%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : ALG227+1 : TPTP v9.0.0. Released v3.4.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 04:58:47 AM UTC 2025
% Result : Theorem 0.12s 0.39s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : ALG227+1 : TPTP v9.0.0. Released v3.4.0.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n003.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 : Thu Jun 19 10:54:09 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.12/0.39 % SZS status Theorem
% 0.12/0.39 % SZS output start Proof
% 0.12/0.39 tff(r2_hidden_type, type, (
% 0.12/0.39 r2_hidden: ( $i * $i ) > $o)).
% 0.12/0.39 tff(k1_closure3_type, type, (
% 0.12/0.39 k1_closure3: ( $i * $i ) > $i)).
% 0.12/0.39 tff(tptp_fun_B_1_type, type, (
% 0.12/0.39 tptp_fun_B_1: $i)).
% 0.12/0.39 tff(tptp_fun_A_0_type, type, (
% 0.12/0.39 tptp_fun_A_0: $i)).
% 0.12/0.39 tff(tptp_fun_C_2_type, type, (
% 0.12/0.39 tptp_fun_C_2: $i)).
% 0.12/0.39 tff(a_2_0_closure3_type, type, (
% 0.12/0.39 a_2_0_closure3: ( $i * $i ) > $i)).
% 0.12/0.39 tff(m1_pboole_type, type, (
% 0.12/0.39 m1_pboole: ( $i * $i ) > $o)).
% 0.12/0.39 tff(v1_xboole_0_type, type, (
% 0.12/0.39 v1_xboole_0: $i > $o)).
% 0.12/0.39 tff(m1_subset_1_type, type, (
% 0.12/0.39 m1_subset_1: ( $i * $i ) > $o)).
% 0.12/0.39 tff(k1_xboole_0_type, type, (
% 0.12/0.39 k1_xboole_0: $i)).
% 0.12/0.39 tff(k1_funct_1_type, type, (
% 0.12/0.39 k1_funct_1: ( $i * $i ) > $i)).
% 0.12/0.39 tff(1,plain,
% 0.12/0.39 (^[A: $i] : refl((v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B))))) <=> (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(2,plain,
% 0.12/0.39 (![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B))))) <=> ![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[1])).
% 0.12/0.39 tff(3,plain,
% 0.12/0.39 (^[A: $i] : rewrite((v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B))))) <=> (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(4,plain,
% 0.12/0.39 (![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B))))) <=> ![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[3])).
% 0.12/0.39 tff(5,plain,
% 0.12/0.39 (![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B))))) <=> ![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[4, 2])).
% 0.12/0.39 tff(6,plain,
% 0.12/0.39 (![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B))))) <=> ![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(7,plain,
% 0.12/0.39 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : rewrite((m1_pboole(B, A) => ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))) <=> ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))), (![B: $i] : (m1_pboole(B, A) => ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))) <=> ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))), (((~v1_xboole_0(A)) => ![B: $i] : (m1_pboole(B, A) => ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B))))) <=> ((~v1_xboole_0(A)) => ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B))))))), rewrite(((~v1_xboole_0(A)) => ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B))))) <=> (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))), (((~v1_xboole_0(A)) => ![B: $i] : (m1_pboole(B, A) => ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B))))) <=> (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(8,plain,
% 0.12/0.39 (![A: $i] : ((~v1_xboole_0(A)) => ![B: $i] : (m1_pboole(B, A) => ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B))))) <=> ![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[7])).
% 0.12/0.40 tff(9,axiom,(![A: $i] : ((~v1_xboole_0(A)) => ![B: $i] : (m1_pboole(B, A) => ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d3_closure3')).
% 0.12/0.40 tff(10,plain,
% 0.12/0.40 (![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[9, 8])).
% 0.12/0.40 tff(11,plain,
% 0.12/0.40 (![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[10, 6])).
% 0.12/0.40 tff(12,plain,(
% 0.12/0.40 ![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))),
% 0.12/0.40 inference(skolemize,[status(sab)],[11])).
% 0.12/0.40 tff(13,plain,
% 0.12/0.40 (![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[12, 5])).
% 0.12/0.40 tff(14,plain,
% 0.12/0.40 (((~v1_xboole_0(A!0)) & (m1_pboole(B!1, A!0) & (~((k1_funct_1(B!1, C!2) = k1_xboole_0) | r2_hidden(C!2, k1_closure3(A!0, B!1)) | (~m1_subset_1(C!2, A!0)))))) <=> ((~v1_xboole_0(A!0)) & m1_pboole(B!1, A!0) & (~((k1_funct_1(B!1, C!2) = k1_xboole_0) | r2_hidden(C!2, k1_closure3(A!0, B!1)) | (~m1_subset_1(C!2, A!0)))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(15,plain,
% 0.12/0.40 ((~((k1_funct_1(B!1, C!2) = k1_xboole_0) | r2_hidden(C!2, k1_closure3(A!0, B!1)) | (~m1_subset_1(C!2, A!0)))) <=> (~((k1_funct_1(B!1, C!2) = k1_xboole_0) | r2_hidden(C!2, k1_closure3(A!0, B!1)) | (~m1_subset_1(C!2, A!0))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(16,plain,
% 0.12/0.40 ((m1_pboole(B!1, A!0) & (~((k1_funct_1(B!1, C!2) = k1_xboole_0) | r2_hidden(C!2, k1_closure3(A!0, B!1)) | (~m1_subset_1(C!2, A!0))))) <=> (m1_pboole(B!1, A!0) & (~((k1_funct_1(B!1, C!2) = k1_xboole_0) | r2_hidden(C!2, k1_closure3(A!0, B!1)) | (~m1_subset_1(C!2, A!0)))))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[15])).
% 0.12/0.40 tff(17,plain,
% 0.12/0.40 (((~v1_xboole_0(A!0)) & (m1_pboole(B!1, A!0) & (~((k1_funct_1(B!1, C!2) = k1_xboole_0) | r2_hidden(C!2, k1_closure3(A!0, B!1)) | (~m1_subset_1(C!2, A!0)))))) <=> ((~v1_xboole_0(A!0)) & (m1_pboole(B!1, A!0) & (~((k1_funct_1(B!1, C!2) = k1_xboole_0) | r2_hidden(C!2, k1_closure3(A!0, B!1)) | (~m1_subset_1(C!2, A!0))))))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[16])).
% 0.12/0.40 tff(18,plain,
% 0.12/0.40 (((~v1_xboole_0(A!0)) & (m1_pboole(B!1, A!0) & (~((k1_funct_1(B!1, C!2) = k1_xboole_0) | r2_hidden(C!2, k1_closure3(A!0, B!1)) | (~m1_subset_1(C!2, A!0)))))) <=> ((~v1_xboole_0(A!0)) & m1_pboole(B!1, A!0) & (~((k1_funct_1(B!1, C!2) = k1_xboole_0) | r2_hidden(C!2, k1_closure3(A!0, B!1)) | (~m1_subset_1(C!2, A!0)))))),
% 0.12/0.40 inference(transitivity,[status(thm)],[17, 14])).
% 0.12/0.40 tff(19,plain,
% 0.12/0.40 ((~![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((k1_funct_1(B, C) = k1_xboole_0) | r2_hidden(C, k1_closure3(A, B)) | (~m1_subset_1(C, A)))))) <=> (~![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((k1_funct_1(B, C) = k1_xboole_0) | r2_hidden(C, k1_closure3(A, B)) | (~m1_subset_1(C, A))))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(20,plain,
% 0.12/0.40 ((~![A: $i] : ((~v1_xboole_0(A)) => ![B: $i] : (m1_pboole(B, A) => ![C: $i] : (m1_subset_1(C, A) => ((~r2_hidden(C, k1_closure3(A, B))) => (k1_funct_1(B, C) = k1_xboole_0)))))) <=> (~![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((k1_funct_1(B, C) = k1_xboole_0) | r2_hidden(C, k1_closure3(A, B)) | (~m1_subset_1(C, A))))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(21,axiom,(~![A: $i] : ((~v1_xboole_0(A)) => ![B: $i] : (m1_pboole(B, A) => ![C: $i] : (m1_subset_1(C, A) => ((~r2_hidden(C, k1_closure3(A, B))) => (k1_funct_1(B, C) = k1_xboole_0)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t8_closure3')).
% 0.12/0.40 tff(22,plain,
% 0.12/0.40 (~![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((k1_funct_1(B, C) = k1_xboole_0) | r2_hidden(C, k1_closure3(A, B)) | (~m1_subset_1(C, A)))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.12/0.40 tff(23,plain,
% 0.12/0.40 (~![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((k1_funct_1(B, C) = k1_xboole_0) | r2_hidden(C, k1_closure3(A, B)) | (~m1_subset_1(C, A)))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[22, 19])).
% 0.12/0.40 tff(24,plain,
% 0.12/0.40 (~![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((k1_funct_1(B, C) = k1_xboole_0) | r2_hidden(C, k1_closure3(A, B)) | (~m1_subset_1(C, A)))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[23, 19])).
% 0.12/0.40 tff(25,plain,
% 0.12/0.40 (~![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((k1_funct_1(B, C) = k1_xboole_0) | r2_hidden(C, k1_closure3(A, B)) | (~m1_subset_1(C, A)))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[24, 19])).
% 0.12/0.40 tff(26,plain,
% 0.12/0.40 ((~v1_xboole_0(A!0)) & m1_pboole(B!1, A!0) & (~((k1_funct_1(B!1, C!2) = k1_xboole_0) | r2_hidden(C!2, k1_closure3(A!0, B!1)) | (~m1_subset_1(C!2, A!0))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[25, 18])).
% 0.12/0.40 tff(27,plain,
% 0.12/0.40 (~v1_xboole_0(A!0)),
% 0.12/0.40 inference(and_elim,[status(thm)],[26])).
% 0.12/0.40 tff(28,plain,
% 0.12/0.40 (((~![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))) | (v1_xboole_0(A!0) | ![B: $i] : ((~m1_pboole(B, A!0)) | ![C: $i] : ((C = k1_closure3(A!0, B)) <=> (C = a_2_0_closure3(A!0, B)))))) <=> ((~![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))) | v1_xboole_0(A!0) | ![B: $i] : ((~m1_pboole(B, A!0)) | ![C: $i] : ((C = k1_closure3(A!0, B)) <=> (C = a_2_0_closure3(A!0, B)))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(29,plain,
% 0.12/0.40 ((~![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))) | (v1_xboole_0(A!0) | ![B: $i] : ((~m1_pboole(B, A!0)) | ![C: $i] : ((C = k1_closure3(A!0, B)) <=> (C = a_2_0_closure3(A!0, B)))))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(30,plain,
% 0.12/0.40 ((~![A: $i] : (v1_xboole_0(A) | ![B: $i] : ((~m1_pboole(B, A)) | ![C: $i] : ((C = k1_closure3(A, B)) <=> (C = a_2_0_closure3(A, B)))))) | v1_xboole_0(A!0) | ![B: $i] : ((~m1_pboole(B, A!0)) | ![C: $i] : ((C = k1_closure3(A!0, B)) <=> (C = a_2_0_closure3(A!0, B))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[29, 28])).
% 0.12/0.40 tff(31,plain,
% 0.12/0.40 (![B: $i] : ((~m1_pboole(B, A!0)) | ![C: $i] : ((C = k1_closure3(A!0, B)) <=> (C = a_2_0_closure3(A!0, B))))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[30, 27, 13])).
% 0.12/0.40 tff(32,plain,
% 0.12/0.40 (m1_pboole(B!1, A!0)),
% 0.12/0.40 inference(and_elim,[status(thm)],[26])).
% 0.12/0.40 tff(33,plain,
% 0.12/0.40 (((~![B: $i] : ((~m1_pboole(B, A!0)) | ![C: $i] : ((C = k1_closure3(A!0, B)) <=> (C = a_2_0_closure3(A!0, B))))) | ((~m1_pboole(B!1, A!0)) | ![C: $i] : ((C = k1_closure3(A!0, B!1)) <=> (C = a_2_0_closure3(A!0, B!1))))) <=> ((~![B: $i] : ((~m1_pboole(B, A!0)) | ![C: $i] : ((C = k1_closure3(A!0, B)) <=> (C = a_2_0_closure3(A!0, B))))) | (~m1_pboole(B!1, A!0)) | ![C: $i] : ((C = k1_closure3(A!0, B!1)) <=> (C = a_2_0_closure3(A!0, B!1))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(34,plain,
% 0.12/0.40 ((~![B: $i] : ((~m1_pboole(B, A!0)) | ![C: $i] : ((C = k1_closure3(A!0, B)) <=> (C = a_2_0_closure3(A!0, B))))) | ((~m1_pboole(B!1, A!0)) | ![C: $i] : ((C = k1_closure3(A!0, B!1)) <=> (C = a_2_0_closure3(A!0, B!1))))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(35,plain,
% 0.12/0.40 ((~![B: $i] : ((~m1_pboole(B, A!0)) | ![C: $i] : ((C = k1_closure3(A!0, B)) <=> (C = a_2_0_closure3(A!0, B))))) | (~m1_pboole(B!1, A!0)) | ![C: $i] : ((C = k1_closure3(A!0, B!1)) <=> (C = a_2_0_closure3(A!0, B!1)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[34, 33])).
% 0.12/0.40 tff(36,plain,
% 0.12/0.40 (![C: $i] : ((C = k1_closure3(A!0, B!1)) <=> (C = a_2_0_closure3(A!0, B!1)))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[35, 32, 31])).
% 0.12/0.40 tff(37,plain,
% 0.12/0.40 (((~![C: $i] : ((C = k1_closure3(A!0, B!1)) <=> (C = a_2_0_closure3(A!0, B!1)))) | (k1_closure3(A!0, B!1) = a_2_0_closure3(A!0, B!1))) <=> ((~![C: $i] : ((C = k1_closure3(A!0, B!1)) <=> (C = a_2_0_closure3(A!0, B!1)))) | (k1_closure3(A!0, B!1) = a_2_0_closure3(A!0, B!1)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(38,plain,
% 0.12/0.40 (($true <=> (k1_closure3(A!0, B!1) = a_2_0_closure3(A!0, B!1))) <=> (k1_closure3(A!0, B!1) = a_2_0_closure3(A!0, B!1))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(39,plain,
% 0.12/0.40 ((k1_closure3(A!0, B!1) = k1_closure3(A!0, B!1)) <=> $true),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(40,plain,
% 0.12/0.40 (((k1_closure3(A!0, B!1) = k1_closure3(A!0, B!1)) <=> (k1_closure3(A!0, B!1) = a_2_0_closure3(A!0, B!1))) <=> ($true <=> (k1_closure3(A!0, B!1) = a_2_0_closure3(A!0, B!1)))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[39])).
% 0.12/0.40 tff(41,plain,
% 0.12/0.40 (((k1_closure3(A!0, B!1) = k1_closure3(A!0, B!1)) <=> (k1_closure3(A!0, B!1) = a_2_0_closure3(A!0, B!1))) <=> (k1_closure3(A!0, B!1) = a_2_0_closure3(A!0, B!1))),
% 0.12/0.40 inference(transitivity,[status(thm)],[40, 38])).
% 0.12/0.40 tff(42,plain,
% 0.12/0.40 (((~![C: $i] : ((C = k1_closure3(A!0, B!1)) <=> (C = a_2_0_closure3(A!0, B!1)))) | ((k1_closure3(A!0, B!1) = k1_closure3(A!0, B!1)) <=> (k1_closure3(A!0, B!1) = a_2_0_closure3(A!0, B!1)))) <=> ((~![C: $i] : ((C = k1_closure3(A!0, B!1)) <=> (C = a_2_0_closure3(A!0, B!1)))) | (k1_closure3(A!0, B!1) = a_2_0_closure3(A!0, B!1)))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[41])).
% 0.12/0.40 tff(43,plain,
% 0.12/0.40 (((~![C: $i] : ((C = k1_closure3(A!0, B!1)) <=> (C = a_2_0_closure3(A!0, B!1)))) | ((k1_closure3(A!0, B!1) = k1_closure3(A!0, B!1)) <=> (k1_closure3(A!0, B!1) = a_2_0_closure3(A!0, B!1)))) <=> ((~![C: $i] : ((C = k1_closure3(A!0, B!1)) <=> (C = a_2_0_closure3(A!0, B!1)))) | (k1_closure3(A!0, B!1) = a_2_0_closure3(A!0, B!1)))),
% 0.12/0.40 inference(transitivity,[status(thm)],[42, 37])).
% 0.12/0.40 tff(44,plain,
% 0.12/0.40 ((~![C: $i] : ((C = k1_closure3(A!0, B!1)) <=> (C = a_2_0_closure3(A!0, B!1)))) | ((k1_closure3(A!0, B!1) = k1_closure3(A!0, B!1)) <=> (k1_closure3(A!0, B!1) = a_2_0_closure3(A!0, B!1)))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(45,plain,
% 0.12/0.40 ((~![C: $i] : ((C = k1_closure3(A!0, B!1)) <=> (C = a_2_0_closure3(A!0, B!1)))) | (k1_closure3(A!0, B!1) = a_2_0_closure3(A!0, B!1))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.12/0.40 tff(46,plain,
% 0.12/0.40 (k1_closure3(A!0, B!1) = a_2_0_closure3(A!0, B!1)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[45, 36])).
% 0.12/0.40 tff(47,plain,
% 0.12/0.40 (r2_hidden(C!2, k1_closure3(A!0, B!1)) <=> r2_hidden(C!2, a_2_0_closure3(A!0, B!1))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[46])).
% 0.12/0.40 tff(48,plain,
% 0.12/0.40 (r2_hidden(C!2, a_2_0_closure3(A!0, B!1)) <=> r2_hidden(C!2, k1_closure3(A!0, B!1))),
% 0.12/0.40 inference(symmetry,[status(thm)],[47])).
% 0.12/0.40 tff(49,plain,
% 0.12/0.40 (^[A: $i, B: $i, C: $i] : refl((v1_xboole_0(B) | (~m1_pboole(C, B)) | (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C))))) <=> (v1_xboole_0(B) | (~m1_pboole(C, B)) | (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(50,plain,
% 0.12/0.40 (![A: $i, B: $i, C: $i] : (v1_xboole_0(B) | (~m1_pboole(C, B)) | (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C))))) <=> ![A: $i, B: $i, C: $i] : (v1_xboole_0(B) | (~m1_pboole(C, B)) | (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C)))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[49])).
% 0.12/0.40 tff(51,plain,
% 0.12/0.40 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~v1_xboole_0(B)) & m1_pboole(C, B)) <=> (~(v1_xboole_0(B) | (~m1_pboole(C, B))))), ((~((~v1_xboole_0(B)) & m1_pboole(C, B))) <=> (~(~(v1_xboole_0(B) | (~m1_pboole(C, B))))))), rewrite((~(~(v1_xboole_0(B) | (~m1_pboole(C, B))))) <=> (v1_xboole_0(B) | (~m1_pboole(C, B)))), ((~((~v1_xboole_0(B)) & m1_pboole(C, B))) <=> (v1_xboole_0(B) | (~m1_pboole(C, B))))), trans(trans(monotonicity(rewrite(((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B)) <=> (~((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))))), ((r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B))) <=> (r2_hidden(A, a_2_0_closure3(B, C)) <=> (~((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))))))), rewrite((r2_hidden(A, a_2_0_closure3(B, C)) <=> (~((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))))) <=> (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C))))), ((r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B))) <=> (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C)))))), rewrite((~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C)))) <=> (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C))))), ((r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B))) <=> (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C)))))), (((~((~v1_xboole_0(B)) & m1_pboole(C, B))) | (r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B)))) <=> ((v1_xboole_0(B) | (~m1_pboole(C, B))) | (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C))))))), rewrite(((v1_xboole_0(B) | (~m1_pboole(C, B))) | (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C))))) <=> (v1_xboole_0(B) | (~m1_pboole(C, B)) | (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C)))))), (((~((~v1_xboole_0(B)) & m1_pboole(C, B))) | (r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B)))) <=> (v1_xboole_0(B) | (~m1_pboole(C, B)) | (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C)))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(52,plain,
% 0.12/0.40 (![A: $i, B: $i, C: $i] : ((~((~v1_xboole_0(B)) & m1_pboole(C, B))) | (r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B)))) <=> ![A: $i, B: $i, C: $i] : (v1_xboole_0(B) | (~m1_pboole(C, B)) | (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C)))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[51])).
% 0.12/0.40 tff(53,plain,
% 0.12/0.40 (![A: $i, B: $i, C: $i] : ((~((~v1_xboole_0(B)) & m1_pboole(C, B))) | (r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B)))) <=> ![A: $i, B: $i, C: $i] : ((~((~v1_xboole_0(B)) & m1_pboole(C, B))) | (r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(54,plain,
% 0.12/0.40 (^[A: $i, B: $i, C: $i] : trans(monotonicity(rewrite((r2_hidden(A, a_2_0_closure3(B, C)) <=> ?[D: $i] : ((m1_subset_1(D, B) & (A = D)) & (~(k1_funct_1(C, D) = k1_xboole_0)))) <=> (r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B)))), ((((~v1_xboole_0(B)) & m1_pboole(C, B)) => (r2_hidden(A, a_2_0_closure3(B, C)) <=> ?[D: $i] : ((m1_subset_1(D, B) & (A = D)) & (~(k1_funct_1(C, D) = k1_xboole_0))))) <=> (((~v1_xboole_0(B)) & m1_pboole(C, B)) => (r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B)))))), rewrite((((~v1_xboole_0(B)) & m1_pboole(C, B)) => (r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B)))) <=> ((~((~v1_xboole_0(B)) & m1_pboole(C, B))) | (r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B))))), ((((~v1_xboole_0(B)) & m1_pboole(C, B)) => (r2_hidden(A, a_2_0_closure3(B, C)) <=> ?[D: $i] : ((m1_subset_1(D, B) & (A = D)) & (~(k1_funct_1(C, D) = k1_xboole_0))))) <=> ((~((~v1_xboole_0(B)) & m1_pboole(C, B))) | (r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B))))))),
% 0.12/0.41 inference(bind,[status(th)],[])).
% 0.12/0.41 tff(55,plain,
% 0.12/0.41 (![A: $i, B: $i, C: $i] : (((~v1_xboole_0(B)) & m1_pboole(C, B)) => (r2_hidden(A, a_2_0_closure3(B, C)) <=> ?[D: $i] : ((m1_subset_1(D, B) & (A = D)) & (~(k1_funct_1(C, D) = k1_xboole_0))))) <=> ![A: $i, B: $i, C: $i] : ((~((~v1_xboole_0(B)) & m1_pboole(C, B))) | (r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B))))),
% 0.12/0.41 inference(quant_intro,[status(thm)],[54])).
% 0.12/0.41 tff(56,axiom,(![A: $i, B: $i, C: $i] : (((~v1_xboole_0(B)) & m1_pboole(C, B)) => (r2_hidden(A, a_2_0_closure3(B, C)) <=> ?[D: $i] : ((m1_subset_1(D, B) & (A = D)) & (~(k1_funct_1(C, D) = k1_xboole_0)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','fraenkel_a_2_0_closure3')).
% 0.12/0.41 tff(57,plain,
% 0.12/0.41 (![A: $i, B: $i, C: $i] : ((~((~v1_xboole_0(B)) & m1_pboole(C, B))) | (r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[56, 55])).
% 0.12/0.41 tff(58,plain,
% 0.12/0.41 (![A: $i, B: $i, C: $i] : ((~((~v1_xboole_0(B)) & m1_pboole(C, B))) | (r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[57, 53])).
% 0.12/0.41 tff(59,plain,(
% 0.12/0.41 ![A: $i, B: $i, C: $i] : ((~((~v1_xboole_0(B)) & m1_pboole(C, B))) | (r2_hidden(A, a_2_0_closure3(B, C)) <=> ((~(k1_funct_1(C, A) = k1_xboole_0)) & m1_subset_1(A, B))))),
% 0.12/0.41 inference(skolemize,[status(sab)],[58])).
% 0.12/0.41 tff(60,plain,
% 0.12/0.41 (![A: $i, B: $i, C: $i] : (v1_xboole_0(B) | (~m1_pboole(C, B)) | (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C)))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[59, 52])).
% 0.12/0.41 tff(61,plain,
% 0.12/0.41 (![A: $i, B: $i, C: $i] : (v1_xboole_0(B) | (~m1_pboole(C, B)) | (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C)))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[60, 50])).
% 0.12/0.41 tff(62,plain,
% 0.12/0.41 (((~![A: $i, B: $i, C: $i] : (v1_xboole_0(B) | (~m1_pboole(C, B)) | (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C)))))) | (v1_xboole_0(A!0) | (~m1_pboole(B!1, A!0)) | (~(((k1_funct_1(B!1, C!2) = k1_xboole_0) | (~m1_subset_1(C!2, A!0))) <=> r2_hidden(C!2, a_2_0_closure3(A!0, B!1)))))) <=> ((~![A: $i, B: $i, C: $i] : (v1_xboole_0(B) | (~m1_pboole(C, B)) | (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C)))))) | v1_xboole_0(A!0) | (~m1_pboole(B!1, A!0)) | (~(((k1_funct_1(B!1, C!2) = k1_xboole_0) | (~m1_subset_1(C!2, A!0))) <=> r2_hidden(C!2, a_2_0_closure3(A!0, B!1)))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(63,plain,
% 0.12/0.41 ((~![A: $i, B: $i, C: $i] : (v1_xboole_0(B) | (~m1_pboole(C, B)) | (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C)))))) | (v1_xboole_0(A!0) | (~m1_pboole(B!1, A!0)) | (~(((k1_funct_1(B!1, C!2) = k1_xboole_0) | (~m1_subset_1(C!2, A!0))) <=> r2_hidden(C!2, a_2_0_closure3(A!0, B!1)))))),
% 0.12/0.41 inference(quant_inst,[status(thm)],[])).
% 0.12/0.41 tff(64,plain,
% 0.12/0.41 ((~![A: $i, B: $i, C: $i] : (v1_xboole_0(B) | (~m1_pboole(C, B)) | (~(((k1_funct_1(C, A) = k1_xboole_0) | (~m1_subset_1(A, B))) <=> r2_hidden(A, a_2_0_closure3(B, C)))))) | v1_xboole_0(A!0) | (~m1_pboole(B!1, A!0)) | (~(((k1_funct_1(B!1, C!2) = k1_xboole_0) | (~m1_subset_1(C!2, A!0))) <=> r2_hidden(C!2, a_2_0_closure3(A!0, B!1))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[63, 62])).
% 0.12/0.41 tff(65,plain,
% 0.12/0.41 (~(((k1_funct_1(B!1, C!2) = k1_xboole_0) | (~m1_subset_1(C!2, A!0))) <=> r2_hidden(C!2, a_2_0_closure3(A!0, B!1)))),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[64, 27, 32, 61])).
% 0.12/0.41 tff(66,plain,
% 0.12/0.41 (~((k1_funct_1(B!1, C!2) = k1_xboole_0) | r2_hidden(C!2, k1_closure3(A!0, B!1)) | (~m1_subset_1(C!2, A!0)))),
% 0.12/0.41 inference(and_elim,[status(thm)],[26])).
% 0.12/0.41 tff(67,plain,
% 0.12/0.41 (m1_subset_1(C!2, A!0)),
% 0.12/0.41 inference(or_elim,[status(thm)],[66])).
% 0.12/0.41 tff(68,plain,
% 0.12/0.41 (~(k1_funct_1(B!1, C!2) = k1_xboole_0)),
% 0.12/0.41 inference(or_elim,[status(thm)],[66])).
% 0.21/0.41 tff(69,plain,
% 0.21/0.41 ((~((k1_funct_1(B!1, C!2) = k1_xboole_0) | (~m1_subset_1(C!2, A!0)))) | (k1_funct_1(B!1, C!2) = k1_xboole_0) | (~m1_subset_1(C!2, A!0))),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(70,plain,
% 0.21/0.41 (~((k1_funct_1(B!1, C!2) = k1_xboole_0) | (~m1_subset_1(C!2, A!0)))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[69, 68, 67])).
% 0.21/0.41 tff(71,plain,
% 0.21/0.41 ((((k1_funct_1(B!1, C!2) = k1_xboole_0) | (~m1_subset_1(C!2, A!0))) <=> r2_hidden(C!2, a_2_0_closure3(A!0, B!1))) | ((k1_funct_1(B!1, C!2) = k1_xboole_0) | (~m1_subset_1(C!2, A!0))) | r2_hidden(C!2, a_2_0_closure3(A!0, B!1))),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(72,plain,
% 0.21/0.41 ((((k1_funct_1(B!1, C!2) = k1_xboole_0) | (~m1_subset_1(C!2, A!0))) <=> r2_hidden(C!2, a_2_0_closure3(A!0, B!1))) | r2_hidden(C!2, a_2_0_closure3(A!0, B!1))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[71, 70])).
% 0.21/0.41 tff(73,plain,
% 0.21/0.41 (r2_hidden(C!2, a_2_0_closure3(A!0, B!1))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[72, 65])).
% 0.21/0.41 tff(74,plain,
% 0.21/0.41 (r2_hidden(C!2, k1_closure3(A!0, B!1))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[73, 48])).
% 0.21/0.41 tff(75,plain,
% 0.21/0.41 (~r2_hidden(C!2, k1_closure3(A!0, B!1))),
% 0.21/0.41 inference(or_elim,[status(thm)],[66])).
% 0.21/0.41 tff(76,plain,
% 0.21/0.41 ($false),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[75, 74])).
% 0.21/0.41 % SZS output end Proof
% 0.21/0.41 % E exiting
%------------------------------------------------------------------------------