TSTP Solution File: ARI616_1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : ARI616_1 : TPTP v8.1.0. Released v5.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n018.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 : Tue Sep 6 17:02:18 EDT 2022
% Result : Theorem 0.13s 0.39s
% Output : Proof 0.13s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ARI616_1 : TPTP v8.1.0. Released v5.1.0.
% 0.12/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n018.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Tue Aug 30 00:55:10 EDT 2022
% 0.13/0.35 % CPUTime :
% 0.13/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.35 Usage: tptp [options] [-file:]file
% 0.13/0.35 -h, -? prints this message.
% 0.13/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.35 -m, -model generate model.
% 0.13/0.35 -p, -proof generate proof.
% 0.13/0.35 -c, -core generate unsat core of named formulas.
% 0.13/0.35 -st, -statistics display statistics.
% 0.13/0.35 -t:timeout set timeout (in second).
% 0.13/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.35 -<param>:<value> configuration parameter and value.
% 0.13/0.35 -o:<output-file> file to place output in.
% 0.13/0.39 % SZS status Theorem
% 0.13/0.39 % SZS output start Proof
% 0.13/0.39 tff(tptp_fun_X_4_type, type, (
% 0.13/0.39 tptp_fun_X_4: $int)).
% 0.13/0.39 tff(tptp_fun_Y2_1_type, type, (
% 0.13/0.39 tptp_fun_Y2_1: $int)).
% 0.13/0.39 tff(tptp_fun_Z2_0_type, type, (
% 0.13/0.39 tptp_fun_Z2_0: $int)).
% 0.13/0.39 tff(p_type, type, (
% 0.13/0.39 p: ( $int * $int * $int ) > $o)).
% 0.13/0.39 tff(tptp_fun_Y1_3_type, type, (
% 0.13/0.39 tptp_fun_Y1_3: $int)).
% 0.13/0.39 tff(tptp_fun_Z1_2_type, type, (
% 0.13/0.39 tptp_fun_Z1_2: $int)).
% 0.13/0.39 tff(1,plain,
% 0.13/0.39 (^[X: $int, Y: $int, Z: $int] : refl(((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z)) <=> ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z)))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(2,plain,
% 0.13/0.39 (![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z)) <=> ![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[1])).
% 0.13/0.39 tff(3,plain,
% 0.13/0.39 (^[X: $int, Y: $int, Z: $int] : rewrite((($greatereq($sum(X, $sum($product(-1, Y), Z)), 0) & $lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)) <=> p(X, Y, Z)) <=> ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z)))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(4,plain,
% 0.13/0.39 (![X: $int, Y: $int, Z: $int] : (($greatereq($sum(X, $sum($product(-1, Y), Z)), 0) & $lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)) <=> p(X, Y, Z)) <=> ![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[3])).
% 0.13/0.39 tff(5,plain,
% 0.13/0.39 (^[X: $int, Y: $int, Z: $int] : rewrite((($lesseq($sum(Y, $sum($product(-1, Z), $product(-1, X))), 0) & $lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)) <=> p(X, Y, Z)) <=> (($greatereq($sum(X, $sum($product(-1, Y), Z)), 0) & $lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)) <=> p(X, Y, Z)))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(6,plain,
% 0.13/0.39 (![X: $int, Y: $int, Z: $int] : (($lesseq($sum(Y, $sum($product(-1, Z), $product(-1, X))), 0) & $lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)) <=> p(X, Y, Z)) <=> ![X: $int, Y: $int, Z: $int] : (($greatereq($sum(X, $sum($product(-1, Y), Z)), 0) & $lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)) <=> p(X, Y, Z))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[5])).
% 0.13/0.39 tff(7,plain,
% 0.13/0.39 (^[X: $int, Y: $int, Z: $int] : rewrite((($lesseq($sum(Y, $product(-1, Z)), X) & $lesseq(X, $sum(Y, Z))) <=> p(X, Y, Z)) <=> (($lesseq($sum(Y, $sum($product(-1, Z), $product(-1, X))), 0) & $lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)) <=> p(X, Y, Z)))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(8,plain,
% 0.13/0.39 (![X: $int, Y: $int, Z: $int] : (($lesseq($sum(Y, $product(-1, Z)), X) & $lesseq(X, $sum(Y, Z))) <=> p(X, Y, Z)) <=> ![X: $int, Y: $int, Z: $int] : (($lesseq($sum(Y, $sum($product(-1, Z), $product(-1, X))), 0) & $lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)) <=> p(X, Y, Z))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[7])).
% 0.13/0.39 tff(9,plain,
% 0.13/0.39 (![X: $int, Y: $int, Z: $int] : (($lesseq($sum(Y, $product(-1, Z)), X) & $lesseq(X, $sum(Y, Z))) <=> p(X, Y, Z)) <=> ![X: $int, Y: $int, Z: $int] : (($lesseq($sum(Y, $product(-1, Z)), X) & $lesseq(X, $sum(Y, Z))) <=> p(X, Y, Z))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(10,plain,
% 0.13/0.39 ((~(![X: $int, Y: $int, Z: $int] : (($lesseq($sum(Y, $uminus(Z)), X) & $lesseq(X, $sum(Y, Z))) <=> p(X, Y, Z)) => ![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : (?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2)) => $lesseq($sum(Y1, $uminus(Y2)), $sum(Z1, Z2))))) <=> (~((~![X: $int, Y: $int, Z: $int] : (($lesseq($sum(Y, $product(-1, Z)), X) & $lesseq(X, $sum(Y, Z))) <=> p(X, Y, Z))) | ![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $lesseq($sum(Y1, $product(-1, Y2)), $sum(Z1, Z2)))))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(11,axiom,(~(![X: $int, Y: $int, Z: $int] : (($lesseq($sum(Y, $uminus(Z)), X) & $lesseq(X, $sum(Y, Z))) <=> p(X, Y, Z)) => ![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : (?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2)) => $lesseq($sum(Y1, $uminus(Y2)), $sum(Z1, Z2))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','sum_of_radii_gt_distance_of_centers')).
% 0.13/0.39 tff(12,plain,
% 0.13/0.39 (~((~![X: $int, Y: $int, Z: $int] : (($lesseq($sum(Y, $product(-1, Z)), X) & $lesseq(X, $sum(Y, Z))) <=> p(X, Y, Z))) | ![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $lesseq($sum(Y1, $product(-1, Y2)), $sum(Z1, Z2))))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.13/0.39 tff(13,plain,
% 0.13/0.39 (![X: $int, Y: $int, Z: $int] : (($lesseq($sum(Y, $product(-1, Z)), X) & $lesseq(X, $sum(Y, Z))) <=> p(X, Y, Z))),
% 0.13/0.39 inference(or_elim,[status(thm)],[12])).
% 0.13/0.39 tff(14,plain,
% 0.13/0.39 (![X: $int, Y: $int, Z: $int] : (($lesseq($sum(Y, $product(-1, Z)), X) & $lesseq(X, $sum(Y, Z))) <=> p(X, Y, Z))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[13, 9])).
% 0.13/0.39 tff(15,plain,
% 0.13/0.39 (![X: $int, Y: $int, Z: $int] : (($lesseq($sum(Y, $sum($product(-1, Z), $product(-1, X))), 0) & $lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)) <=> p(X, Y, Z))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[14, 8])).
% 0.13/0.39 tff(16,plain,
% 0.13/0.39 (![X: $int, Y: $int, Z: $int] : (($greatereq($sum(X, $sum($product(-1, Y), Z)), 0) & $lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)) <=> p(X, Y, Z))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[15, 6])).
% 0.13/0.39 tff(17,plain,(
% 0.13/0.39 ![X: $int, Y: $int, Z: $int] : (($greatereq($sum(X, $sum($product(-1, Y), Z)), 0) & $lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)) <=> p(X, Y, Z))),
% 0.13/0.39 inference(skolemize,[status(sab)],[16])).
% 0.13/0.39 tff(18,plain,
% 0.13/0.39 (![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[17, 4])).
% 0.13/0.39 tff(19,plain,
% 0.13/0.39 (![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[18, 2])).
% 0.13/0.39 tff(20,plain,
% 0.13/0.39 (((~![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))) | ((~((~$greatereq($sum(Z2!0, $sum($product(-1, Y2!1), X!4)), 0)) | (~$greatereq($sum(Z2!0, $sum(Y2!1, $product(-1, X!4))), 0)))) <=> p(X!4, Y2!1, Z2!0))) <=> ((~![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))) | ((~((~$greatereq($sum(Z2!0, $sum($product(-1, Y2!1), X!4)), 0)) | (~$greatereq($sum(Z2!0, $sum(Y2!1, $product(-1, X!4))), 0)))) <=> p(X!4, Y2!1, Z2!0)))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(21,plain,
% 0.13/0.39 (((~((~$greatereq($sum(X!4, $sum($product(-1, Y2!1), Z2!0)), 0)) | (~$lesseq($sum(X!4, $sum($product(-1, Y2!1), $product(-1, Z2!0))), 0)))) <=> p(X!4, Y2!1, Z2!0)) <=> ((~((~$greatereq($sum(Z2!0, $sum($product(-1, Y2!1), X!4)), 0)) | (~$greatereq($sum(Z2!0, $sum(Y2!1, $product(-1, X!4))), 0)))) <=> p(X!4, Y2!1, Z2!0))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(22,plain,
% 0.13/0.39 (((~![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))) | ((~((~$greatereq($sum(X!4, $sum($product(-1, Y2!1), Z2!0)), 0)) | (~$lesseq($sum(X!4, $sum($product(-1, Y2!1), $product(-1, Z2!0))), 0)))) <=> p(X!4, Y2!1, Z2!0))) <=> ((~![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))) | ((~((~$greatereq($sum(Z2!0, $sum($product(-1, Y2!1), X!4)), 0)) | (~$greatereq($sum(Z2!0, $sum(Y2!1, $product(-1, X!4))), 0)))) <=> p(X!4, Y2!1, Z2!0)))),
% 0.13/0.40 inference(monotonicity,[status(thm)],[21])).
% 0.13/0.40 tff(23,plain,
% 0.13/0.40 (((~![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))) | ((~((~$greatereq($sum(X!4, $sum($product(-1, Y2!1), Z2!0)), 0)) | (~$lesseq($sum(X!4, $sum($product(-1, Y2!1), $product(-1, Z2!0))), 0)))) <=> p(X!4, Y2!1, Z2!0))) <=> ((~![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))) | ((~((~$greatereq($sum(Z2!0, $sum($product(-1, Y2!1), X!4)), 0)) | (~$greatereq($sum(Z2!0, $sum(Y2!1, $product(-1, X!4))), 0)))) <=> p(X!4, Y2!1, Z2!0)))),
% 0.13/0.40 inference(transitivity,[status(thm)],[22, 20])).
% 0.13/0.40 tff(24,plain,
% 0.13/0.40 ((~![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))) | ((~((~$greatereq($sum(X!4, $sum($product(-1, Y2!1), Z2!0)), 0)) | (~$lesseq($sum(X!4, $sum($product(-1, Y2!1), $product(-1, Z2!0))), 0)))) <=> p(X!4, Y2!1, Z2!0))),
% 0.13/0.40 inference(quant_inst,[status(thm)],[])).
% 0.13/0.40 tff(25,plain,
% 0.13/0.40 ((~![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))) | ((~((~$greatereq($sum(Z2!0, $sum($product(-1, Y2!1), X!4)), 0)) | (~$greatereq($sum(Z2!0, $sum(Y2!1, $product(-1, X!4))), 0)))) <=> p(X!4, Y2!1, Z2!0))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[24, 23])).
% 0.13/0.40 tff(26,plain,
% 0.13/0.40 ((~((~$greatereq($sum(Z2!0, $sum($product(-1, Y2!1), X!4)), 0)) | (~$greatereq($sum(Z2!0, $sum(Y2!1, $product(-1, X!4))), 0)))) <=> p(X!4, Y2!1, Z2!0)),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[25, 19])).
% 0.13/0.40 tff(27,plain,
% 0.13/0.40 (((p(X!4, Y1!3, Z1!2) & p(X!4, Y2!1, Z2!0)) & (~$greatereq($sum(Z2!0, $sum(Y2!1, $sum(Z1!2, $product(-1, Y1!3)))), 0))) <=> (p(X!4, Y1!3, Z1!2) & p(X!4, Y2!1, Z2!0) & (~$greatereq($sum(Z2!0, $sum(Y2!1, $sum(Z1!2, $product(-1, Y1!3)))), 0)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(28,plain,
% 0.13/0.40 ((~$greatereq($sum(Z1!2, $sum(Y2!1, $sum(Z2!0, $product(-1, Y1!3)))), 0)) <=> (~$greatereq($sum(Z2!0, $sum(Y2!1, $sum(Z1!2, $product(-1, Y1!3)))), 0))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(29,plain,
% 0.13/0.40 (((p(X!4, Y1!3, Z1!2) & p(X!4, Y2!1, Z2!0)) & (~$greatereq($sum(Z1!2, $sum(Y2!1, $sum(Z2!0, $product(-1, Y1!3)))), 0))) <=> ((p(X!4, Y1!3, Z1!2) & p(X!4, Y2!1, Z2!0)) & (~$greatereq($sum(Z2!0, $sum(Y2!1, $sum(Z1!2, $product(-1, Y1!3)))), 0)))),
% 0.13/0.40 inference(monotonicity,[status(thm)],[28])).
% 0.13/0.40 tff(30,plain,
% 0.13/0.40 (((p(X!4, Y1!3, Z1!2) & p(X!4, Y2!1, Z2!0)) & (~$greatereq($sum(Z1!2, $sum(Y2!1, $sum(Z2!0, $product(-1, Y1!3)))), 0))) <=> (p(X!4, Y1!3, Z1!2) & p(X!4, Y2!1, Z2!0) & (~$greatereq($sum(Z2!0, $sum(Y2!1, $sum(Z1!2, $product(-1, Y1!3)))), 0)))),
% 0.13/0.40 inference(transitivity,[status(thm)],[29, 27])).
% 0.13/0.40 tff(31,plain,
% 0.13/0.40 ((~![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $greatereq($sum(Z1, $sum(Y2, $sum(Z2, $product(-1, Y1)))), 0))) <=> (~![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $greatereq($sum(Z1, $sum(Y2, $sum(Z2, $product(-1, Y1)))), 0)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(32,plain,
% 0.13/0.40 ((~![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $lesseq($sum(Y1, $sum($product(-1, Y2), $sum($product(-1, Z1), $product(-1, Z2)))), 0))) <=> (~![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $greatereq($sum(Z1, $sum(Y2, $sum(Z2, $product(-1, Y1)))), 0)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(33,plain,
% 0.13/0.40 ((~![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $lesseq($sum(Y1, $product(-1, Y2)), $sum(Z1, Z2)))) <=> (~![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $lesseq($sum(Y1, $sum($product(-1, Y2), $sum($product(-1, Z1), $product(-1, Z2)))), 0)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(34,plain,
% 0.13/0.40 ((~![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $lesseq($sum(Y1, $product(-1, Y2)), $sum(Z1, Z2)))) <=> (~![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $lesseq($sum(Y1, $product(-1, Y2)), $sum(Z1, Z2))))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(35,plain,
% 0.13/0.40 (~![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $lesseq($sum(Y1, $product(-1, Y2)), $sum(Z1, Z2)))),
% 0.13/0.40 inference(or_elim,[status(thm)],[12])).
% 0.13/0.40 tff(36,plain,
% 0.13/0.40 (~![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $lesseq($sum(Y1, $product(-1, Y2)), $sum(Z1, Z2)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.13/0.40 tff(37,plain,
% 0.13/0.40 (~![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $lesseq($sum(Y1, $product(-1, Y2)), $sum(Z1, Z2)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[36, 34])).
% 0.13/0.40 tff(38,plain,
% 0.13/0.40 (~![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $lesseq($sum(Y1, $product(-1, Y2)), $sum(Z1, Z2)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[37, 34])).
% 0.13/0.40 tff(39,plain,
% 0.13/0.40 (~![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $lesseq($sum(Y1, $sum($product(-1, Y2), $sum($product(-1, Z1), $product(-1, Z2)))), 0))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[38, 33])).
% 0.13/0.40 tff(40,plain,
% 0.13/0.40 (~![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $greatereq($sum(Z1, $sum(Y2, $sum(Z2, $product(-1, Y1)))), 0))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[39, 32])).
% 0.13/0.40 tff(41,plain,
% 0.13/0.40 (~![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $greatereq($sum(Z1, $sum(Y2, $sum(Z2, $product(-1, Y1)))), 0))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[40, 31])).
% 0.13/0.40 tff(42,plain,
% 0.13/0.40 (~![Y1: $int, Z1: $int, Y2: $int, Z2: $int] : ((~?[X: $int] : (p(X, Y1, Z1) & p(X, Y2, Z2))) | $greatereq($sum(Z1, $sum(Y2, $sum(Z2, $product(-1, Y1)))), 0))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[41, 31])).
% 0.13/0.40 tff(43,plain,
% 0.13/0.40 (p(X!4, Y1!3, Z1!2) & p(X!4, Y2!1, Z2!0) & (~$greatereq($sum(Z2!0, $sum(Y2!1, $sum(Z1!2, $product(-1, Y1!3)))), 0))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[42, 30])).
% 0.13/0.40 tff(44,plain,
% 0.13/0.40 (p(X!4, Y2!1, Z2!0)),
% 0.13/0.40 inference(and_elim,[status(thm)],[43])).
% 0.13/0.40 tff(45,plain,
% 0.13/0.40 ((~((~((~$greatereq($sum(Z2!0, $sum($product(-1, Y2!1), X!4)), 0)) | (~$greatereq($sum(Z2!0, $sum(Y2!1, $product(-1, X!4))), 0)))) <=> p(X!4, Y2!1, Z2!0))) | (~((~$greatereq($sum(Z2!0, $sum($product(-1, Y2!1), X!4)), 0)) | (~$greatereq($sum(Z2!0, $sum(Y2!1, $product(-1, X!4))), 0)))) | (~p(X!4, Y2!1, Z2!0))),
% 0.13/0.40 inference(tautology,[status(thm)],[])).
% 0.13/0.40 tff(46,plain,
% 0.13/0.40 ((~((~((~$greatereq($sum(Z2!0, $sum($product(-1, Y2!1), X!4)), 0)) | (~$greatereq($sum(Z2!0, $sum(Y2!1, $product(-1, X!4))), 0)))) <=> p(X!4, Y2!1, Z2!0))) | (~((~$greatereq($sum(Z2!0, $sum($product(-1, Y2!1), X!4)), 0)) | (~$greatereq($sum(Z2!0, $sum(Y2!1, $product(-1, X!4))), 0))))),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[45, 44])).
% 0.13/0.40 tff(47,plain,
% 0.13/0.40 (~((~$greatereq($sum(Z2!0, $sum($product(-1, Y2!1), X!4)), 0)) | (~$greatereq($sum(Z2!0, $sum(Y2!1, $product(-1, X!4))), 0)))),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[46, 26])).
% 0.13/0.40 tff(48,plain,
% 0.13/0.40 (((~$greatereq($sum(Z2!0, $sum($product(-1, Y2!1), X!4)), 0)) | (~$greatereq($sum(Z2!0, $sum(Y2!1, $product(-1, X!4))), 0))) | $greatereq($sum(Z2!0, $sum(Y2!1, $product(-1, X!4))), 0)),
% 0.13/0.40 inference(tautology,[status(thm)],[])).
% 0.13/0.40 tff(49,plain,
% 0.13/0.40 ($greatereq($sum(Z2!0, $sum(Y2!1, $product(-1, X!4))), 0)),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[48, 47])).
% 0.13/0.40 tff(50,plain,
% 0.13/0.40 (~$greatereq($sum(Z2!0, $sum(Y2!1, $sum(Z1!2, $product(-1, Y1!3)))), 0)),
% 0.13/0.40 inference(and_elim,[status(thm)],[43])).
% 0.13/0.40 tff(51,plain,
% 0.13/0.40 (((~![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))) | ((~((~$greatereq($sum(Z1!2, $sum($product(-1, Y1!3), X!4)), 0)) | (~$greatereq($sum(Z1!2, $sum(Y1!3, $product(-1, X!4))), 0)))) <=> p(X!4, Y1!3, Z1!2))) <=> ((~![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))) | ((~((~$greatereq($sum(Z1!2, $sum($product(-1, Y1!3), X!4)), 0)) | (~$greatereq($sum(Z1!2, $sum(Y1!3, $product(-1, X!4))), 0)))) <=> p(X!4, Y1!3, Z1!2)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(52,plain,
% 0.13/0.40 (((~((~$greatereq($sum(X!4, $sum($product(-1, Y1!3), Z1!2)), 0)) | (~$lesseq($sum(X!4, $sum($product(-1, Y1!3), $product(-1, Z1!2))), 0)))) <=> p(X!4, Y1!3, Z1!2)) <=> ((~((~$greatereq($sum(Z1!2, $sum($product(-1, Y1!3), X!4)), 0)) | (~$greatereq($sum(Z1!2, $sum(Y1!3, $product(-1, X!4))), 0)))) <=> p(X!4, Y1!3, Z1!2))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(53,plain,
% 0.13/0.40 (((~![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))) | ((~((~$greatereq($sum(X!4, $sum($product(-1, Y1!3), Z1!2)), 0)) | (~$lesseq($sum(X!4, $sum($product(-1, Y1!3), $product(-1, Z1!2))), 0)))) <=> p(X!4, Y1!3, Z1!2))) <=> ((~![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))) | ((~((~$greatereq($sum(Z1!2, $sum($product(-1, Y1!3), X!4)), 0)) | (~$greatereq($sum(Z1!2, $sum(Y1!3, $product(-1, X!4))), 0)))) <=> p(X!4, Y1!3, Z1!2)))),
% 0.13/0.40 inference(monotonicity,[status(thm)],[52])).
% 0.13/0.40 tff(54,plain,
% 0.13/0.40 (((~![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))) | ((~((~$greatereq($sum(X!4, $sum($product(-1, Y1!3), Z1!2)), 0)) | (~$lesseq($sum(X!4, $sum($product(-1, Y1!3), $product(-1, Z1!2))), 0)))) <=> p(X!4, Y1!3, Z1!2))) <=> ((~![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))) | ((~((~$greatereq($sum(Z1!2, $sum($product(-1, Y1!3), X!4)), 0)) | (~$greatereq($sum(Z1!2, $sum(Y1!3, $product(-1, X!4))), 0)))) <=> p(X!4, Y1!3, Z1!2)))),
% 0.13/0.40 inference(transitivity,[status(thm)],[53, 51])).
% 0.13/0.40 tff(55,plain,
% 0.13/0.40 ((~![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))) | ((~((~$greatereq($sum(X!4, $sum($product(-1, Y1!3), Z1!2)), 0)) | (~$lesseq($sum(X!4, $sum($product(-1, Y1!3), $product(-1, Z1!2))), 0)))) <=> p(X!4, Y1!3, Z1!2))),
% 0.13/0.40 inference(quant_inst,[status(thm)],[])).
% 0.13/0.40 tff(56,plain,
% 0.13/0.40 ((~![X: $int, Y: $int, Z: $int] : ((~((~$greatereq($sum(X, $sum($product(-1, Y), Z)), 0)) | (~$lesseq($sum(X, $sum($product(-1, Y), $product(-1, Z))), 0)))) <=> p(X, Y, Z))) | ((~((~$greatereq($sum(Z1!2, $sum($product(-1, Y1!3), X!4)), 0)) | (~$greatereq($sum(Z1!2, $sum(Y1!3, $product(-1, X!4))), 0)))) <=> p(X!4, Y1!3, Z1!2))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[55, 54])).
% 0.13/0.40 tff(57,plain,
% 0.13/0.40 ((~((~$greatereq($sum(Z1!2, $sum($product(-1, Y1!3), X!4)), 0)) | (~$greatereq($sum(Z1!2, $sum(Y1!3, $product(-1, X!4))), 0)))) <=> p(X!4, Y1!3, Z1!2)),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[56, 19])).
% 0.13/0.40 tff(58,plain,
% 0.13/0.40 (p(X!4, Y1!3, Z1!2)),
% 0.13/0.40 inference(and_elim,[status(thm)],[43])).
% 0.13/0.40 tff(59,plain,
% 0.13/0.40 ((~((~((~$greatereq($sum(Z1!2, $sum($product(-1, Y1!3), X!4)), 0)) | (~$greatereq($sum(Z1!2, $sum(Y1!3, $product(-1, X!4))), 0)))) <=> p(X!4, Y1!3, Z1!2))) | (~((~$greatereq($sum(Z1!2, $sum($product(-1, Y1!3), X!4)), 0)) | (~$greatereq($sum(Z1!2, $sum(Y1!3, $product(-1, X!4))), 0)))) | (~p(X!4, Y1!3, Z1!2))),
% 0.13/0.40 inference(tautology,[status(thm)],[])).
% 0.13/0.40 tff(60,plain,
% 0.13/0.40 ((~((~((~$greatereq($sum(Z1!2, $sum($product(-1, Y1!3), X!4)), 0)) | (~$greatereq($sum(Z1!2, $sum(Y1!3, $product(-1, X!4))), 0)))) <=> p(X!4, Y1!3, Z1!2))) | (~((~$greatereq($sum(Z1!2, $sum($product(-1, Y1!3), X!4)), 0)) | (~$greatereq($sum(Z1!2, $sum(Y1!3, $product(-1, X!4))), 0))))),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[59, 58])).
% 0.13/0.40 tff(61,plain,
% 0.13/0.40 (~((~$greatereq($sum(Z1!2, $sum($product(-1, Y1!3), X!4)), 0)) | (~$greatereq($sum(Z1!2, $sum(Y1!3, $product(-1, X!4))), 0)))),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[60, 57])).
% 0.13/0.40 tff(62,plain,
% 0.13/0.40 (((~$greatereq($sum(Z1!2, $sum($product(-1, Y1!3), X!4)), 0)) | (~$greatereq($sum(Z1!2, $sum(Y1!3, $product(-1, X!4))), 0))) | $greatereq($sum(Z1!2, $sum($product(-1, Y1!3), X!4)), 0)),
% 0.13/0.40 inference(tautology,[status(thm)],[])).
% 0.13/0.40 tff(63,plain,
% 0.13/0.40 ($greatereq($sum(Z1!2, $sum($product(-1, Y1!3), X!4)), 0)),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[62, 61])).
% 0.13/0.40 tff(64,plain,
% 0.13/0.40 ($false),
% 0.13/0.40 inference(theory_lemma,[status(thm)],[63, 50, 49])).
% 0.13/0.40 % SZS output end Proof
%------------------------------------------------------------------------------