TSTP Solution File: ARI617_1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : ARI617_1 : TPTP v8.1.0. Released v5.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n025.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 0.20s 0.39s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : ARI617_1 : TPTP v8.1.0. Released v5.1.0.
% 0.04/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n025.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 01:03:14 EDT 2022
% 0.13/0.34 % 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.
% 0.20/0.39 % SZS status Theorem
% 0.20/0.39 % SZS output start Proof
% 0.20/0.39 tff(g_type, type, (
% 0.20/0.39 g: $int > $int)).
% 0.20/0.39 tff(tptp_fun_X_0_type, type, (
% 0.20/0.39 tptp_fun_X_0: $int)).
% 0.20/0.39 tff(f_type, type, (
% 0.20/0.39 f: $int > $int)).
% 0.20/0.39 tff(1,assumption,(~$greatereq($sum(g(X!0), $product(-1, f(X!0))), 0)), introduced(assumption)).
% 0.20/0.39 tff(2,assumption,($sum(X!0, $product(-1, g(X!0))) = 0), introduced(assumption)).
% 0.20/0.39 tff(3,plain,
% 0.20/0.39 ((~($sum(X!0, $product(-1, g(X!0))) = 0)) | $lesseq($sum(X!0, $product(-1, g(X!0))), 0)),
% 0.20/0.39 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.39 tff(4,plain,
% 0.20/0.39 ($lesseq($sum(X!0, $product(-1, g(X!0))), 0)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[3, 2])).
% 0.20/0.39 tff(5,plain,
% 0.20/0.39 ((~($sum(X!0, $product(-1, g(X!0))) = 0)) | $greatereq($sum(X!0, $product(-1, g(X!0))), 0)),
% 0.20/0.39 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.39 tff(6,plain,
% 0.20/0.39 ($greatereq($sum(X!0, $product(-1, g(X!0))), 0)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[5, 2])).
% 0.20/0.39 tff(7,assumption,($greatereq($sum(X!0, $product(-1, g(X!0))), 0)), introduced(assumption)).
% 0.20/0.39 tff(8,assumption,($lesseq($sum(X!0, f(X!0)), 0)), introduced(assumption)).
% 0.20/0.39 tff(9,plain,
% 0.20/0.39 (^[X: $int] : refl((~((~$greatereq(g(X), 0)) | (~(($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0))))) <=> (~((~$greatereq(g(X), 0)) | (~(($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0))))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(10,plain,
% 0.20/0.39 (![X: $int] : (~((~$greatereq(g(X), 0)) | (~(($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0))))) <=> ![X: $int] : (~((~$greatereq(g(X), 0)) | (~(($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0)))))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[9])).
% 0.20/0.39 tff(11,plain,
% 0.20/0.39 (^[X: $int] : rewrite(($greatereq(g(X), 0) & (($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0))) <=> (~((~$greatereq(g(X), 0)) | (~(($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0))))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(12,plain,
% 0.20/0.39 (![X: $int] : ($greatereq(g(X), 0) & (($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0))) <=> ![X: $int] : (~((~$greatereq(g(X), 0)) | (~(($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0)))))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[11])).
% 0.20/0.39 tff(13,plain,
% 0.20/0.39 (^[X: $int] : rewrite(($lesseq(0, g(X)) & ((g(X) = X) | (g(X) = $product(-1, X)))) <=> ($greatereq(g(X), 0) & (($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(14,plain,
% 0.20/0.39 (![X: $int] : ($lesseq(0, g(X)) & ((g(X) = X) | (g(X) = $product(-1, X)))) <=> ![X: $int] : ($greatereq(g(X), 0) & (($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0)))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[13])).
% 0.20/0.39 tff(15,plain,
% 0.20/0.39 (![X: $int] : ($lesseq(0, g(X)) & ((g(X) = X) | (g(X) = $product(-1, X)))) <=> ![X: $int] : ($lesseq(0, g(X)) & ((g(X) = X) | (g(X) = $product(-1, X))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(16,plain,
% 0.20/0.39 ((~((![X: $int] : (($lesseq(X, f(X)) & $lesseq($uminus(X), f(X))) & ($lesseq(f(X), X) | $lesseq(f(X), $uminus(X)))) & ![X: $int] : ($lesseq(0, g(X)) & ((g(X) = X) | (g(X) = $uminus(X))))) => ![X: $int] : (f(X) = g(X)))) <=> (~((~(![X: $int] : ($lesseq(X, f(X)) & $lesseq($product(-1, X), f(X)) & ($lesseq(f(X), X) | $lesseq(f(X), $product(-1, X)))) & ![X: $int] : ($lesseq(0, g(X)) & ((g(X) = X) | (g(X) = $product(-1, X)))))) | ![X: $int] : (f(X) = g(X))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(17,axiom,(~((![X: $int] : (($lesseq(X, f(X)) & $lesseq($uminus(X), f(X))) & ($lesseq(f(X), X) | $lesseq(f(X), $uminus(X)))) & ![X: $int] : ($lesseq(0, g(X)) & ((g(X) = X) | (g(X) = $uminus(X))))) => ![X: $int] : (f(X) = g(X)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','absolute_value_defs')).
% 0.20/0.39 tff(18,plain,
% 0.20/0.39 (~((~(![X: $int] : ($lesseq(X, f(X)) & $lesseq($product(-1, X), f(X)) & ($lesseq(f(X), X) | $lesseq(f(X), $product(-1, X)))) & ![X: $int] : ($lesseq(0, g(X)) & ((g(X) = X) | (g(X) = $product(-1, X)))))) | ![X: $int] : (f(X) = g(X)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[17, 16])).
% 0.20/0.39 tff(19,plain,
% 0.20/0.39 (![X: $int] : ($lesseq(X, f(X)) & $lesseq($product(-1, X), f(X)) & ($lesseq(f(X), X) | $lesseq(f(X), $product(-1, X)))) & ![X: $int] : ($lesseq(0, g(X)) & ((g(X) = X) | (g(X) = $product(-1, X))))),
% 0.20/0.39 inference(or_elim,[status(thm)],[18])).
% 0.20/0.39 tff(20,plain,
% 0.20/0.39 (![X: $int] : ($lesseq(0, g(X)) & ((g(X) = X) | (g(X) = $product(-1, X))))),
% 0.20/0.39 inference(and_elim,[status(thm)],[19])).
% 0.20/0.39 tff(21,plain,
% 0.20/0.39 (![X: $int] : ($lesseq(0, g(X)) & ((g(X) = X) | (g(X) = $product(-1, X))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[20, 15])).
% 0.20/0.39 tff(22,plain,
% 0.20/0.39 (![X: $int] : ($greatereq(g(X), 0) & (($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[21, 14])).
% 0.20/0.39 tff(23,plain,(
% 0.20/0.39 ![X: $int] : ($greatereq(g(X), 0) & (($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0)))),
% 0.20/0.39 inference(skolemize,[status(sab)],[22])).
% 0.20/0.39 tff(24,plain,
% 0.20/0.39 (![X: $int] : (~((~$greatereq(g(X), 0)) | (~(($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0)))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[23, 12])).
% 0.20/0.39 tff(25,plain,
% 0.20/0.39 (![X: $int] : (~((~$greatereq(g(X), 0)) | (~(($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0)))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[24, 10])).
% 0.20/0.39 tff(26,plain,
% 0.20/0.39 (((~![X: $int] : (~((~$greatereq(g(X), 0)) | (~(($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0)))))) | (~((~$greatereq(g(X!0), 0)) | (~(($sum(X!0, $product(-1, g(X!0))) = 0) | ($sum(X!0, g(X!0)) = 0)))))) <=> ((~![X: $int] : (~((~$greatereq(g(X), 0)) | (~(($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0)))))) | (~((~$greatereq(g(X!0), 0)) | (~(($sum(X!0, $product(-1, g(X!0))) = 0) | ($sum(X!0, g(X!0)) = 0))))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(27,plain,
% 0.20/0.39 ((~((~$greatereq(g(X!0), 0)) | (~(($sum(g(X!0), $product(-1, X!0)) = 0) | ($sum(g(X!0), X!0) = 0))))) <=> (~((~$greatereq(g(X!0), 0)) | (~(($sum(X!0, $product(-1, g(X!0))) = 0) | ($sum(X!0, g(X!0)) = 0)))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(28,plain,
% 0.20/0.39 (((~![X: $int] : (~((~$greatereq(g(X), 0)) | (~(($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0)))))) | (~((~$greatereq(g(X!0), 0)) | (~(($sum(g(X!0), $product(-1, X!0)) = 0) | ($sum(g(X!0), X!0) = 0)))))) <=> ((~![X: $int] : (~((~$greatereq(g(X), 0)) | (~(($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0)))))) | (~((~$greatereq(g(X!0), 0)) | (~(($sum(X!0, $product(-1, g(X!0))) = 0) | ($sum(X!0, g(X!0)) = 0))))))),
% 0.20/0.39 inference(monotonicity,[status(thm)],[27])).
% 0.20/0.39 tff(29,plain,
% 0.20/0.39 (((~![X: $int] : (~((~$greatereq(g(X), 0)) | (~(($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0)))))) | (~((~$greatereq(g(X!0), 0)) | (~(($sum(g(X!0), $product(-1, X!0)) = 0) | ($sum(g(X!0), X!0) = 0)))))) <=> ((~![X: $int] : (~((~$greatereq(g(X), 0)) | (~(($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0)))))) | (~((~$greatereq(g(X!0), 0)) | (~(($sum(X!0, $product(-1, g(X!0))) = 0) | ($sum(X!0, g(X!0)) = 0))))))),
% 0.20/0.39 inference(transitivity,[status(thm)],[28, 26])).
% 0.20/0.39 tff(30,plain,
% 0.20/0.39 ((~![X: $int] : (~((~$greatereq(g(X), 0)) | (~(($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0)))))) | (~((~$greatereq(g(X!0), 0)) | (~(($sum(g(X!0), $product(-1, X!0)) = 0) | ($sum(g(X!0), X!0) = 0)))))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(31,plain,
% 0.20/0.39 ((~![X: $int] : (~((~$greatereq(g(X), 0)) | (~(($sum(g(X), $product(-1, X)) = 0) | ($sum(g(X), X) = 0)))))) | (~((~$greatereq(g(X!0), 0)) | (~(($sum(X!0, $product(-1, g(X!0))) = 0) | ($sum(X!0, g(X!0)) = 0)))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[30, 29])).
% 0.20/0.39 tff(32,plain,
% 0.20/0.39 (~((~$greatereq(g(X!0), 0)) | (~(($sum(X!0, $product(-1, g(X!0))) = 0) | ($sum(X!0, g(X!0)) = 0))))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[31, 25])).
% 0.20/0.39 tff(33,plain,
% 0.20/0.39 (((~$greatereq(g(X!0), 0)) | (~(($sum(X!0, $product(-1, g(X!0))) = 0) | ($sum(X!0, g(X!0)) = 0)))) | $greatereq(g(X!0), 0)),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(34,plain,
% 0.20/0.39 ($greatereq(g(X!0), 0)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[33, 32])).
% 0.20/0.39 tff(35,plain,
% 0.20/0.39 ($false),
% 0.20/0.39 inference(theory_lemma,[status(thm)],[34, 8, 1, 7])).
% 0.20/0.39 tff(36,plain,((~$lesseq($sum(X!0, f(X!0)), 0)) | $greatereq($sum(g(X!0), $product(-1, f(X!0))), 0) | (~$greatereq($sum(X!0, $product(-1, g(X!0))), 0))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.40 tff(37,plain,
% 0.20/0.40 (~$lesseq($sum(X!0, f(X!0)), 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[36, 6, 1])).
% 0.20/0.40 tff(38,plain,
% 0.20/0.40 (^[X: $int] : refl((~((~$lesseq($sum(X, $product(-1, f(X))), 0)) | (~$greatereq($sum(X, f(X)), 0)) | (~($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0))))) <=> (~((~$lesseq($sum(X, $product(-1, f(X))), 0)) | (~$greatereq($sum(X, f(X)), 0)) | (~($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(39,plain,
% 0.20/0.40 (![X: $int] : (~((~$lesseq($sum(X, $product(-1, f(X))), 0)) | (~$greatereq($sum(X, f(X)), 0)) | (~($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0))))) <=> ![X: $int] : (~((~$lesseq($sum(X, $product(-1, f(X))), 0)) | (~$greatereq($sum(X, f(X)), 0)) | (~($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0)))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[38])).
% 0.20/0.40 tff(40,plain,
% 0.20/0.40 (^[X: $int] : trans(monotonicity(rewrite(($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0)) <=> ($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0))), (($lesseq($sum(X, $product(-1, f(X))), 0) & $greatereq($sum(X, f(X)), 0) & ($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0))) <=> ($lesseq($sum(X, $product(-1, f(X))), 0) & $greatereq($sum(X, f(X)), 0) & ($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0))))), rewrite(($lesseq($sum(X, $product(-1, f(X))), 0) & $greatereq($sum(X, f(X)), 0) & ($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0))) <=> (~((~$lesseq($sum(X, $product(-1, f(X))), 0)) | (~$greatereq($sum(X, f(X)), 0)) | (~($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0)))))), (($lesseq($sum(X, $product(-1, f(X))), 0) & $greatereq($sum(X, f(X)), 0) & ($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0))) <=> (~((~$lesseq($sum(X, $product(-1, f(X))), 0)) | (~$greatereq($sum(X, f(X)), 0)) | (~($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0)))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(41,plain,
% 0.20/0.40 (![X: $int] : ($lesseq($sum(X, $product(-1, f(X))), 0) & $greatereq($sum(X, f(X)), 0) & ($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0))) <=> ![X: $int] : (~((~$lesseq($sum(X, $product(-1, f(X))), 0)) | (~$greatereq($sum(X, f(X)), 0)) | (~($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0)))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[40])).
% 0.20/0.40 tff(42,plain,
% 0.20/0.40 (^[X: $int] : rewrite(($lesseq($sum(X, $product(-1, f(X))), 0) & $greatereq($sum(X, f(X)), 0) & ($lesseq($sum(f(X), $product(-1, X)), 0) | $lesseq($sum(f(X), X), 0))) <=> ($lesseq($sum(X, $product(-1, f(X))), 0) & $greatereq($sum(X, f(X)), 0) & ($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(43,plain,
% 0.20/0.40 (![X: $int] : ($lesseq($sum(X, $product(-1, f(X))), 0) & $greatereq($sum(X, f(X)), 0) & ($lesseq($sum(f(X), $product(-1, X)), 0) | $lesseq($sum(f(X), X), 0))) <=> ![X: $int] : ($lesseq($sum(X, $product(-1, f(X))), 0) & $greatereq($sum(X, f(X)), 0) & ($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[42])).
% 0.20/0.40 tff(44,plain,
% 0.20/0.40 (^[X: $int] : rewrite(($lesseq(X, f(X)) & $lesseq($product(-1, X), f(X)) & ($lesseq(f(X), X) | $lesseq(f(X), $product(-1, X)))) <=> ($lesseq($sum(X, $product(-1, f(X))), 0) & $greatereq($sum(X, f(X)), 0) & ($lesseq($sum(f(X), $product(-1, X)), 0) | $lesseq($sum(f(X), X), 0))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(45,plain,
% 0.20/0.40 (![X: $int] : ($lesseq(X, f(X)) & $lesseq($product(-1, X), f(X)) & ($lesseq(f(X), X) | $lesseq(f(X), $product(-1, X)))) <=> ![X: $int] : ($lesseq($sum(X, $product(-1, f(X))), 0) & $greatereq($sum(X, f(X)), 0) & ($lesseq($sum(f(X), $product(-1, X)), 0) | $lesseq($sum(f(X), X), 0)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[44])).
% 0.20/0.40 tff(46,plain,
% 0.20/0.40 (![X: $int] : ($lesseq(X, f(X)) & $lesseq($product(-1, X), f(X)) & ($lesseq(f(X), X) | $lesseq(f(X), $product(-1, X)))) <=> ![X: $int] : ($lesseq(X, f(X)) & $lesseq($product(-1, X), f(X)) & ($lesseq(f(X), X) | $lesseq(f(X), $product(-1, X))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(47,plain,
% 0.20/0.40 (![X: $int] : ($lesseq(X, f(X)) & $lesseq($product(-1, X), f(X)) & ($lesseq(f(X), X) | $lesseq(f(X), $product(-1, X))))),
% 0.20/0.40 inference(and_elim,[status(thm)],[19])).
% 0.20/0.40 tff(48,plain,
% 0.20/0.40 (![X: $int] : ($lesseq(X, f(X)) & $lesseq($product(-1, X), f(X)) & ($lesseq(f(X), X) | $lesseq(f(X), $product(-1, X))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.20/0.40 tff(49,plain,
% 0.20/0.40 (![X: $int] : ($lesseq($sum(X, $product(-1, f(X))), 0) & $greatereq($sum(X, f(X)), 0) & ($lesseq($sum(f(X), $product(-1, X)), 0) | $lesseq($sum(f(X), X), 0)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[48, 45])).
% 0.20/0.40 tff(50,plain,
% 0.20/0.40 (![X: $int] : ($lesseq($sum(X, $product(-1, f(X))), 0) & $greatereq($sum(X, f(X)), 0) & ($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[49, 43])).
% 0.20/0.40 tff(51,plain,(
% 0.20/0.40 ![X: $int] : ($lesseq($sum(X, $product(-1, f(X))), 0) & $greatereq($sum(X, f(X)), 0) & ($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0)))),
% 0.20/0.40 inference(skolemize,[status(sab)],[50])).
% 0.20/0.40 tff(52,plain,
% 0.20/0.40 (![X: $int] : (~((~$lesseq($sum(X, $product(-1, f(X))), 0)) | (~$greatereq($sum(X, f(X)), 0)) | (~($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[51, 41])).
% 0.20/0.40 tff(53,plain,
% 0.20/0.40 (![X: $int] : (~((~$lesseq($sum(X, $product(-1, f(X))), 0)) | (~$greatereq($sum(X, f(X)), 0)) | (~($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[52, 39])).
% 0.20/0.40 tff(54,plain,
% 0.20/0.40 ((~![X: $int] : (~((~$lesseq($sum(X, $product(-1, f(X))), 0)) | (~$greatereq($sum(X, f(X)), 0)) | (~($greatereq($sum(X, $product(-1, f(X))), 0) | $lesseq($sum(X, f(X)), 0)))))) | (~((~$lesseq($sum(X!0, $product(-1, f(X!0))), 0)) | (~$greatereq($sum(X!0, f(X!0)), 0)) | (~($greatereq($sum(X!0, $product(-1, f(X!0))), 0) | $lesseq($sum(X!0, f(X!0)), 0)))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(55,plain,
% 0.20/0.40 (~((~$lesseq($sum(X!0, $product(-1, f(X!0))), 0)) | (~$greatereq($sum(X!0, f(X!0)), 0)) | (~($greatereq($sum(X!0, $product(-1, f(X!0))), 0) | $lesseq($sum(X!0, f(X!0)), 0))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[54, 53])).
% 0.20/0.40 tff(56,plain,
% 0.20/0.40 (((~$lesseq($sum(X!0, $product(-1, f(X!0))), 0)) | (~$greatereq($sum(X!0, f(X!0)), 0)) | (~($greatereq($sum(X!0, $product(-1, f(X!0))), 0) | $lesseq($sum(X!0, f(X!0)), 0)))) | ($greatereq($sum(X!0, $product(-1, f(X!0))), 0) | $lesseq($sum(X!0, f(X!0)), 0))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(57,plain,
% 0.20/0.40 ($greatereq($sum(X!0, $product(-1, f(X!0))), 0) | $lesseq($sum(X!0, f(X!0)), 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[56, 55])).
% 0.20/0.40 tff(58,plain,
% 0.20/0.40 ((~($greatereq($sum(X!0, $product(-1, f(X!0))), 0) | $lesseq($sum(X!0, f(X!0)), 0))) | $greatereq($sum(X!0, $product(-1, f(X!0))), 0) | $lesseq($sum(X!0, f(X!0)), 0)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(59,plain,
% 0.20/0.40 ($greatereq($sum(X!0, $product(-1, f(X!0))), 0) | $lesseq($sum(X!0, f(X!0)), 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[58, 57])).
% 0.20/0.40 tff(60,plain,
% 0.20/0.40 ($greatereq($sum(X!0, $product(-1, f(X!0))), 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[59, 37])).
% 0.20/0.40 tff(61,plain,
% 0.20/0.40 ($false),
% 0.20/0.40 inference(theory_lemma,[status(thm)],[1, 60, 4])).
% 0.20/0.40 tff(62,plain,((~($sum(X!0, $product(-1, g(X!0))) = 0)) | $greatereq($sum(g(X!0), $product(-1, f(X!0))), 0)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.40 tff(63,plain,
% 0.20/0.40 (~($sum(X!0, $product(-1, g(X!0))) = 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[62, 1])).
% 0.20/0.40 tff(64,plain,
% 0.20/0.40 (((~$greatereq(g(X!0), 0)) | (~(($sum(X!0, $product(-1, g(X!0))) = 0) | ($sum(X!0, g(X!0)) = 0)))) | (($sum(X!0, $product(-1, g(X!0))) = 0) | ($sum(X!0, g(X!0)) = 0))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(65,plain,
% 0.20/0.40 (($sum(X!0, $product(-1, g(X!0))) = 0) | ($sum(X!0, g(X!0)) = 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[64, 32])).
% 0.20/0.40 tff(66,plain,
% 0.20/0.40 ((~(($sum(X!0, $product(-1, g(X!0))) = 0) | ($sum(X!0, g(X!0)) = 0))) | ($sum(X!0, $product(-1, g(X!0))) = 0) | ($sum(X!0, g(X!0)) = 0)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(67,plain,
% 0.20/0.40 (($sum(X!0, $product(-1, g(X!0))) = 0) | ($sum(X!0, g(X!0)) = 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[66, 65])).
% 0.20/0.40 tff(68,plain,
% 0.20/0.40 ($sum(X!0, g(X!0)) = 0),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[67, 63])).
% 0.20/0.40 tff(69,plain,
% 0.20/0.40 ((~($sum(X!0, g(X!0)) = 0)) | $greatereq($sum(X!0, g(X!0)), 0)),
% 0.20/0.40 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.40 tff(70,plain,
% 0.20/0.40 ($greatereq($sum(X!0, g(X!0)), 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[69, 68])).
% 0.20/0.40 tff(71,plain,
% 0.20/0.40 ((~($sum(X!0, g(X!0)) = 0)) | $lesseq($sum(X!0, g(X!0)), 0)),
% 0.20/0.40 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.40 tff(72,plain,
% 0.20/0.40 ($lesseq($sum(X!0, g(X!0)), 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[71, 68])).
% 0.20/0.40 tff(73,plain,
% 0.20/0.40 ((~$greatereq($sum(X!0, $product(-1, f(X!0))), 0)) | (~$lesseq($sum(X!0, g(X!0)), 0)) | $greatereq($sum(g(X!0), $product(-1, f(X!0))), 0) | (~$greatereq(g(X!0), 0))),
% 0.20/0.40 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.40 tff(74,plain,
% 0.20/0.40 (~$greatereq($sum(X!0, $product(-1, f(X!0))), 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[73, 1, 34, 72])).
% 0.20/0.40 tff(75,plain,
% 0.20/0.40 ($lesseq($sum(X!0, f(X!0)), 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[59, 74])).
% 0.20/0.40 tff(76,plain,
% 0.20/0.40 ($false),
% 0.20/0.40 inference(theory_lemma,[status(thm)],[1, 75, 70])).
% 0.20/0.40 tff(77,plain,($greatereq($sum(g(X!0), $product(-1, f(X!0))), 0)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.40 tff(78,plain,
% 0.20/0.40 ((~![X: $int] : ($sum(g(X), $product(-1, f(X))) = 0)) <=> (~![X: $int] : ($sum(g(X), $product(-1, f(X))) = 0))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(79,plain,
% 0.20/0.40 ((~![X: $int] : ($sum(f(X), $product(-1, g(X))) = 0)) <=> (~![X: $int] : ($sum(g(X), $product(-1, f(X))) = 0))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(80,plain,
% 0.20/0.40 ((~![X: $int] : (f(X) = g(X))) <=> (~![X: $int] : ($sum(f(X), $product(-1, g(X))) = 0))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(81,plain,
% 0.20/0.40 ((~![X: $int] : (f(X) = g(X))) <=> (~![X: $int] : (f(X) = g(X)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(82,plain,
% 0.20/0.40 (~![X: $int] : (f(X) = g(X))),
% 0.20/0.40 inference(or_elim,[status(thm)],[18])).
% 0.20/0.40 tff(83,plain,
% 0.20/0.40 (~![X: $int] : (f(X) = g(X))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.20/0.40 tff(84,plain,
% 0.20/0.40 (~![X: $int] : (f(X) = g(X))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[83, 81])).
% 0.20/0.40 tff(85,plain,
% 0.20/0.40 (~![X: $int] : (f(X) = g(X))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[84, 81])).
% 0.20/0.40 tff(86,plain,
% 0.20/0.40 (~![X: $int] : ($sum(f(X), $product(-1, g(X))) = 0)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[85, 80])).
% 0.20/0.40 tff(87,plain,
% 0.20/0.40 (~![X: $int] : ($sum(g(X), $product(-1, f(X))) = 0)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[86, 79])).
% 0.20/0.40 tff(88,plain,
% 0.20/0.40 (~![X: $int] : ($sum(g(X), $product(-1, f(X))) = 0)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[87, 78])).
% 0.20/0.40 tff(89,plain,
% 0.20/0.40 (~![X: $int] : ($sum(g(X), $product(-1, f(X))) = 0)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[88, 78])).
% 0.20/0.40 tff(90,plain,(
% 0.20/0.40 ~($sum(g(X!0), $product(-1, f(X!0))) = 0)),
% 0.20/0.40 inference(skolemize,[status(sab)],[89])).
% 0.20/0.40 tff(91,plain,
% 0.20/0.40 (($sum(g(X!0), $product(-1, f(X!0))) = 0) | (~$lesseq($sum(g(X!0), $product(-1, f(X!0))), 0)) | (~$greatereq($sum(g(X!0), $product(-1, f(X!0))), 0))),
% 0.20/0.40 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.40 tff(92,plain,
% 0.20/0.40 ((~$lesseq($sum(g(X!0), $product(-1, f(X!0))), 0)) | (~$greatereq($sum(g(X!0), $product(-1, f(X!0))), 0))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[91, 90])).
% 0.20/0.40 tff(93,plain,
% 0.20/0.40 (~$lesseq($sum(g(X!0), $product(-1, f(X!0))), 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[92, 77])).
% 0.20/0.40 tff(94,plain,
% 0.20/0.40 (((~$lesseq($sum(X!0, $product(-1, f(X!0))), 0)) | (~$greatereq($sum(X!0, f(X!0)), 0)) | (~($greatereq($sum(X!0, $product(-1, f(X!0))), 0) | $lesseq($sum(X!0, f(X!0)), 0)))) | $lesseq($sum(X!0, $product(-1, f(X!0))), 0)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(95,plain,
% 0.20/0.40 ($lesseq($sum(X!0, $product(-1, f(X!0))), 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[94, 55])).
% 0.20/0.40 tff(96,plain,
% 0.20/0.40 ((~$greatereq($sum(X!0, $product(-1, g(X!0))), 0)) | $lesseq($sum(g(X!0), $product(-1, f(X!0))), 0) | (~$lesseq($sum(X!0, $product(-1, f(X!0))), 0))),
% 0.20/0.40 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.40 tff(97,plain,
% 0.20/0.40 (~$greatereq($sum(X!0, $product(-1, g(X!0))), 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[96, 95, 93])).
% 0.20/0.40 tff(98,plain,
% 0.20/0.40 (~($sum(X!0, $product(-1, g(X!0))) = 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[5, 97])).
% 0.20/0.40 tff(99,plain,
% 0.20/0.40 (((~$lesseq($sum(X!0, $product(-1, f(X!0))), 0)) | (~$greatereq($sum(X!0, f(X!0)), 0)) | (~($greatereq($sum(X!0, $product(-1, f(X!0))), 0) | $lesseq($sum(X!0, f(X!0)), 0)))) | $greatereq($sum(X!0, f(X!0)), 0)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(100,plain,
% 0.20/0.40 ($greatereq($sum(X!0, f(X!0)), 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[99, 55])).
% 0.20/0.40 tff(101,plain,
% 0.20/0.40 ((~$lesseq($sum(X!0, g(X!0)), 0)) | $lesseq($sum(g(X!0), $product(-1, f(X!0))), 0) | (~$greatereq($sum(X!0, f(X!0)), 0))),
% 0.20/0.40 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.40 tff(102,plain,
% 0.20/0.40 (~$lesseq($sum(X!0, g(X!0)), 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[101, 100, 93])).
% 0.20/0.40 tff(103,plain,
% 0.20/0.40 (~($sum(X!0, g(X!0)) = 0)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[71, 102])).
% 0.20/0.40 tff(104,plain,
% 0.20/0.40 ($false),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[67, 103, 98])).
% 0.20/0.40 % SZS output end Proof
%------------------------------------------------------------------------------