(78,plain,
% 0.13/0.41 ((~((cartesian_product2(B!5, A!2) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))) | (~(cartesian_product2(B!5, A!2) = empty_set))),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[77, 67])).
% 0.13/0.41 tff(79,plain,
% 0.13/0.41 (~(cartesian_product2(B!5, A!2) = empty_set)),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[78, 76])).
% 0.13/0.41 tff(80,assumption,(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4))), introduced(assumption)).
% 0.13/0.41 tff(81,plain,
% 0.13/0.41 (^[A: $i, B: $i, C: $i, D: $i] : refl(((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D))))) <=> ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D))))))),
% 0.13/0.41 inference(bind,[status(th)],[])).
% 0.13/0.41 tff(82,plain,
% 0.13/0.41 (![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D))))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))),
% 0.13/0.41 inference(quant_intro,[status(thm)],[81])).
% 0.13/0.41 tff(83,plain,
% 0.13/0.41 (^[A: $i, B: $i, C: $i, D: $i] : trans(monotonicity(rewrite((subset(A, C) & subset(B, D)) <=> (~((~subset(A, C)) | (~subset(B, D))))), (((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D)))) <=> ((~((~subset(A, C)) | (~subset(B, D)))) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D)))))), rewrite(((~((~subset(A, C)) | (~subset(B, D)))) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D)))) <=> ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))), (((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D)))) <=> ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))))),
% 0.13/0.41 inference(bind,[status(th)],[])).
% 0.13/0.41 tff(84,plain,
% 0.13/0.41 (![A: $i, B: $i, C: $i, D: $i] : ((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D)))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))),
% 0.13/0.41 inference(quant_intro,[status(thm)],[83])).
% 0.13/0.41 tff(85,plain,
% 0.13/0.41 (![A: $i, B: $i, C: $i, D: $i] : ((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D)))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(86,plain,
% 0.13/0.41 (^[A: $i, B: $i, C: $i, D: $i] : rewrite((subset(cartesian_product2(A, B), cartesian_product2(C, D)) => ((cartesian_product2(A, B) = empty_set) | (subset(A, C) & subset(B, D)))) <=> ((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D)))))),
% 0.13/0.41 inference(bind,[status(th)],[])).
% 0.13/0.41 tff(87,plain,
% 0.13/0.41 (![A: $i, B: $i, C: $i, D: $i] : (subset(cartesian_product2(A, B), cartesian_product2(C, D)) => ((cartesian_product2(A, B) = empty_set) | (subset(A, C) & subset(B, D)))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))))),
% 0.13/0.41 inference(quant_intro,[status(thm)],[86])).
% 0.13/0.41 tff(88,axiom,(![A: $i, B: $i, C: $i, D: $i] : (subset(cartesian_product2(A, B), cartesian_product2(C, D)) => ((cartesian_product2(A, B) = empty_set) | (subset(A, C) & subset(B, D))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t138_zfmisc_1')).
% 0.13/0.41 tff(89,plain,
% 0.13/0.41 (![A: $i, B: $i, C: $i, D: $i] : ((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.13/0.41 tff(90,plain,
% 0.13/0.41 (![A: $i, B: $i, C: $i, D: $i] : ((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[89, 85])).
% 0.13/0.41 tff(91,plain,(
% 0.13/0.41 ![A: $i, B: $i, C: $i, D: $i] : ((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))))),
% 0.13/0.41 inference(skolemize,[status(sab)],[90])).
% 0.13/0.41 tff(92,plain,
% 0.13/0.41 (![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[91, 84])).
% 0.13/0.41 tff(93,plain,
% 0.13/0.41 (![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[92, 82])).
% 0.13/0.41 tff(94,plain,
% 0.13/0.41 (((~![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))) | ((cartesian_product2(B!5, A!2) = empty_set) | (~subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4))) | (~((~subset(B!5, D!3)) | (~subset(A!2, C!4)))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))) | (cartesian_product2(B!5, A!2) = empty_set) | (~subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4))) | (~((~subset(B!5, D!3)) | (~subset(A!2, C!4)))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(95,plain,
% 0.13/0.41 ((~![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))) | ((cartesian_product2(B!5, A!2) = empty_set) | (~subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4))) | (~((~subset(B!5, D!3)) | (~subset(A!2, C!4)))))),
% 0.13/0.41 inference(quant_inst,[status(thm)],[])).
% 0.13/0.41 tff(96,plain,
% 0.13/0.41 ((~![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))) | (cartesian_product2(B!5, A!2) = empty_set) | (~subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4))) | (~((~subset(B!5, D!3)) | (~subset(A!2, C!4))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[95, 94])).
% 0.13/0.41 tff(97,plain,
% 0.13/0.41 ($false),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[96, 93, 80, 79, 72])).
% 0.13/0.41 tff(98,plain,(~subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4))), inference(lemma,lemma(discharge,[]))).
% 0.13/0.41 tff(99,plain,
% 0.13/0.41 (subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3))),
% 0.13/0.41 inference(or_elim,[status(thm)],[16])).
% 0.13/0.41 tff(100,plain,
% 0.13/0.41 (subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3))),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[99, 98])).
% 0.13/0.41 tff(101,plain,
% 0.13/0.41 (((~![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))) | ((cartesian_product2(A!2, B!5) = empty_set) | (~subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3))) | (~((~subset(A!2, C!4)) | (~subset(B!5, D!3)))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))) | (cartesian_product2(A!2, B!5) = empty_set) | (~subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3))) | (~((~subset(A!2, C!4)) | (~subset(B!5, D!3)))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(102,plain,
% 0.13/0.41 ((~![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))) | ((cartesian_product2(A!2, B!5) = empty_set) | (~subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3))) | (~((~subset(A!2, C!4)) | (~subset(B!5, D!3)))))),
% 0.13/0.41 inference(quant_inst,[status(thm)],[])).
% 0.13/0.41 tff(103,plain,
% 0.13/0.41 ((~![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))) | (cartesian_product2(A!2, B!5) = empty_set) | (~subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3))) | (~((~subset(A!2, C!4)) | (~subset(B!5, D!3))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.13/0.41 tff(104,plain,
% 0.13/0.41 ($false),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[103, 93, 100, 70, 19])).
% 0.13/0.41 % SZS output end Proof
%------------------------------------------------------------------------------