%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : REL027+4 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n016.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 12:33:17 PM UTC 2026
% Result : Theorem 32.58s 4.71s
% Output : Proof 34.74s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : REL027+4 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.05 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.14/0.41 % Computer : n016.cluster.edu
% 0.14/0.41 % Model : x86_64 x86_64
% 0.14/0.41 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.41 % Memory : 8046.5625MB
% 0.14/0.41 % OS : Linux 6.8.0-71-generic
% 0.14/0.42 % CPULimit : 300
% 0.14/0.42 % WCLimit : 300
% 0.14/0.42 % DateTime : Sun Sep 27 22:58:32 UTC 2026
% 0.14/0.42 % CPUTime :
% 0.14/0.42 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 32.58/4.71 Command-line arguments: --no-flatten-goal
% 32.58/4.71
% 32.58/4.71 % SZS status Theorem
% 32.58/4.71
% 34.05/4.86 % SZS output start Proof
% 34.05/4.86 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 34.05/4.86 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 34.05/4.86 Axiom 3 (def_top): top = join(X, complement(X)).
% 34.05/4.86 Axiom 4 (goals): join(x0, one) = one.
% 34.05/4.86 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 34.05/4.86 Axiom 6 (composition_identity): composition(X, one) = X.
% 34.05/4.86 Axiom 7 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 34.05/4.86 Axiom 8 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 34.05/4.86 Axiom 9 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 34.05/4.86 Axiom 10 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 34.05/4.86 Axiom 11 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 34.05/4.86 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 34.05/4.86 Axiom 13 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 34.05/4.86 Axiom 14 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 34.05/4.86
% 34.05/4.86 Lemma 15: complement(top) = zero.
% 34.05/4.86 Proof:
% 34.05/4.86 complement(top)
% 34.05/4.86 = { by axiom 3 (def_top) }
% 34.05/4.86 complement(join(complement(X), complement(complement(X))))
% 34.05/4.86 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 34.05/4.86 meet(X, complement(X))
% 34.05/4.86 = { by axiom 5 (def_zero) R->L }
% 34.05/4.86 zero
% 34.05/4.86
% 34.05/4.86 Lemma 16: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 34.05/4.86 Proof:
% 34.05/4.86 join(meet(X, Y), complement(join(complement(X), Y)))
% 34.05/4.86 = { by axiom 7 (maddux4_definiton_of_meet) }
% 34.05/4.86 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 34.05/4.86 = { by axiom 13 (maddux3_a_kind_of_de_Morgan) R->L }
% 34.05/4.86 X
% 34.05/4.86
% 34.05/4.86 Lemma 17: join(meet(X, Y), meet(X, complement(Y))) = X.
% 34.05/4.86 Proof:
% 34.05/4.86 join(meet(X, Y), meet(X, complement(Y)))
% 34.05/4.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.86 join(meet(X, complement(Y)), meet(X, Y))
% 34.05/4.86 = { by axiom 7 (maddux4_definiton_of_meet) }
% 34.05/4.86 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 34.05/4.86 = { by lemma 16 }
% 34.05/4.86 X
% 34.05/4.86
% 34.05/4.86 Lemma 18: meet(Y, X) = meet(X, Y).
% 34.05/4.86 Proof:
% 34.05/4.86 meet(Y, X)
% 34.05/4.86 = { by axiom 7 (maddux4_definiton_of_meet) }
% 34.05/4.86 complement(join(complement(Y), complement(X)))
% 34.05/4.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.86 complement(join(complement(X), complement(Y)))
% 34.05/4.86 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 34.05/4.86 meet(X, Y)
% 34.05/4.86
% 34.05/4.86 Lemma 19: complement(join(zero, complement(X))) = meet(X, top).
% 34.05/4.86 Proof:
% 34.05/4.86 complement(join(zero, complement(X)))
% 34.05/4.86 = { by lemma 15 R->L }
% 34.05/4.86 complement(join(complement(top), complement(X)))
% 34.05/4.86 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 34.05/4.86 meet(top, X)
% 34.05/4.86 = { by lemma 18 R->L }
% 34.05/4.86 meet(X, top)
% 34.05/4.86
% 34.05/4.86 Lemma 20: composition(converse(one), X) = X.
% 34.05/4.86 Proof:
% 34.05/4.86 composition(converse(one), X)
% 34.05/4.86 = { by axiom 1 (converse_idempotence) R->L }
% 34.05/4.86 composition(converse(one), converse(converse(X)))
% 34.05/4.86 = { by axiom 10 (converse_multiplicativity) R->L }
% 34.05/4.86 converse(composition(converse(X), one))
% 34.05/4.86 = { by axiom 6 (composition_identity) }
% 34.05/4.86 converse(converse(X))
% 34.05/4.86 = { by axiom 1 (converse_idempotence) }
% 34.05/4.86 X
% 34.05/4.86
% 34.05/4.86 Lemma 21: composition(one, X) = X.
% 34.05/4.86 Proof:
% 34.05/4.86 composition(one, X)
% 34.05/4.86 = { by lemma 20 R->L }
% 34.05/4.86 composition(converse(one), composition(one, X))
% 34.05/4.86 = { by axiom 11 (composition_associativity) }
% 34.05/4.86 composition(composition(converse(one), one), X)
% 34.05/4.86 = { by axiom 6 (composition_identity) }
% 34.05/4.86 composition(converse(one), X)
% 34.05/4.86 = { by lemma 20 }
% 34.05/4.86 X
% 34.05/4.86
% 34.05/4.86 Lemma 22: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 34.05/4.86 Proof:
% 34.05/4.86 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 34.05/4.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.86 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 34.05/4.86 = { by axiom 14 (converse_cancellativity) }
% 34.05/4.86 complement(X)
% 34.05/4.86
% 34.05/4.86 Lemma 23: join(complement(X), complement(X)) = complement(X).
% 34.05/4.86 Proof:
% 34.05/4.86 join(complement(X), complement(X))
% 34.05/4.86 = { by lemma 20 R->L }
% 34.05/4.86 join(complement(X), composition(converse(one), complement(X)))
% 34.05/4.86 = { by lemma 21 R->L }
% 34.05/4.86 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 34.05/4.86 = { by lemma 22 }
% 34.05/4.86 complement(X)
% 34.05/4.86
% 34.05/4.86 Lemma 24: join(zero, complement(complement(X))) = X.
% 34.05/4.86 Proof:
% 34.05/4.86 join(zero, complement(complement(X)))
% 34.05/4.86 = { by axiom 5 (def_zero) }
% 34.05/4.86 join(meet(X, complement(X)), complement(complement(X)))
% 34.05/4.86 = { by lemma 23 R->L }
% 34.05/4.86 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 34.05/4.86 = { by lemma 16 }
% 34.05/4.86 X
% 34.05/4.86
% 34.05/4.86 Lemma 25: meet(top, complement(X)) = complement(X).
% 34.05/4.86 Proof:
% 34.05/4.86 meet(top, complement(X))
% 34.05/4.86 = { by lemma 18 }
% 34.05/4.86 meet(complement(X), top)
% 34.05/4.86 = { by lemma 19 R->L }
% 34.05/4.86 complement(join(zero, complement(complement(X))))
% 34.05/4.86 = { by lemma 24 }
% 34.05/4.86 complement(X)
% 34.05/4.86
% 34.05/4.86 Lemma 26: complement(zero) = top.
% 34.05/4.86 Proof:
% 34.05/4.86 complement(zero)
% 34.05/4.86 = { by lemma 17 R->L }
% 34.05/4.86 join(meet(complement(zero), top), meet(complement(zero), complement(top)))
% 34.05/4.86 = { by lemma 15 }
% 34.05/4.86 join(meet(complement(zero), top), meet(complement(zero), zero))
% 34.05/4.86 = { by axiom 2 (maddux1_join_commutativity) }
% 34.05/4.86 join(meet(complement(zero), zero), meet(complement(zero), top))
% 34.05/4.86 = { by lemma 18 R->L }
% 34.05/4.86 join(meet(zero, complement(zero)), meet(complement(zero), top))
% 34.05/4.86 = { by axiom 5 (def_zero) R->L }
% 34.05/4.86 join(zero, meet(complement(zero), top))
% 34.05/4.86 = { by lemma 18 R->L }
% 34.05/4.86 join(zero, meet(top, complement(zero)))
% 34.05/4.86 = { by lemma 25 }
% 34.05/4.86 join(zero, complement(zero))
% 34.05/4.86 = { by axiom 3 (def_top) R->L }
% 34.05/4.86 top
% 34.05/4.86
% 34.05/4.86 Lemma 27: join(X, join(Y, complement(X))) = join(Y, top).
% 34.05/4.86 Proof:
% 34.05/4.86 join(X, join(Y, complement(X)))
% 34.05/4.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.86 join(X, join(complement(X), Y))
% 34.05/4.86 = { by axiom 9 (maddux2_join_associativity) }
% 34.05/4.86 join(join(X, complement(X)), Y)
% 34.05/4.86 = { by axiom 3 (def_top) R->L }
% 34.05/4.86 join(top, Y)
% 34.05/4.86 = { by axiom 2 (maddux1_join_commutativity) }
% 34.05/4.86 join(Y, top)
% 34.05/4.86
% 34.05/4.86 Lemma 28: join(top, complement(X)) = top.
% 34.05/4.86 Proof:
% 34.05/4.86 join(top, complement(X))
% 34.05/4.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.86 join(complement(X), top)
% 34.05/4.86 = { by lemma 27 R->L }
% 34.05/4.86 join(X, join(complement(X), complement(X)))
% 34.05/4.86 = { by lemma 23 }
% 34.05/4.86 join(X, complement(X))
% 34.05/4.86 = { by axiom 3 (def_top) R->L }
% 34.05/4.86 top
% 34.05/4.86
% 34.05/4.87 Lemma 29: join(zero, meet(X, top)) = X.
% 34.05/4.87 Proof:
% 34.05/4.87 join(zero, meet(X, top))
% 34.05/4.87 = { by lemma 26 R->L }
% 34.05/4.87 join(zero, meet(X, complement(zero)))
% 34.05/4.87 = { by lemma 15 R->L }
% 34.05/4.87 join(complement(top), meet(X, complement(zero)))
% 34.05/4.87 = { by lemma 28 R->L }
% 34.05/4.87 join(complement(join(top, complement(X))), meet(X, complement(zero)))
% 34.05/4.87 = { by lemma 26 R->L }
% 34.05/4.87 join(complement(join(complement(zero), complement(X))), meet(X, complement(zero)))
% 34.05/4.87 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 34.05/4.87 join(meet(zero, X), meet(X, complement(zero)))
% 34.05/4.87 = { by lemma 18 R->L }
% 34.05/4.87 join(meet(X, zero), meet(X, complement(zero)))
% 34.05/4.87 = { by lemma 17 }
% 34.05/4.87 X
% 34.05/4.87
% 34.05/4.87 Lemma 30: join(zero, complement(X)) = complement(X).
% 34.05/4.87 Proof:
% 34.05/4.87 join(zero, complement(X))
% 34.05/4.87 = { by lemma 25 R->L }
% 34.05/4.87 join(zero, meet(top, complement(X)))
% 34.05/4.87 = { by lemma 18 }
% 34.05/4.87 join(zero, meet(complement(X), top))
% 34.05/4.87 = { by lemma 29 }
% 34.05/4.87 complement(X)
% 34.05/4.87
% 34.05/4.87 Lemma 31: meet(X, top) = X.
% 34.05/4.87 Proof:
% 34.05/4.87 meet(X, top)
% 34.05/4.87 = { by lemma 19 R->L }
% 34.05/4.87 complement(join(zero, complement(X)))
% 34.05/4.87 = { by lemma 30 R->L }
% 34.05/4.87 join(zero, complement(join(zero, complement(X))))
% 34.05/4.87 = { by lemma 19 }
% 34.05/4.87 join(zero, meet(X, top))
% 34.05/4.87 = { by lemma 29 }
% 34.05/4.87 X
% 34.05/4.87
% 34.05/4.87 Lemma 32: join(X, X) = X.
% 34.05/4.87 Proof:
% 34.05/4.87 join(X, X)
% 34.05/4.87 = { by lemma 31 R->L }
% 34.05/4.87 join(X, meet(X, top))
% 34.05/4.87 = { by lemma 31 R->L }
% 34.05/4.87 join(meet(X, top), meet(X, top))
% 34.05/4.87 = { by lemma 18 }
% 34.05/4.87 join(meet(top, X), meet(X, top))
% 34.05/4.87 = { by lemma 18 }
% 34.05/4.87 join(meet(top, X), meet(top, X))
% 34.05/4.87 = { by axiom 7 (maddux4_definiton_of_meet) }
% 34.05/4.87 join(meet(top, X), complement(join(complement(top), complement(X))))
% 34.05/4.87 = { by axiom 7 (maddux4_definiton_of_meet) }
% 34.05/4.87 join(complement(join(complement(top), complement(X))), complement(join(complement(top), complement(X))))
% 34.05/4.87 = { by lemma 23 }
% 34.05/4.87 complement(join(complement(top), complement(X)))
% 34.05/4.87 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 34.05/4.87 meet(top, X)
% 34.05/4.87 = { by lemma 18 R->L }
% 34.05/4.87 meet(X, top)
% 34.05/4.87 = { by lemma 31 }
% 34.05/4.87 X
% 34.05/4.87
% 34.05/4.87 Lemma 33: join(Y, top) = join(X, top).
% 34.05/4.87 Proof:
% 34.05/4.87 join(Y, top)
% 34.05/4.87 = { by lemma 28 R->L }
% 34.05/4.87 join(Y, join(top, complement(Y)))
% 34.05/4.87 = { by lemma 27 }
% 34.05/4.87 join(top, top)
% 34.05/4.87 = { by lemma 27 R->L }
% 34.05/4.87 join(X, join(top, complement(X)))
% 34.05/4.87 = { by lemma 28 }
% 34.05/4.87 join(X, top)
% 34.05/4.87
% 34.05/4.87 Lemma 34: join(X, top) = top.
% 34.05/4.87 Proof:
% 34.05/4.87 join(X, top)
% 34.05/4.87 = { by lemma 33 }
% 34.05/4.87 join(zero, top)
% 34.05/4.87 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.87 join(top, zero)
% 34.05/4.87 = { by lemma 15 R->L }
% 34.05/4.87 join(top, complement(top))
% 34.05/4.87 = { by axiom 3 (def_top) R->L }
% 34.05/4.87 top
% 34.05/4.87
% 34.05/4.87 Lemma 35: complement(complement(X)) = X.
% 34.05/4.87 Proof:
% 34.05/4.87 complement(complement(X))
% 34.05/4.87 = { by lemma 30 R->L }
% 34.05/4.87 join(zero, complement(complement(X)))
% 34.05/4.87 = { by lemma 24 }
% 34.05/4.87 X
% 34.05/4.87
% 34.05/4.87 Lemma 36: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 34.05/4.87 Proof:
% 34.05/4.87 join(Y, join(X, Z))
% 34.05/4.87 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.87 join(join(X, Z), Y)
% 34.05/4.87 = { by axiom 9 (maddux2_join_associativity) R->L }
% 34.05/4.87 join(X, join(Z, Y))
% 34.05/4.87 = { by axiom 2 (maddux1_join_commutativity) }
% 34.05/4.87 join(X, join(Y, Z))
% 34.05/4.87
% 34.05/4.87 Lemma 37: meet(X, join(Y, complement(X))) = meet(X, Y).
% 34.05/4.87 Proof:
% 34.05/4.87 meet(X, join(Y, complement(X)))
% 34.05/4.87 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.87 meet(X, join(complement(X), Y))
% 34.05/4.87 = { by axiom 7 (maddux4_definiton_of_meet) }
% 34.05/4.88 complement(join(complement(X), complement(join(complement(X), Y))))
% 34.05/4.88 = { by lemma 16 R->L }
% 34.05/4.88 complement(join(complement(X), complement(join(complement(X), join(meet(Y, complement(X)), complement(join(complement(Y), complement(X))))))))
% 34.05/4.88 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.88 complement(join(complement(X), complement(join(complement(X), join(complement(join(complement(Y), complement(X))), meet(Y, complement(X)))))))
% 34.05/4.88 = { by lemma 23 R->L }
% 34.05/4.88 complement(join(complement(X), complement(join(complement(X), join(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X)))), meet(Y, complement(X)))))))
% 34.05/4.88 = { by axiom 9 (maddux2_join_associativity) R->L }
% 34.05/4.88 complement(join(complement(X), complement(join(complement(X), join(complement(join(complement(Y), complement(X))), join(complement(join(complement(Y), complement(X))), meet(Y, complement(X))))))))
% 34.05/4.88 = { by axiom 2 (maddux1_join_commutativity) }
% 34.05/4.88 complement(join(complement(X), complement(join(complement(X), join(complement(join(complement(Y), complement(X))), join(meet(Y, complement(X)), complement(join(complement(Y), complement(X)))))))))
% 34.05/4.88 = { by lemma 16 }
% 34.05/4.88 complement(join(complement(X), complement(join(complement(X), join(complement(join(complement(Y), complement(X))), Y)))))
% 34.05/4.88 = { by axiom 2 (maddux1_join_commutativity) }
% 34.05/4.88 complement(join(complement(X), complement(join(complement(X), join(Y, complement(join(complement(Y), complement(X))))))))
% 34.05/4.88 = { by axiom 2 (maddux1_join_commutativity) }
% 34.05/4.88 complement(join(complement(X), complement(join(complement(X), join(Y, complement(join(complement(X), complement(Y))))))))
% 34.05/4.88 = { by axiom 9 (maddux2_join_associativity) }
% 34.05/4.88 complement(join(complement(X), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 34.05/4.88 = { by lemma 31 R->L }
% 34.05/4.88 complement(join(meet(complement(X), top), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 34.05/4.88 = { by lemma 19 R->L }
% 34.05/4.88 complement(join(complement(join(zero, complement(complement(X)))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 34.05/4.88 = { by lemma 16 R->L }
% 34.05/4.88 complement(join(complement(join(meet(join(zero, complement(complement(X))), Y), complement(join(complement(join(zero, complement(complement(X)))), Y)))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 34.05/4.88 = { by axiom 7 (maddux4_definiton_of_meet) }
% 34.05/4.88 complement(join(complement(join(complement(join(complement(join(zero, complement(complement(X)))), complement(Y))), complement(join(complement(join(zero, complement(complement(X)))), Y)))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 34.05/4.88 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 34.05/4.88 complement(join(meet(join(complement(join(zero, complement(complement(X)))), complement(Y)), join(complement(join(zero, complement(complement(X)))), Y)), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 34.05/4.88 = { by lemma 18 R->L }
% 34.05/4.88 complement(join(meet(join(complement(join(zero, complement(complement(X)))), Y), join(complement(join(zero, complement(complement(X)))), complement(Y))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 34.05/4.88 = { by axiom 2 (maddux1_join_commutativity) }
% 34.05/4.88 complement(join(meet(join(Y, complement(join(zero, complement(complement(X))))), join(complement(join(zero, complement(complement(X)))), complement(Y))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 34.05/4.88 = { by axiom 2 (maddux1_join_commutativity) }
% 34.05/4.88 complement(join(meet(join(Y, complement(join(zero, complement(complement(X))))), join(complement(Y), complement(join(zero, complement(complement(X)))))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 34.05/4.88 = { by lemma 19 }
% 34.05/4.88 complement(join(meet(join(Y, meet(complement(X), top)), join(complement(Y), complement(join(zero, complement(complement(X)))))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 34.05/4.88 = { by lemma 31 }
% 34.05/4.88 complement(join(meet(join(Y, complement(X)), join(complement(Y), complement(join(zero, complement(complement(X)))))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 34.05/4.88 = { by lemma 19 }
% 34.05/4.88 complement(join(meet(join(Y, complement(X)), join(complement(Y), meet(complement(X), top))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 34.05/4.88 = { by lemma 31 }
% 34.05/4.88 complement(join(meet(join(Y, complement(X)), join(complement(Y), complement(X))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 34.05/4.88 = { by axiom 2 (maddux1_join_commutativity) }
% 34.05/4.88 complement(join(meet(join(complement(X), Y), join(complement(Y), complement(X))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 34.05/4.88 = { by axiom 2 (maddux1_join_commutativity) }
% 34.05/4.88 complement(join(meet(join(complement(X), Y), join(complement(X), complement(Y))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 34.05/4.88 = { by lemma 18 }
% 34.05/4.88 complement(join(meet(join(complement(X), complement(Y)), join(complement(X), Y)), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 34.05/4.88 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.88 complement(join(meet(join(complement(X), complement(Y)), join(complement(X), Y)), complement(join(complement(join(complement(X), complement(Y))), join(complement(X), Y)))))
% 34.05/4.89 = { by lemma 16 }
% 34.05/4.89 complement(join(complement(X), complement(Y)))
% 34.05/4.89 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 34.05/4.89 meet(X, Y)
% 34.05/4.89
% 34.05/4.89 Lemma 38: join(X, join(complement(X), Y)) = top.
% 34.05/4.89 Proof:
% 34.05/4.89 join(X, join(complement(X), Y))
% 34.05/4.89 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.89 join(X, join(Y, complement(X)))
% 34.05/4.89 = { by lemma 27 }
% 34.05/4.89 join(Y, top)
% 34.05/4.89 = { by lemma 33 R->L }
% 34.05/4.89 join(Z, top)
% 34.05/4.89 = { by lemma 34 }
% 34.05/4.89 top
% 34.05/4.89
% 34.05/4.89 Lemma 39: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 34.05/4.89 Proof:
% 34.05/4.89 complement(join(X, complement(Y)))
% 34.05/4.89 = { by lemma 35 R->L }
% 34.05/4.89 complement(join(X, complement(complement(complement(Y)))))
% 34.05/4.89 = { by lemma 17 R->L }
% 34.05/4.89 complement(join(meet(join(X, complement(complement(complement(Y)))), complement(Y)), meet(join(X, complement(complement(complement(Y)))), complement(complement(Y)))))
% 34.05/4.89 = { by lemma 18 R->L }
% 34.05/4.89 complement(join(meet(complement(Y), join(X, complement(complement(complement(Y))))), meet(join(X, complement(complement(complement(Y)))), complement(complement(Y)))))
% 34.05/4.89 = { by lemma 18 R->L }
% 34.05/4.89 complement(join(meet(complement(Y), join(X, complement(complement(complement(Y))))), meet(complement(complement(Y)), join(X, complement(complement(complement(Y)))))))
% 34.05/4.89 = { by lemma 37 }
% 34.05/4.89 complement(join(meet(complement(Y), join(X, complement(complement(complement(Y))))), meet(complement(complement(Y)), X)))
% 34.05/4.89 = { by lemma 35 }
% 34.05/4.89 complement(join(meet(complement(Y), join(X, complement(Y))), meet(complement(complement(Y)), X)))
% 34.05/4.89 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.89 complement(join(meet(complement(Y), join(complement(Y), X)), meet(complement(complement(Y)), X)))
% 34.05/4.89 = { by lemma 16 R->L }
% 34.05/4.89 complement(join(meet(complement(Y), join(complement(Y), join(meet(X, Z), complement(join(complement(X), Z))))), meet(complement(complement(Y)), X)))
% 34.05/4.89 = { by lemma 36 }
% 34.05/4.89 complement(join(meet(complement(Y), join(meet(X, Z), join(complement(Y), complement(join(complement(X), Z))))), meet(complement(complement(Y)), X)))
% 34.05/4.89 = { by lemma 16 R->L }
% 34.05/4.89 complement(join(join(meet(meet(complement(Y), join(meet(X, Z), join(complement(Y), complement(join(complement(X), Z))))), top), complement(join(complement(meet(complement(Y), join(meet(X, Z), join(complement(Y), complement(join(complement(X), Z)))))), top))), meet(complement(complement(Y)), X)))
% 34.05/4.89 = { by lemma 31 }
% 34.05/4.89 complement(join(join(meet(complement(Y), join(meet(X, Z), join(complement(Y), complement(join(complement(X), Z))))), complement(join(complement(meet(complement(Y), join(meet(X, Z), join(complement(Y), complement(join(complement(X), Z)))))), top))), meet(complement(complement(Y)), X)))
% 34.05/4.89 = { by lemma 34 }
% 34.05/4.89 complement(join(join(meet(complement(Y), join(meet(X, Z), join(complement(Y), complement(join(complement(X), Z))))), complement(top)), meet(complement(complement(Y)), X)))
% 34.05/4.89 = { by lemma 35 R->L }
% 34.05/4.89 complement(join(join(meet(complement(Y), join(meet(X, Z), join(complement(complement(complement(Y))), complement(join(complement(X), Z))))), complement(top)), meet(complement(complement(Y)), X)))
% 34.05/4.89 = { by lemma 38 R->L }
% 34.05/4.89 complement(join(join(meet(complement(Y), join(meet(X, Z), join(complement(complement(complement(Y))), complement(join(complement(X), Z))))), complement(join(complement(complement(Y)), join(complement(complement(complement(Y))), join(meet(X, Z), complement(join(complement(X), Z))))))), meet(complement(complement(Y)), X)))
% 34.05/4.89 = { by lemma 36 R->L }
% 34.05/4.89 complement(join(join(meet(complement(Y), join(meet(X, Z), join(complement(complement(complement(Y))), complement(join(complement(X), Z))))), complement(join(complement(complement(Y)), join(meet(X, Z), join(complement(complement(complement(Y))), complement(join(complement(X), Z))))))), meet(complement(complement(Y)), X)))
% 34.05/4.90 = { by lemma 16 }
% 34.05/4.90 complement(join(complement(Y), meet(complement(complement(Y)), X)))
% 34.05/4.90 = { by axiom 7 (maddux4_definiton_of_meet) }
% 34.05/4.90 complement(join(complement(Y), complement(join(complement(complement(complement(Y))), complement(X)))))
% 34.05/4.90 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 34.05/4.90 meet(Y, join(complement(complement(complement(Y))), complement(X)))
% 34.05/4.90 = { by axiom 2 (maddux1_join_commutativity) }
% 34.05/4.90 meet(Y, join(complement(X), complement(complement(complement(Y)))))
% 34.05/4.90 = { by lemma 35 }
% 34.05/4.90 meet(Y, join(complement(X), complement(Y)))
% 34.05/4.90 = { by lemma 37 }
% 34.05/4.90 meet(Y, complement(X))
% 34.05/4.90
% 34.05/4.90 Lemma 40: composition(join(X, one), Y) = join(Y, composition(X, Y)).
% 34.05/4.90 Proof:
% 34.05/4.90 composition(join(X, one), Y)
% 34.05/4.90 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.90 composition(join(one, X), Y)
% 34.05/4.90 = { by axiom 12 (composition_distributivity) }
% 34.05/4.90 join(composition(one, Y), composition(X, Y))
% 34.05/4.90 = { by lemma 21 }
% 34.05/4.90 join(Y, composition(X, Y))
% 34.05/4.90
% 34.05/4.90 Lemma 41: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 34.05/4.90 Proof:
% 34.05/4.90 converse(join(X, converse(Y)))
% 34.05/4.90 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.90 converse(join(converse(Y), X))
% 34.05/4.90 = { by axiom 8 (converse_additivity) }
% 34.05/4.90 join(converse(converse(Y)), converse(X))
% 34.05/4.90 = { by axiom 1 (converse_idempotence) }
% 34.05/4.90 join(Y, converse(X))
% 34.05/4.90
% 34.05/4.90 Lemma 42: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 34.05/4.90 Proof:
% 34.05/4.90 converse(join(converse(X), Y))
% 34.05/4.90 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.90 converse(join(Y, converse(X)))
% 34.05/4.90 = { by lemma 41 }
% 34.05/4.90 join(X, converse(Y))
% 34.05/4.90
% 34.05/4.90 Lemma 43: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 34.05/4.90 Proof:
% 34.05/4.90 converse(composition(X, converse(Y)))
% 34.05/4.90 = { by axiom 10 (converse_multiplicativity) }
% 34.05/4.90 composition(converse(converse(Y)), converse(X))
% 34.05/4.90 = { by axiom 1 (converse_idempotence) }
% 34.05/4.90 composition(Y, converse(X))
% 34.05/4.90
% 34.05/4.90 Lemma 44: join(complement(one), converse(complement(one))) = complement(one).
% 34.05/4.90 Proof:
% 34.05/4.90 join(complement(one), converse(complement(one)))
% 34.05/4.90 = { by axiom 6 (composition_identity) R->L }
% 34.05/4.90 join(complement(one), composition(converse(complement(one)), one))
% 34.05/4.90 = { by lemma 31 R->L }
% 34.05/4.90 join(complement(one), composition(converse(complement(one)), meet(one, top)))
% 34.05/4.90 = { by lemma 30 R->L }
% 34.05/4.90 join(complement(one), composition(converse(join(zero, complement(one))), meet(one, top)))
% 34.05/4.90 = { by lemma 19 R->L }
% 34.05/4.90 join(complement(one), composition(converse(join(zero, complement(one))), complement(join(zero, complement(one)))))
% 34.05/4.90 = { by axiom 6 (composition_identity) R->L }
% 34.05/4.90 join(complement(one), composition(converse(join(zero, complement(one))), complement(composition(join(zero, complement(one)), one))))
% 34.05/4.90 = { by lemma 22 }
% 34.05/4.90 complement(one)
% 34.05/4.90
% 34.05/4.90 Lemma 45: meet(one, complement(composition(x0, top))) = meet(one, complement(x0)).
% 34.05/4.90 Proof:
% 34.05/4.90 meet(one, complement(composition(x0, top)))
% 34.05/4.90 = { by lemma 35 R->L }
% 34.05/4.90 meet(complement(complement(one)), complement(composition(x0, top)))
% 34.05/4.90 = { by lemma 18 }
% 34.05/4.90 meet(complement(composition(x0, top)), complement(complement(one)))
% 34.05/4.90 = { by lemma 30 R->L }
% 34.05/4.90 meet(join(zero, complement(composition(x0, top))), complement(complement(one)))
% 34.05/4.90 = { by lemma 39 R->L }
% 34.05/4.90 complement(join(complement(one), complement(join(zero, complement(composition(x0, top))))))
% 34.05/4.90 = { by lemma 19 }
% 34.05/4.90 complement(join(complement(one), meet(composition(x0, top), top)))
% 34.05/4.90 = { by lemma 31 }
% 34.05/4.90 complement(join(complement(one), composition(x0, top)))
% 34.05/4.90 = { by lemma 38 R->L }
% 34.05/4.90 complement(join(complement(one), composition(x0, join(converse(X), join(complement(converse(X)), converse(complement(converse(complement(converse(X))))))))))
% 34.05/4.90 = { by lemma 42 R->L }
% 34.05/4.90 complement(join(complement(one), composition(x0, join(converse(X), converse(join(converse(complement(converse(X))), complement(converse(complement(converse(X))))))))))
% 34.05/4.90 = { by axiom 3 (def_top) R->L }
% 34.05/4.90 complement(join(complement(one), composition(x0, join(converse(X), converse(top)))))
% 34.05/4.90 = { by axiom 8 (converse_additivity) R->L }
% 34.05/4.90 complement(join(complement(one), composition(x0, converse(join(X, top)))))
% 34.05/4.90 = { by lemma 34 }
% 34.05/4.90 complement(join(complement(one), composition(x0, converse(top))))
% 34.05/4.90 = { by axiom 3 (def_top) }
% 34.05/4.90 complement(join(complement(one), composition(x0, converse(join(one, complement(one))))))
% 34.05/4.90 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.90 complement(join(complement(one), composition(x0, converse(join(complement(one), one)))))
% 34.05/4.90 = { by lemma 43 R->L }
% 34.05/4.90 complement(join(complement(one), converse(composition(join(complement(one), one), converse(x0)))))
% 34.05/4.90 = { by lemma 40 }
% 34.05/4.90 complement(join(complement(one), converse(join(converse(x0), composition(complement(one), converse(x0))))))
% 34.05/4.90 = { by lemma 42 }
% 34.05/4.90 complement(join(complement(one), join(x0, converse(composition(complement(one), converse(x0))))))
% 34.05/4.90 = { by lemma 43 }
% 34.05/4.91 complement(join(complement(one), join(x0, composition(x0, converse(complement(one))))))
% 34.05/4.91 = { by lemma 44 R->L }
% 34.05/4.91 complement(join(complement(one), join(x0, composition(x0, converse(join(complement(one), converse(complement(one))))))))
% 34.05/4.91 = { by lemma 41 }
% 34.05/4.91 complement(join(complement(one), join(x0, composition(x0, join(complement(one), converse(complement(one)))))))
% 34.05/4.91 = { by lemma 44 }
% 34.05/4.91 complement(join(complement(one), join(x0, composition(x0, complement(one)))))
% 34.05/4.91 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 34.05/4.91 complement(join(complement(one), join(composition(x0, complement(one)), x0)))
% 34.05/4.91 = { by axiom 9 (maddux2_join_associativity) }
% 34.05/4.91 complement(join(join(complement(one), composition(x0, complement(one))), x0))
% 34.05/4.91 = { by lemma 40 R->L }
% 34.05/4.91 complement(join(composition(join(x0, one), complement(one)), x0))
% 34.05/4.91 = { by axiom 4 (goals) }
% 34.05/4.91 complement(join(composition(one, complement(one)), x0))
% 34.05/4.91 = { by lemma 21 }
% 34.05/4.91 complement(join(complement(one), x0))
% 34.05/4.91 = { by axiom 2 (maddux1_join_commutativity) }
% 34.05/4.91 complement(join(x0, complement(one)))
% 34.05/4.91 = { by lemma 39 }
% 34.05/4.91 meet(one, complement(x0))
% 34.05/4.91
% 34.05/4.91 Goal 1 (goals_1): tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(x0), one), meet(complement(composition(x0, top)), one))) = tuple(meet(complement(x0), one), meet(complement(composition(x0, top)), one)).
% 34.05/4.91 Proof:
% 34.05/4.91 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(x0), one), meet(complement(composition(x0, top)), one)))
% 34.05/4.91 = { by axiom 2 (maddux1_join_commutativity) }
% 34.05/4.91 tuple(join(meet(complement(x0), one), meet(complement(composition(x0, top)), one)), join(meet(complement(x0), one), meet(complement(composition(x0, top)), one)))
% 34.05/4.91 = { by lemma 18 R->L }
% 34.05/4.91 tuple(join(meet(one, complement(x0)), meet(complement(composition(x0, top)), one)), join(meet(complement(x0), one), meet(complement(composition(x0, top)), one)))
% 34.05/4.91 = { by lemma 18 R->L }
% 34.05/4.91 tuple(join(meet(one, complement(x0)), meet(complement(composition(x0, top)), one)), join(meet(one, complement(x0)), meet(complement(composition(x0, top)), one)))
% 34.05/4.91 = { by lemma 18 R->L }
% 34.05/4.91 tuple(join(meet(one, complement(x0)), meet(one, complement(composition(x0, top)))), join(meet(one, complement(x0)), meet(complement(composition(x0, top)), one)))
% 34.05/4.91 = { by lemma 18 R->L }
% 34.74/4.91 tuple(join(meet(one, complement(x0)), meet(one, complement(composition(x0, top)))), join(meet(one, complement(x0)), meet(one, complement(composition(x0, top)))))
% 34.74/4.91 = { by lemma 45 }
% 34.74/4.91 tuple(join(meet(one, complement(x0)), meet(one, complement(x0))), join(meet(one, complement(x0)), meet(one, complement(composition(x0, top)))))
% 34.74/4.91 = { by lemma 45 }
% 34.74/4.91 tuple(join(meet(one, complement(x0)), meet(one, complement(x0))), join(meet(one, complement(x0)), meet(one, complement(x0))))
% 34.74/4.91 = { by lemma 32 }
% 34.74/4.91 tuple(meet(one, complement(x0)), join(meet(one, complement(x0)), meet(one, complement(x0))))
% 34.74/4.91 = { by lemma 32 }
% 34.74/4.91 tuple(meet(one, complement(x0)), meet(one, complement(x0)))
% 34.74/4.91 = { by lemma 45 R->L }
% 34.74/4.91 tuple(meet(one, complement(x0)), meet(one, complement(composition(x0, top))))
% 34.74/4.91 = { by lemma 18 }
% 34.74/4.91 tuple(meet(one, complement(x0)), meet(complement(composition(x0, top)), one))
% 34.74/4.91 = { by lemma 18 }
% 34.74/4.91 tuple(meet(complement(x0), one), meet(complement(composition(x0, top)), one))
% 34.74/4.91 % SZS output end Proof
% 34.74/4.91
% 34.74/4.91 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------