TSTP Solution File: SWW677_1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SWW677_1 : TPTP v8.1.0. Released v6.4.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n010.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 : Thu Sep 29 20:59:38 EDT 2022
% Result : Theorem 0.64s 0.66s
% Output : Proof 0.64s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : SWW677_1 : TPTP v8.1.0. Released v6.4.0.
% 0.11/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.34 % Computer : n010.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Sun Sep 4 20:45:27 EDT 2022
% 0.12/0.34 % 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.
% 0.64/0.66 % SZS status Theorem
% 0.64/0.66 % SZS output start Proof
% 0.64/0.66 tff(div2_type, type, (
% 0.64/0.66 div2: $int > $int)).
% 0.64/0.66 tff(tptp_fun_U_5_type, type, (
% 0.64/0.66 tptp_fun_U_5: $int)).
% 0.64/0.66 tff(tptp_fun_L_6_type, type, (
% 0.64/0.66 tptp_fun_L_6: $int)).
% 0.64/0.66 tff(length_type, type, (
% 0.64/0.66 length: 'Array[Int,Int]' > $int)).
% 0.64/0.66 tff(tptp_fun_A_4_type, type, (
% 0.64/0.66 tptp_fun_A_4: 'Array[Int,Int]')).
% 0.64/0.66 tff(sorted_type, type, (
% 0.64/0.66 sorted: ( 'Array[Int,Int]' * $int * $int ) > $o)).
% 0.64/0.66 tff(tptp_fun__select__Array_Int_Int__Int__Int__type, type, (
% 0.64/0.66 tptp_fun__select__Array_Int_Int__Int__Int_: ( 'Array[Int,Int]' * $int ) > $int)).
% 0.64/0.66 tff(tptp_fun_E_3_type, type, (
% 0.64/0.66 tptp_fun_E_3: $int)).
% 0.64/0.66 tff(1,assumption,(~$lesseq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), -1)), introduced(assumption)).
% 0.64/0.66 tff(2,plain,
% 0.64/0.66 (($lesseq($sum(L!6, $product(-1, U!5)), 0) & ($greatereq(L!6, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))) & ![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | ((~($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0)) & (((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) & ![L_2: $int] : (~(($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) | ($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) & ![U_3: $int] : (~(($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L!6, 0) & (~$greatereq($sum(U_3, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))))))) <=> ($lesseq($sum(L!6, $product(-1, U!5)), 0) & $greatereq(L!6, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4))) & ![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | ((~($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0)) & (((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) & ![L_2: $int] : (~(($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) | ($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) & ![U_3: $int] : (~(($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L!6, 0) & (~$greatereq($sum(U_3, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4))))))))))),
% 0.64/0.66 inference(rewrite,[status(thm)],[])).
% 0.64/0.66 tff(3,plain,
% 0.64/0.66 (^[M_1: $int] : rewrite(((~($sum(M_1, $product(-1, div2($sum(U!5, L!6)))) = 0)) | ((~($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0)) & (((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) & ![L_2: $int] : (~(($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$greatereq($sum(U!5, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) | ((~(~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0))) & ![U_3: $int] : (~(($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L!6, 0) & (~$greatereq($sum(U_3, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4))))))))) <=> ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | ((~($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0)) & (((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) & ![L_2: $int] : (~(($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) | ($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) & ![U_3: $int] : (~(($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L!6, 0) & (~$greatereq($sum(U_3, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4))))))))))),
% 0.64/0.66 inference(bind,[status(th)],[])).
% 0.64/0.66 tff(4,plain,
% 0.64/0.66 (![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(U!5, L!6)))) = 0)) | ((~($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0)) & (((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) & ![L_2: $int] : (~(($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$greatereq($sum(U!5, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) | ((~(~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0))) & ![U_3: $int] : (~(($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L!6, 0) & (~$greatereq($sum(U_3, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4))))))))) <=> ![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | ((~($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0)) & (((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) & ![L_2: $int] : (~(($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) | ($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) & ![U_3: $int] : (~(($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L!6, 0) & (~$greatereq($sum(U_3, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))))))),
% 0.64/0.66 inference(quant_intro,[status(thm)],[3])).
% 0.64/0.66 tff(5,plain,
% 0.64/0.66 ((~(~($greatereq(L!6, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) <=> ($greatereq(L!6, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4))))),
% 0.64/0.66 inference(rewrite,[status(thm)],[])).
% 0.64/0.66 tff(6,plain,
% 0.64/0.66 ((~($greatereq(L!6, 0) & (~$greatereq($sum(U!5, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4))))) <=> (~($greatereq(L!6, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))),
% 0.64/0.66 inference(rewrite,[status(thm)],[])).
% 0.64/0.66 tff(7,plain,
% 0.64/0.66 ((~(~($greatereq(L!6, 0) & (~$greatereq($sum(U!5, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) <=> (~(~($greatereq(L!6, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4))))))),
% 0.64/0.66 inference(monotonicity,[status(thm)],[6])).
% 0.64/0.66 tff(8,plain,
% 0.64/0.66 ((~(~($greatereq(L!6, 0) & (~$greatereq($sum(U!5, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) <=> ($greatereq(L!6, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4))))),
% 0.64/0.66 inference(transitivity,[status(thm)],[7, 5])).
% 0.64/0.66 tff(9,plain,
% 0.64/0.66 ((~(~$lesseq($sum(L!6, $product(-1, U!5)), 0))) <=> $lesseq($sum(L!6, $product(-1, U!5)), 0)),
% 0.64/0.66 inference(rewrite,[status(thm)],[])).
% 0.64/0.66 tff(10,plain,
% 0.64/0.66 ((~$greatereq($sum(U!5, $product(-1, L!6)), 0)) <=> (~$lesseq($sum(L!6, $product(-1, U!5)), 0))),
% 0.64/0.66 inference(rewrite,[status(thm)],[])).
% 0.64/0.66 tff(11,plain,
% 0.64/0.66 ((~(~$greatereq($sum(U!5, $product(-1, L!6)), 0))) <=> (~(~$lesseq($sum(L!6, $product(-1, U!5)), 0)))),
% 0.64/0.66 inference(monotonicity,[status(thm)],[10])).
% 0.64/0.66 tff(12,plain,
% 0.64/0.66 ((~(~$greatereq($sum(U!5, $product(-1, L!6)), 0))) <=> $lesseq($sum(L!6, $product(-1, U!5)), 0)),
% 0.64/0.66 inference(transitivity,[status(thm)],[11, 9])).
% 0.64/0.66 tff(13,plain,
% 0.64/0.66 (((~(~$greatereq($sum(U!5, $product(-1, L!6)), 0))) & (~(~($greatereq(L!6, 0) & (~$greatereq($sum(U!5, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) & ![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(U!5, L!6)))) = 0)) | ((~($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0)) & (((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) & ![L_2: $int] : (~(($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$greatereq($sum(U!5, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) | ((~(~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0))) & ![U_3: $int] : (~(($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L!6, 0) & (~$greatereq($sum(U_3, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))))))) <=> ($lesseq($sum(L!6, $product(-1, U!5)), 0) & ($greatereq(L!6, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))) & ![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | ((~($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0)) & (((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) & ![L_2: $int] : (~(($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) | ($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) & ![U_3: $int] : (~(($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L!6, 0) & (~$greatereq($sum(U_3, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4))))))))))),
% 0.64/0.66 inference(monotonicity,[status(thm)],[12, 8, 4])).
% 0.64/0.66 tff(14,plain,
% 0.64/0.66 (((~(~$greatereq($sum(U!5, $product(-1, L!6)), 0))) & (~(~($greatereq(L!6, 0) & (~$greatereq($sum(U!5, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) & ![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(U!5, L!6)))) = 0)) | ((~($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0)) & (((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) & ![L_2: $int] : (~(($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$greatereq($sum(U!5, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) | ((~(~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0))) & ![U_3: $int] : (~(($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L!6, 0) & (~$greatereq($sum(U_3, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))))))) <=> ($lesseq($sum(L!6, $product(-1, U!5)), 0) & $greatereq(L!6, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4))) & ![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | ((~($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0)) & (((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) & ![L_2: $int] : (~(($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) | ($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) & ![U_3: $int] : (~(($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L!6, 0) & (~$greatereq($sum(U_3, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4))))))))))),
% 0.64/0.66 inference(transitivity,[status(thm)],[13, 2])).
% 0.64/0.66 tff(15,plain,
% 0.64/0.66 ((~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$greatereq($sum(U, $product(-1, L)), 0)) | (~($greatereq(L, 0) & (~$greatereq($sum(U, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A))))) | ?[M_1: $int] : (($sum(M_1, $product(-1, div2($sum(U, L)))) = 0) & (($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))) = 0) | (($lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0) | ?[L_2: $int] : (($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$greatereq($sum(U, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0)) | ?[U_3: $int] : (($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L, 0) & (~$greatereq($sum(U_3, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A)))))))))) <=> (~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$greatereq($sum(U, $product(-1, L)), 0)) | (~($greatereq(L, 0) & (~$greatereq($sum(U, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A))))) | ?[M_1: $int] : (($sum(M_1, $product(-1, div2($sum(U, L)))) = 0) & (($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))) = 0) | (($lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0) | ?[L_2: $int] : (($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$greatereq($sum(U, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0)) | ?[U_3: $int] : (($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L, 0) & (~$greatereq($sum(U_3, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A))))))))))),
% 0.64/0.66 inference(rewrite,[status(thm)],[])).
% 0.64/0.66 tff(16,plain,
% 0.64/0.66 ((~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$lesseq($sum(L, $product(-1, U)), 0)) | (~($greatereq(L, 0) & (~$lesseq($sum(length(A), $product(-1, U)), 0)) & sorted(A, 0, $sum(-1, length(A))))) | ?[M_1: $int] : (($sum(M_1, $product(-1, div2($sum(L, U)))) = 0) & (($sum(tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1), $product(-1, E)) = 0) | (($lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0) | ?[L_2: $int] : (($sum(L_2, $product(-1, M_1)) = 1) & $greatereq(L_2, 0) & (~$lesseq($sum(length(A), $product(-1, U)), 0)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0)) | ?[U_3: $int] : (($sum(U_3, $product(-1, M_1)) = -1) & $greatereq(L, 0) & (~$lesseq($sum(length(A), $product(-1, U_3)), 0)) & sorted(A, 0, $sum(-1, length(A)))))))))) <=> (~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$greatereq($sum(U, $product(-1, L)), 0)) | (~($greatereq(L, 0) & (~$greatereq($sum(U, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A))))) | ?[M_1: $int] : (($sum(M_1, $product(-1, div2($sum(U, L)))) = 0) & (($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))) = 0) | (($lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0) | ?[L_2: $int] : (($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$greatereq($sum(U, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0)) | ?[U_3: $int] : (($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L, 0) & (~$greatereq($sum(U_3, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A))))))))))),
% 0.64/0.66 inference(rewrite,[status(thm)],[])).
% 0.64/0.66 tff(17,plain,
% 0.64/0.66 ((~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$lesseq(L, U)) | ?[M_1: $int] : ((M_1 = div2($sum(L, U))) & ((tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1) = E) | (($lesseq(E, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1)) | ?[L_2: $int] : ((L_2 = $sum(1, M_1)) & $lesseq(0, L_2) & (~$lesseq(length(A), U)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq(E, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))) | ?[U_3: $int] : ((U_3 = $sum(-1, M_1)) & $lesseq(0, L) & (~$lesseq(length(A), U_3)) & sorted(A, 0, $sum(-1, length(A)))))))) | (~($lesseq(0, L) & (~$lesseq(length(A), U)) & sorted(A, 0, $sum(-1, length(A))))))) <=> (~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$lesseq($sum(L, $product(-1, U)), 0)) | (~($greatereq(L, 0) & (~$lesseq($sum(length(A), $product(-1, U)), 0)) & sorted(A, 0, $sum(-1, length(A))))) | ?[M_1: $int] : (($sum(M_1, $product(-1, div2($sum(L, U)))) = 0) & (($sum(tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1), $product(-1, E)) = 0) | (($lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0) | ?[L_2: $int] : (($sum(L_2, $product(-1, M_1)) = 1) & $greatereq(L_2, 0) & (~$lesseq($sum(length(A), $product(-1, U)), 0)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0)) | ?[U_3: $int] : (($sum(U_3, $product(-1, M_1)) = -1) & $greatereq(L, 0) & (~$lesseq($sum(length(A), $product(-1, U_3)), 0)) & sorted(A, 0, $sum(-1, length(A))))))))))),
% 0.64/0.66 inference(rewrite,[status(thm)],[])).
% 0.64/0.66 tff(18,plain,
% 0.64/0.66 ((~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$lesseq(L, U)) | ?[M_1: $int] : ((M_1 = div2($sum(L, U))) & ((tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1) = E) | (($lesseq(E, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1)) | ?[L_2: $int] : ((L_2 = $sum(1, M_1)) & $lesseq(0, L_2) & (~$lesseq(length(A), U)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq(E, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))) | ?[U_3: $int] : ((U_3 = $sum(-1, M_1)) & $lesseq(0, L) & (~$lesseq(length(A), U_3)) & sorted(A, 0, $sum(-1, length(A)))))))) | (~($lesseq(0, L) & (~$lesseq(length(A), U)) & sorted(A, 0, $sum(-1, length(A))))))) <=> (~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$lesseq(L, U)) | ?[M_1: $int] : ((M_1 = div2($sum(L, U))) & ((tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1) = E) | (($lesseq(E, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1)) | ?[L_2: $int] : ((L_2 = $sum(1, M_1)) & $lesseq(0, L_2) & (~$lesseq(length(A), U)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq(E, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))) | ?[U_3: $int] : ((U_3 = $sum(-1, M_1)) & $lesseq(0, L) & (~$lesseq(length(A), U_3)) & sorted(A, 0, $sum(-1, length(A)))))))) | (~($lesseq(0, L) & (~$lesseq(length(A), U)) & sorted(A, 0, $sum(-1, length(A)))))))),
% 0.64/0.66 inference(rewrite,[status(thm)],[])).
% 0.64/0.66 tff(19,plain,
% 0.64/0.66 ((~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((($lesseq(0, L) & $less(U, length(A))) & sorted(A, 0, $difference(length(A), 1))) => (($greater(L, U) => $true) & ((~$greater(L, U)) => ?[M_1: $int] : (((M_1 = div2($sum(L, U))) & ((tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1) = E) => $true)) & ((~(tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1) = E)) => (($less(tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1), E) => ?[L_2: $int] : ((((L_2 = $sum(M_1, 1)) & $lesseq(0, L_2)) & $less(U, length(A))) & sorted(A, 0, $difference(length(A), 1)))) & ((~$less(tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1), E)) => ?[U_3: $int] : ((((U_3 = $difference(M_1, 1)) & $lesseq(0, L)) & $less(U_3, length(A))) & sorted(A, 0, $difference(length(A), 1))))))))))) <=> (~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$lesseq(L, U)) | ?[M_1: $int] : ((M_1 = div2($sum(L, U))) & ((tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1) = E) | (($lesseq(E, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1)) | ?[L_2: $int] : ((L_2 = $sum(1, M_1)) & $lesseq(0, L_2) & (~$lesseq(length(A), U)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq(E, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))) | ?[U_3: $int] : ((U_3 = $sum(-1, M_1)) & $lesseq(0, L) & (~$lesseq(length(A), U_3)) & sorted(A, 0, $sum(-1, length(A)))))))) | (~($lesseq(0, L) & (~$lesseq(length(A), U)) & sorted(A, 0, $sum(-1, length(A)))))))),
% 0.64/0.66 inference(rewrite,[status(thm)],[])).
% 0.64/0.66 tff(20,axiom,(~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((($lesseq(0, L) & $less(U, length(A))) & sorted(A, 0, $difference(length(A), 1))) => (($greater(L, U) => $true) & ((~$greater(L, U)) => ?[M_1: $int] : (((M_1 = div2($sum(L, U))) & ((tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1) = E) => $true)) & ((~(tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1) = E)) => (($less(tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1), E) => ?[L_2: $int] : ((((L_2 = $sum(M_1, 1)) & $lesseq(0, L_2)) & $less(U, length(A))) & sorted(A, 0, $difference(length(A), 1)))) & ((~$less(tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1), E)) => ?[U_3: $int] : ((((U_3 = $difference(M_1, 1)) & $lesseq(0, L)) & $less(U_3, length(A))) & sorted(A, 0, $difference(length(A), 1))))))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','formula_007')).
% 0.64/0.66 tff(21,plain,
% 0.64/0.66 (~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$lesseq(L, U)) | ?[M_1: $int] : ((M_1 = div2($sum(L, U))) & ((tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1) = E) | (($lesseq(E, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1)) | ?[L_2: $int] : ((L_2 = $sum(1, M_1)) & $lesseq(0, L_2) & (~$lesseq(length(A), U)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq(E, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))) | ?[U_3: $int] : ((U_3 = $sum(-1, M_1)) & $lesseq(0, L) & (~$lesseq(length(A), U_3)) & sorted(A, 0, $sum(-1, length(A)))))))) | (~($lesseq(0, L) & (~$lesseq(length(A), U)) & sorted(A, 0, $sum(-1, length(A))))))),
% 0.64/0.66 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.64/0.66 tff(22,plain,
% 0.64/0.66 (~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$lesseq(L, U)) | ?[M_1: $int] : ((M_1 = div2($sum(L, U))) & ((tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1) = E) | (($lesseq(E, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1)) | ?[L_2: $int] : ((L_2 = $sum(1, M_1)) & $lesseq(0, L_2) & (~$lesseq(length(A), U)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq(E, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))) | ?[U_3: $int] : ((U_3 = $sum(-1, M_1)) & $lesseq(0, L) & (~$lesseq(length(A), U_3)) & sorted(A, 0, $sum(-1, length(A)))))))) | (~($lesseq(0, L) & (~$lesseq(length(A), U)) & sorted(A, 0, $sum(-1, length(A))))))),
% 0.64/0.66 inference(modus_ponens,[status(thm)],[21, 18])).
% 0.64/0.66 tff(23,plain,
% 0.64/0.66 (~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$lesseq(L, U)) | ?[M_1: $int] : ((M_1 = div2($sum(L, U))) & ((tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1) = E) | (($lesseq(E, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1)) | ?[L_2: $int] : ((L_2 = $sum(1, M_1)) & $lesseq(0, L_2) & (~$lesseq(length(A), U)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq(E, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))) | ?[U_3: $int] : ((U_3 = $sum(-1, M_1)) & $lesseq(0, L) & (~$lesseq(length(A), U_3)) & sorted(A, 0, $sum(-1, length(A)))))))) | (~($lesseq(0, L) & (~$lesseq(length(A), U)) & sorted(A, 0, $sum(-1, length(A))))))),
% 0.64/0.66 inference(modus_ponens,[status(thm)],[22, 18])).
% 0.64/0.66 tff(24,plain,
% 0.64/0.66 (~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$lesseq(L, U)) | ?[M_1: $int] : ((M_1 = div2($sum(L, U))) & ((tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1) = E) | (($lesseq(E, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1)) | ?[L_2: $int] : ((L_2 = $sum(1, M_1)) & $lesseq(0, L_2) & (~$lesseq(length(A), U)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq(E, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))) | ?[U_3: $int] : ((U_3 = $sum(-1, M_1)) & $lesseq(0, L) & (~$lesseq(length(A), U_3)) & sorted(A, 0, $sum(-1, length(A)))))))) | (~($lesseq(0, L) & (~$lesseq(length(A), U)) & sorted(A, 0, $sum(-1, length(A))))))),
% 0.64/0.66 inference(modus_ponens,[status(thm)],[23, 18])).
% 0.64/0.66 tff(25,plain,
% 0.64/0.66 (~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$lesseq($sum(L, $product(-1, U)), 0)) | (~($greatereq(L, 0) & (~$lesseq($sum(length(A), $product(-1, U)), 0)) & sorted(A, 0, $sum(-1, length(A))))) | ?[M_1: $int] : (($sum(M_1, $product(-1, div2($sum(L, U)))) = 0) & (($sum(tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1), $product(-1, E)) = 0) | (($lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0) | ?[L_2: $int] : (($sum(L_2, $product(-1, M_1)) = 1) & $greatereq(L_2, 0) & (~$lesseq($sum(length(A), $product(-1, U)), 0)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0)) | ?[U_3: $int] : (($sum(U_3, $product(-1, M_1)) = -1) & $greatereq(L, 0) & (~$lesseq($sum(length(A), $product(-1, U_3)), 0)) & sorted(A, 0, $sum(-1, length(A)))))))))),
% 0.64/0.66 inference(modus_ponens,[status(thm)],[24, 17])).
% 0.64/0.66 tff(26,plain,
% 0.64/0.66 (~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$greatereq($sum(U, $product(-1, L)), 0)) | (~($greatereq(L, 0) & (~$greatereq($sum(U, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A))))) | ?[M_1: $int] : (($sum(M_1, $product(-1, div2($sum(U, L)))) = 0) & (($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))) = 0) | (($lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0) | ?[L_2: $int] : (($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$greatereq($sum(U, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0)) | ?[U_3: $int] : (($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L, 0) & (~$greatereq($sum(U_3, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A)))))))))),
% 0.64/0.66 inference(modus_ponens,[status(thm)],[25, 16])).
% 0.64/0.66 tff(27,plain,
% 0.64/0.66 (~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$greatereq($sum(U, $product(-1, L)), 0)) | (~($greatereq(L, 0) & (~$greatereq($sum(U, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A))))) | ?[M_1: $int] : (($sum(M_1, $product(-1, div2($sum(U, L)))) = 0) & (($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))) = 0) | (($lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0) | ?[L_2: $int] : (($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$greatereq($sum(U, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0)) | ?[U_3: $int] : (($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L, 0) & (~$greatereq($sum(U_3, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A)))))))))),
% 0.64/0.66 inference(modus_ponens,[status(thm)],[26, 15])).
% 0.64/0.66 tff(28,plain,
% 0.64/0.66 (~![L: $int, U: $int, A: 'Array[Int,Int]', E: $int] : ((~$greatereq($sum(U, $product(-1, L)), 0)) | (~($greatereq(L, 0) & (~$greatereq($sum(U, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A))))) | ?[M_1: $int] : (($sum(M_1, $product(-1, div2($sum(U, L)))) = 0) & (($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))) = 0) | (($lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0) | ?[L_2: $int] : (($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$greatereq($sum(U, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A))))) & ((~$lesseq($sum(E, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A, M_1))), 0)) | ?[U_3: $int] : (($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L, 0) & (~$greatereq($sum(U_3, $product(-1, length(A))), 0)) & sorted(A, 0, $sum(-1, length(A)))))))))),
% 0.64/0.66 inference(modus_ponens,[status(thm)],[27, 15])).
% 0.64/0.66 tff(29,plain,
% 0.64/0.66 ($lesseq($sum(L!6, $product(-1, U!5)), 0) & $greatereq(L!6, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4))) & ![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | ((~($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0)) & (((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) & ![L_2: $int] : (~(($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) | ($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) & ![U_3: $int] : (~(($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L!6, 0) & (~$greatereq($sum(U_3, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))))))),
% 0.64/0.66 inference(modus_ponens,[status(thm)],[28, 14])).
% 0.64/0.66 tff(30,plain,
% 0.64/0.66 ($lesseq($sum(L!6, $product(-1, U!5)), 0)),
% 0.64/0.66 inference(and_elim,[status(thm)],[29])).
% 0.64/0.66 tff(31,plain,
% 0.64/0.66 ($greatereq(L!6, 0)),
% 0.64/0.66 inference(and_elim,[status(thm)],[29])).
% 0.64/0.66 tff(32,assumption,(~$greatereq(div2($sum(L!6, U!5)), 0)), introduced(assumption)).
% 0.64/0.66 tff(33,plain,
% 0.64/0.66 (^[A: $int, Res: $int] : refl(((~((~$greatereq($sum(A, $product(-2, Res)), 0)) | $greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0)) <=> ((~((~$greatereq($sum(A, $product(-2, Res)), 0)) | $greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0)))),
% 0.64/0.66 inference(bind,[status(th)],[])).
% 0.64/0.66 tff(34,plain,
% 0.64/0.66 (![A: $int, Res: $int] : ((~((~$greatereq($sum(A, $product(-2, Res)), 0)) | $greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0)) <=> ![A: $int, Res: $int] : ((~((~$greatereq($sum(A, $product(-2, Res)), 0)) | $greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0))),
% 0.64/0.66 inference(quant_intro,[status(thm)],[33])).
% 0.64/0.66 tff(35,plain,
% 0.64/0.66 (^[A: $int, Res: $int] : rewrite((($greatereq($sum(A, $product(-2, Res)), 0) & (~$greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0)) <=> ((~((~$greatereq($sum(A, $product(-2, Res)), 0)) | $greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0)))),
% 0.64/0.66 inference(bind,[status(th)],[])).
% 0.64/0.66 tff(36,plain,
% 0.64/0.66 (![A: $int, Res: $int] : (($greatereq($sum(A, $product(-2, Res)), 0) & (~$greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0)) <=> ![A: $int, Res: $int] : ((~((~$greatereq($sum(A, $product(-2, Res)), 0)) | $greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0))),
% 0.64/0.66 inference(quant_intro,[status(thm)],[35])).
% 0.64/0.66 tff(37,plain,
% 0.64/0.66 (^[A: $int, Res: $int] : rewrite((($lesseq($sum($product(2, Res), $product(-1, A)), 0) & (~$lesseq($sum($product(2, Res), $product(-1, A)), -2))) <=> ($sum(div2(A), $product(-1, Res)) = 0)) <=> (($greatereq($sum(A, $product(-2, Res)), 0) & (~$greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0)))),
% 0.64/0.66 inference(bind,[status(th)],[])).
% 0.64/0.66 tff(38,plain,
% 0.64/0.66 (![A: $int, Res: $int] : (($lesseq($sum($product(2, Res), $product(-1, A)), 0) & (~$lesseq($sum($product(2, Res), $product(-1, A)), -2))) <=> ($sum(div2(A), $product(-1, Res)) = 0)) <=> ![A: $int, Res: $int] : (($greatereq($sum(A, $product(-2, Res)), 0) & (~$greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0))),
% 0.64/0.66 inference(quant_intro,[status(thm)],[37])).
% 0.64/0.66 tff(39,plain,
% 0.64/0.66 (^[A: $int, Res: $int] : rewrite((($lesseq($product(2, Res), A) & (~$lesseq($product(2, Res), $sum(-2, A)))) <=> (div2(A) = Res)) <=> (($lesseq($sum($product(2, Res), $product(-1, A)), 0) & (~$lesseq($sum($product(2, Res), $product(-1, A)), -2))) <=> ($sum(div2(A), $product(-1, Res)) = 0)))),
% 0.64/0.66 inference(bind,[status(th)],[])).
% 0.64/0.66 tff(40,plain,
% 0.64/0.66 (![A: $int, Res: $int] : (($lesseq($product(2, Res), A) & (~$lesseq($product(2, Res), $sum(-2, A)))) <=> (div2(A) = Res)) <=> ![A: $int, Res: $int] : (($lesseq($sum($product(2, Res), $product(-1, A)), 0) & (~$lesseq($sum($product(2, Res), $product(-1, A)), -2))) <=> ($sum(div2(A), $product(-1, Res)) = 0))),
% 0.64/0.66 inference(quant_intro,[status(thm)],[39])).
% 0.64/0.66 tff(41,plain,
% 0.64/0.66 (![A: $int, Res: $int] : (($lesseq($product(2, Res), A) & (~$lesseq($product(2, Res), $sum(-2, A)))) <=> (div2(A) = Res)) <=> ![A: $int, Res: $int] : (($lesseq($product(2, Res), A) & (~$lesseq($product(2, Res), $sum(-2, A)))) <=> (div2(A) = Res))),
% 0.64/0.66 inference(rewrite,[status(thm)],[])).
% 0.64/0.66 tff(42,plain,
% 0.64/0.66 (^[A: $int, Res: $int] : rewrite((($lesseq($product(2, Res), A) & $greater($product(2, $sum(Res, 1)), A)) <=> (div2(A) = Res)) <=> (($lesseq($product(2, Res), A) & (~$lesseq($product(2, Res), $sum(-2, A)))) <=> (div2(A) = Res)))),
% 0.64/0.66 inference(bind,[status(th)],[])).
% 0.64/0.66 tff(43,plain,
% 0.64/0.66 (![A: $int, Res: $int] : (($lesseq($product(2, Res), A) & $greater($product(2, $sum(Res, 1)), A)) <=> (div2(A) = Res)) <=> ![A: $int, Res: $int] : (($lesseq($product(2, Res), A) & (~$lesseq($product(2, Res), $sum(-2, A)))) <=> (div2(A) = Res))),
% 0.64/0.66 inference(quant_intro,[status(thm)],[42])).
% 0.64/0.66 tff(44,axiom,(![A: $int, Res: $int] : (($lesseq($product(2, Res), A) & $greater($product(2, $sum(Res, 1)), A)) <=> (div2(A) = Res))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','formula_004')).
% 0.64/0.66 tff(45,plain,
% 0.64/0.66 (![A: $int, Res: $int] : (($lesseq($product(2, Res), A) & (~$lesseq($product(2, Res), $sum(-2, A)))) <=> (div2(A) = Res))),
% 0.64/0.66 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.64/0.66 tff(46,plain,
% 0.64/0.66 (![A: $int, Res: $int] : (($lesseq($product(2, Res), A) & (~$lesseq($product(2, Res), $sum(-2, A)))) <=> (div2(A) = Res))),
% 0.64/0.66 inference(modus_ponens,[status(thm)],[45, 41])).
% 0.64/0.66 tff(47,plain,
% 0.64/0.66 (![A: $int, Res: $int] : (($lesseq($sum($product(2, Res), $product(-1, A)), 0) & (~$lesseq($sum($product(2, Res), $product(-1, A)), -2))) <=> ($sum(div2(A), $product(-1, Res)) = 0))),
% 0.64/0.66 inference(modus_ponens,[status(thm)],[46, 40])).
% 0.64/0.66 tff(48,plain,
% 0.64/0.66 (![A: $int, Res: $int] : (($greatereq($sum(A, $product(-2, Res)), 0) & (~$greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0))),
% 0.64/0.66 inference(modus_ponens,[status(thm)],[47, 38])).
% 0.64/0.66 tff(49,plain,(
% 0.64/0.66 ![A: $int, Res: $int] : (($greatereq($sum(A, $product(-2, Res)), 0) & (~$greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0))),
% 0.64/0.66 inference(skolemize,[status(sab)],[48])).
% 0.64/0.66 tff(50,plain,
% 0.64/0.66 (![A: $int, Res: $int] : ((~((~$greatereq($sum(A, $product(-2, Res)), 0)) | $greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0))),
% 0.64/0.66 inference(modus_ponens,[status(thm)],[49, 36])).
% 0.64/0.66 tff(51,plain,
% 0.64/0.66 (![A: $int, Res: $int] : ((~((~$greatereq($sum(A, $product(-2, Res)), 0)) | $greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0))),
% 0.64/0.66 inference(modus_ponens,[status(thm)],[50, 34])).
% 0.64/0.66 tff(52,plain,
% 0.64/0.66 (((~![A: $int, Res: $int] : ((~((~$greatereq($sum(A, $product(-2, Res)), 0)) | $greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0))) | (~((~$greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 0)) | $greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 2)))) <=> ((~![A: $int, Res: $int] : ((~((~$greatereq($sum(A, $product(-2, Res)), 0)) | $greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0))) | (~((~$greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 0)) | $greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 2))))),
% 0.64/0.66 inference(rewrite,[status(thm)],[])).
% 0.64/0.66 tff(53,plain,
% 0.64/0.66 (((~((~$greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 0)) | $greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 2))) <=> $true) <=> (~((~$greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 0)) | $greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 2)))),
% 0.64/0.66 inference(rewrite,[status(thm)],[])).
% 0.64/0.66 tff(54,plain,
% 0.64/0.66 ((0 = 0) <=> $true),
% 0.64/0.66 inference(rewrite,[status(thm)],[])).
% 0.64/0.66 tff(55,plain,
% 0.64/0.66 ($sum(div2($sum(L!6, U!5)), $product(-1, div2($sum(L!6, U!5)))) = 0),
% 0.64/0.66 inference(rewrite,[status(thm)],[])).
% 0.64/0.66 tff(56,plain,
% 0.64/0.66 (($sum(div2($sum(L!6, U!5)), $product(-1, div2($sum(L!6, U!5)))) = 0) <=> (0 = 0)),
% 0.64/0.66 inference(monotonicity,[status(thm)],[55])).
% 0.64/0.66 tff(57,plain,
% 0.64/0.66 (($sum(div2($sum(L!6, U!5)), $product(-1, div2($sum(L!6, U!5)))) = 0) <=> $true),
% 0.64/0.66 inference(transitivity,[status(thm)],[56, 54])).
% 0.64/0.66 tff(58,plain,
% 0.64/0.66 ((~((~$greatereq($sum($sum(L!6, U!5), $product(-2, div2($sum(L!6, U!5)))), 0)) | $greatereq($sum($sum(L!6, U!5), $product(-2, div2($sum(L!6, U!5)))), 2))) <=> (~((~$greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 0)) | $greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 2)))),
% 0.64/0.66 inference(rewrite,[status(thm)],[])).
% 0.64/0.66 tff(59,plain,
% 0.64/0.66 (((~((~$greatereq($sum($sum(L!6, U!5), $product(-2, div2($sum(L!6, U!5)))), 0)) | $greatereq($sum($sum(L!6, U!5), $product(-2, div2($sum(L!6, U!5)))), 2))) <=> ($sum(div2($sum(L!6, U!5)), $product(-1, div2($sum(L!6, U!5)))) = 0)) <=> ((~((~$greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 0)) | $greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 2))) <=> $true)),
% 0.64/0.66 inference(monotonicity,[status(thm)],[58, 57])).
% 0.64/0.66 tff(60,plain,
% 0.64/0.66 (((~((~$greatereq($sum($sum(L!6, U!5), $product(-2, div2($sum(L!6, U!5)))), 0)) | $greatereq($sum($sum(L!6, U!5), $product(-2, div2($sum(L!6, U!5)))), 2))) <=> ($sum(div2($sum(L!6, U!5)), $product(-1, div2($sum(L!6, U!5)))) = 0)) <=> (~((~$greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 0)) | $greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 2)))),
% 0.64/0.66 inference(transitivity,[status(thm)],[59, 53])).
% 0.64/0.66 tff(61,plain,
% 0.64/0.66 (((~![A: $int, Res: $int] : ((~((~$greatereq($sum(A, $product(-2, Res)), 0)) | $greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0))) | ((~((~$greatereq($sum($sum(L!6, U!5), $product(-2, div2($sum(L!6, U!5)))), 0)) | $greatereq($sum($sum(L!6, U!5), $product(-2, div2($sum(L!6, U!5)))), 2))) <=> ($sum(div2($sum(L!6, U!5)), $product(-1, div2($sum(L!6, U!5)))) = 0))) <=> ((~![A: $int, Res: $int] : ((~((~$greatereq($sum(A, $product(-2, Res)), 0)) | $greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0))) | (~((~$greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 0)) | $greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 2))))),
% 0.64/0.67 inference(monotonicity,[status(thm)],[60])).
% 0.64/0.67 tff(62,plain,
% 0.64/0.67 (((~![A: $int, Res: $int] : ((~((~$greatereq($sum(A, $product(-2, Res)), 0)) | $greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0))) | ((~((~$greatereq($sum($sum(L!6, U!5), $product(-2, div2($sum(L!6, U!5)))), 0)) | $greatereq($sum($sum(L!6, U!5), $product(-2, div2($sum(L!6, U!5)))), 2))) <=> ($sum(div2($sum(L!6, U!5)), $product(-1, div2($sum(L!6, U!5)))) = 0))) <=> ((~![A: $int, Res: $int] : ((~((~$greatereq($sum(A, $product(-2, Res)), 0)) | $greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0))) | (~((~$greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 0)) | $greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 2))))),
% 0.64/0.67 inference(transitivity,[status(thm)],[61, 52])).
% 0.64/0.67 tff(63,plain,
% 0.64/0.67 ((~![A: $int, Res: $int] : ((~((~$greatereq($sum(A, $product(-2, Res)), 0)) | $greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0))) | ((~((~$greatereq($sum($sum(L!6, U!5), $product(-2, div2($sum(L!6, U!5)))), 0)) | $greatereq($sum($sum(L!6, U!5), $product(-2, div2($sum(L!6, U!5)))), 2))) <=> ($sum(div2($sum(L!6, U!5)), $product(-1, div2($sum(L!6, U!5)))) = 0))),
% 0.64/0.67 inference(quant_inst,[status(thm)],[])).
% 0.64/0.67 tff(64,plain,
% 0.64/0.67 ((~![A: $int, Res: $int] : ((~((~$greatereq($sum(A, $product(-2, Res)), 0)) | $greatereq($sum(A, $product(-2, Res)), 2))) <=> ($sum(Res, $product(-1, div2(A))) = 0))) | (~((~$greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 0)) | $greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 2)))),
% 0.64/0.67 inference(modus_ponens,[status(thm)],[63, 62])).
% 0.64/0.67 tff(65,plain,
% 0.64/0.67 (~((~$greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 0)) | $greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 2))),
% 0.64/0.67 inference(unit_resolution,[status(thm)],[64, 51])).
% 0.64/0.67 tff(66,plain,
% 0.64/0.67 (((~$greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 0)) | $greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 2)) | (~$greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 2))),
% 0.64/0.67 inference(tautology,[status(thm)],[])).
% 0.64/0.67 tff(67,plain,
% 0.64/0.67 (~$greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 2)),
% 0.64/0.67 inference(unit_resolution,[status(thm)],[66, 65])).
% 0.64/0.67 tff(68,plain,
% 0.64/0.67 ($false),
% 0.64/0.67 inference(theory_lemma,[status(thm)],[67, 32, 31, 30])).
% 0.64/0.67 tff(69,plain,($greatereq(div2($sum(L!6, U!5)), 0)), inference(lemma,lemma(discharge,[]))).
% 0.64/0.67 tff(70,plain,
% 0.64/0.67 ($greatereq(div2($sum(L!6, U!5)), -1) | (~$greatereq(div2($sum(L!6, U!5)), 0))),
% 0.64/0.67 inference(theory_lemma,[status(thm)],[])).
% 0.64/0.67 tff(71,plain,
% 0.64/0.67 ($greatereq(div2($sum(L!6, U!5)), -1)),
% 0.64/0.67 inference(unit_resolution,[status(thm)],[70, 69])).
% 0.64/0.67 tff(72,assumption,($greatereq(div2($sum(L!6, U!5)), -1)), introduced(assumption)).
% 0.64/0.67 tff(73,assumption,(![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))), introduced(assumption)).
% 0.64/0.67 tff(74,plain,
% 0.64/0.67 (sorted(A!4, 0, $sum(-1, length(A!4)))),
% 0.64/0.67 inference(and_elim,[status(thm)],[29])).
% 0.64/0.67 tff(75,plain,
% 0.64/0.67 (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)),
% 0.64/0.67 inference(and_elim,[status(thm)],[29])).
% 0.64/0.67 tff(76,plain,
% 0.64/0.67 (((~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))) | ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(div2($sum(L!6, U!5)), -1)))) <=> ((~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))) | $lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(div2($sum(L!6, U!5)), -1)))),
% 0.64/0.67 inference(rewrite,[status(thm)],[])).
% 0.64/0.67 tff(77,plain,
% 0.64/0.67 (($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(div2($sum(L!6, U!5)), -1)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | $false) <=> ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(div2($sum(L!6, U!5)), -1)))),
% 0.64/0.67 inference(rewrite,[status(thm)],[])).
% 0.64/0.67 tff(78,plain,
% 0.64/0.67 ((~$true) <=> $false),
% 0.64/0.67 inference(rewrite,[status(thm)],[])).
% 0.64/0.67 tff(79,plain,
% 0.64/0.67 ((1 = 1) <=> $true),
% 0.64/0.67 inference(rewrite,[status(thm)],[])).
% 0.64/0.67 tff(80,plain,
% 0.64/0.67 ($sum($sum(1, div2($sum(L!6, U!5))), $product(-1, div2($sum(L!6, U!5)))) = 1),
% 0.64/0.67 inference(rewrite,[status(thm)],[])).
% 0.64/0.67 tff(81,plain,
% 0.64/0.67 ($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))) = $sum(1, div2($sum(L!6, U!5)))),
% 0.64/0.67 inference(rewrite,[status(thm)],[])).
% 0.64/0.67 tff(82,plain,
% 0.64/0.67 ($sum($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = $sum($sum(1, div2($sum(L!6, U!5))), $product(-1, div2($sum(L!6, U!5))))),
% 0.64/0.67 inference(monotonicity,[status(thm)],[81])).
% 0.64/0.67 tff(83,plain,
% 0.64/0.67 ($sum($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 1),
% 0.64/0.67 inference(transitivity,[status(thm)],[82, 80])).
% 0.64/0.67 tff(84,plain,
% 0.64/0.67 (($sum($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 1) <=> (1 = 1)),
% 0.64/0.67 inference(monotonicity,[status(thm)],[83])).
% 0.64/0.67 tff(85,plain,
% 0.64/0.67 (($sum($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 1) <=> $true),
% 0.64/0.67 inference(transitivity,[status(thm)],[84, 79])).
% 0.64/0.67 tff(86,plain,
% 0.64/0.67 ((~($sum($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 1)) <=> (~$true)),
% 0.64/0.67 inference(monotonicity,[status(thm)],[85])).
% 0.64/0.67 tff(87,plain,
% 0.64/0.67 ((~($sum($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 1)) <=> $false),
% 0.64/0.67 inference(transitivity,[status(thm)],[86, 78])).
% 0.64/0.67 tff(88,plain,
% 0.64/0.67 ((~$greatereq($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), 0)) <=> (~$greatereq(div2($sum(L!6, U!5)), -1))),
% 0.64/0.67 inference(rewrite,[status(thm)],[])).
% 0.64/0.67 tff(89,plain,
% 0.64/0.67 (($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 1))) <=> ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(div2($sum(L!6, U!5)), -1)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | $false)),
% 0.64/0.67 inference(monotonicity,[status(thm)],[88, 87])).
% 0.64/0.67 tff(90,plain,
% 0.64/0.67 (($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 1))) <=> ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(div2($sum(L!6, U!5)), -1)))),
% 0.64/0.67 inference(transitivity,[status(thm)],[89, 77])).
% 0.64/0.67 tff(91,plain,
% 0.64/0.67 (((~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))) | ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 1)))) <=> ((~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))) | ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(div2($sum(L!6, U!5)), -1))))),
% 0.64/0.67 inference(monotonicity,[status(thm)],[90])).
% 0.64/0.67 tff(92,plain,
% 0.64/0.67 (((~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))) | ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 1)))) <=> ((~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))) | $lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(div2($sum(L!6, U!5)), -1)))),
% 0.64/0.67 inference(transitivity,[status(thm)],[91, 76])).
% 0.64/0.67 tff(93,plain,
% 0.64/0.67 ((~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))) | ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum($sum(1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 1)))),
% 0.64/0.67 inference(quant_inst,[status(thm)],[])).
% 0.64/0.67 tff(94,plain,
% 0.64/0.67 ((~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))) | $lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(div2($sum(L!6, U!5)), -1))),
% 0.64/0.67 inference(modus_ponens,[status(thm)],[93, 92])).
% 0.64/0.67 tff(95,plain,
% 0.64/0.67 ($false),
% 0.64/0.67 inference(unit_resolution,[status(thm)],[94, 75, 74, 73, 72])).
% 0.64/0.67 tff(96,plain,((~$greatereq(div2($sum(L!6, U!5)), -1)) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1))))), inference(lemma,lemma(discharge,[]))).
% 0.64/0.67 tff(97,plain,
% 0.64/0.67 (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))),
% 0.64/0.67 inference(unit_resolution,[status(thm)],[96, 71])).
% 0.64/0.67 tff(98,plain,
% 0.64/0.67 (($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1))))) | ![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))),
% 0.64/0.67 inference(tautology,[status(thm)],[])).
% 0.64/0.67 tff(99,plain,
% 0.64/0.67 ($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1))))),
% 0.64/0.67 inference(unit_resolution,[status(thm)],[98, 97])).
% 0.64/0.67 tff(100,plain,
% 0.64/0.67 (^[M_1: $int] : refl(((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0))))))))))) <=> ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0))))))))))))),
% 0.64/0.67 inference(bind,[status(th)],[])).
% 0.64/0.67 tff(101,plain,
% 0.64/0.67 (![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0))))))))))) <=> ![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))),
% 0.64/0.67 inference(quant_intro,[status(thm)],[100])).
% 0.64/0.67 tff(102,plain,
% 0.64/0.67 (^[M_1: $int] : rewrite(((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0))))))))))) <=> ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0))))))))))))),
% 0.64/0.67 inference(bind,[status(th)],[])).
% 0.64/0.67 tff(103,plain,
% 0.64/0.67 (![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0))))))))))) <=> ![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))),
% 0.64/0.67 inference(quant_intro,[status(thm)],[102])).
% 0.64/0.67 tff(104,plain,
% 0.64/0.67 (![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0))))))))))) <=> ![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))),
% 0.64/0.67 inference(transitivity,[status(thm)],[103, 101])).
% 0.64/0.67 tff(105,plain,
% 0.64/0.67 (^[M_1: $int] : rewrite(((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | ((~($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0)) & (((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) & ![L_2: $int] : (~(($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) | ($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) & ![U_3: $int] : (~(($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L!6, 0) & (~$greatereq($sum(U_3, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4))))))))) <=> ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0))))))))))))),
% 0.64/0.67 inference(bind,[status(th)],[])).
% 0.64/0.67 tff(106,plain,
% 0.64/0.67 (![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | ((~($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0)) & (((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) & ![L_2: $int] : (~(($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) | ($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) & ![U_3: $int] : (~(($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L!6, 0) & (~$greatereq($sum(U_3, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4))))))))) <=> ![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))),
% 0.64/0.67 inference(quant_intro,[status(thm)],[105])).
% 0.64/0.67 tff(107,plain,
% 0.64/0.67 (![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | ((~($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0)) & (((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) & ![L_2: $int] : (~(($sum(M_1, $product(-1, L_2)) = -1) & $greatereq(L_2, 0) & (~$lesseq($sum(length(A!4), $product(-1, U!5)), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))) | ($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) & ![U_3: $int] : (~(($sum(M_1, $product(-1, U_3)) = 1) & $greatereq(L!6, 0) & (~$greatereq($sum(U_3, $product(-1, length(A!4))), 0)) & sorted(A!4, 0, $sum(-1, length(A!4)))))))))),
% 0.64/0.67 inference(and_elim,[status(thm)],[29])).
% 0.64/0.67 tff(108,plain,
% 0.64/0.67 (![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))),
% 0.64/0.67 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.64/0.67 tff(109,plain,
% 0.64/0.67 (![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))),
% 0.64/0.68 inference(modus_ponens,[status(thm)],[108, 104])).
% 0.64/0.68 tff(110,plain,
% 0.64/0.68 (((~![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))) = 0) | (~((~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1))))))))))) <=> ((~![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))) = 0) | (~((~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))))))))))),
% 0.64/0.68 inference(rewrite,[status(thm)],[])).
% 0.64/0.68 tff(111,plain,
% 0.64/0.68 (($false | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))) = 0) | (~((~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1))))))))))) <=> (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))) = 0) | (~((~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1))))))))))),
% 0.64/0.68 inference(rewrite,[status(thm)],[])).
% 0.64/0.68 tff(112,plain,
% 0.64/0.68 ((~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))) <=> (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))) = 0) | (~((~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1))))))))))),
% 0.64/0.68 inference(rewrite,[status(thm)],[])).
% 0.64/0.68 tff(113,plain,
% 0.64/0.68 ($sum(0, div2($sum(L!6, U!5))) = div2($sum(L!6, U!5))),
% 0.64/0.68 inference(rewrite,[status(thm)],[])).
% 0.64/0.68 tff(114,plain,
% 0.64/0.68 ($product(-1, $product(-1, div2($sum(L!6, U!5)))) = div2($sum(L!6, U!5))),
% 0.64/0.68 inference(rewrite,[status(thm)],[])).
% 0.64/0.68 tff(115,plain,
% 0.64/0.68 ($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))) = $sum(0, div2($sum(L!6, U!5)))),
% 0.64/0.68 inference(monotonicity,[status(thm)],[114])).
% 0.64/0.68 tff(116,plain,
% 0.64/0.68 ($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))) = div2($sum(L!6, U!5))),
% 0.64/0.68 inference(transitivity,[status(thm)],[115, 113])).
% 0.64/0.68 tff(117,plain,
% 0.64/0.68 ($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = $sum(div2($sum(L!6, U!5)), $product(-1, div2($sum(L!6, U!5))))),
% 0.64/0.68 inference(monotonicity,[status(thm)],[116])).
% 0.64/0.68 tff(118,plain,
% 0.64/0.68 ($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 0),
% 0.64/0.68 inference(transitivity,[status(thm)],[117, 55])).
% 0.64/0.68 tff(119,plain,
% 0.64/0.68 (($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 0) <=> (0 = 0)),
% 0.64/0.68 inference(monotonicity,[status(thm)],[118])).
% 0.64/0.68 tff(120,plain,
% 0.64/0.68 (($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 0) <=> $true),
% 0.64/0.68 inference(transitivity,[status(thm)],[119, 54])).
% 0.64/0.68 tff(121,plain,
% 0.64/0.68 ((~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 0)) <=> (~$true)),
% 0.64/0.68 inference(monotonicity,[status(thm)],[120])).
% 0.64/0.68 tff(122,plain,
% 0.64/0.68 ((~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 0)) <=> $false),
% 0.64/0.68 inference(transitivity,[status(thm)],[121, 78])).
% 0.64/0.68 tff(123,plain,
% 0.64/0.68 (((~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0))))))))))) <=> ($false | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))) = 0) | (~((~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))))))))))),
% 0.64/0.68 inference(monotonicity,[status(thm)],[122, 112])).
% 0.64/0.68 tff(124,plain,
% 0.64/0.68 (((~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0))))))))))) <=> (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))) = 0) | (~((~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1))))))))))),
% 0.64/0.68 inference(transitivity,[status(thm)],[123, 111])).
% 0.64/0.68 tff(125,plain,
% 0.64/0.68 (((~![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))) | ((~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))) <=> ((~![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))) = 0) | (~((~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))))))))))),
% 0.64/0.68 inference(monotonicity,[status(thm)],[124])).
% 0.64/0.68 tff(126,plain,
% 0.64/0.68 (((~![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))) | ((~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))) <=> ((~![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))) = 0) | (~((~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))))))))))),
% 0.64/0.68 inference(transitivity,[status(thm)],[125, 110])).
% 0.64/0.68 tff(127,plain,
% 0.64/0.68 ((~![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))) | ((~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, $sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5)))))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum($sum(0, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))),
% 0.64/0.68 inference(quant_inst,[status(thm)],[])).
% 0.64/0.68 tff(128,plain,
% 0.64/0.68 ((~![M_1: $int] : ((~($sum(M_1, $product(-1, div2($sum(L!6, U!5)))) = 0)) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))) = 0) | (~((~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~($sum(M_1, $product(-1, L_2)) = -1)) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, M_1))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(M_1, $product(-1, U_3)) = 1)) | (~$greatereq(L!6, 0)))))))))))) | (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))) = 0) | (~((~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1))))))))))),
% 0.64/0.68 inference(modus_ponens,[status(thm)],[127, 126])).
% 0.64/0.68 tff(129,plain,
% 0.64/0.68 (~(($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))) = 0) | (~((~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))))))))),
% 0.64/0.68 inference(unit_resolution,[status(thm)],[128, 109])).
% 0.64/0.68 tff(130,plain,
% 0.64/0.68 ((($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))) = 0) | (~((~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1))))))))) | ((~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))))))),
% 0.64/0.68 inference(tautology,[status(thm)],[])).
% 0.64/0.68 tff(131,plain,
% 0.64/0.68 ((~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1))))))),
% 0.64/0.69 inference(unit_resolution,[status(thm)],[130, 129])).
% 0.64/0.69 tff(132,plain,
% 0.64/0.69 ((~((~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1)))))))) | (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1))))))),
% 0.64/0.69 inference(tautology,[status(thm)],[])).
% 0.64/0.69 tff(133,plain,
% 0.64/0.69 ((~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))) | (~($lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0) | (~![L_2: $int] : ($lesseq($sum(length(A!4), $product(-1, U!5)), 0) | (~$greatereq(L_2, 0)) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~($sum(L_2, $product(-1, div2($sum(L!6, U!5)))) = 1))))))),
% 0.64/0.69 inference(unit_resolution,[status(thm)],[132, 131])).
% 0.64/0.69 tff(134,plain,
% 0.64/0.69 (~((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))))),
% 0.64/0.69 inference(unit_resolution,[status(thm)],[133, 99])).
% 0.64/0.69 tff(135,plain,
% 0.64/0.69 (((~$lesseq($sum(E!3, $product(-1, tptp_fun__select__Array_Int_Int__Int__Int_(A!4, div2($sum(L!6, U!5))))), 0)) | (~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1))))) | ![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))),
% 0.64/0.69 inference(tautology,[status(thm)],[])).
% 0.64/0.69 tff(136,plain,
% 0.64/0.69 (![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))),
% 0.64/0.69 inference(unit_resolution,[status(thm)],[135, 134])).
% 0.64/0.69 tff(137,plain,
% 0.64/0.69 (((~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))) | ((~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | $lesseq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), -1))) <=> ((~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | $lesseq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), -1))),
% 0.64/0.69 inference(rewrite,[status(thm)],[])).
% 0.64/0.69 tff(138,plain,
% 0.64/0.69 (($lesseq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), -1) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | $false) <=> ((~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | $lesseq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), -1))),
% 0.64/0.69 inference(rewrite,[status(thm)],[])).
% 0.64/0.69 tff(139,plain,
% 0.64/0.69 ((-1 = -1) <=> $true),
% 0.64/0.69 inference(rewrite,[status(thm)],[])).
% 0.64/0.69 tff(140,plain,
% 0.64/0.69 ($sum($sum(-1, div2($sum(L!6, U!5))), $product(-1, div2($sum(L!6, U!5)))) = -1),
% 0.64/0.69 inference(rewrite,[status(thm)],[])).
% 0.64/0.69 tff(141,plain,
% 0.64/0.69 ($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))) = $sum(-1, div2($sum(L!6, U!5)))),
% 0.64/0.69 inference(rewrite,[status(thm)],[])).
% 0.64/0.69 tff(142,plain,
% 0.64/0.69 ($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = $sum($sum(-1, div2($sum(L!6, U!5))), $product(-1, div2($sum(L!6, U!5))))),
% 0.64/0.69 inference(monotonicity,[status(thm)],[141])).
% 0.64/0.69 tff(143,plain,
% 0.64/0.69 ($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = -1),
% 0.64/0.69 inference(transitivity,[status(thm)],[142, 140])).
% 0.64/0.69 tff(144,plain,
% 0.64/0.69 (($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = -1) <=> (-1 = -1)),
% 0.64/0.69 inference(monotonicity,[status(thm)],[143])).
% 0.64/0.69 tff(145,plain,
% 0.64/0.69 (($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = -1) <=> $true),
% 0.64/0.69 inference(transitivity,[status(thm)],[144, 139])).
% 0.64/0.69 tff(146,plain,
% 0.64/0.69 ((~($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = -1)) <=> (~$true)),
% 0.64/0.69 inference(monotonicity,[status(thm)],[145])).
% 0.64/0.69 tff(147,plain,
% 0.64/0.69 ((~($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = -1)) <=> $false),
% 0.64/0.69 inference(transitivity,[status(thm)],[146, 78])).
% 0.64/0.69 tff(148,plain,
% 0.64/0.69 ($greatereq($sum(-1, $sum($product(-1, length(A!4)), div2($sum(L!6, U!5)))), 0) <=> $lesseq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), -1)),
% 0.64/0.69 inference(rewrite,[status(thm)],[])).
% 0.64/0.69 tff(149,plain,
% 0.64/0.69 ($sum($sum(-1, div2($sum(L!6, U!5))), $product(-1, length(A!4))) = $sum(-1, $sum($product(-1, length(A!4)), div2($sum(L!6, U!5))))),
% 0.64/0.69 inference(rewrite,[status(thm)],[])).
% 0.64/0.69 tff(150,plain,
% 0.64/0.69 ($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, length(A!4))) = $sum($sum(-1, div2($sum(L!6, U!5))), $product(-1, length(A!4)))),
% 0.64/0.69 inference(monotonicity,[status(thm)],[141])).
% 0.64/0.69 tff(151,plain,
% 0.64/0.69 ($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, length(A!4))) = $sum(-1, $sum($product(-1, length(A!4)), div2($sum(L!6, U!5))))),
% 0.64/0.69 inference(transitivity,[status(thm)],[150, 149])).
% 0.64/0.69 tff(152,plain,
% 0.64/0.69 ($greatereq($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, length(A!4))), 0) <=> $greatereq($sum(-1, $sum($product(-1, length(A!4)), div2($sum(L!6, U!5)))), 0)),
% 0.64/0.69 inference(monotonicity,[status(thm)],[151])).
% 0.64/0.69 tff(153,plain,
% 0.64/0.69 ($greatereq($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, length(A!4))), 0) <=> $lesseq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), -1)),
% 0.64/0.69 inference(transitivity,[status(thm)],[152, 148])).
% 0.64/0.69 tff(154,plain,
% 0.64/0.69 (($greatereq($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = -1))) <=> ($lesseq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), -1) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | $false)),
% 0.64/0.69 inference(monotonicity,[status(thm)],[153, 147])).
% 0.64/0.69 tff(155,plain,
% 0.64/0.69 (($greatereq($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = -1))) <=> ((~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | $lesseq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), -1))),
% 0.64/0.69 inference(transitivity,[status(thm)],[154, 138])).
% 0.64/0.69 tff(156,plain,
% 0.64/0.69 (((~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))) | ($greatereq($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = -1)))) <=> ((~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))) | ((~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | $lesseq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), -1)))),
% 0.64/0.69 inference(monotonicity,[status(thm)],[155])).
% 0.64/0.69 tff(157,plain,
% 0.64/0.69 (((~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))) | ($greatereq($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = -1)))) <=> ((~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | $lesseq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), -1))),
% 0.64/0.69 inference(transitivity,[status(thm)],[156, 137])).
% 0.64/0.69 tff(158,plain,
% 0.64/0.69 ((~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))) | ($greatereq($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum($sum(-1, $product(-1, $product(-1, div2($sum(L!6, U!5))))), $product(-1, div2($sum(L!6, U!5)))) = -1)))),
% 0.64/0.69 inference(quant_inst,[status(thm)],[])).
% 0.64/0.69 tff(159,plain,
% 0.64/0.69 ((~![U_3: $int] : ($greatereq($sum(U_3, $product(-1, length(A!4))), 0) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | (~($sum(U_3, $product(-1, div2($sum(L!6, U!5)))) = -1)))) | (~sorted(A!4, 0, $sum(-1, length(A!4)))) | (~$greatereq(L!6, 0)) | $lesseq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), -1)),
% 0.64/0.69 inference(modus_ponens,[status(thm)],[158, 157])).
% 0.64/0.69 tff(160,plain,
% 0.64/0.69 ($false),
% 0.64/0.69 inference(unit_resolution,[status(thm)],[159, 31, 74, 136, 1])).
% 0.64/0.69 tff(161,plain,($lesseq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), -1)), inference(lemma,lemma(discharge,[]))).
% 0.64/0.69 tff(162,plain,
% 0.64/0.69 (((~$greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 0)) | $greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 2)) | $greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 0)),
% 0.64/0.69 inference(tautology,[status(thm)],[])).
% 0.64/0.69 tff(163,plain,
% 0.64/0.69 ($greatereq($sum(L!6, $sum(U!5, $product(-2, div2($sum(L!6, U!5))))), 0)),
% 0.64/0.69 inference(unit_resolution,[status(thm)],[162, 65])).
% 0.64/0.69 tff(164,assumption,($lesseq($sum(U!5, $product(-1, div2($sum(L!6, U!5)))), -1)), introduced(assumption)).
% 0.64/0.69 tff(165,plain,
% 0.64/0.69 ($false),
% 0.64/0.69 inference(theory_lemma,[status(thm)],[30, 164, 163])).
% 0.64/0.69 tff(166,plain,(~$lesseq($sum(U!5, $product(-1, div2($sum(L!6, U!5)))), -1)), inference(lemma,lemma(discharge,[]))).
% 0.64/0.69 tff(167,plain,
% 0.64/0.69 ($greatereq($sum(U!5, $product(-1, div2($sum(L!6, U!5)))), -1) | $lesseq($sum(U!5, $product(-1, div2($sum(L!6, U!5)))), -1)),
% 0.64/0.69 inference(theory_lemma,[status(thm)],[])).
% 0.64/0.69 tff(168,plain,
% 0.64/0.69 ($greatereq($sum(U!5, $product(-1, div2($sum(L!6, U!5)))), -1)),
% 0.64/0.69 inference(unit_resolution,[status(thm)],[167, 166])).
% 0.64/0.69 tff(169,plain,
% 0.64/0.69 ($greatereq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), 0) | (~$greatereq($sum(U!5, $product(-1, div2($sum(L!6, U!5)))), -1)) | $lesseq($sum(length(A!4), $product(-1, U!5)), 0)),
% 0.64/0.69 inference(theory_lemma,[status(thm)],[])).
% 0.64/0.69 tff(170,plain,
% 0.64/0.69 ($greatereq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), 0)),
% 0.64/0.69 inference(unit_resolution,[status(thm)],[169, 75, 168])).
% 0.64/0.69 tff(171,plain,
% 0.64/0.69 ((~$lesseq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), -1)) | (~$greatereq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), 0))),
% 0.64/0.69 inference(theory_lemma,[status(thm)],[])).
% 0.64/0.69 tff(172,plain,
% 0.64/0.69 (~$lesseq($sum(length(A!4), $product(-1, div2($sum(L!6, U!5)))), -1)),
% 0.64/0.69 inference(unit_resolution,[status(thm)],[171, 170])).
% 0.64/0.69 tff(173,plain,
% 0.64/0.69 ($false),
% 0.64/0.69 inference(unit_resolution,[status(thm)],[172, 161])).
% 0.64/0.69 % SZS output end Proof
%------------------------------------------------------------------------------