TSTP Solution File: SWW579_2 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SWW579_2 : TPTP v8.1.0. Released v6.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n009.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 : Thu Sep 29 20:59:20 EDT 2022
% Result : Theorem 0.21s 0.41s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SWW579_2 : TPTP v8.1.0. Released v6.1.0.
% 0.07/0.14 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.14/0.34 % Computer : n009.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Sun Sep 4 20:05:35 EDT 2022
% 0.14/0.35 % CPUTime :
% 0.14/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.14/0.35 Usage: tptp [options] [-file:]file
% 0.14/0.35 -h, -? prints this message.
% 0.14/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.14/0.35 -m, -model generate model.
% 0.14/0.35 -p, -proof generate proof.
% 0.14/0.35 -c, -core generate unsat core of named formulas.
% 0.14/0.35 -st, -statistics display statistics.
% 0.14/0.35 -t:timeout set timeout (in second).
% 0.14/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.14/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.14/0.35 -<param>:<value> configuration parameter and value.
% 0.14/0.35 -o:<output-file> file to place output in.
% 0.21/0.41 % SZS status Theorem
% 0.21/0.41 % SZS output start Proof
% 0.21/0.41 tff(tptp_fun_Eps0_0_type, type, (
% 0.21/0.41 tptp_fun_Eps0_0: $real)).
% 0.21/0.41 tff(tptp_fun_N_1_type, type, (
% 0.21/0.41 tptp_fun_N_1: $int)).
% 0.21/0.41 tff(tptp_fun_Eps_2_type, type, (
% 0.21/0.41 tptp_fun_Eps_2: $real)).
% 0.21/0.41 tff(tptp_fun_R_3_type, type, (
% 0.21/0.41 tptp_fun_R_3: $real)).
% 0.21/0.41 tff(max_type, type, (
% 0.21/0.41 max: ( $real * $real ) > $real)).
% 0.21/0.41 tff(1,plain,
% 0.21/0.41 ((0 = $sum($quotient($product(Eps0!0, $to_real(N!1)), Eps0!0), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0))))) <=> ($sum($quotient($product(Eps0!0, $to_real(N!1)), Eps0!0), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))) = 0)),
% 0.21/0.41 inference(commutativity,[status(thm)],[])).
% 0.21/0.41 tff(2,plain,
% 0.21/0.41 (($sum($quotient($product(Eps0!0, $to_real(N!1)), Eps0!0), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))) = 0) <=> (0 = $sum($quotient($product(Eps0!0, $to_real(N!1)), Eps0!0), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))))),
% 0.21/0.41 inference(symmetry,[status(thm)],[1])).
% 0.21/0.41 tff(3,plain,
% 0.21/0.41 (((Eps0!0 = 0) <=> $false) <=> (~(Eps0!0 = 0))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(4,plain,
% 0.21/0.41 ((~(((~$lesseq($sum(Eps!2, $product(-1, R!3)), 0)) & (~$lesseq(Eps!2, 1))) | $lesseq($quotient(1, Eps0!0), 0) | $greatereq($sum($quotient(max(R!3, 1), Eps0!0), $product(-1, $to_real(N!1))), 0) | (~$greatereq($sum($quotient(max(R!3, 1), Eps0!0), $product(-1, $quotient($product(Eps0!0, $to_real(N!1)), Eps0!0))), 0)) | (~$greatereq($sum($product(max(R!3, 1), $quotient(1, Eps0!0)), $product(-1, $product(Eps0!0, $product($to_real(N!1), $quotient(1, Eps0!0))))), 0)) | (~$greatereq($sum(max(R!3, 1), $product(-1, $product(Eps0!0, $to_real(N!1)))), 0)) | (~($greatereq(R!3, 0) & (~$lesseq(Eps0!0, 0)) & $greatereq(N!1, 1) & ($sum(Eps!2, $product(-1, $product(Eps0!0, $to_real(N!1)))) = 0))))) <=> (~(((~$lesseq($sum(Eps!2, $product(-1, R!3)), 0)) & (~$lesseq(Eps!2, 1))) | $lesseq($quotient(1, Eps0!0), 0) | $greatereq($sum($quotient(max(R!3, 1), Eps0!0), $product(-1, $to_real(N!1))), 0) | (~$greatereq($sum($quotient(max(R!3, 1), Eps0!0), $product(-1, $quotient($product(Eps0!0, $to_real(N!1)), Eps0!0))), 0)) | (~$greatereq($sum(max(R!3, 1), $product(-1, $product(Eps0!0, $to_real(N!1)))), 0)) | (~($greatereq(R!3, 0) & (~$lesseq(Eps0!0, 0)) & $greatereq(N!1, 1) & ($sum(Eps!2, $product(-1, $product(Eps0!0, $to_real(N!1)))) = 0))) | (~$greatereq($sum($product($quotient(1, Eps0!0), max(R!3, 1)), $product(-1, $product(Eps0!0, $product($quotient(1, Eps0!0), $to_real(N!1))))), 0))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(5,plain,
% 0.21/0.41 ((~![R: $real, Eps: $real, N: $int, Eps0: $real] : (((~$lesseq($sum(Eps, $product(-1, R)), 0)) & (~$lesseq(Eps, 1))) | $lesseq($quotient(1, Eps0), 0) | $greatereq($sum($quotient(max(R, 1), Eps0), $product(-1, $to_real(N))), 0) | (~$greatereq($sum($quotient(max(R, 1), Eps0), $product(-1, $quotient($product(Eps0, $to_real(N)), Eps0))), 0)) | (~$greatereq($sum($product(max(R, 1), $quotient(1, Eps0)), $product(-1, $product(Eps0, $product($to_real(N), $quotient(1, Eps0))))), 0)) | (~$greatereq($sum(max(R, 1), $product(-1, $product(Eps0, $to_real(N)))), 0)) | (~($greatereq(R, 0) & (~$lesseq(Eps0, 0)) & $greatereq(N, 1) & ($sum(Eps, $product(-1, $product(Eps0, $to_real(N)))) = 0))))) <=> (~![R: $real, Eps: $real, N: $int, Eps0: $real] : (((~$lesseq($sum(Eps, $product(-1, R)), 0)) & (~$lesseq(Eps, 1))) | $lesseq($quotient(1, Eps0), 0) | $greatereq($sum($quotient(max(R, 1), Eps0), $product(-1, $to_real(N))), 0) | (~$greatereq($sum($quotient(max(R, 1), Eps0), $product(-1, $quotient($product(Eps0, $to_real(N)), Eps0))), 0)) | (~$greatereq($sum($product(max(R, 1), $quotient(1, Eps0)), $product(-1, $product(Eps0, $product($to_real(N), $quotient(1, Eps0))))), 0)) | (~$greatereq($sum(max(R, 1), $product(-1, $product(Eps0, $to_real(N)))), 0)) | (~($greatereq(R, 0) & (~$lesseq(Eps0, 0)) & $greatereq(N, 1) & ($sum(Eps, $product(-1, $product(Eps0, $to_real(N)))) = 0)))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(6,plain,
% 0.21/0.41 ((~![R: $real, Eps: $real, N: $int, Eps0: $real] : (((~$lesseq($sum(Eps, $product(-1, R)), 0)) & (~$lesseq(Eps, 1))) | $lesseq($quotient(1, Eps0), 0) | $lesseq($sum($to_real(N), $product(-1, $quotient(max(R, 1), Eps0))), 0) | (~$lesseq($sum($quotient($product($to_real(N), Eps0), Eps0), $product(-1, $quotient(max(R, 1), Eps0))), 0)) | (~$lesseq($sum($product($to_real(N), $product($quotient(1, Eps0), Eps0)), $product(-1, $product($quotient(1, Eps0), max(R, 1)))), 0)) | (~$lesseq($sum($product($to_real(N), Eps0), $product(-1, max(R, 1))), 0)) | (~($greatereq(R, 0) & (~$lesseq(Eps0, 0)) & $greatereq(N, 1) & ($sum(Eps, $product(-1, $product($to_real(N), Eps0))) = 0))))) <=> (~![R: $real, Eps: $real, N: $int, Eps0: $real] : (((~$lesseq($sum(Eps, $product(-1, R)), 0)) & (~$lesseq(Eps, 1))) | $lesseq($quotient(1, Eps0), 0) | $greatereq($sum($quotient(max(R, 1), Eps0), $product(-1, $to_real(N))), 0) | (~$greatereq($sum($quotient(max(R, 1), Eps0), $product(-1, $quotient($product(Eps0, $to_real(N)), Eps0))), 0)) | (~$greatereq($sum($product(max(R, 1), $quotient(1, Eps0)), $product(-1, $product(Eps0, $product($to_real(N), $quotient(1, Eps0))))), 0)) | (~$greatereq($sum(max(R, 1), $product(-1, $product(Eps0, $to_real(N)))), 0)) | (~($greatereq(R, 0) & (~$lesseq(Eps0, 0)) & $greatereq(N, 1) & ($sum(Eps, $product(-1, $product(Eps0, $to_real(N)))) = 0)))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(7,plain,
% 0.21/0.41 ((~![R: $real, Eps: $real, N: $int, Eps0: $real] : ($lesseq($to_real(N), $quotient(max(R, 1), Eps0)) | $lesseq($quotient(1, Eps0), 0) | ((~$lesseq(Eps, R)) & (~$lesseq(Eps, 1))) | (~$lesseq($quotient($product($to_real(N), Eps0), Eps0), $quotient(max(R, 1), Eps0))) | (~$lesseq($product($to_real(N), $product($quotient(1, Eps0), Eps0)), $product($quotient(1, Eps0), max(R, 1)))) | (~$lesseq($product($to_real(N), Eps0), max(R, 1))) | (~($lesseq(0, R) & (~$lesseq(Eps0, 0)) & $lesseq(1, N) & (Eps = $product($to_real(N), Eps0)))))) <=> (~![R: $real, Eps: $real, N: $int, Eps0: $real] : (((~$lesseq($sum(Eps, $product(-1, R)), 0)) & (~$lesseq(Eps, 1))) | $lesseq($quotient(1, Eps0), 0) | $lesseq($sum($to_real(N), $product(-1, $quotient(max(R, 1), Eps0))), 0) | (~$lesseq($sum($quotient($product($to_real(N), Eps0), Eps0), $product(-1, $quotient(max(R, 1), Eps0))), 0)) | (~$lesseq($sum($product($to_real(N), $product($quotient(1, Eps0), Eps0)), $product(-1, $product($quotient(1, Eps0), max(R, 1)))), 0)) | (~$lesseq($sum($product($to_real(N), Eps0), $product(-1, max(R, 1))), 0)) | (~($greatereq(R, 0) & (~$lesseq(Eps0, 0)) & $greatereq(N, 1) & ($sum(Eps, $product(-1, $product($to_real(N), Eps0))) = 0)))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(8,plain,
% 0.21/0.41 ((~![R: $real, Eps: $real, N: $int, Eps0: $real] : ($lesseq($to_real(N), $quotient(max(R, 1), Eps0)) | $lesseq($quotient(1, Eps0), 0) | ((~$lesseq(Eps, R)) & (~$lesseq(Eps, 1))) | (~$lesseq($quotient($product($to_real(N), Eps0), Eps0), $quotient(max(R, 1), Eps0))) | (~$lesseq($product($to_real(N), $product($quotient(1, Eps0), Eps0)), $product($quotient(1, Eps0), max(R, 1)))) | (~$lesseq($product($to_real(N), Eps0), max(R, 1))) | (~($lesseq(0, R) & (~$lesseq(Eps0, 0)) & $lesseq(1, N) & (Eps = $product($to_real(N), Eps0)))))) <=> (~![R: $real, Eps: $real, N: $int, Eps0: $real] : ($lesseq($to_real(N), $quotient(max(R, 1), Eps0)) | $lesseq($quotient(1, Eps0), 0) | ((~$lesseq(Eps, R)) & (~$lesseq(Eps, 1))) | (~$lesseq($quotient($product($to_real(N), Eps0), Eps0), $quotient(max(R, 1), Eps0))) | (~$lesseq($product($to_real(N), $product($quotient(1, Eps0), Eps0)), $product($quotient(1, Eps0), max(R, 1)))) | (~$lesseq($product($to_real(N), Eps0), max(R, 1))) | (~($lesseq(0, R) & (~$lesseq(Eps0, 0)) & $lesseq(1, N) & (Eps = $product($to_real(N), Eps0))))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(9,plain,
% 0.21/0.41 ((~![R: $real, Eps: $real, N: $int, Eps0: $real] : (((($lesseq(0, R) & $less(0, Eps0)) & $lesseq(1, N)) & (Eps = $product($to_real(N), Eps0))) => ((~($less(R, Eps) & $less(1, Eps))) => ($lesseq($product($to_real(N), Eps0), max(R, 1)) => ($less(0, $quotient(1, Eps0)) => ($lesseq($product($product($to_real(N), Eps0), $quotient(1, Eps0)), $product(max(R, 1), $quotient(1, Eps0))) => ($lesseq($quotient($product($to_real(N), Eps0), Eps0), $quotient(max(R, 1), Eps0)) => $lesseq($to_real(N), $quotient(max(R, 1), Eps0))))))))) <=> (~![R: $real, Eps: $real, N: $int, Eps0: $real] : ($lesseq($to_real(N), $quotient(max(R, 1), Eps0)) | $lesseq($quotient(1, Eps0), 0) | ((~$lesseq(Eps, R)) & (~$lesseq(Eps, 1))) | (~$lesseq($quotient($product($to_real(N), Eps0), Eps0), $quotient(max(R, 1), Eps0))) | (~$lesseq($product($to_real(N), $product($quotient(1, Eps0), Eps0)), $product($quotient(1, Eps0), max(R, 1)))) | (~$lesseq($product($to_real(N), Eps0), max(R, 1))) | (~($lesseq(0, R) & (~$lesseq(Eps0, 0)) & $lesseq(1, N) & (Eps = $product($to_real(N), Eps0))))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(10,axiom,(~![R: $real, Eps: $real, N: $int, Eps0: $real] : (((($lesseq(0, R) & $less(0, Eps0)) & $lesseq(1, N)) & (Eps = $product($to_real(N), Eps0))) => ((~($less(R, Eps) & $less(1, Eps))) => ($lesseq($product($to_real(N), Eps0), max(R, 1)) => ($less(0, $quotient(1, Eps0)) => ($lesseq($product($product($to_real(N), Eps0), $quotient(1, Eps0)), $product(max(R, 1), $quotient(1, Eps0))) => ($lesseq($quotient($product($to_real(N), Eps0), Eps0), $quotient(max(R, 1), Eps0)) => $lesseq($to_real(N), $quotient(max(R, 1), Eps0))))))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','wP_parameter_sqrt')).
% 0.21/0.41 tff(11,plain,
% 0.21/0.41 (~![R: $real, Eps: $real, N: $int, Eps0: $real] : ($lesseq($to_real(N), $quotient(max(R, 1), Eps0)) | $lesseq($quotient(1, Eps0), 0) | ((~$lesseq(Eps, R)) & (~$lesseq(Eps, 1))) | (~$lesseq($quotient($product($to_real(N), Eps0), Eps0), $quotient(max(R, 1), Eps0))) | (~$lesseq($product($to_real(N), $product($quotient(1, Eps0), Eps0)), $product($quotient(1, Eps0), max(R, 1)))) | (~$lesseq($product($to_real(N), Eps0), max(R, 1))) | (~($lesseq(0, R) & (~$lesseq(Eps0, 0)) & $lesseq(1, N) & (Eps = $product($to_real(N), Eps0)))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[10, 9])).
% 0.21/0.41 tff(12,plain,
% 0.21/0.41 (~![R: $real, Eps: $real, N: $int, Eps0: $real] : ($lesseq($to_real(N), $quotient(max(R, 1), Eps0)) | $lesseq($quotient(1, Eps0), 0) | ((~$lesseq(Eps, R)) & (~$lesseq(Eps, 1))) | (~$lesseq($quotient($product($to_real(N), Eps0), Eps0), $quotient(max(R, 1), Eps0))) | (~$lesseq($product($to_real(N), $product($quotient(1, Eps0), Eps0)), $product($quotient(1, Eps0), max(R, 1)))) | (~$lesseq($product($to_real(N), Eps0), max(R, 1))) | (~($lesseq(0, R) & (~$lesseq(Eps0, 0)) & $lesseq(1, N) & (Eps = $product($to_real(N), Eps0)))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[11, 8])).
% 0.21/0.41 tff(13,plain,
% 0.21/0.41 (~![R: $real, Eps: $real, N: $int, Eps0: $real] : ($lesseq($to_real(N), $quotient(max(R, 1), Eps0)) | $lesseq($quotient(1, Eps0), 0) | ((~$lesseq(Eps, R)) & (~$lesseq(Eps, 1))) | (~$lesseq($quotient($product($to_real(N), Eps0), Eps0), $quotient(max(R, 1), Eps0))) | (~$lesseq($product($to_real(N), $product($quotient(1, Eps0), Eps0)), $product($quotient(1, Eps0), max(R, 1)))) | (~$lesseq($product($to_real(N), Eps0), max(R, 1))) | (~($lesseq(0, R) & (~$lesseq(Eps0, 0)) & $lesseq(1, N) & (Eps = $product($to_real(N), Eps0)))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.21/0.41 tff(14,plain,
% 0.21/0.41 (~![R: $real, Eps: $real, N: $int, Eps0: $real] : ($lesseq($to_real(N), $quotient(max(R, 1), Eps0)) | $lesseq($quotient(1, Eps0), 0) | ((~$lesseq(Eps, R)) & (~$lesseq(Eps, 1))) | (~$lesseq($quotient($product($to_real(N), Eps0), Eps0), $quotient(max(R, 1), Eps0))) | (~$lesseq($product($to_real(N), $product($quotient(1, Eps0), Eps0)), $product($quotient(1, Eps0), max(R, 1)))) | (~$lesseq($product($to_real(N), Eps0), max(R, 1))) | (~($lesseq(0, R) & (~$lesseq(Eps0, 0)) & $lesseq(1, N) & (Eps = $product($to_real(N), Eps0)))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[13, 8])).
% 0.21/0.41 tff(15,plain,
% 0.21/0.41 (~![R: $real, Eps: $real, N: $int, Eps0: $real] : (((~$lesseq($sum(Eps, $product(-1, R)), 0)) & (~$lesseq(Eps, 1))) | $lesseq($quotient(1, Eps0), 0) | $lesseq($sum($to_real(N), $product(-1, $quotient(max(R, 1), Eps0))), 0) | (~$lesseq($sum($quotient($product($to_real(N), Eps0), Eps0), $product(-1, $quotient(max(R, 1), Eps0))), 0)) | (~$lesseq($sum($product($to_real(N), $product($quotient(1, Eps0), Eps0)), $product(-1, $product($quotient(1, Eps0), max(R, 1)))), 0)) | (~$lesseq($sum($product($to_real(N), Eps0), $product(-1, max(R, 1))), 0)) | (~($greatereq(R, 0) & (~$lesseq(Eps0, 0)) & $greatereq(N, 1) & ($sum(Eps, $product(-1, $product($to_real(N), Eps0))) = 0))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.21/0.41 tff(16,plain,
% 0.21/0.41 (~![R: $real, Eps: $real, N: $int, Eps0: $real] : (((~$lesseq($sum(Eps, $product(-1, R)), 0)) & (~$lesseq(Eps, 1))) | $lesseq($quotient(1, Eps0), 0) | $greatereq($sum($quotient(max(R, 1), Eps0), $product(-1, $to_real(N))), 0) | (~$greatereq($sum($quotient(max(R, 1), Eps0), $product(-1, $quotient($product(Eps0, $to_real(N)), Eps0))), 0)) | (~$greatereq($sum($product(max(R, 1), $quotient(1, Eps0)), $product(-1, $product(Eps0, $product($to_real(N), $quotient(1, Eps0))))), 0)) | (~$greatereq($sum(max(R, 1), $product(-1, $product(Eps0, $to_real(N)))), 0)) | (~($greatereq(R, 0) & (~$lesseq(Eps0, 0)) & $greatereq(N, 1) & ($sum(Eps, $product(-1, $product(Eps0, $to_real(N)))) = 0))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[15, 6])).
% 0.21/0.41 tff(17,plain,
% 0.21/0.41 (~![R: $real, Eps: $real, N: $int, Eps0: $real] : (((~$lesseq($sum(Eps, $product(-1, R)), 0)) & (~$lesseq(Eps, 1))) | $lesseq($quotient(1, Eps0), 0) | $greatereq($sum($quotient(max(R, 1), Eps0), $product(-1, $to_real(N))), 0) | (~$greatereq($sum($quotient(max(R, 1), Eps0), $product(-1, $quotient($product(Eps0, $to_real(N)), Eps0))), 0)) | (~$greatereq($sum($product(max(R, 1), $quotient(1, Eps0)), $product(-1, $product(Eps0, $product($to_real(N), $quotient(1, Eps0))))), 0)) | (~$greatereq($sum(max(R, 1), $product(-1, $product(Eps0, $to_real(N)))), 0)) | (~($greatereq(R, 0) & (~$lesseq(Eps0, 0)) & $greatereq(N, 1) & ($sum(Eps, $product(-1, $product(Eps0, $to_real(N)))) = 0))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[16, 5])).
% 0.21/0.41 tff(18,plain,
% 0.21/0.41 (~![R: $real, Eps: $real, N: $int, Eps0: $real] : (((~$lesseq($sum(Eps, $product(-1, R)), 0)) & (~$lesseq(Eps, 1))) | $lesseq($quotient(1, Eps0), 0) | $greatereq($sum($quotient(max(R, 1), Eps0), $product(-1, $to_real(N))), 0) | (~$greatereq($sum($quotient(max(R, 1), Eps0), $product(-1, $quotient($product(Eps0, $to_real(N)), Eps0))), 0)) | (~$greatereq($sum($product(max(R, 1), $quotient(1, Eps0)), $product(-1, $product(Eps0, $product($to_real(N), $quotient(1, Eps0))))), 0)) | (~$greatereq($sum(max(R, 1), $product(-1, $product(Eps0, $to_real(N)))), 0)) | (~($greatereq(R, 0) & (~$lesseq(Eps0, 0)) & $greatereq(N, 1) & ($sum(Eps, $product(-1, $product(Eps0, $to_real(N)))) = 0))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[17, 5])).
% 0.21/0.41 tff(19,plain,(
% 0.21/0.41 ~(((~$lesseq($sum(Eps!2, $product(-1, R!3)), 0)) & (~$lesseq(Eps!2, 1))) | $lesseq($quotient(1, Eps0!0), 0) | $greatereq($sum($quotient(max(R!3, 1), Eps0!0), $product(-1, $to_real(N!1))), 0) | (~$greatereq($sum($quotient(max(R!3, 1), Eps0!0), $product(-1, $quotient($product(Eps0!0, $to_real(N!1)), Eps0!0))), 0)) | (~$greatereq($sum($product(max(R!3, 1), $quotient(1, Eps0!0)), $product(-1, $product(Eps0!0, $product($to_real(N!1), $quotient(1, Eps0!0))))), 0)) | (~$greatereq($sum(max(R!3, 1), $product(-1, $product(Eps0!0, $to_real(N!1)))), 0)) | (~($greatereq(R!3, 0) & (~$lesseq(Eps0!0, 0)) & $greatereq(N!1, 1) & ($sum(Eps!2, $product(-1, $product(Eps0!0, $to_real(N!1)))) = 0))))),
% 0.21/0.41 inference(skolemize,[status(sab)],[18])).
% 0.21/0.41 tff(20,plain,
% 0.21/0.41 (~(((~$lesseq($sum(Eps!2, $product(-1, R!3)), 0)) & (~$lesseq(Eps!2, 1))) | $lesseq($quotient(1, Eps0!0), 0) | $greatereq($sum($quotient(max(R!3, 1), Eps0!0), $product(-1, $to_real(N!1))), 0) | (~$greatereq($sum($quotient(max(R!3, 1), Eps0!0), $product(-1, $quotient($product(Eps0!0, $to_real(N!1)), Eps0!0))), 0)) | (~$greatereq($sum(max(R!3, 1), $product(-1, $product(Eps0!0, $to_real(N!1)))), 0)) | (~($greatereq(R!3, 0) & (~$lesseq(Eps0!0, 0)) & $greatereq(N!1, 1) & ($sum(Eps!2, $product(-1, $product(Eps0!0, $to_real(N!1)))) = 0))) | (~$greatereq($sum($product($quotient(1, Eps0!0), max(R!3, 1)), $product(-1, $product(Eps0!0, $product($quotient(1, Eps0!0), $to_real(N!1))))), 0)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[19, 4])).
% 0.21/0.41 tff(21,plain,
% 0.21/0.41 ($greatereq(R!3, 0) & (~$lesseq(Eps0!0, 0)) & $greatereq(N!1, 1) & ($sum(Eps!2, $product(-1, $product(Eps0!0, $to_real(N!1)))) = 0)),
% 0.21/0.41 inference(or_elim,[status(thm)],[20])).
% 0.21/0.41 tff(22,plain,
% 0.21/0.41 (~$lesseq(Eps0!0, 0)),
% 0.21/0.41 inference(and_elim,[status(thm)],[21])).
% 0.21/0.41 tff(23,plain,
% 0.21/0.41 ((~(0 = Eps0!0)) | $lesseq(Eps0!0, 0)),
% 0.21/0.41 inference(theory_lemma,[status(thm)],[])).
% 0.21/0.41 tff(24,plain,
% 0.21/0.41 (~(0 = Eps0!0)),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[23, 22])).
% 0.21/0.41 tff(25,plain,
% 0.21/0.41 ((0 = Eps0!0) <=> $false),
% 0.21/0.41 inference(iff_false,[status(thm)],[24])).
% 0.21/0.41 tff(26,plain,
% 0.21/0.41 ((Eps0!0 = 0) <=> (0 = Eps0!0)),
% 0.21/0.41 inference(commutativity,[status(thm)],[])).
% 0.21/0.41 tff(27,plain,
% 0.21/0.41 ((Eps0!0 = 0) <=> $false),
% 0.21/0.41 inference(transitivity,[status(thm)],[26, 25])).
% 0.21/0.41 tff(28,plain,
% 0.21/0.41 (~(Eps0!0 = 0)),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[27, 3])).
% 0.21/0.41 tff(29,plain,
% 0.21/0.41 (^[X: $real, Y: $real, Z: $real] : refl(((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0)) <=> ((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0)))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(30,plain,
% 0.21/0.41 (![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0)) <=> ![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[29])).
% 0.21/0.41 tff(31,plain,
% 0.21/0.41 (^[X: $real, Y: $real, Z: $real] : rewrite(((Z = 0) | ($sum($quotient($product(Y, X), Z), $product(-1, $product($quotient(Y, Z), X))) = 0)) <=> ((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0)))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(32,plain,
% 0.21/0.41 (![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($quotient($product(Y, X), Z), $product(-1, $product($quotient(Y, Z), X))) = 0)) <=> ![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[31])).
% 0.21/0.41 tff(33,plain,
% 0.21/0.41 (^[X: $real, Y: $real, Z: $real] : rewrite(((Z = 0) | ($quotient($product(Y, X), Z) = $product($quotient(Y, Z), X))) <=> ((Z = 0) | ($sum($quotient($product(Y, X), Z), $product(-1, $product($quotient(Y, Z), X))) = 0)))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(34,plain,
% 0.21/0.41 (![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($quotient($product(Y, X), Z) = $product($quotient(Y, Z), X))) <=> ![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($quotient($product(Y, X), Z), $product(-1, $product($quotient(Y, Z), X))) = 0))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[33])).
% 0.21/0.41 tff(35,plain,
% 0.21/0.41 (![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($quotient($product(Y, X), Z) = $product($quotient(Y, Z), X))) <=> ![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($quotient($product(Y, X), Z) = $product($quotient(Y, Z), X)))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(36,plain,
% 0.21/0.41 (^[X: $real, Y: $real, Z: $real] : trans(monotonicity(rewrite(($quotient($product(X, Y), Z) = $product(X, $quotient(Y, Z))) <=> ($quotient($product(Y, X), Z) = $product($quotient(Y, Z), X))), (((~(Z = 0)) => ($quotient($product(X, Y), Z) = $product(X, $quotient(Y, Z)))) <=> ((~(Z = 0)) => ($quotient($product(Y, X), Z) = $product($quotient(Y, Z), X))))), rewrite(((~(Z = 0)) => ($quotient($product(Y, X), Z) = $product($quotient(Y, Z), X))) <=> ((Z = 0) | ($quotient($product(Y, X), Z) = $product($quotient(Y, Z), X)))), (((~(Z = 0)) => ($quotient($product(X, Y), Z) = $product(X, $quotient(Y, Z)))) <=> ((Z = 0) | ($quotient($product(Y, X), Z) = $product($quotient(Y, Z), X)))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(37,plain,
% 0.21/0.41 (![X: $real, Y: $real, Z: $real] : ((~(Z = 0)) => ($quotient($product(X, Y), Z) = $product(X, $quotient(Y, Z)))) <=> ![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($quotient($product(Y, X), Z) = $product($quotient(Y, Z), X)))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[36])).
% 0.21/0.41 tff(38,axiom,(![X: $real, Y: $real, Z: $real] : ((~(Z = 0)) => ($quotient($product(X, Y), Z) = $product(X, $quotient(Y, Z))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','assoc_mul_div')).
% 0.21/0.41 tff(39,plain,
% 0.21/0.41 (![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($quotient($product(Y, X), Z) = $product($quotient(Y, Z), X)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.21/0.41 tff(40,plain,
% 0.21/0.41 (![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($quotient($product(Y, X), Z) = $product($quotient(Y, Z), X)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[39, 35])).
% 0.21/0.41 tff(41,plain,
% 0.21/0.41 (![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($quotient($product(Y, X), Z), $product(-1, $product($quotient(Y, Z), X))) = 0))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[40, 34])).
% 0.21/0.41 tff(42,plain,
% 0.21/0.41 (![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[41, 32])).
% 0.21/0.41 tff(43,plain,(
% 0.21/0.41 ![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0))),
% 0.21/0.41 inference(skolemize,[status(sab)],[42])).
% 0.21/0.41 tff(44,plain,
% 0.21/0.41 (![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[43, 30])).
% 0.21/0.41 tff(45,plain,
% 0.21/0.41 (((~![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0))) | ((Eps0!0 = 0) | ($sum($quotient($product(Eps0!0, $to_real(N!1)), Eps0!0), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))) = 0))) <=> ((~![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0))) | (Eps0!0 = 0) | ($sum($quotient($product(Eps0!0, $to_real(N!1)), Eps0!0), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))) = 0))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(46,plain,
% 0.21/0.41 (((Eps0!0 = 0) | ($sum($product(Eps0!0, $quotient($to_real(N!1), Eps0!0)), $product(-1, $quotient($product(Eps0!0, $to_real(N!1)), Eps0!0))) = 0)) <=> ((Eps0!0 = 0) | ($sum($quotient($product(Eps0!0, $to_real(N!1)), Eps0!0), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))) = 0))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(47,plain,
% 0.21/0.41 (((~![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0))) | ((Eps0!0 = 0) | ($sum($product(Eps0!0, $quotient($to_real(N!1), Eps0!0)), $product(-1, $quotient($product(Eps0!0, $to_real(N!1)), Eps0!0))) = 0))) <=> ((~![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0))) | ((Eps0!0 = 0) | ($sum($quotient($product(Eps0!0, $to_real(N!1)), Eps0!0), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))) = 0)))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[46])).
% 0.21/0.41 tff(48,plain,
% 0.21/0.41 (((~![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0))) | ((Eps0!0 = 0) | ($sum($product(Eps0!0, $quotient($to_real(N!1), Eps0!0)), $product(-1, $quotient($product(Eps0!0, $to_real(N!1)), Eps0!0))) = 0))) <=> ((~![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0))) | (Eps0!0 = 0) | ($sum($quotient($product(Eps0!0, $to_real(N!1)), Eps0!0), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))) = 0))),
% 0.21/0.41 inference(transitivity,[status(thm)],[47, 45])).
% 0.21/0.41 tff(49,plain,
% 0.21/0.41 ((~![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0))) | ((Eps0!0 = 0) | ($sum($product(Eps0!0, $quotient($to_real(N!1), Eps0!0)), $product(-1, $quotient($product(Eps0!0, $to_real(N!1)), Eps0!0))) = 0))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(50,plain,
% 0.21/0.41 ((~![X: $real, Y: $real, Z: $real] : ((Z = 0) | ($sum($product(X, $quotient(Y, Z)), $product(-1, $quotient($product(X, Y), Z))) = 0))) | (Eps0!0 = 0) | ($sum($quotient($product(Eps0!0, $to_real(N!1)), Eps0!0), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))) = 0)),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[49, 48])).
% 0.21/0.42 tff(51,plain,
% 0.21/0.42 ((Eps0!0 = 0) | ($sum($quotient($product(Eps0!0, $to_real(N!1)), Eps0!0), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))) = 0)),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[50, 44])).
% 0.21/0.42 tff(52,plain,
% 0.21/0.42 ($sum($quotient($product(Eps0!0, $to_real(N!1)), Eps0!0), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))) = 0),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[51, 28])).
% 0.21/0.42 tff(53,plain,
% 0.21/0.42 (0 = $sum($quotient($product(Eps0!0, $to_real(N!1)), Eps0!0), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[52, 2])).
% 0.21/0.42 tff(54,plain,
% 0.21/0.42 ((~(0 = $sum($quotient($product(Eps0!0, $to_real(N!1)), Eps0!0), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))))) | $greatereq($sum($quotient($product(Eps0!0, $to_real(N!1)), Eps0!0), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))), 0)),
% 0.21/0.42 inference(theory_lemma,[status(thm)],[])).
% 0.21/0.42 tff(55,plain,
% 0.21/0.42 ($greatereq($sum($quotient($product(Eps0!0, $to_real(N!1)), Eps0!0), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))), 0)),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[54, 53])).
% 0.21/0.42 tff(56,plain,
% 0.21/0.42 ($greatereq($sum($quotient(max(R!3, 1), Eps0!0), $product(-1, $quotient($product(Eps0!0, $to_real(N!1)), Eps0!0))), 0)),
% 0.21/0.42 inference(or_elim,[status(thm)],[20])).
% 0.21/0.42 tff(57,plain,
% 0.21/0.42 ((0 = Eps0!0) | ($to_real(N!1) = $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))),
% 0.21/0.42 inference(theory_lemma,[status(thm)],[])).
% 0.21/0.42 tff(58,plain,
% 0.21/0.42 ($to_real(N!1) = $product(Eps0!0, $quotient($to_real(N!1), Eps0!0))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[57, 24])).
% 0.21/0.42 tff(59,plain,
% 0.21/0.42 ((~($to_real(N!1) = $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))) | $lesseq($sum($to_real(N!1), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))), 0)),
% 0.21/0.42 inference(theory_lemma,[status(thm)],[])).
% 0.21/0.42 tff(60,plain,
% 0.21/0.42 ($lesseq($sum($to_real(N!1), $product(-1, $product(Eps0!0, $quotient($to_real(N!1), Eps0!0)))), 0)),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[59, 58])).
% 0.21/0.42 tff(61,plain,
% 0.21/0.42 (~$greatereq($sum($quotient(max(R!3, 1), Eps0!0), $product(-1, $to_real(N!1))), 0)),
% 0.21/0.42 inference(or_elim,[status(thm)],[20])).
% 0.21/0.42 tff(62,plain,
% 0.21/0.42 ($false),
% 0.21/0.42 inference(theory_lemma,[status(thm)],[61, 60, 56, 55])).
% 0.21/0.42 % SZS output end Proof
%------------------------------------------------------------------------------