%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU362+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n008.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 05:27:42 AM UTC 2025
% Result : Theorem 0.13s 0.40s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.13/0.13 % Problem : SEU362+1 : TPTP v9.0.0. Released v3.3.0.
% 0.13/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n008.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Jun 20 06:52:48 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.13/0.40 % SZS status Theorem
% 0.13/0.40 % SZS output start Proof
% 0.13/0.40 tff(in_type, type, (
% 0.13/0.40 in: ( $i * $i ) > $o)).
% 0.13/0.40 tff(the_InternalRel_type, type, (
% 0.13/0.40 the_InternalRel: $i > $i)).
% 0.13/0.40 tff(tptp_fun_B_11_type, type, (
% 0.13/0.40 tptp_fun_B_11: $i)).
% 0.13/0.40 tff(ordered_pair_type, type, (
% 0.13/0.40 ordered_pair: ( $i * $i ) > $i)).
% 0.13/0.40 tff(tptp_fun_D_13_type, type, (
% 0.13/0.40 tptp_fun_D_13: $i)).
% 0.13/0.40 tff(tptp_fun_C_12_type, type, (
% 0.13/0.40 tptp_fun_C_12: $i)).
% 0.13/0.40 tff(related_type, type, (
% 0.13/0.40 related: ( $i * $i * $i ) > $o)).
% 0.13/0.40 tff(element_type, type, (
% 0.13/0.40 element: ( $i * $i ) > $o)).
% 0.13/0.40 tff(the_carrier_type, type, (
% 0.13/0.40 the_carrier: $i > $i)).
% 0.13/0.40 tff(rel_str_type, type, (
% 0.13/0.40 rel_str: $i > $o)).
% 0.13/0.40 tff(subrelstr_type, type, (
% 0.13/0.40 subrelstr: ( $i * $i ) > $o)).
% 0.13/0.40 tff(tptp_fun_A_10_type, type, (
% 0.13/0.40 tptp_fun_A_10: $i)).
% 0.13/0.40 tff(empty_type, type, (
% 0.13/0.40 empty: $i > $o)).
% 0.13/0.40 tff(powerset_type, type, (
% 0.13/0.40 powerset: $i > $i)).
% 0.13/0.40 tff(subset_type, type, (
% 0.13/0.40 subset: ( $i * $i ) > $o)).
% 0.13/0.40 tff(1,plain,
% 0.13/0.40 ((rel_str(A!10) & (subrelstr(B!11, A!10) & element(C!12, the_carrier(A!10)) & (~((~element(C!12, the_carrier(B!11))) | (~element(D!13, the_carrier(B!11))) | (~element(D!13, the_carrier(A!10))) | related(A!10, C!12, D!13) | (~related(B!11, C!12, D!13)))))) <=> (rel_str(A!10) & subrelstr(B!11, A!10) & element(C!12, the_carrier(A!10)) & (~((~element(C!12, the_carrier(B!11))) | (~element(D!13, the_carrier(B!11))) | (~element(D!13, the_carrier(A!10))) | related(A!10, C!12, D!13) | (~related(B!11, C!12, D!13)))))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(2,plain,
% 0.13/0.40 ((subrelstr(B!11, A!10) & (element(C!12, the_carrier(A!10)) & (~((~element(C!12, the_carrier(B!11))) | (~element(D!13, the_carrier(B!11))) | (~element(D!13, the_carrier(A!10))) | related(A!10, C!12, D!13) | (~related(B!11, C!12, D!13)))))) <=> (subrelstr(B!11, A!10) & element(C!12, the_carrier(A!10)) & (~((~element(C!12, the_carrier(B!11))) | (~element(D!13, the_carrier(B!11))) | (~element(D!13, the_carrier(A!10))) | related(A!10, C!12, D!13) | (~related(B!11, C!12, D!13)))))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(3,plain,
% 0.13/0.40 ((element(C!12, the_carrier(A!10)) & (~((~element(D!13, the_carrier(B!11))) | (~element(D!13, the_carrier(A!10))) | (~element(C!12, the_carrier(B!11))) | related(A!10, C!12, D!13) | (~related(B!11, C!12, D!13))))) <=> (element(C!12, the_carrier(A!10)) & (~((~element(C!12, the_carrier(B!11))) | (~element(D!13, the_carrier(B!11))) | (~element(D!13, the_carrier(A!10))) | related(A!10, C!12, D!13) | (~related(B!11, C!12, D!13)))))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(4,plain,
% 0.13/0.40 ((subrelstr(B!11, A!10) & (element(C!12, the_carrier(A!10)) & (~((~element(D!13, the_carrier(B!11))) | (~element(D!13, the_carrier(A!10))) | (~element(C!12, the_carrier(B!11))) | related(A!10, C!12, D!13) | (~related(B!11, C!12, D!13)))))) <=> (subrelstr(B!11, A!10) & (element(C!12, the_carrier(A!10)) & (~((~element(C!12, the_carrier(B!11))) | (~element(D!13, the_carrier(B!11))) | (~element(D!13, the_carrier(A!10))) | related(A!10, C!12, D!13) | (~related(B!11, C!12, D!13))))))),
% 0.13/0.40 inference(monotonicity,[status(thm)],[3])).
% 0.13/0.40 tff(5,plain,
% 0.13/0.40 ((subrelstr(B!11, A!10) & (element(C!12, the_carrier(A!10)) & (~((~element(D!13, the_carrier(B!11))) | (~element(D!13, the_carrier(A!10))) | (~element(C!12, the_carrier(B!11))) | related(A!10, C!12, D!13) | (~related(B!11, C!12, D!13)))))) <=> (subrelstr(B!11, A!10) & element(C!12, the_carrier(A!10)) & (~((~element(C!12, the_carrier(B!11))) | (~element(D!13, the_carrier(B!11))) | (~element(D!13, the_carrier(A!10))) | related(A!10, C!12, D!13) | (~related(B!11, C!12, D!13)))))),
% 0.13/0.40 inference(transitivity,[status(thm)],[4, 2])).
% 0.13/0.40 tff(6,plain,
% 0.13/0.40 ((rel_str(A!10) & (subrelstr(B!11, A!10) & (element(C!12, the_carrier(A!10)) & (~((~element(D!13, the_carrier(B!11))) | (~element(D!13, the_carrier(A!10))) | (~element(C!12, the_carrier(B!11))) | related(A!10, C!12, D!13) | (~related(B!11, C!12, D!13))))))) <=> (rel_str(A!10) & (subrelstr(B!11, A!10) & element(C!12, the_carrier(A!10)) & (~((~element(C!12, the_carrier(B!11))) | (~element(D!13, the_carrier(B!11))) | (~element(D!13, the_carrier(A!10))) | related(A!10, C!12, D!13) | (~related(B!11, C!12, D!13))))))),
% 0.13/0.41 inference(monotonicity,[status(thm)],[5])).
% 0.13/0.41 tff(7,plain,
% 0.13/0.41 ((rel_str(A!10) & (subrelstr(B!11, A!10) & (element(C!12, the_carrier(A!10)) & (~((~element(D!13, the_carrier(B!11))) | (~element(D!13, the_carrier(A!10))) | (~element(C!12, the_carrier(B!11))) | related(A!10, C!12, D!13) | (~related(B!11, C!12, D!13))))))) <=> (rel_str(A!10) & subrelstr(B!11, A!10) & element(C!12, the_carrier(A!10)) & (~((~element(C!12, the_carrier(B!11))) | (~element(D!13, the_carrier(B!11))) | (~element(D!13, the_carrier(A!10))) | related(A!10, C!12, D!13) | (~related(B!11, C!12, D!13)))))),
% 0.13/0.41 inference(transitivity,[status(thm)],[6, 1])).
% 0.13/0.41 tff(8,plain,
% 0.13/0.41 ((~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~subrelstr(B, A)) | ![C: $i] : ((~element(C, the_carrier(A))) | ![D: $i] : ((~element(D, the_carrier(B))) | (~element(D, the_carrier(A))) | (~element(C, the_carrier(B))) | related(A, C, D) | (~related(B, C, D))))))) <=> (~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~subrelstr(B, A)) | ![C: $i] : ((~element(C, the_carrier(A))) | ![D: $i] : ((~element(D, the_carrier(B))) | (~element(D, the_carrier(A))) | (~element(C, the_carrier(B))) | related(A, C, D) | (~related(B, C, D)))))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(9,plain,
% 0.13/0.41 ((~![A: $i] : (rel_str(A) => ![B: $i] : (subrelstr(B, A) => ![C: $i] : (element(C, the_carrier(A)) => ![D: $i] : (element(D, the_carrier(A)) => ![E: $i] : (element(E, the_carrier(B)) => ![F: $i] : (element(F, the_carrier(B)) => ((((E = C) & (F = D)) & related(B, E, F)) => related(A, C, D))))))))) <=> (~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~subrelstr(B, A)) | ![C: $i] : ((~element(C, the_carrier(A))) | ![D: $i] : ((~element(D, the_carrier(B))) | (~element(D, the_carrier(A))) | (~element(C, the_carrier(B))) | related(A, C, D) | (~related(B, C, D)))))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(10,axiom,(~![A: $i] : (rel_str(A) => ![B: $i] : (subrelstr(B, A) => ![C: $i] : (element(C, the_carrier(A)) => ![D: $i] : (element(D, the_carrier(A)) => ![E: $i] : (element(E, the_carrier(B)) => ![F: $i] : (element(F, the_carrier(B)) => ((((E = C) & (F = D)) & related(B, E, F)) => related(A, C, D))))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t60_yellow_0')).
% 0.13/0.41 tff(11,plain,
% 0.13/0.41 (~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~subrelstr(B, A)) | ![C: $i] : ((~element(C, the_carrier(A))) | ![D: $i] : ((~element(D, the_carrier(B))) | (~element(D, the_carrier(A))) | (~element(C, the_carrier(B))) | related(A, C, D) | (~related(B, C, D))))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[10, 9])).
% 0.13/0.41 tff(12,plain,
% 0.13/0.41 (~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~subrelstr(B, A)) | ![C: $i] : ((~element(C, the_carrier(A))) | ![D: $i] : ((~element(D, the_carrier(B))) | (~element(D, the_carrier(A))) | (~element(C, the_carrier(B))) | related(A, C, D) | (~related(B, C, D))))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[11, 8])).
% 0.13/0.41 tff(13,plain,
% 0.13/0.41 (~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~subrelstr(B, A)) | ![C: $i] : ((~element(C, the_carrier(A))) | ![D: $i] : ((~element(D, the_carrier(B))) | (~element(D, the_carrier(A))) | (~element(C, the_carrier(B))) | related(A, C, D) | (~related(B, C, D))))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.13/0.41 tff(14,plain,
% 0.13/0.41 (~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~subrelstr(B, A)) | ![C: $i] : ((~element(C, the_carrier(A))) | ![D: $i] : ((~element(D, the_carrier(B))) | (~element(D, the_carrier(A))) | (~element(C, the_carrier(B))) | related(A, C, D) | (~related(B, C, D))))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[13, 8])).
% 0.13/0.41 tff(15,plain,
% 0.13/0.41 (rel_str(A!10) & subrelstr(B!11, A!10) & element(C!12, the_carrier(A!10)) & (~((~element(C!12, the_carrier(B!11))) | (~element(D!13, the_carrier(B!11))) | (~element(D!13, the_carrier(A!10))) | related(A!10, C!12, D!13) | (~related(B!11, C!12, D!13))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.13/0.41 tff(16,plain,
% 0.13/0.41 (rel_str(A!10)),
% 0.13/0.41 inference(and_elim,[status(thm)],[15])).
% 0.13/0.41 tff(17,plain,
% 0.13/0.41 (^[A: $i] : refl(((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A)))) <=> ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A)))))),
% 0.13/0.41 inference(bind,[status(th)],[])).
% 0.13/0.41 tff(18,plain,
% 0.13/0.41 (![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A)))) <=> ![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A))))),
% 0.13/0.41 inference(quant_intro,[status(thm)],[17])).
% 0.13/0.41 tff(19,plain,
% 0.13/0.41 (^[A: $i] : rewrite(((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A)))) <=> ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A)))))),
% 0.13/0.41 inference(bind,[status(th)],[])).
% 0.13/0.41 tff(20,plain,
% 0.13/0.41 (![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A)))) <=> ![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A))))),
% 0.13/0.41 inference(quant_intro,[status(thm)],[19])).
% 0.13/0.41 tff(21,plain,
% 0.13/0.41 (![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A)))) <=> ![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A))))),
% 0.13/0.41 inference(transitivity,[status(thm)],[20, 18])).
% 0.13/0.41 tff(22,plain,
% 0.13/0.41 (![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A)))) <=> ![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(23,plain,
% 0.13/0.41 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : rewrite((subrelstr(B, A) => rel_str(B)) <=> (rel_str(B) | (~subrelstr(B, A))))), (![B: $i] : (subrelstr(B, A) => rel_str(B)) <=> ![B: $i] : (rel_str(B) | (~subrelstr(B, A))))), ((rel_str(A) => ![B: $i] : (subrelstr(B, A) => rel_str(B))) <=> (rel_str(A) => ![B: $i] : (rel_str(B) | (~subrelstr(B, A)))))), rewrite((rel_str(A) => ![B: $i] : (rel_str(B) | (~subrelstr(B, A)))) <=> ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A))))), ((rel_str(A) => ![B: $i] : (subrelstr(B, A) => rel_str(B))) <=> ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A))))))),
% 0.13/0.41 inference(bind,[status(th)],[])).
% 0.13/0.41 tff(24,plain,
% 0.13/0.41 (![A: $i] : (rel_str(A) => ![B: $i] : (subrelstr(B, A) => rel_str(B))) <=> ![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A))))),
% 0.13/0.41 inference(quant_intro,[status(thm)],[23])).
% 0.13/0.41 tff(25,axiom,(![A: $i] : (rel_str(A) => ![B: $i] : (subrelstr(B, A) => rel_str(B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_m1_yellow_0')).
% 0.13/0.41 tff(26,plain,
% 0.13/0.41 (![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[25, 24])).
% 0.13/0.41 tff(27,plain,
% 0.13/0.41 (![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[26, 22])).
% 0.13/0.41 tff(28,plain,(
% 0.13/0.41 ![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A))))),
% 0.13/0.41 inference(skolemize,[status(sab)],[27])).
% 0.13/0.41 tff(29,plain,
% 0.13/0.41 (![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[28, 21])).
% 0.13/0.41 tff(30,plain,
% 0.13/0.41 (((~![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A))))) | ((~rel_str(A!10)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A!10))))) <=> ((~![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A))))) | (~rel_str(A!10)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A!10))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(31,plain,
% 0.13/0.41 ((~![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A))))) | ((~rel_str(A!10)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A!10))))),
% 0.13/0.41 inference(quant_inst,[status(thm)],[])).
% 0.13/0.41 tff(32,plain,
% 0.13/0.41 ((~![A: $i] : ((~rel_str(A)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A))))) | (~rel_str(A!10)) | ![B: $i] : (rel_str(B) | (~subrelstr(B, A!10)))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[31, 30])).
% 0.13/0.41 tff(33,plain,
% 0.13/0.41 (![B: $i] : (rel_str(B) | (~subrelstr(B, A!10)))),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[32, 29, 16])).
% 0.13/0.41 tff(34,plain,
% 0.13/0.41 (subrelstr(B!11, A!10)),
% 0.13/0.41 inference(and_elim,[status(thm)],[15])).
% 0.13/0.41 tff(35,plain,
% 0.13/0.41 (((~![B: $i] : (rel_str(B) | (~subrelstr(B, A!10)))) | (rel_str(B!11) | (~subrelstr(B!11, A!10)))) <=> ((~![B: $i] : (rel_str(B) | (~subrelstr(B, A!10)))) | rel_str(B!11) | (~subrelstr(B!11, A!10)))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(36,plain,
% 0.13/0.41 ((~![B: $i] : (rel_str(B) | (~subrelstr(B, A!10)))) | (rel_str(B!11) | (~subrelstr(B!11, A!10)))),
% 0.13/0.41 inference(quant_inst,[status(thm)],[])).
% 0.13/0.41 tff(37,plain,
% 0.13/0.41 ((~![B: $i] : (rel_str(B) | (~subrelstr(B, A!10)))) | rel_str(B!11) | (~subrelstr(B!11, A!10))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.13/0.41 tff(38,plain,
% 0.13/0.41 (rel_str(B!11)),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[37, 34, 33])).
% 0.13/0.41 tff(39,plain,
% 0.13/0.41 (^[A: $i] : refl(((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))) <=> ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))))),
% 0.13/0.41 inference(bind,[status(th)],[])).
% 0.13/0.41 tff(40,plain,
% 0.13/0.41 (![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))) <=> ![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))),
% 0.13/0.41 inference(quant_intro,[status(thm)],[39])).
% 0.13/0.41 tff(41,plain,
% 0.13/0.41 (^[A: $i] : rewrite(((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))) <=> ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))))),
% 0.13/0.41 inference(bind,[status(th)],[])).
% 0.13/0.41 tff(42,plain,
% 0.13/0.41 (![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))) <=> ![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))),
% 0.13/0.41 inference(quant_intro,[status(thm)],[41])).
% 0.13/0.41 tff(43,plain,
% 0.13/0.41 (![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))) <=> ![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))),
% 0.13/0.41 inference(transitivity,[status(thm)],[42, 40])).
% 0.13/0.41 tff(44,plain,
% 0.13/0.41 (![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))) <=> ![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(45,plain,
% 0.13/0.41 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : trans(monotonicity(quant_intro(proof_bind(^[C: $i] : rewrite((element(C, the_carrier(A)) => (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))) <=> ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))), (![C: $i] : (element(C, the_carrier(A)) => (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))) <=> ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))), ((element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))) <=> (element(B, the_carrier(A)) => ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))), rewrite((element(B, the_carrier(A)) => ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))) <=> ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))), ((element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))) <=> ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))))), (![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))) <=> ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))), ((rel_str(A) => ![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))) <=> (rel_str(A) => ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))))), rewrite((rel_str(A) => ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))) <=> ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))), ((rel_str(A) => ![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))) <=> ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))))),
% 0.13/0.41 inference(bind,[status(th)],[])).
% 0.13/0.41 tff(46,plain,
% 0.13/0.41 (![A: $i] : (rel_str(A) => ![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A)))))) <=> ![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))),
% 0.13/0.41 inference(quant_intro,[status(thm)],[45])).
% 0.13/0.41 tff(47,axiom,(![A: $i] : (rel_str(A) => ![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d9_orders_2')).
% 0.13/0.41 tff(48,plain,
% 0.13/0.41 (![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.13/0.41 tff(49,plain,
% 0.13/0.41 (![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[48, 44])).
% 0.13/0.41 tff(50,plain,(
% 0.13/0.41 ![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))),
% 0.13/0.41 inference(skolemize,[status(sab)],[49])).
% 0.13/0.41 tff(51,plain,
% 0.13/0.41 (![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[50, 43])).
% 0.13/0.41 tff(52,plain,
% 0.13/0.41 (((~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))) | ((~rel_str(B!11)) | ![B: $i] : ((~element(B, the_carrier(B!11))) | ![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, B, C) <=> in(ordered_pair(B, C), the_InternalRel(B!11))))))) <=> ((~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))) | (~rel_str(B!11)) | ![B: $i] : ((~element(B, the_carrier(B!11))) | ![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, B, C) <=> in(ordered_pair(B, C), the_InternalRel(B!11))))))),
% 0.13/0.42 inference(rewrite,[status(thm)],[])).
% 0.13/0.42 tff(53,plain,
% 0.13/0.42 ((~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))) | ((~rel_str(B!11)) | ![B: $i] : ((~element(B, the_carrier(B!11))) | ![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, B, C) <=> in(ordered_pair(B, C), the_InternalRel(B!11))))))),
% 0.13/0.42 inference(quant_inst,[status(thm)],[])).
% 0.13/0.42 tff(54,plain,
% 0.13/0.42 ((~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))) | (~rel_str(B!11)) | ![B: $i] : ((~element(B, the_carrier(B!11))) | ![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, B, C) <=> in(ordered_pair(B, C), the_InternalRel(B!11)))))),
% 0.13/0.42 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.13/0.42 tff(55,plain,
% 0.13/0.42 ((~rel_str(B!11)) | ![B: $i] : ((~element(B, the_carrier(B!11))) | ![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, B, C) <=> in(ordered_pair(B, C), the_InternalRel(B!11)))))),
% 0.13/0.42 inference(unit_resolution,[status(thm)],[54, 51])).
% 0.13/0.42 tff(56,plain,
% 0.13/0.42 (![B: $i] : ((~element(B, the_carrier(B!11))) | ![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, B, C) <=> in(ordered_pair(B, C), the_InternalRel(B!11)))))),
% 0.13/0.42 inference(unit_resolution,[status(thm)],[55, 38])).
% 0.13/0.42 tff(57,plain,
% 0.13/0.42 (~((~element(C!12, the_carrier(B!11))) | (~element(D!13, the_carrier(B!11))) | (~element(D!13, the_carrier(A!10))) | related(A!10, C!12, D!13) | (~related(B!11, C!12, D!13)))),
% 0.13/0.42 inference(and_elim,[status(thm)],[15])).
% 0.13/0.42 tff(58,plain,
% 0.13/0.42 (element(C!12, the_carrier(B!11))),
% 0.13/0.42 inference(or_elim,[status(thm)],[57])).
% 0.13/0.42 tff(59,plain,
% 0.13/0.42 (((~![B: $i] : ((~element(B, the_carrier(B!11))) | ![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, B, C) <=> in(ordered_pair(B, C), the_InternalRel(B!11)))))) | ((~element(C!12, the_carrier(B!11))) | ![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(B!11)))))) <=> ((~![B: $i] : ((~element(B, the_carrier(B!11))) | ![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, B, C) <=> in(ordered_pair(B, C), the_InternalRel(B!11)))))) | (~element(C!12, the_carrier(B!11))) | ![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(B!11)))))),
% 0.13/0.42 inference(rewrite,[status(thm)],[])).
% 0.13/0.42 tff(60,plain,
% 0.13/0.42 ((~![B: $i] : ((~element(B, the_carrier(B!11))) | ![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, B, C) <=> in(ordered_pair(B, C), the_InternalRel(B!11)))))) | ((~element(C!12, the_carrier(B!11))) | ![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(B!11)))))),
% 0.13/0.42 inference(quant_inst,[status(thm)],[])).
% 0.13/0.42 tff(61,plain,
% 0.13/0.42 ((~![B: $i] : ((~element(B, the_carrier(B!11))) | ![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, B, C) <=> in(ordered_pair(B, C), the_InternalRel(B!11)))))) | (~element(C!12, the_carrier(B!11))) | ![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(B!11))))),
% 0.13/0.42 inference(modus_ponens,[status(thm)],[60, 59])).
% 0.13/0.42 tff(62,plain,
% 0.13/0.42 (![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(B!11))))),
% 0.13/0.42 inference(unit_resolution,[status(thm)],[61, 58, 56])).
% 0.13/0.42 tff(63,plain,
% 0.13/0.42 (element(D!13, the_carrier(B!11))),
% 0.13/0.42 inference(or_elim,[status(thm)],[57])).
% 0.13/0.42 tff(64,plain,
% 0.13/0.42 (((~![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(B!11))))) | ((~element(D!13, the_carrier(B!11))) | (related(B!11, C!12, D!13) <=> in(ordered_pair(C!12, D!13), the_InternalRel(B!11))))) <=> ((~![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(B!11))))) | (~element(D!13, the_carrier(B!11))) | (related(B!11, C!12, D!13) <=> in(ordered_pair(C!12, D!13), the_InternalRel(B!11))))),
% 0.13/0.42 inference(rewrite,[status(thm)],[])).
% 0.13/0.42 tff(65,plain,
% 0.13/0.42 ((~![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(B!11))))) | ((~element(D!13, the_carrier(B!11))) | (related(B!11, C!12, D!13) <=> in(ordered_pair(C!12, D!13), the_InternalRel(B!11))))),
% 0.13/0.42 inference(quant_inst,[status(thm)],[])).
% 0.13/0.42 tff(66,plain,
% 0.13/0.42 ((~![C: $i] : ((~element(C, the_carrier(B!11))) | (related(B!11, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(B!11))))) | (~element(D!13, the_carrier(B!11))) | (related(B!11, C!12, D!13) <=> in(ordered_pair(C!12, D!13), the_InternalRel(B!11)))),
% 0.13/0.42 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.13/0.42 tff(67,plain,
% 0.13/0.42 (related(B!11, C!12, D!13) <=> in(ordered_pair(C!12, D!13), the_InternalRel(B!11))),
% 0.13/0.42 inference(unit_resolution,[status(thm)],[66, 63, 62])).
% 0.13/0.42 tff(68,plain,
% 0.13/0.42 (related(B!11, C!12, D!13)),
% 0.13/0.42 inference(or_elim,[status(thm)],[57])).
% 0.13/0.42 tff(69,plain,
% 0.13/0.42 ((~(related(B!11, C!12, D!13) <=> in(ordered_pair(C!12, D!13), the_InternalRel(B!11)))) | (~related(B!11, C!12, D!13)) | in(ordered_pair(C!12, D!13), the_InternalRel(B!11))),
% 0.13/0.42 inference(tautology,[status(thm)],[])).
% 0.13/0.42 tff(70,plain,
% 0.13/0.42 ((~(related(B!11, C!12, D!13) <=> in(ordered_pair(C!12, D!13), the_InternalRel(B!11)))) | in(ordered_pair(C!12, D!13), the_InternalRel(B!11))),
% 0.13/0.42 inference(unit_resolution,[status(thm)],[69, 68])).
% 0.13/0.42 tff(71,plain,
% 0.13/0.42 (in(ordered_pair(C!12, D!13), the_InternalRel(B!11))),
% 0.13/0.42 inference(unit_resolution,[status(thm)],[70, 67])).
% 0.13/0.42 tff(72,plain,
% 0.13/0.42 (^[A: $i, B: $i] : refl((element(A, powerset(B)) <=> subset(A, B)) <=> (element(A, powerset(B)) <=> subset(A, B)))),
% 0.13/0.42 inference(bind,[status(th)],[])).
% 0.13/0.42 tff(73,plain,
% 0.13/0.42 (![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B)) <=> ![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))),
% 0.13/0.42 inference(quant_intro,[status(thm)],[72])).
% 0.13/0.42 tff(74,plain,
% 0.13/0.42 (![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B)) <=> ![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))),
% 0.13/0.42 inference(rewrite,[status(thm)],[])).
% 0.13/0.42 tff(75,axiom,(![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t3_subset')).
% 0.13/0.42 tff(76,plain,
% 0.13/0.42 (![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))),
% 0.13/0.42 inference(modus_ponens,[status(thm)],[75, 74])).
% 0.13/0.42 tff(77,plain,(
% 0.13/0.42 ![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))),
% 0.13/0.42 inference(skolemize,[status(sab)],[76])).
% 0.13/0.42 tff(78,plain,
% 0.13/0.42 (![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))),
% 0.13/0.42 inference(modus_ponens,[status(thm)],[77, 73])).
% 0.13/0.42 tff(79,plain,
% 0.13/0.42 ((~![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))) | (element(the_InternalRel(B!11), powerset(the_InternalRel(A!10))) <=> subset(the_InternalRel(B!11), the_InternalRel(A!10)))),
% 0.13/0.42 inference(quant_inst,[status(thm)],[])).
% 0.13/0.42 tff(80,plain,
% 0.13/0.42 (element(the_InternalRel(B!11), powerset(the_InternalRel(A!10))) <=> subset(the_InternalRel(B!11), the_InternalRel(A!10))),
% 0.13/0.42 inference(unit_resolution,[status(thm)],[79, 78])).
% 0.13/0.42 tff(81,plain,
% 0.13/0.42 (^[A: $i] : refl(((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A))))) <=> ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A))))))),
% 0.13/0.42 inference(bind,[status(th)],[])).
% 0.13/0.42 tff(82,plain,
% 0.13/0.42 (![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A))))) <=> ![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A)))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[81])).
% 0.21/0.42 tff(83,plain,
% 0.21/0.42 (^[A: $i] : rewrite(((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A))))) <=> ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A))))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(84,plain,
% 0.21/0.42 (![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A))))) <=> ![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A)))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[83])).
% 0.21/0.42 tff(85,plain,
% 0.21/0.42 (![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A))))) <=> ![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A)))))),
% 0.21/0.42 inference(transitivity,[status(thm)],[84, 82])).
% 0.21/0.42 tff(86,plain,
% 0.21/0.42 (^[A: $i] : rewrite(((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A)))))) <=> ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A))))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(87,plain,
% 0.21/0.42 (![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A)))))) <=> ![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A)))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[86])).
% 0.21/0.42 tff(88,plain,
% 0.21/0.42 (![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A)))))) <=> ![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A))))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(89,plain,
% 0.21/0.42 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : rewrite((rel_str(B) => (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A))))) <=> ((~rel_str(B)) | (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A))))))), (![B: $i] : (rel_str(B) => (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A))))) <=> ![B: $i] : ((~rel_str(B)) | (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A))))))), ((rel_str(A) => ![B: $i] : (rel_str(B) => (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A)))))) <=> (rel_str(A) => ![B: $i] : ((~rel_str(B)) | (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A)))))))), rewrite((rel_str(A) => ![B: $i] : ((~rel_str(B)) | (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A)))))) <=> ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A))))))), ((rel_str(A) => ![B: $i] : (rel_str(B) => (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A)))))) <=> ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A))))))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(90,plain,
% 0.21/0.42 (![A: $i] : (rel_str(A) => ![B: $i] : (rel_str(B) => (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A)))))) <=> ![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A))))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[89])).
% 0.21/0.42 tff(91,axiom,(![A: $i] : (rel_str(A) => ![B: $i] : (rel_str(B) => (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d13_yellow_0')).
% 0.21/0.42 tff(92,plain,
% 0.21/0.42 (![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A))))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.21/0.42 tff(93,plain,
% 0.21/0.42 (![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A))))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[92, 88])).
% 0.21/0.42 tff(94,plain,(
% 0.21/0.42 ![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (subrelstr(B, A) <=> (subset(the_carrier(B), the_carrier(A)) & subset(the_InternalRel(B), the_InternalRel(A))))))),
% 0.21/0.42 inference(skolemize,[status(sab)],[93])).
% 0.21/0.42 tff(95,plain,
% 0.21/0.42 (![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A)))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[94, 87])).
% 0.21/0.42 tff(96,plain,
% 0.21/0.42 (![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A)))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[95, 85])).
% 0.21/0.42 tff(97,plain,
% 0.21/0.42 (((~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A)))))) | ((~rel_str(A!10)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A!10))) | (~subset(the_InternalRel(B), the_InternalRel(A!10)))) <=> subrelstr(B, A!10)))))) <=> ((~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A)))))) | (~rel_str(A!10)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A!10))) | (~subset(the_InternalRel(B), the_InternalRel(A!10)))) <=> subrelstr(B, A!10)))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(98,plain,
% 0.21/0.42 ((~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A)))))) | ((~rel_str(A!10)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A!10))) | (~subset(the_InternalRel(B), the_InternalRel(A!10)))) <=> subrelstr(B, A!10)))))),
% 0.21/0.42 inference(quant_inst,[status(thm)],[])).
% 0.21/0.42 tff(99,plain,
% 0.21/0.42 ((~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A))) | (~subset(the_InternalRel(B), the_InternalRel(A)))) <=> subrelstr(B, A)))))) | (~rel_str(A!10)) | ![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A!10))) | (~subset(the_InternalRel(B), the_InternalRel(A!10)))) <=> subrelstr(B, A!10))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[98, 97])).
% 0.21/0.42 tff(100,plain,
% 0.21/0.42 (![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A!10))) | (~subset(the_InternalRel(B), the_InternalRel(A!10)))) <=> subrelstr(B, A!10))))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[99, 96, 16])).
% 0.21/0.42 tff(101,plain,
% 0.21/0.42 (((~![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A!10))) | (~subset(the_InternalRel(B), the_InternalRel(A!10)))) <=> subrelstr(B, A!10))))) | ((~rel_str(B!11)) | (~(((~subset(the_carrier(B!11), the_carrier(A!10))) | (~subset(the_InternalRel(B!11), the_InternalRel(A!10)))) <=> subrelstr(B!11, A!10))))) <=> ((~![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A!10))) | (~subset(the_InternalRel(B), the_InternalRel(A!10)))) <=> subrelstr(B, A!10))))) | (~rel_str(B!11)) | (~(((~subset(the_carrier(B!11), the_carrier(A!10))) | (~subset(the_InternalRel(B!11), the_InternalRel(A!10)))) <=> subrelstr(B!11, A!10))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(102,plain,
% 0.21/0.42 ((~![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A!10))) | (~subset(the_InternalRel(B), the_InternalRel(A!10)))) <=> subrelstr(B, A!10))))) | ((~rel_str(B!11)) | (~(((~subset(the_carrier(B!11), the_carrier(A!10))) | (~subset(the_InternalRel(B!11), the_InternalRel(A!10)))) <=> subrelstr(B!11, A!10))))),
% 0.21/0.42 inference(quant_inst,[status(thm)],[])).
% 0.21/0.42 tff(103,plain,
% 0.21/0.42 ((~![B: $i] : ((~rel_str(B)) | (~(((~subset(the_carrier(B), the_carrier(A!10))) | (~subset(the_InternalRel(B), the_InternalRel(A!10)))) <=> subrelstr(B, A!10))))) | (~rel_str(B!11)) | (~(((~subset(the_carrier(B!11), the_carrier(A!10))) | (~subset(the_InternalRel(B!11), the_InternalRel(A!10)))) <=> subrelstr(B!11, A!10)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.21/0.42 tff(104,plain,
% 0.21/0.42 (~(((~subset(the_carrier(B!11), the_carrier(A!10))) | (~subset(the_InternalRel(B!11), the_InternalRel(A!10)))) <=> subrelstr(B!11, A!10))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[103, 100, 38])).
% 0.21/0.42 tff(105,plain,
% 0.21/0.42 ((((~subset(the_carrier(B!11), the_carrier(A!10))) | (~subset(the_InternalRel(B!11), the_InternalRel(A!10)))) <=> subrelstr(B!11, A!10)) | (~((~subset(the_carrier(B!11), the_carrier(A!10))) | (~subset(the_InternalRel(B!11), the_InternalRel(A!10))))) | (~subrelstr(B!11, A!10))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(106,plain,
% 0.21/0.42 ((((~subset(the_carrier(B!11), the_carrier(A!10))) | (~subset(the_InternalRel(B!11), the_InternalRel(A!10)))) <=> subrelstr(B!11, A!10)) | (~((~subset(the_carrier(B!11), the_carrier(A!10))) | (~subset(the_InternalRel(B!11), the_InternalRel(A!10)))))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[105, 34])).
% 0.21/0.42 tff(107,plain,
% 0.21/0.42 (~((~subset(the_carrier(B!11), the_carrier(A!10))) | (~subset(the_InternalRel(B!11), the_InternalRel(A!10))))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[106, 104])).
% 0.21/0.42 tff(108,plain,
% 0.21/0.42 (((~subset(the_carrier(B!11), the_carrier(A!10))) | (~subset(the_InternalRel(B!11), the_InternalRel(A!10)))) | subset(the_InternalRel(B!11), the_InternalRel(A!10))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(109,plain,
% 0.21/0.42 (subset(the_InternalRel(B!11), the_InternalRel(A!10))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[108, 107])).
% 0.21/0.42 tff(110,plain,
% 0.21/0.42 ((~(element(the_InternalRel(B!11), powerset(the_InternalRel(A!10))) <=> subset(the_InternalRel(B!11), the_InternalRel(A!10)))) | element(the_InternalRel(B!11), powerset(the_InternalRel(A!10))) | (~subset(the_InternalRel(B!11), the_InternalRel(A!10)))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(111,plain,
% 0.21/0.42 ((~(element(the_InternalRel(B!11), powerset(the_InternalRel(A!10))) <=> subset(the_InternalRel(B!11), the_InternalRel(A!10)))) | element(the_InternalRel(B!11), powerset(the_InternalRel(A!10)))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[110, 109])).
% 0.21/0.42 tff(112,plain,
% 0.21/0.42 (element(the_InternalRel(B!11), powerset(the_InternalRel(A!10)))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[111, 80])).
% 0.21/0.42 tff(113,plain,
% 0.21/0.42 (^[A: $i, B: $i, C: $i] : refl(((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C)))) <=> ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C)))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(114,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C)))) <=> ![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[113])).
% 0.21/0.43 tff(115,plain,
% 0.21/0.43 (^[A: $i, B: $i, C: $i] : trans(monotonicity(rewrite((in(A, B) & element(B, powerset(C)) & empty(C)) <=> (~((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C)))))), ((~(in(A, B) & element(B, powerset(C)) & empty(C))) <=> (~(~((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C)))))))), rewrite((~(~((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C)))))) <=> ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))), ((~(in(A, B) & element(B, powerset(C)) & empty(C))) <=> ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(116,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : (~(in(A, B) & element(B, powerset(C)) & empty(C))) <=> ![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[115])).
% 0.21/0.43 tff(117,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : (~(in(A, B) & element(B, powerset(C)) & empty(C))) <=> ![A: $i, B: $i, C: $i] : (~(in(A, B) & element(B, powerset(C)) & empty(C)))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(118,plain,
% 0.21/0.43 (^[A: $i, B: $i, C: $i] : rewrite((~((in(A, B) & element(B, powerset(C))) & empty(C))) <=> (~(in(A, B) & element(B, powerset(C)) & empty(C))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(119,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : (~((in(A, B) & element(B, powerset(C))) & empty(C))) <=> ![A: $i, B: $i, C: $i] : (~(in(A, B) & element(B, powerset(C)) & empty(C)))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[118])).
% 0.21/0.43 tff(120,axiom,(![A: $i, B: $i, C: $i] : (~((in(A, B) & element(B, powerset(C))) & empty(C)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t5_subset')).
% 0.21/0.43 tff(121,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : (~(in(A, B) & element(B, powerset(C)) & empty(C)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[120, 119])).
% 0.21/0.43 tff(122,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : (~(in(A, B) & element(B, powerset(C)) & empty(C)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[121, 117])).
% 0.21/0.43 tff(123,plain,(
% 0.21/0.43 ![A: $i, B: $i, C: $i] : (~(in(A, B) & element(B, powerset(C)) & empty(C)))),
% 0.21/0.43 inference(skolemize,[status(sab)],[122])).
% 0.21/0.43 tff(124,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[123, 116])).
% 0.21/0.43 tff(125,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[124, 114])).
% 0.21/0.43 tff(126,plain,
% 0.21/0.43 (((~![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))) | ((~empty(the_InternalRel(A!10))) | (~in(ordered_pair(C!12, D!13), the_InternalRel(B!11))) | (~element(the_InternalRel(B!11), powerset(the_InternalRel(A!10)))))) <=> ((~![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))) | (~empty(the_InternalRel(A!10))) | (~in(ordered_pair(C!12, D!13), the_InternalRel(B!11))) | (~element(the_InternalRel(B!11), powerset(the_InternalRel(A!10)))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(127,plain,
% 0.21/0.43 ((~![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))) | ((~empty(the_InternalRel(A!10))) | (~in(ordered_pair(C!12, D!13), the_InternalRel(B!11))) | (~element(the_InternalRel(B!11), powerset(the_InternalRel(A!10)))))),
% 0.21/0.43 inference(quant_inst,[status(thm)],[])).
% 0.21/0.43 tff(128,plain,
% 0.21/0.43 ((~![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))) | (~empty(the_InternalRel(A!10))) | (~in(ordered_pair(C!12, D!13), the_InternalRel(B!11))) | (~element(the_InternalRel(B!11), powerset(the_InternalRel(A!10))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[127, 126])).
% 0.21/0.43 tff(129,plain,
% 0.21/0.43 (~empty(the_InternalRel(A!10))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[128, 125, 112, 71])).
% 0.21/0.43 tff(130,plain,
% 0.21/0.43 (((~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))) | ((~rel_str(A!10)) | ![B: $i] : ((~element(B, the_carrier(A!10))) | ![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A!10))))))) <=> ((~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))) | (~rel_str(A!10)) | ![B: $i] : ((~element(B, the_carrier(A!10))) | ![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A!10))))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(131,plain,
% 0.21/0.43 ((~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))) | ((~rel_str(A!10)) | ![B: $i] : ((~element(B, the_carrier(A!10))) | ![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A!10))))))),
% 0.21/0.43 inference(quant_inst,[status(thm)],[])).
% 0.21/0.43 tff(132,plain,
% 0.21/0.43 ((~![A: $i] : ((~rel_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (related(A, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A))))))) | (~rel_str(A!10)) | ![B: $i] : ((~element(B, the_carrier(A!10))) | ![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A!10)))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[131, 130])).
% 0.21/0.43 tff(133,plain,
% 0.21/0.43 (![B: $i] : ((~element(B, the_carrier(A!10))) | ![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A!10)))))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[132, 51, 16])).
% 0.21/0.43 tff(134,plain,
% 0.21/0.43 (element(C!12, the_carrier(A!10))),
% 0.21/0.43 inference(and_elim,[status(thm)],[15])).
% 0.21/0.43 tff(135,plain,
% 0.21/0.43 (((~![B: $i] : ((~element(B, the_carrier(A!10))) | ![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A!10)))))) | ((~element(C!12, the_carrier(A!10))) | ![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(A!10)))))) <=> ((~![B: $i] : ((~element(B, the_carrier(A!10))) | ![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A!10)))))) | (~element(C!12, the_carrier(A!10))) | ![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(A!10)))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(136,plain,
% 0.21/0.43 ((~![B: $i] : ((~element(B, the_carrier(A!10))) | ![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A!10)))))) | ((~element(C!12, the_carrier(A!10))) | ![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(A!10)))))),
% 0.21/0.43 inference(quant_inst,[status(thm)],[])).
% 0.21/0.43 tff(137,plain,
% 0.21/0.43 ((~![B: $i] : ((~element(B, the_carrier(A!10))) | ![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, B, C) <=> in(ordered_pair(B, C), the_InternalRel(A!10)))))) | (~element(C!12, the_carrier(A!10))) | ![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(A!10))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[136, 135])).
% 0.21/0.43 tff(138,plain,
% 0.21/0.43 (![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(A!10))))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[137, 134, 133])).
% 0.21/0.43 tff(139,plain,
% 0.21/0.43 (element(D!13, the_carrier(A!10))),
% 0.21/0.43 inference(or_elim,[status(thm)],[57])).
% 0.21/0.43 tff(140,plain,
% 0.21/0.43 (((~![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(A!10))))) | ((~element(D!13, the_carrier(A!10))) | (related(A!10, C!12, D!13) <=> in(ordered_pair(C!12, D!13), the_InternalRel(A!10))))) <=> ((~![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(A!10))))) | (~element(D!13, the_carrier(A!10))) | (related(A!10, C!12, D!13) <=> in(ordered_pair(C!12, D!13), the_InternalRel(A!10))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(141,plain,
% 0.21/0.43 ((~![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(A!10))))) | ((~element(D!13, the_carrier(A!10))) | (related(A!10, C!12, D!13) <=> in(ordered_pair(C!12, D!13), the_InternalRel(A!10))))),
% 0.21/0.43 inference(quant_inst,[status(thm)],[])).
% 0.21/0.43 tff(142,plain,
% 0.21/0.43 ((~![C: $i] : ((~element(C, the_carrier(A!10))) | (related(A!10, C!12, C) <=> in(ordered_pair(C!12, C), the_InternalRel(A!10))))) | (~element(D!13, the_carrier(A!10))) | (related(A!10, C!12, D!13) <=> in(ordered_pair(C!12, D!13), the_InternalRel(A!10)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[141, 140])).
% 0.21/0.43 tff(143,plain,
% 0.21/0.43 (related(A!10, C!12, D!13) <=> in(ordered_pair(C!12, D!13), the_InternalRel(A!10))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[142, 139, 138])).
% 0.21/0.43 tff(144,plain,
% 0.21/0.43 (~related(A!10, C!12, D!13)),
% 0.21/0.43 inference(or_elim,[status(thm)],[57])).
% 0.21/0.43 tff(145,plain,
% 0.21/0.43 ((~(related(A!10, C!12, D!13) <=> in(ordered_pair(C!12, D!13), the_InternalRel(A!10)))) | related(A!10, C!12, D!13) | (~in(ordered_pair(C!12, D!13), the_InternalRel(A!10)))),
% 0.21/0.43 inference(tautology,[status(thm)],[])).
% 0.21/0.43 tff(146,plain,
% 0.21/0.43 ((~(related(A!10, C!12, D!13) <=> in(ordered_pair(C!12, D!13), the_InternalRel(A!10)))) | (~in(ordered_pair(C!12, D!13), the_InternalRel(A!10)))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[145, 144])).
% 0.21/0.43 tff(147,plain,
% 0.21/0.43 (~in(ordered_pair(C!12, D!13), the_InternalRel(A!10))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[146, 143])).
% 0.21/0.43 tff(148,plain,
% 0.21/0.43 (^[A: $i, B: $i] : refl((empty(B) | in(A, B) | (~element(A, B))) <=> (empty(B) | in(A, B) | (~element(A, B))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(149,plain,
% 0.21/0.43 (![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B))) <=> ![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[148])).
% 0.21/0.43 tff(150,plain,
% 0.21/0.43 (![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B))) <=> ![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(151,plain,
% 0.21/0.43 (^[A: $i, B: $i] : rewrite((element(A, B) => (empty(B) | in(A, B))) <=> (empty(B) | in(A, B) | (~element(A, B))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(152,plain,
% 0.21/0.43 (![A: $i, B: $i] : (element(A, B) => (empty(B) | in(A, B))) <=> ![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[151])).
% 0.21/0.43 tff(153,axiom,(![A: $i, B: $i] : (element(A, B) => (empty(B) | in(A, B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t2_subset')).
% 0.21/0.43 tff(154,plain,
% 0.21/0.43 (![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[153, 152])).
% 0.21/0.43 tff(155,plain,
% 0.21/0.43 (![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[154, 150])).
% 0.21/0.43 tff(156,plain,(
% 0.21/0.43 ![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))),
% 0.21/0.43 inference(skolemize,[status(sab)],[155])).
% 0.21/0.43 tff(157,plain,
% 0.21/0.43 (![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[156, 149])).
% 0.21/0.43 tff(158,plain,
% 0.21/0.43 (((~![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))) | (in(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | empty(the_InternalRel(A!10)) | (~element(ordered_pair(C!12, D!13), the_InternalRel(A!10))))) <=> ((~![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))) | in(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | empty(the_InternalRel(A!10)) | (~element(ordered_pair(C!12, D!13), the_InternalRel(A!10))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(159,plain,
% 0.21/0.43 ((empty(the_InternalRel(A!10)) | in(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | (~element(ordered_pair(C!12, D!13), the_InternalRel(A!10)))) <=> (in(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | empty(the_InternalRel(A!10)) | (~element(ordered_pair(C!12, D!13), the_InternalRel(A!10))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(160,plain,
% 0.21/0.44 (((~![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))) | (empty(the_InternalRel(A!10)) | in(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | (~element(ordered_pair(C!12, D!13), the_InternalRel(A!10))))) <=> ((~![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))) | (in(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | empty(the_InternalRel(A!10)) | (~element(ordered_pair(C!12, D!13), the_InternalRel(A!10)))))),
% 0.21/0.44 inference(monotonicity,[status(thm)],[159])).
% 0.21/0.44 tff(161,plain,
% 0.21/0.44 (((~![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))) | (empty(the_InternalRel(A!10)) | in(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | (~element(ordered_pair(C!12, D!13), the_InternalRel(A!10))))) <=> ((~![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))) | in(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | empty(the_InternalRel(A!10)) | (~element(ordered_pair(C!12, D!13), the_InternalRel(A!10))))),
% 0.21/0.44 inference(transitivity,[status(thm)],[160, 158])).
% 0.21/0.44 tff(162,plain,
% 0.21/0.44 ((~![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))) | (empty(the_InternalRel(A!10)) | in(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | (~element(ordered_pair(C!12, D!13), the_InternalRel(A!10))))),
% 0.21/0.44 inference(quant_inst,[status(thm)],[])).
% 0.21/0.44 tff(163,plain,
% 0.21/0.44 ((~![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))) | in(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | empty(the_InternalRel(A!10)) | (~element(ordered_pair(C!12, D!13), the_InternalRel(A!10)))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[162, 161])).
% 0.21/0.44 tff(164,plain,
% 0.21/0.44 (empty(the_InternalRel(A!10)) | (~element(ordered_pair(C!12, D!13), the_InternalRel(A!10)))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[163, 157, 147])).
% 0.21/0.44 tff(165,plain,
% 0.21/0.44 (~element(ordered_pair(C!12, D!13), the_InternalRel(A!10))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[164, 129])).
% 0.21/0.44 tff(166,plain,
% 0.21/0.44 (^[A: $i, B: $i, C: $i] : refl((element(A, C) | (~in(A, B)) | (~element(B, powerset(C)))) <=> (element(A, C) | (~in(A, B)) | (~element(B, powerset(C)))))),
% 0.21/0.44 inference(bind,[status(th)],[])).
% 0.21/0.44 tff(167,plain,
% 0.21/0.44 (![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C)))) <=> ![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))),
% 0.21/0.44 inference(quant_intro,[status(thm)],[166])).
% 0.21/0.44 tff(168,plain,
% 0.21/0.44 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(rewrite((in(A, B) & element(B, powerset(C))) <=> (~((~in(A, B)) | (~element(B, powerset(C)))))), ((~(in(A, B) & element(B, powerset(C)))) <=> (~(~((~in(A, B)) | (~element(B, powerset(C)))))))), rewrite((~(~((~in(A, B)) | (~element(B, powerset(C)))))) <=> ((~in(A, B)) | (~element(B, powerset(C))))), ((~(in(A, B) & element(B, powerset(C)))) <=> ((~in(A, B)) | (~element(B, powerset(C)))))), (((~(in(A, B) & element(B, powerset(C)))) | element(A, C)) <=> (((~in(A, B)) | (~element(B, powerset(C)))) | element(A, C)))), rewrite((((~in(A, B)) | (~element(B, powerset(C)))) | element(A, C)) <=> (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))), (((~(in(A, B) & element(B, powerset(C)))) | element(A, C)) <=> (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))))),
% 0.21/0.44 inference(bind,[status(th)],[])).
% 0.21/0.44 tff(169,plain,
% 0.21/0.44 (![A: $i, B: $i, C: $i] : ((~(in(A, B) & element(B, powerset(C)))) | element(A, C)) <=> ![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))),
% 0.21/0.44 inference(quant_intro,[status(thm)],[168])).
% 0.21/0.44 tff(170,plain,
% 0.21/0.44 (![A: $i, B: $i, C: $i] : ((~(in(A, B) & element(B, powerset(C)))) | element(A, C)) <=> ![A: $i, B: $i, C: $i] : ((~(in(A, B) & element(B, powerset(C)))) | element(A, C))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(171,plain,
% 0.21/0.44 (^[A: $i, B: $i, C: $i] : rewrite(((in(A, B) & element(B, powerset(C))) => element(A, C)) <=> ((~(in(A, B) & element(B, powerset(C)))) | element(A, C)))),
% 0.21/0.44 inference(bind,[status(th)],[])).
% 0.21/0.44 tff(172,plain,
% 0.21/0.44 (![A: $i, B: $i, C: $i] : ((in(A, B) & element(B, powerset(C))) => element(A, C)) <=> ![A: $i, B: $i, C: $i] : ((~(in(A, B) & element(B, powerset(C)))) | element(A, C))),
% 0.21/0.44 inference(quant_intro,[status(thm)],[171])).
% 0.21/0.44 tff(173,axiom,(![A: $i, B: $i, C: $i] : ((in(A, B) & element(B, powerset(C))) => element(A, C))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t4_subset')).
% 0.21/0.44 tff(174,plain,
% 0.21/0.44 (![A: $i, B: $i, C: $i] : ((~(in(A, B) & element(B, powerset(C)))) | element(A, C))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[173, 172])).
% 0.21/0.44 tff(175,plain,
% 0.21/0.44 (![A: $i, B: $i, C: $i] : ((~(in(A, B) & element(B, powerset(C)))) | element(A, C))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[174, 170])).
% 0.21/0.44 tff(176,plain,(
% 0.21/0.44 ![A: $i, B: $i, C: $i] : ((~(in(A, B) & element(B, powerset(C)))) | element(A, C))),
% 0.21/0.44 inference(skolemize,[status(sab)],[175])).
% 0.21/0.44 tff(177,plain,
% 0.21/0.44 (![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[176, 169])).
% 0.21/0.44 tff(178,plain,
% 0.21/0.44 (![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[177, 167])).
% 0.21/0.44 tff(179,plain,
% 0.21/0.44 (((~![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))) | ((~element(the_InternalRel(B!11), powerset(the_InternalRel(A!10)))) | element(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | (~in(ordered_pair(C!12, D!13), the_InternalRel(B!11))))) <=> ((~![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))) | (~element(the_InternalRel(B!11), powerset(the_InternalRel(A!10)))) | element(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | (~in(ordered_pair(C!12, D!13), the_InternalRel(B!11))))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(180,plain,
% 0.21/0.44 ((element(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | (~in(ordered_pair(C!12, D!13), the_InternalRel(B!11))) | (~element(the_InternalRel(B!11), powerset(the_InternalRel(A!10))))) <=> ((~element(the_InternalRel(B!11), powerset(the_InternalRel(A!10)))) | element(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | (~in(ordered_pair(C!12, D!13), the_InternalRel(B!11))))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(181,plain,
% 0.21/0.44 (((~![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))) | (element(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | (~in(ordered_pair(C!12, D!13), the_InternalRel(B!11))) | (~element(the_InternalRel(B!11), powerset(the_InternalRel(A!10)))))) <=> ((~![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))) | ((~element(the_InternalRel(B!11), powerset(the_InternalRel(A!10)))) | element(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | (~in(ordered_pair(C!12, D!13), the_InternalRel(B!11)))))),
% 0.21/0.44 inference(monotonicity,[status(thm)],[180])).
% 0.21/0.44 tff(182,plain,
% 0.21/0.44 (((~![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))) | (element(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | (~in(ordered_pair(C!12, D!13), the_InternalRel(B!11))) | (~element(the_InternalRel(B!11), powerset(the_InternalRel(A!10)))))) <=> ((~![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))) | (~element(the_InternalRel(B!11), powerset(the_InternalRel(A!10)))) | element(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | (~in(ordered_pair(C!12, D!13), the_InternalRel(B!11))))),
% 0.21/0.44 inference(transitivity,[status(thm)],[181, 179])).
% 0.21/0.44 tff(183,plain,
% 0.21/0.44 ((~![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))) | (element(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | (~in(ordered_pair(C!12, D!13), the_InternalRel(B!11))) | (~element(the_InternalRel(B!11), powerset(the_InternalRel(A!10)))))),
% 0.21/0.44 inference(quant_inst,[status(thm)],[])).
% 0.21/0.44 tff(184,plain,
% 0.21/0.44 ((~![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))) | (~element(the_InternalRel(B!11), powerset(the_InternalRel(A!10)))) | element(ordered_pair(C!12, D!13), the_InternalRel(A!10)) | (~in(ordered_pair(C!12, D!13), the_InternalRel(B!11)))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[183, 182])).
% 0.21/0.44 tff(185,plain,
% 0.21/0.44 ($false),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[184, 178, 112, 165, 71])).
% 0.21/0.44 % SZS output end Proof
% 0.21/0.44 % E exiting
%------------------------------------------------------------------------------