TSTP Solution File: DAT050_1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : DAT050_1 : TPTP v8.1.0. Released v5.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n023.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 14:36:31 EDT 2022
% Result : Theorem 0.20s 0.50s
% Output : Proof 0.49s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : DAT050_1 : TPTP v8.1.0. Released v5.0.0.
% 0.03/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n023.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 02:06:50 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.
% 0.20/0.50 % SZS status Theorem
% 0.20/0.50 % SZS output start Proof
% 0.20/0.50 tff(count_type, type, (
% 0.20/0.50 count: collection > $int)).
% 0.20/0.50 tff(tptp_fun_U_1_type, type, (
% 0.20/0.50 tptp_fun_U_1: collection)).
% 0.20/0.50 tff(remove_type, type, (
% 0.20/0.50 remove: ( $int * collection ) > collection)).
% 0.20/0.50 tff(tptp_fun_V_0_type, type, (
% 0.20/0.50 tptp_fun_V_0: $int)).
% 0.20/0.50 tff(in_type, type, (
% 0.20/0.50 in: ( $int * collection ) > $o)).
% 0.20/0.50 tff(add_type, type, (
% 0.20/0.50 add: ( $int * collection ) > collection)).
% 0.20/0.50 tff(1,plain,
% 0.20/0.50 (^[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)))),
% 0.20/0.50 inference(bind,[status(th)],[])).
% 0.20/0.50 tff(2,plain,
% 0.20/0.50 (![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))),
% 0.20/0.50 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.50 tff(3,plain,
% 0.20/0.50 (^[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)))),
% 0.20/0.50 inference(bind,[status(th)],[])).
% 0.20/0.50 tff(4,plain,
% 0.20/0.50 (![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))),
% 0.20/0.50 inference(quant_intro,[status(thm)],[3])).
% 0.20/0.50 tff(5,plain,
% 0.20/0.50 (^[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)))),
% 0.20/0.50 inference(bind,[status(th)],[])).
% 0.20/0.50 tff(6,plain,
% 0.20/0.50 (![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))),
% 0.20/0.50 inference(quant_intro,[status(thm)],[5])).
% 0.20/0.50 tff(7,plain,
% 0.20/0.50 (![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))))),
% 0.20/0.50 inference(rewrite,[status(thm)],[])).
% 0.20/0.50 tff(8,plain,
% 0.20/0.50 (^[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)))))),
% 0.20/0.50 inference(bind,[status(th)],[])).
% 0.20/0.50 tff(9,plain,
% 0.20/0.50 (![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))))),
% 0.20/0.50 inference(quant_intro,[status(thm)],[8])).
% 0.20/0.50 tff(10,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')).
% 0.20/0.50 tff(11,plain,
% 0.20/0.50 (![X12: $int, X13: collection] : (in(X12, X13) <=> (count(remove(X12, X13)) = $sum(-1, count(X13))))),
% 0.20/0.50 inference(modus_ponens,[status(thm)],[10, 9])).
% 0.20/0.50 tff(12,plain,
% 0.20/0.50 (![X12: $int, X13: collection] : (in(X12, X13) <=> (count(remove(X12, X13)) = $sum(-1, count(X13))))),
% 0.20/0.50 inference(modus_ponens,[status(thm)],[11, 7])).
% 0.20/0.50 tff(13,plain,
% 0.20/0.50 (![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(remove(X12, X13)), $product(-1, count(X13))) = -1))),
% 0.20/0.50 inference(modus_ponens,[status(thm)],[12, 6])).
% 0.20/0.50 tff(14,plain,
% 0.20/0.50 (![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))),
% 0.20/0.50 inference(modus_ponens,[status(thm)],[13, 4])).
% 0.20/0.50 tff(15,plain,(
% 0.20/0.50 ![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))),
% 0.20/0.50 inference(skolemize,[status(sab)],[14])).
% 0.20/0.50 tff(16,plain,
% 0.20/0.50 (![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))),
% 0.20/0.50 inference(modus_ponens,[status(thm)],[15, 2])).
% 0.20/0.50 tff(17,plain,
% 0.20/0.50 ((~![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))) | (in(V!0, add(V!0, U!1)) <=> ($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1))),
% 0.20/0.50 inference(quant_inst,[status(thm)],[])).
% 0.20/0.50 tff(18,plain,
% 0.20/0.50 (in(V!0, add(V!0, U!1)) <=> ($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1)),
% 0.20/0.50 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.20/0.50 tff(19,plain,
% 0.20/0.50 (^[V: $int, W: collection] : refl(in(V, add(V, W)) <=> in(V, add(V, W)))),
% 0.20/0.50 inference(bind,[status(th)],[])).
% 0.20/0.50 tff(20,plain,
% 0.20/0.50 (![V: $int, W: collection] : in(V, add(V, W)) <=> ![V: $int, W: collection] : in(V, add(V, W))),
% 0.20/0.50 inference(quant_intro,[status(thm)],[19])).
% 0.20/0.50 tff(21,plain,
% 0.20/0.50 (![V: $int, W: collection] : in(V, add(V, W)) <=> ![V: $int, W: collection] : in(V, add(V, W))),
% 0.20/0.50 inference(rewrite,[status(thm)],[])).
% 0.20/0.50 tff(22,axiom,(![V: $int, W: collection] : in(V, add(V, W))), file('/export/starexec/sandbox/benchmark/Axioms/DAT002=0.ax','ax2')).
% 0.20/0.50 tff(23,plain,
% 0.20/0.50 (![V: $int, W: collection] : in(V, add(V, W))),
% 0.20/0.50 inference(modus_ponens,[status(thm)],[22, 21])).
% 0.20/0.50 tff(24,plain,(
% 0.20/0.50 ![V: $int, W: collection] : in(V, add(V, W))),
% 0.20/0.50 inference(skolemize,[status(sab)],[23])).
% 0.20/0.50 tff(25,plain,
% 0.20/0.50 (![V: $int, W: collection] : in(V, add(V, W))),
% 0.20/0.50 inference(modus_ponens,[status(thm)],[24, 20])).
% 0.20/0.50 tff(26,plain,
% 0.20/0.50 ((~![V: $int, W: collection] : in(V, add(V, W))) | in(V!0, add(V!0, U!1))),
% 0.20/0.50 inference(quant_inst,[status(thm)],[])).
% 0.20/0.50 tff(27,plain,
% 0.20/0.50 (in(V!0, add(V!0, U!1))),
% 0.20/0.50 inference(unit_resolution,[status(thm)],[26, 25])).
% 0.20/0.50 tff(28,plain,
% 0.20/0.50 ((~(in(V!0, add(V!0, U!1)) <=> ($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1))) | (~in(V!0, add(V!0, U!1))) | ($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1)),
% 0.20/0.50 inference(tautology,[status(thm)],[])).
% 0.20/0.50 tff(29,plain,
% 0.20/0.50 ((~(in(V!0, add(V!0, U!1)) <=> ($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1))) | ($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1)),
% 0.20/0.50 inference(unit_resolution,[status(thm)],[28, 27])).
% 0.20/0.50 tff(30,plain,
% 0.20/0.50 ($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1),
% 0.20/0.50 inference(unit_resolution,[status(thm)],[29, 18])).
% 0.20/0.50 tff(31,plain,
% 0.20/0.50 ((~($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1)) | $lesseq($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))), 1)),
% 0.20/0.50 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.50 tff(32,plain,
% 0.20/0.50 ($lesseq($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))), 1)),
% 0.20/0.50 inference(unit_resolution,[status(thm)],[31, 30])).
% 0.20/0.50 tff(33,plain,
% 0.20/0.50 (^[X16: $int, X17: collection] : refl(((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17)))) <=> ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17)))))),
% 0.20/0.50 inference(bind,[status(th)],[])).
% 0.20/0.50 tff(34,plain,
% 0.20/0.50 (![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))))),
% 0.20/0.50 inference(quant_intro,[status(thm)],[33])).
% 0.20/0.50 tff(35,plain,
% 0.20/0.50 (![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))))),
% 0.20/0.50 inference(rewrite,[status(thm)],[])).
% 0.20/0.50 tff(36,plain,
% 0.20/0.50 (^[X16: $int, X17: collection] : rewrite((in(X16, X17) => (X17 = add(X16, remove(X16, X17)))) <=> ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17)))))),
% 0.20/0.50 inference(bind,[status(th)],[])).
% 0.20/0.50 tff(37,plain,
% 0.20/0.50 (![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))))),
% 0.20/0.50 inference(quant_intro,[status(thm)],[36])).
% 0.20/0.50 tff(38,axiom,(![X16: $int, X17: collection] : (in(X16, X17) => (X17 = add(X16, remove(X16, X17))))), file('/export/starexec/sandbox/benchmark/Axioms/DAT002=1.ax','ax7')).
% 0.20/0.50 tff(39,plain,
% 0.20/0.50 (![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))),
% 0.20/0.50 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.20/0.50 tff(40,plain,
% 0.20/0.50 (![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))),
% 0.20/0.50 inference(modus_ponens,[status(thm)],[39, 35])).
% 0.20/0.50 tff(41,plain,(
% 0.20/0.50 ![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))),
% 0.20/0.50 inference(skolemize,[status(sab)],[40])).
% 0.20/0.50 tff(42,plain,
% 0.20/0.50 (![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))),
% 0.20/0.50 inference(modus_ponens,[status(thm)],[41, 34])).
% 0.20/0.50 tff(43,plain,
% 0.20/0.50 (((~![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))) | ((~in(V!0, add(V!0, U!1))) | (add(V!0, U!1) = add(V!0, remove(V!0, add(V!0, U!1)))))) <=> ((~![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))) | (~in(V!0, add(V!0, U!1))) | (add(V!0, U!1) = add(V!0, remove(V!0, add(V!0, U!1)))))),
% 0.20/0.50 inference(rewrite,[status(thm)],[])).
% 0.20/0.50 tff(44,plain,
% 0.20/0.50 ((~![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))) | ((~in(V!0, add(V!0, U!1))) | (add(V!0, U!1) = add(V!0, remove(V!0, add(V!0, U!1)))))),
% 0.20/0.50 inference(quant_inst,[status(thm)],[])).
% 0.20/0.50 tff(45,plain,
% 0.20/0.50 ((~![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))) | (~in(V!0, add(V!0, U!1))) | (add(V!0, U!1) = add(V!0, remove(V!0, add(V!0, U!1))))),
% 0.20/0.50 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.20/0.50 tff(46,plain,
% 0.20/0.50 (add(V!0, U!1) = add(V!0, remove(V!0, add(V!0, U!1)))),
% 0.20/0.50 inference(unit_resolution,[status(thm)],[45, 42, 27])).
% 0.20/0.50 tff(47,plain,
% 0.20/0.50 (add(V!0, remove(V!0, add(V!0, U!1))) = add(V!0, U!1)),
% 0.20/0.50 inference(symmetry,[status(thm)],[46])).
% 0.20/0.50 tff(48,plain,
% 0.20/0.50 (remove(V!0, add(V!0, remove(V!0, add(V!0, U!1)))) = remove(V!0, add(V!0, U!1))),
% 0.20/0.50 inference(monotonicity,[status(thm)],[47])).
% 0.20/0.50 tff(49,plain,
% 0.20/0.50 (in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1))))) <=> in(0, remove(V!0, add(V!0, U!1)))),
% 0.20/0.50 inference(monotonicity,[status(thm)],[48])).
% 0.20/0.50 tff(50,plain,
% 0.20/0.50 ((~in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1)))))) <=> (~in(0, remove(V!0, add(V!0, U!1))))),
% 0.20/0.50 inference(monotonicity,[status(thm)],[49])).
% 0.20/0.50 tff(51,plain,
% 0.20/0.50 (in(0, add(V!0, U!1)) <=> in(0, add(V!0, remove(V!0, add(V!0, U!1))))),
% 0.20/0.50 inference(monotonicity,[status(thm)],[46])).
% 0.20/0.50 tff(52,plain,
% 0.20/0.50 (in(0, add(V!0, remove(V!0, add(V!0, U!1)))) <=> in(0, add(V!0, U!1))),
% 0.20/0.50 inference(symmetry,[status(thm)],[51])).
% 0.20/0.50 tff(53,assumption,(~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0))), introduced(assumption)).
% 0.20/0.50 tff(54,plain,
% 0.20/0.50 (((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0)) | in(0, add(V!0, remove(V!0, add(V!0, U!1))))),
% 0.20/0.50 inference(tautology,[status(thm)],[])).
% 0.20/0.50 tff(55,plain,
% 0.20/0.50 (in(0, add(V!0, remove(V!0, add(V!0, U!1))))),
% 0.20/0.50 inference(unit_resolution,[status(thm)],[54, 53])).
% 0.20/0.50 tff(56,plain,
% 0.20/0.50 (in(0, add(V!0, U!1))),
% 0.20/0.50 inference(modus_ponens,[status(thm)],[55, 52])).
% 0.20/0.50 tff(57,plain,
% 0.20/0.50 (^[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))))),
% 0.20/0.50 inference(bind,[status(th)],[])).
% 0.20/0.50 tff(58,plain,
% 0.20/0.50 (![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)))),
% 0.20/0.50 inference(quant_intro,[status(thm)],[57])).
% 0.20/0.50 tff(59,plain,
% 0.20/0.50 (^[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))))),
% 0.20/0.50 inference(bind,[status(th)],[])).
% 0.20/0.50 tff(60,plain,
% 0.20/0.50 (![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)))),
% 0.20/0.51 inference(quant_intro,[status(thm)],[59])).
% 0.20/0.51 tff(61,plain,
% 0.20/0.51 (^[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))))),
% 0.20/0.51 inference(bind,[status(th)],[])).
% 0.20/0.51 tff(62,plain,
% 0.20/0.51 (![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)))),
% 0.20/0.51 inference(quant_intro,[status(thm)],[61])).
% 0.20/0.51 tff(63,plain,
% 0.20/0.51 (![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)))),
% 0.20/0.51 inference(rewrite,[status(thm)],[])).
% 0.20/0.51 tff(64,plain,
% 0.20/0.51 (^[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))))),
% 0.20/0.51 inference(bind,[status(th)],[])).
% 0.20/0.51 tff(65,plain,
% 0.20/0.51 (![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)))),
% 0.20/0.51 inference(quant_intro,[status(thm)],[64])).
% 0.20/0.51 tff(66,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')).
% 0.20/0.51 tff(67,plain,
% 0.20/0.51 (![Z: $int, X1: collection, X2: $int] : (((Z = X2) | in(Z, X1)) <=> in(Z, add(X2, X1)))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[66, 65])).
% 0.20/0.51 tff(68,plain,
% 0.20/0.51 (![Z: $int, X1: collection, X2: $int] : (((Z = X2) | in(Z, X1)) <=> in(Z, add(X2, X1)))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[67, 63])).
% 0.20/0.51 tff(69,plain,
% 0.20/0.51 (![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(Z, $product(-1, X2)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[68, 62])).
% 0.20/0.51 tff(70,plain,
% 0.20/0.51 (![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[69, 60])).
% 0.20/0.51 tff(71,plain,(
% 0.20/0.51 ![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.20/0.51 inference(skolemize,[status(sab)],[70])).
% 0.20/0.51 tff(72,plain,
% 0.20/0.51 (![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[71, 58])).
% 0.20/0.51 tff(73,plain,
% 0.20/0.51 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(0, U!1) | (V!0 = 0)) <=> in(0, add(V!0, U!1)))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(0, U!1) | (V!0 = 0)) <=> in(0, add(V!0, U!1))))),
% 0.20/0.51 inference(rewrite,[status(thm)],[])).
% 0.20/0.51 tff(74,plain,
% 0.20/0.51 (((in(0, U!1) | ($sum(V!0, $product(-1, 0)) = 0)) <=> in(0, add(V!0, U!1))) <=> ((in(0, U!1) | (V!0 = 0)) <=> in(0, add(V!0, U!1)))),
% 0.20/0.51 inference(rewrite,[status(thm)],[])).
% 0.20/0.51 tff(75,plain,
% 0.20/0.51 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(0, U!1) | ($sum(V!0, $product(-1, 0)) = 0)) <=> in(0, add(V!0, U!1)))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(0, U!1) | (V!0 = 0)) <=> in(0, add(V!0, U!1))))),
% 0.20/0.51 inference(monotonicity,[status(thm)],[74])).
% 0.20/0.51 tff(76,plain,
% 0.20/0.51 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(0, U!1) | ($sum(V!0, $product(-1, 0)) = 0)) <=> in(0, add(V!0, U!1)))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(0, U!1) | (V!0 = 0)) <=> in(0, add(V!0, U!1))))),
% 0.20/0.51 inference(transitivity,[status(thm)],[75, 73])).
% 0.20/0.51 tff(77,plain,
% 0.20/0.51 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(0, U!1) | ($sum(V!0, $product(-1, 0)) = 0)) <=> in(0, add(V!0, U!1)))),
% 0.20/0.51 inference(quant_inst,[status(thm)],[])).
% 0.20/0.51 tff(78,plain,
% 0.20/0.51 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(0, U!1) | (V!0 = 0)) <=> in(0, add(V!0, U!1)))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.20/0.51 tff(79,plain,
% 0.20/0.51 ((in(0, U!1) | (V!0 = 0)) <=> in(0, add(V!0, U!1))),
% 0.20/0.51 inference(unit_resolution,[status(thm)],[78, 72])).
% 0.20/0.51 tff(80,plain,
% 0.20/0.51 (((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0)) | (~(V!0 = 0))),
% 0.20/0.51 inference(tautology,[status(thm)],[])).
% 0.20/0.51 tff(81,plain,
% 0.20/0.51 (~(V!0 = 0)),
% 0.20/0.51 inference(unit_resolution,[status(thm)],[80, 53])).
% 0.20/0.51 tff(82,assumption,(~$greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 1)), introduced(assumption)).
% 0.20/0.51 tff(83,assumption,(~in(V!0, U!1)), introduced(assumption)).
% 0.20/0.51 tff(84,plain,
% 0.20/0.51 (^[X14: $int, X15: collection] : refl(((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0)) <=> ((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0)))),
% 0.20/0.51 inference(bind,[status(th)],[])).
% 0.20/0.51 tff(85,plain,
% 0.20/0.51 (![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0)) <=> ![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0))),
% 0.20/0.51 inference(quant_intro,[status(thm)],[84])).
% 0.20/0.51 tff(86,plain,
% 0.20/0.51 (^[X14: $int, X15: collection] : rewrite(((~in(X14, X15)) <=> ($sum(count(remove(X14, X15)), $product(-1, count(X15))) = 0)) <=> ((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0)))),
% 0.20/0.51 inference(bind,[status(th)],[])).
% 0.20/0.51 tff(87,plain,
% 0.20/0.51 (![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(remove(X14, X15)), $product(-1, count(X15))) = 0)) <=> ![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0))),
% 0.20/0.51 inference(quant_intro,[status(thm)],[86])).
% 0.20/0.51 tff(88,plain,
% 0.20/0.51 (^[X14: $int, X15: collection] : rewrite(((~in(X14, X15)) <=> (count(remove(X14, X15)) = count(X15))) <=> ((~in(X14, X15)) <=> ($sum(count(remove(X14, X15)), $product(-1, count(X15))) = 0)))),
% 0.20/0.51 inference(bind,[status(th)],[])).
% 0.20/0.51 tff(89,plain,
% 0.20/0.51 (![X14: $int, X15: collection] : ((~in(X14, X15)) <=> (count(remove(X14, X15)) = count(X15))) <=> ![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(remove(X14, X15)), $product(-1, count(X15))) = 0))),
% 0.20/0.51 inference(quant_intro,[status(thm)],[88])).
% 0.20/0.51 tff(90,plain,
% 0.20/0.51 (![X14: $int, X15: collection] : ((~in(X14, X15)) <=> (count(remove(X14, X15)) = count(X15))) <=> ![X14: $int, X15: collection] : ((~in(X14, X15)) <=> (count(remove(X14, X15)) = count(X15)))),
% 0.20/0.51 inference(rewrite,[status(thm)],[])).
% 0.20/0.51 tff(91,axiom,(![X14: $int, X15: collection] : ((~in(X14, X15)) <=> (count(remove(X14, X15)) = count(X15)))), file('/export/starexec/sandbox/benchmark/Axioms/DAT002=1.ax','ax6')).
% 0.20/0.51 tff(92,plain,
% 0.20/0.51 (![X14: $int, X15: collection] : ((~in(X14, X15)) <=> (count(remove(X14, X15)) = count(X15)))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.20/0.51 tff(93,plain,
% 0.20/0.51 (![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(remove(X14, X15)), $product(-1, count(X15))) = 0))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[92, 89])).
% 0.20/0.51 tff(94,plain,
% 0.20/0.51 (![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[93, 87])).
% 0.20/0.51 tff(95,plain,(
% 0.20/0.51 ![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0))),
% 0.20/0.51 inference(skolemize,[status(sab)],[94])).
% 0.20/0.51 tff(96,plain,
% 0.20/0.51 (![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[95, 85])).
% 0.20/0.51 tff(97,plain,
% 0.20/0.51 (((~![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0))) | ((~in(V!0, U!1)) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = 0))) <=> ((~![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0))) | ((~in(V!0, U!1)) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = 0)))),
% 0.20/0.51 inference(rewrite,[status(thm)],[])).
% 0.20/0.51 tff(98,plain,
% 0.20/0.51 (((~in(V!0, U!1)) <=> ($sum(count(U!1), $product(-1, count(remove(V!0, U!1)))) = 0)) <=> ((~in(V!0, U!1)) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = 0))),
% 0.20/0.51 inference(rewrite,[status(thm)],[])).
% 0.20/0.51 tff(99,plain,
% 0.20/0.51 (((~![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0))) | ((~in(V!0, U!1)) <=> ($sum(count(U!1), $product(-1, count(remove(V!0, U!1)))) = 0))) <=> ((~![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0))) | ((~in(V!0, U!1)) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = 0)))),
% 0.20/0.51 inference(monotonicity,[status(thm)],[98])).
% 0.20/0.51 tff(100,plain,
% 0.20/0.51 (((~![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0))) | ((~in(V!0, U!1)) <=> ($sum(count(U!1), $product(-1, count(remove(V!0, U!1)))) = 0))) <=> ((~![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0))) | ((~in(V!0, U!1)) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = 0)))),
% 0.20/0.51 inference(transitivity,[status(thm)],[99, 97])).
% 0.20/0.51 tff(101,plain,
% 0.20/0.51 ((~![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0))) | ((~in(V!0, U!1)) <=> ($sum(count(U!1), $product(-1, count(remove(V!0, U!1)))) = 0))),
% 0.20/0.51 inference(quant_inst,[status(thm)],[])).
% 0.20/0.51 tff(102,plain,
% 0.20/0.51 ((~![X14: $int, X15: collection] : ((~in(X14, X15)) <=> ($sum(count(X15), $product(-1, count(remove(X14, X15)))) = 0))) | ((~in(V!0, U!1)) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = 0))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.20/0.51 tff(103,plain,
% 0.20/0.51 ((~in(V!0, U!1)) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = 0)),
% 0.20/0.51 inference(unit_resolution,[status(thm)],[102, 96])).
% 0.20/0.51 tff(104,plain,
% 0.20/0.51 ((~((~in(V!0, U!1)) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = 0))) | in(V!0, U!1) | ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = 0)),
% 0.20/0.51 inference(tautology,[status(thm)],[])).
% 0.20/0.51 tff(105,plain,
% 0.20/0.51 (in(V!0, U!1) | ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = 0)),
% 0.20/0.51 inference(unit_resolution,[status(thm)],[104, 103])).
% 0.20/0.51 tff(106,plain,
% 0.20/0.51 ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = 0),
% 0.20/0.51 inference(unit_resolution,[status(thm)],[105, 83])).
% 0.20/0.51 tff(107,plain,
% 0.20/0.51 ((~($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = 0)) | $lesseq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), 0)),
% 0.20/0.51 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.51 tff(108,plain,
% 0.20/0.51 ($lesseq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), 0)),
% 0.20/0.51 inference(unit_resolution,[status(thm)],[107, 106])).
% 0.20/0.51 tff(109,plain,
% 0.20/0.51 ((~($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1)) | $greatereq($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))), 1)),
% 0.20/0.51 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.51 tff(110,plain,
% 0.20/0.51 ($greatereq($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))), 1)),
% 0.20/0.51 inference(unit_resolution,[status(thm)],[109, 30])).
% 0.20/0.51 tff(111,assumption,(~$greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))), 0)), introduced(assumption)).
% 0.20/0.51 tff(112,assumption,($greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 1)), introduced(assumption)).
% 0.20/0.51 tff(113,plain,
% 0.20/0.51 ((~($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = 0)) | $greatereq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), 0)),
% 0.20/0.51 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.52 tff(114,plain,
% 0.20/0.52 ($greatereq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), 0)),
% 0.20/0.52 inference(unit_resolution,[status(thm)],[113, 106])).
% 0.20/0.52 tff(115,plain,
% 0.20/0.52 (^[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)))),
% 0.20/0.52 inference(bind,[status(th)],[])).
% 0.20/0.52 tff(116,plain,
% 0.20/0.52 (![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))),
% 0.20/0.52 inference(quant_intro,[status(thm)],[115])).
% 0.20/0.52 tff(117,plain,
% 0.20/0.52 (^[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)))),
% 0.20/0.52 inference(bind,[status(th)],[])).
% 0.20/0.52 tff(118,plain,
% 0.20/0.52 (![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))),
% 0.20/0.52 inference(quant_intro,[status(thm)],[117])).
% 0.20/0.52 tff(119,plain,
% 0.20/0.52 (^[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)))),
% 0.20/0.52 inference(bind,[status(th)],[])).
% 0.20/0.52 tff(120,plain,
% 0.20/0.52 (![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))),
% 0.20/0.52 inference(quant_intro,[status(thm)],[119])).
% 0.20/0.52 tff(121,plain,
% 0.20/0.52 (![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))))),
% 0.20/0.52 inference(rewrite,[status(thm)],[])).
% 0.20/0.52 tff(122,plain,
% 0.20/0.52 (^[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)))))),
% 0.20/0.52 inference(bind,[status(th)],[])).
% 0.20/0.52 tff(123,plain,
% 0.20/0.52 (![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))))),
% 0.20/0.52 inference(quant_intro,[status(thm)],[122])).
% 0.20/0.52 tff(124,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')).
% 0.20/0.52 tff(125,plain,
% 0.20/0.52 (![X8: $int, X9: collection] : ((~in(X8, X9)) <=> (count(add(X8, X9)) = $sum(1, count(X9))))),
% 0.20/0.52 inference(modus_ponens,[status(thm)],[124, 123])).
% 0.20/0.52 tff(126,plain,
% 0.20/0.52 (![X8: $int, X9: collection] : ((~in(X8, X9)) <=> (count(add(X8, X9)) = $sum(1, count(X9))))),
% 0.20/0.52 inference(modus_ponens,[status(thm)],[125, 121])).
% 0.20/0.52 tff(127,plain,
% 0.20/0.52 (![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(add(X8, X9)), $product(-1, count(X9))) = 1))),
% 0.20/0.52 inference(modus_ponens,[status(thm)],[126, 120])).
% 0.20/0.52 tff(128,plain,
% 0.20/0.52 (![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))),
% 0.20/0.52 inference(modus_ponens,[status(thm)],[127, 118])).
% 0.20/0.52 tff(129,plain,(
% 0.20/0.52 ![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))),
% 0.20/0.52 inference(skolemize,[status(sab)],[128])).
% 0.20/0.52 tff(130,plain,
% 0.20/0.52 (![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))),
% 0.20/0.52 inference(modus_ponens,[status(thm)],[129, 116])).
% 0.20/0.52 tff(131,plain,
% 0.20/0.52 ((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(V!0, U!1)) <=> ($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = -1))),
% 0.20/0.52 inference(quant_inst,[status(thm)],[])).
% 0.20/0.52 tff(132,plain,
% 0.20/0.52 ((~in(V!0, U!1)) <=> ($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = -1)),
% 0.20/0.52 inference(unit_resolution,[status(thm)],[131, 130])).
% 0.20/0.52 tff(133,plain,
% 0.20/0.52 ((~((~in(V!0, U!1)) <=> ($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = -1))) | in(V!0, U!1) | ($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = -1)),
% 0.20/0.52 inference(tautology,[status(thm)],[])).
% 0.20/0.52 tff(134,plain,
% 0.20/0.52 (in(V!0, U!1) | ($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = -1)),
% 0.20/0.52 inference(unit_resolution,[status(thm)],[133, 132])).
% 0.20/0.52 tff(135,plain,
% 0.20/0.52 ($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = -1),
% 0.20/0.52 inference(unit_resolution,[status(thm)],[134, 83])).
% 0.20/0.52 tff(136,plain,
% 0.20/0.52 ((~($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = -1)) | $greatereq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), -1)),
% 0.20/0.52 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.52 tff(137,plain,
% 0.20/0.52 ($greatereq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), -1)),
% 0.20/0.52 inference(unit_resolution,[status(thm)],[136, 135])).
% 0.20/0.52 tff(138,assumption,($greatereq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), 0)), introduced(assumption)).
% 0.20/0.52 tff(139,assumption,($lesseq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 0)), introduced(assumption)).
% 0.20/0.52 tff(140,plain,
% 0.20/0.52 ((~$greatereq($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))), 1)) | $greatereq($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))), 0)),
% 0.20/0.52 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.52 tff(141,plain,
% 0.20/0.52 ($greatereq($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))), 0)),
% 0.20/0.52 inference(unit_resolution,[status(thm)],[140, 110])).
% 0.20/0.52 tff(142,assumption,($greatereq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), -1)), introduced(assumption)).
% 0.20/0.52 tff(143,plain,
% 0.20/0.52 ($false),
% 0.20/0.52 inference(theory_lemma,[status(thm)],[142, 141, 139, 111, 112, 138])).
% 0.20/0.52 tff(144,plain,((~$lesseq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 0)) | (~$greatereq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), -1)) | $greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))), 0) | (~$greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 1)) | (~$greatereq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), 0))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.52 tff(145,plain,
% 0.20/0.52 (~$lesseq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 0)),
% 0.20/0.52 inference(unit_resolution,[status(thm)],[144, 137, 111, 112, 114])).
% 0.20/0.52 tff(146,plain,
% 0.20/0.52 ((~($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 0)) | $lesseq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 0)),
% 0.20/0.52 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.52 tff(147,plain,
% 0.20/0.52 (~($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 0)),
% 0.20/0.52 inference(unit_resolution,[status(thm)],[146, 145])).
% 0.20/0.52 tff(148,plain,
% 0.20/0.52 (^[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)))),
% 0.20/0.52 inference(bind,[status(th)],[])).
% 0.20/0.52 tff(149,plain,
% 0.20/0.52 (![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))),
% 0.20/0.52 inference(quant_intro,[status(thm)],[148])).
% 0.20/0.52 tff(150,plain,
% 0.20/0.52 (^[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)))),
% 0.20/0.52 inference(bind,[status(th)],[])).
% 0.20/0.52 tff(151,plain,
% 0.20/0.52 (![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))),
% 0.20/0.53 inference(quant_intro,[status(thm)],[150])).
% 0.20/0.53 tff(152,plain,
% 0.20/0.53 (^[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)))),
% 0.20/0.53 inference(bind,[status(th)],[])).
% 0.20/0.53 tff(153,plain,
% 0.20/0.53 (![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))),
% 0.20/0.53 inference(quant_intro,[status(thm)],[152])).
% 0.20/0.53 tff(154,plain,
% 0.20/0.53 (![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)))),
% 0.20/0.53 inference(rewrite,[status(thm)],[])).
% 0.20/0.53 tff(155,axiom,(![X10: $int, X11: collection] : (in(X10, X11) <=> (count(add(X10, X11)) = count(X11)))), file('/export/starexec/sandbox/benchmark/Axioms/DAT002=1.ax','ax4')).
% 0.20/0.53 tff(156,plain,
% 0.20/0.53 (![X10: $int, X11: collection] : (in(X10, X11) <=> (count(add(X10, X11)) = count(X11)))),
% 0.20/0.53 inference(modus_ponens,[status(thm)],[155, 154])).
% 0.20/0.53 tff(157,plain,
% 0.20/0.53 (![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(add(X10, X11)), $product(-1, count(X11))) = 0))),
% 0.20/0.53 inference(modus_ponens,[status(thm)],[156, 153])).
% 0.20/0.53 tff(158,plain,
% 0.20/0.53 (![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))),
% 0.20/0.53 inference(modus_ponens,[status(thm)],[157, 151])).
% 0.20/0.53 tff(159,plain,(
% 0.20/0.53 ![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))),
% 0.20/0.53 inference(skolemize,[status(sab)],[158])).
% 0.20/0.53 tff(160,plain,
% 0.20/0.53 (![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))),
% 0.20/0.53 inference(modus_ponens,[status(thm)],[159, 149])).
% 0.20/0.53 tff(161,plain,
% 0.20/0.53 (((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(0, remove(V!0, add(V!0, U!1))) <=> ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 0))) <=> ((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(0, remove(V!0, add(V!0, U!1))) <=> ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 0)))),
% 0.20/0.53 inference(rewrite,[status(thm)],[])).
% 0.20/0.53 tff(162,plain,
% 0.20/0.53 ((in(0, remove(V!0, add(V!0, U!1))) <=> ($sum(count(remove(V!0, add(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))) = 0)) <=> (in(0, remove(V!0, add(V!0, U!1))) <=> ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 0))),
% 0.20/0.53 inference(rewrite,[status(thm)],[])).
% 0.20/0.53 tff(163,plain,
% 0.20/0.53 (((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(0, remove(V!0, add(V!0, U!1))) <=> ($sum(count(remove(V!0, add(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))) = 0))) <=> ((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(0, remove(V!0, add(V!0, U!1))) <=> ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 0)))),
% 0.20/0.53 inference(monotonicity,[status(thm)],[162])).
% 0.20/0.53 tff(164,plain,
% 0.20/0.53 (((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(0, remove(V!0, add(V!0, U!1))) <=> ($sum(count(remove(V!0, add(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))) = 0))) <=> ((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(0, remove(V!0, add(V!0, U!1))) <=> ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 0)))),
% 0.20/0.53 inference(transitivity,[status(thm)],[163, 161])).
% 0.20/0.53 tff(165,plain,
% 0.20/0.53 ((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(0, remove(V!0, add(V!0, U!1))) <=> ($sum(count(remove(V!0, add(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))) = 0))),
% 0.20/0.53 inference(quant_inst,[status(thm)],[])).
% 0.20/0.53 tff(166,plain,
% 0.20/0.53 ((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(0, remove(V!0, add(V!0, U!1))) <=> ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 0))),
% 0.20/0.53 inference(modus_ponens,[status(thm)],[165, 164])).
% 0.20/0.53 tff(167,plain,
% 0.20/0.53 (in(0, remove(V!0, add(V!0, U!1))) <=> ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 0)),
% 0.20/0.53 inference(unit_resolution,[status(thm)],[166, 160])).
% 0.20/0.53 tff(168,plain,
% 0.20/0.53 ((~(in(0, remove(V!0, add(V!0, U!1))) <=> ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 0))) | (~in(0, remove(V!0, add(V!0, U!1)))) | ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 0)),
% 0.20/0.53 inference(tautology,[status(thm)],[])).
% 0.20/0.53 tff(169,plain,
% 0.20/0.53 ((~in(0, remove(V!0, add(V!0, U!1)))) | ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 0)),
% 0.20/0.53 inference(unit_resolution,[status(thm)],[168, 167])).
% 0.20/0.53 tff(170,plain,
% 0.20/0.53 (~in(0, remove(V!0, add(V!0, U!1)))),
% 0.20/0.53 inference(unit_resolution,[status(thm)],[169, 147])).
% 0.20/0.53 tff(171,plain,
% 0.20/0.53 (((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(0, remove(V!0, add(V!0, U!1)))) <=> ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1))) <=> ((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(0, remove(V!0, add(V!0, U!1)))) <=> ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1)))),
% 0.20/0.53 inference(rewrite,[status(thm)],[])).
% 0.20/0.53 tff(172,plain,
% 0.20/0.53 (((~in(0, remove(V!0, add(V!0, U!1)))) <=> ($sum(count(remove(V!0, add(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))) = -1)) <=> ((~in(0, remove(V!0, add(V!0, U!1)))) <=> ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1))),
% 0.20/0.53 inference(rewrite,[status(thm)],[])).
% 0.20/0.53 tff(173,plain,
% 0.20/0.53 (((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(0, remove(V!0, add(V!0, U!1)))) <=> ($sum(count(remove(V!0, add(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))) = -1))) <=> ((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(0, remove(V!0, add(V!0, U!1)))) <=> ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1)))),
% 0.20/0.53 inference(monotonicity,[status(thm)],[172])).
% 0.20/0.53 tff(174,plain,
% 0.20/0.53 (((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(0, remove(V!0, add(V!0, U!1)))) <=> ($sum(count(remove(V!0, add(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))) = -1))) <=> ((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(0, remove(V!0, add(V!0, U!1)))) <=> ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1)))),
% 0.20/0.53 inference(transitivity,[status(thm)],[173, 171])).
% 0.20/0.53 tff(175,plain,
% 0.20/0.53 ((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(0, remove(V!0, add(V!0, U!1)))) <=> ($sum(count(remove(V!0, add(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))) = -1))),
% 0.20/0.53 inference(quant_inst,[status(thm)],[])).
% 0.20/0.53 tff(176,plain,
% 0.20/0.53 ((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(0, remove(V!0, add(V!0, U!1)))) <=> ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1))),
% 0.20/0.54 inference(modus_ponens,[status(thm)],[175, 174])).
% 0.20/0.54 tff(177,plain,
% 0.20/0.54 ((~in(0, remove(V!0, add(V!0, U!1)))) <=> ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1)),
% 0.20/0.54 inference(unit_resolution,[status(thm)],[176, 130])).
% 0.20/0.54 tff(178,plain,
% 0.20/0.54 ((~((~in(0, remove(V!0, add(V!0, U!1)))) <=> ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1))) | in(0, remove(V!0, add(V!0, U!1))) | ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1)),
% 0.20/0.54 inference(tautology,[status(thm)],[])).
% 0.20/0.54 tff(179,plain,
% 0.20/0.54 (in(0, remove(V!0, add(V!0, U!1))) | ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1)),
% 0.20/0.54 inference(unit_resolution,[status(thm)],[178, 177])).
% 0.20/0.54 tff(180,plain,
% 0.20/0.54 ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1),
% 0.20/0.54 inference(unit_resolution,[status(thm)],[179, 170])).
% 0.20/0.54 tff(181,plain,
% 0.20/0.54 ((~($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1)) | $lesseq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 1)),
% 0.20/0.54 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.54 tff(182,plain,
% 0.20/0.54 ($lesseq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 1)),
% 0.20/0.54 inference(unit_resolution,[status(thm)],[181, 180])).
% 0.20/0.54 tff(183,plain,
% 0.20/0.54 ($false),
% 0.20/0.54 inference(theory_lemma,[status(thm)],[137, 110, 182, 111, 112, 114])).
% 0.20/0.54 tff(184,plain,(in(V!0, U!1) | $greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))), 0) | (~$greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 1))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.54 tff(185,plain,
% 0.20/0.54 (in(V!0, U!1)),
% 0.20/0.54 inference(unit_resolution,[status(thm)],[184, 112, 111])).
% 0.20/0.54 tff(186,plain,
% 0.20/0.54 (((~![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))) | (in(V!0, U!1) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = -1))) <=> ((~![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))) | (in(V!0, U!1) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = -1)))),
% 0.20/0.54 inference(rewrite,[status(thm)],[])).
% 0.20/0.54 tff(187,plain,
% 0.20/0.54 ((in(V!0, U!1) <=> ($sum(count(U!1), $product(-1, count(remove(V!0, U!1)))) = 1)) <=> (in(V!0, U!1) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = -1))),
% 0.20/0.54 inference(rewrite,[status(thm)],[])).
% 0.20/0.54 tff(188,plain,
% 0.20/0.54 (((~![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))) | (in(V!0, U!1) <=> ($sum(count(U!1), $product(-1, count(remove(V!0, U!1)))) = 1))) <=> ((~![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))) | (in(V!0, U!1) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = -1)))),
% 0.20/0.54 inference(monotonicity,[status(thm)],[187])).
% 0.20/0.54 tff(189,plain,
% 0.20/0.54 (((~![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))) | (in(V!0, U!1) <=> ($sum(count(U!1), $product(-1, count(remove(V!0, U!1)))) = 1))) <=> ((~![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))) | (in(V!0, U!1) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = -1)))),
% 0.20/0.54 inference(transitivity,[status(thm)],[188, 186])).
% 0.20/0.54 tff(190,plain,
% 0.20/0.54 ((~![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))) | (in(V!0, U!1) <=> ($sum(count(U!1), $product(-1, count(remove(V!0, U!1)))) = 1))),
% 0.20/0.54 inference(quant_inst,[status(thm)],[])).
% 0.20/0.55 tff(191,plain,
% 0.20/0.55 ((~![X12: $int, X13: collection] : (in(X12, X13) <=> ($sum(count(X13), $product(-1, count(remove(X12, X13)))) = 1))) | (in(V!0, U!1) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = -1))),
% 0.20/0.55 inference(modus_ponens,[status(thm)],[190, 189])).
% 0.20/0.55 tff(192,plain,
% 0.20/0.55 (in(V!0, U!1) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = -1)),
% 0.20/0.55 inference(unit_resolution,[status(thm)],[191, 16])).
% 0.20/0.55 tff(193,plain,
% 0.20/0.55 ((~(in(V!0, U!1) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = -1))) | (~in(V!0, U!1)) | ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = -1)),
% 0.20/0.55 inference(tautology,[status(thm)],[])).
% 0.20/0.55 tff(194,plain,
% 0.20/0.55 ((~in(V!0, U!1)) | ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = -1)),
% 0.20/0.55 inference(unit_resolution,[status(thm)],[193, 192])).
% 0.20/0.55 tff(195,plain,
% 0.20/0.55 ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = -1),
% 0.20/0.55 inference(unit_resolution,[status(thm)],[194, 185])).
% 0.20/0.55 tff(196,plain,
% 0.20/0.55 ((~($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = -1)) | $greatereq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), -1)),
% 0.20/0.55 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.55 tff(197,plain,
% 0.20/0.55 ($greatereq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), -1)),
% 0.20/0.55 inference(unit_resolution,[status(thm)],[196, 195])).
% 0.20/0.55 tff(198,plain,
% 0.20/0.55 ((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(V!0, U!1) <=> ($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = 0))),
% 0.20/0.55 inference(quant_inst,[status(thm)],[])).
% 0.20/0.55 tff(199,plain,
% 0.20/0.55 (in(V!0, U!1) <=> ($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = 0)),
% 0.20/0.55 inference(unit_resolution,[status(thm)],[198, 160])).
% 0.20/0.55 tff(200,plain,
% 0.20/0.55 ((~(in(V!0, U!1) <=> ($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = 0))) | (~in(V!0, U!1)) | ($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = 0)),
% 0.20/0.55 inference(tautology,[status(thm)],[])).
% 0.20/0.55 tff(201,plain,
% 0.20/0.55 ((~in(V!0, U!1)) | ($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = 0)),
% 0.20/0.55 inference(unit_resolution,[status(thm)],[200, 199])).
% 0.20/0.55 tff(202,plain,
% 0.20/0.55 ($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = 0),
% 0.20/0.55 inference(unit_resolution,[status(thm)],[201, 185])).
% 0.20/0.55 tff(203,plain,
% 0.20/0.55 ((~($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = 0)) | $greatereq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), 0)),
% 0.20/0.55 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.55 tff(204,plain,
% 0.20/0.55 ($greatereq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), 0)),
% 0.20/0.55 inference(unit_resolution,[status(thm)],[203, 202])).
% 0.20/0.55 tff(205,plain,
% 0.20/0.55 ((~$lesseq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 1)) | (~$greatereq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), -1)) | (~$greatereq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), 0)) | (~$greatereq($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))), 1)) | $greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))), 0) | (~$greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 1))),
% 0.20/0.55 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.55 tff(206,plain,
% 0.20/0.55 (~$lesseq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 1)),
% 0.20/0.55 inference(unit_resolution,[status(thm)],[205, 112, 204, 111, 197, 110])).
% 0.20/0.55 tff(207,plain,
% 0.20/0.55 (~($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1)),
% 0.20/0.55 inference(unit_resolution,[status(thm)],[181, 206])).
% 0.20/0.55 tff(208,plain,
% 0.20/0.55 ($greatereq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), -1) | (~$greatereq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), 0))),
% 0.20/0.55 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.55 tff(209,plain,
% 0.20/0.55 ($greatereq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), -1)),
% 0.20/0.55 inference(unit_resolution,[status(thm)],[208, 204])).
% 0.20/0.55 tff(210,plain,
% 0.20/0.55 ((~$lesseq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 0)) | (~$greatereq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), -1)) | (~$greatereq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), -1)) | (~$greatereq($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))), 1)) | $greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))), 0) | (~$greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 1))),
% 0.47/0.55 inference(theory_lemma,[status(thm)],[])).
% 0.47/0.55 tff(211,plain,
% 0.47/0.55 (~$lesseq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 0)),
% 0.47/0.55 inference(unit_resolution,[status(thm)],[210, 112, 209, 111, 197, 110])).
% 0.47/0.55 tff(212,plain,
% 0.47/0.55 (~($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 0)),
% 0.47/0.55 inference(unit_resolution,[status(thm)],[146, 211])).
% 0.47/0.55 tff(213,plain,
% 0.47/0.55 (~in(0, remove(V!0, add(V!0, U!1)))),
% 0.47/0.55 inference(unit_resolution,[status(thm)],[169, 212])).
% 0.47/0.55 tff(214,plain,
% 0.47/0.55 ($false),
% 0.47/0.55 inference(unit_resolution,[status(thm)],[179, 213, 207])).
% 0.47/0.55 tff(215,plain,((~$greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 1)) | $greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))), 0)), inference(lemma,lemma(discharge,[]))).
% 0.47/0.55 tff(216,plain,
% 0.47/0.55 (~$greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 1)),
% 0.47/0.55 inference(unit_resolution,[status(thm)],[215, 111])).
% 0.47/0.55 tff(217,plain,
% 0.47/0.55 ((~($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 1)) | $greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 1)),
% 0.47/0.55 inference(theory_lemma,[status(thm)],[])).
% 0.47/0.55 tff(218,plain,
% 0.47/0.55 (~($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 1)),
% 0.47/0.55 inference(unit_resolution,[status(thm)],[217, 216])).
% 0.47/0.55 tff(219,plain,
% 0.47/0.55 (((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(0, remove(V!0, U!1))) <=> ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 1))) <=> ((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(0, remove(V!0, U!1))) <=> ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 1)))),
% 0.47/0.55 inference(rewrite,[status(thm)],[])).
% 0.47/0.55 tff(220,plain,
% 0.47/0.55 (((~in(0, remove(V!0, U!1))) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(add(0, remove(V!0, U!1))))) = -1)) <=> ((~in(0, remove(V!0, U!1))) <=> ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 1))),
% 0.47/0.55 inference(rewrite,[status(thm)],[])).
% 0.47/0.55 tff(221,plain,
% 0.47/0.55 (((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(0, remove(V!0, U!1))) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(add(0, remove(V!0, U!1))))) = -1))) <=> ((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(0, remove(V!0, U!1))) <=> ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 1)))),
% 0.47/0.55 inference(monotonicity,[status(thm)],[220])).
% 0.47/0.55 tff(222,plain,
% 0.47/0.55 (((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(0, remove(V!0, U!1))) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(add(0, remove(V!0, U!1))))) = -1))) <=> ((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(0, remove(V!0, U!1))) <=> ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 1)))),
% 0.47/0.55 inference(transitivity,[status(thm)],[221, 219])).
% 0.47/0.55 tff(223,plain,
% 0.47/0.55 ((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(0, remove(V!0, U!1))) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(add(0, remove(V!0, U!1))))) = -1))),
% 0.47/0.55 inference(quant_inst,[status(thm)],[])).
% 0.47/0.55 tff(224,plain,
% 0.47/0.55 ((~![X8: $int, X9: collection] : ((~in(X8, X9)) <=> ($sum(count(X9), $product(-1, count(add(X8, X9)))) = -1))) | ((~in(0, remove(V!0, U!1))) <=> ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 1))),
% 0.47/0.55 inference(modus_ponens,[status(thm)],[223, 222])).
% 0.47/0.55 tff(225,plain,
% 0.47/0.55 ((~in(0, remove(V!0, U!1))) <=> ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 1)),
% 0.47/0.55 inference(unit_resolution,[status(thm)],[224, 130])).
% 0.47/0.55 tff(226,plain,
% 0.47/0.55 ((~((~in(0, remove(V!0, U!1))) <=> ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 1))) | in(0, remove(V!0, U!1)) | ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 1)),
% 0.47/0.55 inference(tautology,[status(thm)],[])).
% 0.47/0.55 tff(227,plain,
% 0.47/0.55 (in(0, remove(V!0, U!1)) | ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 1)),
% 0.47/0.55 inference(unit_resolution,[status(thm)],[226, 225])).
% 0.47/0.55 tff(228,plain,
% 0.47/0.55 (in(0, remove(V!0, U!1))),
% 0.47/0.55 inference(unit_resolution,[status(thm)],[227, 218])).
% 0.47/0.55 tff(229,plain,
% 0.47/0.55 (^[X3: $int, X4: collection, X5: $int] : refl(((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4))) <=> ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4))))),
% 0.47/0.55 inference(bind,[status(th)],[])).
% 0.47/0.55 tff(230,plain,
% 0.47/0.55 (![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4))) <=> ![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))),
% 0.47/0.55 inference(quant_intro,[status(thm)],[229])).
% 0.47/0.55 tff(231,plain,
% 0.47/0.55 (^[X3: $int, X4: collection, X5: $int] : rewrite(((in(X3, X4) & (~($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4))) <=> ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4))))),
% 0.47/0.55 inference(bind,[status(th)],[])).
% 0.47/0.55 tff(232,plain,
% 0.47/0.55 (![X3: $int, X4: collection, X5: $int] : ((in(X3, X4) & (~($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4))) <=> ![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))),
% 0.47/0.55 inference(quant_intro,[status(thm)],[231])).
% 0.47/0.55 tff(233,plain,
% 0.47/0.55 (^[X3: $int, X4: collection, X5: $int] : rewrite(((in(X3, X4) & (~($sum(X3, $product(-1, X5)) = 0))) <=> in(X3, remove(X5, X4))) <=> ((in(X3, X4) & (~($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4))))),
% 0.47/0.55 inference(bind,[status(th)],[])).
% 0.47/0.55 tff(234,plain,
% 0.47/0.55 (![X3: $int, X4: collection, X5: $int] : ((in(X3, X4) & (~($sum(X3, $product(-1, X5)) = 0))) <=> in(X3, remove(X5, X4))) <=> ![X3: $int, X4: collection, X5: $int] : ((in(X3, X4) & (~($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))),
% 0.47/0.55 inference(quant_intro,[status(thm)],[233])).
% 0.47/0.55 tff(235,plain,
% 0.47/0.55 (^[X3: $int, X4: collection, X5: $int] : rewrite(((in(X3, X4) & (~(X3 = X5))) <=> in(X3, remove(X5, X4))) <=> ((in(X3, X4) & (~($sum(X3, $product(-1, X5)) = 0))) <=> in(X3, remove(X5, X4))))),
% 0.47/0.56 inference(bind,[status(th)],[])).
% 0.47/0.56 tff(236,plain,
% 0.47/0.56 (![X3: $int, X4: collection, X5: $int] : ((in(X3, X4) & (~(X3 = X5))) <=> in(X3, remove(X5, X4))) <=> ![X3: $int, X4: collection, X5: $int] : ((in(X3, X4) & (~($sum(X3, $product(-1, X5)) = 0))) <=> in(X3, remove(X5, X4)))),
% 0.47/0.56 inference(quant_intro,[status(thm)],[235])).
% 0.47/0.56 tff(237,plain,
% 0.47/0.56 (![X3: $int, X4: collection, X5: $int] : ((in(X3, X4) & (~(X3 = X5))) <=> in(X3, remove(X5, X4))) <=> ![X3: $int, X4: collection, X5: $int] : ((in(X3, X4) & (~(X3 = X5))) <=> in(X3, remove(X5, X4)))),
% 0.47/0.56 inference(rewrite,[status(thm)],[])).
% 0.47/0.56 tff(238,axiom,(![X3: $int, X4: collection, X5: $int] : ((in(X3, X4) & (~(X3 = X5))) <=> in(X3, remove(X5, X4)))), file('/export/starexec/sandbox/benchmark/Axioms/DAT002=0.ax','ax5')).
% 0.47/0.56 tff(239,plain,
% 0.47/0.56 (![X3: $int, X4: collection, X5: $int] : ((in(X3, X4) & (~(X3 = X5))) <=> in(X3, remove(X5, X4)))),
% 0.47/0.56 inference(modus_ponens,[status(thm)],[238, 237])).
% 0.47/0.56 tff(240,plain,
% 0.47/0.56 (![X3: $int, X4: collection, X5: $int] : ((in(X3, X4) & (~($sum(X3, $product(-1, X5)) = 0))) <=> in(X3, remove(X5, X4)))),
% 0.47/0.56 inference(modus_ponens,[status(thm)],[239, 236])).
% 0.47/0.56 tff(241,plain,
% 0.47/0.56 (![X3: $int, X4: collection, X5: $int] : ((in(X3, X4) & (~($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))),
% 0.47/0.56 inference(modus_ponens,[status(thm)],[240, 234])).
% 0.47/0.56 tff(242,plain,(
% 0.47/0.56 ![X3: $int, X4: collection, X5: $int] : ((in(X3, X4) & (~($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))),
% 0.47/0.56 inference(skolemize,[status(sab)],[241])).
% 0.47/0.56 tff(243,plain,
% 0.47/0.56 (![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))),
% 0.47/0.56 inference(modus_ponens,[status(thm)],[242, 232])).
% 0.47/0.56 tff(244,plain,
% 0.47/0.56 (![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))),
% 0.47/0.56 inference(modus_ponens,[status(thm)],[243, 230])).
% 0.47/0.56 tff(245,plain,
% 0.47/0.56 (((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(0, U!1)) | (V!0 = 0))) <=> in(0, remove(V!0, U!1)))) <=> ((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(0, U!1)) | (V!0 = 0))) <=> in(0, remove(V!0, U!1))))),
% 0.47/0.56 inference(rewrite,[status(thm)],[])).
% 0.47/0.56 tff(246,plain,
% 0.47/0.56 (((~((~in(0, U!1)) | ($sum(V!0, $product(-1, 0)) = 0))) <=> in(0, remove(V!0, U!1))) <=> ((~((~in(0, U!1)) | (V!0 = 0))) <=> in(0, remove(V!0, U!1)))),
% 0.47/0.56 inference(rewrite,[status(thm)],[])).
% 0.47/0.56 tff(247,plain,
% 0.47/0.56 (((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(0, U!1)) | ($sum(V!0, $product(-1, 0)) = 0))) <=> in(0, remove(V!0, U!1)))) <=> ((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(0, U!1)) | (V!0 = 0))) <=> in(0, remove(V!0, U!1))))),
% 0.47/0.56 inference(monotonicity,[status(thm)],[246])).
% 0.47/0.56 tff(248,plain,
% 0.47/0.56 (((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(0, U!1)) | ($sum(V!0, $product(-1, 0)) = 0))) <=> in(0, remove(V!0, U!1)))) <=> ((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(0, U!1)) | (V!0 = 0))) <=> in(0, remove(V!0, U!1))))),
% 0.47/0.56 inference(transitivity,[status(thm)],[247, 245])).
% 0.47/0.56 tff(249,plain,
% 0.47/0.56 ((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(0, U!1)) | ($sum(V!0, $product(-1, 0)) = 0))) <=> in(0, remove(V!0, U!1)))),
% 0.47/0.56 inference(quant_inst,[status(thm)],[])).
% 0.47/0.56 tff(250,plain,
% 0.47/0.56 ((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(0, U!1)) | (V!0 = 0))) <=> in(0, remove(V!0, U!1)))),
% 0.47/0.56 inference(modus_ponens,[status(thm)],[249, 248])).
% 0.47/0.56 tff(251,plain,
% 0.47/0.56 ((~((~in(0, U!1)) | (V!0 = 0))) <=> in(0, remove(V!0, U!1))),
% 0.47/0.56 inference(unit_resolution,[status(thm)],[250, 244])).
% 0.47/0.56 tff(252,plain,
% 0.47/0.56 ((~((~((~in(0, U!1)) | (V!0 = 0))) <=> in(0, remove(V!0, U!1)))) | (~((~in(0, U!1)) | (V!0 = 0))) | (~in(0, remove(V!0, U!1)))),
% 0.47/0.56 inference(tautology,[status(thm)],[])).
% 0.47/0.56 tff(253,plain,
% 0.47/0.56 ((~((~in(0, U!1)) | (V!0 = 0))) | (~in(0, remove(V!0, U!1)))),
% 0.47/0.56 inference(unit_resolution,[status(thm)],[252, 251])).
% 0.47/0.56 tff(254,plain,
% 0.47/0.56 (~((~in(0, U!1)) | (V!0 = 0))),
% 0.47/0.56 inference(unit_resolution,[status(thm)],[253, 228])).
% 0.47/0.56 tff(255,plain,
% 0.47/0.56 (((~in(0, U!1)) | (V!0 = 0)) | (~(V!0 = 0))),
% 0.47/0.56 inference(tautology,[status(thm)],[])).
% 0.47/0.56 tff(256,plain,
% 0.47/0.56 (~(V!0 = 0)),
% 0.47/0.56 inference(unit_resolution,[status(thm)],[255, 254])).
% 0.47/0.56 tff(257,plain,
% 0.47/0.56 (((~in(0, U!1)) | (V!0 = 0)) | in(0, U!1)),
% 0.47/0.56 inference(tautology,[status(thm)],[])).
% 0.47/0.56 tff(258,plain,
% 0.47/0.56 (in(0, U!1)),
% 0.47/0.56 inference(unit_resolution,[status(thm)],[257, 254])).
% 0.47/0.56 tff(259,plain,
% 0.47/0.56 (((~![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))) | ((~in(0, U!1)) | (U!1 = add(0, remove(0, U!1))))) <=> ((~![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))) | (~in(0, U!1)) | (U!1 = add(0, remove(0, U!1))))),
% 0.47/0.56 inference(rewrite,[status(thm)],[])).
% 0.47/0.56 tff(260,plain,
% 0.47/0.56 ((~![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))) | ((~in(0, U!1)) | (U!1 = add(0, remove(0, U!1))))),
% 0.47/0.56 inference(quant_inst,[status(thm)],[])).
% 0.47/0.56 tff(261,plain,
% 0.47/0.56 ((~![X16: $int, X17: collection] : ((~in(X16, X17)) | (X17 = add(X16, remove(X16, X17))))) | (~in(0, U!1)) | (U!1 = add(0, remove(0, U!1)))),
% 0.47/0.56 inference(modus_ponens,[status(thm)],[260, 259])).
% 0.47/0.56 tff(262,plain,
% 0.47/0.56 (U!1 = add(0, remove(0, U!1))),
% 0.47/0.56 inference(unit_resolution,[status(thm)],[261, 42, 258])).
% 0.47/0.56 tff(263,plain,
% 0.47/0.56 (in(0, U!1) <=> in(0, add(0, remove(0, U!1)))),
% 0.47/0.56 inference(monotonicity,[status(thm)],[262])).
% 0.47/0.56 tff(264,plain,
% 0.47/0.56 (in(0, add(0, remove(0, U!1)))),
% 0.47/0.56 inference(modus_ponens,[status(thm)],[258, 263])).
% 0.47/0.56 tff(265,assumption,(U!1 = add(0, remove(0, U!1))), introduced(assumption)).
% 0.47/0.56 tff(266,plain,
% 0.47/0.56 (add(0, remove(0, U!1)) = U!1),
% 0.47/0.56 inference(symmetry,[status(thm)],[265])).
% 0.47/0.56 tff(267,plain,
% 0.47/0.56 (add(V!0, add(0, remove(0, U!1))) = add(V!0, U!1)),
% 0.47/0.56 inference(monotonicity,[status(thm)],[266])).
% 0.47/0.56 tff(268,plain,
% 0.47/0.56 (add(V!0, add(0, remove(0, U!1))) = add(V!0, remove(V!0, add(V!0, U!1)))),
% 0.47/0.56 inference(transitivity,[status(thm)],[267, 46])).
% 0.47/0.56 tff(269,plain,
% 0.47/0.56 (in(0, add(V!0, add(0, remove(0, U!1)))) <=> in(0, add(V!0, remove(V!0, add(V!0, U!1))))),
% 0.47/0.56 inference(monotonicity,[status(thm)],[268])).
% 0.47/0.56 tff(270,plain,
% 0.47/0.56 (in(0, add(V!0, remove(V!0, add(V!0, U!1)))) <=> in(0, add(V!0, add(0, remove(0, U!1))))),
% 0.47/0.56 inference(symmetry,[status(thm)],[269])).
% 0.47/0.56 tff(271,plain,
% 0.47/0.56 ((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) <=> (~in(0, add(V!0, add(0, remove(0, U!1)))))),
% 0.47/0.56 inference(monotonicity,[status(thm)],[270])).
% 0.47/0.56 tff(272,assumption,(~in(0, add(V!0, remove(V!0, add(V!0, U!1))))), introduced(assumption)).
% 0.47/0.56 tff(273,plain,
% 0.47/0.56 (~in(0, add(V!0, add(0, remove(0, U!1))))),
% 0.47/0.56 inference(modus_ponens,[status(thm)],[272, 271])).
% 0.47/0.56 tff(274,plain,
% 0.47/0.56 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | (((V!0 = 0) | in(0, add(0, remove(0, U!1)))) <=> in(0, add(V!0, add(0, remove(0, U!1)))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | (((V!0 = 0) | in(0, add(0, remove(0, U!1)))) <=> in(0, add(V!0, add(0, remove(0, U!1))))))),
% 0.47/0.56 inference(rewrite,[status(thm)],[])).
% 0.47/0.56 tff(275,plain,
% 0.47/0.56 (((in(0, add(0, remove(0, U!1))) | ($sum(V!0, $product(-1, 0)) = 0)) <=> in(0, add(V!0, add(0, remove(0, U!1))))) <=> (((V!0 = 0) | in(0, add(0, remove(0, U!1)))) <=> in(0, add(V!0, add(0, remove(0, U!1)))))),
% 0.47/0.56 inference(rewrite,[status(thm)],[])).
% 0.47/0.56 tff(276,plain,
% 0.47/0.56 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(0, add(0, remove(0, U!1))) | ($sum(V!0, $product(-1, 0)) = 0)) <=> in(0, add(V!0, add(0, remove(0, U!1)))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | (((V!0 = 0) | in(0, add(0, remove(0, U!1)))) <=> in(0, add(V!0, add(0, remove(0, U!1))))))),
% 0.47/0.56 inference(monotonicity,[status(thm)],[275])).
% 0.47/0.56 tff(277,plain,
% 0.47/0.56 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(0, add(0, remove(0, U!1))) | ($sum(V!0, $product(-1, 0)) = 0)) <=> in(0, add(V!0, add(0, remove(0, U!1)))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | (((V!0 = 0) | in(0, add(0, remove(0, U!1)))) <=> in(0, add(V!0, add(0, remove(0, U!1))))))),
% 0.47/0.56 inference(transitivity,[status(thm)],[276, 274])).
% 0.47/0.57 tff(278,plain,
% 0.47/0.57 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(0, add(0, remove(0, U!1))) | ($sum(V!0, $product(-1, 0)) = 0)) <=> in(0, add(V!0, add(0, remove(0, U!1)))))),
% 0.47/0.57 inference(quant_inst,[status(thm)],[])).
% 0.47/0.57 tff(279,plain,
% 0.47/0.57 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | (((V!0 = 0) | in(0, add(0, remove(0, U!1)))) <=> in(0, add(V!0, add(0, remove(0, U!1)))))),
% 0.47/0.57 inference(modus_ponens,[status(thm)],[278, 277])).
% 0.47/0.57 tff(280,plain,
% 0.47/0.57 (((V!0 = 0) | in(0, add(0, remove(0, U!1)))) <=> in(0, add(V!0, add(0, remove(0, U!1))))),
% 0.47/0.57 inference(unit_resolution,[status(thm)],[279, 72])).
% 0.47/0.57 tff(281,assumption,(in(0, add(0, remove(0, U!1)))), introduced(assumption)).
% 0.47/0.57 tff(282,plain,
% 0.47/0.57 (((V!0 = 0) | in(0, add(0, remove(0, U!1)))) | (~in(0, add(0, remove(0, U!1))))),
% 0.47/0.57 inference(tautology,[status(thm)],[])).
% 0.47/0.57 tff(283,plain,
% 0.47/0.57 ((V!0 = 0) | in(0, add(0, remove(0, U!1)))),
% 0.47/0.57 inference(unit_resolution,[status(thm)],[282, 281])).
% 0.47/0.57 tff(284,plain,
% 0.47/0.57 ((~(((V!0 = 0) | in(0, add(0, remove(0, U!1)))) <=> in(0, add(V!0, add(0, remove(0, U!1)))))) | (~((V!0 = 0) | in(0, add(0, remove(0, U!1))))) | in(0, add(V!0, add(0, remove(0, U!1))))),
% 0.47/0.57 inference(tautology,[status(thm)],[])).
% 0.47/0.57 tff(285,plain,
% 0.47/0.57 ((~(((V!0 = 0) | in(0, add(0, remove(0, U!1)))) <=> in(0, add(V!0, add(0, remove(0, U!1)))))) | in(0, add(V!0, add(0, remove(0, U!1))))),
% 0.47/0.57 inference(unit_resolution,[status(thm)],[284, 283])).
% 0.47/0.57 tff(286,plain,
% 0.47/0.57 (in(0, add(V!0, add(0, remove(0, U!1))))),
% 0.47/0.57 inference(unit_resolution,[status(thm)],[285, 280])).
% 0.47/0.57 tff(287,plain,
% 0.47/0.57 ($false),
% 0.47/0.57 inference(unit_resolution,[status(thm)],[286, 273])).
% 0.47/0.57 tff(288,plain,(in(0, add(V!0, remove(V!0, add(V!0, U!1)))) | (~(U!1 = add(0, remove(0, U!1)))) | (~in(0, add(0, remove(0, U!1))))), inference(lemma,lemma(discharge,[]))).
% 0.47/0.57 tff(289,plain,
% 0.47/0.57 (in(0, add(V!0, remove(V!0, add(V!0, U!1))))),
% 0.47/0.57 inference(unit_resolution,[status(thm)],[288, 264, 262])).
% 0.47/0.57 tff(290,plain,
% 0.47/0.57 ((~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0))) | (~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0)),
% 0.47/0.57 inference(tautology,[status(thm)],[])).
% 0.47/0.57 tff(291,plain,
% 0.47/0.57 (~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0))),
% 0.47/0.57 inference(unit_resolution,[status(thm)],[290, 289, 256])).
% 0.47/0.57 tff(292,plain,
% 0.47/0.57 (((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0))) <=> in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1))))))) <=> ((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0))) <=> in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1)))))))),
% 0.47/0.57 inference(rewrite,[status(thm)],[])).
% 0.47/0.57 tff(293,plain,
% 0.47/0.57 (((~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | ($sum(V!0, $product(-1, 0)) = 0))) <=> in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1)))))) <=> ((~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0))) <=> in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1))))))),
% 0.47/0.57 inference(rewrite,[status(thm)],[])).
% 0.47/0.57 tff(294,plain,
% 0.47/0.57 (((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | ($sum(V!0, $product(-1, 0)) = 0))) <=> in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1))))))) <=> ((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0))) <=> in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1)))))))),
% 0.47/0.57 inference(monotonicity,[status(thm)],[293])).
% 0.47/0.57 tff(295,plain,
% 0.47/0.57 (((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | ($sum(V!0, $product(-1, 0)) = 0))) <=> in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1))))))) <=> ((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0))) <=> in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1)))))))),
% 0.49/0.57 inference(transitivity,[status(thm)],[294, 292])).
% 0.49/0.57 tff(296,plain,
% 0.49/0.57 ((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | ($sum(V!0, $product(-1, 0)) = 0))) <=> in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1))))))),
% 0.49/0.57 inference(quant_inst,[status(thm)],[])).
% 0.49/0.57 tff(297,plain,
% 0.49/0.57 ((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0))) <=> in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1))))))),
% 0.49/0.57 inference(modus_ponens,[status(thm)],[296, 295])).
% 0.49/0.57 tff(298,plain,
% 0.49/0.57 ((~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0))) <=> in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1)))))),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[297, 244])).
% 0.49/0.57 tff(299,plain,
% 0.49/0.57 ((~((~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0))) <=> in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1))))))) | ((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0)) | in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1)))))),
% 0.49/0.57 inference(tautology,[status(thm)],[])).
% 0.49/0.57 tff(300,plain,
% 0.49/0.57 (((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0)) | in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1)))))),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[299, 298])).
% 0.49/0.57 tff(301,plain,
% 0.49/0.57 (in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1)))))),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[300, 291])).
% 0.49/0.57 tff(302,plain,
% 0.49/0.57 (in(0, remove(V!0, add(V!0, U!1)))),
% 0.49/0.57 inference(modus_ponens,[status(thm)],[301, 49])).
% 0.49/0.57 tff(303,plain,
% 0.49/0.57 ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 0),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[169, 302])).
% 0.49/0.57 tff(304,plain,
% 0.49/0.57 ($lesseq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 0)),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[146, 303])).
% 0.49/0.57 tff(305,assumption,(~$greatereq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), -1)), introduced(assumption)).
% 0.49/0.57 tff(306,plain,
% 0.49/0.57 (~$greatereq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), 0)),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[208, 305])).
% 0.49/0.57 tff(307,plain,
% 0.49/0.57 (~($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = 0)),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[203, 306])).
% 0.49/0.57 tff(308,plain,
% 0.49/0.57 (~($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = -1)),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[136, 305])).
% 0.49/0.57 tff(309,plain,
% 0.49/0.57 (in(V!0, U!1)),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[134, 308])).
% 0.49/0.57 tff(310,plain,
% 0.49/0.57 ($false),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[201, 309, 307])).
% 0.49/0.57 tff(311,plain,($greatereq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), -1)), inference(lemma,lemma(discharge,[]))).
% 0.49/0.57 tff(312,plain,
% 0.49/0.57 (((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(0, remove(V!0, U!1)) <=> ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 0))) <=> ((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(0, remove(V!0, U!1)) <=> ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 0)))),
% 0.49/0.57 inference(rewrite,[status(thm)],[])).
% 0.49/0.57 tff(313,plain,
% 0.49/0.57 ((in(0, remove(V!0, U!1)) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(add(0, remove(V!0, U!1))))) = 0)) <=> (in(0, remove(V!0, U!1)) <=> ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 0))),
% 0.49/0.57 inference(rewrite,[status(thm)],[])).
% 0.49/0.57 tff(314,plain,
% 0.49/0.57 (((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(0, remove(V!0, U!1)) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(add(0, remove(V!0, U!1))))) = 0))) <=> ((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(0, remove(V!0, U!1)) <=> ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 0)))),
% 0.49/0.57 inference(monotonicity,[status(thm)],[313])).
% 0.49/0.57 tff(315,plain,
% 0.49/0.57 (((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(0, remove(V!0, U!1)) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(add(0, remove(V!0, U!1))))) = 0))) <=> ((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(0, remove(V!0, U!1)) <=> ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 0)))),
% 0.49/0.57 inference(transitivity,[status(thm)],[314, 312])).
% 0.49/0.57 tff(316,plain,
% 0.49/0.57 ((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(0, remove(V!0, U!1)) <=> ($sum(count(remove(V!0, U!1)), $product(-1, count(add(0, remove(V!0, U!1))))) = 0))),
% 0.49/0.57 inference(quant_inst,[status(thm)],[])).
% 0.49/0.57 tff(317,plain,
% 0.49/0.57 ((~![X10: $int, X11: collection] : (in(X10, X11) <=> ($sum(count(X11), $product(-1, count(add(X10, X11)))) = 0))) | (in(0, remove(V!0, U!1)) <=> ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 0))),
% 0.49/0.57 inference(modus_ponens,[status(thm)],[316, 315])).
% 0.49/0.57 tff(318,plain,
% 0.49/0.57 (in(0, remove(V!0, U!1)) <=> ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 0)),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[317, 160])).
% 0.49/0.57 tff(319,plain,
% 0.49/0.57 ((~(in(0, remove(V!0, U!1)) <=> ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 0))) | (~in(0, remove(V!0, U!1))) | ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 0)),
% 0.49/0.57 inference(tautology,[status(thm)],[])).
% 0.49/0.57 tff(320,plain,
% 0.49/0.57 ((~in(0, remove(V!0, U!1))) | ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 0)),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[319, 318])).
% 0.49/0.57 tff(321,plain,
% 0.49/0.57 ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 0),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[320, 228])).
% 0.49/0.57 tff(322,plain,
% 0.49/0.57 ((~($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 0)) | $greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 0)),
% 0.49/0.57 inference(theory_lemma,[status(thm)],[])).
% 0.49/0.57 tff(323,plain,
% 0.49/0.57 ($greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 0)),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[322, 321])).
% 0.49/0.57 tff(324,plain,
% 0.49/0.57 ((~$greatereq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), 0)) | (~$greatereq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), -1)) | (~$greatereq($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))), 1)) | (~$lesseq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 0)) | $greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))), 0) | (~$greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 0))),
% 0.49/0.57 inference(theory_lemma,[status(thm)],[])).
% 0.49/0.57 tff(325,plain,
% 0.49/0.57 (~$greatereq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), 0)),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[324, 323, 311, 304, 111, 110])).
% 0.49/0.57 tff(326,plain,
% 0.49/0.57 (~($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = 0)),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[113, 325])).
% 0.49/0.57 tff(327,plain,
% 0.49/0.57 (in(V!0, U!1)),
% 0.49/0.57 inference(unit_resolution,[status(thm)],[105, 326])).
% 0.49/0.57 tff(328,plain,
% 0.49/0.57 ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = -1),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[194, 327])).
% 0.49/0.58 tff(329,plain,
% 0.49/0.58 ($greatereq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), -1)),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[196, 328])).
% 0.49/0.58 tff(330,plain,
% 0.49/0.58 ($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = 0),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[201, 327])).
% 0.49/0.58 tff(331,plain,
% 0.49/0.58 ($greatereq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), 0)),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[203, 330])).
% 0.49/0.58 tff(332,plain,
% 0.49/0.58 ($false),
% 0.49/0.58 inference(theory_lemma,[status(thm)],[331, 110, 304, 111, 323, 329])).
% 0.49/0.58 tff(333,plain,($greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))), 0)), inference(lemma,lemma(discharge,[]))).
% 0.49/0.58 tff(334,plain,
% 0.49/0.58 ((~![U: collection, V: $int] : ($sum(count(add(0, remove(V, U))), $product(-1, count(add(0, remove(V, add(V, U)))))) = 0)) <=> (~![U: collection, V: $int] : ($sum(count(add(0, remove(V, U))), $product(-1, count(add(0, remove(V, add(V, U)))))) = 0))),
% 0.49/0.58 inference(rewrite,[status(thm)],[])).
% 0.49/0.58 tff(335,plain,
% 0.49/0.58 ((~![U: collection, V: $int] : ($sum(count(add(0, remove(V, add(V, U)))), $product(-1, count(add(0, remove(V, U))))) = 0)) <=> (~![U: collection, V: $int] : ($sum(count(add(0, remove(V, U))), $product(-1, count(add(0, remove(V, add(V, U)))))) = 0))),
% 0.49/0.58 inference(rewrite,[status(thm)],[])).
% 0.49/0.58 tff(336,plain,
% 0.49/0.58 ((~![U: collection, V: $int] : (count(add(0, remove(V, add(V, U)))) = count(add(0, remove(V, U))))) <=> (~![U: collection, V: $int] : ($sum(count(add(0, remove(V, add(V, U)))), $product(-1, count(add(0, remove(V, U))))) = 0))),
% 0.49/0.58 inference(rewrite,[status(thm)],[])).
% 0.49/0.58 tff(337,plain,
% 0.49/0.58 ((~![U: collection, V: $int] : (count(add(0, remove(V, add(V, U)))) = count(add(0, remove(V, U))))) <=> (~![U: collection, V: $int] : (count(add(0, remove(V, add(V, U)))) = count(add(0, remove(V, U)))))),
% 0.49/0.58 inference(rewrite,[status(thm)],[])).
% 0.49/0.58 tff(338,axiom,(~![U: collection, V: $int] : (count(add(0, remove(V, add(V, U)))) = count(add(0, remove(V, U))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','co1')).
% 0.49/0.58 tff(339,plain,
% 0.49/0.58 (~![U: collection, V: $int] : (count(add(0, remove(V, add(V, U)))) = count(add(0, remove(V, U))))),
% 0.49/0.58 inference(modus_ponens,[status(thm)],[338, 337])).
% 0.49/0.58 tff(340,plain,
% 0.49/0.58 (~![U: collection, V: $int] : (count(add(0, remove(V, add(V, U)))) = count(add(0, remove(V, U))))),
% 0.49/0.58 inference(modus_ponens,[status(thm)],[339, 337])).
% 0.49/0.58 tff(341,plain,
% 0.49/0.58 (~![U: collection, V: $int] : (count(add(0, remove(V, add(V, U)))) = count(add(0, remove(V, U))))),
% 0.49/0.58 inference(modus_ponens,[status(thm)],[340, 337])).
% 0.49/0.58 tff(342,plain,
% 0.49/0.58 (~![U: collection, V: $int] : (count(add(0, remove(V, add(V, U)))) = count(add(0, remove(V, U))))),
% 0.49/0.58 inference(modus_ponens,[status(thm)],[341, 337])).
% 0.49/0.58 tff(343,plain,
% 0.49/0.58 (~![U: collection, V: $int] : ($sum(count(add(0, remove(V, add(V, U)))), $product(-1, count(add(0, remove(V, U))))) = 0)),
% 0.49/0.58 inference(modus_ponens,[status(thm)],[342, 336])).
% 0.49/0.58 tff(344,plain,
% 0.49/0.58 (~![U: collection, V: $int] : ($sum(count(add(0, remove(V, U))), $product(-1, count(add(0, remove(V, add(V, U)))))) = 0)),
% 0.49/0.58 inference(modus_ponens,[status(thm)],[343, 335])).
% 0.49/0.58 tff(345,plain,
% 0.49/0.58 (~![U: collection, V: $int] : ($sum(count(add(0, remove(V, U))), $product(-1, count(add(0, remove(V, add(V, U)))))) = 0)),
% 0.49/0.58 inference(modus_ponens,[status(thm)],[344, 334])).
% 0.49/0.58 tff(346,plain,
% 0.49/0.58 (~![U: collection, V: $int] : ($sum(count(add(0, remove(V, U))), $product(-1, count(add(0, remove(V, add(V, U)))))) = 0)),
% 0.49/0.58 inference(modus_ponens,[status(thm)],[345, 334])).
% 0.49/0.58 tff(347,plain,(
% 0.49/0.58 ~($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))) = 0)),
% 0.49/0.58 inference(skolemize,[status(sab)],[346])).
% 0.49/0.58 tff(348,plain,
% 0.49/0.58 (($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))) = 0) | (~$lesseq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))), 0)) | (~$greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))), 0))),
% 0.49/0.58 inference(theory_lemma,[status(thm)],[])).
% 0.49/0.58 tff(349,plain,
% 0.49/0.58 ((~$lesseq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))), 0)) | (~$greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))), 0))),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[348, 347])).
% 0.49/0.58 tff(350,plain,
% 0.49/0.58 (~$lesseq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))), 0)),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[349, 333])).
% 0.49/0.58 tff(351,plain,
% 0.49/0.58 (~($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 1)),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[217, 82])).
% 0.49/0.58 tff(352,plain,
% 0.49/0.58 (in(0, remove(V!0, U!1))),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[227, 351])).
% 0.49/0.58 tff(353,plain,
% 0.49/0.58 (~((~in(0, U!1)) | (V!0 = 0))),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[253, 352])).
% 0.49/0.58 tff(354,plain,
% 0.49/0.58 (~(V!0 = 0)),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[255, 353])).
% 0.49/0.58 tff(355,plain,
% 0.49/0.58 (in(0, U!1)),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[257, 353])).
% 0.49/0.58 tff(356,plain,
% 0.49/0.58 (U!1 = add(0, remove(0, U!1))),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[261, 42, 355])).
% 0.49/0.58 tff(357,plain,
% 0.49/0.58 (in(0, U!1) <=> in(0, add(0, remove(0, U!1)))),
% 0.49/0.58 inference(monotonicity,[status(thm)],[356])).
% 0.49/0.58 tff(358,plain,
% 0.49/0.58 (in(0, add(0, remove(0, U!1)))),
% 0.49/0.58 inference(modus_ponens,[status(thm)],[355, 357])).
% 0.49/0.58 tff(359,plain,
% 0.49/0.58 (in(0, add(V!0, remove(V!0, add(V!0, U!1))))),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[288, 358, 356])).
% 0.49/0.58 tff(360,plain,
% 0.49/0.58 (~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0))),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[290, 359, 354])).
% 0.49/0.58 tff(361,plain,
% 0.49/0.58 (in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1)))))),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[300, 360])).
% 0.49/0.58 tff(362,plain,
% 0.49/0.58 (in(0, remove(V!0, add(V!0, U!1)))),
% 0.49/0.58 inference(modus_ponens,[status(thm)],[361, 49])).
% 0.49/0.58 tff(363,plain,
% 0.49/0.58 ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 0),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[169, 362])).
% 0.49/0.58 tff(364,plain,
% 0.49/0.58 ((~($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 0)) | $greatereq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 0)),
% 0.49/0.58 inference(theory_lemma,[status(thm)],[])).
% 0.49/0.58 tff(365,plain,
% 0.49/0.58 ($greatereq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 0)),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[364, 363])).
% 0.49/0.58 tff(366,plain,
% 0.49/0.58 ((~($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = -1)) | $lesseq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), -1)),
% 0.49/0.58 inference(theory_lemma,[status(thm)],[])).
% 0.49/0.58 tff(367,plain,
% 0.49/0.58 ($lesseq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), -1)),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[366, 135])).
% 0.49/0.58 tff(368,plain,
% 0.49/0.58 ($false),
% 0.49/0.58 inference(theory_lemma,[status(thm)],[367, 32, 365, 350, 82, 108])).
% 0.49/0.58 tff(369,plain,(in(V!0, U!1) | $greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 1)), inference(lemma,lemma(discharge,[]))).
% 0.49/0.58 tff(370,plain,
% 0.49/0.58 (in(V!0, U!1)),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[369, 82])).
% 0.49/0.58 tff(371,plain,
% 0.49/0.58 ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = -1),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[194, 370])).
% 0.49/0.58 tff(372,plain,
% 0.49/0.58 ((~($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = -1)) | $lesseq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), -1)),
% 0.49/0.58 inference(theory_lemma,[status(thm)],[])).
% 0.49/0.58 tff(373,plain,
% 0.49/0.58 ($lesseq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), -1)),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[372, 371])).
% 0.49/0.58 tff(374,plain,
% 0.49/0.58 ($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = 0),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[201, 370])).
% 0.49/0.58 tff(375,plain,
% 0.49/0.58 ((~($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = 0)) | $lesseq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), 0)),
% 0.49/0.58 inference(theory_lemma,[status(thm)],[])).
% 0.49/0.58 tff(376,plain,
% 0.49/0.58 ($lesseq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), 0)),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[375, 374])).
% 0.49/0.58 tff(377,plain,
% 0.49/0.58 ($false),
% 0.49/0.58 inference(theory_lemma,[status(thm)],[376, 32, 365, 350, 82, 373])).
% 0.49/0.58 tff(378,plain,($greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 1)), inference(lemma,lemma(discharge,[]))).
% 0.49/0.58 tff(379,plain,
% 0.49/0.58 ((~$greatereq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 1)) | (~$lesseq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 0))),
% 0.49/0.58 inference(theory_lemma,[status(thm)],[])).
% 0.49/0.58 tff(380,plain,
% 0.49/0.58 (~$lesseq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 0)),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[379, 378])).
% 0.49/0.58 tff(381,plain,
% 0.49/0.58 ((~($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 0)) | $lesseq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 0)),
% 0.49/0.58 inference(theory_lemma,[status(thm)],[])).
% 0.49/0.58 tff(382,plain,
% 0.49/0.58 (~($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 0)),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[381, 380])).
% 0.49/0.58 tff(383,plain,
% 0.49/0.58 (~in(0, remove(V!0, U!1))),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[320, 382])).
% 0.49/0.58 tff(384,plain,
% 0.49/0.58 ((~((~((~in(0, U!1)) | (V!0 = 0))) <=> in(0, remove(V!0, U!1)))) | ((~in(0, U!1)) | (V!0 = 0)) | in(0, remove(V!0, U!1))),
% 0.49/0.58 inference(tautology,[status(thm)],[])).
% 0.49/0.58 tff(385,plain,
% 0.49/0.58 (((~in(0, U!1)) | (V!0 = 0)) | in(0, remove(V!0, U!1))),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[384, 251])).
% 0.49/0.58 tff(386,plain,
% 0.49/0.58 ((~in(0, U!1)) | (V!0 = 0)),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[385, 383])).
% 0.49/0.58 tff(387,plain,
% 0.49/0.58 ((~((~in(0, U!1)) | (V!0 = 0))) | (~in(0, U!1)) | (V!0 = 0)),
% 0.49/0.58 inference(tautology,[status(thm)],[])).
% 0.49/0.58 tff(388,plain,
% 0.49/0.58 ((~in(0, U!1)) | (V!0 = 0)),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[387, 386])).
% 0.49/0.58 tff(389,plain,
% 0.49/0.58 (~in(0, U!1)),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[388, 81])).
% 0.49/0.58 tff(390,plain,
% 0.49/0.58 ((~(in(0, U!1) | (V!0 = 0))) | in(0, U!1) | (V!0 = 0)),
% 0.49/0.58 inference(tautology,[status(thm)],[])).
% 0.49/0.58 tff(391,plain,
% 0.49/0.58 (~(in(0, U!1) | (V!0 = 0))),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[390, 389, 81])).
% 0.49/0.58 tff(392,plain,
% 0.49/0.58 ((~((in(0, U!1) | (V!0 = 0)) <=> in(0, add(V!0, U!1)))) | (in(0, U!1) | (V!0 = 0)) | (~in(0, add(V!0, U!1)))),
% 0.49/0.58 inference(tautology,[status(thm)],[])).
% 0.49/0.58 tff(393,plain,
% 0.49/0.58 ((~((in(0, U!1) | (V!0 = 0)) <=> in(0, add(V!0, U!1)))) | (~in(0, add(V!0, U!1)))),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[392, 391])).
% 0.49/0.58 tff(394,plain,
% 0.49/0.58 (~in(0, add(V!0, U!1))),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[393, 79])).
% 0.49/0.58 tff(395,plain,
% 0.49/0.58 ($false),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[394, 56])).
% 0.49/0.58 tff(396,plain,((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0)), inference(lemma,lemma(discharge,[]))).
% 0.49/0.58 tff(397,plain,
% 0.49/0.58 ((~((~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0))) <=> in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1))))))) | (~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0))) | (~in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1))))))),
% 0.49/0.58 inference(tautology,[status(thm)],[])).
% 0.49/0.58 tff(398,plain,
% 0.49/0.58 ((~((~in(0, add(V!0, remove(V!0, add(V!0, U!1))))) | (V!0 = 0))) | (~in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1))))))),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[397, 298])).
% 0.49/0.58 tff(399,plain,
% 0.49/0.58 (~in(0, remove(V!0, add(V!0, remove(V!0, add(V!0, U!1)))))),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[398, 396])).
% 0.49/0.58 tff(400,plain,
% 0.49/0.58 (~in(0, remove(V!0, add(V!0, U!1)))),
% 0.49/0.58 inference(modus_ponens,[status(thm)],[399, 50])).
% 0.49/0.58 tff(401,plain,
% 0.49/0.58 ($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1),
% 0.49/0.58 inference(unit_resolution,[status(thm)],[179, 400])).
% 0.49/0.58 tff(402,plain,
% 0.49/0.58 ((~($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))) = 1)) | $greatereq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 1)),
% 0.49/0.59 inference(theory_lemma,[status(thm)],[])).
% 0.49/0.59 tff(403,plain,
% 0.49/0.59 ($greatereq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 1)),
% 0.49/0.59 inference(unit_resolution,[status(thm)],[402, 401])).
% 0.49/0.59 tff(404,plain,
% 0.49/0.59 ((~$greatereq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 1)) | (~$lesseq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 0))),
% 0.49/0.59 inference(theory_lemma,[status(thm)],[])).
% 0.49/0.59 tff(405,plain,
% 0.49/0.59 (~$lesseq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 0)),
% 0.49/0.59 inference(unit_resolution,[status(thm)],[404, 403])).
% 0.49/0.59 tff(406,plain,
% 0.49/0.59 ($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 1),
% 0.49/0.59 inference(unit_resolution,[status(thm)],[227, 383])).
% 0.49/0.59 tff(407,plain,
% 0.49/0.59 ((~($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))) = 1)) | $lesseq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 1)),
% 0.49/0.59 inference(theory_lemma,[status(thm)],[])).
% 0.49/0.59 tff(408,plain,
% 0.49/0.59 ($lesseq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 1)),
% 0.49/0.59 inference(unit_resolution,[status(thm)],[407, 406])).
% 0.49/0.59 tff(409,plain,
% 0.49/0.59 ($lesseq($sum(count(add(0, remove(V!0, add(V!0, U!1)))), $product(-1, count(remove(V!0, add(V!0, U!1))))), 0) | (~$lesseq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), 0)) | (~$lesseq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), -1)) | (~$lesseq($sum(count(add(V!0, U!1)), $product(-1, count(remove(V!0, add(V!0, U!1))))), 1)) | $lesseq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(add(0, remove(V!0, add(V!0, U!1)))))), 0) | (~$lesseq($sum(count(add(0, remove(V!0, U!1))), $product(-1, count(remove(V!0, U!1)))), 1))),
% 0.49/0.59 inference(theory_lemma,[status(thm)],[])).
% 0.49/0.59 tff(410,plain,
% 0.49/0.59 ($false),
% 0.49/0.59 inference(unit_resolution,[status(thm)],[409, 108, 367, 408, 350, 405, 32])).
% 0.49/0.59 tff(411,plain,(in(V!0, U!1)), inference(lemma,lemma(discharge,[]))).
% 0.49/0.59 tff(412,plain,
% 0.49/0.59 ($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))) = -1),
% 0.49/0.59 inference(unit_resolution,[status(thm)],[194, 411])).
% 0.49/0.59 tff(413,plain,
% 0.49/0.59 ($lesseq($sum(count(remove(V!0, U!1)), $product(-1, count(U!1))), -1)),
% 0.49/0.59 inference(unit_resolution,[status(thm)],[372, 412])).
% 0.49/0.59 tff(414,plain,
% 0.49/0.59 ($sum(count(U!1), $product(-1, count(add(V!0, U!1)))) = 0),
% 0.49/0.59 inference(unit_resolution,[status(thm)],[201, 411])).
% 0.49/0.59 tff(415,plain,
% 0.49/0.59 ($lesseq($sum(count(U!1), $product(-1, count(add(V!0, U!1)))), 0)),
% 0.49/0.59 inference(unit_resolution,[status(thm)],[375, 414])).
% 0.49/0.59 tff(416,plain,
% 0.49/0.59 ($false),
% 0.49/0.59 inference(theory_lemma,[status(thm)],[415, 32, 403, 350, 408, 413])).
% 0.49/0.59 % SZS output end Proof
%------------------------------------------------------------------------------