%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET635+3 : TPTP v9.0.0. Released v2.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% 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 : Sat Jun 21 05:26:04 AM UTC 2025
% Result : Theorem 0.12s 0.38s
% Output : Proof 0.12s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.13 % Problem : SET635+3 : TPTP v9.0.0. Released v2.2.0.
% 0.06/0.13 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n015.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 : Fri Jun 20 05:32:26 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.12/0.38 % SZS status Theorem
% 0.12/0.38 % SZS output start Proof
% 0.12/0.38 tff(difference_type, type, (
% 0.12/0.38 difference: ( $i * $i ) > $i)).
% 0.12/0.38 tff(intersection_type, type, (
% 0.12/0.38 intersection: ( $i * $i ) > $i)).
% 0.12/0.38 tff(tptp_fun_D_3_type, type, (
% 0.12/0.38 tptp_fun_D_3: $i)).
% 0.12/0.38 tff(tptp_fun_B_5_type, type, (
% 0.12/0.38 tptp_fun_B_5: $i)).
% 0.12/0.38 tff(tptp_fun_C_4_type, type, (
% 0.12/0.38 tptp_fun_C_4: $i)).
% 0.12/0.38 tff(union_type, type, (
% 0.12/0.38 union: ( $i * $i ) > $i)).
% 0.12/0.38 tff(empty_set_type, type, (
% 0.12/0.38 empty_set: $i)).
% 0.12/0.38 tff(subset_type, type, (
% 0.12/0.38 subset: ( $i * $i ) > $o)).
% 0.12/0.38 tff(1,plain,
% 0.12/0.38 (^[B: $i, C: $i, D: $i] : refl((difference(B, intersection(C, D)) = union(difference(B, C), difference(B, D))) <=> (difference(B, intersection(C, D)) = union(difference(B, C), difference(B, D))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(2,plain,
% 0.12/0.38 (![B: $i, C: $i, D: $i] : (difference(B, intersection(C, D)) = union(difference(B, C), difference(B, D))) <=> ![B: $i, C: $i, D: $i] : (difference(B, intersection(C, D)) = union(difference(B, C), difference(B, D)))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[1])).
% 0.12/0.38 tff(3,plain,
% 0.12/0.38 (![B: $i, C: $i, D: $i] : (difference(B, intersection(C, D)) = union(difference(B, C), difference(B, D))) <=> ![B: $i, C: $i, D: $i] : (difference(B, intersection(C, D)) = union(difference(B, C), difference(B, D)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(4,axiom,(![B: $i, C: $i, D: $i] : (difference(B, intersection(C, D)) = union(difference(B, C), difference(B, D)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','difference_and_intersection_and_union')).
% 0.12/0.38 tff(5,plain,
% 0.12/0.38 (![B: $i, C: $i, D: $i] : (difference(B, intersection(C, D)) = union(difference(B, C), difference(B, D)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.12/0.38 tff(6,plain,(
% 0.12/0.38 ![B: $i, C: $i, D: $i] : (difference(B, intersection(C, D)) = union(difference(B, C), difference(B, D)))),
% 0.12/0.38 inference(skolemize,[status(sab)],[5])).
% 0.12/0.38 tff(7,plain,
% 0.12/0.38 (![B: $i, C: $i, D: $i] : (difference(B, intersection(C, D)) = union(difference(B, C), difference(B, D)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.12/0.38 tff(8,plain,
% 0.12/0.38 ((~![B: $i, C: $i, D: $i] : (difference(B, intersection(C, D)) = union(difference(B, C), difference(B, D)))) | (difference(intersection(B!5, C!4), intersection(B!5, D!3)) = union(difference(intersection(B!5, C!4), B!5), difference(intersection(B!5, C!4), D!3)))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(9,plain,
% 0.12/0.38 (difference(intersection(B!5, C!4), intersection(B!5, D!3)) = union(difference(intersection(B!5, C!4), B!5), difference(intersection(B!5, C!4), D!3))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.12/0.38 tff(10,plain,
% 0.12/0.38 (union(difference(intersection(B!5, C!4), B!5), difference(intersection(B!5, C!4), D!3)) = difference(intersection(B!5, C!4), intersection(B!5, D!3))),
% 0.12/0.38 inference(symmetry,[status(thm)],[9])).
% 0.12/0.38 tff(11,plain,
% 0.12/0.38 (^[B: $i, C: $i, D: $i] : refl((intersection(B, difference(C, D)) = difference(intersection(B, C), D)) <=> (intersection(B, difference(C, D)) = difference(intersection(B, C), D)))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(12,plain,
% 0.12/0.38 (![B: $i, C: $i, D: $i] : (intersection(B, difference(C, D)) = difference(intersection(B, C), D)) <=> ![B: $i, C: $i, D: $i] : (intersection(B, difference(C, D)) = difference(intersection(B, C), D))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[11])).
% 0.12/0.38 tff(13,plain,
% 0.12/0.38 (![B: $i, C: $i, D: $i] : (intersection(B, difference(C, D)) = difference(intersection(B, C), D)) <=> ![B: $i, C: $i, D: $i] : (intersection(B, difference(C, D)) = difference(intersection(B, C), D))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(14,axiom,(![B: $i, C: $i, D: $i] : (intersection(B, difference(C, D)) = difference(intersection(B, C), D))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','difference_and_intersection')).
% 0.12/0.38 tff(15,plain,
% 0.12/0.38 (![B: $i, C: $i, D: $i] : (intersection(B, difference(C, D)) = difference(intersection(B, C), D))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.12/0.38 tff(16,plain,(
% 0.12/0.38 ![B: $i, C: $i, D: $i] : (intersection(B, difference(C, D)) = difference(intersection(B, C), D))),
% 0.12/0.38 inference(skolemize,[status(sab)],[15])).
% 0.12/0.39 tff(17,plain,
% 0.12/0.39 (![B: $i, C: $i, D: $i] : (intersection(B, difference(C, D)) = difference(intersection(B, C), D))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.12/0.39 tff(18,plain,
% 0.12/0.39 ((~![B: $i, C: $i, D: $i] : (intersection(B, difference(C, D)) = difference(intersection(B, C), D))) | (intersection(B!5, difference(C!4, D!3)) = difference(intersection(B!5, C!4), D!3))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(19,plain,
% 0.12/0.39 (intersection(B!5, difference(C!4, D!3)) = difference(intersection(B!5, C!4), D!3)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[18, 17])).
% 0.12/0.39 tff(20,plain,
% 0.12/0.39 (union(difference(intersection(B!5, C!4), B!5), intersection(B!5, difference(C!4, D!3))) = union(difference(intersection(B!5, C!4), B!5), difference(intersection(B!5, C!4), D!3))),
% 0.12/0.39 inference(monotonicity,[status(thm)],[19])).
% 0.12/0.39 tff(21,plain,
% 0.12/0.39 (^[B: $i, C: $i] : refl((union(B, C) = union(C, B)) <=> (union(B, C) = union(C, B)))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(22,plain,
% 0.12/0.39 (![B: $i, C: $i] : (union(B, C) = union(C, B)) <=> ![B: $i, C: $i] : (union(B, C) = union(C, B))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[21])).
% 0.12/0.39 tff(23,plain,
% 0.12/0.39 (![B: $i, C: $i] : (union(B, C) = union(C, B)) <=> ![B: $i, C: $i] : (union(B, C) = union(C, B))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(24,axiom,(![B: $i, C: $i] : (union(B, C) = union(C, B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','commutativity_of_union')).
% 0.12/0.39 tff(25,plain,
% 0.12/0.39 (![B: $i, C: $i] : (union(B, C) = union(C, B))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[24, 23])).
% 0.12/0.39 tff(26,plain,(
% 0.12/0.39 ![B: $i, C: $i] : (union(B, C) = union(C, B))),
% 0.12/0.39 inference(skolemize,[status(sab)],[25])).
% 0.12/0.39 tff(27,plain,
% 0.12/0.39 (![B: $i, C: $i] : (union(B, C) = union(C, B))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[26, 22])).
% 0.12/0.39 tff(28,plain,
% 0.12/0.39 ((~![B: $i, C: $i] : (union(B, C) = union(C, B))) | (union(intersection(B!5, difference(C!4, D!3)), difference(intersection(B!5, C!4), B!5)) = union(difference(intersection(B!5, C!4), B!5), intersection(B!5, difference(C!4, D!3))))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(29,plain,
% 0.12/0.39 (union(intersection(B!5, difference(C!4, D!3)), difference(intersection(B!5, C!4), B!5)) = union(difference(intersection(B!5, C!4), B!5), intersection(B!5, difference(C!4, D!3)))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[28, 27])).
% 0.12/0.39 tff(30,plain,
% 0.12/0.39 (^[B: $i, C: $i] : refl(((difference(B, C) = empty_set) <=> subset(B, C)) <=> ((difference(B, C) = empty_set) <=> subset(B, C)))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(31,plain,
% 0.12/0.39 (![B: $i, C: $i] : ((difference(B, C) = empty_set) <=> subset(B, C)) <=> ![B: $i, C: $i] : ((difference(B, C) = empty_set) <=> subset(B, C))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[30])).
% 0.12/0.39 tff(32,plain,
% 0.12/0.39 (![B: $i, C: $i] : ((difference(B, C) = empty_set) <=> subset(B, C)) <=> ![B: $i, C: $i] : ((difference(B, C) = empty_set) <=> subset(B, C))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(33,axiom,(![B: $i, C: $i] : ((difference(B, C) = empty_set) <=> subset(B, C))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','difference_empty_set')).
% 0.12/0.39 tff(34,plain,
% 0.12/0.39 (![B: $i, C: $i] : ((difference(B, C) = empty_set) <=> subset(B, C))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[33, 32])).
% 0.12/0.39 tff(35,plain,(
% 0.12/0.39 ![B: $i, C: $i] : ((difference(B, C) = empty_set) <=> subset(B, C))),
% 0.12/0.39 inference(skolemize,[status(sab)],[34])).
% 0.12/0.39 tff(36,plain,
% 0.12/0.39 (![B: $i, C: $i] : ((difference(B, C) = empty_set) <=> subset(B, C))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[35, 31])).
% 0.12/0.39 tff(37,plain,
% 0.12/0.39 ((~![B: $i, C: $i] : ((difference(B, C) = empty_set) <=> subset(B, C))) | ((difference(intersection(B!5, C!4), B!5) = empty_set) <=> subset(intersection(B!5, C!4), B!5))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(38,plain,
% 0.12/0.39 ((difference(intersection(B!5, C!4), B!5) = empty_set) <=> subset(intersection(B!5, C!4), B!5)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[37, 36])).
% 0.12/0.39 tff(39,plain,
% 0.12/0.39 (^[B: $i, C: $i] : refl(subset(intersection(B, C), B) <=> subset(intersection(B, C), B))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(40,plain,
% 0.12/0.39 (![B: $i, C: $i] : subset(intersection(B, C), B) <=> ![B: $i, C: $i] : subset(intersection(B, C), B)),
% 0.12/0.39 inference(quant_intro,[status(thm)],[39])).
% 0.12/0.39 tff(41,plain,
% 0.12/0.39 (![B: $i, C: $i] : subset(intersection(B, C), B) <=> ![B: $i, C: $i] : subset(intersection(B, C), B)),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(42,axiom,(![B: $i, C: $i] : subset(intersection(B, C), B)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','intersection_is_subset')).
% 0.12/0.39 tff(43,plain,
% 0.12/0.39 (![B: $i, C: $i] : subset(intersection(B, C), B)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.12/0.39 tff(44,plain,(
% 0.12/0.39 ![B: $i, C: $i] : subset(intersection(B, C), B)),
% 0.12/0.39 inference(skolemize,[status(sab)],[43])).
% 0.12/0.39 tff(45,plain,
% 0.12/0.39 (![B: $i, C: $i] : subset(intersection(B, C), B)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[44, 40])).
% 0.12/0.39 tff(46,plain,
% 0.12/0.39 ((~![B: $i, C: $i] : subset(intersection(B, C), B)) | subset(intersection(B!5, C!4), B!5)),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(47,plain,
% 0.12/0.39 (subset(intersection(B!5, C!4), B!5)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[46, 45])).
% 0.12/0.39 tff(48,plain,
% 0.12/0.39 ((~((difference(intersection(B!5, C!4), B!5) = empty_set) <=> subset(intersection(B!5, C!4), B!5))) | (difference(intersection(B!5, C!4), B!5) = empty_set) | (~subset(intersection(B!5, C!4), B!5))),
% 0.12/0.39 inference(tautology,[status(thm)],[])).
% 0.12/0.39 tff(49,plain,
% 0.12/0.39 ((~((difference(intersection(B!5, C!4), B!5) = empty_set) <=> subset(intersection(B!5, C!4), B!5))) | (difference(intersection(B!5, C!4), B!5) = empty_set)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[48, 47])).
% 0.12/0.39 tff(50,plain,
% 0.12/0.39 (difference(intersection(B!5, C!4), B!5) = empty_set),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[49, 38])).
% 0.12/0.39 tff(51,plain,
% 0.12/0.39 (empty_set = difference(intersection(B!5, C!4), B!5)),
% 0.12/0.39 inference(symmetry,[status(thm)],[50])).
% 0.12/0.39 tff(52,plain,
% 0.12/0.39 (union(intersection(B!5, difference(C!4, D!3)), empty_set) = union(intersection(B!5, difference(C!4, D!3)), difference(intersection(B!5, C!4), B!5))),
% 0.12/0.39 inference(monotonicity,[status(thm)],[51])).
% 0.12/0.39 tff(53,plain,
% 0.12/0.39 (^[B: $i] : refl((union(B, empty_set) = B) <=> (union(B, empty_set) = B))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(54,plain,
% 0.12/0.39 (![B: $i] : (union(B, empty_set) = B) <=> ![B: $i] : (union(B, empty_set) = B)),
% 0.12/0.39 inference(quant_intro,[status(thm)],[53])).
% 0.12/0.39 tff(55,plain,
% 0.12/0.39 (![B: $i] : (union(B, empty_set) = B) <=> ![B: $i] : (union(B, empty_set) = B)),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(56,axiom,(![B: $i] : (union(B, empty_set) = B)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','union_empty_set')).
% 0.12/0.39 tff(57,plain,
% 0.12/0.39 (![B: $i] : (union(B, empty_set) = B)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[56, 55])).
% 0.12/0.39 tff(58,plain,(
% 0.12/0.39 ![B: $i] : (union(B, empty_set) = B)),
% 0.12/0.39 inference(skolemize,[status(sab)],[57])).
% 0.12/0.39 tff(59,plain,
% 0.12/0.39 (![B: $i] : (union(B, empty_set) = B)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[58, 54])).
% 0.12/0.39 tff(60,plain,
% 0.12/0.39 ((~![B: $i] : (union(B, empty_set) = B)) | (union(intersection(B!5, difference(C!4, D!3)), empty_set) = intersection(B!5, difference(C!4, D!3)))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(61,plain,
% 0.12/0.39 (union(intersection(B!5, difference(C!4, D!3)), empty_set) = intersection(B!5, difference(C!4, D!3))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[60, 59])).
% 0.12/0.39 tff(62,plain,
% 0.12/0.39 (intersection(B!5, difference(C!4, D!3)) = union(intersection(B!5, difference(C!4, D!3)), empty_set)),
% 0.12/0.39 inference(symmetry,[status(thm)],[61])).
% 0.12/0.39 tff(63,plain,
% 0.12/0.39 (intersection(B!5, difference(C!4, D!3)) = difference(intersection(B!5, C!4), intersection(B!5, D!3))),
% 0.12/0.39 inference(transitivity,[status(thm)],[62, 52, 29, 20, 10])).
% 0.12/0.39 tff(64,plain,
% 0.12/0.39 ((~![B: $i, C: $i, D: $i] : (intersection(B, difference(C, D)) = difference(intersection(B, C), intersection(B, D)))) <=> (~![B: $i, C: $i, D: $i] : (intersection(B, difference(C, D)) = difference(intersection(B, C), intersection(B, D))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(65,axiom,(~![B: $i, C: $i, D: $i] : (intersection(B, difference(C, D)) = difference(intersection(B, C), intersection(B, D)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_th117')).
% 0.12/0.39 tff(66,plain,
% 0.12/0.39 (~![B: $i, C: $i, D: $i] : (intersection(B, difference(C, D)) = difference(intersection(B, C), intersection(B, D)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.12/0.39 tff(67,plain,
% 0.12/0.39 (~![B: $i, C: $i, D: $i] : (intersection(B, difference(C, D)) = difference(intersection(B, C), intersection(B, D)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[66, 64])).
% 0.12/0.39 tff(68,plain,
% 0.12/0.39 (~![B: $i, C: $i, D: $i] : (intersection(B, difference(C, D)) = difference(intersection(B, C), intersection(B, D)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[67, 64])).
% 0.12/0.39 tff(69,plain,
% 0.12/0.39 (~![B: $i, C: $i, D: $i] : (intersection(B, difference(C, D)) = difference(intersection(B, C), intersection(B, D)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[68, 64])).
% 0.12/0.39 tff(70,plain,(
% 0.12/0.39 ~(intersection(B!5, difference(C!4, D!3)) = difference(intersection(B!5, C!4), intersection(B!5, D!3)))),
% 0.12/0.39 inference(skolemize,[status(sab)],[69])).
% 0.12/0.39 tff(71,plain,
% 0.12/0.39 ($false),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[70, 63])).
% 0.12/0.39 % SZS output end Proof
% 0.12/0.39 % E exiting
%------------------------------------------------------------------------------