TSTP Solution File: DAT021_1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : DAT021_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 14:36:24 EDT 2022
% Result : Theorem 0.19s 0.38s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : DAT021_1 : TPTP v8.1.0. Released v5.0.0.
% 0.06/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.33 % Computer : n023.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Wed Aug 31 02:04:23 EDT 2022
% 0.13/0.33 % CPUTime :
% 0.13/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.34 Usage: tptp [options] [-file:]file
% 0.13/0.34 -h, -? prints this message.
% 0.13/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.34 -m, -model generate model.
% 0.13/0.34 -p, -proof generate proof.
% 0.13/0.34 -c, -core generate unsat core of named formulas.
% 0.13/0.34 -st, -statistics display statistics.
% 0.13/0.34 -t:timeout set timeout (in second).
% 0.13/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.34 -<param>:<value> configuration parameter and value.
% 0.13/0.34 -o:<output-file> file to place output in.
% 0.19/0.38 % SZS status Theorem
% 0.19/0.38 % SZS output start Proof
% 0.19/0.38 tff(in_type, type, (
% 0.19/0.38 in: ( $int * collection ) > $o)).
% 0.19/0.38 tff(add_type, type, (
% 0.19/0.38 add: ( $int * collection ) > collection)).
% 0.19/0.38 tff(empty_type, type, (
% 0.19/0.38 empty: collection)).
% 0.19/0.38 tff(tptp_fun_V_1_type, type, (
% 0.19/0.38 tptp_fun_V_1: $int)).
% 0.19/0.38 tff(tptp_fun_W_0_type, type, (
% 0.19/0.38 tptp_fun_W_0: $int)).
% 0.19/0.38 tff(tptp_fun_U_2_type, type, (
% 0.19/0.38 tptp_fun_U_2: collection)).
% 0.19/0.38 tff(1,assumption,(~((in(W!0, add(1, empty)) | (W!0 = 3)) <=> in(W!0, add(3, add(1, empty))))), introduced(assumption)).
% 0.19/0.38 tff(2,plain,
% 0.19/0.38 (^[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))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(3,plain,
% 0.19/0.38 (![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)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[2])).
% 0.19/0.38 tff(4,plain,
% 0.19/0.38 (^[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))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(5,plain,
% 0.19/0.38 (![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)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[4])).
% 0.19/0.38 tff(6,plain,
% 0.19/0.38 (^[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))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(7,plain,
% 0.19/0.38 (![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)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[6])).
% 0.19/0.38 tff(8,plain,
% 0.19/0.38 (![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)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(9,plain,
% 0.19/0.38 (^[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))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(10,plain,
% 0.19/0.38 (![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)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[9])).
% 0.19/0.38 tff(11,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')).
% 0.19/0.38 tff(12,plain,
% 0.19/0.38 (![Z: $int, X1: collection, X2: $int] : (((Z = X2) | in(Z, X1)) <=> in(Z, add(X2, X1)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.19/0.38 tff(13,plain,
% 0.19/0.38 (![Z: $int, X1: collection, X2: $int] : (((Z = X2) | in(Z, X1)) <=> in(Z, add(X2, X1)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.19/0.38 tff(14,plain,
% 0.19/0.38 (![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(Z, $product(-1, X2)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[13, 7])).
% 0.19/0.38 tff(15,plain,
% 0.19/0.38 (![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[14, 5])).
% 0.19/0.38 tff(16,plain,(
% 0.19/0.38 ![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.19/0.38 inference(skolemize,[status(sab)],[15])).
% 0.19/0.38 tff(17,plain,
% 0.19/0.38 (![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[16, 3])).
% 0.19/0.38 tff(18,plain,
% 0.19/0.38 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, add(1, empty)) | (W!0 = 3)) <=> in(W!0, add(3, add(1, empty))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, add(1, empty)) | (W!0 = 3)) <=> in(W!0, add(3, add(1, empty)))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(19,plain,
% 0.19/0.39 (((in(W!0, add(1, empty)) | ($sum(3, $product(-1, W!0)) = 0)) <=> in(W!0, add(3, add(1, empty)))) <=> ((in(W!0, add(1, empty)) | (W!0 = 3)) <=> in(W!0, add(3, add(1, empty))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(20,plain,
% 0.19/0.39 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, add(1, empty)) | ($sum(3, $product(-1, W!0)) = 0)) <=> in(W!0, add(3, add(1, empty))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, add(1, empty)) | (W!0 = 3)) <=> in(W!0, add(3, add(1, empty)))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[19])).
% 0.19/0.39 tff(21,plain,
% 0.19/0.39 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, add(1, empty)) | ($sum(3, $product(-1, W!0)) = 0)) <=> in(W!0, add(3, add(1, empty))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, add(1, empty)) | (W!0 = 3)) <=> in(W!0, add(3, add(1, empty)))))),
% 0.19/0.39 inference(transitivity,[status(thm)],[20, 18])).
% 0.19/0.39 tff(22,plain,
% 0.19/0.39 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, add(1, empty)) | ($sum(3, $product(-1, W!0)) = 0)) <=> in(W!0, add(3, add(1, empty))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(23,plain,
% 0.19/0.39 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, add(1, empty)) | (W!0 = 3)) <=> in(W!0, add(3, add(1, empty))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[22, 21])).
% 0.19/0.39 tff(24,plain,
% 0.19/0.39 ($false),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[23, 17, 1])).
% 0.19/0.39 tff(25,plain,((in(W!0, add(1, empty)) | (W!0 = 3)) <=> in(W!0, add(3, add(1, empty)))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.39 tff(26,plain,
% 0.19/0.39 ((~![U: collection, V: $int, W: $int] : ((~$greatereq($sum(W, V), 9)) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~($sum(W, $product(-1, V)) = 0)))))) <=> (~![U: collection, V: $int, W: $int] : ((~$greatereq($sum(W, V), 9)) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~($sum(W, $product(-1, V)) = 0))))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(27,plain,
% 0.19/0.39 ((~![U: collection, V: $int, W: $int] : ((~$greatereq($sum(V, W), 9)) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~($sum(V, $product(-1, W)) = 0)))))) <=> (~![U: collection, V: $int, W: $int] : ((~$greatereq($sum(W, V), 9)) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~($sum(W, $product(-1, V)) = 0))))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(28,plain,
% 0.19/0.39 ((~![U: collection, V: $int, W: $int] : ((~$lesseq(9, $sum(V, W))) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~(V = W)))))) <=> (~![U: collection, V: $int, W: $int] : ((~$greatereq($sum(V, W), 9)) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~($sum(V, $product(-1, W)) = 0))))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(29,plain,
% 0.19/0.39 ((~![U: collection, V: $int, W: $int] : ((~$lesseq(9, $sum(V, W))) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~(V = W)))))) <=> (~![U: collection, V: $int, W: $int] : ((~$lesseq(9, $sum(V, W))) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~(V = W))))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(30,plain,
% 0.19/0.39 ((~![U: collection, V: $int, W: $int] : (((((U = add(5, add(3, add(1, empty)))) & in(V, U)) & in(W, U)) & (~(V = W))) => $less($sum(V, W), 9))) <=> (~![U: collection, V: $int, W: $int] : ((~$lesseq(9, $sum(V, W))) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~(V = W))))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(31,axiom,(~![U: collection, V: $int, W: $int] : (((((U = add(5, add(3, add(1, empty)))) & in(V, U)) & in(W, U)) & (~(V = W))) => $less($sum(V, W), 9))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','co1')).
% 0.19/0.39 tff(32,plain,
% 0.19/0.39 (~![U: collection, V: $int, W: $int] : ((~$lesseq(9, $sum(V, W))) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~(V = W)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[31, 30])).
% 0.19/0.39 tff(33,plain,
% 0.19/0.39 (~![U: collection, V: $int, W: $int] : ((~$lesseq(9, $sum(V, W))) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~(V = W)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[32, 29])).
% 0.19/0.39 tff(34,plain,
% 0.19/0.39 (~![U: collection, V: $int, W: $int] : ((~$lesseq(9, $sum(V, W))) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~(V = W)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[33, 29])).
% 0.19/0.39 tff(35,plain,
% 0.19/0.39 (~![U: collection, V: $int, W: $int] : ((~$lesseq(9, $sum(V, W))) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~(V = W)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[34, 29])).
% 0.19/0.39 tff(36,plain,
% 0.19/0.39 (~![U: collection, V: $int, W: $int] : ((~$greatereq($sum(V, W), 9)) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~($sum(V, $product(-1, W)) = 0)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[35, 28])).
% 0.19/0.39 tff(37,plain,
% 0.19/0.39 (~![U: collection, V: $int, W: $int] : ((~$greatereq($sum(W, V), 9)) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~($sum(W, $product(-1, V)) = 0)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[36, 27])).
% 0.19/0.39 tff(38,plain,
% 0.19/0.39 (~![U: collection, V: $int, W: $int] : ((~$greatereq($sum(W, V), 9)) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~($sum(W, $product(-1, V)) = 0)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[37, 26])).
% 0.19/0.39 tff(39,plain,
% 0.19/0.39 (~![U: collection, V: $int, W: $int] : ((~$greatereq($sum(W, V), 9)) | (~((U = add(5, add(3, add(1, empty)))) & in(V, U) & in(W, U) & (~($sum(W, $product(-1, V)) = 0)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[38, 26])).
% 0.19/0.39 tff(40,plain,(
% 0.19/0.39 ~((~$greatereq($sum(W!0, V!1), 9)) | (~((U!2 = add(5, add(3, add(1, empty)))) & in(V!1, U!2) & in(W!0, U!2) & (~($sum(W!0, $product(-1, V!1)) = 0)))))),
% 0.19/0.39 inference(skolemize,[status(sab)],[39])).
% 0.19/0.39 tff(41,plain,
% 0.19/0.39 ($greatereq($sum(W!0, V!1), 9)),
% 0.19/0.39 inference(or_elim,[status(thm)],[40])).
% 0.19/0.39 tff(42,assumption,(~$lesseq(V!1, 5)), introduced(assumption)).
% 0.19/0.39 tff(43,plain,
% 0.19/0.39 ((~$lesseq(V!1, 1)) | $lesseq(V!1, 5)),
% 0.19/0.39 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.39 tff(44,plain,
% 0.19/0.39 (~$lesseq(V!1, 1)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[43, 42])).
% 0.19/0.39 tff(45,plain,
% 0.19/0.39 ((~(V!1 = 1)) | $lesseq(V!1, 1)),
% 0.19/0.39 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.39 tff(46,plain,
% 0.19/0.39 (~(V!1 = 1)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[45, 44])).
% 0.19/0.39 tff(47,assumption,(in(V!1, empty)), introduced(assumption)).
% 0.19/0.39 tff(48,plain,
% 0.19/0.39 (^[U: $int] : refl((~in(U, empty)) <=> (~in(U, empty)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(49,plain,
% 0.19/0.39 (![U: $int] : (~in(U, empty)) <=> ![U: $int] : (~in(U, empty))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[48])).
% 0.19/0.39 tff(50,plain,
% 0.19/0.39 (![U: $int] : (~in(U, empty)) <=> ![U: $int] : (~in(U, empty))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(51,axiom,(![U: $int] : (~in(U, empty))), file('/export/starexec/sandbox/benchmark/Axioms/DAT002=0.ax','ax1')).
% 0.19/0.39 tff(52,plain,
% 0.19/0.39 (![U: $int] : (~in(U, empty))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[51, 50])).
% 0.19/0.39 tff(53,plain,(
% 0.19/0.39 ![U: $int] : (~in(U, empty))),
% 0.19/0.39 inference(skolemize,[status(sab)],[52])).
% 0.19/0.39 tff(54,plain,
% 0.19/0.39 (![U: $int] : (~in(U, empty))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[53, 49])).
% 0.19/0.39 tff(55,plain,
% 0.19/0.39 ((~![U: $int] : (~in(U, empty))) | (~in(V!1, empty))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(56,plain,
% 0.19/0.39 ($false),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[55, 54, 47])).
% 0.19/0.39 tff(57,plain,(~in(V!1, empty)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.39 tff(58,plain,
% 0.19/0.39 ((~(in(V!1, empty) | (V!1 = 1))) | in(V!1, empty) | (V!1 = 1)),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(59,plain,
% 0.19/0.39 ((~(in(V!1, empty) | (V!1 = 1))) | (V!1 = 1)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[58, 57])).
% 0.19/0.39 tff(60,plain,
% 0.19/0.39 (~(in(V!1, empty) | (V!1 = 1))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[59, 46])).
% 0.19/0.39 tff(61,assumption,(~((in(V!1, add(1, empty)) | (V!1 = 3)) <=> in(V!1, add(3, add(1, empty))))), introduced(assumption)).
% 0.19/0.39 tff(62,plain,
% 0.19/0.39 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, add(1, empty)) | (V!1 = 3)) <=> in(V!1, add(3, add(1, empty))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, add(1, empty)) | (V!1 = 3)) <=> in(V!1, add(3, add(1, empty)))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(63,plain,
% 0.19/0.39 (((in(V!1, add(1, empty)) | ($sum(3, $product(-1, V!1)) = 0)) <=> in(V!1, add(3, add(1, empty)))) <=> ((in(V!1, add(1, empty)) | (V!1 = 3)) <=> in(V!1, add(3, add(1, empty))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(64,plain,
% 0.19/0.39 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, add(1, empty)) | ($sum(3, $product(-1, V!1)) = 0)) <=> in(V!1, add(3, add(1, empty))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, add(1, empty)) | (V!1 = 3)) <=> in(V!1, add(3, add(1, empty)))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[63])).
% 0.19/0.39 tff(65,plain,
% 0.19/0.39 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, add(1, empty)) | ($sum(3, $product(-1, V!1)) = 0)) <=> in(V!1, add(3, add(1, empty))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, add(1, empty)) | (V!1 = 3)) <=> in(V!1, add(3, add(1, empty)))))),
% 0.19/0.39 inference(transitivity,[status(thm)],[64, 62])).
% 0.19/0.39 tff(66,plain,
% 0.19/0.39 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, add(1, empty)) | ($sum(3, $product(-1, V!1)) = 0)) <=> in(V!1, add(3, add(1, empty))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(67,plain,
% 0.19/0.39 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, add(1, empty)) | (V!1 = 3)) <=> in(V!1, add(3, add(1, empty))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[66, 65])).
% 0.19/0.39 tff(68,plain,
% 0.19/0.39 ($false),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[67, 17, 61])).
% 0.19/0.39 tff(69,plain,((in(V!1, add(1, empty)) | (V!1 = 3)) <=> in(V!1, add(3, add(1, empty)))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.39 tff(70,plain,
% 0.19/0.39 ((~(V!1 = 5)) | $lesseq(V!1, 5)),
% 0.19/0.39 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.39 tff(71,plain,
% 0.19/0.39 (~(V!1 = 5)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[70, 42])).
% 0.19/0.39 tff(72,plain,
% 0.19/0.39 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, add(3, add(1, empty))) | (V!1 = 5)) <=> in(V!1, add(5, add(3, add(1, empty)))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, add(3, add(1, empty))) | (V!1 = 5)) <=> in(V!1, add(5, add(3, add(1, empty))))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(73,plain,
% 0.19/0.39 (((in(V!1, add(3, add(1, empty))) | ($sum(5, $product(-1, V!1)) = 0)) <=> in(V!1, add(5, add(3, add(1, empty))))) <=> ((in(V!1, add(3, add(1, empty))) | (V!1 = 5)) <=> in(V!1, add(5, add(3, add(1, empty)))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(74,plain,
% 0.19/0.39 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, add(3, add(1, empty))) | ($sum(5, $product(-1, V!1)) = 0)) <=> in(V!1, add(5, add(3, add(1, empty)))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, add(3, add(1, empty))) | (V!1 = 5)) <=> in(V!1, add(5, add(3, add(1, empty))))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[73])).
% 0.19/0.39 tff(75,plain,
% 0.19/0.39 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, add(3, add(1, empty))) | ($sum(5, $product(-1, V!1)) = 0)) <=> in(V!1, add(5, add(3, add(1, empty)))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, add(3, add(1, empty))) | (V!1 = 5)) <=> in(V!1, add(5, add(3, add(1, empty))))))),
% 0.19/0.39 inference(transitivity,[status(thm)],[74, 72])).
% 0.19/0.39 tff(76,plain,
% 0.19/0.39 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, add(3, add(1, empty))) | ($sum(5, $product(-1, V!1)) = 0)) <=> in(V!1, add(5, add(3, add(1, empty)))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(77,plain,
% 0.19/0.39 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, add(3, add(1, empty))) | (V!1 = 5)) <=> in(V!1, add(5, add(3, add(1, empty)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.19/0.39 tff(78,plain,
% 0.19/0.39 ((in(V!1, add(3, add(1, empty))) | (V!1 = 5)) <=> in(V!1, add(5, add(3, add(1, empty))))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[77, 17])).
% 0.19/0.39 tff(79,plain,
% 0.19/0.39 ((U!2 = add(5, add(3, add(1, empty)))) & in(V!1, U!2) & in(W!0, U!2) & (~($sum(W!0, $product(-1, V!1)) = 0))),
% 0.19/0.39 inference(or_elim,[status(thm)],[40])).
% 0.19/0.39 tff(80,plain,
% 0.19/0.39 (U!2 = add(5, add(3, add(1, empty)))),
% 0.19/0.39 inference(and_elim,[status(thm)],[79])).
% 0.19/0.39 tff(81,plain,
% 0.19/0.39 (add(5, add(3, add(1, empty))) = U!2),
% 0.19/0.39 inference(symmetry,[status(thm)],[80])).
% 0.19/0.39 tff(82,plain,
% 0.19/0.39 (in(V!1, add(5, add(3, add(1, empty)))) <=> in(V!1, U!2)),
% 0.19/0.39 inference(monotonicity,[status(thm)],[81])).
% 0.19/0.39 tff(83,plain,
% 0.19/0.39 (in(V!1, U!2) <=> in(V!1, add(5, add(3, add(1, empty))))),
% 0.19/0.39 inference(symmetry,[status(thm)],[82])).
% 0.19/0.39 tff(84,plain,
% 0.19/0.39 (in(V!1, U!2)),
% 0.19/0.39 inference(and_elim,[status(thm)],[79])).
% 0.19/0.39 tff(85,plain,
% 0.19/0.39 (in(V!1, add(5, add(3, add(1, empty))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.19/0.39 tff(86,plain,
% 0.19/0.39 ((~((in(V!1, add(3, add(1, empty))) | (V!1 = 5)) <=> in(V!1, add(5, add(3, add(1, empty)))))) | (in(V!1, add(3, add(1, empty))) | (V!1 = 5)) | (~in(V!1, add(5, add(3, add(1, empty)))))),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(87,plain,
% 0.19/0.39 (in(V!1, add(3, add(1, empty))) | (V!1 = 5)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[86, 85, 78])).
% 0.19/0.39 tff(88,plain,
% 0.19/0.39 ((~(in(V!1, add(3, add(1, empty))) | (V!1 = 5))) | in(V!1, add(3, add(1, empty))) | (V!1 = 5)),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(89,plain,
% 0.19/0.39 (in(V!1, add(3, add(1, empty))) | (V!1 = 5)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[88, 87])).
% 0.19/0.39 tff(90,plain,
% 0.19/0.39 (in(V!1, add(3, add(1, empty)))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[89, 71])).
% 0.19/0.39 tff(91,plain,
% 0.19/0.39 ((~((in(V!1, add(1, empty)) | (V!1 = 3)) <=> in(V!1, add(3, add(1, empty))))) | (in(V!1, add(1, empty)) | (V!1 = 3)) | (~in(V!1, add(3, add(1, empty))))),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(92,plain,
% 0.19/0.39 (in(V!1, add(1, empty)) | (V!1 = 3)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[91, 90, 69])).
% 0.19/0.39 tff(93,plain,
% 0.19/0.39 ((~$lesseq(V!1, 3)) | $lesseq(V!1, 5)),
% 0.19/0.39 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.39 tff(94,plain,
% 0.19/0.39 (~$lesseq(V!1, 3)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[93, 42])).
% 0.19/0.39 tff(95,plain,
% 0.19/0.39 ((~(V!1 = 3)) | $lesseq(V!1, 3)),
% 0.19/0.39 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.39 tff(96,plain,
% 0.19/0.39 (~(V!1 = 3)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[95, 94])).
% 0.19/0.39 tff(97,plain,
% 0.19/0.39 ((~(in(V!1, add(1, empty)) | (V!1 = 3))) | in(V!1, add(1, empty)) | (V!1 = 3)),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(98,plain,
% 0.19/0.39 (in(V!1, add(1, empty))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[97, 96, 92])).
% 0.19/0.39 tff(99,assumption,(~((in(V!1, empty) | (V!1 = 1)) <=> in(V!1, add(1, empty)))), introduced(assumption)).
% 0.19/0.40 tff(100,plain,
% 0.19/0.40 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, empty) | (V!1 = 1)) <=> in(V!1, add(1, empty)))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, empty) | (V!1 = 1)) <=> in(V!1, add(1, empty))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(101,plain,
% 0.19/0.40 (((in(V!1, empty) | ($sum(1, $product(-1, V!1)) = 0)) <=> in(V!1, add(1, empty))) <=> ((in(V!1, empty) | (V!1 = 1)) <=> in(V!1, add(1, empty)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(102,plain,
% 0.19/0.40 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, empty) | ($sum(1, $product(-1, V!1)) = 0)) <=> in(V!1, add(1, empty)))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, empty) | (V!1 = 1)) <=> in(V!1, add(1, empty))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[101])).
% 0.19/0.40 tff(103,plain,
% 0.19/0.40 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, empty) | ($sum(1, $product(-1, V!1)) = 0)) <=> in(V!1, add(1, empty)))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, empty) | (V!1 = 1)) <=> in(V!1, add(1, empty))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[102, 100])).
% 0.19/0.40 tff(104,plain,
% 0.19/0.40 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, empty) | ($sum(1, $product(-1, V!1)) = 0)) <=> in(V!1, add(1, empty)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(105,plain,
% 0.19/0.40 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(V!1, empty) | (V!1 = 1)) <=> in(V!1, add(1, empty)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[104, 103])).
% 0.19/0.40 tff(106,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[105, 17, 99])).
% 0.19/0.40 tff(107,plain,((in(V!1, empty) | (V!1 = 1)) <=> in(V!1, add(1, empty))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.40 tff(108,plain,
% 0.19/0.40 ((~((in(V!1, empty) | (V!1 = 1)) <=> in(V!1, add(1, empty)))) | (in(V!1, empty) | (V!1 = 1)) | (~in(V!1, add(1, empty)))),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(109,plain,
% 0.19/0.40 ((in(V!1, empty) | (V!1 = 1)) | (~in(V!1, add(1, empty)))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[108, 107])).
% 0.19/0.40 tff(110,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[109, 98, 60])).
% 0.19/0.40 tff(111,plain,($lesseq(V!1, 5)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.40 tff(112,assumption,($lesseq(W!0, 3)), introduced(assumption)).
% 0.19/0.40 tff(113,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(theory_lemma,[status(thm)],[112, 111, 41])).
% 0.19/0.40 tff(114,plain,(~$lesseq(W!0, 3)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.40 tff(115,plain,
% 0.19/0.40 ((~(W!0 = 3)) | $lesseq(W!0, 3)),
% 0.19/0.40 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.40 tff(116,plain,
% 0.19/0.40 (~(W!0 = 3)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[115, 114])).
% 0.19/0.40 tff(117,assumption,(~((in(W!0, empty) | (W!0 = 1)) <=> in(W!0, add(1, empty)))), introduced(assumption)).
% 0.19/0.40 tff(118,plain,
% 0.19/0.40 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, empty) | (W!0 = 1)) <=> in(W!0, add(1, empty)))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, empty) | (W!0 = 1)) <=> in(W!0, add(1, empty))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(119,plain,
% 0.19/0.40 (((in(W!0, empty) | ($sum(1, $product(-1, W!0)) = 0)) <=> in(W!0, add(1, empty))) <=> ((in(W!0, empty) | (W!0 = 1)) <=> in(W!0, add(1, empty)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(120,plain,
% 0.19/0.40 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, empty) | ($sum(1, $product(-1, W!0)) = 0)) <=> in(W!0, add(1, empty)))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, empty) | (W!0 = 1)) <=> in(W!0, add(1, empty))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[119])).
% 0.19/0.40 tff(121,plain,
% 0.19/0.40 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, empty) | ($sum(1, $product(-1, W!0)) = 0)) <=> in(W!0, add(1, empty)))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, empty) | (W!0 = 1)) <=> in(W!0, add(1, empty))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[120, 118])).
% 0.19/0.40 tff(122,plain,
% 0.19/0.40 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, empty) | ($sum(1, $product(-1, W!0)) = 0)) <=> in(W!0, add(1, empty)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(123,plain,
% 0.19/0.40 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, empty) | (W!0 = 1)) <=> in(W!0, add(1, empty)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[122, 121])).
% 0.19/0.40 tff(124,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[123, 17, 117])).
% 0.19/0.40 tff(125,plain,((in(W!0, empty) | (W!0 = 1)) <=> in(W!0, add(1, empty))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.40 tff(126,plain,
% 0.19/0.40 ((~$lesseq(W!0, 1)) | $lesseq(W!0, 3)),
% 0.19/0.40 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.40 tff(127,plain,
% 0.19/0.40 (~$lesseq(W!0, 1)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[126, 114])).
% 0.19/0.40 tff(128,plain,
% 0.19/0.40 ((~(W!0 = 1)) | $lesseq(W!0, 1)),
% 0.19/0.40 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.40 tff(129,plain,
% 0.19/0.40 (~(W!0 = 1)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[128, 127])).
% 0.19/0.40 tff(130,assumption,(in(W!0, empty)), introduced(assumption)).
% 0.19/0.40 tff(131,plain,
% 0.19/0.40 ((~![U: $int] : (~in(U, empty))) | (~in(W!0, empty))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(132,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[131, 54, 130])).
% 0.19/0.40 tff(133,plain,(~in(W!0, empty)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.40 tff(134,plain,
% 0.19/0.40 ((~(in(W!0, empty) | (W!0 = 1))) | in(W!0, empty) | (W!0 = 1)),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(135,plain,
% 0.19/0.40 (~(in(W!0, empty) | (W!0 = 1))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[134, 133, 129])).
% 0.19/0.40 tff(136,plain,
% 0.19/0.40 ((~((in(W!0, empty) | (W!0 = 1)) <=> in(W!0, add(1, empty)))) | (in(W!0, empty) | (W!0 = 1)) | (~in(W!0, add(1, empty)))),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(137,plain,
% 0.19/0.40 ((~((in(W!0, empty) | (W!0 = 1)) <=> in(W!0, add(1, empty)))) | (~in(W!0, add(1, empty)))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[136, 135])).
% 0.19/0.40 tff(138,plain,
% 0.19/0.40 (~in(W!0, add(1, empty))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[137, 125])).
% 0.19/0.40 tff(139,plain,
% 0.19/0.40 ((~(in(W!0, add(1, empty)) | (W!0 = 3))) | in(W!0, add(1, empty)) | (W!0 = 3)),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(140,plain,
% 0.19/0.40 (~(in(W!0, add(1, empty)) | (W!0 = 3))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[139, 138, 116])).
% 0.19/0.40 tff(141,plain,
% 0.19/0.40 ((~((in(W!0, add(1, empty)) | (W!0 = 3)) <=> in(W!0, add(3, add(1, empty))))) | (in(W!0, add(1, empty)) | (W!0 = 3)) | (~in(W!0, add(3, add(1, empty))))),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(142,plain,
% 0.19/0.40 ((~((in(W!0, add(1, empty)) | (W!0 = 3)) <=> in(W!0, add(3, add(1, empty))))) | (~in(W!0, add(3, add(1, empty))))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[141, 140])).
% 0.19/0.40 tff(143,plain,
% 0.19/0.40 (~in(W!0, add(3, add(1, empty)))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[142, 25])).
% 0.19/0.40 tff(144,plain,
% 0.19/0.40 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, add(3, add(1, empty))) | (W!0 = 5)) <=> in(W!0, add(5, add(3, add(1, empty)))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, add(3, add(1, empty))) | (W!0 = 5)) <=> in(W!0, add(5, add(3, add(1, empty))))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(145,plain,
% 0.19/0.40 (((in(W!0, add(3, add(1, empty))) | ($sum(5, $product(-1, W!0)) = 0)) <=> in(W!0, add(5, add(3, add(1, empty))))) <=> ((in(W!0, add(3, add(1, empty))) | (W!0 = 5)) <=> in(W!0, add(5, add(3, add(1, empty)))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(146,plain,
% 0.19/0.40 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, add(3, add(1, empty))) | ($sum(5, $product(-1, W!0)) = 0)) <=> in(W!0, add(5, add(3, add(1, empty)))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, add(3, add(1, empty))) | (W!0 = 5)) <=> in(W!0, add(5, add(3, add(1, empty))))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[145])).
% 0.19/0.40 tff(147,plain,
% 0.19/0.40 (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, add(3, add(1, empty))) | ($sum(5, $product(-1, W!0)) = 0)) <=> in(W!0, add(5, add(3, add(1, empty)))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, add(3, add(1, empty))) | (W!0 = 5)) <=> in(W!0, add(5, add(3, add(1, empty))))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[146, 144])).
% 0.19/0.40 tff(148,plain,
% 0.19/0.40 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, add(3, add(1, empty))) | ($sum(5, $product(-1, W!0)) = 0)) <=> in(W!0, add(5, add(3, add(1, empty)))))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(149,plain,
% 0.19/0.40 ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(W!0, add(3, add(1, empty))) | (W!0 = 5)) <=> in(W!0, add(5, add(3, add(1, empty)))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[148, 147])).
% 0.19/0.40 tff(150,plain,
% 0.19/0.40 ((in(W!0, add(3, add(1, empty))) | (W!0 = 5)) <=> in(W!0, add(5, add(3, add(1, empty))))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[149, 17])).
% 0.19/0.40 tff(151,plain,
% 0.19/0.40 (in(W!0, add(5, add(3, add(1, empty)))) <=> in(W!0, U!2)),
% 0.19/0.40 inference(monotonicity,[status(thm)],[81])).
% 0.19/0.40 tff(152,plain,
% 0.19/0.40 (in(W!0, U!2) <=> in(W!0, add(5, add(3, add(1, empty))))),
% 0.19/0.40 inference(symmetry,[status(thm)],[151])).
% 0.19/0.40 tff(153,plain,
% 0.19/0.40 (in(W!0, U!2)),
% 0.19/0.40 inference(and_elim,[status(thm)],[79])).
% 0.19/0.40 tff(154,plain,
% 0.19/0.40 (in(W!0, add(5, add(3, add(1, empty))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[153, 152])).
% 0.19/0.40 tff(155,plain,
% 0.19/0.40 ((~((in(W!0, add(3, add(1, empty))) | (W!0 = 5)) <=> in(W!0, add(5, add(3, add(1, empty)))))) | (in(W!0, add(3, add(1, empty))) | (W!0 = 5)) | (~in(W!0, add(5, add(3, add(1, empty)))))),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(156,plain,
% 0.19/0.40 (in(W!0, add(3, add(1, empty))) | (W!0 = 5)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[155, 154, 150])).
% 0.19/0.40 tff(157,plain,
% 0.19/0.40 ((~(in(W!0, add(3, add(1, empty))) | (W!0 = 5))) | in(W!0, add(3, add(1, empty))) | (W!0 = 5)),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(158,plain,
% 0.19/0.40 (in(W!0, add(3, add(1, empty))) | (W!0 = 5)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[157, 156])).
% 0.19/0.40 tff(159,plain,
% 0.19/0.40 (W!0 = 5),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[158, 143])).
% 0.19/0.40 tff(160,plain,
% 0.19/0.40 ((~(W!0 = 5)) | $lesseq(W!0, 5)),
% 0.19/0.40 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.40 tff(161,plain,
% 0.19/0.40 ($lesseq(W!0, 5)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[160, 159])).
% 0.19/0.40 tff(162,plain,
% 0.19/0.40 ((~(W!0 = 5)) | $greatereq(W!0, 5)),
% 0.19/0.40 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.40 tff(163,plain,
% 0.19/0.40 ($greatereq(W!0, 5)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[162, 159])).
% 0.19/0.40 tff(164,assumption,(~$greatereq($sum(W!0, $product(-1, V!1)), 0)), introduced(assumption)).
% 0.19/0.40 tff(165,assumption,($greatereq(W!0, 5)), introduced(assumption)).
% 0.19/0.40 tff(166,assumption,($lesseq(V!1, 5)), introduced(assumption)).
% 0.19/0.40 tff(167,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(theory_lemma,[status(thm)],[166, 165, 164])).
% 0.19/0.40 tff(168,plain,((~$greatereq(W!0, 5)) | (~$lesseq(V!1, 5)) | $greatereq($sum(W!0, $product(-1, V!1)), 0)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.41 tff(169,plain,
% 0.19/0.41 ((~$greatereq(W!0, 5)) | $greatereq($sum(W!0, $product(-1, V!1)), 0)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[168, 111])).
% 0.19/0.41 tff(170,plain,
% 0.19/0.41 ($greatereq($sum(W!0, $product(-1, V!1)), 0)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[169, 163])).
% 0.19/0.41 tff(171,plain,
% 0.19/0.41 (~($sum(W!0, $product(-1, V!1)) = 0)),
% 0.19/0.41 inference(and_elim,[status(thm)],[79])).
% 0.19/0.41 tff(172,plain,
% 0.19/0.41 (($sum(W!0, $product(-1, V!1)) = 0) | (~$lesseq($sum(W!0, $product(-1, V!1)), 0)) | (~$greatereq($sum(W!0, $product(-1, V!1)), 0))),
% 0.19/0.41 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.41 tff(173,plain,
% 0.19/0.41 ((~$lesseq($sum(W!0, $product(-1, V!1)), 0)) | (~$greatereq($sum(W!0, $product(-1, V!1)), 0))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[172, 171])).
% 0.19/0.41 tff(174,plain,
% 0.19/0.41 (~$lesseq($sum(W!0, $product(-1, V!1)), 0)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[173, 170])).
% 0.19/0.41 tff(175,plain,
% 0.19/0.41 ((~$greatereq(V!1, 5)) | $lesseq($sum(W!0, $product(-1, V!1)), 0) | (~$lesseq(W!0, 5))),
% 0.19/0.41 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.41 tff(176,plain,
% 0.19/0.41 (~$greatereq(V!1, 5)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[175, 174, 161])).
% 0.19/0.41 tff(177,plain,
% 0.19/0.41 ((~(V!1 = 5)) | $greatereq(V!1, 5)),
% 0.19/0.41 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.41 tff(178,plain,
% 0.19/0.41 (~(V!1 = 5)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[177, 176])).
% 0.19/0.41 tff(179,plain,
% 0.19/0.41 (in(V!1, add(3, add(1, empty)))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[89, 178])).
% 0.19/0.41 tff(180,plain,
% 0.19/0.41 ((in(V!1, add(1, empty)) | (V!1 = 3)) | (~in(V!1, add(3, add(1, empty))))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[91, 69])).
% 0.19/0.41 tff(181,plain,
% 0.19/0.41 (in(V!1, add(1, empty)) | (V!1 = 3)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[180, 179])).
% 0.19/0.41 tff(182,assumption,($lesseq(V!1, 3)), introduced(assumption)).
% 0.19/0.41 tff(183,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(theory_lemma,[status(thm)],[161, 182, 41])).
% 0.19/0.41 tff(184,plain,(~$lesseq(V!1, 3)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.41 tff(185,plain,
% 0.19/0.41 (~(V!1 = 3)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[95, 184])).
% 0.19/0.41 tff(186,plain,
% 0.19/0.41 (in(V!1, add(1, empty))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[97, 185, 181])).
% 0.19/0.41 tff(187,plain,
% 0.19/0.41 ($lesseq(V!1, 3) | (~$lesseq(V!1, 1))),
% 0.19/0.41 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.41 tff(188,plain,
% 0.19/0.41 (~$lesseq(V!1, 1)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[187, 184])).
% 0.19/0.41 tff(189,plain,
% 0.19/0.41 (~(V!1 = 1)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[45, 188])).
% 0.19/0.41 tff(190,plain,
% 0.19/0.41 (~(in(V!1, empty) | (V!1 = 1))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[59, 189])).
% 0.19/0.41 tff(191,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[109, 190, 186])).
% 0.19/0.41 % SZS output end Proof
%------------------------------------------------------------------------------