TSTP Solution File: DAT039_1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : DAT039_1 : TPTP v8.1.0. Released v5.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n019.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 : Fri Sep 16 14:36:28 EDT 2022
% Result : Theorem 0.22s 0.42s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.13 % Problem : DAT039_1 : TPTP v8.1.0. Released v5.0.0.
% 0.11/0.14 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.15/0.35 % Computer : n019.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Wed Aug 31 01:47:20 EDT 2022
% 0.15/0.35 % CPUTime :
% 0.15/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.15/0.35 Usage: tptp [options] [-file:]file
% 0.15/0.35 -h, -? prints this message.
% 0.15/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.15/0.35 -m, -model generate model.
% 0.15/0.35 -p, -proof generate proof.
% 0.15/0.35 -c, -core generate unsat core of named formulas.
% 0.15/0.35 -st, -statistics display statistics.
% 0.15/0.35 -t:timeout set timeout (in second).
% 0.15/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.15/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.15/0.35 -<param>:<value> configuration parameter and value.
% 0.15/0.35 -o:<output-file> file to place output in.
% 0.22/0.42 % SZS status Theorem
% 0.22/0.42 % SZS output start Proof
% 0.22/0.42 tff(in_type, type, (
% 0.22/0.42 in: ( $int * collection ) > $o)).
% 0.22/0.42 tff(empty_type, type, (
% 0.22/0.42 empty: collection)).
% 0.22/0.42 tff(tptp_fun_V_0_type, type, (
% 0.22/0.42 tptp_fun_V_0: $int)).
% 0.22/0.42 tff(remove_type, type, (
% 0.22/0.42 remove: ( $int * collection ) > collection)).
% 0.22/0.42 tff(tptp_fun_U_1_type, type, (
% 0.22/0.42 tptp_fun_U_1: collection)).
% 0.22/0.42 tff(count_type, type, (
% 0.22/0.42 count: collection > $int)).
% 0.22/0.42 tff(add_type, type, (
% 0.22/0.42 add: ( $int * collection ) > collection)).
% 0.22/0.42 tff(1,plain,
% 0.22/0.42 (^[X7: collection] : refl(((X7 = empty) <=> (count(X7) = 0)) <=> ((X7 = empty) <=> (count(X7) = 0)))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(2,plain,
% 0.22/0.42 (![X7: collection] : ((X7 = empty) <=> (count(X7) = 0)) <=> ![X7: collection] : ((X7 = empty) <=> (count(X7) = 0))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[1])).
% 0.22/0.42 tff(3,plain,
% 0.22/0.42 (![X7: collection] : ((X7 = empty) <=> (count(X7) = 0)) <=> ![X7: collection] : ((X7 = empty) <=> (count(X7) = 0))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(4,axiom,(![X7: collection] : ((X7 = empty) <=> (count(X7) = 0))), file('/export/starexec/sandbox/benchmark/Axioms/DAT002=1.ax','ax2')).
% 0.22/0.42 tff(5,plain,
% 0.22/0.42 (![X7: collection] : ((X7 = empty) <=> (count(X7) = 0))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.22/0.42 tff(6,plain,(
% 0.22/0.42 ![X7: collection] : ((X7 = empty) <=> (count(X7) = 0))),
% 0.22/0.42 inference(skolemize,[status(sab)],[5])).
% 0.22/0.42 tff(7,plain,
% 0.22/0.42 (![X7: collection] : ((X7 = empty) <=> (count(X7) = 0))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.22/0.42 tff(8,plain,
% 0.22/0.42 ((~![X7: collection] : ((X7 = empty) <=> (count(X7) = 0))) | ((remove(2, remove(3, U!1)) = empty) <=> (count(remove(2, remove(3, U!1))) = 0))),
% 0.22/0.42 inference(quant_inst,[status(thm)],[])).
% 0.22/0.42 tff(9,plain,
% 0.22/0.42 ((remove(2, remove(3, U!1)) = empty) <=> (count(remove(2, remove(3, U!1))) = 0)),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.22/0.42 tff(10,plain,
% 0.22/0.42 (^[X12: $int, X13: collection] : refl((in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1)) <=> (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1)))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(11,plain,
% 0.22/0.42 (![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1)) <=> ![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[10])).
% 0.22/0.42 tff(12,plain,
% 0.22/0.42 (^[X12: $int, X13: collection] : rewrite((in(X12, X13) <=> ($sum(count(remove(X12, X13)), $product(-1, count(X13))) = -1)) <=> (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1)))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(13,plain,
% 0.22/0.42 (![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(remove(X12, X13)), $product(-1, count(X13))) = -1)) <=> ![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[12])).
% 0.22/0.42 tff(14,plain,
% 0.22/0.42 (^[X12: $int, X13: collection] : rewrite((in(X12, X13) <=> (count(remove(X12, X13)) = $sum(-1, count(X13)))) <=> (in(X12, X13) <=> ($sum(count(remove(X12, X13)), $product(-1, count(X13))) = -1)))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(15,plain,
% 0.22/0.42 (![X12: $int, X13: collection] : (in(X12, X13) <=> (count(remove(X12, X13)) = $sum(-1, count(X13)))) <=> ![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(remove(X12, X13)), $product(-1, count(X13))) = -1))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[14])).
% 0.22/0.42 tff(16,plain,
% 0.22/0.42 (![X12: $int, X13: collection] : (in(X12, X13) <=> (count(remove(X12, X13)) = $sum(-1, count(X13)))) <=> ![X12: $int, X13: collection] : (in(X12, X13) <=> (count(remove(X12, X13)) = $sum(-1, count(X13))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(17,plain,
% 0.22/0.42 (^[X12: $int, X13: collection] : rewrite((in(X12, X13) <=> (count(remove(X12, X13)) = $difference(count(X13), 1))) <=> (in(X12, X13) <=> (count(remove(X12, X13)) = $sum(-1, count(X13)))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(18,plain,
% 0.22/0.42 (![X12: $int, X13: collection] : (in(X12, X13) <=> (count(remove(X12, X13)) = $difference(count(X13), 1))) <=> ![X12: $int, X13: collection] : (in(X12, X13) <=> (count(remove(X12, X13)) = $sum(-1, count(X13))))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[17])).
% 0.22/0.43 tff(19,axiom,(![X12: $int, X13: collection] : (in(X12, X13) <=> (count(remove(X12, X13)) = $difference(count(X13), 1)))), file('/export/starexec/sandbox/benchmark/Axioms/DAT002=1.ax','ax5')).
% 0.22/0.43 tff(20,plain,
% 0.22/0.43 (![X12: $int, X13: collection] : (in(X12, X13) <=> (count(remove(X12, X13)) = $sum(-1, count(X13))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[19, 18])).
% 0.22/0.43 tff(21,plain,
% 0.22/0.43 (![X12: $int, X13: collection] : (in(X12, X13) <=> (count(remove(X12, X13)) = $sum(-1, count(X13))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[20, 16])).
% 0.22/0.43 tff(22,plain,
% 0.22/0.43 (![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(remove(X12, X13)), $product(-1, count(X13))) = -1))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[21, 15])).
% 0.22/0.43 tff(23,plain,
% 0.22/0.43 (![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[22, 13])).
% 0.22/0.43 tff(24,plain,(
% 0.22/0.43 ![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))),
% 0.22/0.43 inference(skolemize,[status(sab)],[23])).
% 0.22/0.43 tff(25,plain,
% 0.22/0.43 (![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[24, 11])).
% 0.22/0.43 tff(26,plain,
% 0.22/0.43 ((~![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))) | (in(3, U!1) <=> ($sum(count(U!1), $product(-1, count(remove(3, U!1)))) = 1))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(27,plain,
% 0.22/0.43 (in(3, U!1) <=> ($sum(count(U!1), $product(-1, count(remove(3, U!1)))) = 1)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[26, 25])).
% 0.22/0.43 tff(28,plain,
% 0.22/0.43 ((~![U: collection, V: $int] : ((~in(V, U)) | (~(in(2, U) & in(3, U) & (count(U) = 2) & (~$lesseq(V, 3)))))) <=> (~![U: collection, V: $int] : ((~in(V, U)) | (~(in(2, U) & in(3, U) & (count(U) = 2) & (~$lesseq(V, 3))))))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(29,plain,
% 0.22/0.43 ((~![U: collection, V: $int] : ((((in(2, U) & in(3, U)) & (count(U) = 2)) & $greater(V, 3)) => (~in(V, U)))) <=> (~![U: collection, V: $int] : ((~in(V, U)) | (~(in(2, U) & in(3, U) & (count(U) = 2) & (~$lesseq(V, 3))))))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(30,axiom,(~![U: collection, V: $int] : ((((in(2, U) & in(3, U)) & (count(U) = 2)) & $greater(V, 3)) => (~in(V, U)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','co1')).
% 0.22/0.43 tff(31,plain,
% 0.22/0.43 (~![U: collection, V: $int] : ((~in(V, U)) | (~(in(2, U) & in(3, U) & (count(U) = 2) & (~$lesseq(V, 3)))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[30, 29])).
% 0.22/0.43 tff(32,plain,
% 0.22/0.43 (~![U: collection, V: $int] : ((~in(V, U)) | (~(in(2, U) & in(3, U) & (count(U) = 2) & (~$lesseq(V, 3)))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[31, 28])).
% 0.22/0.43 tff(33,plain,
% 0.22/0.43 (~![U: collection, V: $int] : ((~in(V, U)) | (~(in(2, U) & in(3, U) & (count(U) = 2) & (~$lesseq(V, 3)))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[32, 28])).
% 0.22/0.43 tff(34,plain,
% 0.22/0.43 (~![U: collection, V: $int] : ((~in(V, U)) | (~(in(2, U) & in(3, U) & (count(U) = 2) & (~$lesseq(V, 3)))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[33, 28])).
% 0.22/0.43 tff(35,plain,
% 0.22/0.43 (~![U: collection, V: $int] : ((~in(V, U)) | (~(in(2, U) & in(3, U) & (count(U) = 2) & (~$lesseq(V, 3)))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[34, 28])).
% 0.22/0.43 tff(36,plain,
% 0.22/0.43 (~![U: collection, V: $int] : ((~in(V, U)) | (~(in(2, U) & in(3, U) & (count(U) = 2) & (~$lesseq(V, 3)))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[35, 28])).
% 0.22/0.43 tff(37,plain,
% 0.22/0.43 (~![U: collection, V: $int] : ((~in(V, U)) | (~(in(2, U) & in(3, U) & (count(U) = 2) & (~$lesseq(V, 3)))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[36, 28])).
% 0.22/0.43 tff(38,plain,(
% 0.22/0.43 ~((~in(V!0, U!1)) | (~(in(2, U!1) & in(3, U!1) & (count(U!1) = 2) & (~$lesseq(V!0, 3)))))),
% 0.22/0.43 inference(skolemize,[status(sab)],[37])).
% 0.22/0.43 tff(39,plain,
% 0.22/0.43 (in(2, U!1) & in(3, U!1) & (count(U!1) = 2) & (~$lesseq(V!0, 3))),
% 0.22/0.43 inference(or_elim,[status(thm)],[38])).
% 0.22/0.43 tff(40,plain,
% 0.22/0.43 (in(3, U!1)),
% 0.22/0.43 inference(and_elim,[status(thm)],[39])).
% 0.22/0.43 tff(41,plain,
% 0.22/0.43 ((~(in(3, U!1) <=> ($sum(count(U!1), $product(-1, count(remove(3, U!1)))) = 1))) | (~in(3, U!1)) | ($sum(count(U!1), $product(-1, count(remove(3, U!1)))) = 1)),
% 0.22/0.43 inference(tautology,[status(thm)],[])).
% 0.22/0.43 tff(42,plain,
% 0.22/0.43 ((~(in(3, U!1) <=> ($sum(count(U!1), $product(-1, count(remove(3, U!1)))) = 1))) | ($sum(count(U!1), $product(-1, count(remove(3, U!1)))) = 1)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[41, 40])).
% 0.22/0.43 tff(43,plain,
% 0.22/0.43 ($sum(count(U!1), $product(-1, count(remove(3, U!1)))) = 1),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[42, 27])).
% 0.22/0.43 tff(44,plain,
% 0.22/0.43 ((~($sum(count(U!1), $product(-1, count(remove(3, U!1)))) = 1)) | $lesseq($sum(count(U!1), $product(-1, count(remove(3, U!1)))), 1)),
% 0.22/0.43 inference(theory_lemma,[status(thm)],[])).
% 0.22/0.43 tff(45,plain,
% 0.22/0.43 ($lesseq($sum(count(U!1), $product(-1, count(remove(3, U!1)))), 1)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[44, 43])).
% 0.22/0.43 tff(46,plain,
% 0.22/0.43 ((~($sum(count(U!1), $product(-1, count(remove(3, U!1)))) = 1)) | $greatereq($sum(count(U!1), $product(-1, count(remove(3, U!1)))), 1)),
% 0.22/0.43 inference(theory_lemma,[status(thm)],[])).
% 0.22/0.43 tff(47,plain,
% 0.22/0.43 ($greatereq($sum(count(U!1), $product(-1, count(remove(3, U!1)))), 1)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[46, 43])).
% 0.22/0.43 tff(48,plain,
% 0.22/0.43 (count(U!1) = 2),
% 0.22/0.43 inference(and_elim,[status(thm)],[39])).
% 0.22/0.43 tff(49,plain,
% 0.22/0.43 ((~(count(U!1) = 2)) | $lesseq(count(U!1), 2)),
% 0.22/0.43 inference(theory_lemma,[status(thm)],[])).
% 0.22/0.43 tff(50,plain,
% 0.22/0.43 ($lesseq(count(U!1), 2)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[49, 48])).
% 0.22/0.43 tff(51,plain,
% 0.22/0.43 ((~(count(U!1) = 2)) | $greatereq(count(U!1), 2)),
% 0.22/0.43 inference(theory_lemma,[status(thm)],[])).
% 0.22/0.43 tff(52,plain,
% 0.22/0.43 ($greatereq(count(U!1), 2)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[51, 48])).
% 0.22/0.43 tff(53,plain,
% 0.22/0.43 ((~![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))) | (in(2, remove(3, U!1)) <=> ($sum(count(remove(3, U!1)), $product(-1, count(remove(2, remove(3, U!1))))) = 1))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(54,plain,
% 0.22/0.43 (in(2, remove(3, U!1)) <=> ($sum(count(remove(3, U!1)), $product(-1, count(remove(2, remove(3, U!1))))) = 1)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[53, 25])).
% 0.22/0.43 tff(55,plain,
% 0.22/0.43 (^[Z: $int, X1: collection, X2: $int] : refl(((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1))) <=> ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1))))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(56,plain,
% 0.22/0.43 (![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1))) <=> ![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[55])).
% 0.22/0.43 tff(57,plain,
% 0.22/0.43 (^[Z: $int, X1: collection, X2: $int] : rewrite(((in(Z, X1) | ($sum(Z, $product(-1, X2)) = 0)) <=> in(Z, add(X2, X1))) <=> ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1))))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(58,plain,
% 0.22/0.43 (![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(Z, $product(-1, X2)) = 0)) <=> in(Z, add(X2, X1))) <=> ![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[57])).
% 0.22/0.43 tff(59,plain,
% 0.22/0.43 (^[Z: $int, X1: collection, X2: $int] : rewrite((((Z = X2) | in(Z, X1)) <=> in(Z, add(X2, X1))) <=> ((in(Z, X1) | ($sum(Z, $product(-1, X2)) = 0)) <=> in(Z, add(X2, X1))))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(60,plain,
% 0.22/0.43 (![Z: $int, X1: collection, X2: $int] : (((Z = X2) | in(Z, X1)) <=> in(Z, add(X2, X1))) <=> ![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(Z, $product(-1, X2)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[59])).
% 0.22/0.43 tff(61,plain,
% 0.22/0.43 (![Z: $int, X1: collection, X2: $int] : (((Z = X2) | in(Z, X1)) <=> in(Z, add(X2, X1))) <=> ![Z: $int, X1: collection, X2: $int] : (((Z = X2) | in(Z, X1)) <=> in(Z, add(X2, X1)))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(62,plain,
% 0.22/0.43 (^[Z: $int, X1: collection, X2: $int] : rewrite(((in(Z, X1) | (Z = X2)) <=> in(Z, add(X2, X1))) <=> (((Z = X2) | in(Z, X1)) <=> in(Z, add(X2, X1))))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(63,plain,
% 0.22/0.43 (![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | (Z = X2)) <=> in(Z, add(X2, X1))) <=> ![Z: $int, X1: collection, X2: $int] : (((Z = X2) | in(Z, X1)) <=> in(Z, add(X2, X1)))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[62])).
% 0.22/0.43 tff(64,axiom,(![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | (Z = X2)) <=> in(Z, add(X2, X1)))), file('/export/starexec/sandbox/benchmark/Axioms/DAT002=0.ax','ax4')).
% 0.22/0.43 tff(65,plain,
% 0.22/0.43 (![Z: $int, X1: collection, X2: $int] : (((Z = X2) | in(Z, X1)) <=> in(Z, add(X2, X1)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.22/0.43 tff(66,plain,
% 0.22/0.43 (![Z: $int, X1: collection, X2: $int] : (((Z = X2) | in(Z, X1)) <=> in(Z, add(X2, X1)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[65, 61])).
% 0.22/0.43 tff(67,plain,
% 0.22/0.43 (![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(Z, $product(-1, X2)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[66, 60])).
% 0.22/0.43 tff(68,plain,
% 0.22/0.43 (![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[67, 58])).
% 0.22/0.43 tff(69,plain,(
% 0.22/0.43 ![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.22/0.43 inference(skolemize,[status(sab)],[68])).
% 0.22/0.43 tff(70,plain,
% 0.22/0.43 (![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[69, 56])).
% 0.22/0.43 tff(71,plain,
% 0.22/0.43 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | (in(2, remove(3, U!1)) <=> in(2, add(3, remove(3, U!1))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | (in(2, remove(3, U!1)) <=> in(2, add(3, remove(3, U!1)))))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(72,plain,
% 0.22/0.43 (((in(2, remove(3, U!1)) | ($sum(3, $product(-1, 2)) = 0)) <=> in(2, add(3, remove(3, U!1)))) <=> (in(2, remove(3, U!1)) <=> in(2, add(3, remove(3, U!1))))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(73,plain,
% 0.22/0.43 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(2, remove(3, U!1)) | ($sum(3, $product(-1, 2)) = 0)) <=> in(2, add(3, remove(3, U!1))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | (in(2, remove(3, U!1)) <=> in(2, add(3, remove(3, U!1)))))),
% 0.22/0.43 inference(monotonicity,[status(thm)],[72])).
% 0.22/0.43 tff(74,plain,
% 0.22/0.43 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(2, remove(3, U!1)) | ($sum(3, $product(-1, 2)) = 0)) <=> in(2, add(3, remove(3, U!1))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | (in(2, remove(3, U!1)) <=> in(2, add(3, remove(3, U!1)))))),
% 0.22/0.43 inference(transitivity,[status(thm)],[73, 71])).
% 0.22/0.43 tff(75,plain,
% 0.22/0.43 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(2, remove(3, U!1)) | ($sum(3, $product(-1, 2)) = 0)) <=> in(2, add(3, remove(3, U!1))))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(76,plain,
% 0.22/0.43 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | (in(2, remove(3, U!1)) <=> in(2, add(3, remove(3, U!1))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[75, 74])).
% 0.22/0.43 tff(77,plain,
% 0.22/0.43 (in(2, remove(3, U!1)) <=> in(2, add(3, remove(3, U!1)))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[76, 70])).
% 0.22/0.43 tff(78,plain,
% 0.22/0.43 (^[X16: $int, X17: collection] : refl(((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17)))) <=> ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17)))))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(79,plain,
% 0.22/0.43 (![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17)))) <=> ![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[78])).
% 0.22/0.43 tff(80,plain,
% 0.22/0.43 (![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17)))) <=> ![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(81,plain,
% 0.22/0.43 (^[X16: $int, X17: collection] : rewrite((in(X16, X17) => (X17 = add(X16, remove(X16, X17)))) <=> ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17)))))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(82,plain,
% 0.22/0.43 (![X16: $int, X17: collection] : (in(X16, X17) => (X17 = add(X16, remove(X16, X17)))) <=> ![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[81])).
% 0.22/0.43 tff(83,axiom,(![X16: $int, X17: collection] : (in(X16, X17) => (X17 = add(X16, remove(X16, X17))))), file('/export/starexec/sandbox/benchmark/Axioms/DAT002=1.ax','ax7')).
% 0.22/0.43 tff(84,plain,
% 0.22/0.43 (![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[83, 82])).
% 0.22/0.43 tff(85,plain,
% 0.22/0.43 (![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[84, 80])).
% 0.22/0.43 tff(86,plain,(
% 0.22/0.43 ![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))),
% 0.22/0.43 inference(skolemize,[status(sab)],[85])).
% 0.22/0.43 tff(87,plain,
% 0.22/0.43 (![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[86, 79])).
% 0.22/0.43 tff(88,plain,
% 0.22/0.43 (((~![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))) | ((~in(3, U!1)) | (U!1 = add(3, remove(3, U!1))))) <=> ((~![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))) | (~in(3, U!1)) | (U!1 = add(3, remove(3, U!1))))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(89,plain,
% 0.22/0.43 ((~![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))) | ((~in(3, U!1)) | (U!1 = add(3, remove(3, U!1))))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(90,plain,
% 0.22/0.43 ((~![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))) | (~in(3, U!1)) | (U!1 = add(3, remove(3, U!1)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[89, 88])).
% 0.22/0.43 tff(91,plain,
% 0.22/0.43 (U!1 = add(3, remove(3, U!1))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[90, 87, 40])).
% 0.22/0.43 tff(92,plain,
% 0.22/0.43 (add(3, remove(3, U!1)) = U!1),
% 0.22/0.43 inference(symmetry,[status(thm)],[91])).
% 0.22/0.43 tff(93,plain,
% 0.22/0.43 (in(2, add(3, remove(3, U!1))) <=> in(2, U!1)),
% 0.22/0.43 inference(monotonicity,[status(thm)],[92])).
% 0.22/0.43 tff(94,plain,
% 0.22/0.43 (in(2, U!1) <=> in(2, add(3, remove(3, U!1)))),
% 0.22/0.43 inference(symmetry,[status(thm)],[93])).
% 0.22/0.43 tff(95,plain,
% 0.22/0.43 (in(2, U!1)),
% 0.22/0.43 inference(and_elim,[status(thm)],[39])).
% 0.22/0.43 tff(96,plain,
% 0.22/0.43 (in(2, add(3, remove(3, U!1)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[95, 94])).
% 0.22/0.43 tff(97,plain,
% 0.22/0.43 ((~(in(2, remove(3, U!1)) <=> in(2, add(3, remove(3, U!1))))) | in(2, remove(3, U!1)) | (~in(2, add(3, remove(3, U!1))))),
% 0.22/0.43 inference(tautology,[status(thm)],[])).
% 0.22/0.43 tff(98,plain,
% 0.22/0.43 (in(2, remove(3, U!1))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[97, 96, 77])).
% 0.22/0.43 tff(99,plain,
% 0.22/0.43 ((~(in(2, remove(3, U!1)) <=> ($sum(count(remove(3, U!1)), $product(-1, count(remove(2, remove(3, U!1))))) = 1))) | (~in(2, remove(3, U!1))) | ($sum(count(remove(3, U!1)), $product(-1, count(remove(2, remove(3, U!1))))) = 1)),
% 0.22/0.43 inference(tautology,[status(thm)],[])).
% 0.22/0.43 tff(100,plain,
% 0.22/0.43 ($sum(count(remove(3, U!1)), $product(-1, count(remove(2, remove(3, U!1))))) = 1),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[99, 98, 54])).
% 0.22/0.43 tff(101,plain,
% 0.22/0.43 ((~($sum(count(remove(3, U!1)), $product(-1, count(remove(2, remove(3, U!1))))) = 1)) | $lesseq($sum(count(remove(3, U!1)), $product(-1, count(remove(2, remove(3, U!1))))), 1)),
% 0.22/0.43 inference(theory_lemma,[status(thm)],[])).
% 0.22/0.43 tff(102,plain,
% 0.22/0.43 ($lesseq($sum(count(remove(3, U!1)), $product(-1, count(remove(2, remove(3, U!1))))), 1)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[101, 100])).
% 0.22/0.43 tff(103,plain,
% 0.22/0.43 ((~($sum(count(remove(3, U!1)), $product(-1, count(remove(2, remove(3, U!1))))) = 1)) | $greatereq($sum(count(remove(3, U!1)), $product(-1, count(remove(2, remove(3, U!1))))), 1)),
% 0.22/0.43 inference(theory_lemma,[status(thm)],[])).
% 0.22/0.43 tff(104,plain,
% 0.22/0.43 ($greatereq($sum(count(remove(3, U!1)), $product(-1, count(remove(2, remove(3, U!1))))), 1)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[103, 100])).
% 0.22/0.43 tff(105,plain,
% 0.22/0.43 (count(remove(2, remove(3, U!1))) = 0),
% 0.22/0.43 inference(theory_lemma,[status(thm)],[104, 102, 52, 50, 47, 45])).
% 0.22/0.43 tff(106,plain,
% 0.22/0.43 ((~((remove(2, remove(3, U!1)) = empty) <=> (count(remove(2, remove(3, U!1))) = 0))) | (remove(2, remove(3, U!1)) = empty) | (~(count(remove(2, remove(3, U!1))) = 0))),
% 0.22/0.43 inference(tautology,[status(thm)],[])).
% 0.22/0.43 tff(107,plain,
% 0.22/0.43 (remove(2, remove(3, U!1)) = empty),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[106, 105, 9])).
% 0.22/0.43 tff(108,plain,
% 0.22/0.43 (empty = remove(2, remove(3, U!1))),
% 0.22/0.43 inference(symmetry,[status(thm)],[107])).
% 0.22/0.43 tff(109,plain,
% 0.22/0.43 (in(V!0, empty) <=> in(V!0, remove(2, remove(3, U!1)))),
% 0.22/0.43 inference(monotonicity,[status(thm)],[108])).
% 0.22/0.43 tff(110,plain,
% 0.22/0.43 (in(V!0, remove(2, remove(3, U!1))) <=> in(V!0, empty)),
% 0.22/0.43 inference(symmetry,[status(thm)],[109])).
% 0.22/0.43 tff(111,plain,
% 0.22/0.43 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | (((V!0 = 2) | in(V!0, remove(2, remove(3, U!1)))) <=> in(V!0, add(2, remove(2, remove(3, U!1)))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | (((V!0 = 2) | in(V!0, remove(2, remove(3, U!1)))) <=> in(V!0, add(2, remove(2, remove(3, U!1))))))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(112,plain,
% 0.22/0.43 (((in(V!0, remove(2, remove(3, U!1))) | ($sum(2, $product(-1, V!0)) = 0)) <=> in(V!0, add(2, remove(2, remove(3, U!1))))) <=> (((V!0 = 2) | in(V!0, remove(2, remove(3, U!1)))) <=> in(V!0, add(2, remove(2, remove(3, U!1)))))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(113,plain,
% 0.22/0.43 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!0, remove(2, remove(3, U!1))) | ($sum(2, $product(-1, V!0)) = 0)) <=> in(V!0, add(2, remove(2, remove(3, U!1)))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | (((V!0 = 2) | in(V!0, remove(2, remove(3, U!1)))) <=> in(V!0, add(2, remove(2, remove(3, U!1))))))),
% 0.22/0.43 inference(monotonicity,[status(thm)],[112])).
% 0.22/0.43 tff(114,plain,
% 0.22/0.43 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!0, remove(2, remove(3, U!1))) | ($sum(2, $product(-1, V!0)) = 0)) <=> in(V!0, add(2, remove(2, remove(3, U!1)))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | (((V!0 = 2) | in(V!0, remove(2, remove(3, U!1)))) <=> in(V!0, add(2, remove(2, remove(3, U!1))))))),
% 0.22/0.43 inference(transitivity,[status(thm)],[113, 111])).
% 0.22/0.43 tff(115,plain,
% 0.22/0.43 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!0, remove(2, remove(3, U!1))) | ($sum(2, $product(-1, V!0)) = 0)) <=> in(V!0, add(2, remove(2, remove(3, U!1)))))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(116,plain,
% 0.22/0.43 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | (((V!0 = 2) | in(V!0, remove(2, remove(3, U!1)))) <=> in(V!0, add(2, remove(2, remove(3, U!1)))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[115, 114])).
% 0.22/0.43 tff(117,plain,
% 0.22/0.43 (((V!0 = 2) | in(V!0, remove(2, remove(3, U!1)))) <=> in(V!0, add(2, remove(2, remove(3, U!1))))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[116, 70])).
% 0.22/0.44 tff(118,plain,
% 0.22/0.44 (((~![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))) | ((~in(2, remove(3, U!1))) | (remove(3, U!1) = add(2, remove(2, remove(3, U!1)))))) <=> ((~![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))) | (~in(2, remove(3, U!1))) | (remove(3, U!1) = add(2, remove(2, remove(3, U!1)))))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(119,plain,
% 0.22/0.44 ((~![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))) | ((~in(2, remove(3, U!1))) | (remove(3, U!1) = add(2, remove(2, remove(3, U!1)))))),
% 0.22/0.44 inference(quant_inst,[status(thm)],[])).
% 0.22/0.44 tff(120,plain,
% 0.22/0.44 ((~![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))) | (~in(2, remove(3, U!1))) | (remove(3, U!1) = add(2, remove(2, remove(3, U!1))))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[119, 118])).
% 0.22/0.44 tff(121,plain,
% 0.22/0.44 ((~in(2, remove(3, U!1))) | (remove(3, U!1) = add(2, remove(2, remove(3, U!1))))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[120, 87])).
% 0.22/0.44 tff(122,plain,
% 0.22/0.44 (remove(3, U!1) = add(2, remove(2, remove(3, U!1)))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[121, 98])).
% 0.22/0.44 tff(123,plain,
% 0.22/0.44 (add(2, remove(2, remove(3, U!1))) = remove(3, U!1)),
% 0.22/0.44 inference(symmetry,[status(thm)],[122])).
% 0.22/0.44 tff(124,plain,
% 0.22/0.44 (in(V!0, add(2, remove(2, remove(3, U!1)))) <=> in(V!0, remove(3, U!1))),
% 0.22/0.44 inference(monotonicity,[status(thm)],[123])).
% 0.22/0.44 tff(125,plain,
% 0.22/0.44 (in(V!0, remove(3, U!1)) <=> in(V!0, add(2, remove(2, remove(3, U!1))))),
% 0.22/0.44 inference(symmetry,[status(thm)],[124])).
% 0.22/0.44 tff(126,plain,
% 0.22/0.44 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!0, remove(3, U!1)) | (V!0 = 3)) <=> in(V!0, add(3, remove(3, U!1))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!0, remove(3, U!1)) | (V!0 = 3)) <=> in(V!0, add(3, remove(3, U!1)))))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(127,plain,
% 0.22/0.44 (((in(V!0, remove(3, U!1)) | ($sum(3, $product(-1, V!0)) = 0)) <=> in(V!0, add(3, remove(3, U!1)))) <=> ((in(V!0, remove(3, U!1)) | (V!0 = 3)) <=> in(V!0, add(3, remove(3, U!1))))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(128,plain,
% 0.22/0.44 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!0, remove(3, U!1)) | ($sum(3, $product(-1, V!0)) = 0)) <=> in(V!0, add(3, remove(3, U!1))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!0, remove(3, U!1)) | (V!0 = 3)) <=> in(V!0, add(3, remove(3, U!1)))))),
% 0.22/0.44 inference(monotonicity,[status(thm)],[127])).
% 0.22/0.44 tff(129,plain,
% 0.22/0.44 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!0, remove(3, U!1)) | ($sum(3, $product(-1, V!0)) = 0)) <=> in(V!0, add(3, remove(3, U!1))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!0, remove(3, U!1)) | (V!0 = 3)) <=> in(V!0, add(3, remove(3, U!1)))))),
% 0.22/0.44 inference(transitivity,[status(thm)],[128, 126])).
% 0.22/0.44 tff(130,plain,
% 0.22/0.44 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!0, remove(3, U!1)) | ($sum(3, $product(-1, V!0)) = 0)) <=> in(V!0, add(3, remove(3, U!1))))),
% 0.22/0.44 inference(quant_inst,[status(thm)],[])).
% 0.22/0.44 tff(131,plain,
% 0.22/0.44 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!0, remove(3, U!1)) | (V!0 = 3)) <=> in(V!0, add(3, remove(3, U!1))))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[130, 129])).
% 0.22/0.44 tff(132,plain,
% 0.22/0.44 ((in(V!0, remove(3, U!1)) | (V!0 = 3)) <=> in(V!0, add(3, remove(3, U!1)))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[131, 70])).
% 0.22/0.44 tff(133,plain,
% 0.22/0.44 (in(V!0, add(3, remove(3, U!1))) <=> in(V!0, U!1)),
% 0.22/0.44 inference(monotonicity,[status(thm)],[92])).
% 0.22/0.44 tff(134,plain,
% 0.22/0.44 (in(V!0, U!1) <=> in(V!0, add(3, remove(3, U!1)))),
% 0.22/0.44 inference(symmetry,[status(thm)],[133])).
% 0.22/0.44 tff(135,plain,
% 0.22/0.44 (in(V!0, U!1)),
% 0.22/0.44 inference(or_elim,[status(thm)],[38])).
% 0.22/0.44 tff(136,plain,
% 0.22/0.44 (in(V!0, add(3, remove(3, U!1)))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[135, 134])).
% 0.22/0.44 tff(137,plain,
% 0.22/0.44 ((~((in(V!0, remove(3, U!1)) | (V!0 = 3)) <=> in(V!0, add(3, remove(3, U!1))))) | (in(V!0, remove(3, U!1)) | (V!0 = 3)) | (~in(V!0, add(3, remove(3, U!1))))),
% 0.22/0.44 inference(tautology,[status(thm)],[])).
% 0.22/0.44 tff(138,plain,
% 0.22/0.44 (in(V!0, remove(3, U!1)) | (V!0 = 3)),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[137, 136, 132])).
% 0.22/0.44 tff(139,plain,
% 0.22/0.44 (~$lesseq(V!0, 3)),
% 0.22/0.44 inference(and_elim,[status(thm)],[39])).
% 0.22/0.44 tff(140,plain,
% 0.22/0.44 ((~(V!0 = 3)) | $lesseq(V!0, 3)),
% 0.22/0.44 inference(theory_lemma,[status(thm)],[])).
% 0.22/0.44 tff(141,plain,
% 0.22/0.44 (~(V!0 = 3)),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[140, 139])).
% 0.22/0.44 tff(142,plain,
% 0.22/0.44 ((~(in(V!0, remove(3, U!1)) | (V!0 = 3))) | in(V!0, remove(3, U!1)) | (V!0 = 3)),
% 0.22/0.44 inference(tautology,[status(thm)],[])).
% 0.22/0.44 tff(143,plain,
% 0.22/0.44 ((~(in(V!0, remove(3, U!1)) | (V!0 = 3))) | in(V!0, remove(3, U!1))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[142, 141])).
% 0.22/0.44 tff(144,plain,
% 0.22/0.44 (in(V!0, remove(3, U!1))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[143, 138])).
% 0.22/0.44 tff(145,plain,
% 0.22/0.44 (in(V!0, add(2, remove(2, remove(3, U!1))))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[144, 125])).
% 0.22/0.44 tff(146,plain,
% 0.22/0.44 ((~(((V!0 = 2) | in(V!0, remove(2, remove(3, U!1)))) <=> in(V!0, add(2, remove(2, remove(3, U!1)))))) | ((V!0 = 2) | in(V!0, remove(2, remove(3, U!1)))) | (~in(V!0, add(2, remove(2, remove(3, U!1)))))),
% 0.22/0.44 inference(tautology,[status(thm)],[])).
% 0.22/0.44 tff(147,plain,
% 0.22/0.44 ((V!0 = 2) | in(V!0, remove(2, remove(3, U!1)))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[146, 145, 117])).
% 0.22/0.44 tff(148,plain,
% 0.22/0.44 ($greatereq(V!0, 3) | $lesseq(V!0, 3)),
% 0.22/0.44 inference(theory_lemma,[status(thm)],[])).
% 0.22/0.44 tff(149,plain,
% 0.22/0.44 ($greatereq(V!0, 3)),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[148, 139])).
% 0.22/0.44 tff(150,plain,
% 0.22/0.44 ((~$lesseq(V!0, 2)) | (~$greatereq(V!0, 3))),
% 0.22/0.44 inference(theory_lemma,[status(thm)],[])).
% 0.22/0.44 tff(151,plain,
% 0.22/0.44 (~$lesseq(V!0, 2)),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[150, 149])).
% 0.22/0.44 tff(152,plain,
% 0.22/0.44 ((~(V!0 = 2)) | $lesseq(V!0, 2)),
% 0.22/0.44 inference(theory_lemma,[status(thm)],[])).
% 0.22/0.44 tff(153,plain,
% 0.22/0.44 (~(V!0 = 2)),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[152, 151])).
% 0.22/0.44 tff(154,plain,
% 0.22/0.44 ((~((V!0 = 2) | in(V!0, remove(2, remove(3, U!1))))) | (V!0 = 2) | in(V!0, remove(2, remove(3, U!1)))),
% 0.22/0.44 inference(tautology,[status(thm)],[])).
% 0.22/0.44 tff(155,plain,
% 0.22/0.44 ((~((V!0 = 2) | in(V!0, remove(2, remove(3, U!1))))) | in(V!0, remove(2, remove(3, U!1)))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[154, 153])).
% 0.22/0.44 tff(156,plain,
% 0.22/0.44 (in(V!0, remove(2, remove(3, U!1)))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[155, 147])).
% 0.22/0.44 tff(157,plain,
% 0.22/0.44 (in(V!0, empty)),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[156, 110])).
% 0.22/0.44 tff(158,plain,
% 0.22/0.44 (^[U: $int] : refl((~in(U, empty)) <=> (~in(U, empty)))),
% 0.22/0.44 inference(bind,[status(th)],[])).
% 0.22/0.44 tff(159,plain,
% 0.22/0.44 (![U: $int] : (~in(U, empty)) <=> ![U: $int] : (~in(U, empty))),
% 0.22/0.44 inference(quant_intro,[status(thm)],[158])).
% 0.22/0.44 tff(160,plain,
% 0.22/0.44 (![U: $int] : (~in(U, empty)) <=> ![U: $int] : (~in(U, empty))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(161,axiom,(![U: $int] : (~in(U, empty))), file('/export/starexec/sandbox/benchmark/Axioms/DAT002=0.ax','ax1')).
% 0.22/0.44 tff(162,plain,
% 0.22/0.44 (![U: $int] : (~in(U, empty))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[161, 160])).
% 0.22/0.44 tff(163,plain,(
% 0.22/0.44 ![U: $int] : (~in(U, empty))),
% 0.22/0.44 inference(skolemize,[status(sab)],[162])).
% 0.22/0.44 tff(164,plain,
% 0.22/0.44 (![U: $int] : (~in(U, empty))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[163, 159])).
% 0.22/0.44 tff(165,plain,
% 0.22/0.44 ((~![U: $int] : (~in(U, empty))) | (~in(V!0, empty))),
% 0.22/0.44 inference(quant_inst,[status(thm)],[])).
% 0.22/0.44 tff(166,plain,
% 0.22/0.44 (~in(V!0, empty)),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[165, 164])).
% 0.22/0.44 tff(167,plain,
% 0.22/0.44 ($false),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[166, 157])).
% 0.22/0.44 % SZS output end Proof
%------------------------------------------------------------------------------