TSTP Solution File: SWW604_2 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SWW604_2 : TPTP v8.1.0. Released v6.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n013.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 0.19s 0.48s
% Output : Proof 0.48s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SWW604_2 : TPTP v8.1.0. Released v6.1.0.
% 0.11/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n013.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Sun Sep 4 20:15:02 EDT 2022
% 0.12/0.33 % CPUTime :
% 0.12/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34 Usage: tptp [options] [-file:]file
% 0.12/0.34 -h, -? prints this message.
% 0.12/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.34 -m, -model generate model.
% 0.12/0.34 -p, -proof generate proof.
% 0.12/0.34 -c, -core generate unsat core of named formulas.
% 0.12/0.34 -st, -statistics display statistics.
% 0.12/0.34 -t:timeout set timeout (in second).
% 0.12/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34 -<param>:<value> configuration parameter and value.
% 0.12/0.34 -o:<output-file> file to place output in.
% 0.19/0.48 % SZS status Theorem
% 0.19/0.48 % SZS output start Proof
% 0.19/0.48 tff(bucket1_type, type, (
% 0.19/0.48 bucket1: ( key1 * $int ) > $int)).
% 0.19/0.48 tff(tptp_fun_H_4_type, type, (
% 0.19/0.48 tptp_fun_H_4: $int)).
% 0.19/0.48 tff(tptp_fun_K_10_type, type, (
% 0.19/0.48 tptp_fun_K_10: key1)).
% 0.19/0.48 tff(good_hash1_type, type, (
% 0.19/0.48 good_hash1: ( ty * uni * $int ) > $o)).
% 0.19/0.48 tff(tptp_fun_I_8_type, type, (
% 0.19/0.48 tptp_fun_I_8: $int)).
% 0.19/0.48 tff(mk_array1_type, type, (
% 0.19/0.48 mk_array1: ( ty * $int * uni ) > uni)).
% 0.19/0.48 tff(t2tb2_type, type, (
% 0.19/0.48 t2tb2: map_int_lplist_lpkeycm_a1rprp > uni)).
% 0.19/0.48 tff(tptp_fun_O_6_type, type, (
% 0.19/0.48 tptp_fun_O_6: map_int_lplist_lpkeycm_a1rprp)).
% 0.19/0.48 tff(list_type, type, (
% 0.19/0.48 list: ty > ty)).
% 0.19/0.48 tff(tuple2_type, type, (
% 0.19/0.48 tuple2: ( ty * ty ) > ty)).
% 0.19/0.48 tff(a_type, type, (
% 0.19/0.48 a: ty)).
% 0.19/0.48 tff(key_type, type, (
% 0.19/0.48 key: ty)).
% 0.19/0.48 tff(good_data1_type, type, (
% 0.19/0.48 good_data1: ( ty * key1 * uni * uni * uni ) > $o)).
% 0.19/0.48 tff(t2tb3_type, type, (
% 0.19/0.48 t2tb3: map_key_lpoption_a1rp > uni)).
% 0.19/0.48 tff(tptp_fun_Rho1_7_type, type, (
% 0.19/0.48 tptp_fun_Rho1_7: map_key_lpoption_a1rp)).
% 0.19/0.48 tff(t2tb7_type, type, (
% 0.19/0.48 t2tb7: a1 > uni)).
% 0.19/0.48 tff(tptp_fun_V_9_type, type, (
% 0.19/0.48 tptp_fun_V_9: a1)).
% 0.19/0.48 tff(tb2t3_type, type, (
% 0.19/0.48 tb2t3: uni > map_key_lpoption_a1rp)).
% 0.19/0.48 tff(const_type, type, (
% 0.19/0.48 const: ( ty * ty * uni ) > uni)).
% 0.19/0.48 tff(none_type, type, (
% 0.19/0.48 none: ty > uni)).
% 0.19/0.48 tff(option_type, type, (
% 0.19/0.48 option: ty > ty)).
% 0.19/0.48 tff(tb2t6_type, type, (
% 0.19/0.48 tb2t6: uni > list_lpkeycm_a1rp)).
% 0.19/0.48 tff(nil_type, type, (
% 0.19/0.48 nil: ty > uni)).
% 0.19/0.48 tff(get_type, type, (
% 0.19/0.48 get: ( ty * ty * uni * uni ) > uni)).
% 0.19/0.48 tff(t2tb_type, type, (
% 0.19/0.48 t2tb: $int > uni)).
% 0.19/0.48 tff(int_type, type, (
% 0.19/0.48 int: ty)).
% 0.19/0.48 tff(tptp_fun_Rho_5_type, type, (
% 0.19/0.48 tptp_fun_Rho_5: $int)).
% 0.19/0.48 tff(tptp_fun_H1_3_type, type, (
% 0.19/0.48 tptp_fun_H1_3: map_int_lplist_lpkeycm_a1rprp)).
% 0.19/0.48 tff(tptp_fun_H2_2_type, type, (
% 0.19/0.48 tptp_fun_H2_2: map_key_lpoption_a1rp)).
% 0.19/0.48 tff(length1_type, type, (
% 0.19/0.48 length1: ( ty * uni ) > $int)).
% 0.19/0.48 tff(mem_type, type, (
% 0.19/0.48 mem: ( ty * uni * uni ) > $o)).
% 0.19/0.48 tff(tuple21_type, type, (
% 0.19/0.48 tuple21: ( ty * ty * uni * uni ) > uni)).
% 0.19/0.48 tff(t2tb1_type, type, (
% 0.19/0.48 t2tb1: key1 > uni)).
% 0.19/0.48 tff(get2_type, type, (
% 0.19/0.48 get2: ( ty * uni * $int ) > uni)).
% 0.19/0.48 tff(elts_type, type, (
% 0.19/0.48 elts: ( ty * uni ) > uni)).
% 0.19/0.48 tff(t2tb6_type, type, (
% 0.19/0.48 t2tb6: list_lpkeycm_a1rp > uni)).
% 0.19/0.48 tff(sort1_type, type, (
% 0.19/0.48 sort1: ( ty * uni ) > $o)).
% 0.19/0.48 tff(map_type, type, (
% 0.19/0.48 map: ( ty * ty ) > ty)).
% 0.19/0.48 tff(in_data1_type, type, (
% 0.19/0.48 in_data1: ( ty * key1 * uni * uni ) > $o)).
% 0.19/0.48 tff(some_type, type, (
% 0.19/0.48 some: ( ty * uni ) > uni)).
% 0.19/0.48 tff(tptp_fun_V_0_type, type, (
% 0.19/0.48 tptp_fun_V_0: ( $int * uni * ty ) > uni)).
% 0.19/0.48 tff(tptp_fun_K_1_type, type, (
% 0.19/0.48 tptp_fun_K_1: ( $int * uni * ty ) > key1)).
% 0.19/0.48 tff(cons_type, type, (
% 0.19/0.48 cons: ( ty * uni * uni ) > uni)).
% 0.19/0.48 tff(1,plain,
% 0.19/0.48 ((((~$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))) & $greatereq(H!4, 0)) & ((Rho!5 = 0) & ((~$greatereq(H!4, 0)) | ($greatereq(H!4, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a))))) & (Rho1!7 = tb2t3(const(option(a), key, none(a)))) & ($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~((~($greatereq(I!8, 0) & (~$lesseq($sum(H!4, $product(-1, I!8)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)))))))) <=> ((~$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))) & $greatereq(H!4, 0) & (Rho!5 = 0) & ((~$greatereq(H!4, 0)) | ($greatereq(H!4, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a))))) & (Rho1!7 = tb2t3(const(option(a), key, none(a)))) & ($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~((~($greatereq(I!8, 0) & (~$lesseq($sum(H!4, $product(-1, I!8)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)))))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(2,plain,
% 0.19/0.49 (((~(~(Rho!5 = 0))) & ((~$greatereq(H!4, 0)) | (($greatereq(H!4, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) & ((~(~(Rho1!7 = tb2t3(const(option(a), key, none(a)))))) & ((~(~$lesseq(H!4, 0))) | (~((~($greatereq(I!8, 0) & (~$greatereq($sum(I!8, $product(-1, H!4)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a)))))))))) <=> ((Rho!5 = 0) & ((~$greatereq(H!4, 0)) | ($greatereq(H!4, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a))))) & (Rho1!7 = tb2t3(const(option(a), key, none(a)))) & ($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~((~($greatereq(I!8, 0) & (~$lesseq($sum(H!4, $product(-1, I!8)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)))))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(3,plain,
% 0.19/0.49 (((~$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))) & $greatereq(H!4, 0))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(4,plain,
% 0.19/0.49 ((((~$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))) & $greatereq(H!4, 0)) & ((~(~(Rho!5 = 0))) & ((~$greatereq(H!4, 0)) | (($greatereq(H!4, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) & ((~(~(Rho1!7 = tb2t3(const(option(a), key, none(a)))))) & ((~(~$lesseq(H!4, 0))) | (~((~($greatereq(I!8, 0) & (~$greatereq($sum(I!8, $product(-1, H!4)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))))))))) <=> (((~$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))) & $greatereq(H!4, 0)) & ((Rho!5 = 0) & ((~$greatereq(H!4, 0)) | ($greatereq(H!4, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a))))) & (Rho1!7 = tb2t3(const(option(a), key, none(a)))) & ($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~((~($greatereq(I!8, 0) & (~$lesseq($sum(H!4, $product(-1, I!8)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))))))))),
% 0.19/0.49 inference(monotonicity,[status(thm)],[3, 2])).
% 0.19/0.49 tff(5,plain,
% 0.19/0.49 ((((~$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))) & $greatereq(H!4, 0)) & ((~(~(Rho!5 = 0))) & ((~$greatereq(H!4, 0)) | (($greatereq(H!4, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) & ((~(~(Rho1!7 = tb2t3(const(option(a), key, none(a)))))) & ((~(~$lesseq(H!4, 0))) | (~((~($greatereq(I!8, 0) & (~$greatereq($sum(I!8, $product(-1, H!4)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))))))))) <=> ((~$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))) & $greatereq(H!4, 0) & (Rho!5 = 0) & ((~$greatereq(H!4, 0)) | ($greatereq(H!4, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a))))) & (Rho1!7 = tb2t3(const(option(a), key, none(a)))) & ($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~((~($greatereq(I!8, 0) & (~$lesseq($sum(H!4, $product(-1, I!8)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)))))))),
% 0.19/0.49 inference(transitivity,[status(thm)],[4, 1])).
% 0.19/0.49 tff(6,plain,
% 0.19/0.49 ((~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($greatereq(H, 0) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($greatereq(H, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$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(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $greatereq(H, 0) & (Rho1 = tb2t3(const(option(a), key, none(a))))))))))) <=> (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($greatereq(H, 0) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($greatereq(H, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$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(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $greatereq(H, 0) & (Rho1 = tb2t3(const(option(a), key, none(a)))))))))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(7,plain,
% 0.19/0.49 ((~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($greatereq(H, 0) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($greatereq(H, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$lesseq($sum(H, $product(-1, I)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$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(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $greatereq(H, 0) & (Rho1 = tb2t3(const(option(a), key, none(a))))))))))) <=> (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($greatereq(H, 0) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($greatereq(H, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$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(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $greatereq(H, 0) & (Rho1 = tb2t3(const(option(a), key, none(a)))))))))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(8,plain,
% 0.19/0.49 ((~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($lesseq(0, H) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($lesseq(0, H) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$lesseq(H, 0)) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $lesseq(0, H) & (Rho1 = tb2t3(const(option(a), key, none(a))))))))))) <=> (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($greatereq(H, 0) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($greatereq(H, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$lesseq($sum(H, $product(-1, I)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$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(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $greatereq(H, 0) & (Rho1 = tb2t3(const(option(a), key, none(a)))))))))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(9,plain,
% 0.19/0.49 ((~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($lesseq(0, H) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($lesseq(0, H) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$lesseq(H, 0)) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $lesseq(0, H) & (Rho1 = tb2t3(const(option(a), key, none(a))))))))))) <=> (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($lesseq(0, H) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($lesseq(0, H) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$lesseq(H, 0)) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $lesseq(0, H) & (Rho1 = tb2t3(const(option(a), key, none(a)))))))))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(10,plain,
% 0.19/0.49 ((~![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)) => ![Rho: $int] : ((Rho = 0) => ((($lesseq(0, 0) & $lesseq(0, H)) & $lesseq($sum(0, H), H)) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((($lesseq(0, H) & ![I: $int] : ((($lesseq(0, I) & $less(I, 0)) | ($lesseq($sum(0, H), I) & $less(I, H))) => (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(get(list(tuple2(key, a)), int, t2tb2(H1), t2tb(I)))))) & ![I: $int] : (($lesseq(0, I) & $less(I, $sum(0, H))) => (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) => ![Rho1: map_key_lpoption_a1rp] : ((Rho1 = tb2t3(const(option(a), key, none(a)))) => (((($less(0, H) & ![I: $int] : (($lesseq(0, I) & $less(I, H)) => good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(O)), I))) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O)))) & $lesseq(0, H)) & (Rho1 = tb2t3(const(option(a), key, none(a))))))))))) <=> (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($lesseq(0, H) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($lesseq(0, H) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$lesseq(H, 0)) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $lesseq(0, H) & (Rho1 = tb2t3(const(option(a), key, none(a)))))))))))),
% 0.19/0.50 inference(rewrite,[status(thm)],[])).
% 0.19/0.50 tff(11,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)) => ![Rho: $int] : ((Rho = 0) => ((($lesseq(0, 0) & $lesseq(0, H)) & $lesseq($sum(0, H), H)) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((($lesseq(0, H) & ![I: $int] : ((($lesseq(0, I) & $less(I, 0)) | ($lesseq($sum(0, H), I) & $less(I, H))) => (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(get(list(tuple2(key, a)), int, t2tb2(H1), t2tb(I)))))) & ![I: $int] : (($lesseq(0, I) & $less(I, $sum(0, H))) => (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) => ![Rho1: map_key_lpoption_a1rp] : ((Rho1 = tb2t3(const(option(a), key, none(a)))) => (((($less(0, H) & ![I: $int] : (($lesseq(0, I) & $less(I, H)) => good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(O)), I))) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O)))) & $lesseq(0, H)) & (Rho1 = tb2t3(const(option(a), key, none(a))))))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','wP_parameter_clear')).
% 0.19/0.50 tff(12,plain,
% 0.19/0.50 (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($lesseq(0, H) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($lesseq(0, H) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$lesseq(H, 0)) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $lesseq(0, H) & (Rho1 = tb2t3(const(option(a), key, none(a))))))))))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.19/0.50 tff(13,plain,
% 0.19/0.50 (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($lesseq(0, H) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($lesseq(0, H) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$lesseq(H, 0)) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $lesseq(0, H) & (Rho1 = tb2t3(const(option(a), key, none(a))))))))))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[12, 9])).
% 0.19/0.50 tff(14,plain,
% 0.19/0.50 (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($lesseq(0, H) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($lesseq(0, H) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$lesseq(H, 0)) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $lesseq(0, H) & (Rho1 = tb2t3(const(option(a), key, none(a))))))))))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[13, 9])).
% 0.19/0.50 tff(15,plain,
% 0.19/0.50 (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($lesseq(0, H) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($lesseq(0, H) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$lesseq(H, 0)) & ![I: $int] : ((~($lesseq(0, I) & (~$lesseq(H, I)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H, t2tb2(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $lesseq(0, H) & (Rho1 = tb2t3(const(option(a), key, none(a))))))))))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[14, 9])).
% 0.19/0.50 tff(16,plain,
% 0.19/0.50 (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($greatereq(H, 0) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($greatereq(H, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$lesseq($sum(H, $product(-1, I)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$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(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $greatereq(H, 0) & (Rho1 = tb2t3(const(option(a), key, none(a))))))))))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[15, 8])).
% 0.19/0.50 tff(17,plain,
% 0.19/0.50 (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($greatereq(H, 0) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($greatereq(H, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$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(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $greatereq(H, 0) & (Rho1 = tb2t3(const(option(a), key, none(a))))))))))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[16, 7])).
% 0.19/0.50 tff(18,plain,
% 0.19/0.50 (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($greatereq(H, 0) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($greatereq(H, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$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(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $greatereq(H, 0) & (Rho1 = tb2t3(const(option(a), key, none(a))))))))))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[17, 6])).
% 0.19/0.50 tff(19,plain,
% 0.19/0.50 (~![H: $int, H1: map_int_lplist_lpkeycm_a1rprp, H2: map_key_lpoption_a1rp] : ((~((~$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))) | ![Rho: $int] : ((~(Rho = 0)) | ($greatereq(H, 0) & ![O: map_int_lplist_lpkeycm_a1rprp] : ((~($greatereq(H, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![Rho1: map_key_lpoption_a1rp] : ((~(Rho1 = tb2t3(const(option(a), key, none(a))))) | ((~$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(O)), I)) & ![K: key1, V: a1] : good_data1(a, K, t2tb7(V), t2tb3(Rho1), mk_array1(list(tuple2(key, a)), H, t2tb2(O))) & $greatereq(H, 0) & (Rho1 = tb2t3(const(option(a), key, none(a))))))))))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[18, 6])).
% 0.19/0.50 tff(20,plain,
% 0.19/0.50 ((~$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))) & $greatereq(H!4, 0) & (Rho!5 = 0) & ((~$greatereq(H!4, 0)) | ($greatereq(H!4, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a))))) & (Rho1!7 = tb2t3(const(option(a), key, none(a)))) & ($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~((~($greatereq(I!8, 0) & (~$lesseq($sum(H!4, $product(-1, I!8)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))))))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[19, 5])).
% 0.19/0.50 tff(21,plain,
% 0.19/0.50 (~$lesseq(H!4, 0)),
% 0.19/0.50 inference(and_elim,[status(thm)],[20])).
% 0.19/0.50 tff(22,plain,
% 0.19/0.50 (^[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)))))),
% 0.19/0.50 inference(bind,[status(th)],[])).
% 0.19/0.50 tff(23,plain,
% 0.19/0.50 (![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))))),
% 0.19/0.50 inference(quant_intro,[status(thm)],[22])).
% 0.19/0.50 tff(24,plain,
% 0.19/0.50 (![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))))),
% 0.19/0.50 inference(pull_quant,[status(thm)],[])).
% 0.19/0.50 tff(25,plain,
% 0.19/0.50 (^[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)))))),
% 0.19/0.50 inference(bind,[status(th)],[])).
% 0.19/0.50 tff(26,plain,
% 0.19/0.50 (![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))))),
% 0.19/0.50 inference(quant_intro,[status(thm)],[25])).
% 0.19/0.50 tff(27,plain,
% 0.19/0.50 (![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))))),
% 0.19/0.50 inference(transitivity,[status(thm)],[26, 24])).
% 0.19/0.50 tff(28,plain,
% 0.19/0.50 (![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))))),
% 0.19/0.50 inference(transitivity,[status(thm)],[27, 23])).
% 0.19/0.50 tff(29,plain,
% 0.19/0.50 (^[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)))))),
% 0.19/0.50 inference(bind,[status(th)],[])).
% 0.19/0.50 tff(30,plain,
% 0.19/0.50 (![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))))),
% 0.19/0.50 inference(quant_intro,[status(thm)],[29])).
% 0.19/0.50 tff(31,plain,
% 0.19/0.50 (![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))))),
% 0.19/0.50 inference(transitivity,[status(thm)],[30, 28])).
% 0.19/0.50 tff(32,plain,
% 0.19/0.50 (^[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)))))),
% 0.19/0.50 inference(bind,[status(th)],[])).
% 0.19/0.50 tff(33,plain,
% 0.19/0.50 (![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))))),
% 0.19/0.50 inference(quant_intro,[status(thm)],[32])).
% 0.19/0.50 tff(34,plain,
% 0.19/0.50 (^[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)))))),
% 0.19/0.50 inference(bind,[status(th)],[])).
% 0.19/0.50 tff(35,plain,
% 0.19/0.50 (![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))))),
% 0.19/0.50 inference(quant_intro,[status(thm)],[34])).
% 0.19/0.50 tff(36,plain,
% 0.19/0.50 (![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)))))),
% 0.19/0.50 inference(rewrite,[status(thm)],[])).
% 0.19/0.50 tff(37,plain,
% 0.19/0.50 (^[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)))))))),
% 0.19/0.50 inference(bind,[status(th)],[])).
% 0.19/0.50 tff(38,plain,
% 0.19/0.50 (![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)))))),
% 0.19/0.50 inference(quant_intro,[status(thm)],[37])).
% 0.19/0.50 tff(39,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')).
% 0.19/0.50 tff(40,plain,
% 0.19/0.50 (![N: $int] : ($lesseq(N, 0) | ![K: key1] : ($lesseq(0, bucket1(K, N)) & (~$lesseq(N, bucket1(K, N)))))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[39, 38])).
% 0.19/0.50 tff(41,plain,
% 0.19/0.50 (![N: $int] : ($lesseq(N, 0) | ![K: key1] : ($lesseq(0, bucket1(K, N)) & (~$lesseq(N, bucket1(K, N)))))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[40, 36])).
% 0.19/0.50 tff(42,plain,
% 0.19/0.50 (![N: $int] : ($lesseq(N, 0) | ![K: key1] : ($greatereq(bucket1(K, N), 0) & (~$lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[41, 35])).
% 0.19/0.50 tff(43,plain,(
% 0.19/0.50 ![N: $int] : ($lesseq(N, 0) | ![K: key1] : ($greatereq(bucket1(K, N), 0) & (~$lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.19/0.50 inference(skolemize,[status(sab)],[42])).
% 0.19/0.50 tff(44,plain,
% 0.19/0.50 (![N: $int] : ($lesseq(N, 0) | ![K: key1] : (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[43, 33])).
% 0.19/0.50 tff(45,plain,
% 0.19/0.50 (![N: $int, K: key1] : ($lesseq(N, 0) | (~((~$greatereq(bucket1(K, N), 0)) | $lesseq($sum(N, $product(-1, bucket1(K, N))), 0))))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[44, 31])).
% 0.19/0.50 tff(46,plain,
% 0.19/0.50 (((~![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!10, H!4), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!10, 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!10, H!4), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!10, H!4))), 0))))),
% 0.19/0.50 inference(rewrite,[status(thm)],[])).
% 0.19/0.50 tff(47,plain,
% 0.19/0.50 ((~![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!10, H!4), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!10, H!4))), 0))))),
% 0.19/0.50 inference(quant_inst,[status(thm)],[])).
% 0.19/0.50 tff(48,plain,
% 0.19/0.50 ((~![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!10, H!4), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!10, H!4))), 0)))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.19/0.50 tff(49,plain,
% 0.19/0.50 (~((~$greatereq(bucket1(K!10, H!4), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!10, H!4))), 0))),
% 0.19/0.50 inference(unit_resolution,[status(thm)],[48, 45, 21])).
% 0.19/0.50 tff(50,plain,
% 0.19/0.50 (((~$greatereq(bucket1(K!10, H!4), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!10, H!4))), 0)) | (~$lesseq($sum(H!4, $product(-1, bucket1(K!10, H!4))), 0))),
% 0.19/0.50 inference(tautology,[status(thm)],[])).
% 0.19/0.50 tff(51,plain,
% 0.19/0.50 (~$lesseq($sum(H!4, $product(-1, bucket1(K!10, H!4))), 0)),
% 0.19/0.50 inference(unit_resolution,[status(thm)],[50, 49])).
% 0.19/0.50 tff(52,plain,
% 0.19/0.50 (^[A: ty, A1: uni, I: $int] : refl((get2(A, A1, I) = get(A, int, elts(A, A1), t2tb(I))) <=> (get2(A, A1, I) = get(A, int, elts(A, A1), t2tb(I))))),
% 0.19/0.50 inference(bind,[status(th)],[])).
% 0.19/0.50 tff(53,plain,
% 0.19/0.50 (![A: ty, A1: uni, I: $int] : (get2(A, A1, I) = get(A, int, elts(A, A1), t2tb(I))) <=> ![A: ty, A1: uni, I: $int] : (get2(A, A1, I) = get(A, int, elts(A, A1), t2tb(I)))),
% 0.19/0.50 inference(quant_intro,[status(thm)],[52])).
% 0.19/0.50 tff(54,plain,
% 0.19/0.50 (![A: ty, A1: uni, I: $int] : (get2(A, A1, I) = get(A, int, elts(A, A1), t2tb(I))) <=> ![A: ty, A1: uni, I: $int] : (get2(A, A1, I) = get(A, int, elts(A, A1), t2tb(I)))),
% 0.19/0.50 inference(rewrite,[status(thm)],[])).
% 0.19/0.50 tff(55,axiom,(![A: ty, A1: uni, I: $int] : (get2(A, A1, I) = get(A, int, elts(A, A1), t2tb(I)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','get_def')).
% 0.19/0.50 tff(56,plain,
% 0.19/0.50 (![A: ty, A1: uni, I: $int] : (get2(A, A1, I) = get(A, int, elts(A, A1), t2tb(I)))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[55, 54])).
% 0.19/0.50 tff(57,plain,(
% 0.19/0.50 ![A: ty, A1: uni, I: $int] : (get2(A, A1, I) = get(A, int, elts(A, A1), t2tb(I)))),
% 0.19/0.50 inference(skolemize,[status(sab)],[56])).
% 0.19/0.50 tff(58,plain,
% 0.19/0.50 (![A: ty, A1: uni, I: $int] : (get2(A, A1, I) = get(A, int, elts(A, A1), t2tb(I)))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[57, 53])).
% 0.19/0.50 tff(59,plain,
% 0.19/0.50 ((~![A: ty, A1: uni, I: $int] : (get2(A, A1, I) = get(A, int, elts(A, A1), t2tb(I)))) | (get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))) = get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))))),
% 0.19/0.50 inference(quant_inst,[status(thm)],[])).
% 0.19/0.50 tff(60,plain,
% 0.19/0.50 (get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))) = get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))),
% 0.19/0.50 inference(unit_resolution,[status(thm)],[59, 58])).
% 0.19/0.50 tff(61,plain,
% 0.19/0.50 (get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))),
% 0.19/0.51 inference(symmetry,[status(thm)],[60])).
% 0.19/0.51 tff(62,plain,
% 0.19/0.51 (![J: uni] : (t2tb6(tb2t6(J)) = J) <=> ![J: uni] : (t2tb6(tb2t6(J)) = J)),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(63,plain,
% 0.19/0.51 (![J: uni] : (t2tb6(tb2t6(J)) = J) <=> ![J: uni] : (t2tb6(tb2t6(J)) = J)),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(64,axiom,(![J: uni] : (t2tb6(tb2t6(J)) = J)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','bridgeR6')).
% 0.19/0.51 tff(65,plain,
% 0.19/0.51 (![J: uni] : (t2tb6(tb2t6(J)) = J)),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.19/0.51 tff(66,plain,(
% 0.19/0.51 ![J: uni] : (t2tb6(tb2t6(J)) = J)),
% 0.19/0.51 inference(skolemize,[status(sab)],[65])).
% 0.19/0.51 tff(67,plain,
% 0.19/0.51 (![J: uni] : (t2tb6(tb2t6(J)) = J)),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[66, 62])).
% 0.19/0.51 tff(68,plain,
% 0.19/0.51 ((~![J: uni] : (t2tb6(tb2t6(J)) = J)) | (t2tb6(tb2t6(get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))))) = get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))))),
% 0.19/0.51 inference(quant_inst,[status(thm)],[])).
% 0.19/0.51 tff(69,plain,
% 0.19/0.51 (t2tb6(tb2t6(get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))))) = get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))),
% 0.19/0.51 inference(unit_resolution,[status(thm)],[68, 67])).
% 0.19/0.51 tff(70,plain,
% 0.19/0.51 (^[X: map_int_lplist_lpkeycm_a1rprp] : refl(sort1(map(int, list(tuple2(key, a))), t2tb2(X)) <=> sort1(map(int, list(tuple2(key, a))), t2tb2(X)))),
% 0.19/0.51 inference(bind,[status(th)],[])).
% 0.19/0.51 tff(71,plain,
% 0.19/0.51 (![X: map_int_lplist_lpkeycm_a1rprp] : sort1(map(int, list(tuple2(key, a))), t2tb2(X)) <=> ![X: map_int_lplist_lpkeycm_a1rprp] : sort1(map(int, list(tuple2(key, a))), t2tb2(X))),
% 0.19/0.51 inference(quant_intro,[status(thm)],[70])).
% 0.19/0.51 tff(72,plain,
% 0.19/0.51 (![X: map_int_lplist_lpkeycm_a1rprp] : sort1(map(int, list(tuple2(key, a))), t2tb2(X)) <=> ![X: map_int_lplist_lpkeycm_a1rprp] : sort1(map(int, list(tuple2(key, a))), t2tb2(X))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(73,axiom,(![X: map_int_lplist_lpkeycm_a1rprp] : sort1(map(int, list(tuple2(key, a))), t2tb2(X))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t2tb_sort2')).
% 0.19/0.51 tff(74,plain,
% 0.19/0.51 (![X: map_int_lplist_lpkeycm_a1rprp] : sort1(map(int, list(tuple2(key, a))), t2tb2(X))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[73, 72])).
% 0.19/0.51 tff(75,plain,(
% 0.19/0.51 ![X: map_int_lplist_lpkeycm_a1rprp] : sort1(map(int, list(tuple2(key, a))), t2tb2(X))),
% 0.19/0.51 inference(skolemize,[status(sab)],[74])).
% 0.19/0.51 tff(76,plain,
% 0.19/0.51 (![X: map_int_lplist_lpkeycm_a1rprp] : sort1(map(int, list(tuple2(key, a))), t2tb2(X))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[75, 71])).
% 0.19/0.51 tff(77,plain,
% 0.19/0.51 ((~![X: map_int_lplist_lpkeycm_a1rprp] : sort1(map(int, list(tuple2(key, a))), t2tb2(X))) | sort1(map(int, list(tuple2(key, a))), t2tb2(O!6))),
% 0.19/0.51 inference(quant_inst,[status(thm)],[])).
% 0.19/0.51 tff(78,plain,
% 0.19/0.51 (sort1(map(int, list(tuple2(key, a))), t2tb2(O!6))),
% 0.19/0.51 inference(unit_resolution,[status(thm)],[77, 76])).
% 0.19/0.51 tff(79,plain,
% 0.19/0.51 (^[A: ty, U: $int, U1: uni] : refl(((~sort1(map(int, A), U1)) | (elts(A, mk_array1(A, U, U1)) = U1)) <=> ((~sort1(map(int, A), U1)) | (elts(A, mk_array1(A, U, U1)) = U1)))),
% 0.19/0.51 inference(bind,[status(th)],[])).
% 0.19/0.51 tff(80,plain,
% 0.19/0.51 (![A: ty, U: $int, U1: uni] : ((~sort1(map(int, A), U1)) | (elts(A, mk_array1(A, U, U1)) = U1)) <=> ![A: ty, U: $int, U1: uni] : ((~sort1(map(int, A), U1)) | (elts(A, mk_array1(A, U, U1)) = U1))),
% 0.19/0.51 inference(quant_intro,[status(thm)],[79])).
% 0.19/0.51 tff(81,plain,
% 0.19/0.51 (![A: ty, U: $int, U1: uni] : ((~sort1(map(int, A), U1)) | (elts(A, mk_array1(A, U, U1)) = U1)) <=> ![A: ty, U: $int, U1: uni] : ((~sort1(map(int, A), U1)) | (elts(A, mk_array1(A, U, U1)) = U1))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(82,plain,
% 0.19/0.51 (^[A: ty, U: $int, U1: uni] : rewrite((sort1(map(int, A), U1) => (elts(A, mk_array1(A, U, U1)) = U1)) <=> ((~sort1(map(int, A), U1)) | (elts(A, mk_array1(A, U, U1)) = U1)))),
% 0.19/0.51 inference(bind,[status(th)],[])).
% 0.19/0.51 tff(83,plain,
% 0.19/0.51 (![A: ty, U: $int, U1: uni] : (sort1(map(int, A), U1) => (elts(A, mk_array1(A, U, U1)) = U1)) <=> ![A: ty, U: $int, U1: uni] : ((~sort1(map(int, A), U1)) | (elts(A, mk_array1(A, U, U1)) = U1))),
% 0.19/0.51 inference(quant_intro,[status(thm)],[82])).
% 0.19/0.51 tff(84,axiom,(![A: ty, U: $int, U1: uni] : (sort1(map(int, A), U1) => (elts(A, mk_array1(A, U, U1)) = U1))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','elts_def1')).
% 0.19/0.51 tff(85,plain,
% 0.19/0.51 (![A: ty, U: $int, U1: uni] : ((~sort1(map(int, A), U1)) | (elts(A, mk_array1(A, U, U1)) = U1))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.19/0.51 tff(86,plain,
% 0.19/0.51 (![A: ty, U: $int, U1: uni] : ((~sort1(map(int, A), U1)) | (elts(A, mk_array1(A, U, U1)) = U1))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[85, 81])).
% 0.19/0.51 tff(87,plain,(
% 0.19/0.51 ![A: ty, U: $int, U1: uni] : ((~sort1(map(int, A), U1)) | (elts(A, mk_array1(A, U, U1)) = U1))),
% 0.19/0.51 inference(skolemize,[status(sab)],[86])).
% 0.19/0.51 tff(88,plain,
% 0.19/0.51 (![A: ty, U: $int, U1: uni] : ((~sort1(map(int, A), U1)) | (elts(A, mk_array1(A, U, U1)) = U1))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[87, 80])).
% 0.19/0.51 tff(89,plain,
% 0.19/0.51 (((~![A: ty, U: $int, U1: uni] : ((~sort1(map(int, A), U1)) | (elts(A, mk_array1(A, U, U1)) = U1))) | ((~sort1(map(int, list(tuple2(key, a))), t2tb2(O!6))) | (elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))) = t2tb2(O!6)))) <=> ((~![A: ty, U: $int, U1: uni] : ((~sort1(map(int, A), U1)) | (elts(A, mk_array1(A, U, U1)) = U1))) | (~sort1(map(int, list(tuple2(key, a))), t2tb2(O!6))) | (elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))) = t2tb2(O!6)))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(90,plain,
% 0.19/0.51 ((~![A: ty, U: $int, U1: uni] : ((~sort1(map(int, A), U1)) | (elts(A, mk_array1(A, U, U1)) = U1))) | ((~sort1(map(int, list(tuple2(key, a))), t2tb2(O!6))) | (elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))) = t2tb2(O!6)))),
% 0.19/0.51 inference(quant_inst,[status(thm)],[])).
% 0.19/0.51 tff(91,plain,
% 0.19/0.51 ((~![A: ty, U: $int, U1: uni] : ((~sort1(map(int, A), U1)) | (elts(A, mk_array1(A, U, U1)) = U1))) | (~sort1(map(int, list(tuple2(key, a))), t2tb2(O!6))) | (elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))) = t2tb2(O!6))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[90, 89])).
% 0.19/0.51 tff(92,plain,
% 0.19/0.51 ((~sort1(map(int, list(tuple2(key, a))), t2tb2(O!6))) | (elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))) = t2tb2(O!6))),
% 0.19/0.51 inference(unit_resolution,[status(thm)],[91, 88])).
% 0.19/0.51 tff(93,plain,
% 0.19/0.51 (elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))) = t2tb2(O!6)),
% 0.19/0.51 inference(unit_resolution,[status(thm)],[92, 78])).
% 0.19/0.51 tff(94,plain,
% 0.19/0.51 (t2tb2(O!6) = elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))),
% 0.19/0.51 inference(symmetry,[status(thm)],[93])).
% 0.19/0.51 tff(95,plain,
% 0.19/0.51 (get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))),
% 0.19/0.51 inference(monotonicity,[status(thm)],[94])).
% 0.19/0.51 tff(96,plain,
% 0.19/0.51 (get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))),
% 0.19/0.51 inference(symmetry,[status(thm)],[95])).
% 0.19/0.51 tff(97,plain,
% 0.19/0.51 (tb2t6(get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))) = tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))))),
% 0.19/0.51 inference(monotonicity,[status(thm)],[96])).
% 0.19/0.51 tff(98,plain,
% 0.19/0.51 (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))) = tb2t6(get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))))),
% 0.19/0.51 inference(symmetry,[status(thm)],[97])).
% 0.19/0.51 tff(99,assumption,(tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))) = tb2t6(nil(tuple2(key, a)))), introduced(assumption)).
% 0.19/0.51 tff(100,plain,
% 0.19/0.51 (tb2t6(nil(tuple2(key, a))) = tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))))),
% 0.19/0.51 inference(symmetry,[status(thm)],[99])).
% 0.19/0.51 tff(101,plain,
% 0.19/0.51 (tb2t6(nil(tuple2(key, a))) = tb2t6(get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))))),
% 0.19/0.51 inference(transitivity,[status(thm)],[100, 98])).
% 0.19/0.51 tff(102,plain,
% 0.19/0.51 (t2tb6(tb2t6(nil(tuple2(key, a)))) = t2tb6(tb2t6(get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))))),
% 0.19/0.51 inference(monotonicity,[status(thm)],[101])).
% 0.19/0.51 tff(103,plain,
% 0.19/0.51 ((~![J: uni] : (t2tb6(tb2t6(J)) = J)) | (t2tb6(tb2t6(nil(tuple2(key, a)))) = nil(tuple2(key, a)))),
% 0.19/0.51 inference(quant_inst,[status(thm)],[])).
% 0.19/0.51 tff(104,plain,
% 0.19/0.51 (t2tb6(tb2t6(nil(tuple2(key, a)))) = nil(tuple2(key, a))),
% 0.19/0.51 inference(unit_resolution,[status(thm)],[103, 67])).
% 0.19/0.51 tff(105,plain,
% 0.19/0.51 (nil(tuple2(key, a)) = t2tb6(tb2t6(nil(tuple2(key, a))))),
% 0.19/0.51 inference(symmetry,[status(thm)],[104])).
% 0.19/0.51 tff(106,plain,
% 0.19/0.51 (nil(tuple2(key, a)) = get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))),
% 0.19/0.51 inference(transitivity,[status(thm)],[105, 102, 69, 61])).
% 0.19/0.51 tff(107,plain,
% 0.19/0.51 (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), nil(tuple2(key, a))) <=> mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))),
% 0.19/0.51 inference(monotonicity,[status(thm)],[106])).
% 0.19/0.51 tff(108,plain,
% 0.19/0.51 (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) <=> mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), nil(tuple2(key, a)))),
% 0.19/0.51 inference(symmetry,[status(thm)],[107])).
% 0.19/0.51 tff(109,plain,
% 0.19/0.51 (^[A: ty, K: key1, V: uni, D: uni] : refl((in_data1(A, K, V, D) <=> mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, bucket1(K, length1(list(tuple2(key, A)), D))))) <=> (in_data1(A, K, V, D) <=> mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, bucket1(K, length1(list(tuple2(key, A)), D))))))),
% 0.19/0.51 inference(bind,[status(th)],[])).
% 0.19/0.51 tff(110,plain,
% 0.19/0.51 (![A: ty, K: key1, V: uni, D: uni] : (in_data1(A, K, V, D) <=> mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, bucket1(K, length1(list(tuple2(key, A)), D))))) <=> ![A: ty, K: key1, V: uni, D: uni] : (in_data1(A, K, V, D) <=> mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, bucket1(K, length1(list(tuple2(key, A)), D)))))),
% 0.19/0.51 inference(quant_intro,[status(thm)],[109])).
% 0.19/0.51 tff(111,plain,
% 0.19/0.51 (![A: ty, K: key1, V: uni, D: uni] : (in_data1(A, K, V, D) <=> mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, bucket1(K, length1(list(tuple2(key, A)), D))))) <=> ![A: ty, K: key1, V: uni, D: uni] : (in_data1(A, K, V, D) <=> mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, bucket1(K, length1(list(tuple2(key, A)), D)))))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(112,axiom,(![A: ty, K: key1, V: uni, D: uni] : (in_data1(A, K, V, D) <=> mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, bucket1(K, length1(list(tuple2(key, A)), D)))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','in_data_def')).
% 0.19/0.51 tff(113,plain,
% 0.19/0.51 (![A: ty, K: key1, V: uni, D: uni] : (in_data1(A, K, V, D) <=> mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, bucket1(K, length1(list(tuple2(key, A)), D)))))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[112, 111])).
% 0.19/0.51 tff(114,plain,(
% 0.19/0.51 ![A: ty, K: key1, V: uni, D: uni] : (in_data1(A, K, V, D) <=> mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, bucket1(K, length1(list(tuple2(key, A)), D)))))),
% 0.19/0.51 inference(skolemize,[status(sab)],[113])).
% 0.19/0.51 tff(115,plain,
% 0.19/0.51 (![A: ty, K: key1, V: uni, D: uni] : (in_data1(A, K, V, D) <=> mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, bucket1(K, length1(list(tuple2(key, A)), D)))))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[114, 110])).
% 0.19/0.51 tff(116,plain,
% 0.19/0.51 ((~![A: ty, K: key1, V: uni, D: uni] : (in_data1(A, K, V, D) <=> mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, bucket1(K, length1(list(tuple2(key, A)), D)))))) | (in_data1(a, K!10, t2tb7(V!9), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))) <=> mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))))),
% 0.19/0.51 inference(quant_inst,[status(thm)],[])).
% 0.19/0.51 tff(117,plain,
% 0.19/0.51 (in_data1(a, K!10, t2tb7(V!9), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))) <=> mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))),
% 0.19/0.51 inference(unit_resolution,[status(thm)],[116, 115])).
% 0.19/0.51 tff(118,plain,
% 0.19/0.51 (^[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))))),
% 0.19/0.51 inference(bind,[status(th)],[])).
% 0.19/0.51 tff(119,plain,
% 0.19/0.51 (![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)))),
% 0.19/0.51 inference(quant_intro,[status(thm)],[118])).
% 0.19/0.51 tff(120,plain,
% 0.19/0.51 (![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)))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(121,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')).
% 0.19/0.51 tff(122,plain,
% 0.19/0.51 (![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)))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[121, 120])).
% 0.19/0.51 tff(123,plain,(
% 0.19/0.51 ![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)))),
% 0.19/0.51 inference(skolemize,[status(sab)],[122])).
% 0.19/0.51 tff(124,plain,
% 0.19/0.51 (![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)))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[123, 119])).
% 0.19/0.51 tff(125,plain,
% 0.19/0.51 ((~![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!10, t2tb7(V!9), const(option(a), key, none(a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))) <=> ((get(option(a), key, const(option(a), key, none(a)), t2tb1(K!10)) = some(a, t2tb7(V!9))) <=> in_data1(a, K!10, t2tb7(V!9), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))),
% 0.19/0.51 inference(quant_inst,[status(thm)],[])).
% 0.19/0.51 tff(126,plain,
% 0.19/0.51 (good_data1(a, K!10, t2tb7(V!9), const(option(a), key, none(a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))) <=> ((get(option(a), key, const(option(a), key, none(a)), t2tb1(K!10)) = some(a, t2tb7(V!9))) <=> in_data1(a, K!10, t2tb7(V!9), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))),
% 0.19/0.51 inference(unit_resolution,[status(thm)],[125, 124])).
% 0.19/0.51 tff(127,plain,
% 0.19/0.51 ($greatereq(H!4, 0)),
% 0.19/0.51 inference(and_elim,[status(thm)],[20])).
% 0.19/0.51 tff(128,plain,
% 0.19/0.51 (((~$greatereq(H!4, 0)) | (~((~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | (~($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0))))))))) <=> ((~$greatereq(H!4, 0)) | (~((~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0)))))) | (~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))))))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(129,plain,
% 0.19/0.51 (((~$greatereq(H!4, 0)) | (~((~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | (~($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0))))))))) <=> ((~$greatereq(H!4, 0)) | (~((~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | (~($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0)))))))))),
% 0.19/0.52 inference(rewrite,[status(thm)],[])).
% 0.19/0.52 tff(130,plain,
% 0.19/0.52 (((~$greatereq(H!4, 0)) | ($greatereq(H!4, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a))))) & (Rho1!7 = tb2t3(const(option(a), key, none(a)))) & ($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~((~($greatereq(I!8, 0) & (~$lesseq($sum(H!4, $product(-1, I!8)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)))))) <=> ((~$greatereq(H!4, 0)) | (~((~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | (~($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0)))))))))),
% 0.19/0.52 inference(rewrite,[status(thm)],[])).
% 0.19/0.52 tff(131,plain,
% 0.19/0.52 ((~$greatereq(H!4, 0)) | ($greatereq(H!4, 0) & ![I: $int] : ((~($greatereq(I, 0) & (~$greatereq($sum(I, $product(-1, H!4)), 0)))) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a))))) & (Rho1!7 = tb2t3(const(option(a), key, none(a)))) & ($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~((~($greatereq(I!8, 0) & (~$lesseq($sum(H!4, $product(-1, I!8)), 0)))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)))))),
% 0.19/0.52 inference(and_elim,[status(thm)],[20])).
% 0.19/0.52 tff(132,plain,
% 0.19/0.52 ((~$greatereq(H!4, 0)) | (~((~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | (~($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0))))))))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[131, 130])).
% 0.19/0.52 tff(133,plain,
% 0.19/0.52 ((~$greatereq(H!4, 0)) | (~((~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | (~($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0))))))))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[132, 129])).
% 0.19/0.52 tff(134,plain,
% 0.19/0.52 ((~$greatereq(H!4, 0)) | (~((~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0)))))) | (~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[133, 128])).
% 0.19/0.52 tff(135,plain,
% 0.19/0.52 (~((~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0)))))) | (~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))))),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[134, 127])).
% 0.19/0.52 tff(136,plain,
% 0.19/0.52 (((~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0)))))) | (~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | (Rho1!7 = tb2t3(const(option(a), key, none(a))))),
% 0.19/0.52 inference(tautology,[status(thm)],[])).
% 0.19/0.52 tff(137,plain,
% 0.19/0.52 (Rho1!7 = tb2t3(const(option(a), key, none(a)))),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[136, 135])).
% 0.19/0.52 tff(138,plain,
% 0.19/0.52 (t2tb3(Rho1!7) = t2tb3(tb2t3(const(option(a), key, none(a))))),
% 0.19/0.52 inference(monotonicity,[status(thm)],[137])).
% 0.19/0.52 tff(139,plain,
% 0.19/0.52 (t2tb3(tb2t3(const(option(a), key, none(a)))) = t2tb3(Rho1!7)),
% 0.19/0.52 inference(symmetry,[status(thm)],[138])).
% 0.19/0.52 tff(140,plain,
% 0.19/0.52 (^[A: ty, B: ty, X: uni] : refl(sort1(map(A, B), const(B, A, X)) <=> sort1(map(A, B), const(B, A, X)))),
% 0.19/0.52 inference(bind,[status(th)],[])).
% 0.19/0.52 tff(141,plain,
% 0.19/0.52 (![A: ty, B: ty, X: uni] : sort1(map(A, B), const(B, A, X)) <=> ![A: ty, B: ty, X: uni] : sort1(map(A, B), const(B, A, X))),
% 0.19/0.52 inference(quant_intro,[status(thm)],[140])).
% 0.19/0.52 tff(142,plain,
% 0.19/0.52 (![A: ty, B: ty, X: uni] : sort1(map(A, B), const(B, A, X)) <=> ![A: ty, B: ty, X: uni] : sort1(map(A, B), const(B, A, X))),
% 0.19/0.52 inference(rewrite,[status(thm)],[])).
% 0.19/0.52 tff(143,axiom,(![A: ty, B: ty, X: uni] : sort1(map(A, B), const(B, A, X))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','const_sort1')).
% 0.19/0.52 tff(144,plain,
% 0.19/0.52 (![A: ty, B: ty, X: uni] : sort1(map(A, B), const(B, A, X))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[143, 142])).
% 0.19/0.52 tff(145,plain,(
% 0.19/0.52 ![A: ty, B: ty, X: uni] : sort1(map(A, B), const(B, A, X))),
% 0.19/0.52 inference(skolemize,[status(sab)],[144])).
% 0.19/0.52 tff(146,plain,
% 0.19/0.52 (![A: ty, B: ty, X: uni] : sort1(map(A, B), const(B, A, X))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[145, 141])).
% 0.19/0.52 tff(147,plain,
% 0.19/0.52 ((~![A: ty, B: ty, X: uni] : sort1(map(A, B), const(B, A, X))) | sort1(map(key, option(a)), const(option(a), key, none(a)))),
% 0.19/0.52 inference(quant_inst,[status(thm)],[])).
% 0.19/0.52 tff(148,plain,
% 0.19/0.52 (sort1(map(key, option(a)), const(option(a), key, none(a)))),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[147, 146])).
% 0.19/0.52 tff(149,plain,
% 0.19/0.52 (^[J: uni] : refl(((~sort1(map(key, option(a)), J)) | (t2tb3(tb2t3(J)) = J)) <=> ((~sort1(map(key, option(a)), J)) | (t2tb3(tb2t3(J)) = J)))),
% 0.19/0.52 inference(bind,[status(th)],[])).
% 0.19/0.52 tff(150,plain,
% 0.19/0.52 (![J: uni] : ((~sort1(map(key, option(a)), J)) | (t2tb3(tb2t3(J)) = J)) <=> ![J: uni] : ((~sort1(map(key, option(a)), J)) | (t2tb3(tb2t3(J)) = J))),
% 0.19/0.52 inference(quant_intro,[status(thm)],[149])).
% 0.19/0.52 tff(151,plain,
% 0.19/0.52 (![J: uni] : ((~sort1(map(key, option(a)), J)) | (t2tb3(tb2t3(J)) = J)) <=> ![J: uni] : ((~sort1(map(key, option(a)), J)) | (t2tb3(tb2t3(J)) = J))),
% 0.19/0.52 inference(rewrite,[status(thm)],[])).
% 0.19/0.52 tff(152,plain,
% 0.19/0.52 (^[J: uni] : rewrite((sort1(map(key, option(a)), J) => (t2tb3(tb2t3(J)) = J)) <=> ((~sort1(map(key, option(a)), J)) | (t2tb3(tb2t3(J)) = J)))),
% 0.19/0.52 inference(bind,[status(th)],[])).
% 0.19/0.52 tff(153,plain,
% 0.19/0.52 (![J: uni] : (sort1(map(key, option(a)), J) => (t2tb3(tb2t3(J)) = J)) <=> ![J: uni] : ((~sort1(map(key, option(a)), J)) | (t2tb3(tb2t3(J)) = J))),
% 0.19/0.52 inference(quant_intro,[status(thm)],[152])).
% 0.19/0.52 tff(154,axiom,(![J: uni] : (sort1(map(key, option(a)), J) => (t2tb3(tb2t3(J)) = J))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','bridgeR3')).
% 0.19/0.52 tff(155,plain,
% 0.19/0.52 (![J: uni] : ((~sort1(map(key, option(a)), J)) | (t2tb3(tb2t3(J)) = J))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[154, 153])).
% 0.19/0.52 tff(156,plain,
% 0.19/0.52 (![J: uni] : ((~sort1(map(key, option(a)), J)) | (t2tb3(tb2t3(J)) = J))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[155, 151])).
% 0.19/0.52 tff(157,plain,(
% 0.19/0.52 ![J: uni] : ((~sort1(map(key, option(a)), J)) | (t2tb3(tb2t3(J)) = J))),
% 0.19/0.52 inference(skolemize,[status(sab)],[156])).
% 0.19/0.52 tff(158,plain,
% 0.19/0.52 (![J: uni] : ((~sort1(map(key, option(a)), J)) | (t2tb3(tb2t3(J)) = J))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[157, 150])).
% 0.19/0.52 tff(159,plain,
% 0.19/0.52 (((~![J: uni] : ((~sort1(map(key, option(a)), J)) | (t2tb3(tb2t3(J)) = J))) | ((~sort1(map(key, option(a)), const(option(a), key, none(a)))) | (t2tb3(tb2t3(const(option(a), key, none(a)))) = const(option(a), key, none(a))))) <=> ((~![J: uni] : ((~sort1(map(key, option(a)), J)) | (t2tb3(tb2t3(J)) = J))) | (~sort1(map(key, option(a)), const(option(a), key, none(a)))) | (t2tb3(tb2t3(const(option(a), key, none(a)))) = const(option(a), key, none(a))))),
% 0.19/0.52 inference(rewrite,[status(thm)],[])).
% 0.19/0.52 tff(160,plain,
% 0.19/0.52 ((~![J: uni] : ((~sort1(map(key, option(a)), J)) | (t2tb3(tb2t3(J)) = J))) | ((~sort1(map(key, option(a)), const(option(a), key, none(a)))) | (t2tb3(tb2t3(const(option(a), key, none(a)))) = const(option(a), key, none(a))))),
% 0.19/0.52 inference(quant_inst,[status(thm)],[])).
% 0.19/0.52 tff(161,plain,
% 0.19/0.52 ((~![J: uni] : ((~sort1(map(key, option(a)), J)) | (t2tb3(tb2t3(J)) = J))) | (~sort1(map(key, option(a)), const(option(a), key, none(a)))) | (t2tb3(tb2t3(const(option(a), key, none(a)))) = const(option(a), key, none(a)))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[160, 159])).
% 0.19/0.52 tff(162,plain,
% 0.19/0.52 (t2tb3(tb2t3(const(option(a), key, none(a)))) = const(option(a), key, none(a))),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[161, 158, 148])).
% 0.19/0.52 tff(163,plain,
% 0.19/0.52 (const(option(a), key, none(a)) = t2tb3(tb2t3(const(option(a), key, none(a))))),
% 0.19/0.52 inference(symmetry,[status(thm)],[162])).
% 0.19/0.52 tff(164,plain,
% 0.19/0.52 (const(option(a), key, none(a)) = t2tb3(Rho1!7)),
% 0.19/0.52 inference(transitivity,[status(thm)],[163, 139])).
% 0.19/0.52 tff(165,plain,
% 0.19/0.52 (good_data1(a, K!10, t2tb7(V!9), const(option(a), key, none(a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))) <=> good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))),
% 0.19/0.52 inference(monotonicity,[status(thm)],[164])).
% 0.19/0.52 tff(166,plain,
% 0.19/0.52 (good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))) <=> good_data1(a, K!10, t2tb7(V!9), const(option(a), key, none(a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))),
% 0.19/0.52 inference(symmetry,[status(thm)],[165])).
% 0.19/0.52 tff(167,plain,
% 0.19/0.52 ((~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) <=> (~good_data1(a, K!10, t2tb7(V!9), const(option(a), key, none(a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))),
% 0.19/0.52 inference(monotonicity,[status(thm)],[166])).
% 0.19/0.52 tff(168,plain,
% 0.19/0.52 ((~![A: ty, A1: uni, I: $int] : (get2(A, A1, I) = get(A, int, elts(A, A1), t2tb(I)))) | (get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) = get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(I!8)))),
% 0.19/0.52 inference(quant_inst,[status(thm)],[])).
% 0.19/0.52 tff(169,plain,
% 0.19/0.52 (get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) = get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(I!8))),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[168, 58])).
% 0.19/0.52 tff(170,plain,
% 0.19/0.52 (get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(I!8)) = get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)),
% 0.19/0.52 inference(symmetry,[status(thm)],[169])).
% 0.19/0.52 tff(171,plain,
% 0.19/0.52 ((~![J: uni] : (t2tb6(tb2t6(J)) = J)) | (t2tb6(tb2t6(get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(I!8)))) = get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(I!8)))),
% 0.19/0.52 inference(quant_inst,[status(thm)],[])).
% 0.19/0.52 tff(172,plain,
% 0.19/0.52 (t2tb6(tb2t6(get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(I!8)))) = get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(I!8))),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[171, 67])).
% 0.19/0.52 tff(173,plain,
% 0.19/0.52 (get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I!8)) = get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(I!8))),
% 0.19/0.52 inference(monotonicity,[status(thm)],[94])).
% 0.19/0.52 tff(174,plain,
% 0.19/0.52 (get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(I!8)) = get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I!8))),
% 0.19/0.52 inference(symmetry,[status(thm)],[173])).
% 0.19/0.52 tff(175,plain,
% 0.19/0.52 (tb2t6(get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(I!8))) = tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I!8)))),
% 0.19/0.52 inference(monotonicity,[status(thm)],[174])).
% 0.19/0.52 tff(176,plain,
% 0.19/0.52 (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I!8))) = tb2t6(get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(I!8)))),
% 0.19/0.52 inference(symmetry,[status(thm)],[175])).
% 0.19/0.52 tff(177,plain,
% 0.19/0.52 (((~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0)))))) | (~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))),
% 0.19/0.52 inference(tautology,[status(thm)],[])).
% 0.19/0.52 tff(178,plain,
% 0.19/0.52 (![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[177, 135])).
% 0.19/0.52 tff(179,assumption,(~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0)))), introduced(assumption)).
% 0.19/0.52 tff(180,plain,
% 0.19/0.52 ((good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0))) | $greatereq(I!8, 0)),
% 0.19/0.52 inference(tautology,[status(thm)],[])).
% 0.19/0.52 tff(181,plain,
% 0.19/0.52 ($greatereq(I!8, 0)),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[180, 179])).
% 0.19/0.52 tff(182,plain,
% 0.19/0.52 ((good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0))) | (~$lesseq($sum(H!4, $product(-1, I!8)), 0))),
% 0.19/0.52 inference(tautology,[status(thm)],[])).
% 0.19/0.52 tff(183,plain,
% 0.19/0.52 (~$lesseq($sum(H!4, $product(-1, I!8)), 0)),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[182, 179])).
% 0.19/0.52 tff(184,plain,
% 0.19/0.52 (((~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | ((~$greatereq(I!8, 0)) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I!8))) = tb2t6(nil(tuple2(key, a)))))) <=> ((~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | (~$greatereq(I!8, 0)) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I!8))) = tb2t6(nil(tuple2(key, a)))))),
% 0.19/0.53 inference(rewrite,[status(thm)],[])).
% 0.19/0.53 tff(185,plain,
% 0.19/0.53 (((~$greatereq(I!8, 0)) | $greatereq($sum(I!8, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I!8))) = tb2t6(nil(tuple2(key, a))))) <=> ((~$greatereq(I!8, 0)) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I!8))) = tb2t6(nil(tuple2(key, a)))))),
% 0.19/0.53 inference(rewrite,[status(thm)],[])).
% 0.19/0.53 tff(186,plain,
% 0.19/0.53 (((~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | ((~$greatereq(I!8, 0)) | $greatereq($sum(I!8, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I!8))) = tb2t6(nil(tuple2(key, a)))))) <=> ((~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | ((~$greatereq(I!8, 0)) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I!8))) = tb2t6(nil(tuple2(key, a))))))),
% 0.19/0.53 inference(monotonicity,[status(thm)],[185])).
% 0.19/0.53 tff(187,plain,
% 0.19/0.53 (((~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | ((~$greatereq(I!8, 0)) | $greatereq($sum(I!8, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I!8))) = tb2t6(nil(tuple2(key, a)))))) <=> ((~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | (~$greatereq(I!8, 0)) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I!8))) = tb2t6(nil(tuple2(key, a)))))),
% 0.19/0.53 inference(transitivity,[status(thm)],[186, 184])).
% 0.19/0.53 tff(188,plain,
% 0.19/0.53 ((~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | ((~$greatereq(I!8, 0)) | $greatereq($sum(I!8, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I!8))) = tb2t6(nil(tuple2(key, a)))))),
% 0.19/0.53 inference(quant_inst,[status(thm)],[])).
% 0.19/0.53 tff(189,plain,
% 0.19/0.53 ((~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | (~$greatereq(I!8, 0)) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I!8))) = tb2t6(nil(tuple2(key, a))))),
% 0.19/0.53 inference(modus_ponens,[status(thm)],[188, 187])).
% 0.19/0.53 tff(190,plain,
% 0.19/0.53 (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I!8))) = tb2t6(nil(tuple2(key, a)))),
% 0.19/0.53 inference(unit_resolution,[status(thm)],[189, 183, 181, 178])).
% 0.19/0.53 tff(191,plain,
% 0.19/0.53 (tb2t6(nil(tuple2(key, a))) = tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I!8)))),
% 0.19/0.53 inference(symmetry,[status(thm)],[190])).
% 0.19/0.53 tff(192,plain,
% 0.19/0.53 (tb2t6(nil(tuple2(key, a))) = tb2t6(get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(I!8)))),
% 0.19/0.53 inference(transitivity,[status(thm)],[191, 176])).
% 0.19/0.53 tff(193,plain,
% 0.19/0.53 (t2tb6(tb2t6(nil(tuple2(key, a)))) = t2tb6(tb2t6(get(list(tuple2(key, a)), int, elts(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))), t2tb(I!8))))),
% 0.19/0.53 inference(monotonicity,[status(thm)],[192])).
% 0.19/0.53 tff(194,plain,
% 0.19/0.53 (nil(tuple2(key, a)) = get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)),
% 0.19/0.53 inference(transitivity,[status(thm)],[105, 193, 172, 170])).
% 0.19/0.53 tff(195,plain,
% 0.19/0.53 (mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), nil(tuple2(key, a))) <=> mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))),
% 0.19/0.53 inference(monotonicity,[status(thm)],[194])).
% 0.19/0.53 tff(196,plain,
% 0.19/0.53 (mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)) <=> mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), nil(tuple2(key, a)))),
% 0.19/0.53 inference(symmetry,[status(thm)],[195])).
% 0.19/0.53 tff(197,plain,
% 0.19/0.53 (^[A: ty, D: uni, I: $int] : refl((~((~((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))) | (~((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I))))) <=> (~((~((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))) | (~((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I))))))),
% 0.19/0.53 inference(bind,[status(th)],[])).
% 0.19/0.53 tff(198,plain,
% 0.19/0.53 (![A: ty, D: uni, I: $int] : (~((~((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))) | (~((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I))))) <=> ![A: ty, D: uni, I: $int] : (~((~((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))) | (~((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I)))))),
% 0.19/0.53 inference(quant_intro,[status(thm)],[197])).
% 0.19/0.53 tff(199,plain,
% 0.19/0.53 (^[A: ty, D: uni, I: $int] : rewrite((~((~((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))) | (~((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I))))) <=> (~((~((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))) | (~((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I))))))),
% 0.19/0.53 inference(bind,[status(th)],[])).
% 0.19/0.53 tff(200,plain,
% 0.19/0.53 (![A: ty, D: uni, I: $int] : (~((~((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))) | (~((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I))))) <=> ![A: ty, D: uni, I: $int] : (~((~((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))) | (~((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I)))))),
% 0.19/0.53 inference(quant_intro,[status(thm)],[199])).
% 0.19/0.53 tff(201,plain,
% 0.19/0.53 (![A: ty, D: uni, I: $int] : (~((~((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))) | (~((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I))))) <=> ![A: ty, D: uni, I: $int] : (~((~((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))) | (~((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I)))))),
% 0.19/0.53 inference(transitivity,[status(thm)],[200, 198])).
% 0.19/0.53 tff(202,plain,
% 0.19/0.53 (^[A: ty, D: uni, I: $int] : trans(monotonicity(rewrite(((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0))) <=> ((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))), ((((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0))) & ((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I))) <=> (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0))) & ((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I))))), rewrite((((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0))) & ((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I))) <=> (~((~((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))) | (~((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I)))))), ((((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0))) & ((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I))) <=> (~((~((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))) | (~((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I)))))))),
% 0.19/0.53 inference(bind,[status(th)],[])).
% 0.19/0.53 tff(203,plain,
% 0.19/0.53 (![A: ty, D: uni, I: $int] : (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0))) & ((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I))) <=> ![A: ty, D: uni, I: $int] : (~((~((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))) | (~((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I)))))),
% 0.19/0.53 inference(quant_intro,[status(thm)],[202])).
% 0.19/0.53 tff(204,plain,
% 0.19/0.53 (^[A: ty, D: uni, I: $int] : rewrite((((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(bucket1(K, length1(list(tuple2(key, A)), D)), $product(-1, I)) = 0))) & ((~![K: key1, V: uni] : ((~sort1(A, V)) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(bucket1(K, length1(list(tuple2(key, A)), D)), $product(-1, I)) = 0))) | good_hash1(A, D, I))) <=> (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0))) & ((~![K: key1, V: uni] : ((~sort1(A, V)) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I))))),
% 0.19/0.53 inference(bind,[status(th)],[])).
% 0.19/0.53 tff(205,plain,
% 0.19/0.53 (![A: ty, D: uni, I: $int] : (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(bucket1(K, length1(list(tuple2(key, A)), D)), $product(-1, I)) = 0))) & ((~![K: key1, V: uni] : ((~sort1(A, V)) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(bucket1(K, length1(list(tuple2(key, A)), D)), $product(-1, I)) = 0))) | good_hash1(A, D, I))) <=> ![A: ty, D: uni, I: $int] : (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0))) & ((~![K: key1, V: uni] : ((~sort1(A, V)) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I)))),
% 0.19/0.53 inference(quant_intro,[status(thm)],[204])).
% 0.19/0.53 tff(206,plain,
% 0.19/0.53 (^[A: ty, D: uni, I: $int] : rewrite((((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | (bucket1(K, length1(list(tuple2(key, A)), D)) = I))) & ((~![K: key1, V: uni] : ((bucket1(K, length1(list(tuple2(key, A)), D)) = I) | (~sort1(A, V)) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))))) | good_hash1(A, D, I))) <=> (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(bucket1(K, length1(list(tuple2(key, A)), D)), $product(-1, I)) = 0))) & ((~![K: key1, V: uni] : ((~sort1(A, V)) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(bucket1(K, length1(list(tuple2(key, A)), D)), $product(-1, I)) = 0))) | good_hash1(A, D, I))))),
% 0.19/0.53 inference(bind,[status(th)],[])).
% 0.19/0.53 tff(207,plain,
% 0.19/0.53 (![A: ty, D: uni, I: $int] : (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | (bucket1(K, length1(list(tuple2(key, A)), D)) = I))) & ((~![K: key1, V: uni] : ((bucket1(K, length1(list(tuple2(key, A)), D)) = I) | (~sort1(A, V)) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))))) | good_hash1(A, D, I))) <=> ![A: ty, D: uni, I: $int] : (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(bucket1(K, length1(list(tuple2(key, A)), D)), $product(-1, I)) = 0))) & ((~![K: key1, V: uni] : ((~sort1(A, V)) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(bucket1(K, length1(list(tuple2(key, A)), D)), $product(-1, I)) = 0))) | good_hash1(A, D, I)))),
% 0.19/0.53 inference(quant_intro,[status(thm)],[206])).
% 0.19/0.53 tff(208,plain,
% 0.19/0.53 (![A: ty, D: uni, I: $int] : (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | (bucket1(K, length1(list(tuple2(key, A)), D)) = I))) & ((~![K: key1, V: uni] : ((bucket1(K, length1(list(tuple2(key, A)), D)) = I) | (~sort1(A, V)) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))))) | good_hash1(A, D, I))) <=> ![A: ty, D: uni, I: $int] : (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | (bucket1(K, length1(list(tuple2(key, A)), D)) = I))) & ((~![K: key1, V: uni] : ((bucket1(K, length1(list(tuple2(key, A)), D)) = I) | (~sort1(A, V)) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))))) | good_hash1(A, D, I)))),
% 0.19/0.54 inference(rewrite,[status(thm)],[])).
% 0.19/0.54 tff(209,plain,
% 0.19/0.54 (^[A: ty, D: uni, I: $int] : rewrite(((good_hash1(A, D, I) => ![K: key1, V: uni] : (mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I)) => (bucket1(K, length1(list(tuple2(key, A)), D)) = I))) & (![K: key1, V: uni] : (sort1(A, V) => (mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I)) => (bucket1(K, length1(list(tuple2(key, A)), D)) = I))) => good_hash1(A, D, I))) <=> (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | (bucket1(K, length1(list(tuple2(key, A)), D)) = I))) & ((~![K: key1, V: uni] : ((bucket1(K, length1(list(tuple2(key, A)), D)) = I) | (~sort1(A, V)) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))))) | good_hash1(A, D, I))))),
% 0.19/0.54 inference(bind,[status(th)],[])).
% 0.19/0.54 tff(210,plain,
% 0.19/0.54 (![A: ty, D: uni, I: $int] : ((good_hash1(A, D, I) => ![K: key1, V: uni] : (mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I)) => (bucket1(K, length1(list(tuple2(key, A)), D)) = I))) & (![K: key1, V: uni] : (sort1(A, V) => (mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I)) => (bucket1(K, length1(list(tuple2(key, A)), D)) = I))) => good_hash1(A, D, I))) <=> ![A: ty, D: uni, I: $int] : (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | (bucket1(K, length1(list(tuple2(key, A)), D)) = I))) & ((~![K: key1, V: uni] : ((bucket1(K, length1(list(tuple2(key, A)), D)) = I) | (~sort1(A, V)) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))))) | good_hash1(A, D, I)))),
% 0.19/0.54 inference(quant_intro,[status(thm)],[209])).
% 0.19/0.54 tff(211,axiom,(![A: ty, D: uni, I: $int] : ((good_hash1(A, D, I) => ![K: key1, V: uni] : (mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I)) => (bucket1(K, length1(list(tuple2(key, A)), D)) = I))) & (![K: key1, V: uni] : (sort1(A, V) => (mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I)) => (bucket1(K, length1(list(tuple2(key, A)), D)) = I))) => good_hash1(A, D, I)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','good_hash_def')).
% 0.19/0.54 tff(212,plain,
% 0.19/0.54 (![A: ty, D: uni, I: $int] : (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | (bucket1(K, length1(list(tuple2(key, A)), D)) = I))) & ((~![K: key1, V: uni] : ((bucket1(K, length1(list(tuple2(key, A)), D)) = I) | (~sort1(A, V)) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))))) | good_hash1(A, D, I)))),
% 0.19/0.54 inference(modus_ponens,[status(thm)],[211, 210])).
% 0.19/0.54 tff(213,plain,
% 0.19/0.54 (![A: ty, D: uni, I: $int] : (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | (bucket1(K, length1(list(tuple2(key, A)), D)) = I))) & ((~![K: key1, V: uni] : ((bucket1(K, length1(list(tuple2(key, A)), D)) = I) | (~sort1(A, V)) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))))) | good_hash1(A, D, I)))),
% 0.19/0.54 inference(modus_ponens,[status(thm)],[212, 208])).
% 0.19/0.54 tff(214,plain,
% 0.19/0.54 (![A: ty, D: uni, I: $int] : (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(bucket1(K, length1(list(tuple2(key, A)), D)), $product(-1, I)) = 0))) & ((~![K: key1, V: uni] : ((~sort1(A, V)) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(bucket1(K, length1(list(tuple2(key, A)), D)), $product(-1, I)) = 0))) | good_hash1(A, D, I)))),
% 0.19/0.54 inference(modus_ponens,[status(thm)],[213, 207])).
% 0.19/0.54 tff(215,plain,
% 0.19/0.54 (![A: ty, D: uni, I: $int] : (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0))) & ((~![K: key1, V: uni] : ((~sort1(A, V)) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I)))),
% 0.19/0.54 inference(modus_ponens,[status(thm)],[214, 205])).
% 0.19/0.54 tff(216,plain,(
% 0.19/0.54 ![A: ty, D: uni, I: $int] : (((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0))) & ((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I)))),
% 0.19/0.54 inference(skolemize,[status(sab)],[215])).
% 0.19/0.54 tff(217,plain,
% 0.19/0.54 (![A: ty, D: uni, I: $int] : (~((~((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))) | (~((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I)))))),
% 0.19/0.54 inference(modus_ponens,[status(thm)],[216, 203])).
% 0.19/0.54 tff(218,plain,
% 0.19/0.54 (![A: ty, D: uni, I: $int] : (~((~((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))) | (~((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I)))))),
% 0.19/0.54 inference(modus_ponens,[status(thm)],[217, 201])).
% 0.19/0.54 tff(219,plain,
% 0.19/0.54 ((~![A: ty, D: uni, I: $int] : (~((~((~good_hash1(A, D, I)) | ![K: key1, V: uni] : ((~mem(tuple2(key, A), tuple21(key, A, t2tb1(K), V), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(K, length1(list(tuple2(key, A)), D)))) = 0)))) | (~((~((~sort1(A, tptp_fun_V_0(I, D, A))) | (~mem(tuple2(key, A), tuple21(key, A, t2tb1(tptp_fun_K_1(I, D, A)), tptp_fun_V_0(I, D, A)), get2(list(tuple2(key, A)), D, I))) | ($sum(I, $product(-1, bucket1(tptp_fun_K_1(I, D, A), length1(list(tuple2(key, A)), D)))) = 0))) | good_hash1(A, D, I)))))) | (~((~((~good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)) | ![K: key1, V: uni] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), V), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | ($sum(I!8, $product(-1, bucket1(K, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = 0)))) | (~((~((~sort1(a, tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a))) | (~mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | ($sum(I!8, $product(-1, bucket1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a), length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = 0))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)))))),
% 0.19/0.54 inference(quant_inst,[status(thm)],[])).
% 0.19/0.54 tff(220,plain,
% 0.19/0.54 (~((~((~good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)) | ![K: key1, V: uni] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), V), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | ($sum(I!8, $product(-1, bucket1(K, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = 0)))) | (~((~((~sort1(a, tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a))) | (~mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | ($sum(I!8, $product(-1, bucket1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a), length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = 0))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))))),
% 0.19/0.54 inference(unit_resolution,[status(thm)],[219, 218])).
% 0.19/0.54 tff(221,plain,
% 0.19/0.54 (((~((~good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)) | ![K: key1, V: uni] : ((~mem(tuple2(key, a), tuple21(key, a, t2tb1(K), V), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | ($sum(I!8, $product(-1, bucket1(K, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = 0)))) | (~((~((~sort1(a, tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a))) | (~mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | ($sum(I!8, $product(-1, bucket1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a), length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = 0))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)))) | ((~((~sort1(a, tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a))) | (~mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | ($sum(I!8, $product(-1, bucket1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a), length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = 0))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))),
% 0.19/0.54 inference(tautology,[status(thm)],[])).
% 0.19/0.54 tff(222,plain,
% 0.19/0.54 ((~((~sort1(a, tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a))) | (~mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | ($sum(I!8, $product(-1, bucket1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a), length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = 0))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)),
% 0.19/0.54 inference(unit_resolution,[status(thm)],[221, 220])).
% 0.19/0.54 tff(223,plain,
% 0.19/0.54 ((good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0))) | (~good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))),
% 0.19/0.54 inference(tautology,[status(thm)],[])).
% 0.19/0.54 tff(224,plain,
% 0.19/0.54 (~good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)),
% 0.19/0.54 inference(unit_resolution,[status(thm)],[223, 179])).
% 0.19/0.54 tff(225,plain,
% 0.19/0.54 ((~((~((~sort1(a, tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a))) | (~mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | ($sum(I!8, $product(-1, bucket1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a), length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = 0))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | (~((~sort1(a, tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a))) | (~mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | ($sum(I!8, $product(-1, bucket1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a), length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = 0))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8)),
% 0.19/0.54 inference(tautology,[status(thm)],[])).
% 0.19/0.54 tff(226,plain,
% 0.19/0.54 ((~((~((~sort1(a, tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a))) | (~mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | ($sum(I!8, $product(-1, bucket1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a), length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = 0))) | good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | (~((~sort1(a, tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a))) | (~mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | ($sum(I!8, $product(-1, bucket1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a), length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = 0)))),
% 0.19/0.54 inference(unit_resolution,[status(thm)],[225, 224])).
% 0.19/0.54 tff(227,plain,
% 0.19/0.54 (~((~sort1(a, tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a))) | (~mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | ($sum(I!8, $product(-1, bucket1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a), length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = 0))),
% 0.19/0.54 inference(unit_resolution,[status(thm)],[226, 222])).
% 0.19/0.54 tff(228,plain,
% 0.19/0.54 (((~sort1(a, tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a))) | (~mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))) | ($sum(I!8, $product(-1, bucket1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a), length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) = 0)) | mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))),
% 0.19/0.55 inference(tautology,[status(thm)],[])).
% 0.19/0.55 tff(229,plain,
% 0.19/0.55 (mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8))),
% 0.19/0.55 inference(unit_resolution,[status(thm)],[228, 227])).
% 0.19/0.55 tff(230,plain,
% 0.19/0.55 (mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), nil(tuple2(key, a)))),
% 0.19/0.55 inference(modus_ponens,[status(thm)],[229, 196])).
% 0.19/0.55 tff(231,plain,
% 0.19/0.55 (^[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)))),
% 0.19/0.55 inference(bind,[status(th)],[])).
% 0.19/0.55 tff(232,plain,
% 0.19/0.55 (![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))),
% 0.19/0.55 inference(quant_intro,[status(thm)],[231])).
% 0.19/0.55 tff(233,plain,
% 0.19/0.55 (![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))),
% 0.19/0.55 inference(rewrite,[status(thm)],[])).
% 0.19/0.55 tff(234,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_sort1')).
% 0.19/0.55 tff(235,plain,
% 0.19/0.55 (![A: ty, A1: ty, X: uni, X1: uni] : sort1(tuple2(A1, A), tuple21(A1, A, X, X1))),
% 0.19/0.55 inference(modus_ponens,[status(thm)],[234, 233])).
% 0.19/0.55 tff(236,plain,(
% 0.19/0.55 ![A: ty, A1: ty, X: uni, X1: uni] : sort1(tuple2(A1, A), tuple21(A1, A, X, X1))),
% 0.19/0.55 inference(skolemize,[status(sab)],[235])).
% 0.19/0.55 tff(237,plain,
% 0.19/0.55 (![A: ty, A1: ty, X: uni, X1: uni] : sort1(tuple2(A1, A), tuple21(A1, A, X, X1))),
% 0.19/0.55 inference(modus_ponens,[status(thm)],[236, 232])).
% 0.19/0.55 tff(238,plain,
% 0.19/0.55 ((~![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(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)))),
% 0.19/0.55 inference(quant_inst,[status(thm)],[])).
% 0.19/0.55 tff(239,plain,
% 0.19/0.55 (sort1(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)))),
% 0.19/0.55 inference(unit_resolution,[status(thm)],[238, 237])).
% 0.19/0.55 tff(240,plain,
% 0.19/0.55 (^[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)))))))))),
% 0.19/0.55 inference(bind,[status(th)],[])).
% 0.19/0.55 tff(241,plain,
% 0.19/0.55 (![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))))))))),
% 0.19/0.55 inference(quant_intro,[status(thm)],[240])).
% 0.19/0.55 tff(242,plain,
% 0.19/0.55 (^[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)))))))))),
% 0.19/0.55 inference(bind,[status(th)],[])).
% 0.19/0.55 tff(243,plain,
% 0.19/0.55 (![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))))))))),
% 0.19/0.55 inference(quant_intro,[status(thm)],[242])).
% 0.19/0.55 tff(244,plain,
% 0.19/0.55 (![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))))))))),
% 0.19/0.55 inference(transitivity,[status(thm)],[243, 241])).
% 0.19/0.55 tff(245,plain,
% 0.19/0.55 (^[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)))))))))),
% 0.19/0.55 inference(bind,[status(th)],[])).
% 0.19/0.55 tff(246,plain,
% 0.19/0.55 (![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))))))))),
% 0.19/0.55 inference(quant_intro,[status(thm)],[245])).
% 0.19/0.55 tff(247,plain,
% 0.19/0.55 (![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))))))),
% 0.19/0.55 inference(rewrite,[status(thm)],[])).
% 0.19/0.55 tff(248,plain,
% 0.19/0.55 (^[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))))))))),
% 0.19/0.55 inference(bind,[status(th)],[])).
% 0.19/0.55 tff(249,plain,
% 0.19/0.55 (![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))))))),
% 0.19/0.55 inference(quant_intro,[status(thm)],[248])).
% 0.19/0.55 tff(250,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')).
% 0.19/0.55 tff(251,plain,
% 0.19/0.55 (![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))))))),
% 0.19/0.55 inference(modus_ponens,[status(thm)],[250, 249])).
% 0.19/0.55 tff(252,plain,
% 0.19/0.55 (![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))))))),
% 0.19/0.55 inference(modus_ponens,[status(thm)],[251, 247])).
% 0.19/0.55 tff(253,plain,(
% 0.19/0.55 ![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))))))),
% 0.19/0.55 inference(skolemize,[status(sab)],[252])).
% 0.19/0.55 tff(254,plain,
% 0.19/0.55 (![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))))))))),
% 0.19/0.55 inference(modus_ponens,[status(thm)],[253, 246])).
% 0.19/0.55 tff(255,plain,
% 0.19/0.55 (![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))))))))),
% 0.19/0.55 inference(modus_ponens,[status(thm)],[254, 244])).
% 0.19/0.55 tff(256,plain,
% 0.19/0.55 (((~![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(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)))) | (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), 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(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)))) | (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), X2))))))))),
% 0.19/0.55 inference(rewrite,[status(thm)],[])).
% 0.19/0.55 tff(257,plain,
% 0.19/0.55 ((~![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(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)))) | (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), X2))))))))),
% 0.19/0.55 inference(quant_inst,[status(thm)],[])).
% 0.19/0.55 tff(258,plain,
% 0.19/0.55 ((~![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(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)))) | (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), X2)))))))),
% 0.19/0.55 inference(modus_ponens,[status(thm)],[257, 256])).
% 0.19/0.55 tff(259,plain,
% 0.19/0.55 (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), X2))))))),
% 0.19/0.55 inference(unit_resolution,[status(thm)],[258, 255, 239])).
% 0.19/0.55 tff(260,plain,
% 0.19/0.55 ((mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), X2)))))) | (~mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), nil(tuple2(key, a))))),
% 0.19/0.55 inference(tautology,[status(thm)],[])).
% 0.19/0.55 tff(261,plain,
% 0.19/0.55 (~mem(tuple2(key, a), tuple21(key, a, t2tb1(tptp_fun_K_1(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), tptp_fun_V_0(I!8, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), a)), nil(tuple2(key, a)))),
% 0.19/0.56 inference(unit_resolution,[status(thm)],[260, 259])).
% 0.19/0.56 tff(262,plain,
% 0.19/0.56 ($false),
% 0.19/0.56 inference(unit_resolution,[status(thm)],[261, 230])).
% 0.19/0.56 tff(263,plain,(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.56 tff(264,plain,
% 0.19/0.56 (((~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0)))))) | (~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a))))))) | ($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0)))))),
% 0.19/0.56 inference(tautology,[status(thm)],[])).
% 0.19/0.56 tff(265,plain,
% 0.19/0.56 ($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0))))),
% 0.19/0.56 inference(unit_resolution,[status(thm)],[264, 135])).
% 0.19/0.56 tff(266,plain,
% 0.19/0.56 ((~($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0)))))) | $lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0))))),
% 0.19/0.56 inference(tautology,[status(thm)],[])).
% 0.19/0.56 tff(267,plain,
% 0.19/0.56 ((~($lesseq(H!4, 0) | (~$greatereq(H!4, 0)) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0)))))) | (~(Rho1!7 = tb2t3(const(option(a), key, none(a))))) | (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0))))),
% 0.19/0.56 inference(unit_resolution,[status(thm)],[266, 21, 127])).
% 0.19/0.56 tff(268,plain,
% 0.19/0.56 ((~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (~(good_hash1(a, mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), I!8) | $lesseq($sum(H!4, $product(-1, I!8)), 0) | (~$greatereq(I!8, 0))))),
% 0.19/0.56 inference(unit_resolution,[status(thm)],[267, 137, 265])).
% 0.19/0.56 tff(269,plain,
% 0.19/0.56 (~good_data1(a, K!10, t2tb7(V!9), t2tb3(Rho1!7), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))),
% 0.19/0.56 inference(unit_resolution,[status(thm)],[268, 263])).
% 0.19/0.56 tff(270,plain,
% 0.19/0.56 (~good_data1(a, K!10, t2tb7(V!9), const(option(a), key, none(a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))),
% 0.19/0.56 inference(modus_ponens,[status(thm)],[269, 167])).
% 0.19/0.56 tff(271,plain,
% 0.19/0.56 ((~(good_data1(a, K!10, t2tb7(V!9), const(option(a), key, none(a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))) <=> ((get(option(a), key, const(option(a), key, none(a)), t2tb1(K!10)) = some(a, t2tb7(V!9))) <=> in_data1(a, K!10, t2tb7(V!9), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))) | good_data1(a, K!10, t2tb7(V!9), const(option(a), key, none(a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))) | (~((get(option(a), key, const(option(a), key, none(a)), t2tb1(K!10)) = some(a, t2tb7(V!9))) <=> in_data1(a, K!10, t2tb7(V!9), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))),
% 0.19/0.56 inference(tautology,[status(thm)],[])).
% 0.19/0.56 tff(272,plain,
% 0.19/0.56 (~((get(option(a), key, const(option(a), key, none(a)), t2tb1(K!10)) = some(a, t2tb7(V!9))) <=> in_data1(a, K!10, t2tb7(V!9), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))),
% 0.19/0.56 inference(unit_resolution,[status(thm)],[271, 270, 126])).
% 0.19/0.56 tff(273,plain,
% 0.19/0.56 (^[A: ty] : refl(sort1(option(A), none(A)) <=> sort1(option(A), none(A)))),
% 0.19/0.56 inference(bind,[status(th)],[])).
% 0.19/0.56 tff(274,plain,
% 0.19/0.56 (![A: ty] : sort1(option(A), none(A)) <=> ![A: ty] : sort1(option(A), none(A))),
% 0.19/0.56 inference(quant_intro,[status(thm)],[273])).
% 0.19/0.56 tff(275,plain,
% 0.19/0.56 (![A: ty] : sort1(option(A), none(A)) <=> ![A: ty] : sort1(option(A), none(A))),
% 0.19/0.56 inference(rewrite,[status(thm)],[])).
% 0.19/0.56 tff(276,axiom,(![A: ty] : sort1(option(A), none(A))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','none_sort1')).
% 0.19/0.56 tff(277,plain,
% 0.19/0.56 (![A: ty] : sort1(option(A), none(A))),
% 0.19/0.56 inference(modus_ponens,[status(thm)],[276, 275])).
% 0.19/0.56 tff(278,plain,(
% 0.19/0.56 ![A: ty] : sort1(option(A), none(A))),
% 0.19/0.56 inference(skolemize,[status(sab)],[277])).
% 0.19/0.56 tff(279,plain,
% 0.19/0.56 (![A: ty] : sort1(option(A), none(A))),
% 0.19/0.56 inference(modus_ponens,[status(thm)],[278, 274])).
% 0.19/0.56 tff(280,plain,
% 0.19/0.56 ((~![A: ty] : sort1(option(A), none(A))) | sort1(option(a), none(a))),
% 0.19/0.56 inference(quant_inst,[status(thm)],[])).
% 0.19/0.56 tff(281,plain,
% 0.19/0.56 (sort1(option(a), none(a))),
% 0.19/0.56 inference(unit_resolution,[status(thm)],[280, 279])).
% 0.19/0.56 tff(282,plain,
% 0.19/0.56 (^[A: ty, B: ty, B1: uni, A1: uni] : refl(((~sort1(B, B1)) | (get(B, A, const(B, A, B1), A1) = B1)) <=> ((~sort1(B, B1)) | (get(B, A, const(B, A, B1), A1) = B1)))),
% 0.19/0.56 inference(bind,[status(th)],[])).
% 0.19/0.56 tff(283,plain,
% 0.19/0.56 (![A: ty, B: ty, B1: uni, A1: uni] : ((~sort1(B, B1)) | (get(B, A, const(B, A, B1), A1) = B1)) <=> ![A: ty, B: ty, B1: uni, A1: uni] : ((~sort1(B, B1)) | (get(B, A, const(B, A, B1), A1) = B1))),
% 0.19/0.56 inference(quant_intro,[status(thm)],[282])).
% 0.19/0.56 tff(284,plain,
% 0.19/0.56 (![A: ty, B: ty, B1: uni, A1: uni] : ((~sort1(B, B1)) | (get(B, A, const(B, A, B1), A1) = B1)) <=> ![A: ty, B: ty, B1: uni, A1: uni] : ((~sort1(B, B1)) | (get(B, A, const(B, A, B1), A1) = B1))),
% 0.19/0.56 inference(rewrite,[status(thm)],[])).
% 0.19/0.56 tff(285,plain,
% 0.19/0.56 (^[A: ty, B: ty, B1: uni, A1: uni] : rewrite((sort1(B, B1) => (get(B, A, const(B, A, B1), A1) = B1)) <=> ((~sort1(B, B1)) | (get(B, A, const(B, A, B1), A1) = B1)))),
% 0.19/0.56 inference(bind,[status(th)],[])).
% 0.19/0.56 tff(286,plain,
% 0.19/0.56 (![A: ty, B: ty, B1: uni, A1: uni] : (sort1(B, B1) => (get(B, A, const(B, A, B1), A1) = B1)) <=> ![A: ty, B: ty, B1: uni, A1: uni] : ((~sort1(B, B1)) | (get(B, A, const(B, A, B1), A1) = B1))),
% 0.19/0.56 inference(quant_intro,[status(thm)],[285])).
% 0.19/0.56 tff(287,axiom,(![A: ty, B: ty, B1: uni, A1: uni] : (sort1(B, B1) => (get(B, A, const(B, A, B1), A1) = B1))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','const')).
% 0.19/0.56 tff(288,plain,
% 0.19/0.56 (![A: ty, B: ty, B1: uni, A1: uni] : ((~sort1(B, B1)) | (get(B, A, const(B, A, B1), A1) = B1))),
% 0.19/0.56 inference(modus_ponens,[status(thm)],[287, 286])).
% 0.19/0.56 tff(289,plain,
% 0.19/0.56 (![A: ty, B: ty, B1: uni, A1: uni] : ((~sort1(B, B1)) | (get(B, A, const(B, A, B1), A1) = B1))),
% 0.19/0.56 inference(modus_ponens,[status(thm)],[288, 284])).
% 0.19/0.56 tff(290,plain,(
% 0.19/0.56 ![A: ty, B: ty, B1: uni, A1: uni] : ((~sort1(B, B1)) | (get(B, A, const(B, A, B1), A1) = B1))),
% 0.19/0.56 inference(skolemize,[status(sab)],[289])).
% 0.19/0.56 tff(291,plain,
% 0.19/0.56 (![A: ty, B: ty, B1: uni, A1: uni] : ((~sort1(B, B1)) | (get(B, A, const(B, A, B1), A1) = B1))),
% 0.19/0.56 inference(modus_ponens,[status(thm)],[290, 283])).
% 0.19/0.56 tff(292,plain,
% 0.19/0.56 (((~![A: ty, B: ty, B1: uni, A1: uni] : ((~sort1(B, B1)) | (get(B, A, const(B, A, B1), A1) = B1))) | ((~sort1(option(a), none(a))) | (get(option(a), key, const(option(a), key, none(a)), t2tb1(K!10)) = none(a)))) <=> ((~![A: ty, B: ty, B1: uni, A1: uni] : ((~sort1(B, B1)) | (get(B, A, const(B, A, B1), A1) = B1))) | (~sort1(option(a), none(a))) | (get(option(a), key, const(option(a), key, none(a)), t2tb1(K!10)) = none(a)))),
% 0.19/0.56 inference(rewrite,[status(thm)],[])).
% 0.19/0.56 tff(293,plain,
% 0.19/0.56 ((~![A: ty, B: ty, B1: uni, A1: uni] : ((~sort1(B, B1)) | (get(B, A, const(B, A, B1), A1) = B1))) | ((~sort1(option(a), none(a))) | (get(option(a), key, const(option(a), key, none(a)), t2tb1(K!10)) = none(a)))),
% 0.19/0.56 inference(quant_inst,[status(thm)],[])).
% 0.19/0.56 tff(294,plain,
% 0.19/0.56 ((~![A: ty, B: ty, B1: uni, A1: uni] : ((~sort1(B, B1)) | (get(B, A, const(B, A, B1), A1) = B1))) | (~sort1(option(a), none(a))) | (get(option(a), key, const(option(a), key, none(a)), t2tb1(K!10)) = none(a))),
% 0.19/0.56 inference(modus_ponens,[status(thm)],[293, 292])).
% 0.19/0.56 tff(295,plain,
% 0.19/0.56 (get(option(a), key, const(option(a), key, none(a)), t2tb1(K!10)) = none(a)),
% 0.19/0.56 inference(unit_resolution,[status(thm)],[294, 291, 281])).
% 0.19/0.56 tff(296,plain,
% 0.19/0.56 ((get(option(a), key, const(option(a), key, none(a)), t2tb1(K!10)) = some(a, t2tb7(V!9))) <=> (none(a) = some(a, t2tb7(V!9)))),
% 0.19/0.56 inference(monotonicity,[status(thm)],[295])).
% 0.19/0.56 tff(297,plain,
% 0.19/0.56 ((none(a) = some(a, t2tb7(V!9))) <=> (get(option(a), key, const(option(a), key, none(a)), t2tb1(K!10)) = some(a, t2tb7(V!9)))),
% 0.19/0.56 inference(symmetry,[status(thm)],[296])).
% 0.19/0.56 tff(298,plain,
% 0.19/0.56 ((~(none(a) = some(a, t2tb7(V!9)))) <=> (~(get(option(a), key, const(option(a), key, none(a)), t2tb1(K!10)) = some(a, t2tb7(V!9))))),
% 0.19/0.56 inference(monotonicity,[status(thm)],[297])).
% 0.19/0.56 tff(299,plain,
% 0.19/0.56 (^[A: ty, V: uni] : refl((~(none(A) = some(A, V))) <=> (~(none(A) = some(A, V))))),
% 0.19/0.56 inference(bind,[status(th)],[])).
% 0.19/0.56 tff(300,plain,
% 0.19/0.56 (![A: ty, V: uni] : (~(none(A) = some(A, V))) <=> ![A: ty, V: uni] : (~(none(A) = some(A, V)))),
% 0.19/0.56 inference(quant_intro,[status(thm)],[299])).
% 0.19/0.56 tff(301,plain,
% 0.19/0.56 (![A: ty, V: uni] : (~(none(A) = some(A, V))) <=> ![A: ty, V: uni] : (~(none(A) = some(A, V)))),
% 0.19/0.56 inference(rewrite,[status(thm)],[])).
% 0.19/0.56 tff(302,axiom,(![A: ty, V: uni] : (~(none(A) = some(A, V)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','none_Some1')).
% 0.19/0.56 tff(303,plain,
% 0.19/0.56 (![A: ty, V: uni] : (~(none(A) = some(A, V)))),
% 0.19/0.56 inference(modus_ponens,[status(thm)],[302, 301])).
% 0.19/0.56 tff(304,plain,(
% 0.19/0.56 ![A: ty, V: uni] : (~(none(A) = some(A, V)))),
% 0.19/0.56 inference(skolemize,[status(sab)],[303])).
% 0.19/0.56 tff(305,plain,
% 0.19/0.56 (![A: ty, V: uni] : (~(none(A) = some(A, V)))),
% 0.19/0.56 inference(modus_ponens,[status(thm)],[304, 300])).
% 0.19/0.56 tff(306,plain,
% 0.19/0.56 ((~![A: ty, V: uni] : (~(none(A) = some(A, V)))) | (~(none(a) = some(a, t2tb7(V!9))))),
% 0.19/0.56 inference(quant_inst,[status(thm)],[])).
% 0.19/0.56 tff(307,plain,
% 0.19/0.56 (~(none(a) = some(a, t2tb7(V!9)))),
% 0.19/0.56 inference(unit_resolution,[status(thm)],[306, 305])).
% 0.19/0.56 tff(308,plain,
% 0.19/0.56 (~(get(option(a), key, const(option(a), key, none(a)), t2tb1(K!10)) = some(a, t2tb7(V!9)))),
% 0.19/0.56 inference(modus_ponens,[status(thm)],[307, 298])).
% 0.19/0.56 tff(309,plain,
% 0.19/0.56 (((get(option(a), key, const(option(a), key, none(a)), t2tb1(K!10)) = some(a, t2tb7(V!9))) <=> in_data1(a, K!10, t2tb7(V!9), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | (get(option(a), key, const(option(a), key, none(a)), t2tb1(K!10)) = some(a, t2tb7(V!9))) | in_data1(a, K!10, t2tb7(V!9), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))),
% 0.19/0.56 inference(tautology,[status(thm)],[])).
% 0.19/0.56 tff(310,plain,
% 0.19/0.56 (in_data1(a, K!10, t2tb7(V!9), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))),
% 0.19/0.56 inference(unit_resolution,[status(thm)],[309, 308, 272])).
% 0.19/0.56 tff(311,plain,
% 0.19/0.56 ((~(in_data1(a, K!10, t2tb7(V!9), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))) <=> mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))))) | (~in_data1(a, K!10, t2tb7(V!9), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) | mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))),
% 0.19/0.56 inference(tautology,[status(thm)],[])).
% 0.19/0.56 tff(312,plain,
% 0.19/0.56 (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), get2(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)), bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))),
% 0.19/0.56 inference(unit_resolution,[status(thm)],[311, 310, 117])).
% 0.19/0.56 tff(313,plain,
% 0.19/0.56 (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), nil(tuple2(key, a)))),
% 0.19/0.56 inference(modus_ponens,[status(thm)],[312, 108])).
% 0.19/0.56 tff(314,plain,
% 0.19/0.56 ((~![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!10), t2tb7(V!9)))),
% 0.19/0.56 inference(quant_inst,[status(thm)],[])).
% 0.19/0.56 tff(315,plain,
% 0.19/0.56 (sort1(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)))),
% 0.19/0.56 inference(unit_resolution,[status(thm)],[314, 237])).
% 0.19/0.56 tff(316,plain,
% 0.19/0.56 (((~![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!10), t2tb7(V!9)))) | (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(K!10), t2tb7(V!9)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), 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!10), t2tb7(V!9)))) | (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(K!10), t2tb7(V!9)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), X2))))))))),
% 0.19/0.56 inference(rewrite,[status(thm)],[])).
% 0.19/0.56 tff(317,plain,
% 0.19/0.56 ((~![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!10), t2tb7(V!9)))) | (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(K!10), t2tb7(V!9)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), X2))))))))),
% 0.19/0.56 inference(quant_inst,[status(thm)],[])).
% 0.19/0.56 tff(318,plain,
% 0.19/0.56 ((~![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!10), t2tb7(V!9)))) | (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(K!10), t2tb7(V!9)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), X2)))))))),
% 0.19/0.56 inference(modus_ponens,[status(thm)],[317, 316])).
% 0.19/0.56 tff(319,plain,
% 0.19/0.56 (~(mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(K!10), t2tb7(V!9)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), X2))))))),
% 0.19/0.57 inference(unit_resolution,[status(thm)],[318, 255, 315])).
% 0.19/0.57 tff(320,plain,
% 0.19/0.57 ((mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), nil(tuple2(key, a))) | (~![X1: uni, X2: uni] : ((~sort1(tuple2(key, a), X1)) | (mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), cons(tuple2(key, a), X1, X2)) <=> ((tuple21(key, a, t2tb1(K!10), t2tb7(V!9)) = X1) | mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), X2)))))) | (~mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), nil(tuple2(key, a))))),
% 0.19/0.57 inference(tautology,[status(thm)],[])).
% 0.19/0.57 tff(321,plain,
% 0.19/0.57 (~mem(tuple2(key, a), tuple21(key, a, t2tb1(K!10), t2tb7(V!9)), nil(tuple2(key, a)))),
% 0.19/0.57 inference(unit_resolution,[status(thm)],[320, 319])).
% 0.19/0.57 tff(322,plain,
% 0.19/0.57 ($false),
% 0.19/0.57 inference(unit_resolution,[status(thm)],[321, 313])).
% 0.19/0.57 tff(323,plain,(~(tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))) = tb2t6(nil(tuple2(key, a))))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.57 tff(324,plain,
% 0.19/0.57 (((~$greatereq(bucket1(K!10, H!4), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!10, H!4))), 0)) | $greatereq(bucket1(K!10, H!4), 0)),
% 0.19/0.57 inference(tautology,[status(thm)],[])).
% 0.19/0.57 tff(325,plain,
% 0.19/0.57 ($greatereq(bucket1(K!10, H!4), 0)),
% 0.19/0.57 inference(unit_resolution,[status(thm)],[324, 49])).
% 0.19/0.57 tff(326,plain,
% 0.19/0.57 (^[A: ty, U: $int, U1: uni] : refl(($sum(U, $product(-1, length1(A, mk_array1(A, U, U1)))) = 0) <=> ($sum(U, $product(-1, length1(A, mk_array1(A, U, U1)))) = 0))),
% 0.19/0.57 inference(bind,[status(th)],[])).
% 0.19/0.57 tff(327,plain,
% 0.19/0.57 (![A: ty, U: $int, U1: uni] : ($sum(U, $product(-1, length1(A, mk_array1(A, U, U1)))) = 0) <=> ![A: ty, U: $int, U1: uni] : ($sum(U, $product(-1, length1(A, mk_array1(A, U, U1)))) = 0)),
% 0.19/0.57 inference(quant_intro,[status(thm)],[326])).
% 0.19/0.57 tff(328,plain,
% 0.19/0.57 (^[A: ty, U: $int, U1: uni] : trans(monotonicity(rewrite($sum(length1(A, mk_array1(A, U, U1)), $product(-1, U)) = $sum($product(-1, U), length1(A, mk_array1(A, U, U1)))), (($sum(length1(A, mk_array1(A, U, U1)), $product(-1, U)) = 0) <=> ($sum($product(-1, U), length1(A, mk_array1(A, U, U1))) = 0))), rewrite(($sum($product(-1, U), length1(A, mk_array1(A, U, U1))) = 0) <=> ($sum(U, $product(-1, length1(A, mk_array1(A, U, U1)))) = 0)), (($sum(length1(A, mk_array1(A, U, U1)), $product(-1, U)) = 0) <=> ($sum(U, $product(-1, length1(A, mk_array1(A, U, U1)))) = 0)))),
% 0.19/0.57 inference(bind,[status(th)],[])).
% 0.19/0.57 tff(329,plain,
% 0.19/0.57 (![A: ty, U: $int, U1: uni] : ($sum(length1(A, mk_array1(A, U, U1)), $product(-1, U)) = 0) <=> ![A: ty, U: $int, U1: uni] : ($sum(U, $product(-1, length1(A, mk_array1(A, U, U1)))) = 0)),
% 0.19/0.57 inference(quant_intro,[status(thm)],[328])).
% 0.19/0.57 tff(330,plain,
% 0.19/0.57 (^[A: ty, U: $int, U1: uni] : rewrite((length1(A, mk_array1(A, U, U1)) = U) <=> ($sum(length1(A, mk_array1(A, U, U1)), $product(-1, U)) = 0))),
% 0.19/0.57 inference(bind,[status(th)],[])).
% 0.19/0.57 tff(331,plain,
% 0.19/0.57 (![A: ty, U: $int, U1: uni] : (length1(A, mk_array1(A, U, U1)) = U) <=> ![A: ty, U: $int, U1: uni] : ($sum(length1(A, mk_array1(A, U, U1)), $product(-1, U)) = 0)),
% 0.19/0.57 inference(quant_intro,[status(thm)],[330])).
% 0.19/0.57 tff(332,plain,
% 0.19/0.57 (![A: ty, U: $int, U1: uni] : (length1(A, mk_array1(A, U, U1)) = U) <=> ![A: ty, U: $int, U1: uni] : (length1(A, mk_array1(A, U, U1)) = U)),
% 0.19/0.57 inference(rewrite,[status(thm)],[])).
% 0.19/0.57 tff(333,axiom,(![A: ty, U: $int, U1: uni] : (length1(A, mk_array1(A, U, U1)) = U)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','length_def1')).
% 0.19/0.57 tff(334,plain,
% 0.19/0.57 (![A: ty, U: $int, U1: uni] : (length1(A, mk_array1(A, U, U1)) = U)),
% 0.19/0.57 inference(modus_ponens,[status(thm)],[333, 332])).
% 0.19/0.57 tff(335,plain,
% 0.19/0.57 (![A: ty, U: $int, U1: uni] : ($sum(length1(A, mk_array1(A, U, U1)), $product(-1, U)) = 0)),
% 0.19/0.57 inference(modus_ponens,[status(thm)],[334, 331])).
% 0.19/0.57 tff(336,plain,
% 0.19/0.57 (![A: ty, U: $int, U1: uni] : ($sum(U, $product(-1, length1(A, mk_array1(A, U, U1)))) = 0)),
% 0.19/0.57 inference(modus_ponens,[status(thm)],[335, 329])).
% 0.19/0.57 tff(337,plain,(
% 0.19/0.57 ![A: ty, U: $int, U1: uni] : ($sum(U, $product(-1, length1(A, mk_array1(A, U, U1)))) = 0)),
% 0.19/0.57 inference(skolemize,[status(sab)],[336])).
% 0.19/0.57 tff(338,plain,
% 0.19/0.57 (![A: ty, U: $int, U1: uni] : ($sum(U, $product(-1, length1(A, mk_array1(A, U, U1)))) = 0)),
% 0.19/0.57 inference(modus_ponens,[status(thm)],[337, 327])).
% 0.19/0.57 tff(339,plain,
% 0.19/0.57 ((~![A: ty, U: $int, U1: uni] : ($sum(U, $product(-1, length1(A, mk_array1(A, U, U1)))) = 0)) | ($sum(H!4, $product(-1, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))) = 0)),
% 0.19/0.57 inference(quant_inst,[status(thm)],[])).
% 0.19/0.57 tff(340,plain,
% 0.19/0.57 ($sum(H!4, $product(-1, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))) = 0),
% 0.19/0.57 inference(unit_resolution,[status(thm)],[339, 338])).
% 0.19/0.57 tff(341,plain,
% 0.19/0.57 ((~($sum(H!4, $product(-1, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))) = 0)) | $lesseq($sum(H!4, $product(-1, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))), 0)),
% 0.19/0.57 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.57 tff(342,plain,
% 0.19/0.57 ($lesseq($sum(H!4, $product(-1, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))), 0)),
% 0.19/0.57 inference(unit_resolution,[status(thm)],[341, 340])).
% 0.19/0.57 tff(343,plain,
% 0.19/0.57 ((~($sum(H!4, $product(-1, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))) = 0)) | $greatereq($sum(H!4, $product(-1, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))), 0)),
% 0.19/0.57 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.57 tff(344,plain,
% 0.19/0.57 ($greatereq($sum(H!4, $product(-1, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))), 0)),
% 0.19/0.57 inference(unit_resolution,[status(thm)],[343, 340])).
% 0.19/0.57 tff(345,plain,
% 0.19/0.57 (H!4 = length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))),
% 0.19/0.57 inference(theory_lemma,[status(thm)],[344, 342])).
% 0.19/0.57 tff(346,plain,
% 0.19/0.57 (bucket1(K!10, H!4) = bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))),
% 0.19/0.57 inference(monotonicity,[status(thm)],[345])).
% 0.19/0.57 tff(347,plain,
% 0.19/0.57 (bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) = bucket1(K!10, H!4)),
% 0.19/0.57 inference(symmetry,[status(thm)],[346])).
% 0.19/0.57 tff(348,plain,
% 0.19/0.57 ((~(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) = bucket1(K!10, H!4))) | $greatereq($sum(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), $product(-1, bucket1(K!10, H!4))), 0)),
% 0.19/0.57 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.57 tff(349,plain,
% 0.19/0.57 ($greatereq($sum(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), $product(-1, bucket1(K!10, H!4))), 0)),
% 0.19/0.57 inference(unit_resolution,[status(thm)],[348, 347])).
% 0.19/0.57 tff(350,plain,
% 0.19/0.57 ($greatereq(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), 0) | (~$greatereq(bucket1(K!10, H!4), 0)) | (~$greatereq($sum(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), $product(-1, bucket1(K!10, H!4))), 0))),
% 0.19/0.57 inference(theory_lemma,[status(thm)],[])).
% 0.19/0.57 tff(351,plain,
% 0.19/0.57 ($greatereq(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), 0)),
% 0.19/0.57 inference(unit_resolution,[status(thm)],[350, 349, 325])).
% 0.19/0.57 tff(352,plain,
% 0.19/0.57 (((~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | ((~$greatereq(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))) = tb2t6(nil(tuple2(key, a)))))) <=> ((~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | (~$greatereq(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))) = tb2t6(nil(tuple2(key, a)))))),
% 0.19/0.57 inference(rewrite,[status(thm)],[])).
% 0.19/0.57 tff(353,plain,
% 0.19/0.57 (((~$greatereq(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), 0)) | $greatereq($sum(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))) = tb2t6(nil(tuple2(key, a))))) <=> ((~$greatereq(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))) = tb2t6(nil(tuple2(key, a)))))),
% 0.19/0.57 inference(rewrite,[status(thm)],[])).
% 0.19/0.57 tff(354,plain,
% 0.19/0.57 (((~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | ((~$greatereq(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), 0)) | $greatereq($sum(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))) = tb2t6(nil(tuple2(key, a)))))) <=> ((~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | ((~$greatereq(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))) = tb2t6(nil(tuple2(key, a))))))),
% 0.19/0.57 inference(monotonicity,[status(thm)],[353])).
% 0.19/0.57 tff(355,plain,
% 0.19/0.57 (((~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | ((~$greatereq(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), 0)) | $greatereq($sum(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))) = tb2t6(nil(tuple2(key, a)))))) <=> ((~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | (~$greatereq(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))) = tb2t6(nil(tuple2(key, a)))))),
% 0.48/0.58 inference(transitivity,[status(thm)],[354, 352])).
% 0.48/0.58 tff(356,plain,
% 0.48/0.58 ((~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | ((~$greatereq(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), 0)) | $greatereq($sum(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))) = tb2t6(nil(tuple2(key, a)))))),
% 0.48/0.58 inference(quant_inst,[status(thm)],[])).
% 0.48/0.58 tff(357,plain,
% 0.48/0.58 ((~![I: $int] : ((~$greatereq(I, 0)) | $greatereq($sum(I, $product(-1, H!4)), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(I))) = tb2t6(nil(tuple2(key, a)))))) | (~$greatereq(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))) = tb2t6(nil(tuple2(key, a))))),
% 0.48/0.58 inference(modus_ponens,[status(thm)],[356, 355])).
% 0.48/0.58 tff(358,plain,
% 0.48/0.58 ((~$greatereq(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), 0)) | $lesseq($sum(H!4, $product(-1, bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))) = tb2t6(nil(tuple2(key, a))))),
% 0.48/0.58 inference(unit_resolution,[status(thm)],[357, 178])).
% 0.48/0.58 tff(359,plain,
% 0.48/0.58 ($lesseq($sum(H!4, $product(-1, bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))), 0) | (tb2t6(get(list(tuple2(key, a)), int, t2tb2(O!6), t2tb(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6))))))) = tb2t6(nil(tuple2(key, a))))),
% 0.48/0.58 inference(unit_resolution,[status(thm)],[358, 351])).
% 0.48/0.58 tff(360,plain,
% 0.48/0.58 ($lesseq($sum(H!4, $product(-1, bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))))), 0)),
% 0.48/0.58 inference(unit_resolution,[status(thm)],[359, 323])).
% 0.48/0.58 tff(361,plain,
% 0.48/0.58 ((~(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))) = bucket1(K!10, H!4))) | $lesseq($sum(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), $product(-1, bucket1(K!10, H!4))), 0)),
% 0.48/0.58 inference(theory_lemma,[status(thm)],[])).
% 0.48/0.58 tff(362,plain,
% 0.48/0.58 ($lesseq($sum(bucket1(K!10, length1(list(tuple2(key, a)), mk_array1(list(tuple2(key, a)), H!4, t2tb2(O!6)))), $product(-1, bucket1(K!10, H!4))), 0)),
% 0.48/0.58 inference(unit_resolution,[status(thm)],[361, 347])).
% 0.48/0.58 tff(363,plain,
% 0.48/0.58 ($false),
% 0.48/0.58 inference(theory_lemma,[status(thm)],[362, 360, 51])).
% 0.48/0.58 % SZS output end Proof
%------------------------------------------------------------------------------