%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET642+3 : TPTP v9.0.0. Released v2.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n001.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:26:05 AM UTC 2025
% Result : Theorem 0.20s 0.42s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET642+3 : TPTP v9.0.0. Released v2.2.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n001.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 05:36:35 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.20/0.42 % SZS status Theorem
% 0.20/0.42 % SZS output start Proof
% 0.20/0.42 tff(subset_type, type, (
% 0.20/0.42 subset: ( $i * $i ) > $o)).
% 0.20/0.42 tff(cross_product_type, type, (
% 0.20/0.42 cross_product: ( $i * $i ) > $i)).
% 0.20/0.42 tff(tptp_fun_D_10_type, type, (
% 0.20/0.42 tptp_fun_D_10: $i)).
% 0.20/0.42 tff(tptp_fun_C_9_type, type, (
% 0.20/0.42 tptp_fun_C_9: $i)).
% 0.20/0.42 tff(tptp_fun_B_8_type, type, (
% 0.20/0.42 tptp_fun_B_8: $i)).
% 0.20/0.42 tff(ilf_type_type, type, (
% 0.20/0.42 ilf_type: ( $i * $i ) > $o)).
% 0.20/0.42 tff(relation_type_type, type, (
% 0.20/0.42 relation_type: ( $i * $i ) > $i)).
% 0.20/0.42 tff(set_type_type, type, (
% 0.20/0.42 set_type: $i)).
% 0.20/0.42 tff(tptp_fun_E_11_type, type, (
% 0.20/0.42 tptp_fun_E_11: $i)).
% 0.20/0.42 tff(1,plain,
% 0.20/0.42 ((ilf_type(B!8, set_type) & (ilf_type(C!9, set_type) & ilf_type(D!10, set_type) & (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10))))))) <=> (ilf_type(B!8, set_type) & ilf_type(C!9, set_type) & ilf_type(D!10, set_type) & (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10))))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(2,plain,
% 0.20/0.42 ((ilf_type(C!9, set_type) & (ilf_type(D!10, set_type) & (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10))))))) <=> (ilf_type(C!9, set_type) & ilf_type(D!10, set_type) & (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10))))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(3,plain,
% 0.20/0.42 ((~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10))))) <=> (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10)))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(4,plain,
% 0.20/0.42 ((ilf_type(D!10, set_type) & (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10)))))) <=> (ilf_type(D!10, set_type) & (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10))))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[3])).
% 0.20/0.42 tff(5,plain,
% 0.20/0.42 ((ilf_type(C!9, set_type) & (ilf_type(D!10, set_type) & (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10))))))) <=> (ilf_type(C!9, set_type) & (ilf_type(D!10, set_type) & (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10)))))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[4])).
% 0.20/0.42 tff(6,plain,
% 0.20/0.42 ((ilf_type(C!9, set_type) & (ilf_type(D!10, set_type) & (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10))))))) <=> (ilf_type(C!9, set_type) & ilf_type(D!10, set_type) & (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10))))))),
% 0.20/0.42 inference(transitivity,[status(thm)],[5, 2])).
% 0.20/0.42 tff(7,plain,
% 0.20/0.42 ((ilf_type(B!8, set_type) & (ilf_type(C!9, set_type) & (ilf_type(D!10, set_type) & (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10)))))))) <=> (ilf_type(B!8, set_type) & (ilf_type(C!9, set_type) & ilf_type(D!10, set_type) & (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10)))))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[6])).
% 0.20/0.42 tff(8,plain,
% 0.20/0.42 ((ilf_type(B!8, set_type) & (ilf_type(C!9, set_type) & (ilf_type(D!10, set_type) & (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10)))))))) <=> (ilf_type(B!8, set_type) & ilf_type(C!9, set_type) & ilf_type(D!10, set_type) & (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10))))))),
% 0.20/0.42 inference(transitivity,[status(thm)],[7, 1])).
% 0.20/0.42 tff(9,plain,
% 0.20/0.42 ((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (ilf_type(B, relation_type(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))) <=> (~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (ilf_type(B, relation_type(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(10,plain,
% 0.20/0.42 ((~![B: $i] : (ilf_type(B, set_type) => ![C: $i] : (ilf_type(C, set_type) => ![D: $i] : (ilf_type(D, set_type) => ![E: $i] : (ilf_type(E, relation_type(C, D)) => (subset(B, E) => ilf_type(B, relation_type(C, D)))))))) <=> (~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (ilf_type(B, relation_type(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(11,axiom,(~![B: $i] : (ilf_type(B, set_type) => ![C: $i] : (ilf_type(C, set_type) => ![D: $i] : (ilf_type(D, set_type) => ![E: $i] : (ilf_type(E, relation_type(C, D)) => (subset(B, E) => ilf_type(B, relation_type(C, D)))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_relset_1_4')).
% 0.20/0.42 tff(12,plain,
% 0.20/0.42 (~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (ilf_type(B, relation_type(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.20/0.42 tff(13,plain,
% 0.20/0.42 (~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (ilf_type(B, relation_type(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[12, 9])).
% 0.20/0.42 tff(14,plain,
% 0.20/0.42 (~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (ilf_type(B, relation_type(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[13, 9])).
% 0.20/0.42 tff(15,plain,
% 0.20/0.42 (~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (ilf_type(B, relation_type(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[14, 9])).
% 0.20/0.42 tff(16,plain,
% 0.20/0.42 (ilf_type(B!8, set_type) & ilf_type(C!9, set_type) & ilf_type(D!10, set_type) & (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10)))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[15, 8])).
% 0.20/0.42 tff(17,plain,
% 0.20/0.42 (ilf_type(B!8, set_type)),
% 0.20/0.42 inference(and_elim,[status(thm)],[16])).
% 0.20/0.42 tff(18,plain,
% 0.20/0.42 (^[B: $i] : refl(((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D)))))) <=> ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D)))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(19,plain,
% 0.20/0.42 (![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D)))))) <=> ![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[18])).
% 0.20/0.42 tff(20,plain,
% 0.20/0.42 (^[B: $i] : rewrite(((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D)))))) <=> ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D)))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(21,plain,
% 0.20/0.42 (![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D)))))) <=> ![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[20])).
% 0.20/0.43 tff(22,plain,
% 0.20/0.43 (![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D)))))) <=> ![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))),
% 0.20/0.43 inference(transitivity,[status(thm)],[21, 19])).
% 0.20/0.43 tff(23,plain,
% 0.20/0.43 (![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D)))))) <=> ![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(24,plain,
% 0.20/0.43 (^[B: $i] : trans(monotonicity(quant_intro(proof_bind(^[C: $i] : trans(monotonicity(quant_intro(proof_bind(^[D: $i] : trans(monotonicity(rewrite((subset(B, cross_product(C, D)) => ilf_type(B, relation_type(C, D))) <=> ((~subset(B, cross_product(C, D))) | ilf_type(B, relation_type(C, D)))), ((ilf_type(D, set_type) => (subset(B, cross_product(C, D)) => ilf_type(B, relation_type(C, D)))) <=> (ilf_type(D, set_type) => ((~subset(B, cross_product(C, D))) | ilf_type(B, relation_type(C, D)))))), rewrite((ilf_type(D, set_type) => ((~subset(B, cross_product(C, D))) | ilf_type(B, relation_type(C, D)))) <=> (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))), ((ilf_type(D, set_type) => (subset(B, cross_product(C, D)) => ilf_type(B, relation_type(C, D)))) <=> (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))), (![D: $i] : (ilf_type(D, set_type) => (subset(B, cross_product(C, D)) => ilf_type(B, relation_type(C, D)))) <=> ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D)))))), ((ilf_type(C, set_type) => ![D: $i] : (ilf_type(D, set_type) => (subset(B, cross_product(C, D)) => ilf_type(B, relation_type(C, D))))) <=> (ilf_type(C, set_type) => ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))), rewrite((ilf_type(C, set_type) => ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))) <=> ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D)))))), ((ilf_type(C, set_type) => ![D: $i] : (ilf_type(D, set_type) => (subset(B, cross_product(C, D)) => ilf_type(B, relation_type(C, D))))) <=> ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D)))))))), (![C: $i] : (ilf_type(C, set_type) => ![D: $i] : (ilf_type(D, set_type) => (subset(B, cross_product(C, D)) => ilf_type(B, relation_type(C, D))))) <=> ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))), ((ilf_type(B, set_type) => ![C: $i] : (ilf_type(C, set_type) => ![D: $i] : (ilf_type(D, set_type) => (subset(B, cross_product(C, D)) => ilf_type(B, relation_type(C, D)))))) <=> (ilf_type(B, set_type) => ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D)))))))), rewrite((ilf_type(B, set_type) => ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D)))))) <=> ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))), ((ilf_type(B, set_type) => ![C: $i] : (ilf_type(C, set_type) => ![D: $i] : (ilf_type(D, set_type) => (subset(B, cross_product(C, D)) => ilf_type(B, relation_type(C, D)))))) <=> ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(25,plain,
% 0.20/0.43 (![B: $i] : (ilf_type(B, set_type) => ![C: $i] : (ilf_type(C, set_type) => ![D: $i] : (ilf_type(D, set_type) => (subset(B, cross_product(C, D)) => ilf_type(B, relation_type(C, D)))))) <=> ![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[24])).
% 0.20/0.43 tff(26,axiom,(![B: $i] : (ilf_type(B, set_type) => ![C: $i] : (ilf_type(C, set_type) => ![D: $i] : (ilf_type(D, set_type) => (subset(B, cross_product(C, D)) => ilf_type(B, relation_type(C, D))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','p2')).
% 0.20/0.43 tff(27,plain,
% 0.20/0.43 (![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.20/0.43 tff(28,plain,
% 0.20/0.43 (![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[27, 23])).
% 0.20/0.43 tff(29,plain,(
% 0.20/0.43 ![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))),
% 0.20/0.43 inference(skolemize,[status(sab)],[28])).
% 0.20/0.43 tff(30,plain,
% 0.20/0.43 (![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[29, 22])).
% 0.20/0.43 tff(31,plain,
% 0.20/0.43 (((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))) | ((~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C, D)) | (~subset(B!8, cross_product(C, D))))))) <=> ((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))) | (~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C, D)) | (~subset(B!8, cross_product(C, D))))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(32,plain,
% 0.20/0.43 (((~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B!8, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B!8, cross_product(C, D)))))) <=> ((~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C, D)) | (~subset(B!8, cross_product(C, D))))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(33,plain,
% 0.20/0.43 (((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))) | ((~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B!8, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B!8, cross_product(C, D))))))) <=> ((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))) | ((~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C, D)) | (~subset(B!8, cross_product(C, D)))))))),
% 0.20/0.43 inference(monotonicity,[status(thm)],[32])).
% 0.20/0.43 tff(34,plain,
% 0.20/0.43 (((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))) | ((~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B!8, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B!8, cross_product(C, D))))))) <=> ((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))) | (~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C, D)) | (~subset(B!8, cross_product(C, D))))))),
% 0.20/0.43 inference(transitivity,[status(thm)],[33, 31])).
% 0.20/0.43 tff(35,plain,
% 0.20/0.43 ((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))) | ((~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B!8, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B!8, cross_product(C, D))))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(36,plain,
% 0.20/0.43 ((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : (ilf_type(B, relation_type(C, D)) | (~ilf_type(D, set_type)) | (~subset(B, cross_product(C, D))))))) | (~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C, D)) | (~subset(B!8, cross_product(C, D)))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.20/0.43 tff(37,plain,
% 0.20/0.43 (![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C, D)) | (~subset(B!8, cross_product(C, D)))))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[36, 30, 17])).
% 0.20/0.43 tff(38,plain,
% 0.20/0.43 (ilf_type(C!9, set_type)),
% 0.20/0.43 inference(and_elim,[status(thm)],[16])).
% 0.20/0.43 tff(39,plain,
% 0.20/0.43 (((~![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C, D)) | (~subset(B!8, cross_product(C, D)))))) | ((~ilf_type(C!9, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C!9, D)) | (~subset(B!8, cross_product(C!9, D)))))) <=> ((~![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C, D)) | (~subset(B!8, cross_product(C, D)))))) | (~ilf_type(C!9, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C!9, D)) | (~subset(B!8, cross_product(C!9, D)))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(40,plain,
% 0.20/0.43 ((~![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C, D)) | (~subset(B!8, cross_product(C, D)))))) | ((~ilf_type(C!9, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C!9, D)) | (~subset(B!8, cross_product(C!9, D)))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(41,plain,
% 0.20/0.43 ((~![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C, D)) | (~subset(B!8, cross_product(C, D)))))) | (~ilf_type(C!9, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C!9, D)) | (~subset(B!8, cross_product(C!9, D))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[40, 39])).
% 0.20/0.43 tff(42,plain,
% 0.20/0.43 (![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C!9, D)) | (~subset(B!8, cross_product(C!9, D))))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[41, 38, 37])).
% 0.20/0.43 tff(43,plain,
% 0.20/0.43 (~(ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10))))),
% 0.20/0.43 inference(and_elim,[status(thm)],[16])).
% 0.20/0.43 tff(44,plain,
% 0.20/0.43 (~ilf_type(B!8, relation_type(C!9, D!10))),
% 0.20/0.43 inference(or_elim,[status(thm)],[43])).
% 0.20/0.43 tff(45,plain,
% 0.20/0.43 (ilf_type(D!10, set_type)),
% 0.20/0.43 inference(and_elim,[status(thm)],[16])).
% 0.20/0.43 tff(46,plain,
% 0.20/0.43 (((~![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C!9, D)) | (~subset(B!8, cross_product(C!9, D))))) | (ilf_type(B!8, relation_type(C!9, D!10)) | (~ilf_type(D!10, set_type)) | (~subset(B!8, cross_product(C!9, D!10))))) <=> ((~![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C!9, D)) | (~subset(B!8, cross_product(C!9, D))))) | ilf_type(B!8, relation_type(C!9, D!10)) | (~ilf_type(D!10, set_type)) | (~subset(B!8, cross_product(C!9, D!10))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(47,plain,
% 0.20/0.43 (((~ilf_type(D!10, set_type)) | ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, cross_product(C!9, D!10)))) <=> (ilf_type(B!8, relation_type(C!9, D!10)) | (~ilf_type(D!10, set_type)) | (~subset(B!8, cross_product(C!9, D!10))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(48,plain,
% 0.20/0.43 (((~![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C!9, D)) | (~subset(B!8, cross_product(C!9, D))))) | ((~ilf_type(D!10, set_type)) | ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, cross_product(C!9, D!10))))) <=> ((~![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C!9, D)) | (~subset(B!8, cross_product(C!9, D))))) | (ilf_type(B!8, relation_type(C!9, D!10)) | (~ilf_type(D!10, set_type)) | (~subset(B!8, cross_product(C!9, D!10)))))),
% 0.20/0.43 inference(monotonicity,[status(thm)],[47])).
% 0.20/0.43 tff(49,plain,
% 0.20/0.43 (((~![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C!9, D)) | (~subset(B!8, cross_product(C!9, D))))) | ((~ilf_type(D!10, set_type)) | ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, cross_product(C!9, D!10))))) <=> ((~![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C!9, D)) | (~subset(B!8, cross_product(C!9, D))))) | ilf_type(B!8, relation_type(C!9, D!10)) | (~ilf_type(D!10, set_type)) | (~subset(B!8, cross_product(C!9, D!10))))),
% 0.20/0.43 inference(transitivity,[status(thm)],[48, 46])).
% 0.20/0.43 tff(50,plain,
% 0.20/0.43 ((~![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C!9, D)) | (~subset(B!8, cross_product(C!9, D))))) | ((~ilf_type(D!10, set_type)) | ilf_type(B!8, relation_type(C!9, D!10)) | (~subset(B!8, cross_product(C!9, D!10))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(51,plain,
% 0.20/0.43 ((~![D: $i] : ((~ilf_type(D, set_type)) | ilf_type(B!8, relation_type(C!9, D)) | (~subset(B!8, cross_product(C!9, D))))) | ilf_type(B!8, relation_type(C!9, D!10)) | (~ilf_type(D!10, set_type)) | (~subset(B!8, cross_product(C!9, D!10)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.20/0.43 tff(52,plain,
% 0.20/0.43 (~subset(B!8, cross_product(C!9, D!10))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[51, 45, 44, 42])).
% 0.20/0.43 tff(53,plain,
% 0.20/0.43 (^[B: $i] : refl(((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))) <=> ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(54,plain,
% 0.20/0.43 (![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))) <=> ![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[53])).
% 0.20/0.43 tff(55,plain,
% 0.20/0.43 (^[B: $i] : rewrite(((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))) <=> ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(56,plain,
% 0.20/0.44 (![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))) <=> ![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))),
% 0.20/0.44 inference(quant_intro,[status(thm)],[55])).
% 0.20/0.44 tff(57,plain,
% 0.20/0.44 (![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))) <=> ![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))),
% 0.20/0.44 inference(transitivity,[status(thm)],[56, 54])).
% 0.20/0.44 tff(58,plain,
% 0.20/0.44 (![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))) <=> ![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(59,plain,
% 0.20/0.44 (^[B: $i] : trans(monotonicity(quant_intro(proof_bind(^[C: $i] : trans(monotonicity(quant_intro(proof_bind(^[D: $i] : trans(monotonicity(quant_intro(proof_bind(^[E: $i] : trans(monotonicity(rewrite((subset(B, E) => subset(B, cross_product(C, D))) <=> ((~subset(B, E)) | subset(B, cross_product(C, D)))), ((ilf_type(E, relation_type(C, D)) => (subset(B, E) => subset(B, cross_product(C, D)))) <=> (ilf_type(E, relation_type(C, D)) => ((~subset(B, E)) | subset(B, cross_product(C, D)))))), rewrite((ilf_type(E, relation_type(C, D)) => ((~subset(B, E)) | subset(B, cross_product(C, D)))) <=> (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))), ((ilf_type(E, relation_type(C, D)) => (subset(B, E) => subset(B, cross_product(C, D)))) <=> (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))), (![E: $i] : (ilf_type(E, relation_type(C, D)) => (subset(B, E) => subset(B, cross_product(C, D)))) <=> ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))), ((ilf_type(D, set_type) => ![E: $i] : (ilf_type(E, relation_type(C, D)) => (subset(B, E) => subset(B, cross_product(C, D))))) <=> (ilf_type(D, set_type) => ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))), rewrite((ilf_type(D, set_type) => ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))) <=> ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))), ((ilf_type(D, set_type) => ![E: $i] : (ilf_type(E, relation_type(C, D)) => (subset(B, E) => subset(B, cross_product(C, D))))) <=> ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))), (![D: $i] : (ilf_type(D, set_type) => ![E: $i] : (ilf_type(E, relation_type(C, D)) => (subset(B, E) => subset(B, cross_product(C, D))))) <=> ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))), ((ilf_type(C, set_type) => ![D: $i] : (ilf_type(D, set_type) => ![E: $i] : (ilf_type(E, relation_type(C, D)) => (subset(B, E) => subset(B, cross_product(C, D)))))) <=> (ilf_type(C, set_type) => ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))), rewrite((ilf_type(C, set_type) => ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))) <=> ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))), ((ilf_type(C, set_type) => ![D: $i] : (ilf_type(D, set_type) => ![E: $i] : (ilf_type(E, relation_type(C, D)) => (subset(B, E) => subset(B, cross_product(C, D)))))) <=> ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))))), (![C: $i] : (ilf_type(C, set_type) => ![D: $i] : (ilf_type(D, set_type) => ![E: $i] : (ilf_type(E, relation_type(C, D)) => (subset(B, E) => subset(B, cross_product(C, D)))))) <=> ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))), ((ilf_type(B, set_type) => ![C: $i] : (ilf_type(C, set_type) => ![D: $i] : (ilf_type(D, set_type) => ![E: $i] : (ilf_type(E, relation_type(C, D)) => (subset(B, E) => subset(B, cross_product(C, D))))))) <=> (ilf_type(B, set_type) => ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))))), rewrite((ilf_type(B, set_type) => ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D))))))) <=> ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))), ((ilf_type(B, set_type) => ![C: $i] : (ilf_type(C, set_type) => ![D: $i] : (ilf_type(D, set_type) => ![E: $i] : (ilf_type(E, relation_type(C, D)) => (subset(B, E) => subset(B, cross_product(C, D))))))) <=> ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(60,plain,
% 0.20/0.44 (![B: $i] : (ilf_type(B, set_type) => ![C: $i] : (ilf_type(C, set_type) => ![D: $i] : (ilf_type(D, set_type) => ![E: $i] : (ilf_type(E, relation_type(C, D)) => (subset(B, E) => subset(B, cross_product(C, D))))))) <=> ![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))),
% 0.20/0.44 inference(quant_intro,[status(thm)],[59])).
% 0.20/0.44 tff(61,axiom,(![B: $i] : (ilf_type(B, set_type) => ![C: $i] : (ilf_type(C, set_type) => ![D: $i] : (ilf_type(D, set_type) => ![E: $i] : (ilf_type(E, relation_type(C, D)) => (subset(B, E) => subset(B, cross_product(C, D)))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','p1')).
% 0.20/0.44 tff(62,plain,
% 0.20/0.44 (![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[61, 60])).
% 0.20/0.44 tff(63,plain,
% 0.20/0.44 (![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[62, 58])).
% 0.20/0.44 tff(64,plain,(
% 0.20/0.44 ![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))),
% 0.20/0.44 inference(skolemize,[status(sab)],[63])).
% 0.20/0.44 tff(65,plain,
% 0.20/0.44 (![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[64, 57])).
% 0.20/0.44 tff(66,plain,
% 0.20/0.44 (((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))) | ((~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C, D))))))) <=> ((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))) | (~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C, D))))))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(67,plain,
% 0.20/0.44 (((~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B!8, cross_product(C, D)) | (~subset(B!8, E)) | (~ilf_type(E, relation_type(C, D))))))) <=> ((~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C, D))))))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(68,plain,
% 0.20/0.44 (((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))) | ((~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B!8, cross_product(C, D)) | (~subset(B!8, E)) | (~ilf_type(E, relation_type(C, D)))))))) <=> ((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))) | ((~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C, D)))))))),
% 0.20/0.44 inference(monotonicity,[status(thm)],[67])).
% 0.20/0.44 tff(69,plain,
% 0.20/0.44 (((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))) | ((~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B!8, cross_product(C, D)) | (~subset(B!8, E)) | (~ilf_type(E, relation_type(C, D)))))))) <=> ((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))) | (~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C, D))))))),
% 0.20/0.44 inference(transitivity,[status(thm)],[68, 66])).
% 0.20/0.44 tff(70,plain,
% 0.20/0.44 ((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))) | ((~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B!8, cross_product(C, D)) | (~subset(B!8, E)) | (~ilf_type(E, relation_type(C, D)))))))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(71,plain,
% 0.20/0.45 ((~![B: $i] : ((~ilf_type(B, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : (subset(B, cross_product(C, D)) | (~subset(B, E)) | (~ilf_type(E, relation_type(C, D)))))))) | (~ilf_type(B!8, set_type)) | ![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C, D)))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[70, 69])).
% 0.20/0.45 tff(72,plain,
% 0.20/0.45 (![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C, D)))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[71, 65, 17])).
% 0.20/0.45 tff(73,plain,
% 0.20/0.45 (((~![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C, D)))))) | ((~ilf_type(C!9, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D))) | subset(B!8, cross_product(C!9, D)))))) <=> ((~![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C, D)))))) | (~ilf_type(C!9, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D))) | subset(B!8, cross_product(C!9, D)))))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(74,plain,
% 0.20/0.45 (((~ilf_type(C!9, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C!9, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C!9, D))))) <=> ((~ilf_type(C!9, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D))) | subset(B!8, cross_product(C!9, D)))))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(75,plain,
% 0.20/0.45 (((~![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C, D)))))) | ((~ilf_type(C!9, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C!9, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C!9, D)))))) <=> ((~![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C, D)))))) | ((~ilf_type(C!9, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D))) | subset(B!8, cross_product(C!9, D))))))),
% 0.20/0.45 inference(monotonicity,[status(thm)],[74])).
% 0.20/0.45 tff(76,plain,
% 0.20/0.45 (((~![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C, D)))))) | ((~ilf_type(C!9, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C!9, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C!9, D)))))) <=> ((~![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C, D)))))) | (~ilf_type(C!9, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D))) | subset(B!8, cross_product(C!9, D)))))),
% 0.20/0.45 inference(transitivity,[status(thm)],[75, 73])).
% 0.20/0.45 tff(77,plain,
% 0.20/0.45 ((~![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C, D)))))) | ((~ilf_type(C!9, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C!9, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C!9, D)))))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(78,plain,
% 0.20/0.45 ((~![C: $i] : ((~ilf_type(C, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~ilf_type(E, relation_type(C, D))) | (~subset(B!8, E)) | subset(B!8, cross_product(C, D)))))) | (~ilf_type(C!9, set_type)) | ![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D))) | subset(B!8, cross_product(C!9, D))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.20/0.45 tff(79,plain,
% 0.20/0.45 (![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D))) | subset(B!8, cross_product(C!9, D))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[78, 38, 72])).
% 0.20/0.45 tff(80,plain,
% 0.20/0.45 (((~![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D))) | subset(B!8, cross_product(C!9, D))))) | ((~ilf_type(D!10, set_type)) | ![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D!10))) | subset(B!8, cross_product(C!9, D!10))))) <=> ((~![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D))) | subset(B!8, cross_product(C!9, D))))) | (~ilf_type(D!10, set_type)) | ![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D!10))) | subset(B!8, cross_product(C!9, D!10))))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(81,plain,
% 0.20/0.45 ((~![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D))) | subset(B!8, cross_product(C!9, D))))) | ((~ilf_type(D!10, set_type)) | ![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D!10))) | subset(B!8, cross_product(C!9, D!10))))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(82,plain,
% 0.20/0.45 ((~![D: $i] : ((~ilf_type(D, set_type)) | ![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D))) | subset(B!8, cross_product(C!9, D))))) | (~ilf_type(D!10, set_type)) | ![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D!10))) | subset(B!8, cross_product(C!9, D!10)))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[81, 80])).
% 0.20/0.45 tff(83,plain,
% 0.20/0.45 (![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D!10))) | subset(B!8, cross_product(C!9, D!10)))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[82, 45, 79])).
% 0.20/0.45 tff(84,plain,
% 0.20/0.45 (ilf_type(E!11, relation_type(C!9, D!10))),
% 0.20/0.45 inference(or_elim,[status(thm)],[43])).
% 0.20/0.45 tff(85,plain,
% 0.20/0.45 (subset(B!8, E!11)),
% 0.20/0.45 inference(or_elim,[status(thm)],[43])).
% 0.20/0.45 tff(86,plain,
% 0.20/0.45 (((~![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D!10))) | subset(B!8, cross_product(C!9, D!10)))) | ((~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10))) | subset(B!8, cross_product(C!9, D!10)))) <=> ((~![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D!10))) | subset(B!8, cross_product(C!9, D!10)))) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10))) | subset(B!8, cross_product(C!9, D!10)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(87,plain,
% 0.20/0.45 ((~![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D!10))) | subset(B!8, cross_product(C!9, D!10)))) | ((~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10))) | subset(B!8, cross_product(C!9, D!10)))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(88,plain,
% 0.20/0.45 ((~![E: $i] : ((~subset(B!8, E)) | (~ilf_type(E, relation_type(C!9, D!10))) | subset(B!8, cross_product(C!9, D!10)))) | (~subset(B!8, E!11)) | (~ilf_type(E!11, relation_type(C!9, D!10))) | subset(B!8, cross_product(C!9, D!10))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[87, 86])).
% 0.20/0.45 tff(89,plain,
% 0.20/0.45 ($false),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[88, 85, 84, 83, 52])).
% 0.20/0.45 % SZS output end Proof
% 0.20/0.45 % E exiting
%------------------------------------------------------------------------------