%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : REL030+2 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n015.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 12:33:20 PM UTC 2026
% Result : Theorem 111.22s 14.44s
% Output : Proof 112.80s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : REL030+2 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.08/0.35 % Computer : n015.cluster.edu
% 0.08/0.35 % Model : x86_64 x86_64
% 0.08/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.35 % Memory : 8046.5625MB
% 0.08/0.35 % OS : Linux 6.8.0-71-generic
% 0.08/0.35 % CPULimit : 300
% 0.08/0.35 % WCLimit : 300
% 0.08/0.35 % DateTime : Sun Sep 27 22:57:15 UTC 2026
% 0.08/0.35 % CPUTime :
% 0.08/0.35 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 111.22/14.44 Command-line arguments: --flatten --complete-subsets
% 111.22/14.44
% 111.22/14.44 % SZS status Theorem
% 111.22/14.44
% 112.01/14.57 % SZS output start Proof
% 112.01/14.57 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 112.01/14.57 Axiom 2 (def_zero): zero = meet(X, complement(X)).
% 112.01/14.57 Axiom 3 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 112.01/14.57 Axiom 4 (def_top): top = join(X, complement(X)).
% 112.01/14.57 Axiom 5 (goals): join(x0, one) = one.
% 112.01/14.57 Axiom 6 (composition_identity): composition(X, one) = X.
% 112.01/14.57 Axiom 7 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 112.01/14.57 Axiom 8 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 112.01/14.57 Axiom 9 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 112.01/14.57 Axiom 10 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 112.01/14.57 Axiom 11 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 112.01/14.57 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 112.01/14.57 Axiom 13 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 112.01/14.57 Axiom 14 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 112.01/14.57
% 112.01/14.58 Lemma 15: complement(top) = zero.
% 112.01/14.58 Proof:
% 112.01/14.58 complement(top)
% 112.01/14.58 = { by axiom 4 (def_top) }
% 112.01/14.58 complement(join(complement(X), complement(complement(X))))
% 112.01/14.58 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 112.01/14.58 meet(X, complement(X))
% 112.01/14.58 = { by axiom 2 (def_zero) R->L }
% 112.01/14.58 zero
% 112.01/14.58
% 112.01/14.58 Lemma 16: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 112.01/14.58 Proof:
% 112.01/14.58 join(meet(X, Y), complement(join(complement(X), Y)))
% 112.01/14.58 = { by axiom 7 (maddux4_definiton_of_meet) }
% 112.01/14.58 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 112.01/14.58 = { by axiom 13 (maddux3_a_kind_of_de_Morgan) R->L }
% 112.01/14.58 X
% 112.01/14.58
% 112.01/14.58 Lemma 17: join(meet(X, Y), meet(X, complement(Y))) = X.
% 112.01/14.58 Proof:
% 112.01/14.58 join(meet(X, Y), meet(X, complement(Y)))
% 112.01/14.58 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.58 join(meet(X, complement(Y)), meet(X, Y))
% 112.01/14.58 = { by axiom 7 (maddux4_definiton_of_meet) }
% 112.01/14.58 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 112.01/14.58 = { by lemma 16 }
% 112.01/14.58 X
% 112.01/14.58
% 112.01/14.58 Lemma 18: meet(Y, X) = meet(X, Y).
% 112.01/14.58 Proof:
% 112.01/14.58 meet(Y, X)
% 112.01/14.58 = { by axiom 7 (maddux4_definiton_of_meet) }
% 112.01/14.58 complement(join(complement(Y), complement(X)))
% 112.01/14.58 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.58 complement(join(complement(X), complement(Y)))
% 112.01/14.58 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 112.01/14.58 meet(X, Y)
% 112.01/14.58
% 112.01/14.58 Lemma 19: complement(join(zero, complement(X))) = meet(X, top).
% 112.01/14.58 Proof:
% 112.01/14.58 complement(join(zero, complement(X)))
% 112.01/14.58 = { by lemma 15 R->L }
% 112.01/14.58 complement(join(complement(top), complement(X)))
% 112.01/14.58 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 112.01/14.58 meet(top, X)
% 112.01/14.58 = { by lemma 18 R->L }
% 112.01/14.58 meet(X, top)
% 112.01/14.58
% 112.01/14.58 Lemma 20: composition(X, join(x0, one)) = X.
% 112.01/14.58 Proof:
% 112.01/14.58 composition(X, join(x0, one))
% 112.01/14.58 = { by axiom 5 (goals) }
% 112.01/14.58 composition(X, one)
% 112.01/14.58 = { by axiom 6 (composition_identity) }
% 112.01/14.58 X
% 112.01/14.58
% 112.01/14.58 Lemma 21: composition(converse(join(x0, one)), X) = X.
% 112.01/14.58 Proof:
% 112.01/14.58 composition(converse(join(x0, one)), X)
% 112.01/14.58 = { by axiom 1 (converse_idempotence) R->L }
% 112.01/14.58 composition(converse(join(x0, one)), converse(converse(X)))
% 112.01/14.58 = { by axiom 10 (converse_multiplicativity) R->L }
% 112.01/14.58 converse(composition(converse(X), join(x0, one)))
% 112.01/14.58 = { by lemma 20 }
% 112.01/14.58 converse(converse(X))
% 112.01/14.58 = { by axiom 1 (converse_idempotence) }
% 112.01/14.58 X
% 112.01/14.58
% 112.01/14.58 Lemma 22: composition(join(x0, one), X) = X.
% 112.01/14.58 Proof:
% 112.01/14.58 composition(join(x0, one), X)
% 112.01/14.58 = { by lemma 21 R->L }
% 112.01/14.58 composition(converse(join(x0, one)), composition(join(x0, one), X))
% 112.01/14.58 = { by axiom 11 (composition_associativity) }
% 112.01/14.58 composition(composition(converse(join(x0, one)), join(x0, one)), X)
% 112.01/14.58 = { by lemma 20 }
% 112.01/14.58 composition(converse(join(x0, one)), X)
% 112.01/14.58 = { by lemma 21 }
% 112.01/14.58 X
% 112.01/14.58
% 112.01/14.58 Lemma 23: join(complement(X), complement(X)) = complement(X).
% 112.01/14.58 Proof:
% 112.01/14.58 join(complement(X), complement(X))
% 112.01/14.58 = { by lemma 21 R->L }
% 112.01/14.58 join(complement(X), composition(converse(join(x0, one)), complement(X)))
% 112.01/14.58 = { by lemma 22 R->L }
% 112.01/14.58 join(complement(X), composition(converse(join(x0, one)), complement(composition(join(x0, one), X))))
% 112.01/14.58 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.58 join(composition(converse(join(x0, one)), complement(composition(join(x0, one), X))), complement(X))
% 112.01/14.58 = { by axiom 14 (converse_cancellativity) }
% 112.01/14.58 complement(X)
% 112.01/14.58
% 112.01/14.58 Lemma 24: join(zero, complement(complement(X))) = X.
% 112.01/14.58 Proof:
% 112.01/14.58 join(zero, complement(complement(X)))
% 112.01/14.58 = { by axiom 2 (def_zero) }
% 112.01/14.58 join(meet(X, complement(X)), complement(complement(X)))
% 112.01/14.58 = { by lemma 23 R->L }
% 112.01/14.58 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 112.01/14.58 = { by lemma 16 }
% 112.01/14.58 X
% 112.01/14.58
% 112.01/14.58 Lemma 25: meet(top, complement(X)) = complement(X).
% 112.01/14.58 Proof:
% 112.01/14.58 meet(top, complement(X))
% 112.01/14.58 = { by lemma 18 }
% 112.01/14.58 meet(complement(X), top)
% 112.01/14.58 = { by lemma 19 R->L }
% 112.01/14.58 complement(join(zero, complement(complement(X))))
% 112.01/14.58 = { by lemma 24 }
% 112.01/14.58 complement(X)
% 112.01/14.58
% 112.01/14.58 Lemma 26: complement(zero) = top.
% 112.01/14.58 Proof:
% 112.01/14.58 complement(zero)
% 112.01/14.58 = { by lemma 17 R->L }
% 112.01/14.58 join(meet(complement(zero), top), meet(complement(zero), complement(top)))
% 112.01/14.58 = { by lemma 15 }
% 112.01/14.58 join(meet(complement(zero), top), meet(complement(zero), zero))
% 112.01/14.58 = { by lemma 18 R->L }
% 112.01/14.58 join(meet(complement(zero), top), meet(zero, complement(zero)))
% 112.01/14.58 = { by axiom 2 (def_zero) R->L }
% 112.01/14.58 join(meet(complement(zero), top), zero)
% 112.01/14.58 = { by axiom 3 (maddux1_join_commutativity) }
% 112.01/14.58 join(zero, meet(complement(zero), top))
% 112.01/14.58 = { by lemma 18 R->L }
% 112.01/14.58 join(zero, meet(top, complement(zero)))
% 112.01/14.58 = { by lemma 25 }
% 112.01/14.58 join(zero, complement(zero))
% 112.01/14.58 = { by axiom 4 (def_top) R->L }
% 112.01/14.58 top
% 112.01/14.58
% 112.01/14.58 Lemma 27: join(X, join(Y, complement(X))) = join(Y, top).
% 112.01/14.58 Proof:
% 112.01/14.58 join(X, join(Y, complement(X)))
% 112.01/14.58 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.58 join(X, join(complement(X), Y))
% 112.01/14.58 = { by axiom 9 (maddux2_join_associativity) }
% 112.01/14.58 join(join(X, complement(X)), Y)
% 112.01/14.58 = { by axiom 4 (def_top) R->L }
% 112.01/14.58 join(top, Y)
% 112.01/14.58 = { by axiom 3 (maddux1_join_commutativity) }
% 112.01/14.58 join(Y, top)
% 112.01/14.58
% 112.01/14.58 Lemma 28: join(top, complement(X)) = top.
% 112.01/14.58 Proof:
% 112.01/14.58 join(top, complement(X))
% 112.01/14.58 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.58 join(complement(X), top)
% 112.01/14.58 = { by lemma 27 R->L }
% 112.01/14.58 join(X, join(complement(X), complement(X)))
% 112.01/14.58 = { by lemma 23 }
% 112.01/14.58 join(X, complement(X))
% 112.01/14.58 = { by axiom 4 (def_top) R->L }
% 112.01/14.58 top
% 112.01/14.58
% 112.01/14.58 Lemma 29: join(Y, top) = join(X, top).
% 112.01/14.58 Proof:
% 112.01/14.58 join(Y, top)
% 112.01/14.58 = { by lemma 28 R->L }
% 112.01/14.58 join(Y, join(top, complement(Y)))
% 112.01/14.58 = { by lemma 27 }
% 112.01/14.58 join(top, top)
% 112.01/14.58 = { by lemma 27 R->L }
% 112.01/14.58 join(X, join(top, complement(X)))
% 112.01/14.58 = { by lemma 28 }
% 112.01/14.58 join(X, top)
% 112.01/14.58
% 112.01/14.58 Lemma 30: join(X, top) = top.
% 112.01/14.58 Proof:
% 112.01/14.58 join(X, top)
% 112.01/14.58 = { by lemma 29 }
% 112.01/14.58 join(complement(top), top)
% 112.01/14.58 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.58 join(top, complement(top))
% 112.01/14.58 = { by axiom 4 (def_top) R->L }
% 112.01/14.58 top
% 112.01/14.58
% 112.01/14.58 Lemma 31: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 112.01/14.58 Proof:
% 112.01/14.58 converse(join(X, converse(Y)))
% 112.01/14.58 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.58 converse(join(converse(Y), X))
% 112.01/14.58 = { by axiom 8 (converse_additivity) }
% 112.01/14.58 join(converse(converse(Y)), converse(X))
% 112.01/14.58 = { by axiom 1 (converse_idempotence) }
% 112.01/14.58 join(Y, converse(X))
% 112.01/14.58
% 112.01/14.58 Lemma 32: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 112.01/14.58 Proof:
% 112.01/14.58 converse(join(converse(X), Y))
% 112.01/14.58 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.58 converse(join(Y, converse(X)))
% 112.01/14.58 = { by lemma 31 }
% 112.01/14.58 join(X, converse(Y))
% 112.01/14.58
% 112.01/14.58 Lemma 33: join(X, join(complement(X), Y)) = top.
% 112.01/14.58 Proof:
% 112.01/14.58 join(X, join(complement(X), Y))
% 112.01/14.58 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.58 join(X, join(Y, complement(X)))
% 112.01/14.58 = { by lemma 27 }
% 112.01/14.58 join(Y, top)
% 112.01/14.58 = { by lemma 29 R->L }
% 112.01/14.58 join(Z, top)
% 112.01/14.58 = { by lemma 30 }
% 112.01/14.58 top
% 112.01/14.58
% 112.01/14.58 Lemma 34: converse(top) = top.
% 112.01/14.58 Proof:
% 112.01/14.58 converse(top)
% 112.01/14.58 = { by lemma 30 R->L }
% 112.01/14.58 converse(join(X, top))
% 112.01/14.58 = { by axiom 8 (converse_additivity) }
% 112.01/14.58 join(converse(X), converse(top))
% 112.01/14.58 = { by axiom 4 (def_top) }
% 112.01/14.58 join(converse(X), converse(join(converse(complement(converse(X))), complement(converse(complement(converse(X)))))))
% 112.01/14.58 = { by lemma 32 }
% 112.01/14.58 join(converse(X), join(complement(converse(X)), converse(complement(converse(complement(converse(X)))))))
% 112.01/14.58 = { by lemma 33 }
% 112.01/14.58 top
% 112.01/14.58
% 112.01/14.58 Lemma 35: meet(X, zero) = zero.
% 112.01/14.58 Proof:
% 112.01/14.58 meet(X, zero)
% 112.01/14.58 = { by lemma 18 }
% 112.01/14.58 meet(zero, X)
% 112.01/14.58 = { by axiom 7 (maddux4_definiton_of_meet) }
% 112.01/14.58 complement(join(complement(zero), complement(X)))
% 112.01/14.58 = { by lemma 26 }
% 112.01/14.58 complement(join(top, complement(X)))
% 112.01/14.58 = { by lemma 28 }
% 112.01/14.58 complement(top)
% 112.01/14.58 = { by lemma 15 }
% 112.01/14.58 zero
% 112.01/14.58
% 112.01/14.58 Lemma 36: join(zero, meet(X, top)) = X.
% 112.01/14.58 Proof:
% 112.01/14.58 join(zero, meet(X, top))
% 112.01/14.58 = { by lemma 26 R->L }
% 112.01/14.58 join(zero, meet(X, complement(zero)))
% 112.01/14.58 = { by lemma 35 R->L }
% 112.01/14.58 join(meet(X, zero), meet(X, complement(zero)))
% 112.01/14.58 = { by lemma 17 }
% 112.01/14.58 X
% 112.01/14.58
% 112.01/14.58 Lemma 37: join(zero, complement(X)) = complement(X).
% 112.01/14.58 Proof:
% 112.01/14.58 join(zero, complement(X))
% 112.01/14.58 = { by lemma 25 R->L }
% 112.01/14.58 join(zero, meet(top, complement(X)))
% 112.01/14.58 = { by lemma 18 }
% 112.01/14.58 join(zero, meet(complement(X), top))
% 112.01/14.58 = { by lemma 36 }
% 112.01/14.58 complement(X)
% 112.01/14.58
% 112.01/14.58 Lemma 38: meet(X, top) = X.
% 112.01/14.58 Proof:
% 112.01/14.58 meet(X, top)
% 112.01/14.58 = { by lemma 19 R->L }
% 112.01/14.58 complement(join(zero, complement(X)))
% 112.01/14.58 = { by lemma 37 R->L }
% 112.01/14.58 join(zero, complement(join(zero, complement(X))))
% 112.01/14.58 = { by lemma 19 }
% 112.01/14.58 join(zero, meet(X, top))
% 112.01/14.58 = { by lemma 36 }
% 112.01/14.58 X
% 112.01/14.58
% 112.01/14.58 Lemma 39: join(X, zero) = X.
% 112.01/14.58 Proof:
% 112.01/14.58 join(X, zero)
% 112.01/14.58 = { by lemma 15 R->L }
% 112.01/14.58 join(X, complement(top))
% 112.01/14.58 = { by lemma 30 R->L }
% 112.01/14.58 join(X, complement(join(complement(X), top)))
% 112.01/14.58 = { by lemma 38 R->L }
% 112.01/14.58 join(meet(X, top), complement(join(complement(X), top)))
% 112.01/14.58 = { by lemma 16 }
% 112.01/14.58 X
% 112.01/14.58
% 112.01/14.58 Lemma 40: join(meet(complement(X), Y), meet(Y, X)) = Y.
% 112.01/14.58 Proof:
% 112.01/14.58 join(meet(complement(X), Y), meet(Y, X))
% 112.01/14.58 = { by lemma 18 }
% 112.01/14.58 join(meet(Y, complement(X)), meet(Y, X))
% 112.01/14.58 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.58 join(meet(Y, X), meet(Y, complement(X)))
% 112.01/14.58 = { by lemma 17 }
% 112.01/14.58 Y
% 112.01/14.58
% 112.01/14.58 Lemma 41: complement(complement(X)) = X.
% 112.01/14.58 Proof:
% 112.01/14.58 complement(complement(X))
% 112.01/14.58 = { by lemma 40 R->L }
% 112.01/14.58 join(meet(complement(X), complement(complement(X))), meet(complement(complement(X)), X))
% 112.01/14.58 = { by axiom 2 (def_zero) R->L }
% 112.01/14.58 join(zero, meet(complement(complement(X)), X))
% 112.01/14.58 = { by lemma 18 R->L }
% 112.01/14.58 join(zero, meet(X, complement(complement(X))))
% 112.01/14.58 = { by axiom 2 (def_zero) }
% 112.01/14.58 join(meet(X, complement(X)), meet(X, complement(complement(X))))
% 112.01/14.58 = { by lemma 17 }
% 112.01/14.58 X
% 112.01/14.58
% 112.01/14.58 Lemma 42: join(zero, X) = X.
% 112.01/14.58 Proof:
% 112.01/14.58 join(zero, X)
% 112.01/14.58 = { by lemma 41 R->L }
% 112.01/14.58 join(zero, complement(complement(X)))
% 112.01/14.58 = { by lemma 24 }
% 112.01/14.58 X
% 112.01/14.58
% 112.01/14.58 Lemma 43: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 112.01/14.58 Proof:
% 112.01/14.58 complement(join(X, complement(Y)))
% 112.01/14.58 = { by lemma 41 R->L }
% 112.01/14.58 complement(join(complement(complement(X)), complement(Y)))
% 112.01/14.59 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 112.01/14.59 meet(complement(X), Y)
% 112.01/14.59 = { by lemma 18 R->L }
% 112.01/14.59 meet(Y, complement(X))
% 112.01/14.59
% 112.01/14.59 Lemma 44: complement(join(X, Y)) = meet(complement(X), complement(Y)).
% 112.01/14.59 Proof:
% 112.01/14.59 complement(join(X, Y))
% 112.01/14.59 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.59 complement(join(Y, X))
% 112.01/14.59 = { by lemma 38 R->L }
% 112.01/14.59 complement(join(Y, meet(X, top)))
% 112.01/14.59 = { by lemma 19 R->L }
% 112.01/14.59 complement(join(Y, complement(join(zero, complement(X)))))
% 112.01/14.59 = { by lemma 43 }
% 112.01/14.59 meet(join(zero, complement(X)), complement(Y))
% 112.01/14.59 = { by lemma 37 }
% 112.01/14.59 meet(complement(X), complement(Y))
% 112.01/14.59
% 112.01/14.59 Lemma 45: complement(meet(X, Y)) = join(complement(X), complement(Y)).
% 112.01/14.59 Proof:
% 112.01/14.59 complement(meet(X, Y))
% 112.01/14.59 = { by lemma 18 }
% 112.01/14.59 complement(meet(Y, X))
% 112.01/14.59 = { by lemma 39 R->L }
% 112.01/14.59 complement(join(meet(Y, X), zero))
% 112.01/14.59 = { by lemma 15 R->L }
% 112.01/14.59 complement(join(meet(Y, X), complement(top)))
% 112.01/14.59 = { by axiom 7 (maddux4_definiton_of_meet) }
% 112.01/14.59 complement(join(complement(join(complement(Y), complement(X))), complement(top)))
% 112.01/14.59 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 112.01/14.59 meet(join(complement(Y), complement(X)), top)
% 112.01/14.59 = { by lemma 38 }
% 112.01/14.59 join(complement(Y), complement(X))
% 112.01/14.59 = { by axiom 3 (maddux1_join_commutativity) }
% 112.01/14.59 join(complement(X), complement(Y))
% 112.01/14.59
% 112.01/14.59 Lemma 46: join(X, join(Y, Z)) = join(Y, join(X, Z)).
% 112.01/14.59 Proof:
% 112.01/14.59 join(X, join(Y, Z))
% 112.01/14.59 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.59 join(join(Y, Z), X)
% 112.01/14.59 = { by axiom 9 (maddux2_join_associativity) R->L }
% 112.01/14.59 join(Y, join(Z, X))
% 112.01/14.59 = { by axiom 3 (maddux1_join_commutativity) }
% 112.01/14.59 join(Y, join(X, Z))
% 112.01/14.59
% 112.01/14.59 Lemma 47: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 112.01/14.59 Proof:
% 112.01/14.59 complement(join(complement(X), Y))
% 112.01/14.59 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.59 complement(join(Y, complement(X)))
% 112.01/14.59 = { by lemma 43 }
% 112.01/14.59 meet(X, complement(Y))
% 112.01/14.59
% 112.01/14.59 Lemma 48: complement(meet(Y, complement(X))) = join(X, complement(Y)).
% 112.01/14.59 Proof:
% 112.01/14.59 complement(meet(Y, complement(X)))
% 112.01/14.59 = { by lemma 45 }
% 112.01/14.59 join(complement(Y), complement(complement(X)))
% 112.01/14.59 = { by lemma 41 }
% 112.01/14.59 join(complement(Y), X)
% 112.01/14.59 = { by axiom 3 (maddux1_join_commutativity) }
% 112.01/14.59 join(X, complement(Y))
% 112.01/14.59
% 112.01/14.59 Lemma 49: meet(X, join(X, Y)) = X.
% 112.01/14.59 Proof:
% 112.01/14.59 meet(X, join(X, Y))
% 112.01/14.59 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.59 meet(X, join(Y, X))
% 112.01/14.59 = { by lemma 39 R->L }
% 112.01/14.59 join(meet(X, join(Y, X)), zero)
% 112.01/14.59 = { by lemma 15 R->L }
% 112.01/14.59 join(meet(X, join(Y, X)), complement(top))
% 112.01/14.59 = { by lemma 41 R->L }
% 112.01/14.59 join(meet(X, join(Y, complement(complement(X)))), complement(top))
% 112.01/14.59 = { by lemma 30 R->L }
% 112.01/14.59 join(meet(X, join(Y, complement(complement(X)))), complement(join(Y, top)))
% 112.01/14.59 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.59 join(meet(X, join(Y, complement(complement(X)))), complement(join(top, Y)))
% 112.01/14.59 = { by axiom 4 (def_top) }
% 112.01/14.59 join(meet(X, join(Y, complement(complement(X)))), complement(join(join(complement(X), complement(complement(X))), Y)))
% 112.01/14.59 = { by axiom 9 (maddux2_join_associativity) R->L }
% 112.01/14.59 join(meet(X, join(Y, complement(complement(X)))), complement(join(complement(X), join(complement(complement(X)), Y))))
% 112.01/14.59 = { by axiom 3 (maddux1_join_commutativity) }
% 112.01/14.59 join(meet(X, join(Y, complement(complement(X)))), complement(join(complement(X), join(Y, complement(complement(X))))))
% 112.01/14.59 = { by lemma 16 }
% 112.01/14.59 X
% 112.01/14.59
% 112.01/14.59 Lemma 50: join(X, converse(complement(converse(X)))) = top.
% 112.01/14.59 Proof:
% 112.01/14.59 join(X, converse(complement(converse(X))))
% 112.01/14.59 = { by lemma 32 R->L }
% 112.01/14.59 converse(join(converse(X), complement(converse(X))))
% 112.01/14.59 = { by axiom 4 (def_top) R->L }
% 112.01/14.59 converse(top)
% 112.01/14.59 = { by lemma 34 }
% 112.01/14.59 top
% 112.01/14.59
% 112.01/14.59 Lemma 51: meet(X, converse(complement(converse(complement(X))))) = X.
% 112.01/14.59 Proof:
% 112.01/14.59 meet(X, converse(complement(converse(complement(X)))))
% 112.01/14.59 = { by lemma 39 R->L }
% 112.01/14.59 join(meet(X, converse(complement(converse(complement(X))))), zero)
% 112.01/14.59 = { by lemma 15 R->L }
% 112.01/14.59 join(meet(X, converse(complement(converse(complement(X))))), complement(top))
% 112.01/14.59 = { by lemma 50 R->L }
% 112.01/14.59 join(meet(X, converse(complement(converse(complement(X))))), complement(join(complement(X), converse(complement(converse(complement(X)))))))
% 112.01/14.59 = { by lemma 16 }
% 112.01/14.59 X
% 112.01/14.59
% 112.01/14.59 Lemma 52: meet(X, converse(complement(converse(X)))) = zero.
% 112.01/14.59 Proof:
% 112.01/14.59 meet(X, converse(complement(converse(X))))
% 112.01/14.59 = { by lemma 18 }
% 112.01/14.59 meet(converse(complement(converse(X))), X)
% 112.01/14.59 = { by axiom 1 (converse_idempotence) R->L }
% 112.01/14.59 meet(converse(complement(converse(X))), converse(converse(X)))
% 112.01/14.59 = { by lemma 51 R->L }
% 112.01/14.59 meet(converse(complement(converse(X))), converse(meet(converse(X), converse(complement(converse(complement(converse(X))))))))
% 112.01/14.59 = { by lemma 18 }
% 112.01/14.59 meet(converse(complement(converse(X))), converse(meet(converse(complement(converse(complement(converse(X))))), converse(X))))
% 112.01/14.59 = { by lemma 49 R->L }
% 112.01/14.59 meet(converse(complement(converse(X))), meet(converse(meet(converse(complement(converse(complement(converse(X))))), converse(X))), join(converse(meet(converse(complement(converse(complement(converse(X))))), converse(X))), converse(complement(join(complement(converse(complement(converse(complement(converse(X)))))), converse(X)))))))
% 112.01/14.59 = { by axiom 8 (converse_additivity) R->L }
% 112.01/14.59 meet(converse(complement(converse(X))), meet(converse(meet(converse(complement(converse(complement(converse(X))))), converse(X))), converse(join(meet(converse(complement(converse(complement(converse(X))))), converse(X)), complement(join(complement(converse(complement(converse(complement(converse(X)))))), converse(X)))))))
% 112.01/14.59 = { by lemma 16 }
% 112.01/14.59 meet(converse(complement(converse(X))), meet(converse(meet(converse(complement(converse(complement(converse(X))))), converse(X))), converse(converse(complement(converse(complement(converse(X))))))))
% 112.01/14.59 = { by lemma 18 R->L }
% 112.01/14.59 meet(converse(complement(converse(X))), meet(converse(converse(complement(converse(complement(converse(X)))))), converse(meet(converse(complement(converse(complement(converse(X))))), converse(X)))))
% 112.01/14.59 = { by axiom 1 (converse_idempotence) }
% 112.01/14.59 meet(converse(complement(converse(X))), meet(complement(converse(complement(converse(X)))), converse(meet(converse(complement(converse(complement(converse(X))))), converse(X)))))
% 112.01/14.59 = { by lemma 18 R->L }
% 112.01/14.59 meet(converse(complement(converse(X))), meet(complement(converse(complement(converse(X)))), converse(meet(converse(X), converse(complement(converse(complement(converse(X)))))))))
% 112.01/14.59 = { by lemma 51 }
% 112.01/14.59 meet(converse(complement(converse(X))), meet(complement(converse(complement(converse(X)))), converse(converse(X))))
% 112.01/14.59 = { by lemma 18 R->L }
% 112.01/14.59 meet(converse(complement(converse(X))), meet(converse(converse(X)), complement(converse(complement(converse(X))))))
% 112.01/14.59 = { by axiom 1 (converse_idempotence) }
% 112.01/14.59 meet(converse(complement(converse(X))), meet(X, complement(converse(complement(converse(X))))))
% 112.01/14.59 = { by lemma 43 R->L }
% 112.01/14.59 meet(converse(complement(converse(X))), complement(join(converse(complement(converse(X))), complement(X))))
% 112.01/14.59 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.59 meet(converse(complement(converse(X))), complement(join(complement(X), converse(complement(converse(X))))))
% 112.01/14.59 = { by lemma 47 R->L }
% 112.01/14.59 complement(join(complement(converse(complement(converse(X)))), join(complement(X), converse(complement(converse(X))))))
% 112.01/14.59 = { by axiom 9 (maddux2_join_associativity) }
% 112.01/14.59 complement(join(join(complement(converse(complement(converse(X)))), complement(X)), converse(complement(converse(X)))))
% 112.01/14.59 = { by lemma 49 R->L }
% 112.01/14.59 meet(complement(join(join(complement(converse(complement(converse(X)))), complement(X)), converse(complement(converse(X))))), join(complement(join(join(complement(converse(complement(converse(X)))), complement(X)), converse(complement(converse(X))))), complement(join(complement(converse(complement(converse(X)))), complement(X)))))
% 112.01/14.59 = { by axiom 3 (maddux1_join_commutativity) }
% 112.01/14.59 meet(complement(join(join(complement(converse(complement(converse(X)))), complement(X)), converse(complement(converse(X))))), join(complement(join(complement(converse(complement(converse(X)))), complement(X))), complement(join(join(complement(converse(complement(converse(X)))), complement(X)), converse(complement(converse(X)))))))
% 112.01/14.59 = { by lemma 45 R->L }
% 112.01/14.59 meet(complement(join(join(complement(converse(complement(converse(X)))), complement(X)), converse(complement(converse(X))))), complement(meet(join(complement(converse(complement(converse(X)))), complement(X)), join(join(complement(converse(complement(converse(X)))), complement(X)), converse(complement(converse(X)))))))
% 112.01/14.59 = { by lemma 49 }
% 112.01/14.59 meet(complement(join(join(complement(converse(complement(converse(X)))), complement(X)), converse(complement(converse(X))))), complement(join(complement(converse(complement(converse(X)))), complement(X))))
% 112.01/14.60 = { by lemma 18 R->L }
% 112.01/14.60 meet(complement(join(complement(converse(complement(converse(X)))), complement(X))), complement(join(join(complement(converse(complement(converse(X)))), complement(X)), converse(complement(converse(X))))))
% 112.01/14.60 = { by axiom 3 (maddux1_join_commutativity) }
% 112.01/14.60 meet(complement(join(complement(converse(complement(converse(X)))), complement(X))), complement(join(converse(complement(converse(X))), join(complement(converse(complement(converse(X)))), complement(X)))))
% 112.01/14.60 = { by lemma 33 }
% 112.01/14.60 meet(complement(join(complement(converse(complement(converse(X)))), complement(X))), complement(top))
% 112.01/14.60 = { by lemma 47 }
% 112.01/14.60 meet(meet(converse(complement(converse(X))), complement(complement(X))), complement(top))
% 112.01/14.60 = { by lemma 15 }
% 112.01/14.60 meet(meet(converse(complement(converse(X))), complement(complement(X))), zero)
% 112.01/14.60 = { by lemma 35 }
% 112.01/14.60 zero
% 112.01/14.60
% 112.01/14.60 Lemma 53: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 112.01/14.60 Proof:
% 112.01/14.60 join(meet(X, Y), meet(Y, complement(X)))
% 112.01/14.60 = { by lemma 18 }
% 112.01/14.60 join(meet(Y, X), meet(Y, complement(X)))
% 112.01/14.60 = { by lemma 17 }
% 112.01/14.60 Y
% 112.01/14.60
% 112.01/14.60 Lemma 54: join(composition(X, Y), composition(X, complement(Y))) = composition(X, top).
% 112.01/14.60 Proof:
% 112.01/14.60 join(composition(X, Y), composition(X, complement(Y)))
% 112.01/14.60 = { by lemma 51 R->L }
% 112.01/14.60 join(composition(X, Y), composition(X, meet(complement(Y), converse(complement(converse(complement(complement(Y))))))))
% 112.01/14.60 = { by lemma 41 }
% 112.01/14.60 join(composition(X, Y), composition(X, meet(complement(Y), converse(complement(converse(Y))))))
% 112.01/14.60 = { by lemma 39 R->L }
% 112.01/14.60 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), zero)))
% 112.01/14.60 = { by lemma 52 R->L }
% 112.01/14.60 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), meet(Y, converse(complement(converse(Y)))))))
% 112.01/14.60 = { by lemma 18 }
% 112.01/14.60 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), meet(converse(complement(converse(Y))), Y))))
% 112.01/14.60 = { by lemma 40 }
% 112.01/14.60 join(composition(X, Y), composition(X, converse(complement(converse(Y)))))
% 112.01/14.60 = { by axiom 1 (converse_idempotence) R->L }
% 112.01/14.60 converse(converse(join(composition(X, Y), composition(X, converse(complement(converse(Y)))))))
% 112.01/14.60 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.60 converse(converse(join(composition(X, converse(complement(converse(Y)))), composition(X, Y))))
% 112.01/14.60 = { by axiom 8 (converse_additivity) }
% 112.01/14.60 converse(join(converse(composition(X, converse(complement(converse(Y))))), converse(composition(X, Y))))
% 112.01/14.60 = { by axiom 10 (converse_multiplicativity) }
% 112.01/14.60 converse(join(composition(converse(converse(complement(converse(Y)))), converse(X)), converse(composition(X, Y))))
% 112.01/14.60 = { by axiom 1 (converse_idempotence) }
% 112.01/14.60 converse(join(composition(complement(converse(Y)), converse(X)), converse(composition(X, Y))))
% 112.01/14.60 = { by axiom 3 (maddux1_join_commutativity) }
% 112.01/14.60 converse(join(converse(composition(X, Y)), composition(complement(converse(Y)), converse(X))))
% 112.01/14.60 = { by axiom 10 (converse_multiplicativity) }
% 112.01/14.60 converse(join(composition(converse(Y), converse(X)), composition(complement(converse(Y)), converse(X))))
% 112.01/14.60 = { by axiom 12 (composition_distributivity) R->L }
% 112.01/14.60 converse(composition(join(converse(Y), complement(converse(Y))), converse(X)))
% 112.01/14.60 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.60 converse(composition(join(complement(converse(Y)), converse(Y)), converse(X)))
% 112.01/14.60 = { by lemma 31 R->L }
% 112.01/14.60 converse(composition(converse(join(Y, converse(complement(converse(Y))))), converse(X)))
% 112.01/14.60 = { by axiom 10 (converse_multiplicativity) R->L }
% 112.01/14.60 converse(converse(composition(X, join(Y, converse(complement(converse(Y)))))))
% 112.01/14.60 = { by axiom 1 (converse_idempotence) }
% 112.01/14.60 composition(X, join(Y, converse(complement(converse(Y)))))
% 112.01/14.60 = { by lemma 50 }
% 112.01/14.60 composition(X, top)
% 112.01/14.60
% 112.01/14.60 Lemma 55: join(X, complement(meet(Y, complement(composition(x0, X))))) = complement(meet(Y, complement(X))).
% 112.01/14.60 Proof:
% 112.01/14.60 join(X, complement(meet(Y, complement(composition(x0, X)))))
% 112.01/14.60 = { by lemma 48 }
% 112.01/14.60 join(X, join(composition(x0, X), complement(Y)))
% 112.01/14.60 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.01/14.60 join(X, join(complement(Y), composition(x0, X)))
% 112.01/14.60 = { by axiom 9 (maddux2_join_associativity) }
% 112.01/14.60 join(join(X, complement(Y)), composition(x0, X))
% 112.01/14.60 = { by lemma 48 R->L }
% 112.01/14.60 join(complement(meet(Y, complement(X))), composition(x0, X))
% 112.01/14.60 = { by axiom 3 (maddux1_join_commutativity) }
% 112.01/14.60 join(composition(x0, X), complement(meet(Y, complement(X))))
% 112.01/14.60 = { by lemma 48 }
% 112.01/14.60 join(composition(x0, X), join(X, complement(Y)))
% 112.01/14.60 = { by axiom 9 (maddux2_join_associativity) }
% 112.01/14.60 join(join(composition(x0, X), X), complement(Y))
% 112.01/14.60 = { by lemma 22 R->L }
% 112.01/14.60 join(join(composition(x0, X), composition(join(x0, one), X)), complement(Y))
% 112.01/14.60 = { by axiom 5 (goals) }
% 112.01/14.60 join(join(composition(x0, X), composition(one, X)), complement(Y))
% 112.01/14.60 = { by axiom 12 (composition_distributivity) R->L }
% 112.01/14.60 join(composition(join(x0, one), X), complement(Y))
% 112.01/14.60 = { by lemma 22 }
% 112.01/14.60 join(X, complement(Y))
% 112.01/14.60 = { by lemma 48 R->L }
% 112.01/14.60 complement(meet(Y, complement(X)))
% 112.01/14.60
% 112.01/14.60 Lemma 56: meet(meet(X, complement(Y)), complement(meet(X, complement(composition(x0, Y))))) = zero.
% 112.01/14.60 Proof:
% 112.01/14.60 meet(meet(X, complement(Y)), complement(meet(X, complement(composition(x0, Y)))))
% 112.01/14.60 = { by lemma 43 R->L }
% 112.01/14.60 complement(join(meet(X, complement(composition(x0, Y))), complement(meet(X, complement(Y)))))
% 112.01/14.60 = { by lemma 55 R->L }
% 112.01/14.60 complement(join(meet(X, complement(composition(x0, Y))), join(Y, complement(meet(X, complement(composition(x0, Y)))))))
% 112.01/14.60 = { by lemma 27 }
% 112.01/14.60 complement(join(Y, top))
% 112.01/14.60 = { by lemma 29 R->L }
% 112.01/14.60 complement(join(Z, top))
% 112.01/14.60 = { by lemma 30 }
% 112.01/14.60 complement(top)
% 112.01/14.60 = { by lemma 15 }
% 112.01/14.60 zero
% 112.01/14.60
% 112.01/14.60 Lemma 57: complement(join(meet(X, complement(composition(x0, Y))), meet(X, complement(Y)))) = complement(meet(X, complement(composition(x0, Y)))).
% 112.01/14.60 Proof:
% 112.01/14.60 complement(join(meet(X, complement(composition(x0, Y))), meet(X, complement(Y))))
% 112.01/14.60 = { by lemma 44 }
% 112.01/14.60 meet(complement(meet(X, complement(composition(x0, Y)))), complement(meet(X, complement(Y))))
% 112.01/14.60 = { by lemma 42 R->L }
% 112.01/14.60 join(zero, meet(complement(meet(X, complement(composition(x0, Y)))), complement(meet(X, complement(Y)))))
% 112.80/14.60 = { by lemma 56 R->L }
% 112.80/14.60 join(meet(meet(X, complement(Y)), complement(meet(X, complement(composition(x0, Y))))), meet(complement(meet(X, complement(composition(x0, Y)))), complement(meet(X, complement(Y)))))
% 112.80/14.60 = { by lemma 53 }
% 112.80/14.60 complement(meet(X, complement(composition(x0, Y))))
% 112.80/14.61
% 112.80/14.61 Goal 1 (goals_1): tuple(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2))))) = tuple(meet(composition(x0, x1), complement(x2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))).
% 112.80/14.61 Proof:
% 112.80/14.61 tuple(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by lemma 17 R->L }
% 112.80/14.61 tuple(join(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), x2), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by lemma 18 }
% 112.80/14.61 tuple(join(meet(x2, join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2)))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by lemma 41 R->L }
% 112.80/14.61 tuple(join(meet(x2, complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2)))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by lemma 47 R->L }
% 112.80/14.61 tuple(join(complement(join(complement(x2), complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2)))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by axiom 1 (converse_idempotence) R->L }
% 112.80/14.61 tuple(join(complement(converse(converse(join(complement(x2), complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2)))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by lemma 57 }
% 112.80/14.61 tuple(join(complement(converse(converse(join(complement(x2), complement(meet(composition(x0, x1), complement(composition(x0, x2)))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by lemma 48 }
% 112.80/14.61 tuple(join(complement(converse(converse(join(complement(x2), join(composition(x0, x2), complement(composition(x0, x1))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by axiom 9 (maddux2_join_associativity) }
% 112.80/14.61 tuple(join(complement(converse(converse(join(join(complement(x2), composition(x0, x2)), complement(composition(x0, x1)))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by axiom 8 (converse_additivity) }
% 112.80/14.61 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), converse(complement(composition(x0, x1)))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by lemma 41 R->L }
% 112.80/14.61 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(complement(converse(complement(composition(x0, x1)))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by axiom 1 (converse_idempotence) R->L }
% 112.80/14.61 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(converse(complement(converse(complement(composition(x0, x1)))))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by lemma 53 R->L }
% 112.80/14.61 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(join(meet(composition(x0, x1), converse(complement(converse(complement(composition(x0, x1)))))), meet(converse(complement(converse(complement(composition(x0, x1))))), complement(composition(x0, x1))))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by lemma 18 R->L }
% 112.80/14.61 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(join(meet(composition(x0, x1), converse(complement(converse(complement(composition(x0, x1)))))), meet(complement(composition(x0, x1)), converse(complement(converse(complement(composition(x0, x1)))))))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by lemma 52 }
% 112.80/14.61 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(join(meet(composition(x0, x1), converse(complement(converse(complement(composition(x0, x1)))))), zero)))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by lemma 39 }
% 112.80/14.61 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(meet(composition(x0, x1), converse(complement(converse(complement(composition(x0, x1))))))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by lemma 51 }
% 112.80/14.61 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(composition(x0, x1)))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by lemma 49 R->L }
% 112.80/14.61 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(meet(composition(x0, x1), join(composition(x0, x1), join(complement(x2), composition(x0, complement(x1)))))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.61 = { by lemma 46 }
% 112.80/14.61 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(meet(composition(x0, x1), join(complement(x2), join(composition(x0, x1), composition(x0, complement(x1)))))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by lemma 54 }
% 112.80/14.62 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(meet(composition(x0, x1), join(complement(x2), composition(x0, top)))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.80/14.62 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(meet(composition(x0, x1), join(composition(x0, top), complement(x2)))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by lemma 54 R->L }
% 112.80/14.62 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(meet(composition(x0, x1), join(join(composition(x0, x2), composition(x0, complement(x2))), complement(x2)))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by axiom 9 (maddux2_join_associativity) R->L }
% 112.80/14.62 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(meet(composition(x0, x1), join(composition(x0, x2), join(composition(x0, complement(x2)), complement(x2))))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by axiom 3 (maddux1_join_commutativity) }
% 112.80/14.62 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(meet(composition(x0, x1), join(composition(x0, x2), join(complement(x2), composition(x0, complement(x2)))))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by lemma 22 R->L }
% 112.80/14.62 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(meet(composition(x0, x1), join(composition(x0, x2), join(composition(join(x0, one), complement(x2)), composition(x0, complement(x2)))))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by axiom 12 (composition_distributivity) R->L }
% 112.80/14.62 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(meet(composition(x0, x1), join(composition(x0, x2), composition(join(join(x0, one), x0), complement(x2))))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.80/14.62 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(meet(composition(x0, x1), join(composition(x0, x2), composition(join(x0, join(x0, one)), complement(x2))))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by axiom 5 (goals) }
% 112.80/14.62 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(meet(composition(x0, x1), join(composition(x0, x2), composition(join(x0, one), complement(x2))))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by lemma 22 }
% 112.80/14.62 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(meet(composition(x0, x1), join(composition(x0, x2), complement(x2)))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by axiom 3 (maddux1_join_commutativity) }
% 112.80/14.62 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(meet(composition(x0, x1), join(complement(x2), composition(x0, x2)))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by lemma 18 }
% 112.80/14.62 tuple(join(complement(converse(join(converse(join(complement(x2), composition(x0, x2))), complement(converse(meet(join(complement(x2), composition(x0, x2)), composition(x0, x1))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.80/14.62 tuple(join(complement(converse(join(complement(converse(meet(join(complement(x2), composition(x0, x2)), composition(x0, x1)))), converse(join(complement(x2), composition(x0, x2)))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by lemma 17 R->L }
% 112.80/14.62 tuple(join(complement(converse(join(complement(converse(meet(join(complement(x2), composition(x0, x2)), composition(x0, x1)))), converse(join(meet(join(complement(x2), composition(x0, x2)), composition(x0, x1)), meet(join(complement(x2), composition(x0, x2)), complement(composition(x0, x1)))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.80/14.62 tuple(join(complement(converse(join(complement(converse(meet(join(complement(x2), composition(x0, x2)), composition(x0, x1)))), converse(join(meet(join(complement(x2), composition(x0, x2)), complement(composition(x0, x1))), meet(join(complement(x2), composition(x0, x2)), composition(x0, x1))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 112.80/14.62 tuple(join(complement(converse(join(converse(join(meet(join(complement(x2), composition(x0, x2)), complement(composition(x0, x1))), meet(join(complement(x2), composition(x0, x2)), composition(x0, x1)))), complement(converse(meet(join(complement(x2), composition(x0, x2)), composition(x0, x1))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.62 = { by axiom 8 (converse_additivity) }
% 112.80/14.63 tuple(join(complement(converse(join(join(converse(meet(join(complement(x2), composition(x0, x2)), complement(composition(x0, x1)))), converse(meet(join(complement(x2), composition(x0, x2)), composition(x0, x1)))), complement(converse(meet(join(complement(x2), composition(x0, x2)), composition(x0, x1))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by axiom 9 (maddux2_join_associativity) R->L }
% 112.80/14.63 tuple(join(complement(converse(join(converse(meet(join(complement(x2), composition(x0, x2)), complement(composition(x0, x1)))), join(converse(meet(join(complement(x2), composition(x0, x2)), composition(x0, x1))), complement(converse(meet(join(complement(x2), composition(x0, x2)), composition(x0, x1)))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by lemma 46 }
% 112.80/14.63 tuple(join(complement(converse(join(converse(meet(join(complement(x2), composition(x0, x2)), composition(x0, x1))), join(converse(meet(join(complement(x2), composition(x0, x2)), complement(composition(x0, x1)))), complement(converse(meet(join(complement(x2), composition(x0, x2)), composition(x0, x1)))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by axiom 3 (maddux1_join_commutativity) }
% 112.80/14.63 tuple(join(complement(converse(join(converse(meet(join(complement(x2), composition(x0, x2)), composition(x0, x1))), join(complement(converse(meet(join(complement(x2), composition(x0, x2)), composition(x0, x1)))), converse(meet(join(complement(x2), composition(x0, x2)), complement(composition(x0, x1)))))))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by lemma 33 }
% 112.80/14.63 tuple(join(complement(converse(top)), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by lemma 34 }
% 112.80/14.63 tuple(join(complement(top), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by lemma 15 }
% 112.80/14.63 tuple(join(zero, meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by lemma 42 }
% 112.80/14.63 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(x2)), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by lemma 18 }
% 112.80/14.63 tuple(meet(complement(x2), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2)))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by lemma 38 R->L }
% 112.80/14.63 tuple(meet(complement(x2), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), top)), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by lemma 19 R->L }
% 112.80/14.63 tuple(meet(complement(x2), complement(join(zero, complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by lemma 57 }
% 112.80/14.63 tuple(meet(complement(x2), complement(join(zero, complement(meet(composition(x0, x1), complement(composition(x0, x2))))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by lemma 19 }
% 112.80/14.63 tuple(meet(complement(x2), meet(meet(composition(x0, x1), complement(composition(x0, x2))), top)), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by lemma 38 }
% 112.80/14.63 tuple(meet(complement(x2), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by lemma 18 }
% 112.80/14.63 tuple(meet(meet(composition(x0, x1), complement(composition(x0, x2))), complement(x2)), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by lemma 43 R->L }
% 112.80/14.63 tuple(complement(join(x2, complement(meet(composition(x0, x1), complement(composition(x0, x2)))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by lemma 55 }
% 112.80/14.63 tuple(complement(complement(meet(composition(x0, x1), complement(x2)))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by lemma 41 }
% 112.80/14.63 tuple(meet(composition(x0, x1), complement(x2)), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 112.80/14.63 = { by lemma 38 R->L }
% 112.80/14.63 tuple(meet(composition(x0, x1), complement(x2)), meet(join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))), top))
% 112.80/14.63 = { by lemma 19 R->L }
% 112.80/14.63 tuple(meet(composition(x0, x1), complement(x2)), complement(join(zero, complement(join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2))))))))
% 112.80/14.63 = { by axiom 3 (maddux1_join_commutativity) }
% 112.80/14.63 tuple(meet(composition(x0, x1), complement(x2)), complement(join(zero, complement(join(meet(composition(x0, x1), complement(composition(x0, x2_2))), meet(composition(x0, x1), complement(x2_2)))))))
% 112.80/14.63 = { by lemma 44 }
% 112.80/14.63 tuple(meet(composition(x0, x1), complement(x2)), complement(join(zero, meet(complement(meet(composition(x0, x1), complement(composition(x0, x2_2)))), complement(meet(composition(x0, x1), complement(x2_2)))))))
% 112.80/14.63 = { by lemma 42 R->L }
% 112.80/14.63 tuple(meet(composition(x0, x1), complement(x2)), complement(join(zero, join(zero, meet(complement(meet(composition(x0, x1), complement(composition(x0, x2_2)))), complement(meet(composition(x0, x1), complement(x2_2))))))))
% 112.80/14.63 = { by lemma 56 R->L }
% 112.80/14.63 tuple(meet(composition(x0, x1), complement(x2)), complement(join(zero, join(meet(meet(composition(x0, x1), complement(x2_2)), complement(meet(composition(x0, x1), complement(composition(x0, x2_2))))), meet(complement(meet(composition(x0, x1), complement(composition(x0, x2_2)))), complement(meet(composition(x0, x1), complement(x2_2))))))))
% 112.80/14.63 = { by lemma 53 }
% 112.80/14.63 tuple(meet(composition(x0, x1), complement(x2)), complement(join(zero, complement(meet(composition(x0, x1), complement(composition(x0, x2_2)))))))
% 112.80/14.63 = { by lemma 19 }
% 112.80/14.63 tuple(meet(composition(x0, x1), complement(x2)), meet(meet(composition(x0, x1), complement(composition(x0, x2_2))), top))
% 112.80/14.63 = { by lemma 38 }
% 112.80/14.63 tuple(meet(composition(x0, x1), complement(x2)), meet(composition(x0, x1), complement(composition(x0, x2_2))))
% 112.80/14.63 % SZS output end Proof
% 112.80/14.63
% 112.80/14.63 RESULT: Theorem (the conjecture is true).
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