TSTP Solution File: DAT035_1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : DAT035_1 : TPTP v8.1.0. Released v5.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n006.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 14:36:27 EDT 2022
% Result : Theorem 0.13s 0.40s
% Output : Proof 0.13s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : DAT035_1 : TPTP v8.1.0. Released v5.0.0.
% 0.12/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n006.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 : Wed Aug 31 01:43:40 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.
% 0.13/0.40 % SZS status Theorem
% 0.13/0.40 % SZS output start Proof
% 0.13/0.40 tff(count_type, type, (
% 0.13/0.40 count: collection > $int)).
% 0.13/0.40 tff(tptp_fun_U_2_type, type, (
% 0.13/0.40 tptp_fun_U_2: collection)).
% 0.13/0.40 tff(add_type, type, (
% 0.13/0.40 add: ( $int * collection ) > collection)).
% 0.13/0.40 tff(tptp_fun_V_1_type, type, (
% 0.13/0.40 tptp_fun_V_1: $int)).
% 0.13/0.40 tff(in_type, type, (
% 0.13/0.40 in: ( $int * collection ) > $o)).
% 0.13/0.40 tff(tptp_fun_W_0_type, type, (
% 0.13/0.40 tptp_fun_W_0: $int)).
% 0.13/0.40 tff(1,plain,
% 0.13/0.40 (^[X8: $int, X9: collection] : refl(((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1)) <=> ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1)))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(2,plain,
% 0.13/0.40 (![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1)) <=> ![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[1])).
% 0.13/0.40 tff(3,plain,
% 0.13/0.40 (^[X8: $int, X9: collection] : rewrite(((~in(X8, X9)) <=> ($sum(count(add(X8, X9)), $product(-1, count(X9))) = 1)) <=> ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1)))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(4,plain,
% 0.13/0.40 (![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(add(X8, X9)), $product(-1, count(X9))) = 1)) <=> ![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[3])).
% 0.13/0.40 tff(5,plain,
% 0.13/0.40 (^[X8: $int, X9: collection] : rewrite(((~in(X8, X9)) <=> (count(add(X8, X9)) = $sum(1, count(X9)))) <=> ((~in(X8, X9)) <=> ($sum(count(add(X8, X9)), $product(-1, count(X9))) = 1)))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(6,plain,
% 0.13/0.40 (![X8: $int, X9: collection] : ((~in(X8, X9)) <=> (count(add(X8, X9)) = $sum(1, count(X9)))) <=> ![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(add(X8, X9)), $product(-1, count(X9))) = 1))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[5])).
% 0.13/0.40 tff(7,plain,
% 0.13/0.40 (![X8: $int, X9: collection] : ((~in(X8, X9)) <=> (count(add(X8, X9)) = $sum(1, count(X9)))) <=> ![X8: $int, X9: collection] : ((~in(X8, X9)) <=> (count(add(X8, X9)) = $sum(1, count(X9))))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(8,plain,
% 0.13/0.40 (^[X8: $int, X9: collection] : rewrite(((~in(X8, X9)) <=> (count(add(X8, X9)) = $sum(count(X9), 1))) <=> ((~in(X8, X9)) <=> (count(add(X8, X9)) = $sum(1, count(X9)))))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(9,plain,
% 0.13/0.40 (![X8: $int, X9: collection] : ((~in(X8, X9)) <=> (count(add(X8, X9)) = $sum(count(X9), 1))) <=> ![X8: $int, X9: collection] : ((~in(X8, X9)) <=> (count(add(X8, X9)) = $sum(1, count(X9))))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[8])).
% 0.13/0.40 tff(10,axiom,(![X8: $int, X9: collection] : ((~in(X8, X9)) <=> (count(add(X8, X9)) = $sum(count(X9), 1)))), file('/export/starexec/sandbox/benchmark/Axioms/DAT002=1.ax','ax3')).
% 0.13/0.40 tff(11,plain,
% 0.13/0.40 (![X8: $int, X9: collection] : ((~in(X8, X9)) <=> (count(add(X8, X9)) = $sum(1, count(X9))))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[10, 9])).
% 0.13/0.40 tff(12,plain,
% 0.13/0.40 (![X8: $int, X9: collection] : ((~in(X8, X9)) <=> (count(add(X8, X9)) = $sum(1, count(X9))))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[11, 7])).
% 0.13/0.40 tff(13,plain,
% 0.13/0.40 (![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(add(X8, X9)), $product(-1, count(X9))) = 1))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[12, 6])).
% 0.13/0.40 tff(14,plain,
% 0.13/0.40 (![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[13, 4])).
% 0.13/0.40 tff(15,plain,(
% 0.13/0.40 ![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))),
% 0.13/0.40 inference(skolemize,[status(sab)],[14])).
% 0.13/0.40 tff(16,plain,
% 0.13/0.40 (![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[15, 2])).
% 0.13/0.40 tff(17,plain,
% 0.13/0.40 (((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(V!1, U!2)) <=> ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 1))) <=> ((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(V!1, U!2)) <=> ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 1)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(18,plain,
% 0.13/0.40 (((~in(V!1, U!2)) <=> ($sum(count(U!2), $product(-1, count(add(V!1, U!2)))) = -1)) <=> ((~in(V!1, U!2)) <=> ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 1))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(19,plain,
% 0.13/0.40 (((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(V!1, U!2)) <=> ($sum(count(U!2), $product(-1, count(add(V!1, U!2)))) = -1))) <=> ((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(V!1, U!2)) <=> ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 1)))),
% 0.13/0.40 inference(monotonicity,[status(thm)],[18])).
% 0.13/0.40 tff(20,plain,
% 0.13/0.40 (((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(V!1, U!2)) <=> ($sum(count(U!2), $product(-1, count(add(V!1, U!2)))) = -1))) <=> ((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(V!1, U!2)) <=> ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 1)))),
% 0.13/0.40 inference(transitivity,[status(thm)],[19, 17])).
% 0.13/0.40 tff(21,plain,
% 0.13/0.40 ((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(V!1, U!2)) <=> ($sum(count(U!2), $product(-1, count(add(V!1, U!2)))) = -1))),
% 0.13/0.40 inference(quant_inst,[status(thm)],[])).
% 0.13/0.40 tff(22,plain,
% 0.13/0.40 ((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(V!1, U!2)) <=> ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 1))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.13/0.40 tff(23,plain,
% 0.13/0.40 ((~in(V!1, U!2)) <=> ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 1)),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[22, 16])).
% 0.13/0.40 tff(24,plain,
% 0.13/0.40 ((~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(W, $sum($product(-1, count(add(V, U))), count(U))), 0))) <=> (~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(W, $sum($product(-1, count(add(V, U))), count(U))), 0)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(25,plain,
% 0.13/0.40 ((~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(count(U), $sum(W, $product(-1, count(add(V, U))))), 0))) <=> (~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(W, $sum($product(-1, count(add(V, U))), count(U))), 0)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(26,plain,
% 0.13/0.40 ((~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(count(U), W), count(add(V, U))))) <=> (~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(count(U), $sum(W, $product(-1, count(add(V, U))))), 0)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(27,plain,
% 0.13/0.40 ((~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(count(U), W), count(add(V, U))))) <=> (~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(count(U), W), count(add(V, U)))))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(28,plain,
% 0.13/0.40 ((~![U: collection, V: $int, W: $int] : ($greater(W, 0) => $greatereq($sum(count(U), W), count(add(V, U))))) <=> (~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(count(U), W), count(add(V, U)))))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(29,axiom,(~![U: collection, V: $int, W: $int] : ($greater(W, 0) => $greatereq($sum(count(U), W), count(add(V, U))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','co1')).
% 0.13/0.40 tff(30,plain,
% 0.13/0.40 (~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(count(U), W), count(add(V, U))))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[29, 28])).
% 0.13/0.40 tff(31,plain,
% 0.13/0.40 (~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(count(U), W), count(add(V, U))))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[30, 27])).
% 0.13/0.40 tff(32,plain,
% 0.13/0.40 (~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(count(U), W), count(add(V, U))))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[31, 27])).
% 0.13/0.40 tff(33,plain,
% 0.13/0.40 (~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(count(U), W), count(add(V, U))))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[32, 27])).
% 0.13/0.40 tff(34,plain,
% 0.13/0.40 (~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(count(U), $sum(W, $product(-1, count(add(V, U))))), 0))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[33, 26])).
% 0.13/0.40 tff(35,plain,
% 0.13/0.40 (~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(W, $sum($product(-1, count(add(V, U))), count(U))), 0))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[34, 25])).
% 0.13/0.40 tff(36,plain,
% 0.13/0.40 (~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(W, $sum($product(-1, count(add(V, U))), count(U))), 0))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[35, 24])).
% 0.13/0.40 tff(37,plain,
% 0.13/0.40 (~![U: collection, V: $int, W: $int] : ($lesseq(W, 0) | $greatereq($sum(W, $sum($product(-1, count(add(V, U))), count(U))), 0))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[36, 24])).
% 0.13/0.40 tff(38,plain,(
% 0.13/0.40 ~($lesseq(W!0, 0) | $greatereq($sum(W!0, $sum($product(-1, count(add(V!1, U!2))), count(U!2))), 0))),
% 0.13/0.40 inference(skolemize,[status(sab)],[37])).
% 0.13/0.40 tff(39,plain,
% 0.13/0.40 (~$lesseq(W!0, 0)),
% 0.13/0.40 inference(or_elim,[status(thm)],[38])).
% 0.13/0.40 tff(40,plain,
% 0.13/0.40 (~$greatereq($sum(W!0, $sum($product(-1, count(add(V!1, U!2))), count(U!2))), 0)),
% 0.13/0.40 inference(or_elim,[status(thm)],[38])).
% 0.13/0.40 tff(41,plain,
% 0.13/0.40 ((~$lesseq($sum(count(add(V!1, U!2)), $product(-1, count(U!2))), 1)) | $lesseq(W!0, 0) | $greatereq($sum(W!0, $sum($product(-1, count(add(V!1, U!2))), count(U!2))), 0)),
% 0.13/0.40 inference(theory_lemma,[status(thm)],[])).
% 0.13/0.40 tff(42,plain,
% 0.13/0.40 (~$lesseq($sum(count(add(V!1, U!2)), $product(-1, count(U!2))), 1)),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[41, 40, 39])).
% 0.13/0.40 tff(43,plain,
% 0.13/0.40 ((~($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 1)) | $lesseq($sum(count(add(V!1, U!2)), $product(-1, count(U!2))), 1)),
% 0.13/0.40 inference(theory_lemma,[status(thm)],[])).
% 0.13/0.40 tff(44,plain,
% 0.13/0.40 (~($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 1)),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[43, 42])).
% 0.13/0.40 tff(45,plain,
% 0.13/0.40 (^[X10: $int, X11: collection] : refl((in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0)) <=> (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0)))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(46,plain,
% 0.13/0.40 (![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0)) <=> ![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[45])).
% 0.13/0.40 tff(47,plain,
% 0.13/0.40 (^[X10: $int, X11: collection] : rewrite((in(X10, X11) <=> ($sum(count(add(X10, X11)), $product(-1, count(X11))) = 0)) <=> (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0)))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(48,plain,
% 0.13/0.40 (![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(add(X10, X11)), $product(-1, count(X11))) = 0)) <=> ![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[47])).
% 0.13/0.40 tff(49,plain,
% 0.13/0.40 (^[X10: $int, X11: collection] : rewrite((in(X10, X11) <=> (count(add(X10, X11)) = count(X11))) <=> (in(X10, X11) <=> ($sum(count(add(X10, X11)), $product(-1, count(X11))) = 0)))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(50,plain,
% 0.13/0.40 (![X10: $int, X11: collection] : (in(X10, X11) <=> (count(add(X10, X11)) = count(X11))) <=> ![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(add(X10, X11)), $product(-1, count(X11))) = 0))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[49])).
% 0.13/0.40 tff(51,plain,
% 0.13/0.40 (![X10: $int, X11: collection] : (in(X10, X11) <=> (count(add(X10, X11)) = count(X11))) <=> ![X10: $int, X11: collection] : (in(X10, X11) <=> (count(add(X10, X11)) = count(X11)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(52,axiom,(![X10: $int, X11: collection] : (in(X10, X11) <=> (count(add(X10, X11)) = count(X11)))), file('/export/starexec/sandbox/benchmark/Axioms/DAT002=1.ax','ax4')).
% 0.13/0.40 tff(53,plain,
% 0.13/0.40 (![X10: $int, X11: collection] : (in(X10, X11) <=> (count(add(X10, X11)) = count(X11)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.13/0.40 tff(54,plain,
% 0.13/0.40 (![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(add(X10, X11)), $product(-1, count(X11))) = 0))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[53, 50])).
% 0.13/0.40 tff(55,plain,
% 0.13/0.40 (![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[54, 48])).
% 0.13/0.40 tff(56,plain,(
% 0.13/0.40 ![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))),
% 0.13/0.40 inference(skolemize,[status(sab)],[55])).
% 0.13/0.40 tff(57,plain,
% 0.13/0.40 (![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[56, 46])).
% 0.13/0.40 tff(58,plain,
% 0.13/0.40 (((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(V!1, U!2) <=> ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 0))) <=> ((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(V!1, U!2) <=> ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 0)))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(59,plain,
% 0.13/0.40 ((in(V!1, U!2) <=> ($sum(count(U!2), $product(-1, count(add(V!1, U!2)))) = 0)) <=> (in(V!1, U!2) <=> ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 0))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(60,plain,
% 0.13/0.40 (((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(V!1, U!2) <=> ($sum(count(U!2), $product(-1, count(add(V!1, U!2)))) = 0))) <=> ((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(V!1, U!2) <=> ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 0)))),
% 0.13/0.40 inference(monotonicity,[status(thm)],[59])).
% 0.13/0.40 tff(61,plain,
% 0.13/0.40 (((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(V!1, U!2) <=> ($sum(count(U!2), $product(-1, count(add(V!1, U!2)))) = 0))) <=> ((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(V!1, U!2) <=> ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 0)))),
% 0.13/0.40 inference(transitivity,[status(thm)],[60, 58])).
% 0.13/0.40 tff(62,plain,
% 0.13/0.40 ((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(V!1, U!2) <=> ($sum(count(U!2), $product(-1, count(add(V!1, U!2)))) = 0))),
% 0.13/0.40 inference(quant_inst,[status(thm)],[])).
% 0.13/0.40 tff(63,plain,
% 0.13/0.40 ((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(V!1, U!2) <=> ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 0))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.13/0.40 tff(64,plain,
% 0.13/0.40 (in(V!1, U!2) <=> ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 0)),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[63, 57])).
% 0.13/0.40 tff(65,plain,
% 0.13/0.40 ((~$lesseq($sum(count(add(V!1, U!2)), $product(-1, count(U!2))), 0)) | $lesseq(W!0, 0) | $greatereq($sum(W!0, $sum($product(-1, count(add(V!1, U!2))), count(U!2))), 0)),
% 0.13/0.40 inference(theory_lemma,[status(thm)],[])).
% 0.13/0.40 tff(66,plain,
% 0.13/0.40 (~$lesseq($sum(count(add(V!1, U!2)), $product(-1, count(U!2))), 0)),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[65, 40, 39])).
% 0.13/0.40 tff(67,plain,
% 0.13/0.40 ((~($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 0)) | $lesseq($sum(count(add(V!1, U!2)), $product(-1, count(U!2))), 0)),
% 0.13/0.40 inference(theory_lemma,[status(thm)],[])).
% 0.13/0.40 tff(68,plain,
% 0.13/0.40 (~($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 0)),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[67, 66])).
% 0.13/0.41 tff(69,plain,
% 0.13/0.41 ((~(in(V!1, U!2) <=> ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 0))) | (~in(V!1, U!2)) | ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 0)),
% 0.13/0.41 inference(tautology,[status(thm)],[])).
% 0.13/0.41 tff(70,plain,
% 0.13/0.41 (~in(V!1, U!2)),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[69, 68, 64])).
% 0.13/0.41 tff(71,plain,
% 0.13/0.41 ((~((~in(V!1, U!2)) <=> ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 1))) | in(V!1, U!2) | ($sum(count(add(V!1, U!2)), $product(-1, count(U!2))) = 1)),
% 0.13/0.41 inference(tautology,[status(thm)],[])).
% 0.13/0.41 tff(72,plain,
% 0.13/0.41 ($false),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[71, 70, 44, 23])).
% 0.13/0.41 % SZS output end Proof
%------------------------------------------------------------------------------