TSTP Solution File: SWW605_2 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SWW605_2 : TPTP v8.1.0. Released v6.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n003.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 20:59:25 EDT 2022
% Result : Theorem 0.20s 0.49s
% Output : Proof 0.54s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SWW605_2 : TPTP v8.1.0. Released v6.1.0.
% 0.07/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n003.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Sun Sep 4 20:17:11 EDT 2022
% 0.13/0.34 % CPUTime :
% 0.13/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.34 Usage: tptp [options] [-file:]file
% 0.13/0.34 -h, -? prints this message.
% 0.13/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.34 -m, -model generate model.
% 0.13/0.34 -p, -proof generate proof.
% 0.13/0.34 -c, -core generate unsat core of named formulas.
% 0.13/0.34 -st, -statistics display statistics.
% 0.13/0.34 -t:timeout set timeout (in second).
% 0.13/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.34 -<param>:<value> configuration parameter and value.
% 0.13/0.34 -o:<output-file> file to place output in.
% 0.20/0.49 % SZS status Theorem
% 0.20/0.49 % SZS output start Proof
% 0.20/0.49 tff(some_type, type, (
% 0.20/0.49 some: ( ty * uni ) > uni)).
% 0.20/0.49 tff(t2tb7_type, type, (
% 0.20/0.49 t2tb7: a1 > uni)).
% 0.20/0.49 tff(tptp_fun_V_11_type, type, (
% 0.20/0.49 tptp_fun_V_11: a1)).
% 0.20/0.49 tff(a_type, type, (
% 0.20/0.49 a: ty)).
% 0.20/0.49 tff(get_type, type, (
% 0.20/0.49 get: ( ty * ty * uni * uni ) > uni)).
% 0.20/0.49 tff(t2tb1_type, type, (
% 0.20/0.49 t2tb1: key1 > uni)).
% 0.20/0.49 tff(tptp_fun_K_12_type, type, (
% 0.20/0.49 tptp_fun_K_12: key1)).
% 0.20/0.49 tff(t2tb3_type, type, (
% 0.20/0.49 t2tb3: map_key_lpoption_a1rp > uni)).
% 0.20/0.49 tff(tptp_fun_Rho_7_type, type, (
% 0.20/0.49 tptp_fun_Rho_7: map_key_lpoption_a1rp)).
% 0.20/0.49 tff(key_type, type, (
% 0.20/0.49 key: ty)).
% 0.20/0.49 tff(option_type, type, (
% 0.20/0.49 option: ty > ty)).
% 0.20/0.49 tff(t2tb4_type, type, (
% 0.20/0.49 t2tb4: option_a1 > uni)).
% 0.20/0.49 tff(tb2t4_type, type, (
% 0.20/0.49 tb2t4: uni > option_a1)).
% 0.20/0.49 tff(sort1_type, type, (
% 0.20/0.49 sort1: ( ty * uni ) > $o)).
% 0.20/0.49 tff(in_data1_type, type, (
% 0.20/0.49 in_data1: ( ty * key1 * uni * uni ) > $o)).
% 0.20/0.49 tff(mk_array1_type, type, (
% 0.20/0.49 mk_array1: ( ty * $int * uni ) > uni)).
% 0.20/0.49 tff(t2tb2_type, type, (
% 0.20/0.49 t2tb2: map_int_lplist_lpkeycm_a1rprp > uni)).
% 0.20/0.49 tff(tptp_fun_Ndata_8_type, type, (
% 0.20/0.49 tptp_fun_Ndata_8: map_int_lplist_lpkeycm_a1rprp)).
% 0.20/0.49 tff(tptp_fun_H_4_type, type, (
% 0.20/0.49 tptp_fun_H_4: $int)).
% 0.20/0.49 tff(list_type, type, (
% 0.20/0.49 list: ty > ty)).
% 0.20/0.49 tff(tuple2_type, type, (
% 0.20/0.49 tuple2: ( ty * ty ) > ty)).
% 0.20/0.49 tff(mem_type, type, (
% 0.20/0.49 mem: ( ty * uni * uni ) > $o)).
% 0.20/0.49 tff(t2tb6_type, type, (
% 0.20/0.49 t2tb6: list_lpkeycm_a1rp > uni)).
% 0.20/0.49 tff(tptp_fun_L_9_type, type, (
% 0.20/0.49 tptp_fun_L_9: list_lpkeycm_a1rp)).
% 0.20/0.49 tff(tuple21_type, type, (
% 0.20/0.49 tuple21: ( ty * ty * uni * uni ) > uni)).
% 0.20/0.49 tff(good_data1_type, type, (
% 0.20/0.49 good_data1: ( ty * key1 * uni * uni * uni ) > $o)).
% 0.20/0.49 tff(bucket1_type, type, (
% 0.20/0.49 bucket1: ( key1 * $int ) > $int)).
% 0.20/0.49 tff(tptp_fun_I_10_type, type, (
% 0.20/0.49 tptp_fun_I_10: $int)).
% 0.20/0.49 tff(good_hash1_type, type, (
% 0.20/0.49 good_hash1: ( ty * uni * $int ) > $o)).
% 0.20/0.49 tff(tptp_fun_Rho1_6_type, type, (
% 0.20/0.49 tptp_fun_Rho1_6: $int)).
% 0.20/0.49 tff(tptp_fun_Rho2_5_type, type, (
% 0.20/0.49 tptp_fun_Rho2_5: map_int_lplist_lpkeycm_a1rprp)).
% 0.20/0.49 tff(tb2t6_type, type, (
% 0.20/0.49 tb2t6: uni > list_lpkeycm_a1rp)).
% 0.20/0.49 tff(nil_type, type, (
% 0.20/0.49 nil: ty > uni)).
% 0.20/0.49 tff(tptp_fun_H1_3_type, type, (
% 0.20/0.49 tptp_fun_H1_3: map_int_lplist_lpkeycm_a1rprp)).
% 0.20/0.49 tff(tptp_fun_H2_2_type, type, (
% 0.20/0.49 tptp_fun_H2_2: map_key_lpoption_a1rp)).
% 0.20/0.49 tff(cons_type, type, (
% 0.20/0.49 cons: ( ty * uni * uni ) > uni)).
% 0.20/0.49 tff(1,plain,
% 0.20/0.49 (^[A: ty, X: uni] : refl(sort1(option(A), some(A, X)) <=> sort1(option(A), some(A, X)))),
% 0.20/0.49 inference(bind,[status(th)],[])).
% 0.20/0.49 tff(2,plain,
% 0.20/0.49 (![A: ty, X: uni] : sort1(option(A), some(A, X)) <=> ![A: ty, X: uni] : sort1(option(A), some(A, X))),
% 0.20/0.49 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.49 tff(3,plain,
% 0.20/0.49 (![A: ty, X: uni] : sort1(option(A), some(A, X)) <=> ![A: ty, X: uni] : sort1(option(A), some(A, X))),
% 0.20/0.49 inference(rewrite,[status(thm)],[])).
% 0.20/0.49 tff(4,axiom,(![A: ty, X: uni] : sort1(option(A), some(A, X))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','some_sort2')).
% 0.20/0.49 tff(5,plain,
% 0.20/0.49 (![A: ty, X: uni] : sort1(option(A), some(A, X))),
% 0.20/0.49 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.20/0.49 tff(6,plain,(
% 0.20/0.49 ![A: ty, X: uni] : sort1(option(A), some(A, X))),
% 0.20/0.49 inference(skolemize,[status(sab)],[5])).
% 0.20/0.49 tff(7,plain,
% 0.20/0.49 (![A: ty, X: uni] : sort1(option(A), some(A, X))),
% 0.20/0.49 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.20/0.49 tff(8,plain,
% 0.20/0.49 ((~![A: ty, X: uni] : sort1(option(A), some(A, X))) | sort1(option(a), some(a, t2tb7(V!11)))),
% 0.20/0.49 inference(quant_inst,[status(thm)],[])).
% 0.20/0.49 tff(9,plain,
% 0.20/0.49 (sort1(option(a), some(a, t2tb7(V!11)))),
% 0.20/0.49 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.20/0.49 tff(10,plain,
% 0.20/0.49 (^[J: uni] : refl(((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J)) <=> ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J)))),
% 0.20/0.49 inference(bind,[status(th)],[])).
% 0.20/0.49 tff(11,plain,
% 0.20/0.49 (![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J)) <=> ![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J))),
% 0.20/0.49 inference(quant_intro,[status(thm)],[10])).
% 0.20/0.49 tff(12,plain,
% 0.20/0.49 (![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J)) <=> ![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J))),
% 0.20/0.49 inference(rewrite,[status(thm)],[])).
% 0.20/0.49 tff(13,plain,
% 0.20/0.49 (^[J: uni] : rewrite((sort1(option(a), J) => (t2tb4(tb2t4(J)) = J)) <=> ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J)))),
% 0.20/0.49 inference(bind,[status(th)],[])).
% 0.20/0.49 tff(14,plain,
% 0.20/0.49 (![J: uni] : (sort1(option(a), J) => (t2tb4(tb2t4(J)) = J)) <=> ![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J))),
% 0.20/0.49 inference(quant_intro,[status(thm)],[13])).
% 0.20/0.49 tff(15,axiom,(![J: uni] : (sort1(option(a), J) => (t2tb4(tb2t4(J)) = J))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','bridgeR4')).
% 0.20/0.49 tff(16,plain,
% 0.20/0.49 (![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J))),
% 0.20/0.49 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.20/0.49 tff(17,plain,
% 0.20/0.49 (![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J))),
% 0.20/0.49 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.20/0.49 tff(18,plain,(
% 0.20/0.49 ![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J))),
% 0.20/0.49 inference(skolemize,[status(sab)],[17])).
% 0.20/0.49 tff(19,plain,
% 0.20/0.49 (![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J))),
% 0.20/0.49 inference(modus_ponens,[status(thm)],[18, 11])).
% 0.20/0.49 tff(20,plain,
% 0.20/0.49 (((~![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J))) | ((~sort1(option(a), some(a, t2tb7(V!11)))) | (t2tb4(tb2t4(some(a, t2tb7(V!11)))) = some(a, t2tb7(V!11))))) <=> ((~![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J))) | (~sort1(option(a), some(a, t2tb7(V!11)))) | (t2tb4(tb2t4(some(a, t2tb7(V!11)))) = some(a, t2tb7(V!11))))),
% 0.20/0.49 inference(rewrite,[status(thm)],[])).
% 0.20/0.49 tff(21,plain,
% 0.20/0.49 ((~![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J))) | ((~sort1(option(a), some(a, t2tb7(V!11)))) | (t2tb4(tb2t4(some(a, t2tb7(V!11)))) = some(a, t2tb7(V!11))))),
% 0.20/0.49 inference(quant_inst,[status(thm)],[])).
% 0.20/0.49 tff(22,plain,
% 0.20/0.49 ((~![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J))) | (~sort1(option(a), some(a, t2tb7(V!11)))) | (t2tb4(tb2t4(some(a, t2tb7(V!11)))) = some(a, t2tb7(V!11)))),
% 0.20/0.49 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.20/0.49 tff(23,plain,
% 0.20/0.49 (t2tb4(tb2t4(some(a, t2tb7(V!11)))) = some(a, t2tb7(V!11))),
% 0.20/0.49 inference(unit_resolution,[status(thm)],[22, 19, 9])).
% 0.20/0.49 tff(24,assumption,(~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))), introduced(assumption)).
% 0.20/0.49 tff(25,plain,
% 0.20/0.49 (^[A: ty, K: key1, V: uni, M: uni, D: uni] : refl((good_data1(A, K, V, M, D) <=> ((get(option(A), key, M, t2tb1(K)) = some(A, V)) <=> in_data1(A, K, V, D))) <=> (good_data1(A, K, V, M, D) <=> ((get(option(A), key, M, t2tb1(K)) = some(A, V)) <=> in_data1(A, K, V, D))))),
% 0.20/0.49 inference(bind,[status(th)],[])).
% 0.20/0.49 tff(26,plain,
% 0.20/0.49 (![A: ty, K: key1, V: uni, M: uni, D: uni] : (good_data1(A, K, V, M, D) <=> ((get(option(A), key, M, t2tb1(K)) = some(A, V)) <=> in_data1(A, K, V, D))) <=> ![A: ty, K: key1, V: uni, M: uni, D: uni] : (good_data1(A, K, V, M, D) <=> ((get(option(A), key, M, t2tb1(K)) = some(A, V)) <=> in_data1(A, K, V, D)))),
% 0.20/0.49 inference(quant_intro,[status(thm)],[25])).
% 0.20/0.49 tff(27,plain,
% 0.20/0.49 (![A: ty, K: key1, V: uni, M: uni, D: uni] : (good_data1(A, K, V, M, D) <=> ((get(option(A), key, M, t2tb1(K)) = some(A, V)) <=> in_data1(A, K, V, D))) <=> ![A: ty, K: key1, V: uni, M: uni, D: uni] : (good_data1(A, K, V, M, D) <=> ((get(option(A), key, M, t2tb1(K)) = some(A, V)) <=> in_data1(A, K, V, D)))),
% 0.20/0.49 inference(rewrite,[status(thm)],[])).
% 0.20/0.49 tff(28,axiom,(![A: ty, K: key1, V: uni, M: uni, D: uni] : (good_data1(A, K, V, M, D) <=> ((get(option(A), key, M, t2tb1(K)) = some(A, V)) <=> in_data1(A, K, V, D)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','good_data_def')).
% 0.20/0.49 tff(29,plain,
% 0.20/0.49 (![A: ty, K: key1, V: uni, M: uni, D: uni] : (good_data1(A, K, V, M, D) <=> ((get(option(A), key, M, t2tb1(K)) = some(A, V)) <=> in_data1(A, K, V, D)))),
% 0.20/0.49 inference(modus_ponens,[status(thm)],[28, 27])).
% 0.20/0.49 tff(30,plain,(
% 0.20/0.49 ![A: ty, K: key1, V: uni, M: uni, D: uni] : (good_data1(A, K, V, M, D) <=> ((get(option(A), key, M, t2tb1(K)) = some(A, V)) <=> in_data1(A, K, V, D)))),
% 0.20/0.49 inference(skolemize,[status(sab)],[29])).
% 0.20/0.49 tff(31,plain,
% 0.20/0.49 (![A: ty, K: key1, V: uni, M: uni, D: uni] : (good_data1(A, K, V, M, D) <=> ((get(option(A), key, M, t2tb1(K)) = some(A, V)) <=> in_data1(A, K, V, D)))),
% 0.20/0.49 inference(modus_ponens,[status(thm)],[30, 26])).
% 0.20/0.49 tff(32,plain,
% 0.20/0.49 ((~![A: ty, K: key1, V: uni, M: uni, D: uni] : (good_data1(A, K, V, M, D) <=> ((get(option(A), key, M, t2tb1(K)) = some(A, V)) <=> in_data1(A, K, V, D)))) | (good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) <=> ((get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = some(a, t2tb7(V!11))) <=> in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))),
% 0.20/0.49 inference(quant_inst,[status(thm)],[])).
% 0.20/0.49 tff(33,plain,
% 0.20/0.49 (good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) <=> ((get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = some(a, t2tb7(V!11))) <=> in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))),
% 0.20/0.49 inference(unit_resolution,[status(thm)],[32, 31])).
% 0.20/0.49 tff(34,assumption,(~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)), introduced(assumption)).
% 0.20/0.49 tff(35,plain,
% 0.20/0.49 ((good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)),
% 0.20/0.49 inference(tautology,[status(thm)],[])).
% 0.20/0.49 tff(36,plain,
% 0.20/0.49 (good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))),
% 0.20/0.49 inference(unit_resolution,[status(thm)],[35, 34])).
% 0.20/0.49 tff(37,plain,
% 0.20/0.49 ((~(~((~(good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))) | (~((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))))))) <=> ((~(good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))) | (~((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))))))),
% 0.20/0.49 inference(rewrite,[status(thm)],[])).
% 0.20/0.49 tff(38,plain,
% 0.20/0.49 (((good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))))) <=> (~((~(good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))) | (~((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))))))),
% 0.20/0.49 inference(rewrite,[status(thm)],[])).
% 0.20/0.49 tff(39,plain,
% 0.20/0.49 (((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) <=> ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))))),
% 0.20/0.49 inference(rewrite,[status(thm)],[])).
% 0.20/0.49 tff(40,plain,
% 0.20/0.49 ((((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) <=> (good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))),
% 0.20/0.49 inference(rewrite,[status(thm)],[])).
% 0.20/0.49 tff(41,plain,
% 0.20/0.49 ((~(~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))) <=> ((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))),
% 0.20/0.49 inference(rewrite,[status(thm)],[])).
% 0.20/0.49 tff(42,plain,
% 0.20/0.49 (($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)) <=> (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))),
% 0.20/0.49 inference(rewrite,[status(thm)],[])).
% 0.20/0.49 tff(43,plain,
% 0.20/0.49 ((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) <=> (~(~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))))),
% 0.20/0.49 inference(monotonicity,[status(thm)],[42])).
% 0.20/0.49 tff(44,plain,
% 0.20/0.49 ((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) <=> ((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))),
% 0.20/0.49 inference(transitivity,[status(thm)],[43, 41])).
% 0.20/0.49 tff(45,plain,
% 0.20/0.49 (((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) <=> (((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))),
% 0.20/0.49 inference(monotonicity,[status(thm)],[44])).
% 0.20/0.49 tff(46,plain,
% 0.20/0.49 (((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) <=> (good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))),
% 0.20/0.49 inference(transitivity,[status(thm)],[45, 40])).
% 0.20/0.49 tff(47,plain,
% 0.20/0.49 ((((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))) <=> ((good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))))),
% 0.20/0.49 inference(monotonicity,[status(thm)],[46, 39])).
% 0.20/0.49 tff(48,plain,
% 0.20/0.49 ((((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))) <=> (~((~(good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))) | (~((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))))))),
% 0.20/0.50 inference(transitivity,[status(thm)],[47, 38])).
% 0.20/0.50 tff(49,plain,
% 0.20/0.50 ((~(((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))) <=> (~(~((~(good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))) | (~((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))))))))),
% 0.20/0.50 inference(monotonicity,[status(thm)],[48])).
% 0.20/0.50 tff(50,plain,
% 0.20/0.50 ((~(((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))) <=> ((~(good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))) | (~((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))))))),
% 0.20/0.50 inference(transitivity,[status(thm)],[49, 37])).
% 0.20/0.50 tff(51,plain,
% 0.20/0.50 (($greatereq(H!4, 0) & ((~$lesseq(H!4, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2!2), mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3))) & $greatereq(H!4, 0)) & ((L!9 = tb2t6(nil(tuple2(key, a)))) & (~$lesseq(Rho1!6, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$greatereq($sum(I1, $product(-1, Rho1!6)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5))) & $greatereq(Rho1!6, 0) & $greatereq(H!4, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9))) | ($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H!4)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))) & (~(((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))))) <=> ($greatereq(H!4, 0) & (~$lesseq(H!4, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2!2), mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3))) & (L!9 = tb2t6(nil(tuple2(key, a)))) & (~$lesseq(Rho1!6, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$greatereq($sum(I1, $product(-1, Rho1!6)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5))) & $greatereq(Rho1!6, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9))) | ($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H!4)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))) & (~(((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))))),
% 0.20/0.50 inference(rewrite,[status(thm)],[])).
% 0.20/0.50 tff(52,plain,
% 0.20/0.50 (((L!9 = tb2t6(nil(tuple2(key, a)))) & ((~$lesseq(Rho1!6, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$greatereq($sum(I1, $product(-1, Rho1!6)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5))) & $greatereq(Rho1!6, 0) & $greatereq(H!4, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9))) | ($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H!4)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))) & (~(((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))))) <=> ((L!9 = tb2t6(nil(tuple2(key, a)))) & (~$lesseq(Rho1!6, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$greatereq($sum(I1, $product(-1, Rho1!6)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5))) & $greatereq(Rho1!6, 0) & $greatereq(H!4, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9))) | ($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H!4)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))) & (~(((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))))),
% 0.20/0.51 inference(rewrite,[status(thm)],[])).
% 0.20/0.51 tff(53,plain,
% 0.20/0.51 ((~(((~($greatereq(bucket1(K!12, H!4), 0) & $lesseq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $lesseq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0))))) <=> (~(((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))))),
% 0.20/0.51 inference(rewrite,[status(thm)],[])).
% 0.20/0.51 tff(54,plain,
% 0.20/0.51 (((~$lesseq(Rho1!6, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$lesseq($sum(Rho1!6, $product(-1, I1)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5))) & $greatereq(Rho1!6, 0) & $greatereq(H!4, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9))) | ($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H!4)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))) <=> ((~$lesseq(Rho1!6, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$greatereq($sum(I1, $product(-1, Rho1!6)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5))) & $greatereq(Rho1!6, 0) & $greatereq(H!4, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9))) | ($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H!4)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))))),
% 0.20/0.51 inference(rewrite,[status(thm)],[])).
% 0.20/0.51 tff(55,plain,
% 0.20/0.51 ((~(~(L!9 = tb2t6(nil(tuple2(key, a)))))) <=> (L!9 = tb2t6(nil(tuple2(key, a))))),
% 0.20/0.51 inference(rewrite,[status(thm)],[])).
% 0.20/0.51 tff(56,plain,
% 0.20/0.51 (((~(~(L!9 = tb2t6(nil(tuple2(key, a)))))) & ((~$lesseq(Rho1!6, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$lesseq($sum(Rho1!6, $product(-1, I1)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5))) & $greatereq(Rho1!6, 0) & $greatereq(H!4, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9))) | ($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H!4)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))) & (~(((~($greatereq(bucket1(K!12, H!4), 0) & $lesseq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $lesseq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0)))))) <=> ((L!9 = tb2t6(nil(tuple2(key, a)))) & ((~$lesseq(Rho1!6, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$greatereq($sum(I1, $product(-1, Rho1!6)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5))) & $greatereq(Rho1!6, 0) & $greatereq(H!4, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9))) | ($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H!4)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))) & (~(((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))))),
% 0.20/0.52 inference(monotonicity,[status(thm)],[55, 54, 53])).
% 0.20/0.52 tff(57,plain,
% 0.20/0.52 (((~(~(L!9 = tb2t6(nil(tuple2(key, a)))))) & ((~$lesseq(Rho1!6, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$lesseq($sum(Rho1!6, $product(-1, I1)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5))) & $greatereq(Rho1!6, 0) & $greatereq(H!4, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9))) | ($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H!4)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))) & (~(((~($greatereq(bucket1(K!12, H!4), 0) & $lesseq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $lesseq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0)))))) <=> ((L!9 = tb2t6(nil(tuple2(key, a)))) & (~$lesseq(Rho1!6, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$greatereq($sum(I1, $product(-1, Rho1!6)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5))) & $greatereq(Rho1!6, 0) & $greatereq(H!4, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9))) | ($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H!4)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))) & (~(((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))))),
% 0.20/0.52 inference(transitivity,[status(thm)],[56, 52])).
% 0.20/0.52 tff(58,plain,
% 0.20/0.52 (((~$lesseq(H!4, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2!2), mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3))) & $greatereq(H!4, 0)) <=> ((~$lesseq(H!4, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2!2), mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3))) & $greatereq(H!4, 0))),
% 0.20/0.52 inference(rewrite,[status(thm)],[])).
% 0.20/0.52 tff(59,plain,
% 0.20/0.52 ((~(~$greatereq(H!4, 0))) <=> $greatereq(H!4, 0)),
% 0.20/0.52 inference(rewrite,[status(thm)],[])).
% 0.20/0.52 tff(60,plain,
% 0.20/0.52 (((~(~$greatereq(H!4, 0))) & ((~$lesseq(H!4, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2!2), mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3))) & $greatereq(H!4, 0)) & ((~(~(L!9 = tb2t6(nil(tuple2(key, a)))))) & ((~$lesseq(Rho1!6, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$lesseq($sum(Rho1!6, $product(-1, I1)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5))) & $greatereq(Rho1!6, 0) & $greatereq(H!4, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9))) | ($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H!4)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))) & (~(((~($greatereq(bucket1(K!12, H!4), 0) & $lesseq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $lesseq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0))))))) <=> ($greatereq(H!4, 0) & ((~$lesseq(H!4, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2!2), mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3))) & $greatereq(H!4, 0)) & ((L!9 = tb2t6(nil(tuple2(key, a)))) & (~$lesseq(Rho1!6, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$greatereq($sum(I1, $product(-1, Rho1!6)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5))) & $greatereq(Rho1!6, 0) & $greatereq(H!4, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9))) | ($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H!4)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))) & (~(((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))))))),
% 0.20/0.53 inference(monotonicity,[status(thm)],[59, 58, 57])).
% 0.20/0.53 tff(61,plain,
% 0.20/0.53 (((~(~$greatereq(H!4, 0))) & ((~$lesseq(H!4, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2!2), mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3))) & $greatereq(H!4, 0)) & ((~(~(L!9 = tb2t6(nil(tuple2(key, a)))))) & ((~$lesseq(Rho1!6, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$lesseq($sum(Rho1!6, $product(-1, I1)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5))) & $greatereq(Rho1!6, 0) & $greatereq(H!4, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9))) | ($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H!4)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))) & (~(((~($greatereq(bucket1(K!12, H!4), 0) & $lesseq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $lesseq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0))))))) <=> ($greatereq(H!4, 0) & (~$lesseq(H!4, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2!2), mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3))) & (L!9 = tb2t6(nil(tuple2(key, a)))) & (~$lesseq(Rho1!6, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$greatereq($sum(I1, $product(-1, Rho1!6)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5))) & $greatereq(Rho1!6, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9))) | ($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H!4)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))) & (~(((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))))),
% 0.43/0.53 inference(transitivity,[status(thm)],[60, 51])).
% 0.43/0.53 tff(62,plain,
% 0.43/0.53 ((~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $greatereq(H, 0))) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$lesseq($sum(Rho1, $product(-1, I1)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $greatereq(Rho1, 0) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | ($sum(bucket1(K, H), $product(-1, I)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H), 0) & (~$greatereq($sum(bucket1(K, H), $product(-1, I)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($greatereq(bucket1(K, H), 0) & (~$greatereq($sum(bucket1(K, H), $product(-1, I)), 0)))) & ((($sum(bucket1(K, H), $product(-1, I)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~($sum(bucket1(K, H), $product(-1, I)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & ((~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0))))))) <=> (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $greatereq(H, 0))) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$lesseq($sum(Rho1, $product(-1, I1)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $greatereq(Rho1, 0) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | ($sum(bucket1(K, H), $product(-1, I)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H), 0) & (~$greatereq($sum(bucket1(K, H), $product(-1, I)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($greatereq(bucket1(K, H), 0) & (~$greatereq($sum(bucket1(K, H), $product(-1, I)), 0)))) & ((($sum(bucket1(K, H), $product(-1, I)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~($sum(bucket1(K, H), $product(-1, I)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & ((~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0)))))))),
% 0.43/0.54 inference(rewrite,[status(thm)],[])).
% 0.43/0.54 tff(63,plain,
% 0.43/0.54 ((~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$lesseq($sum(Rho1, $product(-1, I1)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $greatereq(Rho1, 0) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | ($sum(bucket1(K, H), $product(-1, I)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$lesseq($sum($product(2, H), $product(-1, J)), -1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H), 0) & (~$lesseq($sum(I, $product(-1, bucket1(K, H))), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($greatereq(bucket1(K, H), 0) & (~$lesseq($sum(I, $product(-1, bucket1(K, H))), 0)))) & ((($sum(bucket1(K, H), $product(-1, I)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~($sum(bucket1(K, H), $product(-1, I)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & ((~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0))))) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$lesseq($sum(H, $product(-1, I)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $greatereq(H, 0))))) <=> (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $greatereq(H, 0))) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$lesseq($sum(Rho1, $product(-1, I1)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $greatereq(Rho1, 0) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | ($sum(bucket1(K, H), $product(-1, I)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H), 0) & (~$greatereq($sum(bucket1(K, H), $product(-1, I)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($greatereq(bucket1(K, H), 0) & (~$greatereq($sum(bucket1(K, H), $product(-1, I)), 0)))) & ((($sum(bucket1(K, H), $product(-1, I)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~($sum(bucket1(K, H), $product(-1, I)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & ((~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0)))))))),
% 0.43/0.54 inference(rewrite,[status(thm)],[])).
% 0.43/0.54 tff(64,plain,
% 0.43/0.54 ((~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($lesseq(0, I1) & (~$lesseq(Rho1, I1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $lesseq(0, Rho1) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | (bucket1(K, H) = I)) & ![J: $int] : ((~($lesseq(0, J) & (~$lesseq($product(2, H), $sum(-1, J))))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($lesseq(0, bucket1(K, H)) & (~$lesseq(I, bucket1(K, H))) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($lesseq(0, bucket1(K, H)) & (~$lesseq(I, bucket1(K, H))))) & (((bucket1(K, H) = I) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~(bucket1(K, H) = I)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & (($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I)) | (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata))))))) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $lesseq(0, H))))) <=> (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$lesseq($sum(Rho1, $product(-1, I1)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $greatereq(Rho1, 0) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | ($sum(bucket1(K, H), $product(-1, I)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$lesseq($sum($product(2, H), $product(-1, J)), -1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H), 0) & (~$lesseq($sum(I, $product(-1, bucket1(K, H))), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($greatereq(bucket1(K, H), 0) & (~$lesseq($sum(I, $product(-1, bucket1(K, H))), 0)))) & ((($sum(bucket1(K, H), $product(-1, I)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~($sum(bucket1(K, H), $product(-1, I)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & ((~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0))))) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$lesseq($sum(H, $product(-1, I)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $greatereq(H, 0)))))),
% 0.43/0.54 inference(rewrite,[status(thm)],[])).
% 0.43/0.54 tff(65,plain,
% 0.43/0.54 ((~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($lesseq(0, I1) & (~$lesseq(Rho1, I1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $lesseq(0, Rho1) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | (bucket1(K, H) = I)) & ![J: $int] : ((~($lesseq(0, J) & (~$lesseq($product(2, H), $sum(-1, J))))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($lesseq(0, bucket1(K, H)) & (~$lesseq(I, bucket1(K, H))) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($lesseq(0, bucket1(K, H)) & (~$lesseq(I, bucket1(K, H))))) & (((bucket1(K, H) = I) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~(bucket1(K, H) = I)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & (($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I)) | (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata))))))) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $lesseq(0, H))))) <=> (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($lesseq(0, I1) & (~$lesseq(Rho1, I1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $lesseq(0, Rho1) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | (bucket1(K, H) = I)) & ![J: $int] : ((~($lesseq(0, J) & (~$lesseq($product(2, H), $sum(-1, J))))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($lesseq(0, bucket1(K, H)) & (~$lesseq(I, bucket1(K, H))) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($lesseq(0, bucket1(K, H)) & (~$lesseq(I, bucket1(K, H))))) & (((bucket1(K, H) = I) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~(bucket1(K, H) = I)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & (($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I)) | (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata))))))) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $lesseq(0, H)))))),
% 0.43/0.55 inference(rewrite,[status(thm)],[])).
% 0.43/0.55 tff(66,plain,
% 0.43/0.55 ((~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : (((($less(0, H) & ![I: $int] : (($lesseq(0, I) & $less(I, H)) => good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I))) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1)))) & $lesseq(0, H)) => ($lesseq(0, $sum($product(2, H), 1)) => ($lesseq(0, $sum($product(2, H), 1)) => ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : (((((((($less(0, Rho1) & ![I1: $int] : (($lesseq(0, I1) & $less(I1, Rho1)) => good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1))) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)))) & $lesseq(0, Rho1)) & $lesseq(0, $sum($product(2, H), 1))) & ![K: key1, V: a1] : (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) => (bucket1(K, H) = I))) & ![J: $int] : (($lesseq(0, J) & $less(J, $sum($product(2, H), 1))) => good_hash1(a, mk_array1(list(tuple2(key, a)), $sum($product(2, H), 1), t2tb2(Ndata)), J))) & ![K: key1, V: a1] : ((($lesseq(0, bucket1(K, H)) & $less(bucket1(K, H), I)) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum($product(2, H), 1), t2tb2(Ndata)))) | ((~($lesseq(0, bucket1(K, H)) & $less(bucket1(K, H), I))) & (((bucket1(K, H) = I) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum($product(2, H), 1), t2tb2(Ndata)))))) | ((~(bucket1(K, H) = I)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum($product(2, H), 1), t2tb2(Ndata))))))))) => ((L = tb2t6(nil(tuple2(key, a)))) => ![K: key1, V: a1] : ((($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I)) => good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum($product(2, H), 1), t2tb2(Ndata)))) & ((~($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I))) => (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum($product(2, H), 1), t2tb2(Ndata)))))))))))) <=> (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($lesseq(0, I1) & (~$lesseq(Rho1, I1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $lesseq(0, Rho1) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | (bucket1(K, H) = I)) & ![J: $int] : ((~($lesseq(0, J) & (~$lesseq($product(2, H), $sum(-1, J))))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($lesseq(0, bucket1(K, H)) & (~$lesseq(I, bucket1(K, H))) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($lesseq(0, bucket1(K, H)) & (~$lesseq(I, bucket1(K, H))))) & (((bucket1(K, H) = I) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~(bucket1(K, H) = I)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & (($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I)) | (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata))))))) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $lesseq(0, H)))))),
% 0.45/0.55 inference(rewrite,[status(thm)],[])).
% 0.45/0.55 tff(67,axiom,(~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : (((($less(0, H) & ![I: $int] : (($lesseq(0, I) & $less(I, H)) => good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I))) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1)))) & $lesseq(0, H)) => ($lesseq(0, $sum($product(2, H), 1)) => ($lesseq(0, $sum($product(2, H), 1)) => ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : (((((((($less(0, Rho1) & ![I1: $int] : (($lesseq(0, I1) & $less(I1, Rho1)) => good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1))) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)))) & $lesseq(0, Rho1)) & $lesseq(0, $sum($product(2, H), 1))) & ![K: key1, V: a1] : (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) => (bucket1(K, H) = I))) & ![J: $int] : (($lesseq(0, J) & $less(J, $sum($product(2, H), 1))) => good_hash1(a, mk_array1(list(tuple2(key, a)), $sum($product(2, H), 1), t2tb2(Ndata)), J))) & ![K: key1, V: a1] : ((($lesseq(0, bucket1(K, H)) & $less(bucket1(K, H), I)) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum($product(2, H), 1), t2tb2(Ndata)))) | ((~($lesseq(0, bucket1(K, H)) & $less(bucket1(K, H), I))) & (((bucket1(K, H) = I) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum($product(2, H), 1), t2tb2(Ndata)))))) | ((~(bucket1(K, H) = I)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum($product(2, H), 1), t2tb2(Ndata))))))))) => ((L = tb2t6(nil(tuple2(key, a)))) => ![K: key1, V: a1] : ((($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I)) => good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum($product(2, H), 1), t2tb2(Ndata)))) & ((~($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I))) => (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum($product(2, H), 1), t2tb2(Ndata)))))))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','wP_parameter_resize')).
% 0.45/0.55 tff(68,plain,
% 0.45/0.55 (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($lesseq(0, I1) & (~$lesseq(Rho1, I1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $lesseq(0, Rho1) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | (bucket1(K, H) = I)) & ![J: $int] : ((~($lesseq(0, J) & (~$lesseq($product(2, H), $sum(-1, J))))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($lesseq(0, bucket1(K, H)) & (~$lesseq(I, bucket1(K, H))) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($lesseq(0, bucket1(K, H)) & (~$lesseq(I, bucket1(K, H))))) & (((bucket1(K, H) = I) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~(bucket1(K, H) = I)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & (($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I)) | (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata))))))) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $lesseq(0, H))))),
% 0.45/0.55 inference(modus_ponens,[status(thm)],[67, 66])).
% 0.45/0.55 tff(69,plain,
% 0.45/0.55 (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($lesseq(0, I1) & (~$lesseq(Rho1, I1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $lesseq(0, Rho1) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | (bucket1(K, H) = I)) & ![J: $int] : ((~($lesseq(0, J) & (~$lesseq($product(2, H), $sum(-1, J))))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($lesseq(0, bucket1(K, H)) & (~$lesseq(I, bucket1(K, H))) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($lesseq(0, bucket1(K, H)) & (~$lesseq(I, bucket1(K, H))))) & (((bucket1(K, H) = I) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~(bucket1(K, H) = I)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & (($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I)) | (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata))))))) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $lesseq(0, H))))),
% 0.45/0.55 inference(modus_ponens,[status(thm)],[68, 65])).
% 0.45/0.55 tff(70,plain,
% 0.45/0.55 (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($lesseq(0, I1) & (~$lesseq(Rho1, I1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $lesseq(0, Rho1) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | (bucket1(K, H) = I)) & ![J: $int] : ((~($lesseq(0, J) & (~$lesseq($product(2, H), $sum(-1, J))))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($lesseq(0, bucket1(K, H)) & (~$lesseq(I, bucket1(K, H))) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($lesseq(0, bucket1(K, H)) & (~$lesseq(I, bucket1(K, H))))) & (((bucket1(K, H) = I) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~(bucket1(K, H) = I)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & (($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I)) | (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata))))))) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $lesseq(0, H))))),
% 0.45/0.56 inference(modus_ponens,[status(thm)],[69, 65])).
% 0.45/0.56 tff(71,plain,
% 0.45/0.56 (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($lesseq(0, I1) & (~$lesseq(Rho1, I1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $lesseq(0, Rho1) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | (bucket1(K, H) = I)) & ![J: $int] : ((~($lesseq(0, J) & (~$lesseq($product(2, H), $sum(-1, J))))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($lesseq(0, bucket1(K, H)) & (~$lesseq(I, bucket1(K, H))) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($lesseq(0, bucket1(K, H)) & (~$lesseq(I, bucket1(K, H))))) & (((bucket1(K, H) = I) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~(bucket1(K, H) = I)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & (($lesseq(0, bucket1(K, H)) & $lesseq(bucket1(K, H), I)) | (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata))))))) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $lesseq(0, H))))),
% 0.45/0.56 inference(modus_ponens,[status(thm)],[70, 65])).
% 0.45/0.56 tff(72,plain,
% 0.45/0.56 (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$lesseq($sum(Rho1, $product(-1, I1)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $greatereq(Rho1, 0) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | ($sum(bucket1(K, H), $product(-1, I)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$lesseq($sum($product(2, H), $product(-1, J)), -1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H), 0) & (~$lesseq($sum(I, $product(-1, bucket1(K, H))), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($greatereq(bucket1(K, H), 0) & (~$lesseq($sum(I, $product(-1, bucket1(K, H))), 0)))) & ((($sum(bucket1(K, H), $product(-1, I)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~($sum(bucket1(K, H), $product(-1, I)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & ((~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0))))) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$lesseq($sum(H, $product(-1, I)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $greatereq(H, 0))))),
% 0.45/0.56 inference(modus_ponens,[status(thm)],[71, 64])).
% 0.45/0.56 tff(73,plain,
% 0.45/0.56 (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $greatereq(H, 0))) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$lesseq($sum(Rho1, $product(-1, I1)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $greatereq(Rho1, 0) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | ($sum(bucket1(K, H), $product(-1, I)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H), 0) & (~$greatereq($sum(bucket1(K, H), $product(-1, I)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($greatereq(bucket1(K, H), 0) & (~$greatereq($sum(bucket1(K, H), $product(-1, I)), 0)))) & ((($sum(bucket1(K, H), $product(-1, I)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~($sum(bucket1(K, H), $product(-1, I)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & ((~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0))))))),
% 0.45/0.56 inference(modus_ponens,[status(thm)],[72, 63])).
% 0.45/0.56 tff(74,plain,
% 0.45/0.56 (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $greatereq(H, 0))) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$lesseq($sum(Rho1, $product(-1, I1)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $greatereq(Rho1, 0) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | ($sum(bucket1(K, H), $product(-1, I)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H), 0) & (~$greatereq($sum(bucket1(K, H), $product(-1, I)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($greatereq(bucket1(K, H), 0) & (~$greatereq($sum(bucket1(K, H), $product(-1, I)), 0)))) & ((($sum(bucket1(K, H), $product(-1, I)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~($sum(bucket1(K, H), $product(-1, I)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & ((~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0))))))),
% 0.45/0.56 inference(modus_ponens,[status(thm)],[73, 62])).
% 0.45/0.56 tff(75,plain,
% 0.45/0.56 (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~$greatereq(H, 0)) | (~((~$lesseq(H, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(H1)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2), mk_array1(list(tuple2(key, a)), H, t2tb2(H1))) & $greatereq(H, 0))) | ![I: $int, L: list_lpkeycm_a1rp, Ndata: map_int_lplist_lpkeycm_a1rprp, Rho: map_key_lpoption_a1rp, Rho1: $int, Rho2: map_int_lplist_lpkeycm_a1rprp] : ((~(L = tb2t6(nil(tuple2(key, a))))) | (~((~$lesseq(Rho1, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$lesseq($sum(Rho1, $product(-1, I1)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), Rho1, t2tb2(Rho2))) & $greatereq(Rho1, 0) & $greatereq(H, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L))) | ($sum(bucket1(K, H), $product(-1, I)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H), 0) & (~$greatereq($sum(bucket1(K, H), $product(-1, I)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ((~($greatereq(bucket1(K, H), 0) & (~$greatereq($sum(bucket1(K, H), $product(-1, I)), 0)))) & ((($sum(bucket1(K, H), $product(-1, I)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))) | ((~($sum(bucket1(K, H), $product(-1, I)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))))))))) | ![K: key1, V: a1] : (((~($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0))) | good_data1(a, K, t2tb7(V), t2tb3(Rho), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) & ((~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H)), t2tb2(Ndata)))) | ($greatereq(bucket1(K, H), 0) & $lesseq($sum(bucket1(K, H), $product(-1, I)), 0))))))),
% 0.45/0.57 inference(modus_ponens,[status(thm)],[74, 62])).
% 0.45/0.57 tff(76,plain,
% 0.45/0.57 ($greatereq(H!4, 0) & (~$lesseq(H!4, 0)) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(H2!2), mk_array1(list(tuple2(key, a)), H!4, t2tb2(H1!3))) & (L!9 = tb2t6(nil(tuple2(key, a)))) & (~$lesseq(Rho1!6, 0)) & ![I1: $int] : ((~($greatereq(I1, 0) & (~$greatereq($sum(I1, $product(-1, Rho1!6)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5)), I1)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), Rho1!6, t2tb2(Rho2!5))) & $greatereq(Rho1!6, 0) & ![K: key1, V: a1] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9))) | ($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & ![J: $int] : ((~($greatereq(J, 0) & (~$greatereq($sum(J, $product(-2, H!4)), 1)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)), J)) & ![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))) & (~(((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))))),
% 0.45/0.57 inference(modus_ponens,[status(thm)],[75, 61])).
% 0.45/0.57 tff(77,plain,
% 0.45/0.57 (~(((~($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) & ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ($greatereq(bucket1(K!12, H!4), 0) & $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))),
% 0.45/0.57 inference(and_elim,[status(thm)],[76])).
% 0.45/0.57 tff(78,plain,
% 0.45/0.57 ((~(good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))) | (~((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))))),
% 0.45/0.57 inference(modus_ponens,[status(thm)],[77, 50])).
% 0.45/0.57 tff(79,plain,
% 0.45/0.57 (~((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))))),
% 0.45/0.57 inference(unit_resolution,[status(thm)],[78, 36])).
% 0.45/0.57 tff(80,plain,
% 0.45/0.57 ($lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0) | $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)),
% 0.45/0.57 inference(theory_lemma,[status(thm)],[])).
% 0.45/0.57 tff(81,plain,
% 0.45/0.57 ($lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)),
% 0.45/0.57 inference(unit_resolution,[status(thm)],[80, 34])).
% 0.45/0.57 tff(82,plain,
% 0.45/0.57 (((~$greatereq(bucket1(K!12, H!4), 0)) | (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)) | (~$lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))),
% 0.45/0.57 inference(tautology,[status(thm)],[])).
% 0.45/0.57 tff(83,plain,
% 0.45/0.57 ((~$greatereq(bucket1(K!12, H!4), 0)) | (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)),
% 0.45/0.57 inference(unit_resolution,[status(thm)],[82, 81])).
% 0.45/0.57 tff(84,plain,
% 0.45/0.57 (^[K: key1, V: a1] : refl(((~($greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K, H!4), 0)) | (~good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K, H!4), 0)) | $greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))))) <=> ((~($greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K, H!4), 0)) | (~good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K, H!4), 0)) | $greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))))))),
% 0.45/0.57 inference(bind,[status(th)],[])).
% 0.45/0.57 tff(85,plain,
% 0.45/0.57 (![K: key1, V: a1] : ((~($greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K, H!4), 0)) | (~good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K, H!4), 0)) | $greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))))) <=> ![K: key1, V: a1] : ((~($greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K, H!4), 0)) | (~good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K, H!4), 0)) | $greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))),
% 0.45/0.58 inference(quant_intro,[status(thm)],[84])).
% 0.45/0.58 tff(86,plain,
% 0.45/0.58 (^[K: key1, V: a1] : rewrite((($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))) <=> ((~($greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K, H!4), 0)) | (~good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K, H!4), 0)) | $greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))))))),
% 0.45/0.58 inference(bind,[status(th)],[])).
% 0.45/0.58 tff(87,plain,
% 0.45/0.58 (![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))) <=> ![K: key1, V: a1] : ((~($greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K, H!4), 0)) | (~good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K, H!4), 0)) | $greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))),
% 0.45/0.58 inference(quant_intro,[status(thm)],[86])).
% 0.45/0.58 tff(88,plain,
% 0.45/0.58 (![K: key1, V: a1] : (($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)) & good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | ((~($greatereq(bucket1(K, H!4), 0) & (~$greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0)))) & ((($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) & ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V)))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) & (~in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))),
% 0.45/0.58 inference(and_elim,[status(thm)],[76])).
% 0.45/0.58 tff(89,plain,
% 0.45/0.58 (![K: key1, V: a1] : ((~($greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K, H!4), 0)) | (~good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K, H!4), 0)) | $greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))),
% 0.45/0.58 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.45/0.58 tff(90,plain,
% 0.45/0.58 (![K: key1, V: a1] : ((~($greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K, H!4), 0)) | (~good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K, H!4), 0)) | $greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))),
% 0.45/0.58 inference(modus_ponens,[status(thm)],[89, 85])).
% 0.45/0.58 tff(91,plain,
% 0.45/0.58 (((~![K: key1, V: a1] : ((~($greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K, H!4), 0)) | (~good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K, H!4), 0)) | $greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))) | ((~((~$greatereq(bucket1(K!12, H!4), 0)) | (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))) <=> ((~![K: key1, V: a1] : ((~($greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K, H!4), 0)) | (~good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K, H!4), 0)) | $greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))),
% 0.45/0.58 inference(rewrite,[status(thm)],[])).
% 0.45/0.58 tff(92,plain,
% 0.45/0.58 (((~($greatereq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K!12, H!4), 0)) | $greatereq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K!12, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K!12, H!4), $product(-1, I!10)) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))))) <=> ((~((~$greatereq(bucket1(K!12, H!4), 0)) | (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))),
% 0.45/0.58 inference(rewrite,[status(thm)],[])).
% 0.45/0.58 tff(93,plain,
% 0.45/0.58 (((~![K: key1, V: a1] : ((~($greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K, H!4), 0)) | (~good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K, H!4), 0)) | $greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))) | ((~($greatereq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K!12, H!4), 0)) | $greatereq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K!12, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K!12, H!4), $product(-1, I!10)) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))) <=> ((~![K: key1, V: a1] : ((~($greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K, H!4), 0)) | (~good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K, H!4), 0)) | $greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))) | ((~((~$greatereq(bucket1(K!12, H!4), 0)) | (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))))))),
% 0.45/0.59 inference(monotonicity,[status(thm)],[92])).
% 0.45/0.59 tff(94,plain,
% 0.45/0.59 (((~![K: key1, V: a1] : ((~($greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K, H!4), 0)) | (~good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K, H!4), 0)) | $greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))) | ((~($greatereq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K!12, H!4), 0)) | $greatereq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K!12, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K!12, H!4), $product(-1, I!10)) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))) <=> ((~![K: key1, V: a1] : ((~($greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K, H!4), 0)) | (~good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K, H!4), 0)) | $greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))),
% 0.45/0.59 inference(transitivity,[status(thm)],[93, 91])).
% 0.45/0.59 tff(95,plain,
% 0.45/0.59 ((~![K: key1, V: a1] : ((~($greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K, H!4), 0)) | (~good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K, H!4), 0)) | $greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))) | ((~($greatereq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K!12, H!4), 0)) | $greatereq($sum(bucket1(K!12, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K!12, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K!12, H!4), $product(-1, I!10)) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))),
% 0.45/0.60 inference(quant_inst,[status(thm)],[])).
% 0.45/0.60 tff(96,plain,
% 0.45/0.60 ((~![K: key1, V: a1] : ((~($greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0) | (~$greatereq(bucket1(K, H!4), 0)) | (~good_data1(a, K, t2tb7(V), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~((~$greatereq(bucket1(K, H!4), 0)) | $greatereq($sum(bucket1(K, H!4), $product(-1, I!10)), 0))) | (~((~((~($sum(bucket1(K, H!4), $product(-1, I!10)) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K))) = tb2t4(some(a, t2tb7(V))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K), t2tb7(V)), t2tb6(L!9)) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(bucket1(K, H!4), $product(-1, I!10)) = 0) | in_data1(a, K, t2tb7(V), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))))),
% 0.45/0.60 inference(modus_ponens,[status(thm)],[95, 94])).
% 0.45/0.60 tff(97,plain,
% 0.45/0.60 ((~((~$greatereq(bucket1(K!12, H!4), 0)) | (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))))),
% 0.45/0.60 inference(unit_resolution,[status(thm)],[96, 90])).
% 0.45/0.60 tff(98,plain,
% 0.45/0.60 (~((~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))),
% 0.45/0.60 inference(unit_resolution,[status(thm)],[97, 83])).
% 0.45/0.60 tff(99,plain,
% 0.45/0.60 ((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | $greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)),
% 0.45/0.60 inference(theory_lemma,[status(thm)],[])).
% 0.45/0.60 tff(100,plain,
% 0.45/0.60 (~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)),
% 0.45/0.60 inference(unit_resolution,[status(thm)],[99, 34])).
% 0.45/0.60 tff(101,plain,
% 0.45/0.60 (((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | ($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)),
% 0.45/0.60 inference(tautology,[status(thm)],[])).
% 0.45/0.60 tff(102,plain,
% 0.45/0.60 ((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))),
% 0.45/0.60 inference(unit_resolution,[status(thm)],[101, 100])).
% 0.45/0.60 tff(103,assumption,(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))), introduced(assumption)).
% 0.45/0.60 tff(104,plain,
% 0.45/0.60 ((~((~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))),
% 0.45/0.60 inference(tautology,[status(thm)],[])).
% 0.45/0.60 tff(105,plain,
% 0.45/0.60 ((~((~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))),
% 0.45/0.60 inference(unit_resolution,[status(thm)],[104, 103])).
% 0.45/0.60 tff(106,plain,
% 0.45/0.60 (~((~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))),
% 0.45/0.60 inference(unit_resolution,[status(thm)],[105, 102])).
% 0.45/0.60 tff(107,plain,
% 0.45/0.60 (((~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))) | ((~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))),
% 0.45/0.61 inference(tautology,[status(thm)],[])).
% 0.45/0.61 tff(108,plain,
% 0.45/0.61 ($false),
% 0.45/0.61 inference(unit_resolution,[status(thm)],[107, 106, 98])).
% 0.45/0.61 tff(109,plain,($greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))), inference(lemma,lemma(discharge,[]))).
% 0.45/0.61 tff(110,plain,
% 0.45/0.61 (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))),
% 0.45/0.61 inference(unit_resolution,[status(thm)],[109, 34])).
% 0.45/0.61 tff(111,plain,
% 0.45/0.61 ((($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))),
% 0.45/0.61 inference(tautology,[status(thm)],[])).
% 0.45/0.61 tff(112,plain,
% 0.45/0.61 (~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))),
% 0.45/0.61 inference(unit_resolution,[status(thm)],[111, 110])).
% 0.45/0.61 tff(113,plain,
% 0.45/0.61 (((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))),
% 0.45/0.61 inference(tautology,[status(thm)],[])).
% 0.45/0.61 tff(114,plain,
% 0.45/0.61 ($false),
% 0.45/0.61 inference(unit_resolution,[status(thm)],[113, 112, 79])).
% 0.45/0.61 tff(115,plain,($greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)), inference(lemma,lemma(discharge,[]))).
% 0.45/0.61 tff(116,assumption,((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!12, H!4))), 0)), introduced(assumption)).
% 0.45/0.61 tff(117,plain,
% 0.45/0.61 (~$lesseq(H!4, 0)),
% 0.45/0.61 inference(and_elim,[status(thm)],[76])).
% 0.45/0.61 tff(118,plain,
% 0.45/0.61 (^[N: $int, K: key1] : refl(($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))) <=> ($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))))),
% 0.45/0.61 inference(bind,[status(th)],[])).
% 0.45/0.61 tff(119,plain,
% 0.45/0.61 (![N: $int, K: key1] : ($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))) <=> ![N: $int, K: key1] : ($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.45/0.61 inference(quant_intro,[status(thm)],[118])).
% 0.45/0.61 tff(120,plain,
% 0.45/0.61 (![N: $int] : ![K: key1] : ($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))) <=> ![N: $int, K: key1] : ($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.45/0.61 inference(pull_quant,[status(thm)],[])).
% 0.45/0.61 tff(121,plain,
% 0.45/0.61 (^[N: $int] : pull_quant(($lesseq(N, 0) | ![K: key1] : (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))) <=> ![K: key1] : ($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))))),
% 0.45/0.61 inference(bind,[status(th)],[])).
% 0.45/0.61 tff(122,plain,
% 0.45/0.61 (![N: $int] : ($lesseq(N, 0) | ![K: key1] : (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))) <=> ![N: $int] : ![K: key1] : ($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.45/0.61 inference(quant_intro,[status(thm)],[121])).
% 0.45/0.61 tff(123,plain,
% 0.45/0.61 (![N: $int] : ($lesseq(N, 0) | ![K: key1] : (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))) <=> ![N: $int, K: key1] : ($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.45/0.61 inference(transitivity,[status(thm)],[122, 120])).
% 0.45/0.61 tff(124,plain,
% 0.45/0.61 (![N: $int] : ($lesseq(N, 0) | ![K: key1] : (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))) <=> ![N: $int, K: key1] : ($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.45/0.61 inference(transitivity,[status(thm)],[123, 119])).
% 0.45/0.61 tff(125,plain,
% 0.45/0.61 (^[N: $int] : rewrite(($lesseq(N, 0) | ![K: key1] : (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))) <=> ($lesseq(N, 0) | ![K: key1] : (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))))),
% 0.45/0.61 inference(bind,[status(th)],[])).
% 0.45/0.61 tff(126,plain,
% 0.45/0.61 (![N: $int] : ($lesseq(N, 0) | ![K: key1] : (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))) <=> ![N: $int] : ($lesseq(N, 0) | ![K: key1] : (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.45/0.61 inference(quant_intro,[status(thm)],[125])).
% 0.45/0.61 tff(127,plain,
% 0.45/0.61 (![N: $int] : ($lesseq(N, 0) | ![K: key1] : (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))) <=> ![N: $int, K: key1] : ($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.45/0.61 inference(transitivity,[status(thm)],[126, 124])).
% 0.45/0.61 tff(128,plain,
% 0.45/0.61 (^[N: $int] : rewrite(($lesseq(N, 0) | ![K: key1] : ($greatereq(bucket1(K, N), 0) & (~$lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))) <=> ($lesseq(N, 0) | ![K: key1] : (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))))),
% 0.45/0.61 inference(bind,[status(th)],[])).
% 0.45/0.61 tff(129,plain,
% 0.45/0.61 (![N: $int] : ($lesseq(N, 0) | ![K: key1] : ($greatereq(bucket1(K, N), 0) & (~$lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))) <=> ![N: $int] : ($lesseq(N, 0) | ![K: key1] : (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.45/0.61 inference(quant_intro,[status(thm)],[128])).
% 0.45/0.61 tff(130,plain,
% 0.45/0.61 (^[N: $int] : rewrite(($lesseq(N, 0) | ![K: key1] : ($lesseq(0, bucket1(K, N)) & (~$lesseq(N, bucket1(K, N))))) <=> ($lesseq(N, 0) | ![K: key1] : ($greatereq(bucket1(K, N), 0) & (~$lesseq($sum(N, $product(-1, bucket1(K, N))), 0)))))),
% 0.45/0.61 inference(bind,[status(th)],[])).
% 0.45/0.61 tff(131,plain,
% 0.45/0.61 (![N: $int] : ($lesseq(N, 0) | ![K: key1] : ($lesseq(0, bucket1(K, N)) & (~$lesseq(N, bucket1(K, N))))) <=> ![N: $int] : ($lesseq(N, 0) | ![K: key1] : ($greatereq(bucket1(K, N), 0) & (~$lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.45/0.61 inference(quant_intro,[status(thm)],[130])).
% 0.45/0.61 tff(132,plain,
% 0.45/0.61 (![N: $int] : ($lesseq(N, 0) | ![K: key1] : ($lesseq(0, bucket1(K, N)) & (~$lesseq(N, bucket1(K, N))))) <=> ![N: $int] : ($lesseq(N, 0) | ![K: key1] : ($lesseq(0, bucket1(K, N)) & (~$lesseq(N, bucket1(K, N)))))),
% 0.45/0.61 inference(rewrite,[status(thm)],[])).
% 0.45/0.61 tff(133,plain,
% 0.45/0.61 (^[N: $int] : trans(monotonicity(rewrite($less(0, N) <=> (~$lesseq(N, 0))), quant_intro(proof_bind(^[K: key1] : rewrite(($lesseq(0, bucket1(K, N)) & $less(bucket1(K, N), N)) <=> ($lesseq(0, bucket1(K, N)) & (~$lesseq(N, bucket1(K, N)))))), (![K: key1] : ($lesseq(0, bucket1(K, N)) & $less(bucket1(K, N), N)) <=> ![K: key1] : ($lesseq(0, bucket1(K, N)) & (~$lesseq(N, bucket1(K, N)))))), (($less(0, N) => ![K: key1] : ($lesseq(0, bucket1(K, N)) & $less(bucket1(K, N), N))) <=> ((~$lesseq(N, 0)) => ![K: key1] : ($lesseq(0, bucket1(K, N)) & (~$lesseq(N, bucket1(K, N))))))), rewrite(((~$lesseq(N, 0)) => ![K: key1] : ($lesseq(0, bucket1(K, N)) & (~$lesseq(N, bucket1(K, N))))) <=> ($lesseq(N, 0) | ![K: key1] : ($lesseq(0, bucket1(K, N)) & (~$lesseq(N, bucket1(K, N)))))), (($less(0, N) => ![K: key1] : ($lesseq(0, bucket1(K, N)) & $less(bucket1(K, N), N))) <=> ($lesseq(N, 0) | ![K: key1] : ($lesseq(0, bucket1(K, N)) & (~$lesseq(N, bucket1(K, N)))))))),
% 0.45/0.61 inference(bind,[status(th)],[])).
% 0.45/0.61 tff(134,plain,
% 0.45/0.61 (![N: $int] : ($less(0, N) => ![K: key1] : ($lesseq(0, bucket1(K, N)) & $less(bucket1(K, N), N))) <=> ![N: $int] : ($lesseq(N, 0) | ![K: key1] : ($lesseq(0, bucket1(K, N)) & (~$lesseq(N, bucket1(K, N)))))),
% 0.45/0.61 inference(quant_intro,[status(thm)],[133])).
% 0.45/0.61 tff(135,axiom,(![N: $int] : ($less(0, N) => ![K: key1] : ($lesseq(0, bucket1(K, N)) & $less(bucket1(K, N), N)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','bucket_bounds')).
% 0.45/0.61 tff(136,plain,
% 0.45/0.61 (![N: $int] : ($lesseq(N, 0) | ![K: key1] : ($lesseq(0, bucket1(K, N)) & (~$lesseq(N, bucket1(K, N)))))),
% 0.45/0.61 inference(modus_ponens,[status(thm)],[135, 134])).
% 0.45/0.61 tff(137,plain,
% 0.45/0.61 (![N: $int] : ($lesseq(N, 0) | ![K: key1] : ($lesseq(0, bucket1(K, N)) & (~$lesseq(N, bucket1(K, N)))))),
% 0.45/0.61 inference(modus_ponens,[status(thm)],[136, 132])).
% 0.45/0.61 tff(138,plain,
% 0.45/0.61 (![N: $int] : ($lesseq(N, 0) | ![K: key1] : ($greatereq(bucket1(K, N), 0) & (~$lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.45/0.61 inference(modus_ponens,[status(thm)],[137, 131])).
% 0.45/0.61 tff(139,plain,(
% 0.45/0.61 ![N: $int] : ($lesseq(N, 0) | ![K: key1] : ($greatereq(bucket1(K, N), 0) & (~$lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.45/0.61 inference(skolemize,[status(sab)],[138])).
% 0.45/0.61 tff(140,plain,
% 0.45/0.61 (![N: $int] : ($lesseq(N, 0) | ![K: key1] : (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.45/0.61 inference(modus_ponens,[status(thm)],[139, 129])).
% 0.45/0.61 tff(141,plain,
% 0.45/0.61 (![N: $int, K: key1] : ($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.45/0.61 inference(modus_ponens,[status(thm)],[140, 127])).
% 0.45/0.61 tff(142,plain,
% 0.45/0.61 (((~![N: $int, K: key1] : ($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))) | ($lesseq(H!4, 0) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!12, H!4))), 0))))) <=> ((~![N: $int, K: key1] : ($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))) | $lesseq(H!4, 0) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!12, H!4))), 0))))),
% 0.45/0.61 inference(rewrite,[status(thm)],[])).
% 0.45/0.61 tff(143,plain,
% 0.45/0.61 ((~![N: $int, K: key1] : ($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))) | ($lesseq(H!4, 0) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!12, H!4))), 0))))),
% 0.45/0.61 inference(quant_inst,[status(thm)],[])).
% 0.45/0.61 tff(144,plain,
% 0.45/0.61 ((~![N: $int, K: key1] : ($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))) | $lesseq(H!4, 0) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!12, H!4))), 0)))),
% 0.45/0.61 inference(modus_ponens,[status(thm)],[143, 142])).
% 0.45/0.61 tff(145,plain,
% 0.45/0.61 ($false),
% 0.45/0.61 inference(unit_resolution,[status(thm)],[144, 141, 117, 116])).
% 0.45/0.61 tff(146,plain,(~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!12, H!4))), 0))), inference(lemma,lemma(discharge,[]))).
% 0.45/0.61 tff(147,plain,
% 0.45/0.61 (((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!12, H!4))), 0)) | $greatereq(bucket1(K!12, H!4), 0)),
% 0.45/0.61 inference(tautology,[status(thm)],[])).
% 0.45/0.61 tff(148,plain,
% 0.45/0.61 ($greatereq(bucket1(K!12, H!4), 0)),
% 0.45/0.61 inference(unit_resolution,[status(thm)],[147, 146])).
% 0.45/0.61 tff(149,plain,
% 0.45/0.61 ((~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))),
% 0.45/0.61 inference(tautology,[status(thm)],[])).
% 0.45/0.61 tff(150,plain,
% 0.45/0.61 ((~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))),
% 0.45/0.61 inference(unit_resolution,[status(thm)],[149, 148])).
% 0.45/0.61 tff(151,plain,
% 0.45/0.61 (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))),
% 0.45/0.61 inference(unit_resolution,[status(thm)],[150, 115])).
% 0.45/0.61 tff(152,plain,
% 0.45/0.61 (((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))) | ((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))),
% 0.45/0.61 inference(tautology,[status(thm)],[])).
% 0.45/0.61 tff(153,plain,
% 0.45/0.61 ((~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~((~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))))),
% 0.45/0.61 inference(unit_resolution,[status(thm)],[152, 151])).
% 0.45/0.61 tff(154,plain,
% 0.45/0.61 (~(good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)))),
% 0.45/0.61 inference(unit_resolution,[status(thm)],[78, 153])).
% 0.45/0.61 tff(155,plain,
% 0.45/0.61 ((good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) | (~$greatereq(bucket1(K!12, H!4), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))),
% 0.45/0.61 inference(tautology,[status(thm)],[])).
% 0.45/0.61 tff(156,plain,
% 0.45/0.61 (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))),
% 0.45/0.61 inference(unit_resolution,[status(thm)],[155, 154])).
% 0.45/0.61 tff(157,plain,
% 0.45/0.61 ((~(good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) <=> ((get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = some(a, t2tb7(V!11))) <=> in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) | (~((get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = some(a, t2tb7(V!11))) <=> in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))),
% 0.45/0.61 inference(tautology,[status(thm)],[])).
% 0.45/0.61 tff(158,plain,
% 0.45/0.61 ((~(good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))) <=> ((get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = some(a, t2tb7(V!11))) <=> in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = some(a, t2tb7(V!11))) <=> in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))),
% 0.45/0.61 inference(unit_resolution,[status(thm)],[157, 156])).
% 0.45/0.61 tff(159,plain,
% 0.45/0.61 (~((get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = some(a, t2tb7(V!11))) <=> in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))),
% 0.45/0.61 inference(unit_resolution,[status(thm)],[158, 33])).
% 0.45/0.61 tff(160,plain,
% 0.45/0.61 (((get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = some(a, t2tb7(V!11))) <=> in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = some(a, t2tb7(V!11))) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))),
% 0.45/0.61 inference(tautology,[status(thm)],[])).
% 0.45/0.62 tff(161,plain,
% 0.45/0.62 ((get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = some(a, t2tb7(V!11))) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))),
% 0.45/0.62 inference(unit_resolution,[status(thm)],[160, 159])).
% 0.45/0.62 tff(162,plain,
% 0.45/0.62 (get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = some(a, t2tb7(V!11))),
% 0.45/0.62 inference(unit_resolution,[status(thm)],[161, 24])).
% 0.45/0.62 tff(163,plain,
% 0.45/0.62 (tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11)))),
% 0.45/0.62 inference(monotonicity,[status(thm)],[162])).
% 0.45/0.62 tff(164,plain,
% 0.45/0.62 (L!9 = tb2t6(nil(tuple2(key, a)))),
% 0.45/0.62 inference(and_elim,[status(thm)],[76])).
% 0.45/0.62 tff(165,plain,
% 0.45/0.62 (t2tb6(L!9) = t2tb6(tb2t6(nil(tuple2(key, a))))),
% 0.45/0.62 inference(monotonicity,[status(thm)],[164])).
% 0.45/0.62 tff(166,plain,
% 0.45/0.62 (t2tb6(tb2t6(nil(tuple2(key, a)))) = t2tb6(L!9)),
% 0.45/0.62 inference(symmetry,[status(thm)],[165])).
% 0.45/0.62 tff(167,plain,
% 0.45/0.62 (![J: uni] : (t2tb6(tb2t6(J)) = J) <=> ![J: uni] : (t2tb6(tb2t6(J)) = J)),
% 0.45/0.62 inference(rewrite,[status(thm)],[])).
% 0.45/0.62 tff(168,plain,
% 0.45/0.62 (![J: uni] : (t2tb6(tb2t6(J)) = J) <=> ![J: uni] : (t2tb6(tb2t6(J)) = J)),
% 0.45/0.62 inference(rewrite,[status(thm)],[])).
% 0.45/0.62 tff(169,axiom,(![J: uni] : (t2tb6(tb2t6(J)) = J)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','bridgeR6')).
% 0.45/0.62 tff(170,plain,
% 0.45/0.62 (![J: uni] : (t2tb6(tb2t6(J)) = J)),
% 0.45/0.62 inference(modus_ponens,[status(thm)],[169, 168])).
% 0.45/0.62 tff(171,plain,(
% 0.45/0.62 ![J: uni] : (t2tb6(tb2t6(J)) = J)),
% 0.45/0.62 inference(skolemize,[status(sab)],[170])).
% 0.45/0.62 tff(172,plain,
% 0.45/0.62 (![J: uni] : (t2tb6(tb2t6(J)) = J)),
% 0.45/0.62 inference(modus_ponens,[status(thm)],[171, 167])).
% 0.45/0.62 tff(173,plain,
% 0.45/0.62 ((~![J: uni] : (t2tb6(tb2t6(J)) = J)) | (t2tb6(tb2t6(nil(tuple2(key, a)))) = nil(tuple2(key, a)))),
% 0.45/0.62 inference(quant_inst,[status(thm)],[])).
% 0.45/0.62 tff(174,plain,
% 0.45/0.62 (t2tb6(tb2t6(nil(tuple2(key, a)))) = nil(tuple2(key, a))),
% 0.45/0.62 inference(unit_resolution,[status(thm)],[173, 172])).
% 0.45/0.62 tff(175,plain,
% 0.45/0.62 (nil(tuple2(key, a)) = t2tb6(tb2t6(nil(tuple2(key, a))))),
% 0.45/0.62 inference(symmetry,[status(thm)],[174])).
% 0.45/0.62 tff(176,plain,
% 0.45/0.62 (nil(tuple2(key, a)) = t2tb6(L!9)),
% 0.45/0.62 inference(transitivity,[status(thm)],[175, 166])).
% 0.45/0.62 tff(177,plain,
% 0.45/0.62 (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), nil(tuple2(key, a))) <=> mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9))),
% 0.45/0.62 inference(monotonicity,[status(thm)],[176])).
% 0.45/0.62 tff(178,plain,
% 0.45/0.62 ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), nil(tuple2(key, a)))) <=> (~mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)))),
% 0.45/0.62 inference(monotonicity,[status(thm)],[177])).
% 0.45/0.62 tff(179,plain,
% 0.45/0.62 (^[A: ty, A1: ty, X: uni, X1: uni] : refl(sort1(tuple2(A1, A), tuple21(A1, A, X, X1)) <=> sort1(tuple2(A1, A), tuple21(A1, A, X, X1)))),
% 0.45/0.62 inference(bind,[status(th)],[])).
% 0.45/0.62 tff(180,plain,
% 0.45/0.62 (![A: ty, A1: ty, X: uni, X1: uni] : sort1(tuple2(A1, A), tuple21(A1, A, X, X1)) <=> ![A: ty, A1: ty, X: uni, X1: uni] : sort1(tuple2(A1, A), tuple21(A1, A, X, X1))),
% 0.45/0.62 inference(quant_intro,[status(thm)],[179])).
% 0.45/0.62 tff(181,plain,
% 0.45/0.62 (![A: ty, A1: ty, X: uni, X1: uni] : sort1(tuple2(A1, A), tuple21(A1, A, X, X1)) <=> ![A: ty, A1: ty, X: uni, X1: uni] : sort1(tuple2(A1, A), tuple21(A1, A, X, X1))),
% 0.45/0.62 inference(rewrite,[status(thm)],[])).
% 0.45/0.62 tff(182,axiom,(![A: ty, A1: ty, X: uni, X1: uni] : sort1(tuple2(A1, A), tuple21(A1, A, X, X1))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','tuple2_sort2')).
% 0.45/0.62 tff(183,plain,
% 0.45/0.62 (![A: ty, A1: ty, X: uni, X1: uni] : sort1(tuple2(A1, A), tuple21(A1, A, X, X1))),
% 0.45/0.62 inference(modus_ponens,[status(thm)],[182, 181])).
% 0.45/0.62 tff(184,plain,(
% 0.45/0.62 ![A: ty, A1: ty, X: uni, X1: uni] : sort1(tuple2(A1, A), tuple21(A1, A, X, X1))),
% 0.45/0.62 inference(skolemize,[status(sab)],[183])).
% 0.45/0.62 tff(185,plain,
% 0.45/0.62 (![A: ty, A1: ty, X: uni, X1: uni] : sort1(tuple2(A1, A), tuple21(A1, A, X, X1))),
% 0.45/0.62 inference(modus_ponens,[status(thm)],[184, 180])).
% 0.45/0.62 tff(186,plain,
% 0.45/0.62 ((~![A: ty, A1: ty, X: uni, X1: uni] : sort1(tuple2(A1, A), tuple21(A1, A, X, X1))) | sort1(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)))),
% 0.45/0.62 inference(quant_inst,[status(thm)],[])).
% 0.45/0.62 tff(187,plain,
% 0.45/0.62 (sort1(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)))),
% 0.45/0.62 inference(unit_resolution,[status(thm)],[186, 185])).
% 0.45/0.62 tff(188,plain,
% 0.45/0.62 (^[A: ty, X: uni] : refl(((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))))) <=> ((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))))))),
% 0.45/0.62 inference(bind,[status(th)],[])).
% 0.45/0.62 tff(189,plain,
% 0.45/0.62 (![A: ty, X: uni] : ((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))))) <=> ![A: ty, X: uni] : ((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))))),
% 0.45/0.62 inference(quant_intro,[status(thm)],[188])).
% 0.45/0.62 tff(190,plain,
% 0.45/0.62 (^[A: ty, X: uni] : rewrite(((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))))) <=> ((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))))))),
% 0.45/0.62 inference(bind,[status(th)],[])).
% 0.45/0.62 tff(191,plain,
% 0.45/0.62 (![A: ty, X: uni] : ((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))))) <=> ![A: ty, X: uni] : ((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))))),
% 0.45/0.62 inference(quant_intro,[status(thm)],[190])).
% 0.45/0.62 tff(192,plain,
% 0.45/0.62 (![A: ty, X: uni] : ((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))))) <=> ![A: ty, X: uni] : ((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))))),
% 0.45/0.62 inference(transitivity,[status(thm)],[191, 189])).
% 0.45/0.62 tff(193,plain,
% 0.45/0.62 (^[A: ty, X: uni] : rewrite(((~sort1(A, X)) | ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))) <=> ((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))))))),
% 0.45/0.62 inference(bind,[status(th)],[])).
% 0.45/0.62 tff(194,plain,
% 0.45/0.62 (![A: ty, X: uni] : ((~sort1(A, X)) | ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))) <=> ![A: ty, X: uni] : ((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))))),
% 0.45/0.62 inference(quant_intro,[status(thm)],[193])).
% 0.45/0.62 tff(195,plain,
% 0.45/0.62 (![A: ty, X: uni] : ((~sort1(A, X)) | ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))) <=> ![A: ty, X: uni] : ((~sort1(A, X)) | ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))),
% 0.45/0.62 inference(rewrite,[status(thm)],[])).
% 0.45/0.62 tff(196,plain,
% 0.45/0.62 (^[A: ty, X: uni] : trans(monotonicity(rewrite(((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : (sort1(A, X1) => (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))) <=> ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))), ((sort1(A, X) => ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : (sort1(A, X1) => (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))) <=> (sort1(A, X) => ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))))), rewrite((sort1(A, X) => ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))) <=> ((~sort1(A, X)) | ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))), ((sort1(A, X) => ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : (sort1(A, X1) => (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))) <=> ((~sort1(A, X)) | ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))))),
% 0.45/0.62 inference(bind,[status(th)],[])).
% 0.45/0.62 tff(197,plain,
% 0.45/0.62 (![A: ty, X: uni] : (sort1(A, X) => ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : (sort1(A, X1) => (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2)))))) <=> ![A: ty, X: uni] : ((~sort1(A, X)) | ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))),
% 0.45/0.62 inference(quant_intro,[status(thm)],[196])).
% 0.45/0.62 tff(198,axiom,(![A: ty, X: uni] : (sort1(A, X) => ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : (sort1(A, X1) => (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','mem_def')).
% 0.45/0.62 tff(199,plain,
% 0.45/0.62 (![A: ty, X: uni] : ((~sort1(A, X)) | ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))),
% 0.45/0.62 inference(modus_ponens,[status(thm)],[198, 197])).
% 0.45/0.62 tff(200,plain,
% 0.45/0.62 (![A: ty, X: uni] : ((~sort1(A, X)) | ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))),
% 0.45/0.62 inference(modus_ponens,[status(thm)],[199, 195])).
% 0.45/0.62 tff(201,plain,(
% 0.45/0.62 ![A: ty, X: uni] : ((~sort1(A, X)) | ((~mem(A, X, nil(A))) & ![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))),
% 0.45/0.62 inference(skolemize,[status(sab)],[200])).
% 0.45/0.62 tff(202,plain,
% 0.45/0.62 (![A: ty, X: uni] : ((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))))),
% 0.45/0.62 inference(modus_ponens,[status(thm)],[201, 194])).
% 0.45/0.62 tff(203,plain,
% 0.45/0.62 (![A: ty, X: uni] : ((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))))),
% 0.45/0.62 inference(modus_ponens,[status(thm)],[202, 192])).
% 0.45/0.62 tff(204,plain,
% 0.45/0.62 (((~![A: ty, X: uni] : ((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))))) | ((~sort1(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)))) | (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(K!12), t2tb7(V!11)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), X2))))))))) <=> ((~![A: ty, X: uni] : ((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))))) | (~sort1(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)))) | (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(K!12), t2tb7(V!11)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), X2))))))))),
% 0.45/0.62 inference(rewrite,[status(thm)],[])).
% 0.45/0.62 tff(205,plain,
% 0.45/0.62 ((~![A: ty, X: uni] : ((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))))) | ((~sort1(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)))) | (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(K!12), t2tb7(V!11)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), X2))))))))),
% 0.45/0.63 inference(quant_inst,[status(thm)],[])).
% 0.45/0.63 tff(206,plain,
% 0.45/0.63 ((~![A: ty, X: uni] : ((~sort1(A, X)) | (~(mem(A, X, nil(A)) | (~![X1: uni, X2: uni] : ((~sort1(A, X1)) | (mem(A, X, cons(A, X1, X2)) <=> ((X = X1) | mem(A, X, X2))))))))) | (~sort1(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)))) | (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(K!12), t2tb7(V!11)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), X2)))))))),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[205, 204])).
% 0.45/0.63 tff(207,plain,
% 0.45/0.63 (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(K!12), t2tb7(V!11)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), X2))))))),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[206, 203, 187])).
% 0.45/0.63 tff(208,plain,
% 0.45/0.63 ((mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(K!12), t2tb7(V!11)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), X2)))))) | (~mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), nil(tuple2(key, a))))),
% 0.45/0.63 inference(tautology,[status(thm)],[])).
% 0.45/0.63 tff(209,plain,
% 0.45/0.63 (~mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), nil(tuple2(key, a)))),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[208, 207])).
% 0.45/0.63 tff(210,plain,
% 0.45/0.63 (~mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9))),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[209, 178])).
% 0.45/0.63 tff(211,plain,
% 0.45/0.63 ((~(mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))) | mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))),
% 0.45/0.63 inference(tautology,[status(thm)],[])).
% 0.45/0.63 tff(212,plain,
% 0.45/0.63 ((~(mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[211, 210])).
% 0.45/0.63 tff(213,plain,
% 0.45/0.63 (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[212, 24])).
% 0.45/0.63 tff(214,plain,
% 0.45/0.63 (((~$greatereq(bucket1(K!12, H!4), 0)) | (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)) | good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))),
% 0.45/0.63 inference(tautology,[status(thm)],[])).
% 0.45/0.63 tff(215,plain,
% 0.45/0.63 ((~$greatereq(bucket1(K!12, H!4), 0)) | (~good_data1(a, K!12, t2tb7(V!11), t2tb3(Rho!7), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[214, 156])).
% 0.45/0.63 tff(216,plain,
% 0.45/0.63 (~((~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))))),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[96, 90, 215])).
% 0.45/0.63 tff(217,plain,
% 0.45/0.63 ((~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[107, 216])).
% 0.45/0.63 tff(218,plain,
% 0.45/0.63 (((~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~((~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))) | (~(($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))))) | ((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))),
% 0.45/0.63 inference(tautology,[status(thm)],[])).
% 0.45/0.63 tff(219,plain,
% 0.45/0.63 ((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[218, 216])).
% 0.45/0.63 tff(220,plain,
% 0.45/0.63 ((~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | (~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)),
% 0.45/0.63 inference(tautology,[status(thm)],[])).
% 0.45/0.63 tff(221,plain,
% 0.45/0.63 ((~((~$greatereq(bucket1(K!12, H!4), 0)) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))) | $lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[220, 148])).
% 0.45/0.63 tff(222,plain,
% 0.45/0.63 ($lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[221, 219])).
% 0.45/0.63 tff(223,plain,
% 0.45/0.63 (($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | (~$lesseq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0)) | (~$greatereq($sum(I!10, $product(-1, bucket1(K!12, H!4))), 0))),
% 0.45/0.63 inference(theory_lemma,[status(thm)],[])).
% 0.45/0.63 tff(224,plain,
% 0.45/0.63 ($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[223, 222, 115])).
% 0.45/0.63 tff(225,plain,
% 0.45/0.63 ((($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0))),
% 0.45/0.63 inference(tautology,[status(thm)],[])).
% 0.45/0.63 tff(226,plain,
% 0.45/0.63 (($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[225, 224])).
% 0.45/0.63 tff(227,plain,
% 0.45/0.63 (~((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))),
% 0.54/0.64 inference(unit_resolution,[status(thm)],[104, 226, 217])).
% 0.54/0.64 tff(228,plain,
% 0.54/0.64 (((~($sum(I!10, $product(-1, bucket1(K!12, H!4))) = 0)) | ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))) | (~((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))))),
% 0.54/0.64 inference(tautology,[status(thm)],[])).
% 0.54/0.64 tff(229,plain,
% 0.54/0.64 (~((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))),
% 0.54/0.64 inference(unit_resolution,[status(thm)],[228, 227])).
% 0.54/0.64 tff(230,plain,
% 0.54/0.64 (((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))) | (~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))),
% 0.54/0.64 inference(tautology,[status(thm)],[])).
% 0.54/0.64 tff(231,plain,
% 0.54/0.64 ((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))),
% 0.54/0.64 inference(unit_resolution,[status(thm)],[230, 229])).
% 0.54/0.64 tff(232,plain,
% 0.54/0.64 (~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))),
% 0.54/0.64 inference(unit_resolution,[status(thm)],[231, 213])).
% 0.54/0.64 tff(233,plain,
% 0.54/0.64 ($false),
% 0.54/0.64 inference(unit_resolution,[status(thm)],[232, 163])).
% 0.54/0.64 tff(234,plain,(in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))), inference(lemma,lemma(discharge,[]))).
% 0.54/0.64 tff(235,plain,
% 0.54/0.64 ((mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))),
% 0.54/0.64 inference(tautology,[status(thm)],[])).
% 0.54/0.64 tff(236,plain,
% 0.54/0.64 (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))),
% 0.54/0.64 inference(unit_resolution,[status(thm)],[235, 234])).
% 0.54/0.64 tff(237,plain,
% 0.54/0.64 (((~(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11))))) <=> (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))) | (tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11)))) | (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))),
% 0.54/0.64 inference(tautology,[status(thm)],[])).
% 0.54/0.64 tff(238,plain,
% 0.54/0.64 ((tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11)))) | (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(K!12), t2tb7(V!11)), t2tb6(L!9)) | in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))))),
% 0.54/0.64 inference(unit_resolution,[status(thm)],[237, 229])).
% 0.54/0.64 tff(239,plain,
% 0.54/0.64 (tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))) = tb2t4(some(a, t2tb7(V!11)))),
% 0.54/0.64 inference(unit_resolution,[status(thm)],[238, 236])).
% 0.54/0.64 tff(240,plain,
% 0.54/0.64 (t2tb4(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)))) = t2tb4(tb2t4(some(a, t2tb7(V!11))))),
% 0.54/0.64 inference(monotonicity,[status(thm)],[239])).
% 0.54/0.64 tff(241,plain,
% 0.54/0.64 (^[A: ty, B: ty, X: uni, X1: uni] : refl(sort1(B, get(B, A, X, X1)) <=> sort1(B, get(B, A, X, X1)))),
% 0.54/0.64 inference(bind,[status(th)],[])).
% 0.54/0.64 tff(242,plain,
% 0.54/0.64 (![A: ty, B: ty, X: uni, X1: uni] : sort1(B, get(B, A, X, X1)) <=> ![A: ty, B: ty, X: uni, X1: uni] : sort1(B, get(B, A, X, X1))),
% 0.54/0.64 inference(quant_intro,[status(thm)],[241])).
% 0.54/0.64 tff(243,plain,
% 0.54/0.64 (![A: ty, B: ty, X: uni, X1: uni] : sort1(B, get(B, A, X, X1)) <=> ![A: ty, B: ty, X: uni, X1: uni] : sort1(B, get(B, A, X, X1))),
% 0.54/0.64 inference(rewrite,[status(thm)],[])).
% 0.54/0.64 tff(244,axiom,(![A: ty, B: ty, X: uni, X1: uni] : sort1(B, get(B, A, X, X1))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','get_sort4')).
% 0.54/0.64 tff(245,plain,
% 0.54/0.64 (![A: ty, B: ty, X: uni, X1: uni] : sort1(B, get(B, A, X, X1))),
% 0.54/0.64 inference(modus_ponens,[status(thm)],[244, 243])).
% 0.54/0.64 tff(246,plain,(
% 0.54/0.64 ![A: ty, B: ty, X: uni, X1: uni] : sort1(B, get(B, A, X, X1))),
% 0.54/0.64 inference(skolemize,[status(sab)],[245])).
% 0.54/0.64 tff(247,plain,
% 0.54/0.64 (![A: ty, B: ty, X: uni, X1: uni] : sort1(B, get(B, A, X, X1))),
% 0.54/0.64 inference(modus_ponens,[status(thm)],[246, 242])).
% 0.54/0.64 tff(248,plain,
% 0.54/0.64 ((~![A: ty, B: ty, X: uni, X1: uni] : sort1(B, get(B, A, X, X1))) | sort1(option(a), get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)))),
% 0.54/0.64 inference(quant_inst,[status(thm)],[])).
% 0.54/0.64 tff(249,plain,
% 0.54/0.64 (sort1(option(a), get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)))),
% 0.54/0.64 inference(unit_resolution,[status(thm)],[248, 247])).
% 0.54/0.64 tff(250,plain,
% 0.54/0.64 (((~![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J))) | ((~sort1(option(a), get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)))) | (t2tb4(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)))) = get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))))) <=> ((~![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J))) | (~sort1(option(a), get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)))) | (t2tb4(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)))) = get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))))),
% 0.54/0.64 inference(rewrite,[status(thm)],[])).
% 0.54/0.64 tff(251,plain,
% 0.54/0.64 ((~![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J))) | ((~sort1(option(a), get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)))) | (t2tb4(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)))) = get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))))),
% 0.54/0.64 inference(quant_inst,[status(thm)],[])).
% 0.54/0.64 tff(252,plain,
% 0.54/0.64 ((~![J: uni] : ((~sort1(option(a), J)) | (t2tb4(tb2t4(J)) = J))) | (~sort1(option(a), get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)))) | (t2tb4(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)))) = get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)))),
% 0.54/0.64 inference(modus_ponens,[status(thm)],[251, 250])).
% 0.54/0.64 tff(253,plain,
% 0.54/0.64 (t2tb4(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)))) = get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))),
% 0.54/0.64 inference(unit_resolution,[status(thm)],[252, 19, 249])).
% 0.54/0.64 tff(254,plain,
% 0.54/0.64 (get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = t2tb4(tb2t4(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12))))),
% 0.54/0.64 inference(symmetry,[status(thm)],[253])).
% 0.54/0.64 tff(255,plain,
% 0.54/0.64 (get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = some(a, t2tb7(V!11))),
% 0.54/0.64 inference(transitivity,[status(thm)],[254, 240, 23])).
% 0.54/0.64 tff(256,plain,
% 0.54/0.64 (((get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = some(a, t2tb7(V!11))) <=> in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8)))) | (~(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = some(a, t2tb7(V!11)))) | (~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))),
% 0.54/0.64 inference(tautology,[status(thm)],[])).
% 0.54/0.64 tff(257,plain,
% 0.54/0.64 ((~(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = some(a, t2tb7(V!11)))) | (~in_data1(a, K!12, t2tb7(V!11), mk_array1(list(tuple2(key, a)), $sum(1, $product(2, H!4)), t2tb2(Ndata!8))))),
% 0.54/0.65 inference(unit_resolution,[status(thm)],[256, 159])).
% 0.54/0.65 tff(258,plain,
% 0.54/0.65 (~(get(option(a), key, t2tb3(Rho!7), t2tb1(K!12)) = some(a, t2tb7(V!11)))),
% 0.54/0.65 inference(unit_resolution,[status(thm)],[257, 234])).
% 0.54/0.65 tff(259,plain,
% 0.54/0.65 ($false),
% 0.54/0.65 inference(unit_resolution,[status(thm)],[258, 255])).
% 0.54/0.65 % SZS output end Proof
%------------------------------------------------------------------------------