%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET617+3 : TPTP v9.0.0. Released v2.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% 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 Jun 21 05:26:01 AM UTC 2025
% Result : Theorem 0.11s 0.35s
% Output : Proof 0.11s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.11 % Problem : SET617+3 : TPTP v9.0.0. Released v2.2.0.
% 0.06/0.11 % Command : run_E %s %d THM
% 0.11/0.31 % Computer : n014.cluster.edu
% 0.11/0.31 % Model : x86_64 x86_64
% 0.11/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.31 % Memory : 8042.1875MB
% 0.11/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.31 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Fri Jun 20 05:31:39 EDT 2025
% 0.11/0.32 % CPUTime :
% 0.11/0.35 % SZS status Theorem
% 0.11/0.35 % SZS output start Proof
% 0.11/0.35 tff(tptp_fun_B_3_type, type, (
% 0.11/0.35 tptp_fun_B_3: $i)).
% 0.11/0.35 tff(symmetric_difference_type, type, (
% 0.11/0.35 symmetric_difference: ( $i * $i ) > $i)).
% 0.11/0.35 tff(empty_set_type, type, (
% 0.11/0.35 empty_set: $i)).
% 0.11/0.35 tff(union_type, type, (
% 0.11/0.35 union: ( $i * $i ) > $i)).
% 0.11/0.35 tff(difference_type, type, (
% 0.11/0.35 difference: ( $i * $i ) > $i)).
% 0.11/0.35 tff(1,plain,
% 0.11/0.35 (^[B: $i] : refl((union(B, empty_set) = B) <=> (union(B, empty_set) = B))),
% 0.11/0.35 inference(bind,[status(th)],[])).
% 0.11/0.35 tff(2,plain,
% 0.11/0.35 (![B: $i] : (union(B, empty_set) = B) <=> ![B: $i] : (union(B, empty_set) = B)),
% 0.11/0.35 inference(quant_intro,[status(thm)],[1])).
% 0.11/0.35 tff(3,plain,
% 0.11/0.35 (![B: $i] : (union(B, empty_set) = B) <=> ![B: $i] : (union(B, empty_set) = B)),
% 0.11/0.35 inference(rewrite,[status(thm)],[])).
% 0.11/0.35 tff(4,axiom,(![B: $i] : (union(B, empty_set) = B)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','union_empty_set')).
% 0.11/0.35 tff(5,plain,
% 0.11/0.35 (![B: $i] : (union(B, empty_set) = B)),
% 0.11/0.35 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.11/0.35 tff(6,plain,(
% 0.11/0.35 ![B: $i] : (union(B, empty_set) = B)),
% 0.11/0.35 inference(skolemize,[status(sab)],[5])).
% 0.11/0.35 tff(7,plain,
% 0.11/0.35 (![B: $i] : (union(B, empty_set) = B)),
% 0.11/0.35 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.11/0.35 tff(8,plain,
% 0.11/0.35 ((~![B: $i] : (union(B, empty_set) = B)) | (union(B!3, empty_set) = B!3)),
% 0.11/0.35 inference(quant_inst,[status(thm)],[])).
% 0.11/0.35 tff(9,plain,
% 0.11/0.35 (union(B!3, empty_set) = B!3),
% 0.11/0.35 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.11/0.35 tff(10,plain,
% 0.11/0.35 (^[B: $i] : refl((difference(empty_set, B) = empty_set) <=> (difference(empty_set, B) = empty_set))),
% 0.11/0.35 inference(bind,[status(th)],[])).
% 0.11/0.35 tff(11,plain,
% 0.11/0.35 (![B: $i] : (difference(empty_set, B) = empty_set) <=> ![B: $i] : (difference(empty_set, B) = empty_set)),
% 0.11/0.35 inference(quant_intro,[status(thm)],[10])).
% 0.11/0.35 tff(12,plain,
% 0.11/0.35 (![B: $i] : (difference(empty_set, B) = empty_set) <=> ![B: $i] : (difference(empty_set, B) = empty_set)),
% 0.11/0.35 inference(rewrite,[status(thm)],[])).
% 0.11/0.35 tff(13,axiom,(![B: $i] : (difference(empty_set, B) = empty_set)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','no_difference_with_empty_set2')).
% 0.11/0.35 tff(14,plain,
% 0.11/0.35 (![B: $i] : (difference(empty_set, B) = empty_set)),
% 0.11/0.35 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.11/0.35 tff(15,plain,(
% 0.11/0.35 ![B: $i] : (difference(empty_set, B) = empty_set)),
% 0.11/0.35 inference(skolemize,[status(sab)],[14])).
% 0.11/0.35 tff(16,plain,
% 0.11/0.35 (![B: $i] : (difference(empty_set, B) = empty_set)),
% 0.11/0.35 inference(modus_ponens,[status(thm)],[15, 11])).
% 0.11/0.35 tff(17,plain,
% 0.11/0.35 ((~![B: $i] : (difference(empty_set, B) = empty_set)) | (difference(empty_set, B!3) = empty_set)),
% 0.11/0.35 inference(quant_inst,[status(thm)],[])).
% 0.11/0.35 tff(18,plain,
% 0.11/0.35 (difference(empty_set, B!3) = empty_set),
% 0.11/0.35 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.11/0.35 tff(19,plain,
% 0.11/0.35 (empty_set = difference(empty_set, B!3)),
% 0.11/0.35 inference(symmetry,[status(thm)],[18])).
% 0.11/0.35 tff(20,plain,
% 0.11/0.35 (^[B: $i] : refl((difference(B, empty_set) = B) <=> (difference(B, empty_set) = B))),
% 0.11/0.35 inference(bind,[status(th)],[])).
% 0.11/0.35 tff(21,plain,
% 0.11/0.35 (![B: $i] : (difference(B, empty_set) = B) <=> ![B: $i] : (difference(B, empty_set) = B)),
% 0.11/0.35 inference(quant_intro,[status(thm)],[20])).
% 0.11/0.35 tff(22,plain,
% 0.11/0.35 (![B: $i] : (difference(B, empty_set) = B) <=> ![B: $i] : (difference(B, empty_set) = B)),
% 0.11/0.35 inference(rewrite,[status(thm)],[])).
% 0.11/0.35 tff(23,axiom,(![B: $i] : (difference(B, empty_set) = B)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','no_difference_with_empty_set1')).
% 0.11/0.35 tff(24,plain,
% 0.11/0.35 (![B: $i] : (difference(B, empty_set) = B)),
% 0.11/0.35 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.11/0.35 tff(25,plain,(
% 0.11/0.35 ![B: $i] : (difference(B, empty_set) = B)),
% 0.11/0.35 inference(skolemize,[status(sab)],[24])).
% 0.11/0.35 tff(26,plain,
% 0.11/0.35 (![B: $i] : (difference(B, empty_set) = B)),
% 0.11/0.35 inference(modus_ponens,[status(thm)],[25, 21])).
% 0.11/0.35 tff(27,plain,
% 0.11/0.35 ((~![B: $i] : (difference(B, empty_set) = B)) | (difference(B!3, empty_set) = B!3)),
% 0.11/0.35 inference(quant_inst,[status(thm)],[])).
% 0.11/0.35 tff(28,plain,
% 0.11/0.35 (difference(B!3, empty_set) = B!3),
% 0.11/0.35 inference(unit_resolution,[status(thm)],[27, 26])).
% 0.11/0.35 tff(29,plain,
% 0.11/0.35 (B!3 = difference(B!3, empty_set)),
% 0.11/0.35 inference(symmetry,[status(thm)],[28])).
% 0.11/0.35 tff(30,plain,
% 0.11/0.35 (union(B!3, empty_set) = union(difference(B!3, empty_set), difference(empty_set, B!3))),
% 0.11/0.35 inference(monotonicity,[status(thm)],[29, 19])).
% 0.11/0.35 tff(31,plain,
% 0.11/0.35 (union(difference(B!3, empty_set), difference(empty_set, B!3)) = union(B!3, empty_set)),
% 0.11/0.35 inference(symmetry,[status(thm)],[30])).
% 0.11/0.35 tff(32,plain,
% 0.11/0.35 (^[B: $i, C: $i] : refl((symmetric_difference(B, C) = union(difference(B, C), difference(C, B))) <=> (symmetric_difference(B, C) = union(difference(B, C), difference(C, B))))),
% 0.11/0.35 inference(bind,[status(th)],[])).
% 0.11/0.35 tff(33,plain,
% 0.11/0.35 (![B: $i, C: $i] : (symmetric_difference(B, C) = union(difference(B, C), difference(C, B))) <=> ![B: $i, C: $i] : (symmetric_difference(B, C) = union(difference(B, C), difference(C, B)))),
% 0.11/0.35 inference(quant_intro,[status(thm)],[32])).
% 0.11/0.35 tff(34,plain,
% 0.11/0.35 (![B: $i, C: $i] : (symmetric_difference(B, C) = union(difference(B, C), difference(C, B))) <=> ![B: $i, C: $i] : (symmetric_difference(B, C) = union(difference(B, C), difference(C, B)))),
% 0.11/0.35 inference(rewrite,[status(thm)],[])).
% 0.11/0.35 tff(35,axiom,(![B: $i, C: $i] : (symmetric_difference(B, C) = union(difference(B, C), difference(C, B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','symmetric_difference_defn')).
% 0.11/0.35 tff(36,plain,
% 0.11/0.35 (![B: $i, C: $i] : (symmetric_difference(B, C) = union(difference(B, C), difference(C, B)))),
% 0.11/0.35 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.11/0.35 tff(37,plain,(
% 0.11/0.35 ![B: $i, C: $i] : (symmetric_difference(B, C) = union(difference(B, C), difference(C, B)))),
% 0.11/0.35 inference(skolemize,[status(sab)],[36])).
% 0.11/0.35 tff(38,plain,
% 0.11/0.35 (![B: $i, C: $i] : (symmetric_difference(B, C) = union(difference(B, C), difference(C, B)))),
% 0.11/0.35 inference(modus_ponens,[status(thm)],[37, 33])).
% 0.11/0.35 tff(39,plain,
% 0.11/0.35 ((~![B: $i, C: $i] : (symmetric_difference(B, C) = union(difference(B, C), difference(C, B)))) | (symmetric_difference(B!3, empty_set) = union(difference(B!3, empty_set), difference(empty_set, B!3)))),
% 0.11/0.35 inference(quant_inst,[status(thm)],[])).
% 0.11/0.35 tff(40,plain,
% 0.11/0.35 (symmetric_difference(B!3, empty_set) = union(difference(B!3, empty_set), difference(empty_set, B!3))),
% 0.11/0.35 inference(unit_resolution,[status(thm)],[39, 38])).
% 0.11/0.35 tff(41,plain,
% 0.11/0.35 (^[B: $i, C: $i] : refl((symmetric_difference(B, C) = symmetric_difference(C, B)) <=> (symmetric_difference(B, C) = symmetric_difference(C, B)))),
% 0.11/0.35 inference(bind,[status(th)],[])).
% 0.11/0.35 tff(42,plain,
% 0.11/0.35 (![B: $i, C: $i] : (symmetric_difference(B, C) = symmetric_difference(C, B)) <=> ![B: $i, C: $i] : (symmetric_difference(B, C) = symmetric_difference(C, B))),
% 0.11/0.35 inference(quant_intro,[status(thm)],[41])).
% 0.11/0.35 tff(43,plain,
% 0.11/0.35 (![B: $i, C: $i] : (symmetric_difference(B, C) = symmetric_difference(C, B)) <=> ![B: $i, C: $i] : (symmetric_difference(B, C) = symmetric_difference(C, B))),
% 0.11/0.35 inference(rewrite,[status(thm)],[])).
% 0.11/0.35 tff(44,axiom,(![B: $i, C: $i] : (symmetric_difference(B, C) = symmetric_difference(C, B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','commutativity_of_symmetric_difference')).
% 0.11/0.35 tff(45,plain,
% 0.11/0.35 (![B: $i, C: $i] : (symmetric_difference(B, C) = symmetric_difference(C, B))),
% 0.11/0.35 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.11/0.35 tff(46,plain,(
% 0.11/0.35 ![B: $i, C: $i] : (symmetric_difference(B, C) = symmetric_difference(C, B))),
% 0.11/0.35 inference(skolemize,[status(sab)],[45])).
% 0.11/0.35 tff(47,plain,
% 0.11/0.35 (![B: $i, C: $i] : (symmetric_difference(B, C) = symmetric_difference(C, B))),
% 0.11/0.35 inference(modus_ponens,[status(thm)],[46, 42])).
% 0.11/0.35 tff(48,plain,
% 0.11/0.35 ((~![B: $i, C: $i] : (symmetric_difference(B, C) = symmetric_difference(C, B))) | (symmetric_difference(B!3, empty_set) = symmetric_difference(empty_set, B!3))),
% 0.11/0.35 inference(quant_inst,[status(thm)],[])).
% 0.11/0.35 tff(49,plain,
% 0.11/0.35 (symmetric_difference(B!3, empty_set) = symmetric_difference(empty_set, B!3)),
% 0.11/0.35 inference(unit_resolution,[status(thm)],[48, 47])).
% 0.11/0.35 tff(50,plain,
% 0.11/0.35 (symmetric_difference(empty_set, B!3) = symmetric_difference(B!3, empty_set)),
% 0.11/0.35 inference(symmetry,[status(thm)],[49])).
% 0.11/0.35 tff(51,plain,
% 0.11/0.35 (symmetric_difference(empty_set, B!3) = B!3),
% 0.11/0.36 inference(transitivity,[status(thm)],[50, 40, 31, 9])).
% 0.11/0.36 tff(52,assumption,(~(symmetric_difference(empty_set, B!3) = B!3)), introduced(assumption)).
% 0.11/0.36 tff(53,plain,
% 0.11/0.36 ($false),
% 0.11/0.36 inference(unit_resolution,[status(thm)],[52, 51])).
% 0.11/0.36 tff(54,plain,(symmetric_difference(empty_set, B!3) = B!3), inference(lemma,lemma(discharge,[]))).
% 0.11/0.36 tff(55,plain,
% 0.11/0.36 (symmetric_difference(B!3, empty_set) = B!3),
% 0.11/0.36 inference(transitivity,[status(thm)],[49, 54])).
% 0.11/0.36 tff(56,plain,
% 0.11/0.36 ((~(~((~(symmetric_difference(B!3, empty_set) = B!3)) | (~(symmetric_difference(empty_set, B!3) = B!3))))) <=> ((~(symmetric_difference(B!3, empty_set) = B!3)) | (~(symmetric_difference(empty_set, B!3) = B!3)))),
% 0.11/0.36 inference(rewrite,[status(thm)],[])).
% 0.11/0.36 tff(57,plain,
% 0.11/0.36 (((symmetric_difference(B!3, empty_set) = B!3) & (symmetric_difference(empty_set, B!3) = B!3)) <=> (~((~(symmetric_difference(B!3, empty_set) = B!3)) | (~(symmetric_difference(empty_set, B!3) = B!3))))),
% 0.11/0.36 inference(rewrite,[status(thm)],[])).
% 0.11/0.36 tff(58,plain,
% 0.11/0.36 ((~((symmetric_difference(B!3, empty_set) = B!3) & (symmetric_difference(empty_set, B!3) = B!3))) <=> (~(~((~(symmetric_difference(B!3, empty_set) = B!3)) | (~(symmetric_difference(empty_set, B!3) = B!3)))))),
% 0.11/0.36 inference(monotonicity,[status(thm)],[57])).
% 0.11/0.36 tff(59,plain,
% 0.11/0.36 ((~((symmetric_difference(B!3, empty_set) = B!3) & (symmetric_difference(empty_set, B!3) = B!3))) <=> ((~(symmetric_difference(B!3, empty_set) = B!3)) | (~(symmetric_difference(empty_set, B!3) = B!3)))),
% 0.11/0.36 inference(transitivity,[status(thm)],[58, 56])).
% 0.11/0.36 tff(60,plain,
% 0.11/0.36 ((~![B: $i] : ((symmetric_difference(B, empty_set) = B) & (symmetric_difference(empty_set, B) = B))) <=> (~![B: $i] : ((symmetric_difference(B, empty_set) = B) & (symmetric_difference(empty_set, B) = B)))),
% 0.11/0.36 inference(rewrite,[status(thm)],[])).
% 0.11/0.36 tff(61,axiom,(~![B: $i] : ((symmetric_difference(B, empty_set) = B) & (symmetric_difference(empty_set, B) = B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_th92')).
% 0.11/0.36 tff(62,plain,
% 0.11/0.36 (~![B: $i] : ((symmetric_difference(B, empty_set) = B) & (symmetric_difference(empty_set, B) = B))),
% 0.11/0.36 inference(modus_ponens,[status(thm)],[61, 60])).
% 0.11/0.36 tff(63,plain,
% 0.11/0.36 (~![B: $i] : ((symmetric_difference(B, empty_set) = B) & (symmetric_difference(empty_set, B) = B))),
% 0.11/0.36 inference(modus_ponens,[status(thm)],[62, 60])).
% 0.11/0.36 tff(64,plain,
% 0.11/0.36 (~![B: $i] : ((symmetric_difference(B, empty_set) = B) & (symmetric_difference(empty_set, B) = B))),
% 0.11/0.36 inference(modus_ponens,[status(thm)],[63, 60])).
% 0.11/0.36 tff(65,plain,
% 0.11/0.36 (~![B: $i] : ((symmetric_difference(B, empty_set) = B) & (symmetric_difference(empty_set, B) = B))),
% 0.11/0.36 inference(modus_ponens,[status(thm)],[64, 60])).
% 0.11/0.36 tff(66,plain,(
% 0.11/0.36 ~((symmetric_difference(B!3, empty_set) = B!3) & (symmetric_difference(empty_set, B!3) = B!3))),
% 0.11/0.36 inference(skolemize,[status(sab)],[65])).
% 0.11/0.36 tff(67,plain,
% 0.11/0.36 ((~(symmetric_difference(B!3, empty_set) = B!3)) | (~(symmetric_difference(empty_set, B!3) = B!3))),
% 0.11/0.36 inference(modus_ponens,[status(thm)],[66, 59])).
% 0.11/0.36 tff(68,plain,
% 0.11/0.36 (~(symmetric_difference(B!3, empty_set) = B!3)),
% 0.11/0.36 inference(unit_resolution,[status(thm)],[67, 54])).
% 0.11/0.36 tff(69,plain,
% 0.11/0.36 ($false),
% 0.11/0.36 inference(unit_resolution,[status(thm)],[68, 55])).
% 0.11/0.36 % SZS output end Proof
% 0.11/0.36 % E exiting
%------------------------------------------------------------------------------