%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : NUM412+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n016.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 05:20:54 AM UTC 2025
% Result : Theorem 0.19s 0.37s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : NUM412+1 : TPTP v9.0.0. Released v3.2.0.
% 0.00/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n016.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 01:02:49 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.19/0.37 % SZS status Theorem
% 0.19/0.37 % SZS output start Proof
% 0.19/0.37 tff(transfinite_sequence_of_type, type, (
% 0.19/0.37 transfinite_sequence_of: ( $i * $i ) > $o)).
% 0.19/0.37 tff(tptp_fun_A_17_type, type, (
% 0.19/0.37 tptp_fun_A_17: $i)).
% 0.19/0.37 tff(relation_dom_restriction_type, type, (
% 0.19/0.37 relation_dom_restriction: ( $i * $i ) > $i)).
% 0.19/0.37 tff(tptp_fun_C_18_type, type, (
% 0.19/0.37 tptp_fun_C_18: $i)).
% 0.19/0.37 tff(tptp_fun_B_16_type, type, (
% 0.19/0.37 tptp_fun_B_16: $i)).
% 0.19/0.37 tff(tseq_dom_restriction_type, type, (
% 0.19/0.37 tseq_dom_restriction: ( $i * $i ) > $i)).
% 0.19/0.37 tff(relation_type, type, (
% 0.19/0.37 relation: $i > $o)).
% 0.19/0.37 tff(transfinite_sequence_type, type, (
% 0.19/0.37 transfinite_sequence: $i > $o)).
% 0.19/0.37 tff(function_type, type, (
% 0.19/0.37 function: $i > $o)).
% 0.19/0.37 tff(ordinal_type, type, (
% 0.19/0.37 ordinal: $i > $o)).
% 0.19/0.37 tff(relation_rng_type, type, (
% 0.19/0.37 relation_rng: $i > $i)).
% 0.19/0.37 tff(subset_type, type, (
% 0.19/0.37 subset: ( $i * $i ) > $o)).
% 0.19/0.37 tff(1,plain,
% 0.19/0.37 ((~![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | ![C: $i] : ((~ordinal(C)) | transfinite_sequence_of(tseq_dom_restriction(B, C), A)))) <=> (~![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | ![C: $i] : ((~ordinal(C)) | transfinite_sequence_of(tseq_dom_restriction(B, C), A))))),
% 0.19/0.37 inference(rewrite,[status(thm)],[])).
% 0.19/0.37 tff(2,plain,
% 0.19/0.37 ((~![A: $i, B: $i] : (transfinite_sequence_of(B, A) => ![C: $i] : (ordinal(C) => transfinite_sequence_of(tseq_dom_restriction(B, C), A)))) <=> (~![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | ![C: $i] : ((~ordinal(C)) | transfinite_sequence_of(tseq_dom_restriction(B, C), A))))),
% 0.19/0.37 inference(rewrite,[status(thm)],[])).
% 0.19/0.37 tff(3,axiom,(~![A: $i, B: $i] : (transfinite_sequence_of(B, A) => ![C: $i] : (ordinal(C) => transfinite_sequence_of(tseq_dom_restriction(B, C), A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t48_ordinal1')).
% 0.19/0.37 tff(4,plain,
% 0.19/0.37 (~![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | ![C: $i] : ((~ordinal(C)) | transfinite_sequence_of(tseq_dom_restriction(B, C), A)))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[3, 2])).
% 0.19/0.37 tff(5,plain,
% 0.19/0.37 (~![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | ![C: $i] : ((~ordinal(C)) | transfinite_sequence_of(tseq_dom_restriction(B, C), A)))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[4, 1])).
% 0.19/0.37 tff(6,plain,
% 0.19/0.37 (~![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | ![C: $i] : ((~ordinal(C)) | transfinite_sequence_of(tseq_dom_restriction(B, C), A)))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[5, 1])).
% 0.19/0.37 tff(7,plain,
% 0.19/0.37 (~![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | ![C: $i] : ((~ordinal(C)) | transfinite_sequence_of(tseq_dom_restriction(B, C), A)))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[6, 1])).
% 0.19/0.37 tff(8,plain,
% 0.19/0.37 (transfinite_sequence_of(B!16, A!17)),
% 0.19/0.37 inference(and_elim,[status(thm)],[7])).
% 0.19/0.37 tff(9,plain,
% 0.19/0.37 (^[A: $i, B: $i] : refl(((~transfinite_sequence_of(B, A)) | (~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B))))) <=> ((~transfinite_sequence_of(B, A)) | (~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B))))))),
% 0.19/0.37 inference(bind,[status(th)],[])).
% 0.19/0.37 tff(10,plain,
% 0.19/0.37 (![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | (~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B))))) <=> ![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | (~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))))),
% 0.19/0.37 inference(quant_intro,[status(thm)],[9])).
% 0.19/0.37 tff(11,plain,
% 0.19/0.37 (^[A: $i, B: $i] : rewrite(((~transfinite_sequence_of(B, A)) | (relation(B) & function(B) & transfinite_sequence(B))) <=> ((~transfinite_sequence_of(B, A)) | (~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B))))))),
% 0.19/0.37 inference(bind,[status(th)],[])).
% 0.19/0.37 tff(12,plain,
% 0.19/0.37 (![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | (relation(B) & function(B) & transfinite_sequence(B))) <=> ![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | (~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))))),
% 0.19/0.37 inference(quant_intro,[status(thm)],[11])).
% 0.19/0.37 tff(13,plain,
% 0.19/0.37 (![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | (relation(B) & function(B) & transfinite_sequence(B))) <=> ![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | (relation(B) & function(B) & transfinite_sequence(B)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(14,plain,
% 0.19/0.38 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((relation(B) & function(B)) & transfinite_sequence(B)) <=> (relation(B) & function(B) & transfinite_sequence(B))), ((transfinite_sequence_of(B, A) => ((relation(B) & function(B)) & transfinite_sequence(B))) <=> (transfinite_sequence_of(B, A) => (relation(B) & function(B) & transfinite_sequence(B))))), rewrite((transfinite_sequence_of(B, A) => (relation(B) & function(B) & transfinite_sequence(B))) <=> ((~transfinite_sequence_of(B, A)) | (relation(B) & function(B) & transfinite_sequence(B)))), ((transfinite_sequence_of(B, A) => ((relation(B) & function(B)) & transfinite_sequence(B))) <=> ((~transfinite_sequence_of(B, A)) | (relation(B) & function(B) & transfinite_sequence(B)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(15,plain,
% 0.19/0.38 (![A: $i, B: $i] : (transfinite_sequence_of(B, A) => ((relation(B) & function(B)) & transfinite_sequence(B))) <=> ![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | (relation(B) & function(B) & transfinite_sequence(B)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[14])).
% 0.19/0.38 tff(16,axiom,(![A: $i, B: $i] : (transfinite_sequence_of(B, A) => ((relation(B) & function(B)) & transfinite_sequence(B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_m1_ordinal1')).
% 0.19/0.38 tff(17,plain,
% 0.19/0.38 (![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | (relation(B) & function(B) & transfinite_sequence(B)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.19/0.38 tff(18,plain,
% 0.19/0.38 (![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | (relation(B) & function(B) & transfinite_sequence(B)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.19/0.38 tff(19,plain,(
% 0.19/0.38 ![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | (relation(B) & function(B) & transfinite_sequence(B)))),
% 0.19/0.38 inference(skolemize,[status(sab)],[18])).
% 0.19/0.38 tff(20,plain,
% 0.19/0.38 (![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | (~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[19, 12])).
% 0.19/0.38 tff(21,plain,
% 0.19/0.38 (![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | (~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[20, 10])).
% 0.19/0.38 tff(22,plain,
% 0.19/0.38 (((~![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | (~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))))) | ((~transfinite_sequence_of(B!16, A!17)) | (~((~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)))))) <=> ((~![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | (~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))))) | (~transfinite_sequence_of(B!16, A!17)) | (~((~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(23,plain,
% 0.19/0.38 ((~![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | (~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))))) | ((~transfinite_sequence_of(B!16, A!17)) | (~((~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)))))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(24,plain,
% 0.19/0.38 ((~![A: $i, B: $i] : ((~transfinite_sequence_of(B, A)) | (~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))))) | (~transfinite_sequence_of(B!16, A!17)) | (~((~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.19/0.38 tff(25,plain,
% 0.19/0.38 (~((~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[24, 21, 8])).
% 0.19/0.38 tff(26,plain,
% 0.19/0.38 (((~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16))) | relation(B!16)),
% 0.19/0.38 inference(tautology,[status(thm)],[])).
% 0.19/0.38 tff(27,plain,
% 0.19/0.38 (relation(B!16)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[26, 25])).
% 0.19/0.38 tff(28,plain,
% 0.19/0.38 (((~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16))) | transfinite_sequence(B!16)),
% 0.19/0.38 inference(tautology,[status(thm)],[])).
% 0.19/0.38 tff(29,plain,
% 0.19/0.38 (transfinite_sequence(B!16)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[28, 25])).
% 0.19/0.38 tff(30,plain,
% 0.19/0.38 (((~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16))) | function(B!16)),
% 0.19/0.38 inference(tautology,[status(thm)],[])).
% 0.19/0.38 tff(31,plain,
% 0.19/0.38 (function(B!16)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[30, 25])).
% 0.19/0.38 tff(32,plain,
% 0.19/0.38 (~((~ordinal(C!18)) | transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), A!17))),
% 0.19/0.38 inference(and_elim,[status(thm)],[7])).
% 0.19/0.38 tff(33,plain,
% 0.19/0.38 (ordinal(C!18)),
% 0.19/0.38 inference(or_elim,[status(thm)],[32])).
% 0.19/0.38 tff(34,plain,
% 0.19/0.38 (^[A: $i, B: $i] : refl(((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))) <=> ((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(35,plain,
% 0.19/0.38 (![A: $i, B: $i] : ((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))) <=> ![A: $i, B: $i] : ((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[34])).
% 0.19/0.38 tff(36,plain,
% 0.19/0.38 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((relation(A) & function(A) & transfinite_sequence(A) & ordinal(B)) <=> (~((~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))))), ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) <=> (~(~((~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))))))), rewrite((~(~((~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))))) <=> ((~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))), ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) <=> ((~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))))), (((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B))) <=> (((~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))) | (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B))))), rewrite((((~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))) | (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B))) <=> ((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))), (((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B))) <=> ((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(37,plain,
% 0.19/0.38 (![A: $i, B: $i] : ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B))) <=> ![A: $i, B: $i] : ((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[36])).
% 0.19/0.38 tff(38,plain,
% 0.19/0.38 (![A: $i, B: $i] : ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B))) <=> ![A: $i, B: $i] : ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(39,plain,
% 0.19/0.38 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite(((relation(A) & function(A)) & transfinite_sequence(A)) <=> (relation(A) & function(A) & transfinite_sequence(A))), ((((relation(A) & function(A)) & transfinite_sequence(A)) & ordinal(B)) <=> ((relation(A) & function(A) & transfinite_sequence(A)) & ordinal(B)))), rewrite(((relation(A) & function(A) & transfinite_sequence(A)) & ordinal(B)) <=> (relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))), ((((relation(A) & function(A)) & transfinite_sequence(A)) & ordinal(B)) <=> (relation(A) & function(A) & transfinite_sequence(A) & ordinal(B)))), (((((relation(A) & function(A)) & transfinite_sequence(A)) & ordinal(B)) => (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B))) <=> ((relation(A) & function(A) & transfinite_sequence(A) & ordinal(B)) => (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B))))), rewrite(((relation(A) & function(A) & transfinite_sequence(A) & ordinal(B)) => (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B))) <=> ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)))), (((((relation(A) & function(A)) & transfinite_sequence(A)) & ordinal(B)) => (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B))) <=> ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(40,plain,
% 0.19/0.38 (![A: $i, B: $i] : ((((relation(A) & function(A)) & transfinite_sequence(A)) & ordinal(B)) => (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B))) <=> ![A: $i, B: $i] : ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[39])).
% 0.19/0.38 tff(41,axiom,(![A: $i, B: $i] : ((((relation(A) & function(A)) & transfinite_sequence(A)) & ordinal(B)) => (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','redefinition_k2_ordinal1')).
% 0.19/0.38 tff(42,plain,
% 0.19/0.38 (![A: $i, B: $i] : ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[41, 40])).
% 0.19/0.38 tff(43,plain,
% 0.19/0.38 (![A: $i, B: $i] : ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[42, 38])).
% 0.19/0.38 tff(44,plain,(
% 0.19/0.38 ![A: $i, B: $i] : ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | (tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)))),
% 0.19/0.38 inference(skolemize,[status(sab)],[43])).
% 0.19/0.38 tff(45,plain,
% 0.19/0.38 (![A: $i, B: $i] : ((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[44, 37])).
% 0.19/0.38 tff(46,plain,
% 0.19/0.38 (![A: $i, B: $i] : ((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[45, 35])).
% 0.19/0.38 tff(47,plain,
% 0.19/0.38 (((~![A: $i, B: $i] : ((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))) | ((~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)) | (tseq_dom_restriction(B!16, C!18) = relation_dom_restriction(B!16, C!18)))) <=> ((~![A: $i, B: $i] : ((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))) | (~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)) | (tseq_dom_restriction(B!16, C!18) = relation_dom_restriction(B!16, C!18)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(48,plain,
% 0.19/0.38 (((tseq_dom_restriction(B!16, C!18) = relation_dom_restriction(B!16, C!18)) | (~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16))) <=> ((~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)) | (tseq_dom_restriction(B!16, C!18) = relation_dom_restriction(B!16, C!18)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(49,plain,
% 0.19/0.38 (((~![A: $i, B: $i] : ((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))) | ((tseq_dom_restriction(B!16, C!18) = relation_dom_restriction(B!16, C!18)) | (~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)))) <=> ((~![A: $i, B: $i] : ((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))) | ((~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)) | (tseq_dom_restriction(B!16, C!18) = relation_dom_restriction(B!16, C!18))))),
% 0.19/0.38 inference(monotonicity,[status(thm)],[48])).
% 0.19/0.38 tff(50,plain,
% 0.19/0.38 (((~![A: $i, B: $i] : ((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))) | ((tseq_dom_restriction(B!16, C!18) = relation_dom_restriction(B!16, C!18)) | (~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)))) <=> ((~![A: $i, B: $i] : ((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))) | (~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)) | (tseq_dom_restriction(B!16, C!18) = relation_dom_restriction(B!16, C!18)))),
% 0.19/0.38 inference(transitivity,[status(thm)],[49, 47])).
% 0.19/0.38 tff(51,plain,
% 0.19/0.38 ((~![A: $i, B: $i] : ((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))) | ((tseq_dom_restriction(B!16, C!18) = relation_dom_restriction(B!16, C!18)) | (~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(52,plain,
% 0.19/0.38 ((~![A: $i, B: $i] : ((tseq_dom_restriction(A, B) = relation_dom_restriction(A, B)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))) | (~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)) | (tseq_dom_restriction(B!16, C!18) = relation_dom_restriction(B!16, C!18))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[51, 50])).
% 0.19/0.38 tff(53,plain,
% 0.19/0.38 ((~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)) | (tseq_dom_restriction(B!16, C!18) = relation_dom_restriction(B!16, C!18))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[52, 46, 33])).
% 0.19/0.38 tff(54,plain,
% 0.19/0.38 (tseq_dom_restriction(B!16, C!18) = relation_dom_restriction(B!16, C!18)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[53, 31, 29, 27])).
% 0.19/0.38 tff(55,plain,
% 0.19/0.38 (relation_dom_restriction(B!16, C!18) = tseq_dom_restriction(B!16, C!18)),
% 0.19/0.38 inference(symmetry,[status(thm)],[54])).
% 0.19/0.38 tff(56,plain,
% 0.19/0.38 (transfinite_sequence_of(relation_dom_restriction(B!16, C!18), A!17) <=> transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), A!17)),
% 0.19/0.38 inference(monotonicity,[status(thm)],[55])).
% 0.19/0.38 tff(57,plain,
% 0.19/0.38 (transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), A!17) <=> transfinite_sequence_of(relation_dom_restriction(B!16, C!18), A!17)),
% 0.19/0.38 inference(symmetry,[status(thm)],[56])).
% 0.19/0.38 tff(58,plain,
% 0.19/0.38 ((~transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), A!17)) <=> (~transfinite_sequence_of(relation_dom_restriction(B!16, C!18), A!17))),
% 0.19/0.38 inference(monotonicity,[status(thm)],[57])).
% 0.19/0.38 tff(59,plain,
% 0.19/0.38 (~transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), A!17)),
% 0.19/0.38 inference(or_elim,[status(thm)],[32])).
% 0.19/0.38 tff(60,plain,
% 0.19/0.38 (~transfinite_sequence_of(relation_dom_restriction(B!16, C!18), A!17)),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.19/0.38 tff(61,plain,
% 0.19/0.38 (^[A: $i, B: $i] : refl(((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B))) <=> ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(62,plain,
% 0.19/0.38 (![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B))) <=> ![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[61])).
% 0.19/0.38 tff(63,plain,
% 0.19/0.38 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((relation(B) & function(B) & transfinite_sequence(B)) <=> (~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B))))), ((~(relation(B) & function(B) & transfinite_sequence(B))) <=> (~(~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B))))))), rewrite((~(~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B))))) <=> ((~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))), ((~(relation(B) & function(B) & transfinite_sequence(B))) <=> ((~relation(B)) | (~function(B)) | (~transfinite_sequence(B))))), (((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> (((~relation(B)) | (~function(B)) | (~transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))))), rewrite((((~relation(B)) | (~function(B)) | (~transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))), (((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(64,plain,
% 0.19/0.38 (![A: $i, B: $i] : ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> ![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[63])).
% 0.19/0.38 tff(65,plain,
% 0.19/0.38 (![A: $i, B: $i] : ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> ![A: $i, B: $i] : ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(66,plain,
% 0.19/0.38 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((relation(B) & function(B)) & transfinite_sequence(B)) <=> (relation(B) & function(B) & transfinite_sequence(B))), ((((relation(B) & function(B)) & transfinite_sequence(B)) => (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> ((relation(B) & function(B) & transfinite_sequence(B)) => (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))))), rewrite(((relation(B) & function(B) & transfinite_sequence(B)) => (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)))), ((((relation(B) & function(B)) & transfinite_sequence(B)) => (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(67,plain,
% 0.19/0.38 (![A: $i, B: $i] : (((relation(B) & function(B)) & transfinite_sequence(B)) => (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> ![A: $i, B: $i] : ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[66])).
% 0.19/0.38 tff(68,axiom,(![A: $i, B: $i] : (((relation(B) & function(B)) & transfinite_sequence(B)) => (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d8_ordinal1')).
% 0.19/0.38 tff(69,plain,
% 0.19/0.38 (![A: $i, B: $i] : ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[68, 67])).
% 0.19/0.39 tff(70,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[69, 65])).
% 0.19/0.39 tff(71,plain,(
% 0.19/0.39 ![A: $i, B: $i] : ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[70])).
% 0.19/0.39 tff(72,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[71, 64])).
% 0.19/0.39 tff(73,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[72, 62])).
% 0.19/0.39 tff(74,plain,
% 0.19/0.39 (((~![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))) | ((transfinite_sequence_of(B!16, A!17) <=> subset(relation_rng(B!16), A!17)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)))) <=> ((~![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))) | (transfinite_sequence_of(B!16, A!17) <=> subset(relation_rng(B!16), A!17)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(75,plain,
% 0.19/0.39 ((~![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))) | ((transfinite_sequence_of(B!16, A!17) <=> subset(relation_rng(B!16), A!17)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(76,plain,
% 0.19/0.39 ((~![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))) | (transfinite_sequence_of(B!16, A!17) <=> subset(relation_rng(B!16), A!17)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[75, 74])).
% 0.19/0.39 tff(77,plain,
% 0.19/0.39 ((transfinite_sequence_of(B!16, A!17) <=> subset(relation_rng(B!16), A!17)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[76, 73])).
% 0.19/0.39 tff(78,plain,
% 0.19/0.39 (transfinite_sequence_of(B!16, A!17) <=> subset(relation_rng(B!16), A!17)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[77, 27, 31, 29])).
% 0.19/0.39 tff(79,plain,
% 0.19/0.39 ((~(transfinite_sequence_of(B!16, A!17) <=> subset(relation_rng(B!16), A!17))) | (~transfinite_sequence_of(B!16, A!17)) | subset(relation_rng(B!16), A!17)),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(80,plain,
% 0.19/0.39 ((~(transfinite_sequence_of(B!16, A!17) <=> subset(relation_rng(B!16), A!17))) | subset(relation_rng(B!16), A!17)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[79, 8])).
% 0.19/0.39 tff(81,plain,
% 0.19/0.39 (subset(relation_rng(B!16), A!17)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[80, 78])).
% 0.19/0.39 tff(82,plain,
% 0.19/0.39 (^[A: $i, B: $i] : refl(((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B))) <=> ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(83,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B))) <=> ![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[82])).
% 0.19/0.39 tff(84,plain,
% 0.19/0.39 (^[A: $i, B: $i] : rewrite(((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B))) <=> ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(85,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B))) <=> ![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[84])).
% 0.19/0.39 tff(86,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B))) <=> ![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))),
% 0.19/0.39 inference(transitivity,[status(thm)],[85, 83])).
% 0.19/0.39 tff(87,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B))) <=> ![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(88,plain,
% 0.19/0.39 (^[A: $i, B: $i] : trans(monotonicity(quant_intro(proof_bind(^[C: $i] : rewrite((transfinite_sequence_of(C, A) => transfinite_sequence_of(C, B)) <=> ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))), (![C: $i] : (transfinite_sequence_of(C, A) => transfinite_sequence_of(C, B)) <=> ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))), ((subset(A, B) => ![C: $i] : (transfinite_sequence_of(C, A) => transfinite_sequence_of(C, B))) <=> (subset(A, B) => ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B))))), rewrite((subset(A, B) => ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B))) <=> ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))), ((subset(A, B) => ![C: $i] : (transfinite_sequence_of(C, A) => transfinite_sequence_of(C, B))) <=> ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(89,plain,
% 0.19/0.39 (![A: $i, B: $i] : (subset(A, B) => ![C: $i] : (transfinite_sequence_of(C, A) => transfinite_sequence_of(C, B))) <=> ![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[88])).
% 0.19/0.39 tff(90,axiom,(![A: $i, B: $i] : (subset(A, B) => ![C: $i] : (transfinite_sequence_of(C, A) => transfinite_sequence_of(C, B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t47_ordinal1')).
% 0.19/0.39 tff(91,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[90, 89])).
% 0.19/0.39 tff(92,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[91, 87])).
% 0.19/0.39 tff(93,plain,(
% 0.19/0.39 ![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[92])).
% 0.19/0.39 tff(94,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[93, 86])).
% 0.19/0.39 tff(95,plain,
% 0.19/0.39 (((~![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))) | ((~subset(relation_rng(B!16), A!17)) | ![C: $i] : ((~transfinite_sequence_of(C, relation_rng(B!16))) | transfinite_sequence_of(C, A!17)))) <=> ((~![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))) | (~subset(relation_rng(B!16), A!17)) | ![C: $i] : ((~transfinite_sequence_of(C, relation_rng(B!16))) | transfinite_sequence_of(C, A!17)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(96,plain,
% 0.19/0.39 ((~![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))) | ((~subset(relation_rng(B!16), A!17)) | ![C: $i] : ((~transfinite_sequence_of(C, relation_rng(B!16))) | transfinite_sequence_of(C, A!17)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(97,plain,
% 0.19/0.39 ((~![A: $i, B: $i] : ((~subset(A, B)) | ![C: $i] : ((~transfinite_sequence_of(C, A)) | transfinite_sequence_of(C, B)))) | (~subset(relation_rng(B!16), A!17)) | ![C: $i] : ((~transfinite_sequence_of(C, relation_rng(B!16))) | transfinite_sequence_of(C, A!17))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[96, 95])).
% 0.19/0.39 tff(98,plain,
% 0.19/0.39 (![C: $i] : ((~transfinite_sequence_of(C, relation_rng(B!16))) | transfinite_sequence_of(C, A!17))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[97, 94, 81])).
% 0.19/0.39 tff(99,plain,
% 0.19/0.39 (transfinite_sequence_of(relation_dom_restriction(B!16, C!18), relation_rng(B!16)) <=> transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), relation_rng(B!16))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[55])).
% 0.19/0.39 tff(100,plain,
% 0.19/0.39 (transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), relation_rng(B!16)) <=> transfinite_sequence_of(relation_dom_restriction(B!16, C!18), relation_rng(B!16))),
% 0.19/0.39 inference(symmetry,[status(thm)],[99])).
% 0.19/0.39 tff(101,plain,
% 0.19/0.39 (^[A: $i, B: $i] : refl((transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))) <=> (transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(102,plain,
% 0.19/0.39 (![A: $i, B: $i] : (transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))) <=> ![A: $i, B: $i] : (transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[101])).
% 0.19/0.39 tff(103,plain,
% 0.19/0.39 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((relation(A) & function(A) & transfinite_sequence(A) & ordinal(B)) <=> (~((~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))))), ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) <=> (~(~((~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))))))), rewrite((~(~((~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))))) <=> ((~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))), ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) <=> ((~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))))), (((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A))) <=> (((~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))) | transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A))))), rewrite((((~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A))) | transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A))) <=> (transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))), (((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A))) <=> (transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(104,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A))) <=> ![A: $i, B: $i] : (transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[103])).
% 0.19/0.39 tff(105,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A))) <=> ![A: $i, B: $i] : ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(106,plain,
% 0.19/0.39 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite(((relation(A) & function(A)) & transfinite_sequence(A)) <=> (relation(A) & function(A) & transfinite_sequence(A))), ((((relation(A) & function(A)) & transfinite_sequence(A)) & ordinal(B)) <=> ((relation(A) & function(A) & transfinite_sequence(A)) & ordinal(B)))), rewrite(((relation(A) & function(A) & transfinite_sequence(A)) & ordinal(B)) <=> (relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))), ((((relation(A) & function(A)) & transfinite_sequence(A)) & ordinal(B)) <=> (relation(A) & function(A) & transfinite_sequence(A) & ordinal(B)))), (((((relation(A) & function(A)) & transfinite_sequence(A)) & ordinal(B)) => transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A))) <=> ((relation(A) & function(A) & transfinite_sequence(A) & ordinal(B)) => transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A))))), rewrite(((relation(A) & function(A) & transfinite_sequence(A) & ordinal(B)) => transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A))) <=> ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)))), (((((relation(A) & function(A)) & transfinite_sequence(A)) & ordinal(B)) => transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A))) <=> ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(107,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((((relation(A) & function(A)) & transfinite_sequence(A)) & ordinal(B)) => transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A))) <=> ![A: $i, B: $i] : ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[106])).
% 0.19/0.39 tff(108,axiom,(![A: $i, B: $i] : ((((relation(A) & function(A)) & transfinite_sequence(A)) & ordinal(B)) => transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_k2_ordinal1')).
% 0.19/0.39 tff(109,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[108, 107])).
% 0.19/0.39 tff(110,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[109, 105])).
% 0.19/0.39 tff(111,plain,(
% 0.19/0.39 ![A: $i, B: $i] : ((~(relation(A) & function(A) & transfinite_sequence(A) & ordinal(B))) | transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[110])).
% 0.19/0.39 tff(112,plain,
% 0.19/0.39 (![A: $i, B: $i] : (transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[111, 104])).
% 0.19/0.39 tff(113,plain,
% 0.19/0.39 (![A: $i, B: $i] : (transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[112, 102])).
% 0.19/0.39 tff(114,plain,
% 0.19/0.39 (((~![A: $i, B: $i] : (transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))) | ((~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)) | transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), relation_rng(B!16)))) <=> ((~![A: $i, B: $i] : (transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))) | (~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)) | transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), relation_rng(B!16)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(115,plain,
% 0.19/0.39 ((transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), relation_rng(B!16)) | (~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16))) <=> ((~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)) | transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), relation_rng(B!16)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(116,plain,
% 0.19/0.39 (((~![A: $i, B: $i] : (transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))) | (transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), relation_rng(B!16)) | (~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)))) <=> ((~![A: $i, B: $i] : (transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))) | ((~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)) | transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), relation_rng(B!16))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[115])).
% 0.19/0.39 tff(117,plain,
% 0.19/0.39 (((~![A: $i, B: $i] : (transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))) | (transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), relation_rng(B!16)) | (~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)))) <=> ((~![A: $i, B: $i] : (transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))) | (~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)) | transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), relation_rng(B!16)))),
% 0.19/0.39 inference(transitivity,[status(thm)],[116, 114])).
% 0.19/0.39 tff(118,plain,
% 0.19/0.39 ((~![A: $i, B: $i] : (transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))) | (transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), relation_rng(B!16)) | (~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(119,plain,
% 0.19/0.39 ((~![A: $i, B: $i] : (transfinite_sequence_of(tseq_dom_restriction(A, B), relation_rng(A)) | (~ordinal(B)) | (~relation(A)) | (~function(A)) | (~transfinite_sequence(A)))) | (~ordinal(C!18)) | (~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)) | transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), relation_rng(B!16))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[118, 117])).
% 0.19/0.39 tff(120,plain,
% 0.19/0.39 ((~relation(B!16)) | (~function(B!16)) | (~transfinite_sequence(B!16)) | transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), relation_rng(B!16))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[119, 113, 33])).
% 0.19/0.39 tff(121,plain,
% 0.19/0.39 (transfinite_sequence_of(tseq_dom_restriction(B!16, C!18), relation_rng(B!16))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[120, 31, 29, 27])).
% 0.19/0.39 tff(122,plain,
% 0.19/0.39 (transfinite_sequence_of(relation_dom_restriction(B!16, C!18), relation_rng(B!16))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[121, 100])).
% 0.19/0.39 tff(123,plain,
% 0.19/0.39 (((~![C: $i] : ((~transfinite_sequence_of(C, relation_rng(B!16))) | transfinite_sequence_of(C, A!17))) | ((~transfinite_sequence_of(relation_dom_restriction(B!16, C!18), relation_rng(B!16))) | transfinite_sequence_of(relation_dom_restriction(B!16, C!18), A!17))) <=> ((~![C: $i] : ((~transfinite_sequence_of(C, relation_rng(B!16))) | transfinite_sequence_of(C, A!17))) | (~transfinite_sequence_of(relation_dom_restriction(B!16, C!18), relation_rng(B!16))) | transfinite_sequence_of(relation_dom_restriction(B!16, C!18), A!17))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(124,plain,
% 0.19/0.40 ((~![C: $i] : ((~transfinite_sequence_of(C, relation_rng(B!16))) | transfinite_sequence_of(C, A!17))) | ((~transfinite_sequence_of(relation_dom_restriction(B!16, C!18), relation_rng(B!16))) | transfinite_sequence_of(relation_dom_restriction(B!16, C!18), A!17))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(125,plain,
% 0.19/0.40 ((~![C: $i] : ((~transfinite_sequence_of(C, relation_rng(B!16))) | transfinite_sequence_of(C, A!17))) | (~transfinite_sequence_of(relation_dom_restriction(B!16, C!18), relation_rng(B!16))) | transfinite_sequence_of(relation_dom_restriction(B!16, C!18), A!17)),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[124, 123])).
% 0.19/0.40 tff(126,plain,
% 0.19/0.40 (transfinite_sequence_of(relation_dom_restriction(B!16, C!18), A!17)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[125, 122, 98])).
% 0.19/0.40 tff(127,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[126, 60])).
% 0.19/0.40 % SZS output end Proof
% 0.19/0.40 % E exiting
%------------------------------------------------------------------------------