TSTP Solution File: REL010+2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL010+2 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n018.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 13:43:51 EDT 2023
% Result : Theorem 6.45s 1.23s
% Output : Proof 6.45s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL010+2 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n018.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 22:13:44 EDT 2023
% 0.13/0.34 % CPUTime :
% 6.45/1.23 Command-line arguments: --flatten
% 6.45/1.23
% 6.45/1.23 % SZS status Theorem
% 6.45/1.23
% 6.45/1.26 % SZS output start Proof
% 6.45/1.26 Axiom 1 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 6.45/1.26 Axiom 2 (converse_idempotence): converse(converse(X)) = X.
% 6.45/1.26 Axiom 3 (composition_identity): composition(X, one) = X.
% 6.45/1.26 Axiom 4 (def_top): top = join(X, complement(X)).
% 6.45/1.26 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 6.45/1.26 Axiom 6 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 6.45/1.26 Axiom 7 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 6.45/1.26 Axiom 8 (goals): meet(composition(x0, x1), x2) = zero.
% 6.45/1.26 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 6.45/1.26 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 6.45/1.26 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 6.45/1.26 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 6.45/1.26 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 6.45/1.26 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 6.45/1.26
% 6.45/1.26 Lemma 15: join(X, join(Y, complement(X))) = join(Y, top).
% 6.45/1.26 Proof:
% 6.45/1.26 join(X, join(Y, complement(X)))
% 6.45/1.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.26 join(X, join(complement(X), Y))
% 6.45/1.26 = { by axiom 7 (maddux2_join_associativity) }
% 6.45/1.26 join(join(X, complement(X)), Y)
% 6.45/1.26 = { by axiom 4 (def_top) R->L }
% 6.45/1.26 join(top, Y)
% 6.45/1.26 = { by axiom 1 (maddux1_join_commutativity) }
% 6.45/1.26 join(Y, top)
% 6.45/1.26
% 6.45/1.26 Lemma 16: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 6.45/1.26 Proof:
% 6.45/1.26 converse(composition(converse(X), Y))
% 6.45/1.26 = { by axiom 9 (converse_multiplicativity) }
% 6.45/1.26 composition(converse(Y), converse(converse(X)))
% 6.45/1.26 = { by axiom 2 (converse_idempotence) }
% 6.45/1.26 composition(converse(Y), X)
% 6.45/1.26
% 6.45/1.26 Lemma 17: composition(converse(one), X) = X.
% 6.45/1.26 Proof:
% 6.45/1.27 composition(converse(one), X)
% 6.45/1.27 = { by lemma 16 R->L }
% 6.45/1.27 converse(composition(converse(X), one))
% 6.45/1.27 = { by axiom 3 (composition_identity) }
% 6.45/1.27 converse(converse(X))
% 6.45/1.27 = { by axiom 2 (converse_idempotence) }
% 6.45/1.27 X
% 6.45/1.27
% 6.45/1.27 Lemma 18: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 6.45/1.27 Proof:
% 6.45/1.27 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.27 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 6.45/1.27 = { by axiom 13 (converse_cancellativity) }
% 6.45/1.27 complement(X)
% 6.45/1.27
% 6.45/1.27 Lemma 19: join(complement(X), complement(X)) = complement(X).
% 6.45/1.27 Proof:
% 6.45/1.27 join(complement(X), complement(X))
% 6.45/1.27 = { by lemma 17 R->L }
% 6.45/1.27 join(complement(X), composition(converse(one), complement(X)))
% 6.45/1.27 = { by lemma 17 R->L }
% 6.45/1.27 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 6.45/1.27 = { by axiom 3 (composition_identity) R->L }
% 6.45/1.27 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 6.45/1.27 = { by axiom 10 (composition_associativity) R->L }
% 6.45/1.27 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 6.45/1.27 = { by lemma 17 }
% 6.45/1.27 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 6.45/1.27 = { by lemma 18 }
% 6.45/1.27 complement(X)
% 6.45/1.27
% 6.45/1.27 Lemma 20: join(top, complement(X)) = top.
% 6.45/1.27 Proof:
% 6.45/1.27 join(top, complement(X))
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.27 join(complement(X), top)
% 6.45/1.27 = { by lemma 15 R->L }
% 6.45/1.27 join(X, join(complement(X), complement(X)))
% 6.45/1.27 = { by lemma 19 }
% 6.45/1.27 join(X, complement(X))
% 6.45/1.27 = { by axiom 4 (def_top) R->L }
% 6.45/1.27 top
% 6.45/1.27
% 6.45/1.27 Lemma 21: meet(composition(x0, x1), x2) = complement(top).
% 6.45/1.27 Proof:
% 6.45/1.27 meet(composition(x0, x1), x2)
% 6.45/1.27 = { by axiom 8 (goals) }
% 6.45/1.27 zero
% 6.45/1.27 = { by axiom 5 (def_zero) }
% 6.45/1.27 meet(X, complement(X))
% 6.45/1.27 = { by axiom 11 (maddux4_definiton_of_meet) }
% 6.45/1.27 complement(join(complement(X), complement(complement(X))))
% 6.45/1.27 = { by axiom 4 (def_top) R->L }
% 6.45/1.27 complement(top)
% 6.45/1.27
% 6.45/1.27 Lemma 22: join(X, top) = top.
% 6.45/1.27 Proof:
% 6.45/1.27 join(X, top)
% 6.45/1.27 = { by lemma 20 R->L }
% 6.45/1.27 join(X, join(top, complement(X)))
% 6.45/1.27 = { by lemma 15 }
% 6.45/1.27 join(top, top)
% 6.45/1.27 = { by lemma 15 R->L }
% 6.45/1.27 join(join(meet(composition(x0, x1), x2), meet(composition(x0, x1), x2)), join(top, complement(join(meet(composition(x0, x1), x2), meet(composition(x0, x1), x2)))))
% 6.45/1.27 = { by lemma 20 }
% 6.45/1.27 join(join(meet(composition(x0, x1), x2), meet(composition(x0, x1), x2)), top)
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.27 join(top, join(meet(composition(x0, x1), x2), meet(composition(x0, x1), x2)))
% 6.45/1.27 = { by lemma 21 }
% 6.45/1.27 join(top, join(meet(composition(x0, x1), x2), complement(top)))
% 6.45/1.27 = { by lemma 21 }
% 6.45/1.27 join(top, join(complement(top), complement(top)))
% 6.45/1.27 = { by lemma 19 }
% 6.45/1.27 join(top, complement(top))
% 6.45/1.27 = { by axiom 4 (def_top) R->L }
% 6.45/1.27 top
% 6.45/1.27
% 6.45/1.27 Lemma 23: meet(Y, X) = meet(X, Y).
% 6.45/1.27 Proof:
% 6.45/1.27 meet(Y, X)
% 6.45/1.27 = { by axiom 11 (maddux4_definiton_of_meet) }
% 6.45/1.27 complement(join(complement(Y), complement(X)))
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.27 complement(join(complement(X), complement(Y)))
% 6.45/1.27 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 6.45/1.27 meet(X, Y)
% 6.45/1.27
% 6.45/1.27 Lemma 24: complement(join(meet(composition(x0, x1), x2), complement(X))) = meet(X, top).
% 6.45/1.27 Proof:
% 6.45/1.27 complement(join(meet(composition(x0, x1), x2), complement(X)))
% 6.45/1.27 = { by lemma 21 }
% 6.45/1.27 complement(join(complement(top), complement(X)))
% 6.45/1.27 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 6.45/1.27 meet(top, X)
% 6.45/1.27 = { by lemma 23 R->L }
% 6.45/1.27 meet(X, top)
% 6.45/1.27
% 6.45/1.27 Lemma 25: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 6.45/1.27 Proof:
% 6.45/1.27 join(meet(X, Y), complement(join(complement(X), Y)))
% 6.45/1.27 = { by axiom 11 (maddux4_definiton_of_meet) }
% 6.45/1.27 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 6.45/1.27 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 6.45/1.27 X
% 6.45/1.27
% 6.45/1.27 Lemma 26: join(meet(composition(x0, x1), x2), complement(complement(X))) = X.
% 6.45/1.27 Proof:
% 6.45/1.27 join(meet(composition(x0, x1), x2), complement(complement(X)))
% 6.45/1.27 = { by axiom 8 (goals) }
% 6.45/1.27 join(zero, complement(complement(X)))
% 6.45/1.27 = { by axiom 5 (def_zero) }
% 6.45/1.27 join(meet(X, complement(X)), complement(complement(X)))
% 6.45/1.27 = { by lemma 19 R->L }
% 6.45/1.27 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 6.45/1.27 = { by lemma 25 }
% 6.45/1.27 X
% 6.45/1.27
% 6.45/1.27 Lemma 27: join(meet(composition(x0, x1), x2), meet(X, X)) = X.
% 6.45/1.27 Proof:
% 6.45/1.27 join(meet(composition(x0, x1), x2), meet(X, X))
% 6.45/1.27 = { by axiom 11 (maddux4_definiton_of_meet) }
% 6.45/1.27 join(meet(composition(x0, x1), x2), complement(join(complement(X), complement(X))))
% 6.45/1.27 = { by axiom 8 (goals) }
% 6.45/1.27 join(zero, complement(join(complement(X), complement(X))))
% 6.45/1.27 = { by axiom 5 (def_zero) }
% 6.45/1.27 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 6.45/1.27 = { by lemma 25 }
% 6.45/1.27 X
% 6.45/1.27
% 6.45/1.27 Lemma 28: join(meet(composition(x0, x1), x2), join(X, complement(complement(Y)))) = join(X, Y).
% 6.45/1.27 Proof:
% 6.45/1.27 join(meet(composition(x0, x1), x2), join(X, complement(complement(Y))))
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.27 join(meet(composition(x0, x1), x2), join(complement(complement(Y)), X))
% 6.45/1.27 = { by lemma 19 R->L }
% 6.45/1.27 join(meet(composition(x0, x1), x2), join(complement(join(complement(Y), complement(Y))), X))
% 6.45/1.27 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 6.45/1.27 join(meet(composition(x0, x1), x2), join(meet(Y, Y), X))
% 6.45/1.27 = { by axiom 7 (maddux2_join_associativity) }
% 6.45/1.27 join(join(meet(composition(x0, x1), x2), meet(Y, Y)), X)
% 6.45/1.27 = { by lemma 27 }
% 6.45/1.27 join(Y, X)
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) }
% 6.45/1.27 join(X, Y)
% 6.45/1.27
% 6.45/1.27 Lemma 29: join(meet(composition(x0, x1), x2), complement(X)) = complement(X).
% 6.45/1.27 Proof:
% 6.45/1.27 join(meet(composition(x0, x1), x2), complement(X))
% 6.45/1.27 = { by lemma 26 R->L }
% 6.45/1.27 join(meet(composition(x0, x1), x2), join(meet(composition(x0, x1), x2), complement(complement(complement(X)))))
% 6.45/1.27 = { by lemma 19 R->L }
% 6.45/1.27 join(meet(composition(x0, x1), x2), join(meet(composition(x0, x1), x2), join(complement(complement(complement(X))), complement(complement(complement(X))))))
% 6.45/1.27 = { by lemma 28 }
% 6.45/1.27 join(meet(composition(x0, x1), x2), join(complement(complement(complement(X))), complement(X)))
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) }
% 6.45/1.27 join(meet(composition(x0, x1), x2), join(complement(X), complement(complement(complement(X)))))
% 6.45/1.27 = { by lemma 28 }
% 6.45/1.27 join(complement(X), complement(X))
% 6.45/1.27 = { by lemma 19 }
% 6.45/1.27 complement(X)
% 6.45/1.27
% 6.45/1.27 Lemma 30: join(X, complement(meet(composition(x0, x1), x2))) = top.
% 6.45/1.27 Proof:
% 6.45/1.27 join(X, complement(meet(composition(x0, x1), x2)))
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.27 join(complement(meet(composition(x0, x1), x2)), X)
% 6.45/1.27 = { by lemma 28 R->L }
% 6.45/1.27 join(meet(composition(x0, x1), x2), join(complement(meet(composition(x0, x1), x2)), complement(complement(X))))
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.27 join(meet(composition(x0, x1), x2), join(complement(complement(X)), complement(meet(composition(x0, x1), x2))))
% 6.45/1.27 = { by lemma 15 }
% 6.45/1.27 join(complement(complement(X)), top)
% 6.45/1.27 = { by lemma 22 }
% 6.45/1.27 top
% 6.45/1.27
% 6.45/1.27 Lemma 31: meet(X, meet(composition(x0, x1), x2)) = meet(composition(x0, x1), x2).
% 6.45/1.27 Proof:
% 6.45/1.27 meet(X, meet(composition(x0, x1), x2))
% 6.45/1.27 = { by axiom 11 (maddux4_definiton_of_meet) }
% 6.45/1.27 complement(join(complement(X), complement(meet(composition(x0, x1), x2))))
% 6.45/1.27 = { by lemma 30 }
% 6.45/1.27 complement(top)
% 6.45/1.27 = { by lemma 21 R->L }
% 6.45/1.27 meet(composition(x0, x1), x2)
% 6.45/1.27
% 6.45/1.27 Lemma 32: meet(X, top) = X.
% 6.45/1.27 Proof:
% 6.45/1.27 meet(X, top)
% 6.45/1.27 = { by lemma 24 R->L }
% 6.45/1.27 complement(join(meet(composition(x0, x1), x2), complement(X)))
% 6.45/1.27 = { by lemma 29 R->L }
% 6.45/1.27 join(meet(composition(x0, x1), x2), complement(join(meet(composition(x0, x1), x2), complement(X))))
% 6.45/1.27 = { by lemma 24 }
% 6.45/1.27 join(meet(composition(x0, x1), x2), meet(X, top))
% 6.45/1.27 = { by lemma 30 R->L }
% 6.45/1.27 join(meet(composition(x0, x1), x2), meet(X, join(complement(meet(composition(x0, x1), x2)), complement(meet(composition(x0, x1), x2)))))
% 6.45/1.27 = { by lemma 19 }
% 6.45/1.27 join(meet(composition(x0, x1), x2), meet(X, complement(meet(composition(x0, x1), x2))))
% 6.45/1.27 = { by lemma 31 R->L }
% 6.45/1.27 join(meet(X, meet(composition(x0, x1), x2)), meet(X, complement(meet(composition(x0, x1), x2))))
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.27 join(meet(X, complement(meet(composition(x0, x1), x2))), meet(X, meet(composition(x0, x1), x2)))
% 6.45/1.27 = { by axiom 11 (maddux4_definiton_of_meet) }
% 6.45/1.27 join(meet(X, complement(meet(composition(x0, x1), x2))), complement(join(complement(X), complement(meet(composition(x0, x1), x2)))))
% 6.45/1.27 = { by lemma 25 }
% 6.45/1.27 X
% 6.45/1.27
% 6.45/1.27 Lemma 33: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 6.45/1.27 Proof:
% 6.45/1.27 meet(X, join(complement(Y), complement(Z)))
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.27 meet(X, join(complement(Z), complement(Y)))
% 6.45/1.27 = { by lemma 23 }
% 6.45/1.27 meet(join(complement(Z), complement(Y)), X)
% 6.45/1.27 = { by axiom 11 (maddux4_definiton_of_meet) }
% 6.45/1.27 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 6.45/1.27 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 6.45/1.27 complement(join(meet(Z, Y), complement(X)))
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) }
% 6.45/1.27 complement(join(complement(X), meet(Z, Y)))
% 6.45/1.27 = { by lemma 23 R->L }
% 6.45/1.27 complement(join(complement(X), meet(Y, Z)))
% 6.45/1.27
% 6.45/1.27 Lemma 34: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 6.45/1.27 Proof:
% 6.45/1.27 complement(join(X, complement(Y)))
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.27 complement(join(complement(Y), X))
% 6.45/1.27 = { by lemma 32 R->L }
% 6.45/1.27 complement(join(complement(Y), meet(X, top)))
% 6.45/1.27 = { by lemma 23 R->L }
% 6.45/1.27 complement(join(complement(Y), meet(top, X)))
% 6.45/1.27 = { by lemma 33 R->L }
% 6.45/1.27 meet(Y, join(complement(top), complement(X)))
% 6.45/1.27 = { by lemma 21 R->L }
% 6.45/1.27 meet(Y, join(meet(composition(x0, x1), x2), complement(X)))
% 6.45/1.27 = { by lemma 29 }
% 6.45/1.27 meet(Y, complement(X))
% 6.45/1.27
% 6.45/1.27 Lemma 35: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 6.45/1.27 Proof:
% 6.45/1.27 complement(join(complement(X), Y))
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.27 complement(join(Y, complement(X)))
% 6.45/1.27 = { by lemma 34 }
% 6.45/1.27 meet(X, complement(Y))
% 6.45/1.27
% 6.45/1.27 Lemma 36: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 6.45/1.27 Proof:
% 6.45/1.27 converse(join(X, converse(Y)))
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.27 converse(join(converse(Y), X))
% 6.45/1.27 = { by axiom 6 (converse_additivity) }
% 6.45/1.27 join(converse(converse(Y)), converse(X))
% 6.45/1.27 = { by axiom 2 (converse_idempotence) }
% 6.45/1.27 join(Y, converse(X))
% 6.45/1.27
% 6.45/1.27 Lemma 37: join(X, complement(meet(X, Y))) = top.
% 6.45/1.27 Proof:
% 6.45/1.27 join(X, complement(meet(X, Y)))
% 6.45/1.27 = { by lemma 23 }
% 6.45/1.27 join(X, complement(meet(Y, X)))
% 6.45/1.27 = { by axiom 11 (maddux4_definiton_of_meet) }
% 6.45/1.27 join(X, complement(complement(join(complement(Y), complement(X)))))
% 6.45/1.27 = { by lemma 19 R->L }
% 6.45/1.27 join(X, complement(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))))
% 6.45/1.27 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 6.45/1.27 join(X, complement(join(meet(Y, X), complement(join(complement(Y), complement(X))))))
% 6.45/1.27 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 6.45/1.27 join(X, complement(join(meet(Y, X), meet(Y, X))))
% 6.45/1.27 = { by lemma 23 R->L }
% 6.45/1.27 join(X, complement(join(meet(Y, X), meet(X, Y))))
% 6.45/1.27 = { by lemma 23 R->L }
% 6.45/1.27 join(X, complement(join(meet(X, Y), meet(X, Y))))
% 6.45/1.27 = { by axiom 11 (maddux4_definiton_of_meet) }
% 6.45/1.27 join(X, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y))))
% 6.45/1.27 = { by lemma 29 R->L }
% 6.45/1.27 join(X, join(meet(composition(x0, x1), x2), complement(join(complement(join(complement(X), complement(Y))), meet(X, Y)))))
% 6.45/1.27 = { by lemma 33 R->L }
% 6.45/1.27 join(X, join(meet(composition(x0, x1), x2), meet(join(complement(X), complement(Y)), join(complement(X), complement(Y)))))
% 6.45/1.27 = { by lemma 27 }
% 6.45/1.27 join(X, join(complement(X), complement(Y)))
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) }
% 6.45/1.27 join(X, join(complement(Y), complement(X)))
% 6.45/1.27 = { by lemma 15 }
% 6.45/1.27 join(complement(Y), top)
% 6.45/1.27 = { by lemma 22 }
% 6.45/1.27 top
% 6.45/1.27
% 6.45/1.27 Lemma 38: join(X, meet(composition(x0, x1), x2)) = X.
% 6.45/1.27 Proof:
% 6.45/1.27 join(X, meet(composition(x0, x1), x2))
% 6.45/1.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.27 join(meet(composition(x0, x1), x2), X)
% 6.45/1.27 = { by lemma 28 R->L }
% 6.45/1.27 join(meet(composition(x0, x1), x2), join(meet(composition(x0, x1), x2), complement(complement(X))))
% 6.45/1.27 = { by lemma 29 }
% 6.45/1.27 join(meet(composition(x0, x1), x2), complement(complement(X)))
% 6.45/1.27 = { by lemma 26 }
% 6.45/1.27 X
% 6.45/1.27
% 6.45/1.27 Goal 1 (goals_1): meet(x1, composition(converse(x0), x2)) = zero.
% 6.45/1.27 Proof:
% 6.45/1.27 meet(x1, composition(converse(x0), x2))
% 6.45/1.28 = { by lemma 25 R->L }
% 6.45/1.28 join(meet(meet(x1, composition(converse(x0), x2)), complement(join(composition(converse(x0), complement(composition(x0, x1))), composition(converse(x0), x2)))), complement(join(complement(meet(x1, composition(converse(x0), x2))), complement(join(composition(converse(x0), complement(composition(x0, x1))), composition(converse(x0), x2))))))
% 6.45/1.28 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.28 join(meet(meet(x1, composition(converse(x0), x2)), complement(join(composition(converse(x0), x2), composition(converse(x0), complement(composition(x0, x1)))))), complement(join(complement(meet(x1, composition(converse(x0), x2))), complement(join(composition(converse(x0), complement(composition(x0, x1))), composition(converse(x0), x2))))))
% 6.45/1.28 = { by lemma 34 R->L }
% 6.45/1.28 join(complement(join(join(composition(converse(x0), x2), composition(converse(x0), complement(composition(x0, x1)))), complement(meet(x1, composition(converse(x0), x2))))), complement(join(complement(meet(x1, composition(converse(x0), x2))), complement(join(composition(converse(x0), complement(composition(x0, x1))), composition(converse(x0), x2))))))
% 6.45/1.28 = { by axiom 7 (maddux2_join_associativity) R->L }
% 6.45/1.28 join(complement(join(composition(converse(x0), x2), join(composition(converse(x0), complement(composition(x0, x1))), complement(meet(x1, composition(converse(x0), x2)))))), complement(join(complement(meet(x1, composition(converse(x0), x2))), complement(join(composition(converse(x0), complement(composition(x0, x1))), composition(converse(x0), x2))))))
% 6.45/1.28 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.28 join(complement(join(join(composition(converse(x0), complement(composition(x0, x1))), complement(meet(x1, composition(converse(x0), x2)))), composition(converse(x0), x2))), complement(join(complement(meet(x1, composition(converse(x0), x2))), complement(join(composition(converse(x0), complement(composition(x0, x1))), composition(converse(x0), x2))))))
% 6.45/1.28 = { by axiom 7 (maddux2_join_associativity) R->L }
% 6.45/1.28 join(complement(join(composition(converse(x0), complement(composition(x0, x1))), join(complement(meet(x1, composition(converse(x0), x2))), composition(converse(x0), x2)))), complement(join(complement(meet(x1, composition(converse(x0), x2))), complement(join(composition(converse(x0), complement(composition(x0, x1))), composition(converse(x0), x2))))))
% 6.45/1.28 = { by axiom 1 (maddux1_join_commutativity) }
% 6.45/1.28 join(complement(join(composition(converse(x0), complement(composition(x0, x1))), join(composition(converse(x0), x2), complement(meet(x1, composition(converse(x0), x2)))))), complement(join(complement(meet(x1, composition(converse(x0), x2))), complement(join(composition(converse(x0), complement(composition(x0, x1))), composition(converse(x0), x2))))))
% 6.45/1.28 = { by lemma 23 R->L }
% 6.45/1.28 join(complement(join(composition(converse(x0), complement(composition(x0, x1))), join(composition(converse(x0), x2), complement(meet(composition(converse(x0), x2), x1))))), complement(join(complement(meet(x1, composition(converse(x0), x2))), complement(join(composition(converse(x0), complement(composition(x0, x1))), composition(converse(x0), x2))))))
% 6.45/1.28 = { by lemma 37 }
% 6.45/1.28 join(complement(join(composition(converse(x0), complement(composition(x0, x1))), top)), complement(join(complement(meet(x1, composition(converse(x0), x2))), complement(join(composition(converse(x0), complement(composition(x0, x1))), composition(converse(x0), x2))))))
% 6.45/1.28 = { by lemma 22 }
% 6.45/1.28 join(complement(top), complement(join(complement(meet(x1, composition(converse(x0), x2))), complement(join(composition(converse(x0), complement(composition(x0, x1))), composition(converse(x0), x2))))))
% 6.45/1.28 = { by lemma 21 R->L }
% 6.45/1.28 join(meet(composition(x0, x1), x2), complement(join(complement(meet(x1, composition(converse(x0), x2))), complement(join(composition(converse(x0), complement(composition(x0, x1))), composition(converse(x0), x2))))))
% 6.45/1.28 = { by lemma 29 }
% 6.45/1.28 complement(join(complement(meet(x1, composition(converse(x0), x2))), complement(join(composition(converse(x0), complement(composition(x0, x1))), composition(converse(x0), x2)))))
% 6.45/1.28 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), join(composition(converse(x0), complement(composition(x0, x1))), composition(converse(x0), x2)))
% 6.45/1.28 = { by axiom 1 (maddux1_join_commutativity) }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), join(composition(converse(x0), x2), composition(converse(x0), complement(composition(x0, x1)))))
% 6.45/1.28 = { by lemma 16 R->L }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), join(composition(converse(x0), x2), converse(composition(converse(complement(composition(x0, x1))), x0))))
% 6.45/1.28 = { by lemma 36 R->L }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), converse(join(composition(converse(complement(composition(x0, x1))), x0), converse(composition(converse(x0), x2)))))
% 6.45/1.28 = { by axiom 1 (maddux1_join_commutativity) }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), converse(join(converse(composition(converse(x0), x2)), composition(converse(complement(composition(x0, x1))), x0))))
% 6.45/1.28 = { by axiom 2 (converse_idempotence) R->L }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), converse(join(converse(composition(converse(x0), x2)), composition(converse(complement(composition(x0, x1))), converse(converse(x0))))))
% 6.45/1.28 = { by axiom 9 (converse_multiplicativity) }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), converse(join(composition(converse(x2), converse(converse(x0))), composition(converse(complement(composition(x0, x1))), converse(converse(x0))))))
% 6.45/1.28 = { by axiom 12 (composition_distributivity) R->L }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), converse(composition(join(converse(x2), converse(complement(composition(x0, x1)))), converse(converse(x0)))))
% 6.45/1.28 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), converse(composition(join(converse(complement(composition(x0, x1))), converse(x2)), converse(converse(x0)))))
% 6.45/1.28 = { by lemma 36 R->L }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), converse(composition(converse(join(x2, converse(converse(complement(composition(x0, x1)))))), converse(converse(x0)))))
% 6.45/1.28 = { by axiom 9 (converse_multiplicativity) R->L }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), converse(converse(composition(converse(x0), join(x2, converse(converse(complement(composition(x0, x1)))))))))
% 6.45/1.28 = { by lemma 16 }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), converse(composition(converse(join(x2, converse(converse(complement(composition(x0, x1)))))), x0)))
% 6.45/1.28 = { by lemma 36 }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), converse(composition(join(converse(complement(composition(x0, x1))), converse(x2)), x0)))
% 6.45/1.28 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), converse(composition(join(converse(x2), converse(complement(composition(x0, x1)))), x0)))
% 6.45/1.28 = { by axiom 9 (converse_multiplicativity) }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), composition(converse(x0), converse(join(converse(x2), converse(complement(composition(x0, x1)))))))
% 6.45/1.28 = { by lemma 36 }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), composition(converse(x0), join(complement(composition(x0, x1)), converse(converse(x2)))))
% 6.45/1.28 = { by axiom 2 (converse_idempotence) }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), composition(converse(x0), join(complement(composition(x0, x1)), x2)))
% 6.45/1.28 = { by axiom 1 (maddux1_join_commutativity) }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), composition(converse(x0), join(x2, complement(composition(x0, x1)))))
% 6.45/1.28 = { by lemma 32 R->L }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), composition(converse(x0), meet(join(x2, complement(composition(x0, x1))), top)))
% 6.45/1.28 = { by lemma 24 R->L }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), composition(converse(x0), complement(join(meet(composition(x0, x1), x2), complement(join(x2, complement(composition(x0, x1))))))))
% 6.45/1.28 = { by lemma 38 R->L }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), composition(converse(x0), complement(join(meet(composition(x0, x1), x2), join(complement(join(x2, complement(composition(x0, x1)))), meet(composition(x0, x1), x2))))))
% 6.45/1.28 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), composition(converse(x0), complement(join(meet(composition(x0, x1), x2), join(meet(composition(x0, x1), x2), complement(join(x2, complement(composition(x0, x1)))))))))
% 6.45/1.28 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), composition(converse(x0), complement(join(meet(composition(x0, x1), x2), join(meet(composition(x0, x1), x2), complement(join(complement(composition(x0, x1)), x2)))))))
% 6.45/1.28 = { by lemma 25 }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), composition(converse(x0), complement(join(meet(composition(x0, x1), x2), composition(x0, x1)))))
% 6.45/1.28 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), composition(converse(x0), complement(join(composition(x0, x1), meet(composition(x0, x1), x2)))))
% 6.45/1.28 = { by lemma 38 }
% 6.45/1.28 meet(meet(x1, composition(converse(x0), x2)), composition(converse(x0), complement(composition(x0, x1))))
% 6.45/1.28 = { by lemma 23 }
% 6.45/1.28 meet(composition(converse(x0), complement(composition(x0, x1))), meet(x1, composition(converse(x0), x2)))
% 6.45/1.28 = { by lemma 32 R->L }
% 6.45/1.28 meet(composition(converse(x0), complement(composition(x0, x1))), meet(x1, meet(composition(converse(x0), x2), top)))
% 6.45/1.28 = { by lemma 24 R->L }
% 6.45/1.28 meet(composition(converse(x0), complement(composition(x0, x1))), meet(x1, complement(join(meet(composition(x0, x1), x2), complement(composition(converse(x0), x2))))))
% 6.45/1.28 = { by lemma 35 R->L }
% 6.45/1.28 meet(composition(converse(x0), complement(composition(x0, x1))), complement(join(complement(x1), join(meet(composition(x0, x1), x2), complement(composition(converse(x0), x2))))))
% 6.45/1.28 = { by lemma 35 R->L }
% 6.45/1.28 complement(join(complement(composition(converse(x0), complement(composition(x0, x1)))), join(complement(x1), join(meet(composition(x0, x1), x2), complement(composition(converse(x0), x2))))))
% 6.45/1.28 = { by axiom 7 (maddux2_join_associativity) }
% 6.45/1.28 complement(join(join(complement(composition(converse(x0), complement(composition(x0, x1)))), complement(x1)), join(meet(composition(x0, x1), x2), complement(composition(converse(x0), x2)))))
% 6.45/1.28 = { by lemma 32 R->L }
% 6.45/1.28 complement(join(join(complement(composition(converse(x0), complement(composition(x0, x1)))), complement(x1)), meet(join(meet(composition(x0, x1), x2), complement(composition(converse(x0), x2))), top)))
% 6.45/1.28 = { by lemma 24 R->L }
% 6.45/1.28 complement(join(join(complement(composition(converse(x0), complement(composition(x0, x1)))), complement(x1)), complement(join(meet(composition(x0, x1), x2), complement(join(meet(composition(x0, x1), x2), complement(composition(converse(x0), x2))))))))
% 6.45/1.28 = { by lemma 34 }
% 6.45/1.28 meet(join(meet(composition(x0, x1), x2), complement(join(meet(composition(x0, x1), x2), complement(composition(converse(x0), x2))))), complement(join(complement(composition(converse(x0), complement(composition(x0, x1)))), complement(x1))))
% 6.45/1.28 = { by lemma 29 }
% 6.45/1.28 meet(complement(join(meet(composition(x0, x1), x2), complement(composition(converse(x0), x2)))), complement(join(complement(composition(converse(x0), complement(composition(x0, x1)))), complement(x1))))
% 6.45/1.28 = { by lemma 23 R->L }
% 6.45/1.28 meet(complement(join(complement(composition(converse(x0), complement(composition(x0, x1)))), complement(x1))), complement(join(meet(composition(x0, x1), x2), complement(composition(converse(x0), x2)))))
% 6.45/1.28 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 6.45/1.28 meet(meet(composition(converse(x0), complement(composition(x0, x1))), x1), complement(join(meet(composition(x0, x1), x2), complement(composition(converse(x0), x2)))))
% 6.45/1.28 = { by lemma 23 R->L }
% 6.45/1.28 meet(complement(join(meet(composition(x0, x1), x2), complement(composition(converse(x0), x2)))), meet(composition(converse(x0), complement(composition(x0, x1))), x1))
% 6.45/1.28 = { by lemma 23 R->L }
% 6.45/1.28 meet(complement(join(meet(composition(x0, x1), x2), complement(composition(converse(x0), x2)))), meet(x1, composition(converse(x0), complement(composition(x0, x1)))))
% 6.45/1.28 = { by lemma 24 }
% 6.45/1.28 meet(meet(composition(converse(x0), x2), top), meet(x1, composition(converse(x0), complement(composition(x0, x1)))))
% 6.45/1.28 = { by lemma 32 }
% 6.45/1.28 meet(composition(converse(x0), x2), meet(x1, composition(converse(x0), complement(composition(x0, x1)))))
% 6.45/1.28 = { by lemma 23 }
% 6.45/1.28 meet(composition(converse(x0), x2), meet(composition(converse(x0), complement(composition(x0, x1))), x1))
% 6.45/1.28 = { by lemma 26 R->L }
% 6.45/1.28 meet(composition(converse(x0), x2), meet(composition(converse(x0), complement(composition(x0, x1))), join(meet(composition(x0, x1), x2), complement(complement(x1)))))
% 6.45/1.28 = { by lemma 29 }
% 6.45/1.28 meet(composition(converse(x0), x2), meet(composition(converse(x0), complement(composition(x0, x1))), complement(complement(x1))))
% 6.45/1.28 = { by lemma 18 R->L }
% 6.45/1.28 meet(composition(converse(x0), x2), meet(composition(converse(x0), complement(composition(x0, x1))), complement(join(complement(x1), composition(converse(x0), complement(composition(x0, x1)))))))
% 6.45/1.28 = { by lemma 35 }
% 6.45/1.28 meet(composition(converse(x0), x2), meet(composition(converse(x0), complement(composition(x0, x1))), meet(x1, complement(composition(converse(x0), complement(composition(x0, x1)))))))
% 6.45/1.28 = { by lemma 23 }
% 6.45/1.28 meet(composition(converse(x0), x2), meet(composition(converse(x0), complement(composition(x0, x1))), meet(complement(composition(converse(x0), complement(composition(x0, x1)))), x1)))
% 6.45/1.28 = { by axiom 11 (maddux4_definiton_of_meet) }
% 6.45/1.28 meet(composition(converse(x0), x2), complement(join(complement(composition(converse(x0), complement(composition(x0, x1)))), complement(meet(complement(composition(converse(x0), complement(composition(x0, x1)))), x1)))))
% 6.45/1.28 = { by lemma 37 }
% 6.45/1.28 meet(composition(converse(x0), x2), complement(top))
% 6.45/1.28 = { by lemma 21 R->L }
% 6.45/1.28 meet(composition(converse(x0), x2), meet(composition(x0, x1), x2))
% 6.45/1.28 = { by lemma 31 }
% 6.45/1.28 meet(composition(x0, x1), x2)
% 6.45/1.28 = { by axiom 8 (goals) }
% 6.45/1.28 zero
% 6.45/1.28 % SZS output end Proof
% 6.45/1.28
% 6.45/1.28 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------