TSTP Solution File: DAT089_1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : DAT089_1 : TPTP v8.1.0. Released v6.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri Sep 16 14:36:41 EDT 2022
% Result : Theorem 0.20s 0.43s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : DAT089_1 : TPTP v8.1.0. Released v6.1.0.
% 0.03/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.35 % Computer : n005.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Wed Aug 31 01:49:48 EDT 2022
% 0.13/0.35 % CPUTime :
% 0.13/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.35 Usage: tptp [options] [-file:]file
% 0.13/0.35 -h, -? prints this message.
% 0.13/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.35 -m, -model generate model.
% 0.13/0.35 -p, -proof generate proof.
% 0.13/0.35 -c, -core generate unsat core of named formulas.
% 0.13/0.35 -st, -statistics display statistics.
% 0.13/0.35 -t:timeout set timeout (in second).
% 0.13/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.35 -<param>:<value> configuration parameter and value.
% 0.13/0.35 -o:<output-file> file to place output in.
% 0.20/0.43 % SZS status Theorem
% 0.20/0.43 % SZS output start Proof
% 0.20/0.43 tff(count_type, type, (
% 0.20/0.43 count: ( $int * list ) > $int)).
% 0.20/0.43 tff(nil_type, type, (
% 0.20/0.43 nil: list)).
% 0.20/0.43 tff(cons_type, type, (
% 0.20/0.43 cons: ( $int * list ) > list)).
% 0.20/0.43 tff(tail_type, type, (
% 0.20/0.43 tail: list > list)).
% 0.20/0.43 tff(head_type, type, (
% 0.20/0.43 head: list > $int)).
% 0.20/0.43 tff(length_type, type, (
% 0.20/0.43 length: list > $int)).
% 0.20/0.43 tff(1,plain,
% 0.20/0.43 (^[K: $int, H: $int, T: list] : refl((($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0)) <=> (($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0)))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(2,plain,
% 0.20/0.43 (![K: $int, H: $int, T: list] : (($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0)) <=> ![K: $int, H: $int, T: list] : (($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.43 tff(3,plain,
% 0.20/0.43 (^[K: $int, H: $int, T: list] : rewrite((($sum(K, $product(-1, H)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0)) <=> (($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0)))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(4,plain,
% 0.20/0.43 (![K: $int, H: $int, T: list] : (($sum(K, $product(-1, H)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0)) <=> ![K: $int, H: $int, T: list] : (($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[3])).
% 0.20/0.43 tff(5,plain,
% 0.20/0.43 (^[K: $int, H: $int, T: list] : rewrite(((K = H) | (count(K, cons(H, T)) = count(K, T))) <=> (($sum(K, $product(-1, H)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0)))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(6,plain,
% 0.20/0.43 (![K: $int, H: $int, T: list] : ((K = H) | (count(K, cons(H, T)) = count(K, T))) <=> ![K: $int, H: $int, T: list] : (($sum(K, $product(-1, H)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[5])).
% 0.20/0.43 tff(7,plain,
% 0.20/0.43 (![K: $int, H: $int, T: list] : ((K = H) | (count(K, cons(H, T)) = count(K, T))) <=> ![K: $int, H: $int, T: list] : ((K = H) | (count(K, cons(H, T)) = count(K, T)))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(8,plain,
% 0.20/0.43 (![K: $int, H: $int, T: list, N: $int] : ((K = H) | (count(K, cons(H, T)) = count(K, T))) <=> ![K: $int, H: $int, T: list] : ((K = H) | (count(K, cons(H, T)) = count(K, T)))),
% 0.20/0.43 inference(elim_unused_vars,[status(thm)],[])).
% 0.20/0.43 tff(9,plain,
% 0.20/0.43 (^[K: $int, H: $int, T: list, N: $int] : rewrite(((~(K = H)) => (count(K, cons(H, T)) = count(K, T))) <=> ((K = H) | (count(K, cons(H, T)) = count(K, T))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(10,plain,
% 0.20/0.43 (![K: $int, H: $int, T: list, N: $int] : ((~(K = H)) => (count(K, cons(H, T)) = count(K, T))) <=> ![K: $int, H: $int, T: list, N: $int] : ((K = H) | (count(K, cons(H, T)) = count(K, T)))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[9])).
% 0.20/0.43 tff(11,plain,
% 0.20/0.43 (![K: $int, H: $int, T: list, N: $int] : ((~(K = H)) => (count(K, cons(H, T)) = count(K, T))) <=> ![K: $int, H: $int, T: list] : ((K = H) | (count(K, cons(H, T)) = count(K, T)))),
% 0.20/0.43 inference(transitivity,[status(thm)],[10, 8])).
% 0.20/0.43 tff(12,axiom,(![K: $int, H: $int, T: list, N: $int] : ((~(K = H)) => (count(K, cons(H, T)) = count(K, T)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','a_3')).
% 0.20/0.43 tff(13,plain,
% 0.20/0.43 (![K: $int, H: $int, T: list] : ((K = H) | (count(K, cons(H, T)) = count(K, T)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[12, 11])).
% 0.20/0.43 tff(14,plain,
% 0.20/0.43 (![K: $int, H: $int, T: list] : ((K = H) | (count(K, cons(H, T)) = count(K, T)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[13, 7])).
% 0.20/0.43 tff(15,plain,
% 0.20/0.43 (![K: $int, H: $int, T: list] : (($sum(K, $product(-1, H)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[14, 6])).
% 0.20/0.43 tff(16,plain,
% 0.20/0.43 (![K: $int, H: $int, T: list] : (($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[15, 4])).
% 0.20/0.43 tff(17,plain,(
% 0.20/0.43 ![K: $int, H: $int, T: list] : (($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0))),
% 0.20/0.43 inference(skolemize,[status(sab)],[16])).
% 0.20/0.43 tff(18,plain,
% 0.20/0.43 (![K: $int, H: $int, T: list] : (($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[17, 2])).
% 0.20/0.43 tff(19,plain,
% 0.20/0.43 (((~![K: $int, H: $int, T: list] : (($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0))) | ($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))) = 0)) <=> ((~![K: $int, H: $int, T: list] : (($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0))) | ($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))) = 0))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(20,plain,
% 0.20/0.43 (($false | ($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))) = 0)) <=> ($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))) = 0)),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(21,plain,
% 0.20/0.43 ((1 = 0) <=> $false),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(22,plain,
% 0.20/0.43 ($sum(0, 1) = 1),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(23,plain,
% 0.20/0.43 ($product(-1, -1) = 1),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(24,plain,
% 0.20/0.43 ($sum(0, $product(-1, -1)) = $sum(0, 1)),
% 0.20/0.43 inference(monotonicity,[status(thm)],[23])).
% 0.20/0.43 tff(25,plain,
% 0.20/0.43 ($sum(0, $product(-1, -1)) = 1),
% 0.20/0.43 inference(transitivity,[status(thm)],[24, 22])).
% 0.20/0.43 tff(26,plain,
% 0.20/0.43 (($sum(0, $product(-1, -1)) = 0) <=> (1 = 0)),
% 0.20/0.43 inference(monotonicity,[status(thm)],[25])).
% 0.20/0.43 tff(27,plain,
% 0.20/0.43 (($sum(0, $product(-1, -1)) = 0) <=> $false),
% 0.20/0.43 inference(transitivity,[status(thm)],[26, 21])).
% 0.20/0.43 tff(28,plain,
% 0.20/0.43 ((($sum(0, $product(-1, -1)) = 0) | ($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))) = 0)) <=> ($false | ($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))) = 0))),
% 0.20/0.43 inference(monotonicity,[status(thm)],[27])).
% 0.20/0.43 tff(29,plain,
% 0.20/0.43 ((($sum(0, $product(-1, -1)) = 0) | ($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))) = 0)) <=> ($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))) = 0)),
% 0.20/0.43 inference(transitivity,[status(thm)],[28, 20])).
% 0.20/0.43 tff(30,plain,
% 0.20/0.43 (((~![K: $int, H: $int, T: list] : (($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0))) | (($sum(0, $product(-1, -1)) = 0) | ($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))) = 0))) <=> ((~![K: $int, H: $int, T: list] : (($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0))) | ($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))) = 0))),
% 0.20/0.43 inference(monotonicity,[status(thm)],[29])).
% 0.20/0.43 tff(31,plain,
% 0.20/0.43 (((~![K: $int, H: $int, T: list] : (($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0))) | (($sum(0, $product(-1, -1)) = 0) | ($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))) = 0))) <=> ((~![K: $int, H: $int, T: list] : (($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0))) | ($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))) = 0))),
% 0.20/0.43 inference(transitivity,[status(thm)],[30, 19])).
% 0.20/0.43 tff(32,plain,
% 0.20/0.43 ((~![K: $int, H: $int, T: list] : (($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0))) | (($sum(0, $product(-1, -1)) = 0) | ($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))) = 0))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(33,plain,
% 0.20/0.43 ((~![K: $int, H: $int, T: list] : (($sum(H, $product(-1, K)) = 0) | ($sum(count(K, cons(H, T)), $product(-1, count(K, T))) = 0))) | ($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))) = 0)),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.20/0.43 tff(34,plain,
% 0.20/0.43 ($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))) = 0),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[33, 18])).
% 0.20/0.43 tff(35,plain,
% 0.20/0.43 ((~($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))) = 0)) | $lesseq($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))), 0)),
% 0.20/0.43 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.43 tff(36,plain,
% 0.20/0.43 ($lesseq($sum(count(-1, cons(0, nil)), $product(-1, count(-1, nil))), 0)),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[35, 34])).
% 0.20/0.43 tff(37,plain,
% 0.20/0.43 (^[K: $int, L: list] : refl((~(cons(K, L) = nil)) <=> (~(cons(K, L) = nil)))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(38,plain,
% 0.20/0.43 (![K: $int, L: list] : (~(cons(K, L) = nil)) <=> ![K: $int, L: list] : (~(cons(K, L) = nil))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[37])).
% 0.20/0.43 tff(39,plain,
% 0.20/0.43 (![K: $int, L: list] : (~(cons(K, L) = nil)) <=> ![K: $int, L: list] : (~(cons(K, L) = nil))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(40,axiom,(![K: $int, L: list] : (~(cons(K, L) = nil))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','l4')).
% 0.20/0.43 tff(41,plain,
% 0.20/0.43 (![K: $int, L: list] : (~(cons(K, L) = nil))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[40, 39])).
% 0.20/0.43 tff(42,plain,(
% 0.20/0.43 ![K: $int, L: list] : (~(cons(K, L) = nil))),
% 0.20/0.43 inference(skolemize,[status(sab)],[41])).
% 0.20/0.43 tff(43,plain,
% 0.20/0.43 (![K: $int, L: list] : (~(cons(K, L) = nil))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[42, 38])).
% 0.20/0.43 tff(44,plain,
% 0.20/0.43 ((~![K: $int, L: list] : (~(cons(K, L) = nil))) | (~(cons(0, nil) = nil))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(45,plain,
% 0.20/0.43 (~(cons(0, nil) = nil)),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[44, 43])).
% 0.20/0.43 tff(46,plain,
% 0.20/0.43 (^[L: list] : refl(((L = nil) | (L = cons(head(L), tail(L)))) <=> ((L = nil) | (L = cons(head(L), tail(L)))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(47,plain,
% 0.20/0.43 (![L: list] : ((L = nil) | (L = cons(head(L), tail(L)))) <=> ![L: list] : ((L = nil) | (L = cons(head(L), tail(L))))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[46])).
% 0.20/0.43 tff(48,plain,
% 0.20/0.43 (![L: list] : ((L = nil) | (L = cons(head(L), tail(L)))) <=> ![L: list] : ((L = nil) | (L = cons(head(L), tail(L))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(49,axiom,(![L: list] : ((L = nil) | (L = cons(head(L), tail(L))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','l3')).
% 0.20/0.43 tff(50,plain,
% 0.20/0.43 (![L: list] : ((L = nil) | (L = cons(head(L), tail(L))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[49, 48])).
% 0.20/0.43 tff(51,plain,(
% 0.20/0.43 ![L: list] : ((L = nil) | (L = cons(head(L), tail(L))))),
% 0.20/0.43 inference(skolemize,[status(sab)],[50])).
% 0.20/0.43 tff(52,plain,
% 0.20/0.43 (![L: list] : ((L = nil) | (L = cons(head(L), tail(L))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[51, 47])).
% 0.20/0.43 tff(53,plain,
% 0.20/0.43 (((~![L: list] : ((L = nil) | (L = cons(head(L), tail(L))))) | ((cons(0, nil) = nil) | (cons(0, nil) = cons(head(cons(0, nil)), tail(cons(0, nil)))))) <=> ((~![L: list] : ((L = nil) | (L = cons(head(L), tail(L))))) | (cons(0, nil) = nil) | (cons(0, nil) = cons(head(cons(0, nil)), tail(cons(0, nil)))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(54,plain,
% 0.20/0.43 ((~![L: list] : ((L = nil) | (L = cons(head(L), tail(L))))) | ((cons(0, nil) = nil) | (cons(0, nil) = cons(head(cons(0, nil)), tail(cons(0, nil)))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(55,plain,
% 0.20/0.43 ((~![L: list] : ((L = nil) | (L = cons(head(L), tail(L))))) | (cons(0, nil) = nil) | (cons(0, nil) = cons(head(cons(0, nil)), tail(cons(0, nil))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[54, 53])).
% 0.20/0.43 tff(56,plain,
% 0.20/0.43 (cons(0, nil) = cons(head(cons(0, nil)), tail(cons(0, nil)))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[55, 52, 45])).
% 0.20/0.43 tff(57,plain,
% 0.20/0.43 (cons(head(cons(0, nil)), tail(cons(0, nil))) = cons(0, nil)),
% 0.20/0.43 inference(symmetry,[status(thm)],[56])).
% 0.20/0.43 tff(58,plain,
% 0.20/0.43 (count(-1, cons(head(cons(0, nil)), tail(cons(0, nil)))) = count(-1, cons(0, nil))),
% 0.20/0.43 inference(monotonicity,[status(thm)],[57])).
% 0.20/0.43 tff(59,plain,
% 0.20/0.43 ((~(count(-1, cons(head(cons(0, nil)), tail(cons(0, nil)))) = count(-1, cons(0, nil)))) | $lesseq($sum(count(-1, cons(head(cons(0, nil)), tail(cons(0, nil)))), $product(-1, count(-1, cons(0, nil)))), 0)),
% 0.20/0.43 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.43 tff(60,plain,
% 0.20/0.43 ($lesseq($sum(count(-1, cons(head(cons(0, nil)), tail(cons(0, nil)))), $product(-1, count(-1, cons(0, nil)))), 0)),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[59, 58])).
% 0.20/0.44 tff(61,plain,
% 0.20/0.44 (^[N: $int, L: list] : refl($lesseq($sum(length(L), $product(-1, count(N, L))), 0) <=> $lesseq($sum(length(L), $product(-1, count(N, L))), 0))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(62,plain,
% 0.20/0.44 (![N: $int, L: list] : $lesseq($sum(length(L), $product(-1, count(N, L))), 0) <=> ![N: $int, L: list] : $lesseq($sum(length(L), $product(-1, count(N, L))), 0)),
% 0.20/0.44 inference(quant_intro,[status(thm)],[61])).
% 0.20/0.44 tff(63,plain,
% 0.20/0.44 (^[N: $int, L: list] : trans(monotonicity(rewrite($sum(count(N, L), $product(-1, length(L))) = $sum($product(-1, length(L)), count(N, L))), ($greatereq($sum(count(N, L), $product(-1, length(L))), 0) <=> $greatereq($sum($product(-1, length(L)), count(N, L)), 0))), rewrite($greatereq($sum($product(-1, length(L)), count(N, L)), 0) <=> $lesseq($sum(length(L), $product(-1, count(N, L))), 0)), ($greatereq($sum(count(N, L), $product(-1, length(L))), 0) <=> $lesseq($sum(length(L), $product(-1, count(N, L))), 0)))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(64,plain,
% 0.20/0.44 (![N: $int, L: list] : $greatereq($sum(count(N, L), $product(-1, length(L))), 0) <=> ![N: $int, L: list] : $lesseq($sum(length(L), $product(-1, count(N, L))), 0)),
% 0.20/0.44 inference(quant_intro,[status(thm)],[63])).
% 0.20/0.44 tff(65,plain,
% 0.20/0.44 (^[N: $int, L: list] : rewrite($greatereq(count(N, L), length(L)) <=> $greatereq($sum(count(N, L), $product(-1, length(L))), 0))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(66,plain,
% 0.20/0.44 (![N: $int, L: list] : $greatereq(count(N, L), length(L)) <=> ![N: $int, L: list] : $greatereq($sum(count(N, L), $product(-1, length(L))), 0)),
% 0.20/0.44 inference(quant_intro,[status(thm)],[65])).
% 0.20/0.44 tff(67,plain,
% 0.20/0.44 (![N: $int, L: list] : $greatereq(count(N, L), length(L)) <=> ![N: $int, L: list] : $greatereq(count(N, L), length(L))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(68,plain,
% 0.20/0.44 ((~(~![N: $int, L: list] : $greatereq(count(N, L), length(L)))) <=> ![N: $int, L: list] : $greatereq(count(N, L), length(L))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(69,plain,
% 0.20/0.44 ((~![N: $int, L: list] : $greatereq(count(N, L), length(L))) <=> (~![N: $int, L: list] : $greatereq(count(N, L), length(L)))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(70,plain,
% 0.20/0.44 ((~(~![N: $int, L: list] : $greatereq(count(N, L), length(L)))) <=> (~(~![N: $int, L: list] : $greatereq(count(N, L), length(L))))),
% 0.20/0.44 inference(monotonicity,[status(thm)],[69])).
% 0.20/0.44 tff(71,plain,
% 0.20/0.44 ((~(~![N: $int, L: list] : $greatereq(count(N, L), length(L)))) <=> ![N: $int, L: list] : $greatereq(count(N, L), length(L))),
% 0.20/0.44 inference(transitivity,[status(thm)],[70, 68])).
% 0.20/0.44 tff(72,axiom,(~(~![N: $int, L: list] : $greatereq(count(N, L), length(L)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','c')).
% 0.20/0.44 tff(73,plain,
% 0.20/0.44 (![N: $int, L: list] : $greatereq(count(N, L), length(L))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[72, 71])).
% 0.20/0.44 tff(74,plain,
% 0.20/0.44 (![N: $int, L: list] : $greatereq(count(N, L), length(L))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[73, 67])).
% 0.20/0.44 tff(75,plain,
% 0.20/0.44 (![N: $int, L: list] : $greatereq($sum(count(N, L), $product(-1, length(L))), 0)),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[74, 66])).
% 0.20/0.44 tff(76,plain,
% 0.20/0.44 (![N: $int, L: list] : $lesseq($sum(length(L), $product(-1, count(N, L))), 0)),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[75, 64])).
% 0.20/0.44 tff(77,plain,(
% 0.20/0.44 ![N: $int, L: list] : $lesseq($sum(length(L), $product(-1, count(N, L))), 0)),
% 0.20/0.44 inference(skolemize,[status(sab)],[76])).
% 0.20/0.44 tff(78,plain,
% 0.20/0.44 (![N: $int, L: list] : $lesseq($sum(length(L), $product(-1, count(N, L))), 0)),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[77, 62])).
% 0.20/0.44 tff(79,plain,
% 0.20/0.44 ((~![N: $int, L: list] : $lesseq($sum(length(L), $product(-1, count(N, L))), 0)) | $lesseq($sum(length(cons(head(cons(0, nil)), tail(cons(0, nil)))), $product(-1, count(-1, cons(head(cons(0, nil)), tail(cons(0, nil)))))), 0)),
% 0.20/0.44 inference(quant_inst,[status(thm)],[])).
% 0.20/0.44 tff(80,plain,
% 0.20/0.44 ($lesseq($sum(length(cons(head(cons(0, nil)), tail(cons(0, nil)))), $product(-1, count(-1, cons(head(cons(0, nil)), tail(cons(0, nil)))))), 0)),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[79, 78])).
% 0.20/0.44 tff(81,plain,
% 0.20/0.44 (^[H: $int, T: list] : refl(($sum(length(cons(H, T)), $product(-1, length(T))) = 1) <=> ($sum(length(cons(H, T)), $product(-1, length(T))) = 1))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(82,plain,
% 0.20/0.44 (![H: $int, T: list] : ($sum(length(cons(H, T)), $product(-1, length(T))) = 1) <=> ![H: $int, T: list] : ($sum(length(cons(H, T)), $product(-1, length(T))) = 1)),
% 0.20/0.44 inference(quant_intro,[status(thm)],[81])).
% 0.20/0.44 tff(83,plain,
% 0.20/0.44 (^[H: $int, T: list] : rewrite((length(cons(H, T)) = $sum(1, length(T))) <=> ($sum(length(cons(H, T)), $product(-1, length(T))) = 1))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(84,plain,
% 0.20/0.44 (![H: $int, T: list] : (length(cons(H, T)) = $sum(1, length(T))) <=> ![H: $int, T: list] : ($sum(length(cons(H, T)), $product(-1, length(T))) = 1)),
% 0.20/0.44 inference(quant_intro,[status(thm)],[83])).
% 0.20/0.44 tff(85,plain,
% 0.20/0.44 (![H: $int, T: list] : (length(cons(H, T)) = $sum(1, length(T))) <=> ![H: $int, T: list] : (length(cons(H, T)) = $sum(1, length(T)))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(86,axiom,(![H: $int, T: list] : (length(cons(H, T)) = $sum(1, length(T)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','l_1')).
% 0.20/0.44 tff(87,plain,
% 0.20/0.44 (![H: $int, T: list] : (length(cons(H, T)) = $sum(1, length(T)))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[86, 85])).
% 0.20/0.44 tff(88,plain,
% 0.20/0.44 (![H: $int, T: list] : ($sum(length(cons(H, T)), $product(-1, length(T))) = 1)),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[87, 84])).
% 0.20/0.44 tff(89,plain,(
% 0.20/0.44 ![H: $int, T: list] : ($sum(length(cons(H, T)), $product(-1, length(T))) = 1)),
% 0.20/0.44 inference(skolemize,[status(sab)],[88])).
% 0.20/0.44 tff(90,plain,
% 0.20/0.44 (![H: $int, T: list] : ($sum(length(cons(H, T)), $product(-1, length(T))) = 1)),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[89, 82])).
% 0.20/0.44 tff(91,plain,
% 0.20/0.44 (((~![H: $int, T: list] : ($sum(length(cons(H, T)), $product(-1, length(T))) = 1)) | ($sum(length(nil), $product(-1, length(cons(0, nil)))) = -1)) <=> ((~![H: $int, T: list] : ($sum(length(cons(H, T)), $product(-1, length(T))) = 1)) | ($sum(length(nil), $product(-1, length(cons(0, nil)))) = -1))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(92,plain,
% 0.20/0.44 (($sum($product(-1, length(nil)), length(cons(0, nil))) = 1) <=> ($sum(length(nil), $product(-1, length(cons(0, nil)))) = -1)),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(93,plain,
% 0.20/0.44 ($sum(length(cons(0, nil)), $product(-1, length(nil))) = $sum($product(-1, length(nil)), length(cons(0, nil)))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(94,plain,
% 0.20/0.44 (($sum(length(cons(0, nil)), $product(-1, length(nil))) = 1) <=> ($sum($product(-1, length(nil)), length(cons(0, nil))) = 1)),
% 0.20/0.44 inference(monotonicity,[status(thm)],[93])).
% 0.20/0.44 tff(95,plain,
% 0.20/0.44 (($sum(length(cons(0, nil)), $product(-1, length(nil))) = 1) <=> ($sum(length(nil), $product(-1, length(cons(0, nil)))) = -1)),
% 0.20/0.44 inference(transitivity,[status(thm)],[94, 92])).
% 0.20/0.44 tff(96,plain,
% 0.20/0.44 (((~![H: $int, T: list] : ($sum(length(cons(H, T)), $product(-1, length(T))) = 1)) | ($sum(length(cons(0, nil)), $product(-1, length(nil))) = 1)) <=> ((~![H: $int, T: list] : ($sum(length(cons(H, T)), $product(-1, length(T))) = 1)) | ($sum(length(nil), $product(-1, length(cons(0, nil)))) = -1))),
% 0.20/0.44 inference(monotonicity,[status(thm)],[95])).
% 0.20/0.44 tff(97,plain,
% 0.20/0.44 (((~![H: $int, T: list] : ($sum(length(cons(H, T)), $product(-1, length(T))) = 1)) | ($sum(length(cons(0, nil)), $product(-1, length(nil))) = 1)) <=> ((~![H: $int, T: list] : ($sum(length(cons(H, T)), $product(-1, length(T))) = 1)) | ($sum(length(nil), $product(-1, length(cons(0, nil)))) = -1))),
% 0.20/0.44 inference(transitivity,[status(thm)],[96, 91])).
% 0.20/0.44 tff(98,plain,
% 0.20/0.44 ((~![H: $int, T: list] : ($sum(length(cons(H, T)), $product(-1, length(T))) = 1)) | ($sum(length(cons(0, nil)), $product(-1, length(nil))) = 1)),
% 0.20/0.44 inference(quant_inst,[status(thm)],[])).
% 0.20/0.44 tff(99,plain,
% 0.20/0.44 ((~![H: $int, T: list] : ($sum(length(cons(H, T)), $product(-1, length(T))) = 1)) | ($sum(length(nil), $product(-1, length(cons(0, nil)))) = -1)),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[98, 97])).
% 0.20/0.44 tff(100,plain,
% 0.20/0.44 ($sum(length(nil), $product(-1, length(cons(0, nil)))) = -1),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[99, 90])).
% 0.20/0.44 tff(101,plain,
% 0.20/0.44 ((~($sum(length(nil), $product(-1, length(cons(0, nil)))) = -1)) | $lesseq($sum(length(nil), $product(-1, length(cons(0, nil)))), -1)),
% 0.20/0.44 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.44 tff(102,plain,
% 0.20/0.44 ($lesseq($sum(length(nil), $product(-1, length(cons(0, nil)))), -1)),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[101, 100])).
% 0.20/0.44 tff(103,plain,
% 0.20/0.44 ((length(nil) = 0) <=> (length(nil) = 0)),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(104,axiom,(length(nil) = 0), file('/export/starexec/sandbox/benchmark/theBenchmark.p','l')).
% 0.20/0.44 tff(105,plain,
% 0.20/0.44 (length(nil) = 0),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[104, 103])).
% 0.20/0.44 tff(106,plain,
% 0.20/0.44 ((~(length(nil) = 0)) | $greatereq(length(nil), 0)),
% 0.20/0.44 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.44 tff(107,plain,
% 0.20/0.44 ($greatereq(length(nil), 0)),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[106, 105])).
% 0.20/0.44 tff(108,plain,
% 0.20/0.44 (length(cons(head(cons(0, nil)), tail(cons(0, nil)))) = length(cons(0, nil))),
% 0.20/0.44 inference(monotonicity,[status(thm)],[57])).
% 0.20/0.44 tff(109,plain,
% 0.20/0.44 (length(cons(0, nil)) = length(cons(head(cons(0, nil)), tail(cons(0, nil))))),
% 0.20/0.44 inference(symmetry,[status(thm)],[108])).
% 0.20/0.44 tff(110,plain,
% 0.20/0.44 ((~(length(cons(0, nil)) = length(cons(head(cons(0, nil)), tail(cons(0, nil)))))) | $lesseq($sum(length(cons(0, nil)), $product(-1, length(cons(head(cons(0, nil)), tail(cons(0, nil)))))), 0)),
% 0.20/0.44 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.44 tff(111,plain,
% 0.20/0.44 ($lesseq($sum(length(cons(0, nil)), $product(-1, length(cons(head(cons(0, nil)), tail(cons(0, nil)))))), 0)),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[110, 109])).
% 0.20/0.44 tff(112,assumption,($lesseq(count(-1, cons(head(cons(0, nil)), tail(cons(0, nil)))), 0)), introduced(assumption)).
% 0.20/0.44 tff(113,plain,
% 0.20/0.44 ($false),
% 0.20/0.44 inference(theory_lemma,[status(thm)],[112, 111, 107, 102, 80])).
% 0.20/0.44 tff(114,plain,(~$lesseq(count(-1, cons(head(cons(0, nil)), tail(cons(0, nil)))), 0)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.44 tff(115,plain,
% 0.20/0.44 (^[K: $int] : refl((count(K, nil) = 0) <=> (count(K, nil) = 0))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(116,plain,
% 0.20/0.44 (![K: $int] : (count(K, nil) = 0) <=> ![K: $int] : (count(K, nil) = 0)),
% 0.20/0.44 inference(quant_intro,[status(thm)],[115])).
% 0.20/0.44 tff(117,plain,
% 0.20/0.44 (![K: $int] : (count(K, nil) = 0) <=> ![K: $int] : (count(K, nil) = 0)),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(118,axiom,(![K: $int] : (count(K, nil) = 0)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','a')).
% 0.20/0.44 tff(119,plain,
% 0.20/0.44 (![K: $int] : (count(K, nil) = 0)),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[118, 117])).
% 0.20/0.44 tff(120,plain,(
% 0.20/0.44 ![K: $int] : (count(K, nil) = 0)),
% 0.20/0.44 inference(skolemize,[status(sab)],[119])).
% 0.20/0.44 tff(121,plain,
% 0.20/0.44 (![K: $int] : (count(K, nil) = 0)),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[120, 116])).
% 0.20/0.44 tff(122,plain,
% 0.20/0.44 ((~![K: $int] : (count(K, nil) = 0)) | (count(-1, nil) = 0)),
% 0.20/0.44 inference(quant_inst,[status(thm)],[])).
% 0.20/0.44 tff(123,plain,
% 0.20/0.44 (count(-1, nil) = 0),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[122, 121])).
% 0.20/0.44 tff(124,plain,
% 0.20/0.44 ((~(count(-1, nil) = 0)) | $lesseq(count(-1, nil), 0)),
% 0.20/0.44 inference(theory_lemma,[status(thm)],[])).
% 0.20/0.44 tff(125,plain,
% 0.20/0.44 ($lesseq(count(-1, nil), 0)),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[124, 123])).
% 0.20/0.44 tff(126,plain,
% 0.20/0.44 ($false),
% 0.20/0.44 inference(theory_lemma,[status(thm)],[125, 114, 60, 36])).
% 0.20/0.44 % SZS output end Proof
%------------------------------------------------------------------------------