TSTP Solution File: NUM386+1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : NUM386+1 : TPTP v8.1.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n014.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 : Sun Sep 18 13:09:27 EDT 2022
% Result : Theorem 0.21s 0.40s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : NUM386+1 : TPTP v8.1.0. Released v3.2.0.
% 0.03/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n014.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Sep 2 10:09:46 EDT 2022
% 0.13/0.34 % CPUTime :
% 0.13/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.35 Usage: tptp [options] [-file:]file
% 0.13/0.35 -h, -? prints this message.
% 0.13/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.35 -m, -model generate model.
% 0.13/0.35 -p, -proof generate proof.
% 0.13/0.35 -c, -core generate unsat core of named formulas.
% 0.13/0.35 -st, -statistics display statistics.
% 0.13/0.35 -t:timeout set timeout (in second).
% 0.13/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.35 -<param>:<value> configuration parameter and value.
% 0.13/0.35 -o:<output-file> file to place output in.
% 0.21/0.40 % SZS status Theorem
% 0.21/0.40 % SZS output start Proof
% 0.21/0.40 tff(tptp_fun_B_13_type, type, (
% 0.21/0.40 tptp_fun_B_13: $i)).
% 0.21/0.40 tff(tptp_fun_A_14_type, type, (
% 0.21/0.40 tptp_fun_A_14: $i)).
% 0.21/0.40 tff(in_type, type, (
% 0.21/0.40 in: ( $i * $i ) > $o)).
% 0.21/0.40 tff(singleton_type, type, (
% 0.21/0.40 singleton: $i > $i)).
% 0.21/0.40 tff(tptp_fun_C_0_type, type, (
% 0.21/0.40 tptp_fun_C_0: ( $i * $i ) > $i)).
% 0.21/0.40 tff(set_union2_type, type, (
% 0.21/0.40 set_union2: ( $i * $i ) > $i)).
% 0.21/0.40 tff(succ_type, type, (
% 0.21/0.40 succ: $i > $i)).
% 0.21/0.40 tff(tptp_fun_D_1_type, type, (
% 0.21/0.40 tptp_fun_D_1: ( $i * $i * $i ) > $i)).
% 0.21/0.40 tff(1,assumption,(~(in(A!14, singleton(B!13)) <=> (A!14 = B!13))), introduced(assumption)).
% 0.21/0.40 tff(2,plain,
% 0.21/0.40 (^[A: $i, B: $i, C: $i] : refl((~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))) <=> (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(3,plain,
% 0.21/0.40 (![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[2])).
% 0.21/0.40 tff(4,plain,
% 0.21/0.40 (![A: $i, B: $i] : ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))),
% 0.21/0.40 inference(pull_quant,[status(thm)],[])).
% 0.21/0.40 tff(5,plain,
% 0.21/0.40 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(pull_quant(((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) <=> ![C: $i] : ((~(B = singleton(A))) | (in(C, B) <=> (C = A)))), ((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) <=> (~![C: $i] : ((~(B = singleton(A))) | (in(C, B) <=> (C = A)))))), pull_quant((~![C: $i] : ((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) <=> ?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A))))), ((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) <=> ?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))))), (((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))) <=> (?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))), pull_quant((?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))) <=> ?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))), (((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))) <=> ?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))), ((~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))) <=> (~?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))))), pull_quant((~?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))), ((~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(6,plain,
% 0.21/0.40 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ![A: $i, B: $i] : ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[5])).
% 0.21/0.40 tff(7,plain,
% 0.21/0.40 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))),
% 0.21/0.40 inference(transitivity,[status(thm)],[6, 4])).
% 0.21/0.40 tff(8,plain,
% 0.21/0.40 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))),
% 0.21/0.40 inference(transitivity,[status(thm)],[7, 3])).
% 0.21/0.40 tff(9,plain,
% 0.21/0.40 (^[A: $i, B: $i] : rewrite((~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))) <=> (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(10,plain,
% 0.21/0.40 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[9])).
% 0.21/0.40 tff(11,plain,
% 0.21/0.40 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))),
% 0.21/0.40 inference(transitivity,[status(thm)],[10, 8])).
% 0.21/0.40 tff(12,plain,
% 0.21/0.40 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) <=> ((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))), ((((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))) <=> (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))), rewrite((((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))) <=> (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))), ((((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))) <=> (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(13,plain,
% 0.21/0.40 (![A: $i, B: $i] : (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))) <=> ![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[12])).
% 0.21/0.40 tff(14,plain,
% 0.21/0.40 (^[A: $i, B: $i] : rewrite((((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A)))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(15,plain,
% 0.21/0.40 (![A: $i, B: $i] : (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> ![A: $i, B: $i] : (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[14])).
% 0.21/0.40 tff(16,plain,
% 0.21/0.40 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A))) <=> ![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(17,axiom,(![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d1_tarski')).
% 0.21/0.40 tff(18,plain,
% 0.21/0.40 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[17, 16])).
% 0.21/0.40 tff(19,plain,(
% 0.21/0.40 ![A: $i, B: $i] : (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))),
% 0.21/0.40 inference(skolemize,[status(sab)],[18])).
% 0.21/0.40 tff(20,plain,
% 0.21/0.40 (![A: $i, B: $i] : (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[19, 15])).
% 0.21/0.40 tff(21,plain,
% 0.21/0.40 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[20, 13])).
% 0.21/0.40 tff(22,plain,
% 0.21/0.40 (![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[21, 11])).
% 0.21/0.40 tff(23,plain,
% 0.21/0.40 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(24,plain,
% 0.21/0.40 ((~((~in(A!14, singleton(B!13))) <=> (A!14 = B!13))) <=> (in(A!14, singleton(B!13)) <=> (A!14 = B!13))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(25,plain,
% 0.21/0.40 ((((~in(A!14, singleton(B!13))) <=> (A!14 = B!13)) | $false) <=> ((~in(A!14, singleton(B!13))) <=> (A!14 = B!13))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(26,plain,
% 0.21/0.40 ((~$true) <=> $false),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(27,plain,
% 0.21/0.40 (($true | ((~in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13))) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13))) <=> $true),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(28,plain,
% 0.21/0.40 ((singleton(B!13) = singleton(B!13)) <=> $true),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(29,plain,
% 0.21/0.40 (((singleton(B!13) = singleton(B!13)) | ((~in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13))) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13))) <=> ($true | ((~in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13))) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13)))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[28])).
% 0.21/0.41 tff(30,plain,
% 0.21/0.41 (((singleton(B!13) = singleton(B!13)) | ((~in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13))) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13))) <=> $true),
% 0.21/0.41 inference(transitivity,[status(thm)],[29, 27])).
% 0.21/0.41 tff(31,plain,
% 0.21/0.41 ((~((singleton(B!13) = singleton(B!13)) | ((~in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13))) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13)))) <=> (~$true)),
% 0.21/0.41 inference(monotonicity,[status(thm)],[30])).
% 0.21/0.41 tff(32,plain,
% 0.21/0.41 ((~((singleton(B!13) = singleton(B!13)) | ((~in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13))) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13)))) <=> $false),
% 0.21/0.41 inference(transitivity,[status(thm)],[31, 26])).
% 0.21/0.41 tff(33,plain,
% 0.21/0.41 ((~(in(A!14, singleton(B!13)) <=> (A!14 = B!13))) <=> ((~in(A!14, singleton(B!13))) <=> (A!14 = B!13))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(34,plain,
% 0.21/0.41 (($false | (in(A!14, singleton(B!13)) <=> (A!14 = B!13))) <=> (in(A!14, singleton(B!13)) <=> (A!14 = B!13))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(35,plain,
% 0.21/0.41 ((~(singleton(B!13) = singleton(B!13))) <=> (~$true)),
% 0.21/0.41 inference(monotonicity,[status(thm)],[28])).
% 0.21/0.41 tff(36,plain,
% 0.21/0.41 ((~(singleton(B!13) = singleton(B!13))) <=> $false),
% 0.21/0.41 inference(transitivity,[status(thm)],[35, 26])).
% 0.21/0.41 tff(37,plain,
% 0.21/0.41 (((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13))) <=> ($false | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[36])).
% 0.21/0.41 tff(38,plain,
% 0.21/0.41 (((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13))) <=> (in(A!14, singleton(B!13)) <=> (A!14 = B!13))),
% 0.21/0.41 inference(transitivity,[status(thm)],[37, 34])).
% 0.21/0.41 tff(39,plain,
% 0.21/0.41 ((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) <=> (~(in(A!14, singleton(B!13)) <=> (A!14 = B!13)))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[38])).
% 0.21/0.41 tff(40,plain,
% 0.21/0.41 ((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) <=> ((~in(A!14, singleton(B!13))) <=> (A!14 = B!13))),
% 0.21/0.41 inference(transitivity,[status(thm)],[39, 33])).
% 0.21/0.41 tff(41,plain,
% 0.21/0.41 (((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) | (~((singleton(B!13) = singleton(B!13)) | ((~in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13))) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13))))) <=> (((~in(A!14, singleton(B!13))) <=> (A!14 = B!13)) | $false)),
% 0.21/0.41 inference(monotonicity,[status(thm)],[40, 32])).
% 0.21/0.41 tff(42,plain,
% 0.21/0.41 (((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) | (~((singleton(B!13) = singleton(B!13)) | ((~in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13))) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13))))) <=> ((~in(A!14, singleton(B!13))) <=> (A!14 = B!13))),
% 0.21/0.41 inference(transitivity,[status(thm)],[41, 25])).
% 0.21/0.41 tff(43,plain,
% 0.21/0.41 ((~((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) | (~((singleton(B!13) = singleton(B!13)) | ((~in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13))) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13)))))) <=> (~((~in(A!14, singleton(B!13))) <=> (A!14 = B!13)))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[42])).
% 0.21/0.41 tff(44,plain,
% 0.21/0.41 ((~((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) | (~((singleton(B!13) = singleton(B!13)) | ((~in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13))) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13)))))) <=> (in(A!14, singleton(B!13)) <=> (A!14 = B!13))),
% 0.21/0.41 inference(transitivity,[status(thm)],[43, 24])).
% 0.21/0.41 tff(45,plain,
% 0.21/0.41 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))) | (~((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) | (~((singleton(B!13) = singleton(B!13)) | ((~in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13))) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13))))))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[44])).
% 0.21/0.41 tff(46,plain,
% 0.21/0.41 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))) | (~((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) | (~((singleton(B!13) = singleton(B!13)) | ((~in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13))) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13))))))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))),
% 0.21/0.41 inference(transitivity,[status(thm)],[45, 23])).
% 0.21/0.41 tff(47,plain,
% 0.21/0.41 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))) | (~((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) | (~((singleton(B!13) = singleton(B!13)) | ((~in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13))) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13))))))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(48,plain,
% 0.21/0.41 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | ((~in(tptp_fun_C_0(B, A), B)) <=> (tptp_fun_C_0(B, A) = A))))))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.21/0.41 tff(49,plain,
% 0.21/0.41 ($false),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[48, 22, 1])).
% 0.21/0.41 tff(50,plain,(in(A!14, singleton(B!13)) <=> (A!14 = B!13)), inference(lemma,lemma(discharge,[]))).
% 0.21/0.41 tff(51,plain,
% 0.21/0.41 (^[A: $i] : refl((succ(A) = set_union2(A, singleton(A))) <=> (succ(A) = set_union2(A, singleton(A))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(52,plain,
% 0.21/0.41 (![A: $i] : (succ(A) = set_union2(A, singleton(A))) <=> ![A: $i] : (succ(A) = set_union2(A, singleton(A)))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[51])).
% 0.21/0.41 tff(53,plain,
% 0.21/0.41 (![A: $i] : (succ(A) = set_union2(A, singleton(A))) <=> ![A: $i] : (succ(A) = set_union2(A, singleton(A)))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(54,axiom,(![A: $i] : (succ(A) = set_union2(A, singleton(A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d1_ordinal1')).
% 0.21/0.41 tff(55,plain,
% 0.21/0.41 (![A: $i] : (succ(A) = set_union2(A, singleton(A)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[54, 53])).
% 0.21/0.41 tff(56,plain,(
% 0.21/0.41 ![A: $i] : (succ(A) = set_union2(A, singleton(A)))),
% 0.21/0.41 inference(skolemize,[status(sab)],[55])).
% 0.21/0.41 tff(57,plain,
% 0.21/0.41 (![A: $i] : (succ(A) = set_union2(A, singleton(A)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[56, 52])).
% 0.21/0.41 tff(58,plain,
% 0.21/0.41 ((~![A: $i] : (succ(A) = set_union2(A, singleton(A)))) | (succ(B!13) = set_union2(B!13, singleton(B!13)))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(59,plain,
% 0.21/0.41 (succ(B!13) = set_union2(B!13, singleton(B!13))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[58, 57])).
% 0.21/0.41 tff(60,plain,
% 0.21/0.41 (set_union2(B!13, singleton(B!13)) = succ(B!13)),
% 0.21/0.41 inference(symmetry,[status(thm)],[59])).
% 0.21/0.41 tff(61,plain,
% 0.21/0.41 (in(A!14, set_union2(B!13, singleton(B!13))) <=> in(A!14, succ(B!13))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[60])).
% 0.21/0.41 tff(62,plain,
% 0.21/0.41 (in(A!14, succ(B!13)) <=> in(A!14, set_union2(B!13, singleton(B!13)))),
% 0.21/0.41 inference(symmetry,[status(thm)],[61])).
% 0.21/0.41 tff(63,plain,
% 0.21/0.41 ((~in(A!14, succ(B!13))) <=> (~in(A!14, set_union2(B!13, singleton(B!13))))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[62])).
% 0.21/0.41 tff(64,assumption,(~(in(A!14, B!13) | (A!14 = B!13))), introduced(assumption)).
% 0.21/0.41 tff(65,plain,
% 0.21/0.41 ((~(in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13)))) <=> ((~in(A!14, succ(B!13))) <=> (in(A!14, B!13) | (A!14 = B!13)))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(66,plain,
% 0.21/0.41 ((in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13))) <=> (in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13)))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(67,plain,
% 0.21/0.41 ((~(in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13)))) <=> (~(in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13))))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[66])).
% 0.21/0.41 tff(68,plain,
% 0.21/0.41 ((~(in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13)))) <=> ((~in(A!14, succ(B!13))) <=> (in(A!14, B!13) | (A!14 = B!13)))),
% 0.21/0.41 inference(transitivity,[status(thm)],[67, 65])).
% 0.21/0.41 tff(69,plain,
% 0.21/0.41 ((~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B)))) <=> (~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(70,axiom,(~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t13_ordinal1')).
% 0.21/0.41 tff(71,plain,
% 0.21/0.41 (~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[70, 69])).
% 0.21/0.41 tff(72,plain,
% 0.21/0.41 (~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[71, 69])).
% 0.21/0.41 tff(73,plain,
% 0.21/0.41 (~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[72, 69])).
% 0.21/0.41 tff(74,plain,
% 0.21/0.41 (~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[73, 69])).
% 0.21/0.41 tff(75,plain,
% 0.21/0.41 (~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[74, 69])).
% 0.21/0.41 tff(76,plain,
% 0.21/0.41 (~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[75, 69])).
% 0.21/0.41 tff(77,plain,
% 0.21/0.41 (~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[76, 69])).
% 0.21/0.41 tff(78,plain,(
% 0.21/0.41 ~(in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13)))),
% 0.21/0.41 inference(skolemize,[status(sab)],[77])).
% 0.21/0.41 tff(79,plain,
% 0.21/0.41 ((~in(A!14, succ(B!13))) <=> (in(A!14, B!13) | (A!14 = B!13))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[78, 68])).
% 0.21/0.41 tff(80,plain,
% 0.21/0.41 (in(A!14, succ(B!13)) | (in(A!14, B!13) | (A!14 = B!13)) | (~((~in(A!14, succ(B!13))) <=> (in(A!14, B!13) | (A!14 = B!13))))),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(81,plain,
% 0.21/0.41 (in(A!14, succ(B!13)) | (in(A!14, B!13) | (A!14 = B!13))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[80, 79])).
% 0.21/0.41 tff(82,plain,
% 0.21/0.41 (in(A!14, succ(B!13))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[81, 64])).
% 0.21/0.41 tff(83,plain,
% 0.21/0.41 (in(A!14, set_union2(B!13, singleton(B!13)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[82, 62])).
% 0.21/0.41 tff(84,plain,
% 0.21/0.41 ((in(A!14, B!13) | (A!14 = B!13)) | (~in(A!14, B!13))),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(85,plain,
% 0.21/0.41 (~in(A!14, B!13)),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[84, 64])).
% 0.21/0.41 tff(86,plain,
% 0.21/0.41 ((in(A!14, B!13) | (A!14 = B!13)) | (~(A!14 = B!13))),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(87,plain,
% 0.21/0.41 (~(A!14 = B!13)),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[86, 64])).
% 0.21/0.41 tff(88,plain,
% 0.21/0.41 ((~(in(A!14, singleton(B!13)) <=> (A!14 = B!13))) | (~in(A!14, singleton(B!13))) | (A!14 = B!13)),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(89,plain,
% 0.21/0.41 (~in(A!14, singleton(B!13))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[88, 87, 50])).
% 0.21/0.41 tff(90,plain,
% 0.21/0.41 ((~(in(A!14, B!13) | in(A!14, singleton(B!13)))) | in(A!14, B!13) | in(A!14, singleton(B!13))),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(91,plain,
% 0.21/0.41 (~(in(A!14, B!13) | in(A!14, singleton(B!13)))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[90, 89, 85])).
% 0.21/0.41 tff(92,plain,
% 0.21/0.41 (^[A: $i, B: $i, C: $i, D: $i] : refl((~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))) <=> (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(93,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[92])).
% 0.21/0.42 tff(94,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : ![D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))),
% 0.21/0.42 inference(pull_quant,[status(thm)],[])).
% 0.21/0.42 tff(95,plain,
% 0.21/0.42 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(pull_quant(((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) <=> ![D: $i] : ((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))), ((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) <=> (~![D: $i] : ((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))))), pull_quant((~![D: $i] : ((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) <=> ?[D: $i] : (~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B)))))), ((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) <=> ?[D: $i] : (~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))))), (((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))) <=> (?[D: $i] : (~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))), pull_quant((?[D: $i] : (~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))) <=> ?[D: $i] : ((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))), (((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))) <=> ?[D: $i] : ((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))), ((~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))) <=> (~?[D: $i] : ((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))), pull_quant((~?[D: $i] : ((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))) <=> ![D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))), ((~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))) <=> ![D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(96,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))) <=> ![A: $i, B: $i, C: $i] : ![D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[95])).
% 0.21/0.42 tff(97,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))),
% 0.21/0.42 inference(transitivity,[status(thm)],[96, 94])).
% 0.21/0.42 tff(98,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))),
% 0.21/0.42 inference(transitivity,[status(thm)],[97, 93])).
% 0.21/0.42 tff(99,plain,
% 0.21/0.42 (^[A: $i, B: $i, C: $i] : rewrite((~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))) <=> (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(100,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[99])).
% 0.21/0.42 tff(101,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))),
% 0.21/0.42 inference(transitivity,[status(thm)],[100, 98])).
% 0.21/0.42 tff(102,plain,
% 0.21/0.42 (^[A: $i, B: $i, C: $i] : trans(monotonicity(rewrite(((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) <=> ((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))), monotonicity(rewrite(((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))) <=> ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))), (((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))) <=> ((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))), ((((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))) <=> (((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))), rewrite((((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))) <=> (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))), ((((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))) <=> (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(103,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : (((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[102])).
% 0.21/0.42 tff(104,plain,
% 0.21/0.42 (^[A: $i, B: $i, C: $i] : rewrite((((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))) <=> (((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(105,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : (((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))) <=> ![A: $i, B: $i, C: $i] : (((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[104])).
% 0.21/0.42 tff(106,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : ((C = set_union2(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) <=> ![A: $i, B: $i, C: $i] : ((C = set_union2(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(107,axiom,(![A: $i, B: $i, C: $i] : ((C = set_union2(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d2_xboole_0')).
% 0.21/0.42 tff(108,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : ((C = set_union2(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.21/0.42 tff(109,plain,(
% 0.21/0.42 ![A: $i, B: $i, C: $i] : (((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))),
% 0.21/0.42 inference(skolemize,[status(sab)],[108])).
% 0.21/0.42 tff(110,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : (((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[109, 105])).
% 0.21/0.42 tff(111,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[110, 103])).
% 0.21/0.42 tff(112,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[111, 101])).
% 0.21/0.42 tff(113,plain,
% 0.21/0.42 (((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(114,plain,
% 0.21/0.42 ((~((~in(A!14, set_union2(B!13, singleton(B!13)))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) <=> (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(115,plain,
% 0.21/0.42 ((((~in(A!14, set_union2(B!13, singleton(B!13)))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))) | $false) <=> ((~in(A!14, set_union2(B!13, singleton(B!13)))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(116,plain,
% 0.21/0.42 (($true | ((~in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13)))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13))))) <=> $true),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(117,plain,
% 0.21/0.42 ((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) <=> $true),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(118,plain,
% 0.21/0.42 (((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | ((~in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13)))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13))))) <=> ($true | ((~in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13)))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13)))))),
% 0.21/0.42 inference(monotonicity,[status(thm)],[117])).
% 0.21/0.42 tff(119,plain,
% 0.21/0.42 (((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | ((~in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13)))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13))))) <=> $true),
% 0.21/0.43 inference(transitivity,[status(thm)],[118, 116])).
% 0.21/0.43 tff(120,plain,
% 0.21/0.43 ((~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | ((~in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13)))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13)))))) <=> (~$true)),
% 0.21/0.43 inference(monotonicity,[status(thm)],[119])).
% 0.21/0.43 tff(121,plain,
% 0.21/0.43 ((~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | ((~in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13)))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13)))))) <=> $false),
% 0.21/0.43 inference(transitivity,[status(thm)],[120, 26])).
% 0.21/0.43 tff(122,plain,
% 0.21/0.43 ((~(in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) <=> ((~in(A!14, set_union2(B!13, singleton(B!13)))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(123,plain,
% 0.21/0.43 (($false | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) <=> (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(124,plain,
% 0.21/0.43 ((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) <=> (~$true)),
% 0.21/0.43 inference(monotonicity,[status(thm)],[117])).
% 0.21/0.43 tff(125,plain,
% 0.21/0.43 ((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) <=> $false),
% 0.21/0.43 inference(transitivity,[status(thm)],[124, 26])).
% 0.21/0.43 tff(126,plain,
% 0.21/0.43 (((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) <=> ($false | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[125])).
% 0.21/0.43 tff(127,plain,
% 0.21/0.43 (((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) <=> (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[126, 123])).
% 0.21/0.43 tff(128,plain,
% 0.21/0.43 ((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) <=> (~(in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[127])).
% 0.21/0.43 tff(129,plain,
% 0.21/0.43 ((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) <=> ((~in(A!14, set_union2(B!13, singleton(B!13)))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[128, 122])).
% 0.21/0.43 tff(130,plain,
% 0.21/0.43 (((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) | (~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | ((~in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13)))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13))))))) <=> (((~in(A!14, set_union2(B!13, singleton(B!13)))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))) | $false)),
% 0.21/0.43 inference(monotonicity,[status(thm)],[129, 121])).
% 0.21/0.43 tff(131,plain,
% 0.21/0.43 (((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) | (~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | ((~in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13)))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13))))))) <=> ((~in(A!14, set_union2(B!13, singleton(B!13)))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[130, 115])).
% 0.21/0.43 tff(132,plain,
% 0.21/0.43 ((~((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) | (~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | ((~in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13)))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13)))))))) <=> (~((~in(A!14, set_union2(B!13, singleton(B!13)))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[131])).
% 0.21/0.43 tff(133,plain,
% 0.21/0.43 ((~((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) | (~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | ((~in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13)))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13)))))))) <=> (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[132, 114])).
% 0.21/0.43 tff(134,plain,
% 0.21/0.43 (((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) | (~((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) | (~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | ((~in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13)))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13))))))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[133])).
% 0.21/0.43 tff(135,plain,
% 0.21/0.43 (((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) | (~((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) | (~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | ((~in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13)))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13))))))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[134, 113])).
% 0.21/0.43 tff(136,plain,
% 0.21/0.43 ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) | (~((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) | (~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | ((~in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13)))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13))))))))),
% 0.21/0.43 inference(quant_inst,[status(thm)],[])).
% 0.21/0.43 tff(137,plain,
% 0.21/0.43 ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | ((~in(tptp_fun_D_1(C, B, A), C)) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[136, 135])).
% 0.21/0.43 tff(138,plain,
% 0.21/0.43 (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[137, 112])).
% 0.21/0.43 tff(139,plain,
% 0.21/0.43 ((~(in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) | (~in(A!14, set_union2(B!13, singleton(B!13)))) | (in(A!14, B!13) | in(A!14, singleton(B!13)))),
% 0.21/0.43 inference(tautology,[status(thm)],[])).
% 0.21/0.43 tff(140,plain,
% 0.21/0.43 ((~in(A!14, set_union2(B!13, singleton(B!13)))) | (in(A!14, B!13) | in(A!14, singleton(B!13)))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[139, 138])).
% 0.21/0.43 tff(141,plain,
% 0.21/0.43 (~in(A!14, set_union2(B!13, singleton(B!13)))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[140, 91])).
% 0.21/0.43 tff(142,plain,
% 0.21/0.43 ($false),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[141, 83])).
% 0.21/0.43 tff(143,plain,(in(A!14, B!13) | (A!14 = B!13)), inference(lemma,lemma(discharge,[]))).
% 0.21/0.43 tff(144,plain,
% 0.21/0.43 ((~in(A!14, succ(B!13))) | (~(in(A!14, B!13) | (A!14 = B!13))) | (~((~in(A!14, succ(B!13))) <=> (in(A!14, B!13) | (A!14 = B!13))))),
% 0.21/0.43 inference(tautology,[status(thm)],[])).
% 0.21/0.43 tff(145,plain,
% 0.21/0.43 ((~in(A!14, succ(B!13))) | (~(in(A!14, B!13) | (A!14 = B!13)))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[144, 79])).
% 0.21/0.43 tff(146,plain,
% 0.21/0.43 (~in(A!14, succ(B!13))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[145, 143])).
% 0.21/0.43 tff(147,plain,
% 0.21/0.43 (~in(A!14, set_union2(B!13, singleton(B!13)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[146, 63])).
% 0.21/0.43 tff(148,plain,
% 0.21/0.43 ((~(in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) | in(A!14, set_union2(B!13, singleton(B!13))) | (~(in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.21/0.43 inference(tautology,[status(thm)],[])).
% 0.21/0.43 tff(149,plain,
% 0.21/0.43 (in(A!14, set_union2(B!13, singleton(B!13))) | (~(in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[148, 138])).
% 0.21/0.43 tff(150,plain,
% 0.21/0.43 (~(in(A!14, B!13) | in(A!14, singleton(B!13)))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[149, 147])).
% 0.21/0.43 tff(151,plain,
% 0.21/0.43 ((in(A!14, B!13) | in(A!14, singleton(B!13))) | (~in(A!14, singleton(B!13)))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(152,plain,
% 0.21/0.44 (~in(A!14, singleton(B!13))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[151, 150])).
% 0.21/0.44 tff(153,plain,
% 0.21/0.44 ((in(A!14, B!13) | in(A!14, singleton(B!13))) | (~in(A!14, B!13))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(154,plain,
% 0.21/0.44 (~in(A!14, B!13)),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[153, 150])).
% 0.21/0.44 tff(155,plain,
% 0.21/0.44 ((~(in(A!14, B!13) | (A!14 = B!13))) | in(A!14, B!13) | (A!14 = B!13)),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(156,plain,
% 0.21/0.44 (A!14 = B!13),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[155, 154, 143])).
% 0.21/0.44 tff(157,plain,
% 0.21/0.44 ((~(in(A!14, singleton(B!13)) <=> (A!14 = B!13))) | in(A!14, singleton(B!13)) | (~(A!14 = B!13))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(158,plain,
% 0.21/0.44 ($false),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[157, 156, 152, 50])).
% 0.21/0.44 % SZS output end Proof
%------------------------------------------------------------------------------