TSTP Solution File: DAT027_1 by Z3---4.8.9.0

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Z3---4.8.9.0
% Problem  : DAT027_1 : TPTP v8.1.0. Released v5.0.0.
% Transfm  : none
% Format   : tptp
% Command  : z3_tptp -proof -model -t:%d -file:%s

% Computer : n005.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 14:36:25 EDT 2022

% Result   : Theorem 0.12s 0.37s
% Output   : Proof 0.12s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.11  % Problem  : DAT027_1 : TPTP v8.1.0. Released v5.0.0.
% 0.06/0.12  % Command  : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33  % Computer : n005.cluster.edu
% 0.12/0.33  % Model    : x86_64 x86_64
% 0.12/0.33  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33  % Memory   : 8042.1875MB
% 0.12/0.33  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33  % CPULimit : 300
% 0.12/0.33  % WCLimit  : 300
% 0.12/0.33  % DateTime : Wed Aug 31 01:43:35 EDT 2022
% 0.12/0.33  % CPUTime  : 
% 0.12/0.34  Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34  Usage: tptp [options] [-file:]file
% 0.12/0.34    -h, -?       prints this message.
% 0.12/0.34    -smt2        print SMT-LIB2 benchmark.
% 0.12/0.34    -m, -model   generate model.
% 0.12/0.34    -p, -proof   generate proof.
% 0.12/0.34    -c, -core    generate unsat core of named formulas.
% 0.12/0.34    -st, -statistics display statistics.
% 0.12/0.34    -t:timeout   set timeout (in second).
% 0.12/0.34    -smt2status  display status in smt2 format instead of SZS.
% 0.12/0.34    -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34    -<param>:<value> configuration parameter and value.
% 0.12/0.34    -o:<output-file> file to place output in.
% 0.12/0.37  % SZS status Theorem
% 0.12/0.37  % SZS output start Proof
% 0.12/0.37  tff(in_type, type, (
% 0.12/0.37     in: ( $int * collection ) > $o)).
% 0.12/0.37  tff(tptp_fun_V_2_type, type, (
% 0.12/0.37     tptp_fun_V_2: collection)).
% 0.12/0.37  tff(tptp_fun_Z_4_type, type, (
% 0.12/0.37     tptp_fun_Z_4: $int)).
% 0.12/0.37  tff(tptp_fun_W_1_type, type, (
% 0.12/0.37     tptp_fun_W_1: $int)).
% 0.12/0.37  tff(remove_type, type, (
% 0.12/0.37     remove: ( $int * collection ) > collection)).
% 0.12/0.37  tff(tptp_fun_X_0_type, type, (
% 0.12/0.37     tptp_fun_X_0: $int)).
% 0.12/0.37  tff(add_type, type, (
% 0.12/0.37     add: ( $int * collection ) > collection)).
% 0.12/0.37  tff(tptp_fun_U_3_type, type, (
% 0.12/0.37     tptp_fun_U_3: collection)).
% 0.12/0.37  tff(1,plain,
% 0.12/0.37      (^[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))))),
% 0.12/0.37      inference(bind,[status(th)],[])).
% 0.12/0.37  tff(2,plain,
% 0.12/0.37      (![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)))),
% 0.12/0.37      inference(quant_intro,[status(thm)],[1])).
% 0.12/0.37  tff(3,plain,
% 0.12/0.37      (^[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))))),
% 0.12/0.37      inference(bind,[status(th)],[])).
% 0.12/0.37  tff(4,plain,
% 0.12/0.37      (![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)))),
% 0.12/0.37      inference(quant_intro,[status(thm)],[3])).
% 0.12/0.37  tff(5,plain,
% 0.12/0.37      (^[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))))),
% 0.12/0.37      inference(bind,[status(th)],[])).
% 0.12/0.37  tff(6,plain,
% 0.12/0.37      (![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)))),
% 0.12/0.37      inference(quant_intro,[status(thm)],[5])).
% 0.12/0.37  tff(7,plain,
% 0.12/0.37      (^[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))))),
% 0.12/0.37      inference(bind,[status(th)],[])).
% 0.12/0.37  tff(8,plain,
% 0.12/0.37      (![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)))),
% 0.12/0.37      inference(quant_intro,[status(thm)],[7])).
% 0.12/0.37  tff(9,plain,
% 0.12/0.37      (![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)))),
% 0.12/0.37      inference(rewrite,[status(thm)],[])).
% 0.12/0.37  tff(10,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')).
% 0.12/0.37  tff(11,plain,
% 0.12/0.37      (![X3: $int, X4: collection, X5: $int] : ((in(X3, X4) & (~(X3 = X5))) <=> in(X3, remove(X5, X4)))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[10, 9])).
% 0.12/0.37  tff(12,plain,
% 0.12/0.37      (![X3: $int, X4: collection, X5: $int] : ((in(X3, X4) & (~($sum(X3, $product(-1, X5)) = 0))) <=> in(X3, remove(X5, X4)))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[11, 8])).
% 0.12/0.37  tff(13,plain,
% 0.12/0.37      (![X3: $int, X4: collection, X5: $int] : ((in(X3, X4) & (~($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[12, 6])).
% 0.12/0.37  tff(14,plain,(
% 0.12/0.37      ![X3: $int, X4: collection, X5: $int] : ((in(X3, X4) & (~($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))),
% 0.12/0.37      inference(skolemize,[status(sab)],[13])).
% 0.12/0.37  tff(15,plain,
% 0.12/0.37      (![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[14, 4])).
% 0.12/0.37  tff(16,plain,
% 0.12/0.37      (![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[15, 2])).
% 0.12/0.37  tff(17,plain,
% 0.12/0.37      ((~![X3: $int, X4: collection, X5: $int] : ((~((~in(X3, X4)) | ($sum(X5, $product(-1, X3)) = 0))) <=> in(X3, remove(X5, X4)))) | ((~((~in(Z!4, V!2)) | ($sum(W!1, $product(-1, Z!4)) = 0))) <=> in(Z!4, remove(W!1, V!2)))),
% 0.12/0.37      inference(quant_inst,[status(thm)],[])).
% 0.12/0.37  tff(18,plain,
% 0.12/0.37      ((~((~in(Z!4, V!2)) | ($sum(W!1, $product(-1, Z!4)) = 0))) <=> in(Z!4, remove(W!1, V!2))),
% 0.12/0.37      inference(unit_resolution,[status(thm)],[17, 16])).
% 0.12/0.37  tff(19,plain,
% 0.12/0.37      (^[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))))),
% 0.12/0.37      inference(bind,[status(th)],[])).
% 0.12/0.37  tff(20,plain,
% 0.12/0.37      (![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)))),
% 0.12/0.37      inference(quant_intro,[status(thm)],[19])).
% 0.12/0.37  tff(21,plain,
% 0.12/0.37      (^[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))))),
% 0.12/0.37      inference(bind,[status(th)],[])).
% 0.12/0.37  tff(22,plain,
% 0.12/0.37      (![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)))),
% 0.12/0.37      inference(quant_intro,[status(thm)],[21])).
% 0.12/0.37  tff(23,plain,
% 0.12/0.37      (^[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))))),
% 0.12/0.37      inference(bind,[status(th)],[])).
% 0.12/0.37  tff(24,plain,
% 0.12/0.37      (![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)))),
% 0.12/0.37      inference(quant_intro,[status(thm)],[23])).
% 0.12/0.37  tff(25,plain,
% 0.12/0.37      (![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)))),
% 0.12/0.37      inference(rewrite,[status(thm)],[])).
% 0.12/0.37  tff(26,plain,
% 0.12/0.37      (^[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))))),
% 0.12/0.37      inference(bind,[status(th)],[])).
% 0.12/0.37  tff(27,plain,
% 0.12/0.37      (![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)))),
% 0.12/0.37      inference(quant_intro,[status(thm)],[26])).
% 0.12/0.37  tff(28,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')).
% 0.12/0.37  tff(29,plain,
% 0.12/0.37      (![Z: $int, X1: collection, X2: $int] : (((Z = X2) | in(Z, X1)) <=> in(Z, add(X2, X1)))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[28, 27])).
% 0.12/0.37  tff(30,plain,
% 0.12/0.37      (![Z: $int, X1: collection, X2: $int] : (((Z = X2) | in(Z, X1)) <=> in(Z, add(X2, X1)))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[29, 25])).
% 0.12/0.37  tff(31,plain,
% 0.12/0.37      (![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(Z, $product(-1, X2)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[30, 24])).
% 0.12/0.37  tff(32,plain,
% 0.12/0.37      (![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[31, 22])).
% 0.12/0.37  tff(33,plain,(
% 0.12/0.37      ![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.12/0.37      inference(skolemize,[status(sab)],[32])).
% 0.12/0.37  tff(34,plain,
% 0.12/0.37      (![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[33, 20])).
% 0.12/0.37  tff(35,plain,
% 0.12/0.37      (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(Z!4, remove(W!1, V!2)) | ($sum(X!0, $sum(W!1, $product(-1, Z!4))) = 0)) <=> in(Z!4, add($sum(X!0, W!1), remove(W!1, V!2))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(Z!4, remove(W!1, V!2)) | ($sum(X!0, $sum(W!1, $product(-1, Z!4))) = 0)) <=> in(Z!4, add($sum(X!0, W!1), remove(W!1, V!2)))))),
% 0.12/0.37      inference(rewrite,[status(thm)],[])).
% 0.12/0.37  tff(36,plain,
% 0.12/0.37      (((in(Z!4, remove(W!1, V!2)) | ($sum($sum(X!0, W!1), $product(-1, Z!4)) = 0)) <=> in(Z!4, add($sum(X!0, W!1), remove(W!1, V!2)))) <=> ((in(Z!4, remove(W!1, V!2)) | ($sum(X!0, $sum(W!1, $product(-1, Z!4))) = 0)) <=> in(Z!4, add($sum(X!0, W!1), remove(W!1, V!2))))),
% 0.12/0.37      inference(rewrite,[status(thm)],[])).
% 0.12/0.37  tff(37,plain,
% 0.12/0.37      (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(Z!4, remove(W!1, V!2)) | ($sum($sum(X!0, W!1), $product(-1, Z!4)) = 0)) <=> in(Z!4, add($sum(X!0, W!1), remove(W!1, V!2))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(Z!4, remove(W!1, V!2)) | ($sum(X!0, $sum(W!1, $product(-1, Z!4))) = 0)) <=> in(Z!4, add($sum(X!0, W!1), remove(W!1, V!2)))))),
% 0.12/0.37      inference(monotonicity,[status(thm)],[36])).
% 0.12/0.37  tff(38,plain,
% 0.12/0.37      (((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(Z!4, remove(W!1, V!2)) | ($sum($sum(X!0, W!1), $product(-1, Z!4)) = 0)) <=> in(Z!4, add($sum(X!0, W!1), remove(W!1, V!2))))) <=> ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(Z!4, remove(W!1, V!2)) | ($sum(X!0, $sum(W!1, $product(-1, Z!4))) = 0)) <=> in(Z!4, add($sum(X!0, W!1), remove(W!1, V!2)))))),
% 0.12/0.37      inference(transitivity,[status(thm)],[37, 35])).
% 0.12/0.37  tff(39,plain,
% 0.12/0.37      ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(Z!4, remove(W!1, V!2)) | ($sum($sum(X!0, W!1), $product(-1, Z!4)) = 0)) <=> in(Z!4, add($sum(X!0, W!1), remove(W!1, V!2))))),
% 0.12/0.37      inference(quant_inst,[status(thm)],[])).
% 0.12/0.37  tff(40,plain,
% 0.12/0.37      ((~![Z: $int, X1: collection, X2: $int] : ((in(Z, X1) | ($sum(X2, $product(-1, Z)) = 0)) <=> in(Z, add(X2, X1)))) | ((in(Z!4, remove(W!1, V!2)) | ($sum(X!0, $sum(W!1, $product(-1, Z!4))) = 0)) <=> in(Z!4, add($sum(X!0, W!1), remove(W!1, V!2))))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[39, 38])).
% 0.12/0.37  tff(41,plain,
% 0.12/0.37      ((in(Z!4, remove(W!1, V!2)) | ($sum(X!0, $sum(W!1, $product(-1, Z!4))) = 0)) <=> in(Z!4, add($sum(X!0, W!1), remove(W!1, V!2)))),
% 0.12/0.37      inference(unit_resolution,[status(thm)],[40, 34])).
% 0.12/0.37  tff(42,plain,
% 0.12/0.37      (((![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2))) & in(W!1, V!2) & $greatereq(X!0, 0) & (U!3 = add($sum(X!0, W!1), remove(W!1, V!2)))) & (~((~$lesseq(Z!4, 0)) | (~in(Z!4, U!3))))) <=> (![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2))) & in(W!1, V!2) & $greatereq(X!0, 0) & (U!3 = add($sum(X!0, W!1), remove(W!1, V!2))) & (~((~$lesseq(Z!4, 0)) | (~in(Z!4, U!3)))))),
% 0.12/0.37      inference(rewrite,[status(thm)],[])).
% 0.12/0.37  tff(43,plain,
% 0.12/0.37      ((![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2))) & in(W!1, V!2) & $greatereq(X!0, 0) & (U!3 = add($sum(X!0, W!1), remove(W!1, V!2)))) <=> (![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2))) & in(W!1, V!2) & $greatereq(X!0, 0) & (U!3 = add($sum(X!0, W!1), remove(W!1, V!2))))),
% 0.12/0.37      inference(rewrite,[status(thm)],[])).
% 0.12/0.37  tff(44,plain,
% 0.12/0.37      (((![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2))) & in(W!1, V!2) & $greatereq(X!0, 0) & (U!3 = add($sum(X!0, W!1), remove(W!1, V!2)))) & (~((~$lesseq(Z!4, 0)) | (~in(Z!4, U!3))))) <=> ((![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2))) & in(W!1, V!2) & $greatereq(X!0, 0) & (U!3 = add($sum(X!0, W!1), remove(W!1, V!2)))) & (~((~$lesseq(Z!4, 0)) | (~in(Z!4, U!3)))))),
% 0.12/0.37      inference(monotonicity,[status(thm)],[43])).
% 0.12/0.37  tff(45,plain,
% 0.12/0.37      (((![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2))) & in(W!1, V!2) & $greatereq(X!0, 0) & (U!3 = add($sum(X!0, W!1), remove(W!1, V!2)))) & (~((~$lesseq(Z!4, 0)) | (~in(Z!4, U!3))))) <=> (![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2))) & in(W!1, V!2) & $greatereq(X!0, 0) & (U!3 = add($sum(X!0, W!1), remove(W!1, V!2))) & (~((~$lesseq(Z!4, 0)) | (~in(Z!4, U!3)))))),
% 0.12/0.37      inference(transitivity,[status(thm)],[44, 42])).
% 0.12/0.37  tff(46,plain,
% 0.12/0.37      ((~![U: collection, V: collection, W: $int, X: $int] : ((~(![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V))) & in(W, V) & $greatereq(X, 0) & (U = add($sum(X, W), remove(W, V))))) | ![Z: $int] : ((~$lesseq(Z, 0)) | (~in(Z, U))))) <=> (~![U: collection, V: collection, W: $int, X: $int] : ((~(![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V))) & in(W, V) & $greatereq(X, 0) & (U = add($sum(X, W), remove(W, V))))) | ![Z: $int] : ((~$lesseq(Z, 0)) | (~in(Z, U)))))),
% 0.12/0.37      inference(rewrite,[status(thm)],[])).
% 0.12/0.37  tff(47,plain,
% 0.12/0.37      ((~![U: collection, V: collection, W: $int, X: $int] : ((~(![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V))) & in(W, V) & $greatereq(X, 0) & (U = add($sum(W, X), remove(W, V))))) | ![Z: $int] : ((~$lesseq(Z, 0)) | (~in(Z, U))))) <=> (~![U: collection, V: collection, W: $int, X: $int] : ((~(![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V))) & in(W, V) & $greatereq(X, 0) & (U = add($sum(X, W), remove(W, V))))) | ![Z: $int] : ((~$lesseq(Z, 0)) | (~in(Z, U)))))),
% 0.12/0.37      inference(rewrite,[status(thm)],[])).
% 0.12/0.37  tff(48,plain,
% 0.12/0.37      ((~![U: collection, V: collection, W: $int, X: $int] : ((~(![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V))) & in(W, V) & $greatereq(X, 0) & (U = add($sum(W, X), remove(W, V))))) | ![Z: $int] : ((~$lesseq(Z, 0)) | (~in(Z, U))))) <=> (~![U: collection, V: collection, W: $int, X: $int] : ((~(![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V))) & in(W, V) & $greatereq(X, 0) & (U = add($sum(W, X), remove(W, V))))) | ![Z: $int] : ((~$lesseq(Z, 0)) | (~in(Z, U)))))),
% 0.12/0.37      inference(rewrite,[status(thm)],[])).
% 0.12/0.37  tff(49,plain,
% 0.12/0.37      ((~![U: collection, V: collection, W: $int, X: $int] : ((((![Y: $int] : (in(Y, V) => $greater(Y, 0)) & in(W, V)) & $greatereq(X, 0)) & (U = add($sum(W, X), remove(W, V)))) => ![Z: $int] : (in(Z, U) => $greater(Z, 0)))) <=> (~![U: collection, V: collection, W: $int, X: $int] : ((~(![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V))) & in(W, V) & $greatereq(X, 0) & (U = add($sum(W, X), remove(W, V))))) | ![Z: $int] : ((~$lesseq(Z, 0)) | (~in(Z, U)))))),
% 0.12/0.37      inference(rewrite,[status(thm)],[])).
% 0.12/0.37  tff(50,axiom,(~![U: collection, V: collection, W: $int, X: $int] : ((((![Y: $int] : (in(Y, V) => $greater(Y, 0)) & in(W, V)) & $greatereq(X, 0)) & (U = add($sum(W, X), remove(W, V)))) => ![Z: $int] : (in(Z, U) => $greater(Z, 0)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','co1')).
% 0.12/0.37  tff(51,plain,
% 0.12/0.37      (~![U: collection, V: collection, W: $int, X: $int] : ((~(![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V))) & in(W, V) & $greatereq(X, 0) & (U = add($sum(W, X), remove(W, V))))) | ![Z: $int] : ((~$lesseq(Z, 0)) | (~in(Z, U))))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[50, 49])).
% 0.12/0.37  tff(52,plain,
% 0.12/0.37      (~![U: collection, V: collection, W: $int, X: $int] : ((~(![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V))) & in(W, V) & $greatereq(X, 0) & (U = add($sum(W, X), remove(W, V))))) | ![Z: $int] : ((~$lesseq(Z, 0)) | (~in(Z, U))))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[51, 48])).
% 0.12/0.37  tff(53,plain,
% 0.12/0.37      (~![U: collection, V: collection, W: $int, X: $int] : ((~(![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V))) & in(W, V) & $greatereq(X, 0) & (U = add($sum(W, X), remove(W, V))))) | ![Z: $int] : ((~$lesseq(Z, 0)) | (~in(Z, U))))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[52, 48])).
% 0.12/0.37  tff(54,plain,
% 0.12/0.37      (~![U: collection, V: collection, W: $int, X: $int] : ((~(![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V))) & in(W, V) & $greatereq(X, 0) & (U = add($sum(W, X), remove(W, V))))) | ![Z: $int] : ((~$lesseq(Z, 0)) | (~in(Z, U))))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[53, 48])).
% 0.12/0.37  tff(55,plain,
% 0.12/0.37      (~![U: collection, V: collection, W: $int, X: $int] : ((~(![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V))) & in(W, V) & $greatereq(X, 0) & (U = add($sum(W, X), remove(W, V))))) | ![Z: $int] : ((~$lesseq(Z, 0)) | (~in(Z, U))))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[54, 48])).
% 0.12/0.37  tff(56,plain,
% 0.12/0.37      (~![U: collection, V: collection, W: $int, X: $int] : ((~(![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V))) & in(W, V) & $greatereq(X, 0) & (U = add($sum(X, W), remove(W, V))))) | ![Z: $int] : ((~$lesseq(Z, 0)) | (~in(Z, U))))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[55, 47])).
% 0.12/0.37  tff(57,plain,
% 0.12/0.37      (~![U: collection, V: collection, W: $int, X: $int] : ((~(![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V))) & in(W, V) & $greatereq(X, 0) & (U = add($sum(X, W), remove(W, V))))) | ![Z: $int] : ((~$lesseq(Z, 0)) | (~in(Z, U))))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[56, 46])).
% 0.12/0.37  tff(58,plain,
% 0.12/0.37      (~![U: collection, V: collection, W: $int, X: $int] : ((~(![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V))) & in(W, V) & $greatereq(X, 0) & (U = add($sum(X, W), remove(W, V))))) | ![Z: $int] : ((~$lesseq(Z, 0)) | (~in(Z, U))))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[57, 46])).
% 0.12/0.37  tff(59,plain,
% 0.12/0.37      (![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2))) & in(W!1, V!2) & $greatereq(X!0, 0) & (U!3 = add($sum(X!0, W!1), remove(W!1, V!2))) & (~((~$lesseq(Z!4, 0)) | (~in(Z!4, U!3))))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[58, 45])).
% 0.12/0.37  tff(60,plain,
% 0.12/0.37      (U!3 = add($sum(X!0, W!1), remove(W!1, V!2))),
% 0.12/0.37      inference(and_elim,[status(thm)],[59])).
% 0.12/0.37  tff(61,plain,
% 0.12/0.37      (add($sum(X!0, W!1), remove(W!1, V!2)) = U!3),
% 0.12/0.37      inference(symmetry,[status(thm)],[60])).
% 0.12/0.37  tff(62,plain,
% 0.12/0.37      (in(Z!4, add($sum(X!0, W!1), remove(W!1, V!2))) <=> in(Z!4, U!3)),
% 0.12/0.37      inference(monotonicity,[status(thm)],[61])).
% 0.12/0.37  tff(63,plain,
% 0.12/0.37      (in(Z!4, U!3) <=> in(Z!4, add($sum(X!0, W!1), remove(W!1, V!2)))),
% 0.12/0.37      inference(symmetry,[status(thm)],[62])).
% 0.12/0.37  tff(64,plain,
% 0.12/0.37      (~((~$lesseq(Z!4, 0)) | (~in(Z!4, U!3)))),
% 0.12/0.37      inference(and_elim,[status(thm)],[59])).
% 0.12/0.37  tff(65,plain,
% 0.12/0.37      (in(Z!4, U!3)),
% 0.12/0.37      inference(or_elim,[status(thm)],[64])).
% 0.12/0.37  tff(66,plain,
% 0.12/0.37      (in(Z!4, add($sum(X!0, W!1), remove(W!1, V!2)))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[65, 63])).
% 0.12/0.37  tff(67,plain,
% 0.12/0.37      ((~((in(Z!4, remove(W!1, V!2)) | ($sum(X!0, $sum(W!1, $product(-1, Z!4))) = 0)) <=> in(Z!4, add($sum(X!0, W!1), remove(W!1, V!2))))) | (in(Z!4, remove(W!1, V!2)) | ($sum(X!0, $sum(W!1, $product(-1, Z!4))) = 0)) | (~in(Z!4, add($sum(X!0, W!1), remove(W!1, V!2))))),
% 0.12/0.37      inference(tautology,[status(thm)],[])).
% 0.12/0.37  tff(68,plain,
% 0.12/0.37      (in(Z!4, remove(W!1, V!2)) | ($sum(X!0, $sum(W!1, $product(-1, Z!4))) = 0)),
% 0.12/0.37      inference(unit_resolution,[status(thm)],[67, 66, 41])).
% 0.12/0.37  tff(69,plain,
% 0.12/0.37      (in(W!1, V!2)),
% 0.12/0.37      inference(and_elim,[status(thm)],[59])).
% 0.12/0.37  tff(70,plain,
% 0.12/0.37      (^[Y: $int] : refl(((~$lesseq(Y, 0)) | (~in(Y, V!2))) <=> ((~$lesseq(Y, 0)) | (~in(Y, V!2))))),
% 0.12/0.37      inference(bind,[status(th)],[])).
% 0.12/0.37  tff(71,plain,
% 0.12/0.37      (![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2))) <=> ![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2)))),
% 0.12/0.37      inference(quant_intro,[status(thm)],[70])).
% 0.12/0.37  tff(72,plain,
% 0.12/0.37      (![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2)))),
% 0.12/0.37      inference(and_elim,[status(thm)],[59])).
% 0.12/0.37  tff(73,plain,
% 0.12/0.37      (![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2)))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[72, 71])).
% 0.12/0.37  tff(74,plain,
% 0.12/0.37      (((~![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2)))) | ((~$lesseq(W!1, 0)) | (~in(W!1, V!2)))) <=> ((~![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2)))) | (~$lesseq(W!1, 0)) | (~in(W!1, V!2)))),
% 0.12/0.37      inference(rewrite,[status(thm)],[])).
% 0.12/0.37  tff(75,plain,
% 0.12/0.37      ((~![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2)))) | ((~$lesseq(W!1, 0)) | (~in(W!1, V!2)))),
% 0.12/0.37      inference(quant_inst,[status(thm)],[])).
% 0.12/0.37  tff(76,plain,
% 0.12/0.37      ((~![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2)))) | (~$lesseq(W!1, 0)) | (~in(W!1, V!2))),
% 0.12/0.37      inference(modus_ponens,[status(thm)],[75, 74])).
% 0.12/0.37  tff(77,plain,
% 0.12/0.37      (~$lesseq(W!1, 0)),
% 0.12/0.37      inference(unit_resolution,[status(thm)],[76, 73, 69])).
% 0.12/0.37  tff(78,plain,
% 0.12/0.37      ($greatereq(X!0, 0)),
% 0.12/0.37      inference(and_elim,[status(thm)],[59])).
% 0.12/0.38  tff(79,plain,
% 0.12/0.38      ($lesseq(Z!4, 0)),
% 0.12/0.38      inference(or_elim,[status(thm)],[64])).
% 0.12/0.38  tff(80,plain,
% 0.12/0.38      ((~$lesseq($sum(X!0, $sum(W!1, $product(-1, Z!4))), 0)) | (~$greatereq(X!0, 0)) | $lesseq(W!1, 0) | (~$lesseq(Z!4, 0))),
% 0.12/0.38      inference(theory_lemma,[status(thm)],[])).
% 0.12/0.38  tff(81,plain,
% 0.12/0.38      (~$lesseq($sum(X!0, $sum(W!1, $product(-1, Z!4))), 0)),
% 0.12/0.38      inference(unit_resolution,[status(thm)],[80, 79, 78, 77])).
% 0.12/0.38  tff(82,plain,
% 0.12/0.38      ((~($sum(X!0, $sum(W!1, $product(-1, Z!4))) = 0)) | $lesseq($sum(X!0, $sum(W!1, $product(-1, Z!4))), 0)),
% 0.12/0.38      inference(theory_lemma,[status(thm)],[])).
% 0.12/0.38  tff(83,plain,
% 0.12/0.38      (~($sum(X!0, $sum(W!1, $product(-1, Z!4))) = 0)),
% 0.12/0.38      inference(unit_resolution,[status(thm)],[82, 81])).
% 0.12/0.38  tff(84,plain,
% 0.12/0.38      ((~(in(Z!4, remove(W!1, V!2)) | ($sum(X!0, $sum(W!1, $product(-1, Z!4))) = 0))) | in(Z!4, remove(W!1, V!2)) | ($sum(X!0, $sum(W!1, $product(-1, Z!4))) = 0)),
% 0.12/0.38      inference(tautology,[status(thm)],[])).
% 0.12/0.38  tff(85,plain,
% 0.12/0.38      (in(Z!4, remove(W!1, V!2))),
% 0.12/0.38      inference(unit_resolution,[status(thm)],[84, 83, 68])).
% 0.12/0.38  tff(86,plain,
% 0.12/0.38      ((~((~((~in(Z!4, V!2)) | ($sum(W!1, $product(-1, Z!4)) = 0))) <=> in(Z!4, remove(W!1, V!2)))) | (~((~in(Z!4, V!2)) | ($sum(W!1, $product(-1, Z!4)) = 0))) | (~in(Z!4, remove(W!1, V!2)))),
% 0.12/0.38      inference(tautology,[status(thm)],[])).
% 0.12/0.38  tff(87,plain,
% 0.12/0.38      ((~((~((~in(Z!4, V!2)) | ($sum(W!1, $product(-1, Z!4)) = 0))) <=> in(Z!4, remove(W!1, V!2)))) | (~((~in(Z!4, V!2)) | ($sum(W!1, $product(-1, Z!4)) = 0)))),
% 0.12/0.38      inference(unit_resolution,[status(thm)],[86, 85])).
% 0.12/0.38  tff(88,plain,
% 0.12/0.38      (~((~in(Z!4, V!2)) | ($sum(W!1, $product(-1, Z!4)) = 0))),
% 0.12/0.38      inference(unit_resolution,[status(thm)],[87, 18])).
% 0.12/0.38  tff(89,plain,
% 0.12/0.38      (((~in(Z!4, V!2)) | ($sum(W!1, $product(-1, Z!4)) = 0)) | in(Z!4, V!2)),
% 0.12/0.38      inference(tautology,[status(thm)],[])).
% 0.12/0.38  tff(90,plain,
% 0.12/0.38      (in(Z!4, V!2)),
% 0.12/0.38      inference(unit_resolution,[status(thm)],[89, 88])).
% 0.12/0.38  tff(91,plain,
% 0.12/0.38      (((~![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2)))) | ((~$lesseq(Z!4, 0)) | (~in(Z!4, V!2)))) <=> ((~![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2)))) | (~$lesseq(Z!4, 0)) | (~in(Z!4, V!2)))),
% 0.12/0.38      inference(rewrite,[status(thm)],[])).
% 0.12/0.38  tff(92,plain,
% 0.12/0.38      ((~![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2)))) | ((~$lesseq(Z!4, 0)) | (~in(Z!4, V!2)))),
% 0.12/0.38      inference(quant_inst,[status(thm)],[])).
% 0.12/0.38  tff(93,plain,
% 0.12/0.38      ((~![Y: $int] : ((~$lesseq(Y, 0)) | (~in(Y, V!2)))) | (~$lesseq(Z!4, 0)) | (~in(Z!4, V!2))),
% 0.12/0.38      inference(modus_ponens,[status(thm)],[92, 91])).
% 0.12/0.38  tff(94,plain,
% 0.12/0.38      ($false),
% 0.12/0.38      inference(unit_resolution,[status(thm)],[93, 73, 79, 90])).
% 0.12/0.38  % SZS output end Proof
%------------------------------------------------------------------------------