%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : REL029+2 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n002.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 12:33:19 PM UTC 2026
% Result : Theorem 114.45s 14.96s
% Output : Proof 117.56s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.04 % Problem : REL029+2 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.06 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.16/0.42 % Computer : n002.cluster.edu
% 0.16/0.42 % Model : x86_64 x86_64
% 0.16/0.42 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.42 % Memory : 8046.5625MB
% 0.16/0.42 % OS : Linux 6.8.0-71-generic
% 0.16/0.42 % CPULimit : 300
% 0.16/0.42 % WCLimit : 300
% 0.16/0.42 % DateTime : Sun Sep 27 22:55:51 UTC 2026
% 0.16/0.43 % CPUTime :
% 0.16/0.43 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 114.45/14.96 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 114.45/14.96
% 114.45/14.96 % SZS status Theorem
% 114.45/14.96
% 116.77/15.22 % SZS output start Proof
% 116.77/15.22 Axiom 1 (composition_identity): composition(X, one) = X.
% 116.77/15.22 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 116.77/15.22 Axiom 3 (def_top): top = join(X, complement(X)).
% 116.77/15.22 Axiom 4 (goals): join(x1, one) = one.
% 116.77/15.22 Axiom 5 (goals_1): join(x0, one) = one.
% 116.77/15.22 Axiom 6 (def_zero): zero = meet(X, complement(X)).
% 116.77/15.22 Axiom 7 (converse_idempotence): converse(converse(X)) = X.
% 116.77/15.22 Axiom 8 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 116.77/15.22 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 116.77/15.22 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 116.77/15.22 Axiom 11 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 116.77/15.22 Axiom 12 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 116.77/15.22 Axiom 13 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 116.77/15.22 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 116.77/15.22 Axiom 15 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 116.77/15.22
% 116.77/15.22 Lemma 16: complement(top) = zero.
% 116.77/15.22 Proof:
% 116.77/15.22 complement(top)
% 116.77/15.22 = { by axiom 3 (def_top) }
% 116.77/15.22 complement(join(complement(X), complement(complement(X))))
% 116.77/15.22 = { by axiom 8 (maddux4_definiton_of_meet) R->L }
% 116.77/15.22 meet(X, complement(X))
% 116.77/15.22 = { by axiom 6 (def_zero) R->L }
% 116.77/15.22 zero
% 116.77/15.22
% 116.77/15.22 Lemma 17: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 116.77/15.22 Proof:
% 116.77/15.22 converse(composition(converse(X), Y))
% 116.77/15.22 = { by axiom 9 (converse_multiplicativity) }
% 116.77/15.22 composition(converse(Y), converse(converse(X)))
% 116.77/15.22 = { by axiom 7 (converse_idempotence) }
% 116.77/15.22 composition(converse(Y), X)
% 116.77/15.22
% 116.77/15.22 Lemma 18: composition(converse(one), X) = X.
% 116.77/15.22 Proof:
% 116.77/15.22 composition(converse(one), X)
% 116.77/15.22 = { by lemma 17 R->L }
% 116.77/15.22 converse(composition(converse(X), one))
% 116.77/15.22 = { by axiom 1 (composition_identity) }
% 116.77/15.22 converse(converse(X))
% 116.77/15.22 = { by axiom 7 (converse_idempotence) }
% 116.77/15.22 X
% 116.77/15.22
% 116.77/15.22 Lemma 19: converse(one) = one.
% 116.77/15.22 Proof:
% 116.77/15.22 converse(one)
% 116.77/15.22 = { by axiom 1 (composition_identity) R->L }
% 116.77/15.22 composition(converse(one), one)
% 116.77/15.22 = { by lemma 18 }
% 116.77/15.22 one
% 116.77/15.22
% 116.77/15.22 Lemma 20: join(X, join(Y, complement(X))) = join(Y, top).
% 116.77/15.22 Proof:
% 116.77/15.22 join(X, join(Y, complement(X)))
% 116.77/15.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.22 join(X, join(complement(X), Y))
% 116.77/15.22 = { by axiom 12 (maddux2_join_associativity) }
% 116.77/15.22 join(join(X, complement(X)), Y)
% 116.77/15.22 = { by axiom 3 (def_top) R->L }
% 116.77/15.22 join(top, Y)
% 116.77/15.22 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.22 join(Y, top)
% 116.77/15.22
% 116.77/15.22 Lemma 21: composition(one, X) = X.
% 116.77/15.22 Proof:
% 116.77/15.22 composition(one, X)
% 116.77/15.22 = { by lemma 18 R->L }
% 116.77/15.22 composition(converse(one), composition(one, X))
% 116.77/15.22 = { by axiom 10 (composition_associativity) }
% 116.77/15.22 composition(composition(converse(one), one), X)
% 116.77/15.22 = { by axiom 1 (composition_identity) }
% 116.77/15.22 composition(converse(one), X)
% 116.77/15.22 = { by lemma 18 }
% 116.77/15.22 X
% 116.77/15.22
% 116.77/15.22 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 116.77/15.22 Proof:
% 116.77/15.22 join(complement(X), complement(X))
% 116.77/15.22 = { by lemma 18 R->L }
% 116.77/15.22 join(complement(X), composition(converse(one), complement(X)))
% 116.77/15.22 = { by lemma 21 R->L }
% 116.77/15.22 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 116.77/15.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.22 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 116.77/15.22 = { by axiom 15 (converse_cancellativity) }
% 116.77/15.22 complement(X)
% 116.77/15.22
% 116.77/15.22 Lemma 23: join(top, complement(X)) = top.
% 116.77/15.22 Proof:
% 116.77/15.22 join(top, complement(X))
% 116.77/15.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.22 join(complement(X), top)
% 116.77/15.22 = { by lemma 20 R->L }
% 116.77/15.22 join(X, join(complement(X), complement(X)))
% 116.77/15.22 = { by lemma 22 }
% 116.77/15.22 join(X, complement(X))
% 116.77/15.22 = { by axiom 3 (def_top) R->L }
% 116.77/15.22 top
% 116.77/15.22
% 116.77/15.22 Lemma 24: join(X, top) = top.
% 116.77/15.22 Proof:
% 116.77/15.22 join(X, top)
% 116.77/15.22 = { by lemma 23 R->L }
% 116.77/15.22 join(X, join(top, complement(X)))
% 116.77/15.22 = { by lemma 20 }
% 116.77/15.22 join(top, top)
% 116.77/15.22 = { by axiom 3 (def_top) }
% 116.77/15.22 join(top, join(zero, complement(zero)))
% 116.77/15.22 = { by axiom 12 (maddux2_join_associativity) }
% 116.77/15.22 join(join(top, zero), complement(zero))
% 116.77/15.22 = { by lemma 16 R->L }
% 116.77/15.22 join(join(top, complement(top)), complement(zero))
% 116.77/15.22 = { by axiom 3 (def_top) R->L }
% 116.77/15.22 join(top, complement(zero))
% 116.77/15.22 = { by lemma 23 }
% 116.77/15.22 top
% 116.77/15.22
% 116.77/15.22 Lemma 25: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 116.77/15.22 Proof:
% 116.77/15.22 join(meet(X, Y), complement(join(complement(X), Y)))
% 116.77/15.22 = { by axiom 8 (maddux4_definiton_of_meet) }
% 116.77/15.23 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 116.77/15.23 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 116.77/15.23 X
% 116.77/15.23
% 116.77/15.23 Lemma 26: join(zero, complement(complement(X))) = X.
% 116.77/15.23 Proof:
% 116.77/15.23 join(zero, complement(complement(X)))
% 116.77/15.23 = { by axiom 6 (def_zero) }
% 116.77/15.23 join(meet(X, complement(X)), complement(complement(X)))
% 116.77/15.23 = { by lemma 22 R->L }
% 116.77/15.23 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 116.77/15.23 = { by lemma 25 }
% 116.77/15.23 X
% 116.77/15.23
% 116.77/15.23 Lemma 27: meet(Y, X) = meet(X, Y).
% 116.77/15.23 Proof:
% 116.77/15.23 meet(Y, X)
% 116.77/15.23 = { by axiom 8 (maddux4_definiton_of_meet) }
% 116.77/15.23 complement(join(complement(Y), complement(X)))
% 116.77/15.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.23 complement(join(complement(X), complement(Y)))
% 116.77/15.23 = { by axiom 8 (maddux4_definiton_of_meet) R->L }
% 116.77/15.23 meet(X, Y)
% 116.77/15.23
% 116.77/15.23 Lemma 28: complement(join(zero, complement(X))) = meet(X, top).
% 116.77/15.23 Proof:
% 116.77/15.23 complement(join(zero, complement(X)))
% 116.77/15.23 = { by lemma 16 R->L }
% 116.77/15.23 complement(join(complement(top), complement(X)))
% 116.77/15.23 = { by axiom 8 (maddux4_definiton_of_meet) R->L }
% 116.77/15.23 meet(top, X)
% 116.77/15.23 = { by lemma 27 R->L }
% 116.77/15.23 meet(X, top)
% 116.77/15.23
% 116.77/15.23 Lemma 29: join(zero, meet(X, top)) = X.
% 116.77/15.23 Proof:
% 116.77/15.23 join(zero, meet(X, top))
% 116.77/15.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.23 join(meet(X, top), zero)
% 116.77/15.23 = { by lemma 16 R->L }
% 116.77/15.23 join(meet(X, top), complement(top))
% 116.77/15.23 = { by lemma 24 R->L }
% 116.77/15.23 join(meet(X, top), complement(join(complement(X), top)))
% 116.77/15.23 = { by lemma 25 }
% 116.77/15.23 X
% 116.77/15.23
% 116.77/15.23 Lemma 30: join(zero, complement(X)) = complement(X).
% 116.77/15.23 Proof:
% 116.77/15.23 join(zero, complement(X))
% 116.77/15.23 = { by lemma 26 R->L }
% 116.77/15.23 join(zero, complement(join(zero, complement(complement(X)))))
% 116.77/15.23 = { by lemma 28 }
% 116.77/15.23 join(zero, meet(complement(X), top))
% 116.77/15.23 = { by lemma 29 }
% 116.77/15.23 complement(X)
% 116.77/15.23
% 116.77/15.23 Lemma 31: complement(complement(X)) = X.
% 116.77/15.23 Proof:
% 116.77/15.23 complement(complement(X))
% 116.77/15.23 = { by lemma 30 R->L }
% 116.77/15.23 join(zero, complement(complement(X)))
% 116.77/15.23 = { by lemma 26 }
% 116.77/15.23 X
% 116.77/15.23
% 116.77/15.23 Lemma 32: join(zero, X) = X.
% 116.77/15.23 Proof:
% 116.77/15.23 join(zero, X)
% 116.77/15.23 = { by lemma 31 R->L }
% 116.77/15.23 join(zero, complement(complement(X)))
% 116.77/15.23 = { by lemma 26 }
% 116.77/15.23 X
% 116.77/15.23
% 116.77/15.23 Lemma 33: meet(X, top) = X.
% 116.77/15.23 Proof:
% 116.77/15.23 meet(X, top)
% 116.77/15.23 = { by lemma 28 R->L }
% 116.77/15.23 complement(join(zero, complement(X)))
% 116.77/15.23 = { by lemma 30 R->L }
% 116.77/15.23 join(zero, complement(join(zero, complement(X))))
% 116.77/15.23 = { by lemma 28 }
% 116.77/15.23 join(zero, meet(X, top))
% 116.77/15.23 = { by lemma 29 }
% 116.77/15.23 X
% 116.77/15.23
% 116.77/15.23 Lemma 34: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 116.77/15.23 Proof:
% 116.77/15.23 complement(join(X, complement(Y)))
% 116.77/15.23 = { by lemma 31 R->L }
% 116.77/15.23 complement(join(complement(complement(X)), complement(Y)))
% 116.77/15.23 = { by axiom 8 (maddux4_definiton_of_meet) R->L }
% 116.77/15.23 meet(complement(X), Y)
% 116.77/15.23 = { by lemma 27 R->L }
% 116.77/15.23 meet(Y, complement(X))
% 116.77/15.23
% 116.77/15.23 Lemma 35: complement(join(X, Y)) = meet(complement(X), complement(Y)).
% 116.77/15.23 Proof:
% 116.77/15.23 complement(join(X, Y))
% 116.77/15.23 = { by lemma 33 R->L }
% 116.77/15.23 complement(join(X, meet(Y, top)))
% 116.77/15.23 = { by lemma 28 R->L }
% 116.77/15.23 complement(join(X, complement(join(zero, complement(Y)))))
% 116.77/15.23 = { by lemma 34 }
% 116.77/15.23 meet(join(zero, complement(Y)), complement(X))
% 116.77/15.23 = { by lemma 30 }
% 116.77/15.23 meet(complement(Y), complement(X))
% 116.77/15.23 = { by lemma 27 R->L }
% 116.77/15.23 meet(complement(X), complement(Y))
% 116.77/15.23
% 116.77/15.23 Lemma 36: complement(join(meet(X, Y), complement(Z))) = meet(Z, join(complement(X), complement(Y))).
% 116.77/15.23 Proof:
% 116.77/15.23 complement(join(meet(X, Y), complement(Z)))
% 116.77/15.23 = { by axiom 8 (maddux4_definiton_of_meet) }
% 116.77/15.23 complement(join(complement(join(complement(X), complement(Y))), complement(Z)))
% 116.77/15.23 = { by axiom 8 (maddux4_definiton_of_meet) R->L }
% 116.77/15.23 meet(join(complement(X), complement(Y)), Z)
% 116.77/15.23 = { by lemma 27 R->L }
% 116.77/15.23 meet(Z, join(complement(X), complement(Y)))
% 116.77/15.23
% 116.77/15.23 Lemma 37: complement(meet(X, Y)) = join(complement(X), complement(Y)).
% 116.77/15.23 Proof:
% 116.77/15.23 complement(meet(X, Y))
% 116.77/15.23 = { by lemma 27 }
% 116.77/15.23 complement(meet(Y, X))
% 116.77/15.23 = { by lemma 25 R->L }
% 116.77/15.23 complement(join(meet(meet(Y, X), top), complement(join(complement(meet(Y, X)), top))))
% 116.77/15.23 = { by lemma 33 }
% 116.77/15.23 complement(join(meet(Y, X), complement(join(complement(meet(Y, X)), top))))
% 116.77/15.23 = { by lemma 24 }
% 116.77/15.23 complement(join(meet(Y, X), complement(top)))
% 116.77/15.23 = { by lemma 36 }
% 116.77/15.23 meet(top, join(complement(Y), complement(X)))
% 116.77/15.23 = { by lemma 27 }
% 116.77/15.23 meet(join(complement(Y), complement(X)), top)
% 116.77/15.23 = { by lemma 33 }
% 116.77/15.23 join(complement(Y), complement(X))
% 116.77/15.23 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.23 join(complement(X), complement(Y))
% 116.77/15.23
% 116.77/15.23 Lemma 38: complement(meet(Y, X)) = join(complement(X), complement(Y)).
% 116.77/15.23 Proof:
% 116.77/15.23 complement(meet(Y, X))
% 116.77/15.23 = { by lemma 37 }
% 116.77/15.23 join(complement(Y), complement(X))
% 116.77/15.23 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.23 join(complement(X), complement(Y))
% 116.77/15.23
% 116.77/15.23 Lemma 39: join(complement(X), join(Y, complement(X))) = join(Y, complement(X)).
% 116.77/15.23 Proof:
% 116.77/15.23 join(complement(X), join(Y, complement(X)))
% 116.77/15.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.23 join(complement(X), join(complement(X), Y))
% 116.77/15.23 = { by axiom 12 (maddux2_join_associativity) }
% 116.77/15.23 join(join(complement(X), complement(X)), Y)
% 116.77/15.23 = { by lemma 22 }
% 116.77/15.23 join(complement(X), Y)
% 116.77/15.23 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.23 join(Y, complement(X))
% 116.77/15.23
% 116.77/15.23 Lemma 40: join(X, meet(X, Y)) = X.
% 116.77/15.23 Proof:
% 116.77/15.23 join(X, meet(X, Y))
% 116.77/15.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.23 join(meet(X, Y), X)
% 116.77/15.23 = { by lemma 25 R->L }
% 116.77/15.23 join(meet(X, Y), join(meet(X, Y), complement(join(complement(X), Y))))
% 116.77/15.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.23 join(meet(X, Y), join(complement(join(complement(X), Y)), meet(X, Y)))
% 116.77/15.23 = { by lemma 33 R->L }
% 116.77/15.23 join(meet(X, Y), join(complement(join(complement(X), Y)), meet(meet(X, Y), top)))
% 116.77/15.23 = { by lemma 28 R->L }
% 116.77/15.23 join(meet(X, Y), join(complement(join(complement(X), Y)), complement(join(zero, complement(meet(X, Y))))))
% 116.77/15.23 = { by lemma 33 R->L }
% 116.77/15.23 join(meet(meet(X, Y), top), join(complement(join(complement(X), Y)), complement(join(zero, complement(meet(X, Y))))))
% 116.77/15.23 = { by lemma 28 R->L }
% 116.77/15.23 join(complement(join(zero, complement(meet(X, Y)))), join(complement(join(complement(X), Y)), complement(join(zero, complement(meet(X, Y))))))
% 116.77/15.23 = { by lemma 39 }
% 116.77/15.23 join(complement(join(complement(X), Y)), complement(join(zero, complement(meet(X, Y)))))
% 116.77/15.23 = { by lemma 28 }
% 116.77/15.23 join(complement(join(complement(X), Y)), meet(meet(X, Y), top))
% 116.77/15.23 = { by lemma 33 }
% 116.77/15.23 join(complement(join(complement(X), Y)), meet(X, Y))
% 116.77/15.23 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.23 join(meet(X, Y), complement(join(complement(X), Y)))
% 116.77/15.23 = { by lemma 25 }
% 116.77/15.23 X
% 116.77/15.23
% 116.77/15.23 Lemma 41: join(X, meet(Y, X)) = X.
% 116.77/15.23 Proof:
% 116.77/15.23 join(X, meet(Y, X))
% 116.77/15.23 = { by lemma 27 }
% 116.77/15.23 join(X, meet(X, Y))
% 116.77/15.23 = { by lemma 40 }
% 116.77/15.23 X
% 116.77/15.23
% 116.77/15.23 Lemma 42: join(X, composition(Y, X)) = composition(join(Y, one), X).
% 116.77/15.23 Proof:
% 116.77/15.23 join(X, composition(Y, X))
% 116.77/15.23 = { by lemma 21 R->L }
% 116.77/15.23 join(composition(one, X), composition(Y, X))
% 116.77/15.23 = { by axiom 13 (composition_distributivity) R->L }
% 116.77/15.23 composition(join(one, Y), X)
% 116.77/15.23 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.23 composition(join(Y, one), X)
% 116.77/15.23
% 116.77/15.23 Lemma 43: join(Z, join(X, Y)) = join(X, join(Y, Z)).
% 116.77/15.23 Proof:
% 116.77/15.23 join(Z, join(X, Y))
% 116.77/15.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.23 join(join(X, Y), Z)
% 116.77/15.23 = { by axiom 12 (maddux2_join_associativity) R->L }
% 116.77/15.23 join(X, join(Y, Z))
% 116.77/15.23
% 116.77/15.23 Lemma 44: meet(X, join(X, Y)) = X.
% 116.77/15.23 Proof:
% 116.77/15.23 meet(X, join(X, Y))
% 116.77/15.23 = { by lemma 33 R->L }
% 116.77/15.23 meet(X, join(X, meet(Y, top)))
% 116.77/15.23 = { by lemma 28 R->L }
% 116.77/15.23 meet(X, join(X, complement(join(zero, complement(Y)))))
% 116.77/15.23 = { by lemma 31 R->L }
% 116.77/15.23 meet(X, join(complement(complement(X)), complement(join(zero, complement(Y)))))
% 116.77/15.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.23 meet(X, join(complement(join(zero, complement(Y))), complement(complement(X))))
% 116.77/15.23 = { by axiom 8 (maddux4_definiton_of_meet) }
% 116.77/15.23 complement(join(complement(X), complement(join(complement(join(zero, complement(Y))), complement(complement(X))))))
% 116.77/15.23 = { by axiom 8 (maddux4_definiton_of_meet) R->L }
% 116.77/15.23 complement(join(complement(X), meet(join(zero, complement(Y)), complement(X))))
% 116.77/15.23 = { by lemma 27 R->L }
% 116.77/15.23 complement(join(complement(X), meet(complement(X), join(zero, complement(Y)))))
% 116.77/15.23 = { by lemma 40 }
% 116.77/15.23 complement(complement(X))
% 116.77/15.23 = { by lemma 31 }
% 116.77/15.23 X
% 116.77/15.23
% 116.77/15.23 Lemma 45: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 116.77/15.23 Proof:
% 116.77/15.23 converse(join(X, converse(Y)))
% 116.77/15.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.23 converse(join(converse(Y), X))
% 116.77/15.23 = { by axiom 11 (converse_additivity) }
% 116.77/15.23 join(converse(converse(Y)), converse(X))
% 116.77/15.23 = { by axiom 7 (converse_idempotence) }
% 116.77/15.23 join(Y, converse(X))
% 116.77/15.23
% 116.77/15.23 Lemma 46: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 116.77/15.24 Proof:
% 116.77/15.24 converse(composition(X, converse(Y)))
% 116.77/15.24 = { by axiom 9 (converse_multiplicativity) }
% 116.77/15.24 composition(converse(converse(Y)), converse(X))
% 116.77/15.24 = { by axiom 7 (converse_idempotence) }
% 116.77/15.24 composition(Y, converse(X))
% 116.77/15.24
% 116.77/15.24 Lemma 47: join(X, join(complement(X), Y)) = top.
% 116.77/15.24 Proof:
% 116.77/15.24 join(X, join(complement(X), Y))
% 116.77/15.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.24 join(X, join(Y, complement(X)))
% 116.77/15.24 = { by lemma 20 }
% 116.77/15.24 join(Y, top)
% 116.77/15.24 = { by lemma 24 }
% 116.77/15.24 top
% 116.77/15.24
% 116.77/15.24 Lemma 48: join(X, join(Y, composition(X, x0))) = join(X, Y).
% 116.77/15.24 Proof:
% 116.77/15.24 join(X, join(Y, composition(X, x0)))
% 116.77/15.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.24 join(X, join(composition(X, x0), Y))
% 116.77/15.24 = { by axiom 12 (maddux2_join_associativity) }
% 116.77/15.24 join(join(X, composition(X, x0)), Y)
% 116.77/15.24 = { by axiom 7 (converse_idempotence) R->L }
% 116.77/15.24 join(join(X, composition(X, converse(converse(x0)))), Y)
% 116.77/15.24 = { by lemma 46 R->L }
% 116.77/15.24 join(join(X, converse(composition(converse(x0), converse(X)))), Y)
% 116.77/15.24 = { by lemma 45 R->L }
% 116.77/15.24 join(converse(join(composition(converse(x0), converse(X)), converse(X))), Y)
% 116.77/15.24 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.24 join(converse(join(converse(X), composition(converse(x0), converse(X)))), Y)
% 116.77/15.24 = { by lemma 42 }
% 116.77/15.24 join(converse(composition(join(converse(x0), one), converse(X))), Y)
% 116.77/15.24 = { by lemma 46 }
% 116.77/15.24 join(composition(X, converse(join(converse(x0), one))), Y)
% 116.77/15.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.24 join(composition(X, converse(join(one, converse(x0)))), Y)
% 116.77/15.24 = { by axiom 11 (converse_additivity) }
% 116.77/15.24 join(composition(X, join(converse(one), converse(converse(x0)))), Y)
% 116.77/15.24 = { by lemma 19 }
% 116.77/15.24 join(composition(X, join(one, converse(converse(x0)))), Y)
% 116.77/15.24 = { by axiom 7 (converse_idempotence) }
% 116.77/15.24 join(composition(X, join(one, x0)), Y)
% 116.77/15.24 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.24 join(composition(X, join(x0, one)), Y)
% 116.77/15.24 = { by axiom 5 (goals_1) }
% 116.77/15.24 join(composition(X, one), Y)
% 116.77/15.24 = { by axiom 1 (composition_identity) }
% 116.77/15.24 join(X, Y)
% 116.77/15.24
% 116.77/15.24 Lemma 49: join(composition(join(Z, one), X), Y) = join(X, join(Y, composition(Z, X))).
% 116.77/15.24 Proof:
% 116.77/15.24 join(composition(join(Z, one), X), Y)
% 116.77/15.24 = { by lemma 42 R->L }
% 116.77/15.24 join(join(X, composition(Z, X)), Y)
% 116.77/15.24 = { by axiom 12 (maddux2_join_associativity) R->L }
% 116.77/15.24 join(X, join(composition(Z, X), Y))
% 116.77/15.24 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.24 join(X, join(Y, composition(Z, X)))
% 116.77/15.24
% 116.77/15.24 Lemma 50: join(X, join(Y, composition(x1, X))) = join(X, Y).
% 116.77/15.24 Proof:
% 116.77/15.24 join(X, join(Y, composition(x1, X)))
% 116.77/15.24 = { by lemma 49 R->L }
% 116.77/15.24 join(composition(join(x1, one), X), Y)
% 116.77/15.24 = { by axiom 4 (goals) }
% 116.77/15.24 join(composition(one, X), Y)
% 116.77/15.24 = { by lemma 21 }
% 116.77/15.24 join(X, Y)
% 116.77/15.24
% 116.77/15.24 Lemma 51: join(X, composition(x1, join(X, Y))) = join(X, composition(x1, Y)).
% 116.77/15.24 Proof:
% 116.77/15.24 join(X, composition(x1, join(X, Y)))
% 116.77/15.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.24 join(X, composition(x1, join(Y, X)))
% 116.77/15.24 = { by axiom 7 (converse_idempotence) R->L }
% 116.77/15.24 join(X, composition(x1, join(Y, converse(converse(X)))))
% 116.77/15.24 = { by axiom 7 (converse_idempotence) R->L }
% 116.77/15.24 join(X, composition(converse(converse(x1)), join(Y, converse(converse(X)))))
% 116.77/15.24 = { by lemma 45 R->L }
% 116.77/15.24 join(X, composition(converse(converse(x1)), converse(join(converse(X), converse(Y)))))
% 116.77/15.24 = { by axiom 9 (converse_multiplicativity) R->L }
% 116.77/15.24 join(X, converse(composition(join(converse(X), converse(Y)), converse(x1))))
% 116.77/15.24 = { by axiom 13 (composition_distributivity) }
% 116.77/15.24 join(X, converse(join(composition(converse(X), converse(x1)), composition(converse(Y), converse(x1)))))
% 116.77/15.24 = { by axiom 9 (converse_multiplicativity) R->L }
% 116.77/15.24 join(X, converse(join(converse(composition(x1, X)), composition(converse(Y), converse(x1)))))
% 116.77/15.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.24 join(X, converse(join(composition(converse(Y), converse(x1)), converse(composition(x1, X)))))
% 116.77/15.24 = { by axiom 11 (converse_additivity) }
% 116.77/15.24 join(X, join(converse(composition(converse(Y), converse(x1))), converse(converse(composition(x1, X)))))
% 116.77/15.24 = { by lemma 17 }
% 116.77/15.24 join(X, join(composition(converse(converse(x1)), Y), converse(converse(composition(x1, X)))))
% 116.77/15.24 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.24 join(X, join(converse(converse(composition(x1, X))), composition(converse(converse(x1)), Y)))
% 116.77/15.24 = { by axiom 7 (converse_idempotence) }
% 116.77/15.24 join(X, join(composition(x1, X), composition(converse(converse(x1)), Y)))
% 116.77/15.24 = { by axiom 7 (converse_idempotence) }
% 116.77/15.24 join(X, join(composition(x1, X), composition(x1, Y)))
% 116.77/15.24 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.24 join(X, join(composition(x1, Y), composition(x1, X)))
% 116.77/15.24 = { by lemma 50 }
% 116.77/15.24 join(X, composition(x1, Y))
% 116.77/15.24
% 116.77/15.24 Lemma 52: join(complement(X), join(Y, complement(Z))) = join(Y, complement(meet(Z, X))).
% 116.77/15.24 Proof:
% 116.77/15.24 join(complement(X), join(Y, complement(Z)))
% 116.77/15.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.24 join(complement(X), join(complement(Z), Y))
% 116.77/15.24 = { by axiom 12 (maddux2_join_associativity) }
% 116.77/15.24 join(join(complement(X), complement(Z)), Y)
% 116.77/15.24 = { by lemma 38 R->L }
% 116.77/15.24 join(complement(meet(Z, X)), Y)
% 116.77/15.24 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.24 join(Y, complement(meet(Z, X)))
% 116.77/15.24
% 116.77/15.24 Lemma 53: join(composition(X, Y), composition(Z, Y)) = composition(join(Z, X), Y).
% 116.77/15.24 Proof:
% 116.77/15.24 join(composition(X, Y), composition(Z, Y))
% 116.77/15.24 = { by axiom 13 (composition_distributivity) R->L }
% 116.77/15.24 composition(join(X, Z), Y)
% 116.77/15.24 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.24 composition(join(Z, X), Y)
% 116.77/15.25
% 116.77/15.25 Lemma 54: join(complement(meet(X, Y)), complement(X)) = complement(meet(X, Y)).
% 116.77/15.25 Proof:
% 116.77/15.25 join(complement(meet(X, Y)), complement(X))
% 116.77/15.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.25 join(complement(X), complement(meet(X, Y)))
% 116.77/15.25 = { by lemma 38 }
% 116.77/15.25 join(complement(X), join(complement(Y), complement(X)))
% 116.77/15.25 = { by lemma 39 }
% 116.77/15.25 join(complement(Y), complement(X))
% 116.77/15.25 = { by lemma 38 R->L }
% 116.77/15.25 complement(meet(X, Y))
% 116.77/15.25
% 116.77/15.25 Lemma 55: join(meet(X, Y), meet(X, complement(Y))) = X.
% 116.77/15.25 Proof:
% 116.77/15.25 join(meet(X, Y), meet(X, complement(Y)))
% 116.77/15.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.25 join(meet(X, complement(Y)), meet(X, Y))
% 116.77/15.25 = { by axiom 8 (maddux4_definiton_of_meet) }
% 116.77/15.25 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 116.77/15.25 = { by lemma 25 }
% 116.77/15.25 X
% 116.77/15.25
% 116.77/15.25 Lemma 56: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 116.77/15.25 Proof:
% 116.77/15.25 join(meet(X, Y), meet(Y, complement(X)))
% 116.77/15.25 = { by lemma 27 }
% 116.77/15.25 join(meet(Y, X), meet(Y, complement(X)))
% 116.77/15.25 = { by lemma 55 }
% 116.77/15.25 Y
% 116.77/15.25
% 116.77/15.25 Lemma 57: join(join(meet(X, composition(Y, Z)), composition(meet(W, Y), Z)), composition(Y, Z)) = composition(Y, Z).
% 116.77/15.25 Proof:
% 116.77/15.25 join(join(meet(X, composition(Y, Z)), composition(meet(W, Y), Z)), composition(Y, Z))
% 116.77/15.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.25 join(composition(Y, Z), join(meet(X, composition(Y, Z)), composition(meet(W, Y), Z)))
% 116.77/15.25 = { by lemma 43 }
% 116.77/15.25 join(meet(X, composition(Y, Z)), join(composition(meet(W, Y), Z), composition(Y, Z)))
% 116.77/15.25 = { by lemma 53 }
% 116.77/15.25 join(meet(X, composition(Y, Z)), composition(join(Y, meet(W, Y)), Z))
% 116.77/15.25 = { by lemma 41 }
% 116.77/15.25 join(meet(X, composition(Y, Z)), composition(Y, Z))
% 116.77/15.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.25 join(composition(Y, Z), meet(X, composition(Y, Z)))
% 116.77/15.25 = { by lemma 27 R->L }
% 116.77/15.25 join(composition(Y, Z), meet(composition(Y, Z), X))
% 116.77/15.25 = { by lemma 40 }
% 116.77/15.25 composition(Y, Z)
% 116.77/15.25
% 116.77/15.25 Lemma 58: complement(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y))) = complement(meet(composition(X, Y), composition(Z, Y))).
% 116.77/15.25 Proof:
% 116.77/15.25 complement(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)))
% 116.77/15.25 = { by lemma 44 R->L }
% 116.77/15.25 complement(meet(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)), join(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)), composition(Z, Y))))
% 116.77/15.25 = { by lemma 57 }
% 116.77/15.25 complement(meet(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)), composition(Z, Y)))
% 116.77/15.25 = { by lemma 37 }
% 116.77/15.25 join(complement(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y))), complement(composition(Z, Y)))
% 116.77/15.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.25 join(complement(composition(Z, Y)), complement(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y))))
% 116.77/15.25 = { by lemma 44 R->L }
% 116.77/15.25 join(complement(composition(Z, Y)), complement(meet(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)), join(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)), composition(X, Y)))))
% 116.77/15.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.25 join(complement(composition(Z, Y)), complement(meet(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)), join(composition(X, Y), join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y))))))
% 116.77/15.25 = { by lemma 43 }
% 116.77/15.25 join(complement(composition(Z, Y)), complement(meet(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)), join(meet(composition(X, Y), composition(Z, Y)), join(composition(meet(X, Z), Y), composition(X, Y))))))
% 116.77/15.25 = { by lemma 53 }
% 116.77/15.25 join(complement(composition(Z, Y)), complement(meet(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)), join(meet(composition(X, Y), composition(Z, Y)), composition(join(X, meet(X, Z)), Y)))))
% 116.77/15.25 = { by lemma 40 }
% 116.77/15.25 join(complement(composition(Z, Y)), complement(meet(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)), join(meet(composition(X, Y), composition(Z, Y)), composition(X, Y)))))
% 116.77/15.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.25 join(complement(composition(Z, Y)), complement(meet(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)), join(composition(X, Y), meet(composition(X, Y), composition(Z, Y))))))
% 116.77/15.25 = { by lemma 40 }
% 116.77/15.25 join(complement(composition(Z, Y)), complement(meet(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)), composition(X, Y))))
% 116.77/15.25 = { by lemma 37 }
% 116.77/15.25 join(complement(composition(Z, Y)), join(complement(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y))), complement(composition(X, Y))))
% 116.77/15.25 = { by lemma 52 }
% 116.77/15.25 join(complement(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y))), complement(meet(composition(X, Y), composition(Z, Y))))
% 116.77/15.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), complement(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y))))
% 116.77/15.25 = { by lemma 27 }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), complement(join(meet(composition(Z, Y), composition(X, Y)), composition(meet(X, Z), Y))))
% 116.77/15.25 = { by axiom 8 (maddux4_definiton_of_meet) }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), complement(join(complement(join(complement(composition(Z, Y)), complement(composition(X, Y)))), composition(meet(X, Z), Y))))
% 116.77/15.25 = { by lemma 22 R->L }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), complement(join(join(complement(join(complement(composition(Z, Y)), complement(composition(X, Y)))), complement(join(complement(composition(Z, Y)), complement(composition(X, Y))))), composition(meet(X, Z), Y))))
% 116.77/15.25 = { by axiom 8 (maddux4_definiton_of_meet) R->L }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), complement(join(join(meet(composition(Z, Y), composition(X, Y)), complement(join(complement(composition(Z, Y)), complement(composition(X, Y))))), composition(meet(X, Z), Y))))
% 116.77/15.25 = { by axiom 8 (maddux4_definiton_of_meet) R->L }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), complement(join(join(meet(composition(Z, Y), composition(X, Y)), meet(composition(Z, Y), composition(X, Y))), composition(meet(X, Z), Y))))
% 116.77/15.25 = { by lemma 27 R->L }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), complement(join(join(meet(composition(Z, Y), composition(X, Y)), meet(composition(X, Y), composition(Z, Y))), composition(meet(X, Z), Y))))
% 116.77/15.25 = { by lemma 27 R->L }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), complement(join(join(meet(composition(X, Y), composition(Z, Y)), meet(composition(X, Y), composition(Z, Y))), composition(meet(X, Z), Y))))
% 116.77/15.25 = { by axiom 12 (maddux2_join_associativity) R->L }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), complement(join(meet(composition(X, Y), composition(Z, Y)), join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)))))
% 116.77/15.25 = { by lemma 33 R->L }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), complement(join(meet(composition(X, Y), composition(Z, Y)), meet(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)), top))))
% 116.77/15.25 = { by lemma 28 R->L }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), complement(join(meet(composition(X, Y), composition(Z, Y)), complement(join(zero, complement(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y))))))))
% 116.77/15.25 = { by lemma 31 R->L }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), complement(join(complement(complement(meet(composition(X, Y), composition(Z, Y)))), complement(join(zero, complement(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y))))))))
% 116.77/15.25 = { by lemma 54 R->L }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), complement(join(complement(join(complement(meet(composition(X, Y), composition(Z, Y))), complement(composition(X, Y)))), complement(join(zero, complement(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y))))))))
% 116.77/15.25 = { by axiom 8 (maddux4_definiton_of_meet) R->L }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), complement(join(meet(meet(composition(X, Y), composition(Z, Y)), composition(X, Y)), complement(join(zero, complement(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y))))))))
% 116.77/15.25 = { by lemma 36 }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), meet(join(zero, complement(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)))), join(complement(meet(composition(X, Y), composition(Z, Y))), complement(composition(X, Y)))))
% 116.77/15.25 = { by lemma 54 }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), meet(join(zero, complement(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)))), complement(meet(composition(X, Y), composition(Z, Y)))))
% 116.77/15.25 = { by lemma 30 }
% 116.77/15.25 join(complement(meet(composition(X, Y), composition(Z, Y))), meet(complement(join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y))), complement(meet(composition(X, Y), composition(Z, Y)))))
% 116.77/15.25 = { by lemma 41 }
% 116.77/15.25 complement(meet(composition(X, Y), composition(Z, Y)))
% 116.77/15.25
% 116.77/15.25 Lemma 59: join(W, complement(join(meet(composition(Z, Y), composition(X, Y)), composition(meet(Z, X), Y)))) = join(complement(composition(X, Y)), join(complement(composition(Z, Y)), W)).
% 116.77/15.25 Proof:
% 116.77/15.25 join(W, complement(join(meet(composition(Z, Y), composition(X, Y)), composition(meet(Z, X), Y))))
% 116.77/15.25 = { by lemma 58 }
% 116.77/15.25 join(W, complement(meet(composition(Z, Y), composition(X, Y))))
% 116.77/15.25 = { by lemma 52 R->L }
% 116.77/15.25 join(complement(composition(X, Y)), join(W, complement(composition(Z, Y))))
% 116.77/15.25 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.26 join(complement(composition(X, Y)), join(complement(composition(Z, Y)), W))
% 116.77/15.26
% 116.77/15.26 Goal 1 (goals_2): tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2)))) = tuple(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2))).
% 116.77/15.26 Proof:
% 116.77/15.26 tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2))))
% 116.77/15.26 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.26 tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.26 = { by lemma 55 R->L }
% 116.77/15.26 tuple(join(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.26 = { by lemma 27 }
% 116.77/15.26 tuple(join(meet(complement(composition(meet(x0, x1), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.26 = { by lemma 31 R->L }
% 116.77/15.26 tuple(join(meet(complement(composition(meet(x0, x1), x2)), complement(complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.26 = { by lemma 35 R->L }
% 116.77/15.26 tuple(join(complement(join(composition(meet(x0, x1), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.26 = { by lemma 55 R->L }
% 116.77/15.26 tuple(join(complement(join(composition(join(meet(meet(x0, x1), complement(composition(x1, x0))), meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.26 = { by lemma 31 R->L }
% 116.77/15.26 tuple(join(complement(join(composition(join(meet(complement(complement(meet(x0, x1))), complement(composition(x1, x0))), meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.26 = { by lemma 35 R->L }
% 116.77/15.26 tuple(join(complement(join(composition(join(complement(join(complement(meet(x0, x1)), composition(x1, x0))), meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.26 = { by lemma 48 R->L }
% 116.77/15.26 tuple(join(complement(join(composition(join(complement(join(complement(meet(x0, x1)), join(composition(x1, x0), composition(complement(meet(x0, x1)), x0)))), meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by axiom 13 (composition_distributivity) R->L }
% 116.77/15.27 tuple(join(complement(join(composition(join(complement(join(complement(meet(x0, x1)), composition(join(x1, complement(meet(x0, x1))), x0))), meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by lemma 41 R->L }
% 116.77/15.27 tuple(join(complement(join(composition(join(complement(join(complement(meet(x0, x1)), composition(join(join(x1, meet(x0, x1)), complement(meet(x0, x1))), x0))), meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by axiom 12 (maddux2_join_associativity) R->L }
% 116.77/15.27 tuple(join(complement(join(composition(join(complement(join(complement(meet(x0, x1)), composition(join(x1, join(meet(x0, x1), complement(meet(x0, x1)))), x0))), meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by axiom 3 (def_top) R->L }
% 116.77/15.27 tuple(join(complement(join(composition(join(complement(join(complement(meet(x0, x1)), composition(join(x1, top), x0))), meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by lemma 24 }
% 116.77/15.27 tuple(join(complement(join(composition(join(complement(join(complement(meet(x0, x1)), composition(top, x0))), meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.27 tuple(join(complement(join(composition(join(complement(join(composition(top, x0), complement(meet(x0, x1)))), meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by lemma 52 R->L }
% 116.77/15.27 tuple(join(complement(join(composition(join(complement(join(complement(x1), join(composition(top, x0), complement(x0)))), meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by lemma 24 R->L }
% 116.77/15.27 tuple(join(complement(join(composition(join(complement(join(complement(x1), join(composition(join(one, top), x0), complement(x0)))), meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.27 tuple(join(complement(join(composition(join(complement(join(complement(x1), join(composition(join(top, one), x0), complement(x0)))), meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by lemma 49 }
% 116.77/15.27 tuple(join(complement(join(composition(join(complement(join(complement(x1), join(x0, join(complement(x0), composition(top, x0))))), meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by lemma 47 }
% 116.77/15.27 tuple(join(complement(join(composition(join(complement(join(complement(x1), top)), meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by lemma 24 }
% 116.77/15.27 tuple(join(complement(join(composition(join(complement(top), meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by lemma 16 }
% 116.77/15.27 tuple(join(complement(join(composition(join(zero, meet(meet(x0, x1), complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by lemma 32 }
% 116.77/15.27 tuple(join(complement(join(composition(meet(meet(x0, x1), complement(complement(composition(x1, x0)))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by lemma 34 R->L }
% 116.77/15.27 tuple(join(complement(join(composition(complement(join(complement(composition(x1, x0)), complement(meet(x0, x1)))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by lemma 52 R->L }
% 116.77/15.27 tuple(join(complement(join(composition(complement(join(complement(x1), join(complement(composition(x1, x0)), complement(x0)))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by lemma 35 }
% 116.77/15.27 tuple(join(complement(join(composition(meet(complement(complement(x1)), complement(join(complement(composition(x1, x0)), complement(x0)))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.27 = { by lemma 31 }
% 116.77/15.27 tuple(join(complement(join(composition(meet(x1, complement(join(complement(composition(x1, x0)), complement(x0)))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by lemma 34 }
% 116.77/15.28 tuple(join(complement(join(composition(meet(x1, meet(x0, complement(complement(composition(x1, x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by lemma 31 }
% 116.77/15.28 tuple(join(complement(join(composition(meet(x1, meet(x0, composition(x1, x0))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by lemma 27 }
% 116.77/15.28 tuple(join(complement(join(composition(meet(x1, meet(composition(x1, x0), x0)), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by lemma 31 R->L }
% 116.77/15.28 tuple(join(complement(join(composition(meet(x1, meet(composition(x1, x0), complement(complement(x0)))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by lemma 32 R->L }
% 116.77/15.28 tuple(join(complement(join(composition(meet(x1, join(zero, meet(composition(x1, x0), complement(complement(x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by lemma 16 R->L }
% 116.77/15.28 tuple(join(complement(join(composition(meet(x1, join(complement(top), meet(composition(x1, x0), complement(complement(x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by lemma 24 R->L }
% 116.77/15.28 tuple(join(complement(join(composition(meet(x1, join(complement(join(x0, top)), meet(composition(x1, x0), complement(complement(x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by axiom 3 (def_top) }
% 116.77/15.28 tuple(join(complement(join(composition(meet(x1, join(complement(join(x0, join(composition(x1, x0), complement(composition(x1, x0))))), meet(composition(x1, x0), complement(complement(x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.28 tuple(join(complement(join(composition(meet(x1, join(complement(join(x0, join(complement(composition(x1, x0)), composition(x1, x0)))), meet(composition(x1, x0), complement(complement(x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by lemma 50 }
% 116.77/15.28 tuple(join(complement(join(composition(meet(x1, join(complement(join(x0, complement(composition(x1, x0)))), meet(composition(x1, x0), complement(complement(x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by lemma 35 }
% 116.77/15.28 tuple(join(complement(join(composition(meet(x1, join(meet(complement(x0), complement(complement(composition(x1, x0)))), meet(composition(x1, x0), complement(complement(x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by lemma 31 }
% 116.77/15.28 tuple(join(complement(join(composition(meet(x1, join(meet(complement(x0), composition(x1, x0)), meet(composition(x1, x0), complement(complement(x0))))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by lemma 56 }
% 116.77/15.28 tuple(join(complement(join(composition(meet(x1, composition(x1, x0)), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by lemma 27 }
% 116.77/15.28 tuple(join(complement(join(composition(meet(composition(x1, x0), x1), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by lemma 31 R->L }
% 116.77/15.28 tuple(join(complement(join(composition(meet(composition(x1, x0), complement(complement(x1))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by lemma 32 R->L }
% 116.77/15.28 tuple(join(complement(join(composition(join(zero, meet(composition(x1, x0), complement(complement(x1)))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.28 = { by lemma 16 R->L }
% 116.77/15.28 tuple(join(complement(join(composition(join(complement(top), meet(composition(x1, x0), complement(complement(x1)))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by lemma 24 R->L }
% 116.77/15.29 tuple(join(complement(join(composition(join(complement(join(x1, top)), meet(composition(x1, x0), complement(complement(x1)))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by axiom 3 (def_top) }
% 116.77/15.29 tuple(join(complement(join(composition(join(complement(join(x1, join(composition(x1, x0), complement(composition(x1, x0))))), meet(composition(x1, x0), complement(complement(x1)))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.29 tuple(join(complement(join(composition(join(complement(join(x1, join(complement(composition(x1, x0)), composition(x1, x0)))), meet(composition(x1, x0), complement(complement(x1)))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by lemma 48 }
% 116.77/15.29 tuple(join(complement(join(composition(join(complement(join(x1, complement(composition(x1, x0)))), meet(composition(x1, x0), complement(complement(x1)))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by lemma 35 }
% 116.77/15.29 tuple(join(complement(join(composition(join(meet(complement(x1), complement(complement(composition(x1, x0)))), meet(composition(x1, x0), complement(complement(x1)))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by lemma 31 }
% 116.77/15.29 tuple(join(complement(join(composition(join(meet(complement(x1), composition(x1, x0)), meet(composition(x1, x0), complement(complement(x1)))), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by lemma 56 }
% 116.77/15.29 tuple(join(complement(join(composition(composition(x1, x0), x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by axiom 10 (composition_associativity) R->L }
% 116.77/15.29 tuple(join(complement(join(composition(x1, composition(x0, x2)), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by lemma 31 R->L }
% 116.77/15.29 tuple(join(complement(join(composition(x1, complement(complement(composition(x0, x2)))), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by lemma 59 }
% 116.77/15.29 tuple(join(complement(join(complement(composition(x1, x2)), join(complement(composition(x0, x2)), composition(x1, complement(complement(composition(x0, x2))))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by lemma 51 R->L }
% 116.77/15.29 tuple(join(complement(join(complement(composition(x1, x2)), join(complement(composition(x0, x2)), composition(x1, join(complement(composition(x0, x2)), complement(complement(composition(x0, x2)))))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by axiom 3 (def_top) R->L }
% 116.77/15.29 tuple(join(complement(join(complement(composition(x1, x2)), join(complement(composition(x0, x2)), composition(x1, top)))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by lemma 59 R->L }
% 116.77/15.29 tuple(join(complement(join(composition(x1, top), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by lemma 24 R->L }
% 116.77/15.29 tuple(join(complement(join(composition(x1, join(x2, top)), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by axiom 3 (def_top) }
% 116.77/15.29 tuple(join(complement(join(composition(x1, join(x2, join(composition(x0, x2), complement(composition(x0, x2))))), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by axiom 12 (maddux2_join_associativity) }
% 116.77/15.29 tuple(join(complement(join(composition(x1, join(join(x2, composition(x0, x2)), complement(composition(x0, x2)))), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.29 tuple(join(complement(join(composition(x1, join(join(composition(x0, x2), x2), complement(composition(x0, x2)))), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.29 = { by lemma 18 R->L }
% 116.77/15.29 tuple(join(complement(join(composition(x1, join(join(composition(x0, x2), composition(converse(one), x2)), complement(composition(x0, x2)))), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.30 = { by lemma 53 }
% 116.77/15.30 tuple(join(complement(join(composition(x1, join(composition(join(converse(one), x0), x2), complement(composition(x0, x2)))), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.30 = { by lemma 19 }
% 116.77/15.30 tuple(join(complement(join(composition(x1, join(composition(join(one, x0), x2), complement(composition(x0, x2)))), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.30 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.30 tuple(join(complement(join(composition(x1, join(composition(join(x0, one), x2), complement(composition(x0, x2)))), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.30 = { by axiom 5 (goals_1) }
% 116.77/15.30 tuple(join(complement(join(composition(x1, join(composition(one, x2), complement(composition(x0, x2)))), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.30 = { by lemma 21 }
% 116.77/15.30 tuple(join(complement(join(composition(x1, join(x2, complement(composition(x0, x2)))), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.30 = { by lemma 59 }
% 116.77/15.30 tuple(join(complement(join(complement(composition(x1, x2)), join(complement(composition(x0, x2)), composition(x1, join(x2, complement(composition(x0, x2))))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.30 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 116.77/15.30 tuple(join(complement(join(complement(composition(x1, x2)), join(complement(composition(x0, x2)), composition(x1, join(complement(composition(x0, x2)), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.30 = { by lemma 51 }
% 116.77/15.30 tuple(join(complement(join(complement(composition(x1, x2)), join(complement(composition(x0, x2)), composition(x1, x2)))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.30 = { by lemma 59 R->L }
% 116.77/15.30 tuple(join(complement(join(composition(x1, x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.30 = { by lemma 57 R->L }
% 116.77/15.30 tuple(join(complement(join(join(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(x1, x2)), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.30 = { by axiom 12 (maddux2_join_associativity) R->L }
% 116.77/15.30 tuple(join(complement(join(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(x1, x2), complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 116.77/15.30 = { by axiom 2 (maddux1_join_commutativity) }
% 116.77/15.30 tuple(join(complement(join(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))), composition(x1, x2)))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 117.56/15.30 = { by lemma 47 }
% 117.56/15.30 tuple(join(complement(top), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 117.56/15.30 = { by lemma 16 }
% 117.56/15.30 tuple(join(zero, meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 117.56/15.30 = { by lemma 32 }
% 117.56/15.30 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 117.56/15.30 = { by lemma 31 }
% 117.56/15.30 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(meet(x0, x1), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 117.56/15.30 = { by lemma 27 }
% 117.56/15.30 tuple(meet(composition(meet(x0, x1), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 117.56/15.30 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 117.56/15.30 tuple(meet(composition(meet(x0, x1), x2), join(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 117.56/15.30 = { by lemma 44 }
% 117.56/15.30 tuple(composition(meet(x0, x1), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 117.56/15.30 = { by lemma 31 R->L }
% 117.56/15.30 tuple(composition(meet(x0, x1), x2), complement(complement(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))))
% 117.56/15.30 = { by lemma 58 }
% 117.56/15.30 tuple(composition(meet(x0, x1), x2), complement(complement(meet(composition(x0, x2), composition(x1, x2)))))
% 117.56/15.30 = { by lemma 31 }
% 117.56/15.30 tuple(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2)))
% 117.56/15.30 % SZS output end Proof
% 117.56/15.30
% 117.56/15.30 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------