%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU321+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n013.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 05:27:35 AM UTC 2025
% Result : Theorem 0.23s 0.44s
% Output : Proof 0.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.14 % Problem : SEU321+1 : TPTP v9.0.0. Released v3.3.0.
% 0.04/0.14 % Command : run_E %s %d THM
% 0.14/0.36 % Computer : n013.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Fri Jun 20 06:44:23 EDT 2025
% 0.14/0.36 % CPUTime :
% 0.23/0.44 % SZS status Theorem
% 0.23/0.44 % SZS output start Proof
% 0.23/0.44 tff(in_type, type, (
% 0.23/0.44 in: ( $i * $i ) > $o)).
% 0.23/0.44 tff(subset_complement_type, type, (
% 0.23/0.44 subset_complement: ( $i * $i ) > $i)).
% 0.23/0.44 tff(tptp_fun_B_6_type, type, (
% 0.23/0.44 tptp_fun_B_6: $i)).
% 0.23/0.44 tff(the_carrier_type, type, (
% 0.23/0.44 the_carrier: $i > $i)).
% 0.23/0.44 tff(tptp_fun_A_5_type, type, (
% 0.23/0.44 tptp_fun_A_5: $i)).
% 0.23/0.44 tff(tptp_fun_C_7_type, type, (
% 0.23/0.44 tptp_fun_C_7: $i)).
% 0.23/0.44 tff(element_type, type, (
% 0.23/0.44 element: ( $i * $i ) > $o)).
% 0.23/0.44 tff(powerset_type, type, (
% 0.23/0.44 powerset: $i > $i)).
% 0.23/0.44 tff(empty_set_type, type, (
% 0.23/0.44 empty_set: $i)).
% 0.23/0.44 tff(empty_type, type, (
% 0.23/0.44 empty: $i > $o)).
% 0.23/0.44 tff(v1_membered_type, type, (
% 0.23/0.44 v1_membered: $i > $o)).
% 0.23/0.44 tff(v2_membered_type, type, (
% 0.23/0.44 v2_membered: $i > $o)).
% 0.23/0.44 tff(v3_membered_type, type, (
% 0.23/0.44 v3_membered: $i > $o)).
% 0.23/0.44 tff(v4_membered_type, type, (
% 0.23/0.44 v4_membered: $i > $o)).
% 0.23/0.44 tff(v5_membered_type, type, (
% 0.23/0.44 v5_membered: $i > $o)).
% 0.23/0.44 tff(one_sorted_str_type, type, (
% 0.23/0.44 one_sorted_str: $i > $o)).
% 0.23/0.44 tff(empty_carrier_type, type, (
% 0.23/0.44 empty_carrier: $i > $o)).
% 0.23/0.44 tff(1,assumption,(the_carrier(A!5) = empty_set), introduced(assumption)).
% 0.23/0.44 tff(2,plain,
% 0.23/0.44 (empty(the_carrier(A!5)) <=> empty(empty_set)),
% 0.23/0.44 inference(monotonicity,[status(thm)],[1])).
% 0.23/0.44 tff(3,plain,
% 0.23/0.44 (empty(empty_set) <=> empty(the_carrier(A!5))),
% 0.23/0.44 inference(symmetry,[status(thm)],[2])).
% 0.23/0.44 tff(4,plain,
% 0.23/0.44 (empty(empty_set) <=> empty(empty_set)),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(5,axiom,(((((empty(empty_set) & v1_membered(empty_set)) & v2_membered(empty_set)) & v3_membered(empty_set)) & v4_membered(empty_set)) & v5_membered(empty_set)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','fc6_membered')).
% 0.23/0.44 tff(6,plain,
% 0.23/0.44 ((((empty(empty_set) & v1_membered(empty_set)) & v2_membered(empty_set)) & v3_membered(empty_set)) & v4_membered(empty_set)),
% 0.23/0.44 inference(and_elim,[status(thm)],[5])).
% 0.23/0.44 tff(7,plain,
% 0.23/0.44 (((empty(empty_set) & v1_membered(empty_set)) & v2_membered(empty_set)) & v3_membered(empty_set)),
% 0.23/0.44 inference(and_elim,[status(thm)],[6])).
% 0.23/0.44 tff(8,plain,
% 0.23/0.44 ((empty(empty_set) & v1_membered(empty_set)) & v2_membered(empty_set)),
% 0.23/0.44 inference(and_elim,[status(thm)],[7])).
% 0.23/0.44 tff(9,plain,
% 0.23/0.44 (empty(empty_set) & v1_membered(empty_set)),
% 0.23/0.44 inference(and_elim,[status(thm)],[8])).
% 0.23/0.44 tff(10,plain,
% 0.23/0.44 (empty(empty_set)),
% 0.23/0.44 inference(and_elim,[status(thm)],[9])).
% 0.23/0.44 tff(11,plain,
% 0.23/0.44 (empty(empty_set)),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[10, 4])).
% 0.23/0.44 tff(12,plain,
% 0.23/0.44 (empty(the_carrier(A!5))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[11, 3])).
% 0.23/0.44 tff(13,plain,
% 0.23/0.44 ((((~empty_carrier(A!5)) & one_sorted_str(A!5)) & (element(B!6, powerset(the_carrier(A!5))) & (~((~element(C!7, the_carrier(A!5))) | (~(in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6)))))))) <=> ((~empty_carrier(A!5)) & one_sorted_str(A!5) & element(B!6, powerset(the_carrier(A!5))) & (~((~element(C!7, the_carrier(A!5))) | (~(in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6)))))))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(14,plain,
% 0.23/0.44 (((~element(C!7, the_carrier(A!5))) | (~(in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6))))) <=> ((~element(C!7, the_carrier(A!5))) | (~(in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6)))))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(15,plain,
% 0.23/0.44 ((~((~element(C!7, the_carrier(A!5))) | (~(in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6)))))) <=> (~((~element(C!7, the_carrier(A!5))) | (~(in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6))))))),
% 0.23/0.44 inference(monotonicity,[status(thm)],[14])).
% 0.23/0.44 tff(16,plain,
% 0.23/0.44 ((element(B!6, powerset(the_carrier(A!5))) & (~((~element(C!7, the_carrier(A!5))) | (~(in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6))))))) <=> (element(B!6, powerset(the_carrier(A!5))) & (~((~element(C!7, the_carrier(A!5))) | (~(in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6)))))))),
% 0.23/0.44 inference(monotonicity,[status(thm)],[15])).
% 0.23/0.44 tff(17,plain,
% 0.23/0.44 ((((~empty_carrier(A!5)) & one_sorted_str(A!5)) & (element(B!6, powerset(the_carrier(A!5))) & (~((~element(C!7, the_carrier(A!5))) | (~(in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6)))))))) <=> (((~empty_carrier(A!5)) & one_sorted_str(A!5)) & (element(B!6, powerset(the_carrier(A!5))) & (~((~element(C!7, the_carrier(A!5))) | (~(in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6))))))))),
% 0.23/0.44 inference(monotonicity,[status(thm)],[16])).
% 0.23/0.44 tff(18,plain,
% 0.23/0.44 ((((~empty_carrier(A!5)) & one_sorted_str(A!5)) & (element(B!6, powerset(the_carrier(A!5))) & (~((~element(C!7, the_carrier(A!5))) | (~(in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6)))))))) <=> ((~empty_carrier(A!5)) & one_sorted_str(A!5) & element(B!6, powerset(the_carrier(A!5))) & (~((~element(C!7, the_carrier(A!5))) | (~(in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6)))))))),
% 0.23/0.44 inference(transitivity,[status(thm)],[17, 13])).
% 0.23/0.44 tff(19,plain,
% 0.23/0.44 ((~![A: $i] : ((~((~empty_carrier(A)) & one_sorted_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~element(C, the_carrier(A))) | (~(in(C, B) <=> in(C, subset_complement(the_carrier(A), B)))))))) <=> (~![A: $i] : ((~((~empty_carrier(A)) & one_sorted_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~element(C, the_carrier(A))) | (~(in(C, B) <=> in(C, subset_complement(the_carrier(A), B))))))))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(20,plain,
% 0.23/0.44 ((~![A: $i] : (((~empty_carrier(A)) & one_sorted_str(A)) => ![B: $i] : (element(B, powerset(the_carrier(A))) => ![C: $i] : (element(C, the_carrier(A)) => (in(C, subset_complement(the_carrier(A), B)) <=> (~in(C, B))))))) <=> (~![A: $i] : ((~((~empty_carrier(A)) & one_sorted_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~element(C, the_carrier(A))) | (~(in(C, B) <=> in(C, subset_complement(the_carrier(A), B))))))))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(21,axiom,(~![A: $i] : (((~empty_carrier(A)) & one_sorted_str(A)) => ![B: $i] : (element(B, powerset(the_carrier(A))) => ![C: $i] : (element(C, the_carrier(A)) => (in(C, subset_complement(the_carrier(A), B)) <=> (~in(C, B))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','l40_tops_1')).
% 0.23/0.44 tff(22,plain,
% 0.23/0.44 (~![A: $i] : ((~((~empty_carrier(A)) & one_sorted_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~element(C, the_carrier(A))) | (~(in(C, B) <=> in(C, subset_complement(the_carrier(A), B)))))))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.23/0.44 tff(23,plain,
% 0.23/0.44 (~![A: $i] : ((~((~empty_carrier(A)) & one_sorted_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~element(C, the_carrier(A))) | (~(in(C, B) <=> in(C, subset_complement(the_carrier(A), B)))))))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[22, 19])).
% 0.23/0.44 tff(24,plain,
% 0.23/0.44 (~![A: $i] : ((~((~empty_carrier(A)) & one_sorted_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~element(C, the_carrier(A))) | (~(in(C, B) <=> in(C, subset_complement(the_carrier(A), B)))))))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[23, 19])).
% 0.23/0.44 tff(25,plain,
% 0.23/0.44 (~![A: $i] : ((~((~empty_carrier(A)) & one_sorted_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~element(C, the_carrier(A))) | (~(in(C, B) <=> in(C, subset_complement(the_carrier(A), B)))))))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[24, 19])).
% 0.23/0.44 tff(26,plain,
% 0.23/0.44 ((~empty_carrier(A!5)) & one_sorted_str(A!5) & element(B!6, powerset(the_carrier(A!5))) & (~((~element(C!7, the_carrier(A!5))) | (~(in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6))))))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[25, 18])).
% 0.23/0.44 tff(27,plain,
% 0.23/0.44 (one_sorted_str(A!5)),
% 0.23/0.44 inference(and_elim,[status(thm)],[26])).
% 0.23/0.44 tff(28,plain,
% 0.23/0.44 (~empty_carrier(A!5)),
% 0.23/0.44 inference(and_elim,[status(thm)],[26])).
% 0.23/0.44 tff(29,plain,
% 0.23/0.44 (^[A: $i] : refl((empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A))) <=> (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A))))),
% 0.23/0.44 inference(bind,[status(th)],[])).
% 0.23/0.44 tff(30,plain,
% 0.23/0.44 (![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A))) <=> ![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))),
% 0.23/0.44 inference(quant_intro,[status(thm)],[29])).
% 0.23/0.44 tff(31,plain,
% 0.23/0.44 (^[A: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~empty_carrier(A)) & one_sorted_str(A)) <=> (~(empty_carrier(A) | (~one_sorted_str(A))))), ((~((~empty_carrier(A)) & one_sorted_str(A))) <=> (~(~(empty_carrier(A) | (~one_sorted_str(A))))))), rewrite((~(~(empty_carrier(A) | (~one_sorted_str(A))))) <=> (empty_carrier(A) | (~one_sorted_str(A)))), ((~((~empty_carrier(A)) & one_sorted_str(A))) <=> (empty_carrier(A) | (~one_sorted_str(A))))), (((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A)))) <=> ((~empty(the_carrier(A))) | (empty_carrier(A) | (~one_sorted_str(A)))))), rewrite(((~empty(the_carrier(A))) | (empty_carrier(A) | (~one_sorted_str(A)))) <=> (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))), (((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A)))) <=> (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))))),
% 0.23/0.44 inference(bind,[status(th)],[])).
% 0.23/0.44 tff(32,plain,
% 0.23/0.44 (![A: $i] : ((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A)))) <=> ![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))),
% 0.23/0.44 inference(quant_intro,[status(thm)],[31])).
% 0.23/0.44 tff(33,plain,
% 0.23/0.44 (![A: $i] : ((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A)))) <=> ![A: $i] : ((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A))))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(34,plain,
% 0.23/0.44 (^[A: $i] : rewrite((((~empty_carrier(A)) & one_sorted_str(A)) => (~empty(the_carrier(A)))) <=> ((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A)))))),
% 0.23/0.44 inference(bind,[status(th)],[])).
% 0.23/0.44 tff(35,plain,
% 0.23/0.44 (![A: $i] : (((~empty_carrier(A)) & one_sorted_str(A)) => (~empty(the_carrier(A)))) <=> ![A: $i] : ((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A))))),
% 0.23/0.44 inference(quant_intro,[status(thm)],[34])).
% 0.23/0.44 tff(36,axiom,(![A: $i] : (((~empty_carrier(A)) & one_sorted_str(A)) => (~empty(the_carrier(A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','fc1_struct_0')).
% 0.23/0.44 tff(37,plain,
% 0.23/0.44 (![A: $i] : ((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A))))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.23/0.44 tff(38,plain,
% 0.23/0.44 (![A: $i] : ((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A))))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[37, 33])).
% 0.23/0.44 tff(39,plain,(
% 0.23/0.44 ![A: $i] : ((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A))))),
% 0.23/0.44 inference(skolemize,[status(sab)],[38])).
% 0.23/0.44 tff(40,plain,
% 0.23/0.44 (![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[39, 32])).
% 0.23/0.44 tff(41,plain,
% 0.23/0.44 (![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[40, 30])).
% 0.23/0.44 tff(42,plain,
% 0.23/0.44 (((~![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))) | (empty_carrier(A!5) | (~one_sorted_str(A!5)) | (~empty(the_carrier(A!5))))) <=> ((~![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))) | empty_carrier(A!5) | (~one_sorted_str(A!5)) | (~empty(the_carrier(A!5))))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(43,plain,
% 0.23/0.44 ((empty_carrier(A!5) | (~empty(the_carrier(A!5))) | (~one_sorted_str(A!5))) <=> (empty_carrier(A!5) | (~one_sorted_str(A!5)) | (~empty(the_carrier(A!5))))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(44,plain,
% 0.23/0.44 (((~![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))) | (empty_carrier(A!5) | (~empty(the_carrier(A!5))) | (~one_sorted_str(A!5)))) <=> ((~![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))) | (empty_carrier(A!5) | (~one_sorted_str(A!5)) | (~empty(the_carrier(A!5)))))),
% 0.23/0.44 inference(monotonicity,[status(thm)],[43])).
% 0.23/0.44 tff(45,plain,
% 0.23/0.44 (((~![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))) | (empty_carrier(A!5) | (~empty(the_carrier(A!5))) | (~one_sorted_str(A!5)))) <=> ((~![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))) | empty_carrier(A!5) | (~one_sorted_str(A!5)) | (~empty(the_carrier(A!5))))),
% 0.23/0.44 inference(transitivity,[status(thm)],[44, 42])).
% 0.23/0.44 tff(46,plain,
% 0.23/0.44 ((~![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))) | (empty_carrier(A!5) | (~empty(the_carrier(A!5))) | (~one_sorted_str(A!5)))),
% 0.23/0.44 inference(quant_inst,[status(thm)],[])).
% 0.23/0.44 tff(47,plain,
% 0.23/0.44 ((~![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))) | empty_carrier(A!5) | (~one_sorted_str(A!5)) | (~empty(the_carrier(A!5)))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[46, 45])).
% 0.23/0.44 tff(48,plain,
% 0.23/0.44 (~empty(the_carrier(A!5))),
% 0.23/0.44 inference(unit_resolution,[status(thm)],[47, 41, 28, 27])).
% 0.23/0.44 tff(49,plain,
% 0.23/0.44 ($false),
% 0.23/0.44 inference(unit_resolution,[status(thm)],[48, 12])).
% 0.23/0.44 tff(50,plain,(~(the_carrier(A!5) = empty_set)), inference(lemma,lemma(discharge,[]))).
% 0.23/0.44 tff(51,plain,
% 0.23/0.44 (^[A: $i] : refl(((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A))))) <=> ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A))))))),
% 0.23/0.44 inference(bind,[status(th)],[])).
% 0.23/0.44 tff(52,plain,
% 0.23/0.44 (![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A))))) <=> ![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))),
% 0.23/0.44 inference(quant_intro,[status(thm)],[51])).
% 0.23/0.44 tff(53,plain,
% 0.23/0.44 (^[A: $i] : rewrite(((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A))))) <=> ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A))))))),
% 0.23/0.44 inference(bind,[status(th)],[])).
% 0.23/0.44 tff(54,plain,
% 0.23/0.44 (![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A))))) <=> ![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))),
% 0.23/0.44 inference(quant_intro,[status(thm)],[53])).
% 0.23/0.44 tff(55,plain,
% 0.23/0.44 (![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A))))) <=> ![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))),
% 0.23/0.44 inference(transitivity,[status(thm)],[54, 52])).
% 0.23/0.44 tff(56,plain,
% 0.23/0.44 (![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A))))) <=> ![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(57,plain,
% 0.23/0.44 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : trans(monotonicity(quant_intro(proof_bind(^[C: $i] : trans(monotonicity(rewrite(((~in(C, B)) => in(C, subset_complement(A, B))) <=> (in(C, B) | in(C, subset_complement(A, B)))), ((element(C, A) => ((~in(C, B)) => in(C, subset_complement(A, B)))) <=> (element(C, A) => (in(C, B) | in(C, subset_complement(A, B)))))), rewrite((element(C, A) => (in(C, B) | in(C, subset_complement(A, B)))) <=> (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))), ((element(C, A) => ((~in(C, B)) => in(C, subset_complement(A, B)))) <=> (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))), (![C: $i] : (element(C, A) => ((~in(C, B)) => in(C, subset_complement(A, B)))) <=> ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A))))), ((element(B, powerset(A)) => ![C: $i] : (element(C, A) => ((~in(C, B)) => in(C, subset_complement(A, B))))) <=> (element(B, powerset(A)) => ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))), rewrite((element(B, powerset(A)) => ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))) <=> ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A))))), ((element(B, powerset(A)) => ![C: $i] : (element(C, A) => ((~in(C, B)) => in(C, subset_complement(A, B))))) <=> ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A))))))), (![B: $i] : (element(B, powerset(A)) => ![C: $i] : (element(C, A) => ((~in(C, B)) => in(C, subset_complement(A, B))))) <=> ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))), (((~(A = empty_set)) => ![B: $i] : (element(B, powerset(A)) => ![C: $i] : (element(C, A) => ((~in(C, B)) => in(C, subset_complement(A, B)))))) <=> ((~(A = empty_set)) => ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A))))))), rewrite(((~(A = empty_set)) => ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A))))) <=> ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))), (((~(A = empty_set)) => ![B: $i] : (element(B, powerset(A)) => ![C: $i] : (element(C, A) => ((~in(C, B)) => in(C, subset_complement(A, B)))))) <=> ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))))),
% 0.23/0.45 inference(bind,[status(th)],[])).
% 0.23/0.45 tff(58,plain,
% 0.23/0.45 (![A: $i] : ((~(A = empty_set)) => ![B: $i] : (element(B, powerset(A)) => ![C: $i] : (element(C, A) => ((~in(C, B)) => in(C, subset_complement(A, B)))))) <=> ![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))),
% 0.23/0.45 inference(quant_intro,[status(thm)],[57])).
% 0.23/0.45 tff(59,axiom,(![A: $i] : ((~(A = empty_set)) => ![B: $i] : (element(B, powerset(A)) => ![C: $i] : (element(C, A) => ((~in(C, B)) => in(C, subset_complement(A, B))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t50_subset_1')).
% 0.23/0.45 tff(60,plain,
% 0.23/0.45 (![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.23/0.45 tff(61,plain,
% 0.23/0.45 (![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[60, 56])).
% 0.23/0.45 tff(62,plain,(
% 0.23/0.45 ![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))),
% 0.23/0.45 inference(skolemize,[status(sab)],[61])).
% 0.23/0.45 tff(63,plain,
% 0.23/0.45 (![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[62, 55])).
% 0.23/0.45 tff(64,plain,
% 0.23/0.45 (((~![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))) | ((the_carrier(A!5) = empty_set) | ![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B)))))) <=> ((~![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))) | (the_carrier(A!5) = empty_set) | ![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B)))))),
% 0.23/0.45 inference(rewrite,[status(thm)],[])).
% 0.23/0.45 tff(65,plain,
% 0.23/0.45 (((the_carrier(A!5) = empty_set) | ![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | in(C, subset_complement(the_carrier(A!5), B)) | (~element(C, the_carrier(A!5)))))) <=> ((the_carrier(A!5) = empty_set) | ![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B)))))),
% 0.23/0.45 inference(rewrite,[status(thm)],[])).
% 0.23/0.45 tff(66,plain,
% 0.23/0.45 (((~![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))) | ((the_carrier(A!5) = empty_set) | ![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | in(C, subset_complement(the_carrier(A!5), B)) | (~element(C, the_carrier(A!5))))))) <=> ((~![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))) | ((the_carrier(A!5) = empty_set) | ![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B))))))),
% 0.23/0.45 inference(monotonicity,[status(thm)],[65])).
% 0.23/0.45 tff(67,plain,
% 0.23/0.45 (((~![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))) | ((the_carrier(A!5) = empty_set) | ![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | in(C, subset_complement(the_carrier(A!5), B)) | (~element(C, the_carrier(A!5))))))) <=> ((~![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))) | (the_carrier(A!5) = empty_set) | ![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B)))))),
% 0.23/0.45 inference(transitivity,[status(thm)],[66, 64])).
% 0.23/0.45 tff(68,plain,
% 0.23/0.45 ((~![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))) | ((the_carrier(A!5) = empty_set) | ![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | in(C, subset_complement(the_carrier(A!5), B)) | (~element(C, the_carrier(A!5))))))),
% 0.23/0.45 inference(quant_inst,[status(thm)],[])).
% 0.23/0.45 tff(69,plain,
% 0.23/0.45 ((~![A: $i] : ((A = empty_set) | ![B: $i] : ((~element(B, powerset(A))) | ![C: $i] : (in(C, B) | in(C, subset_complement(A, B)) | (~element(C, A)))))) | (the_carrier(A!5) = empty_set) | ![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B))))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[68, 67])).
% 0.23/0.45 tff(70,plain,
% 0.23/0.45 ((the_carrier(A!5) = empty_set) | ![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B))))),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[69, 63])).
% 0.23/0.45 tff(71,plain,
% 0.23/0.45 (![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B))))),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[70, 50])).
% 0.23/0.45 tff(72,assumption,(~![C: $i] : ((~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B!6)) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))))), introduced(assumption)).
% 0.23/0.45 tff(73,plain,
% 0.23/0.45 (element(B!6, powerset(the_carrier(A!5)))),
% 0.23/0.45 inference(and_elim,[status(thm)],[26])).
% 0.23/0.45 tff(74,plain,
% 0.23/0.45 (^[A: $i, B: $i] : refl(((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))) <=> ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))))),
% 0.23/0.45 inference(bind,[status(th)],[])).
% 0.23/0.45 tff(75,plain,
% 0.23/0.45 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.23/0.45 inference(quant_intro,[status(thm)],[74])).
% 0.23/0.45 tff(76,plain,
% 0.23/0.45 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.23/0.45 inference(rewrite,[status(thm)],[])).
% 0.23/0.45 tff(77,plain,
% 0.23/0.45 (^[A: $i, B: $i] : rewrite((element(B, powerset(A)) => element(subset_complement(A, B), powerset(A))) <=> ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))))),
% 0.23/0.45 inference(bind,[status(th)],[])).
% 0.23/0.45 tff(78,plain,
% 0.23/0.45 (![A: $i, B: $i] : (element(B, powerset(A)) => element(subset_complement(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.23/0.45 inference(quant_intro,[status(thm)],[77])).
% 0.23/0.45 tff(79,axiom,(![A: $i, B: $i] : (element(B, powerset(A)) => element(subset_complement(A, B), powerset(A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_k3_subset_1')).
% 0.23/0.45 tff(80,plain,
% 0.23/0.45 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[79, 78])).
% 0.23/0.45 tff(81,plain,
% 0.23/0.45 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[80, 76])).
% 0.23/0.45 tff(82,plain,(
% 0.23/0.45 ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.23/0.45 inference(skolemize,[status(sab)],[81])).
% 0.23/0.45 tff(83,plain,
% 0.23/0.45 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[82, 75])).
% 0.23/0.45 tff(84,plain,
% 0.23/0.45 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | ((~element(B!6, powerset(the_carrier(A!5)))) | element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | (~element(B!6, powerset(the_carrier(A!5)))) | element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5))))),
% 0.23/0.45 inference(rewrite,[status(thm)],[])).
% 0.23/0.45 tff(85,plain,
% 0.23/0.45 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | ((~element(B!6, powerset(the_carrier(A!5)))) | element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5))))),
% 0.23/0.45 inference(quant_inst,[status(thm)],[])).
% 0.23/0.45 tff(86,plain,
% 0.23/0.45 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | (~element(B!6, powerset(the_carrier(A!5)))) | element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.23/0.45 tff(87,plain,
% 0.23/0.45 (element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[86, 83, 73])).
% 0.23/0.45 tff(88,assumption,(![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B))))), introduced(assumption)).
% 0.23/0.45 tff(89,plain,
% 0.23/0.45 (((~![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B))))) | ((~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | ![C: $i] : ((~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B!6)) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6)))))) <=> ((~![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B))))) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | ![C: $i] : ((~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B!6)) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6)))))),
% 0.23/0.45 inference(rewrite,[status(thm)],[])).
% 0.23/0.45 tff(90,plain,
% 0.23/0.45 (((~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, subset_complement(the_carrier(A!5), B!6)) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))))) <=> ((~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | ![C: $i] : ((~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B!6)) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6)))))),
% 0.23/0.45 inference(rewrite,[status(thm)],[])).
% 0.23/0.45 tff(91,plain,
% 0.23/0.45 (((~![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B))))) | ((~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, subset_complement(the_carrier(A!5), B!6)) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6)))))) <=> ((~![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B))))) | ((~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | ![C: $i] : ((~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B!6)) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))))))),
% 0.23/0.45 inference(monotonicity,[status(thm)],[90])).
% 0.23/0.45 tff(92,plain,
% 0.23/0.45 (((~![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B))))) | ((~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, subset_complement(the_carrier(A!5), B!6)) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6)))))) <=> ((~![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B))))) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | ![C: $i] : ((~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B!6)) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6)))))),
% 0.23/0.45 inference(transitivity,[status(thm)],[91, 89])).
% 0.23/0.45 tff(93,plain,
% 0.23/0.45 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B))))) | ((~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, subset_complement(the_carrier(A!5), B!6)) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6)))))),
% 0.23/0.45 inference(quant_inst,[status(thm)],[])).
% 0.23/0.45 tff(94,plain,
% 0.23/0.45 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B))))) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | ![C: $i] : ((~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B!6)) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[93, 92])).
% 0.23/0.45 tff(95,plain,
% 0.23/0.45 ($false),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[94, 88, 87, 72])).
% 0.23/0.45 tff(96,plain,((~![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | ![C: $i] : (in(C, B) | (~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B))))) | ![C: $i] : ((~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B!6)) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))))), inference(lemma,lemma(discharge,[]))).
% 0.23/0.45 tff(97,plain,
% 0.23/0.45 (![C: $i] : ((~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B!6)) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))))),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[96, 71])).
% 0.23/0.46 tff(98,assumption,(![C: $i] : ((~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B!6)) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))))), introduced(assumption)).
% 0.23/0.46 tff(99,plain,
% 0.23/0.46 (^[A: $i, B: $i] : refl(((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)) <=> ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)))),
% 0.23/0.46 inference(bind,[status(th)],[])).
% 0.23/0.46 tff(100,plain,
% 0.23/0.46 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.23/0.46 inference(quant_intro,[status(thm)],[99])).
% 0.23/0.46 tff(101,plain,
% 0.23/0.46 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.23/0.46 inference(rewrite,[status(thm)],[])).
% 0.23/0.46 tff(102,plain,
% 0.23/0.46 (^[A: $i, B: $i] : rewrite((element(B, powerset(A)) => (subset_complement(A, subset_complement(A, B)) = B)) <=> ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)))),
% 0.23/0.46 inference(bind,[status(th)],[])).
% 0.23/0.46 tff(103,plain,
% 0.23/0.46 (![A: $i, B: $i] : (element(B, powerset(A)) => (subset_complement(A, subset_complement(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.23/0.46 inference(quant_intro,[status(thm)],[102])).
% 0.23/0.46 tff(104,axiom,(![A: $i, B: $i] : (element(B, powerset(A)) => (subset_complement(A, subset_complement(A, B)) = B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','involutiveness_k3_subset_1')).
% 0.23/0.46 tff(105,plain,
% 0.23/0.46 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.23/0.46 inference(modus_ponens,[status(thm)],[104, 103])).
% 0.23/0.46 tff(106,plain,
% 0.23/0.46 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.23/0.46 inference(modus_ponens,[status(thm)],[105, 101])).
% 0.23/0.46 tff(107,plain,(
% 0.23/0.46 ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.23/0.46 inference(skolemize,[status(sab)],[106])).
% 0.23/0.46 tff(108,plain,
% 0.23/0.46 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.23/0.46 inference(modus_ponens,[status(thm)],[107, 100])).
% 0.23/0.46 tff(109,plain,
% 0.23/0.46 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | ((~element(B!6, powerset(the_carrier(A!5)))) | (subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6)) = B!6))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | (~element(B!6, powerset(the_carrier(A!5)))) | (subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6)) = B!6))),
% 0.23/0.46 inference(rewrite,[status(thm)],[])).
% 0.23/0.46 tff(110,plain,
% 0.23/0.46 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | ((~element(B!6, powerset(the_carrier(A!5)))) | (subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6)) = B!6))),
% 0.23/0.46 inference(quant_inst,[status(thm)],[])).
% 0.23/0.46 tff(111,plain,
% 0.23/0.46 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | (~element(B!6, powerset(the_carrier(A!5)))) | (subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6)) = B!6)),
% 0.23/0.46 inference(modus_ponens,[status(thm)],[110, 109])).
% 0.23/0.46 tff(112,plain,
% 0.23/0.46 (subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6)) = B!6),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[111, 108, 73])).
% 0.23/0.46 tff(113,plain,
% 0.23/0.46 (in(C!7, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))) <=> in(C!7, B!6)),
% 0.23/0.46 inference(monotonicity,[status(thm)],[112])).
% 0.23/0.46 tff(114,plain,
% 0.23/0.46 (in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6)))),
% 0.23/0.46 inference(symmetry,[status(thm)],[113])).
% 0.23/0.46 tff(115,plain,
% 0.23/0.46 ((~in(C!7, B!6)) <=> (~in(C!7, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))))),
% 0.23/0.46 inference(monotonicity,[status(thm)],[114])).
% 0.23/0.46 tff(116,assumption,(~in(C!7, subset_complement(the_carrier(A!5), B!6))), introduced(assumption)).
% 0.23/0.46 tff(117,plain,
% 0.23/0.46 (~((~element(C!7, the_carrier(A!5))) | (~(in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6)))))),
% 0.23/0.46 inference(and_elim,[status(thm)],[26])).
% 0.23/0.46 tff(118,plain,
% 0.23/0.46 (in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6))),
% 0.23/0.46 inference(or_elim,[status(thm)],[117])).
% 0.23/0.46 tff(119,plain,
% 0.23/0.46 ((~in(C!7, B!6)) | in(C!7, subset_complement(the_carrier(A!5), B!6)) | (~(in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6))))),
% 0.23/0.46 inference(tautology,[status(thm)],[])).
% 0.23/0.46 tff(120,plain,
% 0.23/0.46 ((~in(C!7, B!6)) | in(C!7, subset_complement(the_carrier(A!5), B!6))),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[119, 118])).
% 0.23/0.46 tff(121,plain,
% 0.23/0.46 (~in(C!7, B!6)),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[120, 116])).
% 0.23/0.46 tff(122,plain,
% 0.23/0.46 (~in(C!7, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6)))),
% 0.23/0.46 inference(modus_ponens,[status(thm)],[121, 115])).
% 0.23/0.46 tff(123,plain,
% 0.23/0.46 (element(C!7, the_carrier(A!5))),
% 0.23/0.46 inference(or_elim,[status(thm)],[117])).
% 0.23/0.46 tff(124,plain,
% 0.23/0.46 (((~![C: $i] : ((~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B!6)) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))))) | ((~element(C!7, the_carrier(A!5))) | in(C!7, subset_complement(the_carrier(A!5), B!6)) | in(C!7, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))))) <=> ((~![C: $i] : ((~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B!6)) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))))) | (~element(C!7, the_carrier(A!5))) | in(C!7, subset_complement(the_carrier(A!5), B!6)) | in(C!7, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))))),
% 0.23/0.46 inference(rewrite,[status(thm)],[])).
% 0.23/0.46 tff(125,plain,
% 0.23/0.46 ((~![C: $i] : ((~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B!6)) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))))) | ((~element(C!7, the_carrier(A!5))) | in(C!7, subset_complement(the_carrier(A!5), B!6)) | in(C!7, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))))),
% 0.23/0.46 inference(quant_inst,[status(thm)],[])).
% 0.23/0.46 tff(126,plain,
% 0.23/0.46 ((~![C: $i] : ((~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B!6)) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))))) | (~element(C!7, the_carrier(A!5))) | in(C!7, subset_complement(the_carrier(A!5), B!6)) | in(C!7, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6)))),
% 0.23/0.46 inference(modus_ponens,[status(thm)],[125, 124])).
% 0.23/0.46 tff(127,plain,
% 0.23/0.46 ($false),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[126, 123, 116, 122, 98])).
% 0.23/0.46 tff(128,plain,((~![C: $i] : ((~element(C, the_carrier(A!5))) | in(C, subset_complement(the_carrier(A!5), B!6)) | in(C, subset_complement(the_carrier(A!5), subset_complement(the_carrier(A!5), B!6))))) | in(C!7, subset_complement(the_carrier(A!5), B!6))), inference(lemma,lemma(discharge,[]))).
% 0.23/0.46 tff(129,plain,
% 0.23/0.46 (in(C!7, subset_complement(the_carrier(A!5), B!6))),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[128, 97])).
% 0.23/0.46 tff(130,plain,
% 0.23/0.46 (in(C!7, B!6) | (~in(C!7, subset_complement(the_carrier(A!5), B!6))) | (~(in(C!7, B!6) <=> in(C!7, subset_complement(the_carrier(A!5), B!6))))),
% 0.23/0.46 inference(tautology,[status(thm)],[])).
% 0.23/0.46 tff(131,plain,
% 0.23/0.46 (in(C!7, B!6) | (~in(C!7, subset_complement(the_carrier(A!5), B!6)))),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[130, 118])).
% 0.23/0.46 tff(132,plain,
% 0.23/0.46 (in(C!7, B!6)),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[131, 129])).
% 0.23/0.46 tff(133,plain,
% 0.23/0.46 (^[A: $i, B: $i, C: $i] : refl(((~in(B, C)) | (~element(C, powerset(A))) | (~in(B, subset_complement(A, C)))) <=> ((~in(B, C)) | (~element(C, powerset(A))) | (~in(B, subset_complement(A, C)))))),
% 0.23/0.46 inference(bind,[status(th)],[])).
% 0.23/0.46 tff(134,plain,
% 0.23/0.46 (![A: $i, B: $i, C: $i] : ((~in(B, C)) | (~element(C, powerset(A))) | (~in(B, subset_complement(A, C)))) <=> ![A: $i, B: $i, C: $i] : ((~in(B, C)) | (~element(C, powerset(A))) | (~in(B, subset_complement(A, C))))),
% 0.23/0.46 inference(quant_intro,[status(thm)],[133])).
% 0.23/0.46 tff(135,plain,
% 0.23/0.46 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(rewrite((in(B, subset_complement(A, C)) & in(B, C)) <=> (~((~in(B, C)) | (~in(B, subset_complement(A, C)))))), ((~(in(B, subset_complement(A, C)) & in(B, C))) <=> (~(~((~in(B, C)) | (~in(B, subset_complement(A, C)))))))), rewrite((~(~((~in(B, C)) | (~in(B, subset_complement(A, C)))))) <=> ((~in(B, C)) | (~in(B, subset_complement(A, C))))), ((~(in(B, subset_complement(A, C)) & in(B, C))) <=> ((~in(B, C)) | (~in(B, subset_complement(A, C)))))), (((~element(C, powerset(A))) | (~(in(B, subset_complement(A, C)) & in(B, C)))) <=> ((~element(C, powerset(A))) | ((~in(B, C)) | (~in(B, subset_complement(A, C))))))), rewrite(((~element(C, powerset(A))) | ((~in(B, C)) | (~in(B, subset_complement(A, C))))) <=> ((~in(B, C)) | (~element(C, powerset(A))) | (~in(B, subset_complement(A, C))))), (((~element(C, powerset(A))) | (~(in(B, subset_complement(A, C)) & in(B, C)))) <=> ((~in(B, C)) | (~element(C, powerset(A))) | (~in(B, subset_complement(A, C))))))),
% 0.23/0.46 inference(bind,[status(th)],[])).
% 0.23/0.46 tff(136,plain,
% 0.23/0.46 (![A: $i, B: $i, C: $i] : ((~element(C, powerset(A))) | (~(in(B, subset_complement(A, C)) & in(B, C)))) <=> ![A: $i, B: $i, C: $i] : ((~in(B, C)) | (~element(C, powerset(A))) | (~in(B, subset_complement(A, C))))),
% 0.23/0.46 inference(quant_intro,[status(thm)],[135])).
% 0.23/0.46 tff(137,plain,
% 0.23/0.46 (![A: $i, B: $i, C: $i] : ((~element(C, powerset(A))) | (~(in(B, subset_complement(A, C)) & in(B, C)))) <=> ![A: $i, B: $i, C: $i] : ((~element(C, powerset(A))) | (~(in(B, subset_complement(A, C)) & in(B, C))))),
% 0.23/0.46 inference(rewrite,[status(thm)],[])).
% 0.23/0.46 tff(138,plain,
% 0.23/0.46 (^[A: $i, B: $i, C: $i] : rewrite((element(C, powerset(A)) => (~(in(B, subset_complement(A, C)) & in(B, C)))) <=> ((~element(C, powerset(A))) | (~(in(B, subset_complement(A, C)) & in(B, C)))))),
% 0.23/0.46 inference(bind,[status(th)],[])).
% 0.23/0.46 tff(139,plain,
% 0.23/0.46 (![A: $i, B: $i, C: $i] : (element(C, powerset(A)) => (~(in(B, subset_complement(A, C)) & in(B, C)))) <=> ![A: $i, B: $i, C: $i] : ((~element(C, powerset(A))) | (~(in(B, subset_complement(A, C)) & in(B, C))))),
% 0.23/0.46 inference(quant_intro,[status(thm)],[138])).
% 0.23/0.46 tff(140,axiom,(![A: $i, B: $i, C: $i] : (element(C, powerset(A)) => (~(in(B, subset_complement(A, C)) & in(B, C))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t54_subset_1')).
% 0.23/0.46 tff(141,plain,
% 0.23/0.46 (![A: $i, B: $i, C: $i] : ((~element(C, powerset(A))) | (~(in(B, subset_complement(A, C)) & in(B, C))))),
% 0.23/0.46 inference(modus_ponens,[status(thm)],[140, 139])).
% 0.23/0.46 tff(142,plain,
% 0.23/0.46 (![A: $i, B: $i, C: $i] : ((~element(C, powerset(A))) | (~(in(B, subset_complement(A, C)) & in(B, C))))),
% 0.23/0.46 inference(modus_ponens,[status(thm)],[141, 137])).
% 0.23/0.46 tff(143,plain,(
% 0.23/0.46 ![A: $i, B: $i, C: $i] : ((~element(C, powerset(A))) | (~(in(B, subset_complement(A, C)) & in(B, C))))),
% 0.23/0.46 inference(skolemize,[status(sab)],[142])).
% 0.23/0.46 tff(144,plain,
% 0.23/0.46 (![A: $i, B: $i, C: $i] : ((~in(B, C)) | (~element(C, powerset(A))) | (~in(B, subset_complement(A, C))))),
% 0.23/0.46 inference(modus_ponens,[status(thm)],[143, 136])).
% 0.23/0.46 tff(145,plain,
% 0.23/0.46 (![A: $i, B: $i, C: $i] : ((~in(B, C)) | (~element(C, powerset(A))) | (~in(B, subset_complement(A, C))))),
% 0.23/0.46 inference(modus_ponens,[status(thm)],[144, 134])).
% 0.23/0.46 tff(146,plain,
% 0.23/0.46 (((~![A: $i, B: $i, C: $i] : ((~in(B, C)) | (~element(C, powerset(A))) | (~in(B, subset_complement(A, C))))) | ((~in(C!7, B!6)) | (~element(B!6, powerset(the_carrier(A!5)))) | (~in(C!7, subset_complement(the_carrier(A!5), B!6))))) <=> ((~![A: $i, B: $i, C: $i] : ((~in(B, C)) | (~element(C, powerset(A))) | (~in(B, subset_complement(A, C))))) | (~in(C!7, B!6)) | (~element(B!6, powerset(the_carrier(A!5)))) | (~in(C!7, subset_complement(the_carrier(A!5), B!6))))),
% 0.23/0.47 inference(rewrite,[status(thm)],[])).
% 0.23/0.47 tff(147,plain,
% 0.23/0.47 ((~![A: $i, B: $i, C: $i] : ((~in(B, C)) | (~element(C, powerset(A))) | (~in(B, subset_complement(A, C))))) | ((~in(C!7, B!6)) | (~element(B!6, powerset(the_carrier(A!5)))) | (~in(C!7, subset_complement(the_carrier(A!5), B!6))))),
% 0.23/0.47 inference(quant_inst,[status(thm)],[])).
% 0.23/0.47 tff(148,plain,
% 0.23/0.47 ((~![A: $i, B: $i, C: $i] : ((~in(B, C)) | (~element(C, powerset(A))) | (~in(B, subset_complement(A, C))))) | (~in(C!7, B!6)) | (~element(B!6, powerset(the_carrier(A!5)))) | (~in(C!7, subset_complement(the_carrier(A!5), B!6)))),
% 0.23/0.47 inference(modus_ponens,[status(thm)],[147, 146])).
% 0.23/0.47 tff(149,plain,
% 0.23/0.47 ((~in(C!7, B!6)) | (~in(C!7, subset_complement(the_carrier(A!5), B!6)))),
% 0.23/0.47 inference(unit_resolution,[status(thm)],[148, 73, 145])).
% 0.23/0.47 tff(150,plain,
% 0.23/0.47 ($false),
% 0.23/0.47 inference(unit_resolution,[status(thm)],[149, 132, 129])).
% 0.23/0.47 % SZS output end Proof
% 0.23/0.47 % E exiting
%------------------------------------------------------------------------------