%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : REL005+2 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 12:33:00 PM UTC 2026
% Result : Theorem 3.62s 1.08s
% Output : Proof 4.99s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : REL005+2 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.10/0.35 % Computer : n003.cluster.edu
% 0.10/0.35 % Model : x86_64 x86_64
% 0.10/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.35 % Memory : 8046.5625MB
% 0.10/0.35 % OS : Linux 6.8.0-71-generic
% 0.10/0.35 % CPULimit : 300
% 0.10/0.35 % WCLimit : 300
% 0.10/0.35 % DateTime : Sun Sep 27 22:51:41 UTC 2026
% 0.10/0.35 % CPUTime :
% 0.10/0.35 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 3.62/1.08 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 3.62/1.08
% 3.62/1.08 % SZS status Theorem
% 3.62/1.08
% 4.99/1.13 % SZS output start Proof
% 4.99/1.13 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 4.99/1.13 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 4.99/1.13 Axiom 3 (composition_identity): composition(X, one) = X.
% 4.99/1.13 Axiom 4 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 4.99/1.13 Axiom 5 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 4.99/1.13 Axiom 6 (def_zero): zero = meet(X, complement(X)).
% 4.99/1.13 Axiom 7 (def_top): top = join(X, complement(X)).
% 4.99/1.13 Axiom 8 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 4.99/1.13 Axiom 9 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 4.99/1.13 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 4.99/1.13 Axiom 11 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 4.99/1.13 Axiom 12 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 4.99/1.13
% 4.99/1.13 Lemma 13: complement(top) = zero.
% 4.99/1.13 Proof:
% 4.99/1.13 complement(top)
% 4.99/1.13 = { by axiom 7 (def_top) }
% 4.99/1.13 complement(join(complement(X), complement(complement(X))))
% 4.99/1.13 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 4.99/1.13 meet(X, complement(X))
% 4.99/1.13 = { by axiom 6 (def_zero) R->L }
% 4.99/1.13 zero
% 4.99/1.13
% 4.99/1.13 Lemma 14: join(X, join(Y, complement(X))) = join(Y, top).
% 4.99/1.13 Proof:
% 4.99/1.13 join(X, join(Y, complement(X)))
% 4.99/1.13 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.99/1.13 join(X, join(complement(X), Y))
% 4.99/1.13 = { by axiom 8 (maddux2_join_associativity) }
% 4.99/1.13 join(join(X, complement(X)), Y)
% 4.99/1.13 = { by axiom 7 (def_top) R->L }
% 4.99/1.13 join(top, Y)
% 4.99/1.13 = { by axiom 2 (maddux1_join_commutativity) }
% 4.99/1.13 join(Y, top)
% 4.99/1.13
% 4.99/1.13 Lemma 15: composition(converse(one), X) = X.
% 4.99/1.13 Proof:
% 4.99/1.13 composition(converse(one), X)
% 4.99/1.13 = { by axiom 1 (converse_idempotence) R->L }
% 4.99/1.13 composition(converse(one), converse(converse(X)))
% 4.99/1.13 = { by axiom 5 (converse_multiplicativity) R->L }
% 4.99/1.13 converse(composition(converse(X), one))
% 4.99/1.13 = { by axiom 3 (composition_identity) }
% 4.99/1.13 converse(converse(X))
% 4.99/1.13 = { by axiom 1 (converse_idempotence) }
% 4.99/1.13 X
% 4.99/1.13
% 4.99/1.13 Lemma 16: join(complement(X), complement(X)) = complement(X).
% 4.99/1.13 Proof:
% 4.99/1.13 join(complement(X), complement(X))
% 4.99/1.13 = { by lemma 15 R->L }
% 4.99/1.13 join(complement(X), composition(converse(one), complement(X)))
% 4.99/1.13 = { by lemma 15 R->L }
% 4.99/1.13 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 4.99/1.13 = { by axiom 3 (composition_identity) R->L }
% 4.99/1.13 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 4.99/1.13 = { by axiom 9 (composition_associativity) R->L }
% 4.99/1.13 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 4.99/1.13 = { by lemma 15 }
% 4.99/1.13 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 4.99/1.13 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.99/1.13 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 4.99/1.13 = { by axiom 11 (converse_cancellativity) }
% 4.99/1.13 complement(X)
% 4.99/1.13
% 4.99/1.13 Lemma 17: join(top, complement(X)) = top.
% 4.99/1.13 Proof:
% 4.99/1.13 join(top, complement(X))
% 4.99/1.13 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.99/1.13 join(complement(X), top)
% 4.99/1.13 = { by lemma 14 R->L }
% 4.99/1.13 join(X, join(complement(X), complement(X)))
% 4.99/1.13 = { by lemma 16 }
% 4.99/1.13 join(X, complement(X))
% 4.99/1.13 = { by axiom 7 (def_top) R->L }
% 4.99/1.13 top
% 4.99/1.13
% 4.99/1.13 Lemma 18: join(X, top) = top.
% 4.99/1.13 Proof:
% 4.99/1.13 join(X, top)
% 4.99/1.13 = { by lemma 17 R->L }
% 4.99/1.13 join(X, join(top, complement(X)))
% 4.99/1.13 = { by lemma 14 }
% 4.99/1.13 join(top, top)
% 4.99/1.13 = { by axiom 7 (def_top) }
% 4.99/1.13 join(top, join(zero, complement(zero)))
% 4.99/1.13 = { by axiom 8 (maddux2_join_associativity) }
% 4.99/1.13 join(join(top, zero), complement(zero))
% 4.99/1.13 = { by lemma 13 R->L }
% 4.99/1.13 join(join(top, complement(top)), complement(zero))
% 4.99/1.13 = { by axiom 7 (def_top) R->L }
% 4.99/1.13 join(top, complement(zero))
% 4.99/1.13 = { by lemma 17 }
% 4.99/1.13 top
% 4.99/1.13
% 4.99/1.13 Lemma 19: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 4.99/1.13 Proof:
% 4.99/1.13 join(meet(X, Y), complement(join(complement(X), Y)))
% 4.99/1.14 = { by axiom 10 (maddux4_definiton_of_meet) }
% 4.99/1.14 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 4.99/1.14 = { by axiom 12 (maddux3_a_kind_of_de_Morgan) R->L }
% 4.99/1.14 X
% 4.99/1.14
% 4.99/1.14 Lemma 20: meet(Y, X) = meet(X, Y).
% 4.99/1.14 Proof:
% 4.99/1.14 meet(Y, X)
% 4.99/1.14 = { by axiom 10 (maddux4_definiton_of_meet) }
% 4.99/1.14 complement(join(complement(Y), complement(X)))
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.99/1.14 complement(join(complement(X), complement(Y)))
% 4.99/1.14 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 4.99/1.14 meet(X, Y)
% 4.99/1.14
% 4.99/1.14 Lemma 21: complement(complement(X)) = meet(X, X).
% 4.99/1.14 Proof:
% 4.99/1.14 complement(complement(X))
% 4.99/1.14 = { by lemma 16 R->L }
% 4.99/1.14 complement(join(complement(X), complement(X)))
% 4.99/1.14 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 4.99/1.14 meet(X, X)
% 4.99/1.14
% 4.99/1.14 Lemma 22: complement(join(zero, complement(X))) = meet(X, top).
% 4.99/1.14 Proof:
% 4.99/1.14 complement(join(zero, complement(X)))
% 4.99/1.14 = { by lemma 13 R->L }
% 4.99/1.14 complement(join(complement(top), complement(X)))
% 4.99/1.14 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 4.99/1.14 meet(top, X)
% 4.99/1.14 = { by lemma 20 R->L }
% 4.99/1.14 meet(X, top)
% 4.99/1.14
% 4.99/1.14 Lemma 23: join(zero, meet(X, X)) = X.
% 4.99/1.14 Proof:
% 4.99/1.14 join(zero, meet(X, X))
% 4.99/1.14 = { by axiom 10 (maddux4_definiton_of_meet) }
% 4.99/1.14 join(zero, complement(join(complement(X), complement(X))))
% 4.99/1.14 = { by axiom 6 (def_zero) }
% 4.99/1.14 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 4.99/1.14 = { by lemma 19 }
% 4.99/1.14 X
% 4.99/1.14
% 4.99/1.14 Lemma 24: meet(top, complement(X)) = complement(X).
% 4.99/1.14 Proof:
% 4.99/1.14 meet(top, complement(X))
% 4.99/1.14 = { by lemma 20 }
% 4.99/1.14 meet(complement(X), top)
% 4.99/1.14 = { by lemma 22 R->L }
% 4.99/1.14 complement(join(zero, complement(complement(X))))
% 4.99/1.14 = { by lemma 21 }
% 4.99/1.14 complement(join(zero, meet(X, X)))
% 4.99/1.14 = { by lemma 23 }
% 4.99/1.14 complement(X)
% 4.99/1.14
% 4.99/1.14 Lemma 25: meet(X, X) = X.
% 4.99/1.14 Proof:
% 4.99/1.14 meet(X, X)
% 4.99/1.14 = { by lemma 19 R->L }
% 4.99/1.14 join(meet(meet(X, X), top), complement(join(complement(meet(X, X)), top)))
% 4.99/1.14 = { by lemma 18 }
% 4.99/1.14 join(meet(meet(X, X), top), complement(top))
% 4.99/1.14 = { by lemma 13 }
% 4.99/1.14 join(meet(meet(X, X), top), zero)
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) }
% 4.99/1.14 join(zero, meet(meet(X, X), top))
% 4.99/1.14 = { by lemma 20 R->L }
% 4.99/1.14 join(zero, meet(top, meet(X, X)))
% 4.99/1.14 = { by lemma 21 R->L }
% 4.99/1.14 join(zero, meet(top, complement(complement(X))))
% 4.99/1.14 = { by lemma 24 }
% 4.99/1.14 join(zero, complement(complement(X)))
% 4.99/1.14 = { by lemma 21 }
% 4.99/1.14 join(zero, meet(X, X))
% 4.99/1.14 = { by lemma 23 }
% 4.99/1.14 X
% 4.99/1.14
% 4.99/1.14 Lemma 26: join(X, zero) = X.
% 4.99/1.14 Proof:
% 4.99/1.14 join(X, zero)
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.99/1.14 join(zero, X)
% 4.99/1.14 = { by lemma 25 R->L }
% 4.99/1.14 join(zero, meet(X, X))
% 4.99/1.14 = { by lemma 23 }
% 4.99/1.14 X
% 4.99/1.14
% 4.99/1.14 Lemma 27: join(zero, X) = X.
% 4.99/1.14 Proof:
% 4.99/1.14 join(zero, X)
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.99/1.14 join(X, zero)
% 4.99/1.14 = { by lemma 26 }
% 4.99/1.14 X
% 4.99/1.14
% 4.99/1.14 Lemma 28: complement(complement(X)) = X.
% 4.99/1.14 Proof:
% 4.99/1.14 complement(complement(X))
% 4.99/1.14 = { by lemma 21 }
% 4.99/1.14 meet(X, X)
% 4.99/1.14 = { by lemma 25 }
% 4.99/1.14 X
% 4.99/1.14
% 4.99/1.14 Lemma 29: meet(X, top) = X.
% 4.99/1.14 Proof:
% 4.99/1.14 meet(X, top)
% 4.99/1.14 = { by lemma 20 }
% 4.99/1.14 meet(top, X)
% 4.99/1.14 = { by lemma 25 R->L }
% 4.99/1.14 meet(top, meet(X, X))
% 4.99/1.14 = { by axiom 10 (maddux4_definiton_of_meet) }
% 4.99/1.14 meet(top, complement(join(complement(X), complement(X))))
% 4.99/1.14 = { by lemma 24 }
% 4.99/1.14 complement(join(complement(X), complement(X)))
% 4.99/1.14 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 4.99/1.14 meet(X, X)
% 4.99/1.14 = { by lemma 25 }
% 4.99/1.14 X
% 4.99/1.14
% 4.99/1.14 Lemma 30: join(X, meet(X, Y)) = X.
% 4.99/1.14 Proof:
% 4.99/1.14 join(X, meet(X, Y))
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.99/1.14 join(meet(X, Y), X)
% 4.99/1.14 = { by lemma 19 R->L }
% 4.99/1.14 join(meet(X, Y), join(meet(X, Y), complement(join(complement(X), Y))))
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.99/1.14 join(meet(X, Y), join(complement(join(complement(X), Y)), meet(X, Y)))
% 4.99/1.14 = { by lemma 28 R->L }
% 4.99/1.14 join(meet(X, Y), join(complement(join(complement(X), Y)), complement(complement(meet(X, Y)))))
% 4.99/1.14 = { by lemma 25 R->L }
% 4.99/1.14 join(meet(meet(X, Y), meet(X, Y)), join(complement(join(complement(X), Y)), complement(complement(meet(X, Y)))))
% 4.99/1.14 = { by lemma 21 R->L }
% 4.99/1.14 join(complement(complement(meet(X, Y))), join(complement(join(complement(X), Y)), complement(complement(meet(X, Y)))))
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.99/1.14 join(complement(complement(meet(X, Y))), join(complement(complement(meet(X, Y))), complement(join(complement(X), Y))))
% 4.99/1.14 = { by axiom 8 (maddux2_join_associativity) }
% 4.99/1.14 join(join(complement(complement(meet(X, Y))), complement(complement(meet(X, Y)))), complement(join(complement(X), Y)))
% 4.99/1.14 = { by lemma 16 }
% 4.99/1.14 join(complement(complement(meet(X, Y))), complement(join(complement(X), Y)))
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) }
% 4.99/1.14 join(complement(join(complement(X), Y)), complement(complement(meet(X, Y))))
% 4.99/1.14 = { by lemma 28 }
% 4.99/1.14 join(complement(join(complement(X), Y)), meet(X, Y))
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) }
% 4.99/1.14 join(meet(X, Y), complement(join(complement(X), Y)))
% 4.99/1.14 = { by lemma 19 }
% 4.99/1.14 X
% 4.99/1.14
% 4.99/1.14 Lemma 31: join(X, meet(Y, X)) = X.
% 4.99/1.14 Proof:
% 4.99/1.14 join(X, meet(Y, X))
% 4.99/1.14 = { by lemma 20 }
% 4.99/1.14 join(X, meet(X, Y))
% 4.99/1.14 = { by lemma 30 }
% 4.99/1.14 X
% 4.99/1.14
% 4.99/1.14 Lemma 32: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 4.99/1.14 Proof:
% 4.99/1.14 join(complement(X), complement(Y))
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.99/1.14 join(complement(Y), complement(X))
% 4.99/1.14 = { by lemma 29 R->L }
% 4.99/1.14 meet(join(complement(Y), complement(X)), top)
% 4.99/1.14 = { by lemma 20 R->L }
% 4.99/1.14 meet(top, join(complement(Y), complement(X)))
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.99/1.14 meet(top, join(complement(X), complement(Y)))
% 4.99/1.14 = { by lemma 20 }
% 4.99/1.14 meet(join(complement(X), complement(Y)), top)
% 4.99/1.14 = { by axiom 10 (maddux4_definiton_of_meet) }
% 4.99/1.14 complement(join(complement(join(complement(X), complement(Y))), complement(top)))
% 4.99/1.14 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 4.99/1.14 complement(join(meet(X, Y), complement(top)))
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) }
% 4.99/1.14 complement(join(complement(top), meet(X, Y)))
% 4.99/1.14 = { by lemma 20 R->L }
% 4.99/1.14 complement(join(complement(top), meet(Y, X)))
% 4.99/1.14 = { by lemma 13 }
% 4.99/1.14 complement(join(zero, meet(Y, X)))
% 4.99/1.14 = { by lemma 27 }
% 4.99/1.14 complement(meet(Y, X))
% 4.99/1.14 = { by lemma 20 R->L }
% 4.99/1.14 complement(meet(X, Y))
% 4.99/1.14
% 4.99/1.14 Lemma 33: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 4.99/1.14 Proof:
% 4.99/1.14 complement(meet(X, complement(Y)))
% 4.99/1.14 = { by lemma 20 }
% 4.99/1.14 complement(meet(complement(Y), X))
% 4.99/1.14 = { by lemma 27 R->L }
% 4.99/1.14 complement(meet(join(zero, complement(Y)), X))
% 4.99/1.14 = { by lemma 32 R->L }
% 4.99/1.14 join(complement(join(zero, complement(Y))), complement(X))
% 4.99/1.14 = { by lemma 22 }
% 4.99/1.14 join(meet(Y, top), complement(X))
% 4.99/1.14 = { by lemma 29 }
% 4.99/1.14 join(Y, complement(X))
% 4.99/1.14
% 4.99/1.14 Lemma 34: meet(X, join(X, Y)) = X.
% 4.99/1.14 Proof:
% 4.99/1.14 meet(X, join(X, Y))
% 4.99/1.14 = { by lemma 29 R->L }
% 4.99/1.14 meet(X, join(X, meet(Y, top)))
% 4.99/1.14 = { by lemma 22 R->L }
% 4.99/1.14 meet(X, join(X, complement(join(zero, complement(Y)))))
% 4.99/1.14 = { by lemma 26 R->L }
% 4.99/1.14 join(meet(X, join(X, complement(join(zero, complement(Y))))), zero)
% 4.99/1.14 = { by lemma 13 R->L }
% 4.99/1.14 join(meet(X, join(X, complement(join(zero, complement(Y))))), complement(top))
% 4.99/1.14 = { by lemma 33 R->L }
% 4.99/1.14 join(meet(X, complement(meet(join(zero, complement(Y)), complement(X)))), complement(top))
% 4.99/1.14 = { by lemma 20 R->L }
% 4.99/1.14 join(meet(X, complement(meet(complement(X), join(zero, complement(Y))))), complement(top))
% 4.99/1.14 = { by lemma 18 R->L }
% 4.99/1.14 join(meet(X, complement(meet(complement(X), join(zero, complement(Y))))), complement(join(complement(join(zero, complement(Y))), top)))
% 4.99/1.14 = { by lemma 14 R->L }
% 4.99/1.14 join(meet(X, complement(meet(complement(X), join(zero, complement(Y))))), complement(join(complement(X), join(complement(join(zero, complement(Y))), complement(complement(X))))))
% 4.99/1.14 = { by lemma 32 }
% 4.99/1.14 join(meet(X, complement(meet(complement(X), join(zero, complement(Y))))), complement(join(complement(X), complement(meet(join(zero, complement(Y)), complement(X))))))
% 4.99/1.14 = { by lemma 20 R->L }
% 4.99/1.14 join(meet(X, complement(meet(complement(X), join(zero, complement(Y))))), complement(join(complement(X), complement(meet(complement(X), join(zero, complement(Y)))))))
% 4.99/1.14 = { by lemma 19 }
% 4.99/1.14 X
% 4.99/1.14
% 4.99/1.14 Lemma 35: meet(X, join(Y, X)) = X.
% 4.99/1.14 Proof:
% 4.99/1.14 meet(X, join(Y, X))
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.99/1.14 meet(X, join(X, Y))
% 4.99/1.14 = { by lemma 34 }
% 4.99/1.14 X
% 4.99/1.14
% 4.99/1.14 Lemma 36: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 4.99/1.14 Proof:
% 4.99/1.14 meet(Y, meet(X, Z))
% 4.99/1.14 = { by lemma 20 }
% 4.99/1.14 meet(Y, meet(Z, X))
% 4.99/1.14 = { by lemma 29 R->L }
% 4.99/1.14 meet(meet(Y, meet(Z, X)), top)
% 4.99/1.14 = { by lemma 22 R->L }
% 4.99/1.14 complement(join(zero, complement(meet(Y, meet(Z, X)))))
% 4.99/1.14 = { by lemma 20 }
% 4.99/1.14 complement(join(zero, complement(meet(Y, meet(X, Z)))))
% 4.99/1.14 = { by axiom 10 (maddux4_definiton_of_meet) }
% 4.99/1.14 complement(join(zero, complement(meet(Y, complement(join(complement(X), complement(Z)))))))
% 4.99/1.14 = { by lemma 33 }
% 4.99/1.14 complement(join(zero, join(join(complement(X), complement(Z)), complement(Y))))
% 4.99/1.14 = { by axiom 8 (maddux2_join_associativity) R->L }
% 4.99/1.14 complement(join(zero, join(complement(X), join(complement(Z), complement(Y)))))
% 4.99/1.14 = { by lemma 32 }
% 4.99/1.14 complement(join(zero, join(complement(X), complement(meet(Z, Y)))))
% 4.99/1.14 = { by lemma 32 }
% 4.99/1.14 complement(join(zero, complement(meet(X, meet(Z, Y)))))
% 4.99/1.14 = { by lemma 20 R->L }
% 4.99/1.14 complement(join(zero, complement(meet(X, meet(Y, Z)))))
% 4.99/1.14 = { by lemma 22 }
% 4.99/1.14 meet(meet(X, meet(Y, Z)), top)
% 4.99/1.14 = { by lemma 29 }
% 4.99/1.14 meet(X, meet(Y, Z))
% 4.99/1.14
% 4.99/1.14 Lemma 37: meet(meet(converse(X), converse(Y)), converse(meet(X, Y))) = converse(meet(X, Y)).
% 4.99/1.14 Proof:
% 4.99/1.14 meet(meet(converse(X), converse(Y)), converse(meet(X, Y)))
% 4.99/1.14 = { by lemma 20 }
% 4.99/1.14 meet(converse(meet(X, Y)), meet(converse(X), converse(Y)))
% 4.99/1.14 = { by lemma 36 R->L }
% 4.99/1.14 meet(converse(X), meet(converse(meet(X, Y)), converse(Y)))
% 4.99/1.14 = { by lemma 31 R->L }
% 4.99/1.14 meet(converse(X), meet(converse(meet(X, Y)), converse(join(Y, meet(X, Y)))))
% 4.99/1.14 = { by axiom 4 (converse_additivity) }
% 4.99/1.14 meet(converse(X), meet(converse(meet(X, Y)), join(converse(Y), converse(meet(X, Y)))))
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) }
% 4.99/1.14 meet(converse(X), meet(converse(meet(X, Y)), join(converse(meet(X, Y)), converse(Y))))
% 4.99/1.14 = { by lemma 34 }
% 4.99/1.14 meet(converse(X), converse(meet(X, Y)))
% 4.99/1.14 = { by lemma 20 R->L }
% 4.99/1.14 meet(converse(meet(X, Y)), converse(X))
% 4.99/1.14 = { by lemma 30 R->L }
% 4.99/1.14 meet(converse(meet(X, Y)), converse(join(X, meet(X, Y))))
% 4.99/1.14 = { by axiom 4 (converse_additivity) }
% 4.99/1.14 meet(converse(meet(X, Y)), join(converse(X), converse(meet(X, Y))))
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) }
% 4.99/1.14 meet(converse(meet(X, Y)), join(converse(meet(X, Y)), converse(X)))
% 4.99/1.14 = { by lemma 34 }
% 4.99/1.14 converse(meet(X, Y))
% 4.99/1.14
% 4.99/1.14 Lemma 38: join(meet(converse(X), converse(Y)), converse(meet(X, Y))) = meet(converse(X), converse(Y)).
% 4.99/1.14 Proof:
% 4.99/1.14 join(meet(converse(X), converse(Y)), converse(meet(X, Y)))
% 4.99/1.14 = { by lemma 37 R->L }
% 4.99/1.14 join(meet(converse(X), converse(Y)), meet(meet(converse(X), converse(Y)), converse(meet(X, Y))))
% 4.99/1.14 = { by lemma 30 }
% 4.99/1.14 meet(converse(X), converse(Y))
% 4.99/1.14
% 4.99/1.14 Goal 1 (goals): tuple(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))) = tuple(converse(meet(x0, x1)), meet(converse(x0_2), converse(x1_2))).
% 4.99/1.14 Proof:
% 4.99/1.14 tuple(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) }
% 4.99/1.14 tuple(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))), join(meet(converse(x0_2), converse(x1_2)), converse(meet(x0_2, x1_2))))
% 4.99/1.14 = { by lemma 37 R->L }
% 4.99/1.14 tuple(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))), join(meet(converse(x0_2), converse(x1_2)), meet(meet(converse(x0_2), converse(x1_2)), converse(meet(x0_2, x1_2)))))
% 4.99/1.14 = { by lemma 30 }
% 4.99/1.14 tuple(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.14 = { by axiom 1 (converse_idempotence) R->L }
% 4.99/1.14 tuple(converse(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.14 = { by lemma 38 }
% 4.99/1.14 tuple(converse(converse(meet(converse(x0), converse(x1)))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.14 = { by lemma 35 R->L }
% 4.99/1.14 tuple(converse(meet(converse(meet(converse(x0), converse(x1))), join(x0, converse(meet(converse(x0), converse(x1)))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.99/1.14 tuple(converse(meet(converse(meet(converse(x0), converse(x1))), join(converse(meet(converse(x0), converse(x1))), x0))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.14 = { by axiom 1 (converse_idempotence) R->L }
% 4.99/1.14 tuple(converse(meet(converse(meet(converse(x0), converse(x1))), join(converse(meet(converse(x0), converse(x1))), converse(converse(x0))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.14 = { by axiom 4 (converse_additivity) R->L }
% 4.99/1.14 tuple(converse(meet(converse(meet(converse(x0), converse(x1))), converse(join(meet(converse(x0), converse(x1)), converse(x0))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.99/1.14 tuple(converse(meet(converse(meet(converse(x0), converse(x1))), converse(join(converse(x0), meet(converse(x0), converse(x1)))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.14 = { by lemma 30 }
% 4.99/1.14 tuple(converse(meet(converse(meet(converse(x0), converse(x1))), converse(converse(x0)))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.14 = { by axiom 1 (converse_idempotence) }
% 4.99/1.14 tuple(converse(meet(converse(meet(converse(x0), converse(x1))), x0)), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by lemma 20 R->L }
% 4.99/1.15 tuple(converse(meet(x0, converse(meet(converse(x0), converse(x1))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by lemma 35 R->L }
% 4.99/1.15 tuple(converse(meet(x0, meet(converse(meet(converse(x0), converse(x1))), join(x1, converse(meet(converse(x0), converse(x1))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.99/1.15 tuple(converse(meet(x0, meet(converse(meet(converse(x0), converse(x1))), join(converse(meet(converse(x0), converse(x1))), x1)))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by axiom 1 (converse_idempotence) R->L }
% 4.99/1.15 tuple(converse(meet(x0, meet(converse(meet(converse(x0), converse(x1))), join(converse(meet(converse(x0), converse(x1))), converse(converse(x1)))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by axiom 4 (converse_additivity) R->L }
% 4.99/1.15 tuple(converse(meet(x0, meet(converse(meet(converse(x0), converse(x1))), converse(join(meet(converse(x0), converse(x1)), converse(x1)))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.99/1.15 tuple(converse(meet(x0, meet(converse(meet(converse(x0), converse(x1))), converse(join(converse(x1), meet(converse(x0), converse(x1))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by lemma 31 }
% 4.99/1.15 tuple(converse(meet(x0, meet(converse(meet(converse(x0), converse(x1))), converse(converse(x1))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by axiom 1 (converse_idempotence) }
% 4.99/1.15 tuple(converse(meet(x0, meet(converse(meet(converse(x0), converse(x1))), x1))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by lemma 36 }
% 4.99/1.15 tuple(converse(meet(converse(meet(converse(x0), converse(x1))), meet(x0, x1))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by lemma 38 R->L }
% 4.99/1.15 tuple(converse(meet(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1)))), meet(x0, x1))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by lemma 20 R->L }
% 4.99/1.15 tuple(converse(meet(meet(x0, x1), converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1)))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by axiom 10 (maddux4_definiton_of_meet) }
% 4.99/1.15 tuple(converse(complement(join(complement(meet(x0, x1)), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1)))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by lemma 27 R->L }
% 4.99/1.15 tuple(converse(join(zero, complement(join(complement(meet(x0, x1)), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by lemma 13 R->L }
% 4.99/1.15 tuple(converse(join(complement(top), complement(join(complement(meet(x0, x1)), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by lemma 18 R->L }
% 4.99/1.15 tuple(converse(join(complement(join(converse(meet(converse(x0), converse(x1))), top)), complement(join(complement(meet(x0, x1)), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by axiom 1 (converse_idempotence) R->L }
% 4.99/1.15 tuple(converse(join(complement(join(converse(meet(converse(x0), converse(x1))), converse(converse(top)))), complement(join(complement(meet(x0, x1)), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by axiom 4 (converse_additivity) R->L }
% 4.99/1.15 tuple(converse(join(complement(converse(join(meet(converse(x0), converse(x1)), converse(top)))), complement(join(complement(meet(x0, x1)), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by axiom 7 (def_top) }
% 4.99/1.15 tuple(converse(join(complement(converse(join(meet(converse(x0), converse(x1)), converse(join(meet(x0, x1), complement(meet(x0, x1))))))), complement(join(complement(meet(x0, x1)), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by axiom 4 (converse_additivity) }
% 4.99/1.15 tuple(converse(join(complement(converse(join(meet(converse(x0), converse(x1)), join(converse(meet(x0, x1)), converse(complement(meet(x0, x1))))))), complement(join(complement(meet(x0, x1)), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by axiom 8 (maddux2_join_associativity) }
% 4.99/1.15 tuple(converse(join(complement(converse(join(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))), converse(complement(meet(x0, x1)))))), complement(join(complement(meet(x0, x1)), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by axiom 4 (converse_additivity) }
% 4.99/1.15 tuple(converse(join(complement(join(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1)))), converse(converse(complement(meet(x0, x1)))))), complement(join(complement(meet(x0, x1)), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by axiom 1 (converse_idempotence) }
% 4.99/1.15 tuple(converse(join(complement(join(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1)))), complement(meet(x0, x1)))), complement(join(complement(meet(x0, x1)), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by lemma 27 R->L }
% 4.99/1.15 tuple(converse(join(complement(join(zero, join(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1)))), complement(meet(x0, x1))))), complement(join(complement(meet(x0, x1)), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by lemma 33 R->L }
% 4.99/1.15 tuple(converse(join(complement(join(zero, complement(meet(meet(x0, x1), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))))))), complement(join(complement(meet(x0, x1)), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by lemma 22 }
% 4.99/1.15 tuple(converse(join(meet(meet(meet(x0, x1), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1)))))), top), complement(join(complement(meet(x0, x1)), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by lemma 29 }
% 4.99/1.15 tuple(converse(join(meet(meet(x0, x1), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1)))))), complement(join(complement(meet(x0, x1)), complement(converse(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))))))), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 = { by lemma 19 }
% 4.99/1.15 tuple(converse(meet(x0, x1)), meet(converse(x0_2), converse(x1_2)))
% 4.99/1.15 % SZS output end Proof
% 4.99/1.15
% 4.99/1.15 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------