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 : n022.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 : Unsatisfiable 5.16s 1.09s
% Output : Proof 5.84s
% 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
% 0.13/0.35 % Computer : n022.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Aug 25 20:01:10 EDT 2023
% 0.13/0.35 % CPUTime :
% 5.16/1.09 Command-line arguments: --flatten
% 5.16/1.09
% 5.16/1.09 % SZS status Unsatisfiable
% 5.16/1.09
% 5.84/1.13 % SZS output start Proof
% 5.84/1.13 Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 5.84/1.13 Axiom 2 (converse_idempotence_8): converse(converse(X)) = X.
% 5.84/1.13 Axiom 3 (composition_identity_6): composition(X, one) = X.
% 5.84/1.13 Axiom 4 (def_top_12): top = join(X, complement(X)).
% 5.84/1.13 Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 5.84/1.13 Axiom 6 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 5.84/1.13 Axiom 7 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 5.84/1.13 Axiom 8 (goals_17): meet(composition(sk1, sk2), sk3) = zero.
% 5.84/1.13 Axiom 9 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 5.84/1.13 Axiom 10 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 5.84/1.13 Axiom 11 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 5.84/1.13 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 5.84/1.13 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 5.84/1.13 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 5.84/1.13
% 5.84/1.13 Lemma 15: join(X, join(Y, complement(X))) = join(Y, top).
% 5.84/1.13 Proof:
% 5.84/1.13 join(X, join(Y, complement(X)))
% 5.84/1.13 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.13 join(X, join(complement(X), Y))
% 5.84/1.13 = { by axiom 7 (maddux2_join_associativity_2) }
% 5.84/1.13 join(join(X, complement(X)), Y)
% 5.84/1.13 = { by axiom 4 (def_top_12) R->L }
% 5.84/1.13 join(top, Y)
% 5.84/1.13 = { by axiom 1 (maddux1_join_commutativity_1) }
% 5.84/1.13 join(Y, top)
% 5.84/1.13
% 5.84/1.13 Lemma 16: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 5.84/1.13 Proof:
% 5.84/1.13 converse(composition(converse(X), Y))
% 5.84/1.13 = { by axiom 9 (converse_multiplicativity_10) }
% 5.84/1.13 composition(converse(Y), converse(converse(X)))
% 5.84/1.13 = { by axiom 2 (converse_idempotence_8) }
% 5.84/1.13 composition(converse(Y), X)
% 5.84/1.13
% 5.84/1.13 Lemma 17: composition(converse(one), X) = X.
% 5.84/1.13 Proof:
% 5.84/1.13 composition(converse(one), X)
% 5.84/1.13 = { by lemma 16 R->L }
% 5.84/1.13 converse(composition(converse(X), one))
% 5.84/1.13 = { by axiom 3 (composition_identity_6) }
% 5.84/1.13 converse(converse(X))
% 5.84/1.13 = { by axiom 2 (converse_idempotence_8) }
% 5.84/1.13 X
% 5.84/1.13
% 5.84/1.13 Lemma 18: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 5.84/1.13 Proof:
% 5.84/1.13 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 5.84/1.13 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.13 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 5.84/1.13 = { by axiom 13 (converse_cancellativity_11) }
% 5.84/1.13 complement(X)
% 5.84/1.13
% 5.84/1.13 Lemma 19: join(complement(X), complement(X)) = complement(X).
% 5.84/1.13 Proof:
% 5.84/1.13 join(complement(X), complement(X))
% 5.84/1.13 = { by lemma 17 R->L }
% 5.84/1.13 join(complement(X), composition(converse(one), complement(X)))
% 5.84/1.13 = { by lemma 17 R->L }
% 5.84/1.13 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 5.84/1.13 = { by axiom 3 (composition_identity_6) R->L }
% 5.84/1.13 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 5.84/1.13 = { by axiom 10 (composition_associativity_5) R->L }
% 5.84/1.13 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 5.84/1.13 = { by lemma 17 }
% 5.84/1.13 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 5.84/1.13 = { by lemma 18 }
% 5.84/1.13 complement(X)
% 5.84/1.13
% 5.84/1.13 Lemma 20: join(top, complement(X)) = top.
% 5.84/1.13 Proof:
% 5.84/1.13 join(top, complement(X))
% 5.84/1.13 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.13 join(complement(X), top)
% 5.84/1.13 = { by lemma 15 R->L }
% 5.84/1.13 join(X, join(complement(X), complement(X)))
% 5.84/1.13 = { by lemma 19 }
% 5.84/1.13 join(X, complement(X))
% 5.84/1.13 = { by axiom 4 (def_top_12) R->L }
% 5.84/1.13 top
% 5.84/1.13
% 5.84/1.13 Lemma 21: meet(composition(sk1, sk2), sk3) = complement(top).
% 5.84/1.13 Proof:
% 5.84/1.13 meet(composition(sk1, sk2), sk3)
% 5.84/1.13 = { by axiom 8 (goals_17) }
% 5.84/1.13 zero
% 5.84/1.13 = { by axiom 5 (def_zero_13) }
% 5.84/1.13 meet(X, complement(X))
% 5.84/1.13 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 5.84/1.13 complement(join(complement(X), complement(complement(X))))
% 5.84/1.13 = { by axiom 4 (def_top_12) R->L }
% 5.84/1.13 complement(top)
% 5.84/1.13
% 5.84/1.13 Lemma 22: join(X, top) = top.
% 5.84/1.13 Proof:
% 5.84/1.13 join(X, top)
% 5.84/1.13 = { by lemma 20 R->L }
% 5.84/1.13 join(X, join(top, complement(X)))
% 5.84/1.13 = { by lemma 15 }
% 5.84/1.13 join(top, top)
% 5.84/1.13 = { by lemma 15 R->L }
% 5.84/1.13 join(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, sk2), sk3)), join(top, complement(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, sk2), sk3)))))
% 5.84/1.13 = { by lemma 20 }
% 5.84/1.13 join(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, sk2), sk3)), top)
% 5.84/1.13 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.13 join(top, join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, sk2), sk3)))
% 5.84/1.13 = { by lemma 21 }
% 5.84/1.13 join(top, join(meet(composition(sk1, sk2), sk3), complement(top)))
% 5.84/1.13 = { by lemma 21 }
% 5.84/1.13 join(top, join(complement(top), complement(top)))
% 5.84/1.13 = { by lemma 19 }
% 5.84/1.13 join(top, complement(top))
% 5.84/1.13 = { by axiom 4 (def_top_12) R->L }
% 5.84/1.13 top
% 5.84/1.13
% 5.84/1.13 Lemma 23: meet(Y, X) = meet(X, Y).
% 5.84/1.13 Proof:
% 5.84/1.13 meet(Y, X)
% 5.84/1.13 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 5.84/1.13 complement(join(complement(Y), complement(X)))
% 5.84/1.13 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.13 complement(join(complement(X), complement(Y)))
% 5.84/1.13 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 5.84/1.13 meet(X, Y)
% 5.84/1.13
% 5.84/1.13 Lemma 24: complement(join(meet(composition(sk1, sk2), sk3), complement(X))) = meet(X, top).
% 5.84/1.13 Proof:
% 5.84/1.13 complement(join(meet(composition(sk1, sk2), sk3), complement(X)))
% 5.84/1.13 = { by lemma 21 }
% 5.84/1.13 complement(join(complement(top), complement(X)))
% 5.84/1.13 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 5.84/1.13 meet(top, X)
% 5.84/1.13 = { by lemma 23 R->L }
% 5.84/1.13 meet(X, top)
% 5.84/1.13
% 5.84/1.13 Lemma 25: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 5.84/1.13 Proof:
% 5.84/1.13 join(meet(X, Y), complement(join(complement(X), Y)))
% 5.84/1.13 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 5.84/1.13 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 5.84/1.13 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 5.84/1.13 X
% 5.84/1.13
% 5.84/1.13 Lemma 26: join(meet(composition(sk1, sk2), sk3), complement(complement(X))) = X.
% 5.84/1.13 Proof:
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), complement(complement(X)))
% 5.84/1.13 = { by axiom 8 (goals_17) }
% 5.84/1.13 join(zero, complement(complement(X)))
% 5.84/1.13 = { by axiom 5 (def_zero_13) }
% 5.84/1.13 join(meet(X, complement(X)), complement(complement(X)))
% 5.84/1.13 = { by lemma 19 R->L }
% 5.84/1.13 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 5.84/1.13 = { by lemma 25 }
% 5.84/1.13 X
% 5.84/1.13
% 5.84/1.13 Lemma 27: join(meet(composition(sk1, sk2), sk3), meet(X, X)) = X.
% 5.84/1.13 Proof:
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), meet(X, X))
% 5.84/1.13 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), complement(join(complement(X), complement(X))))
% 5.84/1.13 = { by axiom 8 (goals_17) }
% 5.84/1.13 join(zero, complement(join(complement(X), complement(X))))
% 5.84/1.13 = { by axiom 5 (def_zero_13) }
% 5.84/1.13 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 5.84/1.13 = { by lemma 25 }
% 5.84/1.13 X
% 5.84/1.13
% 5.84/1.13 Lemma 28: join(meet(composition(sk1, sk2), sk3), join(X, complement(complement(Y)))) = join(X, Y).
% 5.84/1.13 Proof:
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), join(X, complement(complement(Y))))
% 5.84/1.13 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), join(complement(complement(Y)), X))
% 5.84/1.13 = { by lemma 19 R->L }
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), join(complement(join(complement(Y), complement(Y))), X))
% 5.84/1.13 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), join(meet(Y, Y), X))
% 5.84/1.13 = { by axiom 7 (maddux2_join_associativity_2) }
% 5.84/1.13 join(join(meet(composition(sk1, sk2), sk3), meet(Y, Y)), X)
% 5.84/1.13 = { by lemma 27 }
% 5.84/1.13 join(Y, X)
% 5.84/1.13 = { by axiom 1 (maddux1_join_commutativity_1) }
% 5.84/1.13 join(X, Y)
% 5.84/1.13
% 5.84/1.13 Lemma 29: join(meet(composition(sk1, sk2), sk3), complement(X)) = complement(X).
% 5.84/1.13 Proof:
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), complement(X))
% 5.84/1.13 = { by lemma 26 R->L }
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), join(meet(composition(sk1, sk2), sk3), complement(complement(complement(X)))))
% 5.84/1.13 = { by lemma 19 R->L }
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), join(meet(composition(sk1, sk2), sk3), join(complement(complement(complement(X))), complement(complement(complement(X))))))
% 5.84/1.13 = { by lemma 28 }
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), join(complement(complement(complement(X))), complement(X)))
% 5.84/1.13 = { by axiom 1 (maddux1_join_commutativity_1) }
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), join(complement(X), complement(complement(complement(X)))))
% 5.84/1.13 = { by lemma 28 }
% 5.84/1.13 join(complement(X), complement(X))
% 5.84/1.13 = { by lemma 19 }
% 5.84/1.13 complement(X)
% 5.84/1.13
% 5.84/1.13 Lemma 30: join(X, complement(meet(composition(sk1, sk2), sk3))) = top.
% 5.84/1.13 Proof:
% 5.84/1.13 join(X, complement(meet(composition(sk1, sk2), sk3)))
% 5.84/1.13 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.13 join(complement(meet(composition(sk1, sk2), sk3)), X)
% 5.84/1.13 = { by lemma 28 R->L }
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), join(complement(meet(composition(sk1, sk2), sk3)), complement(complement(X))))
% 5.84/1.13 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), join(complement(complement(X)), complement(meet(composition(sk1, sk2), sk3))))
% 5.84/1.13 = { by lemma 15 }
% 5.84/1.13 join(complement(complement(X)), top)
% 5.84/1.13 = { by lemma 22 }
% 5.84/1.13 top
% 5.84/1.13
% 5.84/1.13 Lemma 31: meet(X, meet(composition(sk1, sk2), sk3)) = meet(composition(sk1, sk2), sk3).
% 5.84/1.13 Proof:
% 5.84/1.13 meet(X, meet(composition(sk1, sk2), sk3))
% 5.84/1.13 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 5.84/1.13 complement(join(complement(X), complement(meet(composition(sk1, sk2), sk3))))
% 5.84/1.13 = { by lemma 30 }
% 5.84/1.13 complement(top)
% 5.84/1.13 = { by lemma 21 R->L }
% 5.84/1.13 meet(composition(sk1, sk2), sk3)
% 5.84/1.13
% 5.84/1.13 Lemma 32: meet(X, top) = X.
% 5.84/1.13 Proof:
% 5.84/1.13 meet(X, top)
% 5.84/1.13 = { by lemma 24 R->L }
% 5.84/1.13 complement(join(meet(composition(sk1, sk2), sk3), complement(X)))
% 5.84/1.13 = { by lemma 29 R->L }
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), complement(join(meet(composition(sk1, sk2), sk3), complement(X))))
% 5.84/1.13 = { by lemma 24 }
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), meet(X, top))
% 5.84/1.13 = { by lemma 30 R->L }
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), meet(X, join(complement(meet(composition(sk1, sk2), sk3)), complement(meet(composition(sk1, sk2), sk3)))))
% 5.84/1.13 = { by lemma 19 }
% 5.84/1.13 join(meet(composition(sk1, sk2), sk3), meet(X, complement(meet(composition(sk1, sk2), sk3))))
% 5.84/1.13 = { by lemma 31 R->L }
% 5.84/1.13 join(meet(X, meet(composition(sk1, sk2), sk3)), meet(X, complement(meet(composition(sk1, sk2), sk3))))
% 5.84/1.13 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.14 join(meet(X, complement(meet(composition(sk1, sk2), sk3))), meet(X, meet(composition(sk1, sk2), sk3)))
% 5.84/1.14 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 5.84/1.14 join(meet(X, complement(meet(composition(sk1, sk2), sk3))), complement(join(complement(X), complement(meet(composition(sk1, sk2), sk3)))))
% 5.84/1.14 = { by lemma 25 }
% 5.84/1.14 X
% 5.84/1.14
% 5.84/1.14 Lemma 33: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 5.84/1.14 Proof:
% 5.84/1.14 meet(X, join(complement(Y), complement(Z)))
% 5.84/1.14 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.14 meet(X, join(complement(Z), complement(Y)))
% 5.84/1.14 = { by lemma 23 }
% 5.84/1.14 meet(join(complement(Z), complement(Y)), X)
% 5.84/1.14 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 5.84/1.14 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 5.84/1.14 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 5.84/1.14 complement(join(meet(Z, Y), complement(X)))
% 5.84/1.14 = { by axiom 1 (maddux1_join_commutativity_1) }
% 5.84/1.14 complement(join(complement(X), meet(Z, Y)))
% 5.84/1.14 = { by lemma 23 R->L }
% 5.84/1.14 complement(join(complement(X), meet(Y, Z)))
% 5.84/1.14
% 5.84/1.14 Lemma 34: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 5.84/1.14 Proof:
% 5.84/1.14 complement(join(X, complement(Y)))
% 5.84/1.14 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.14 complement(join(complement(Y), X))
% 5.84/1.14 = { by lemma 32 R->L }
% 5.84/1.14 complement(join(complement(Y), meet(X, top)))
% 5.84/1.14 = { by lemma 23 R->L }
% 5.84/1.14 complement(join(complement(Y), meet(top, X)))
% 5.84/1.14 = { by lemma 33 R->L }
% 5.84/1.14 meet(Y, join(complement(top), complement(X)))
% 5.84/1.14 = { by lemma 21 R->L }
% 5.84/1.14 meet(Y, join(meet(composition(sk1, sk2), sk3), complement(X)))
% 5.84/1.14 = { by lemma 29 }
% 5.84/1.14 meet(Y, complement(X))
% 5.84/1.14
% 5.84/1.14 Lemma 35: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 5.84/1.14 Proof:
% 5.84/1.14 complement(join(complement(X), Y))
% 5.84/1.14 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.14 complement(join(Y, complement(X)))
% 5.84/1.14 = { by lemma 34 }
% 5.84/1.14 meet(X, complement(Y))
% 5.84/1.14
% 5.84/1.14 Lemma 36: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 5.84/1.14 Proof:
% 5.84/1.14 converse(join(X, converse(Y)))
% 5.84/1.14 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.14 converse(join(converse(Y), X))
% 5.84/1.14 = { by axiom 6 (converse_additivity_9) }
% 5.84/1.14 join(converse(converse(Y)), converse(X))
% 5.84/1.14 = { by axiom 2 (converse_idempotence_8) }
% 5.84/1.14 join(Y, converse(X))
% 5.84/1.14
% 5.84/1.14 Lemma 37: join(X, complement(meet(X, Y))) = top.
% 5.84/1.14 Proof:
% 5.84/1.14 join(X, complement(meet(X, Y)))
% 5.84/1.14 = { by lemma 23 }
% 5.84/1.14 join(X, complement(meet(Y, X)))
% 5.84/1.14 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 5.84/1.14 join(X, complement(complement(join(complement(Y), complement(X)))))
% 5.84/1.14 = { by lemma 19 R->L }
% 5.84/1.14 join(X, complement(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))))
% 5.84/1.14 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 5.84/1.14 join(X, complement(join(meet(Y, X), complement(join(complement(Y), complement(X))))))
% 5.84/1.14 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 5.84/1.14 join(X, complement(join(meet(Y, X), meet(Y, X))))
% 5.84/1.14 = { by lemma 23 R->L }
% 5.84/1.14 join(X, complement(join(meet(Y, X), meet(X, Y))))
% 5.84/1.14 = { by lemma 23 R->L }
% 5.84/1.14 join(X, complement(join(meet(X, Y), meet(X, Y))))
% 5.84/1.14 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 5.84/1.14 join(X, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y))))
% 5.84/1.14 = { by lemma 29 R->L }
% 5.84/1.14 join(X, join(meet(composition(sk1, sk2), sk3), complement(join(complement(join(complement(X), complement(Y))), meet(X, Y)))))
% 5.84/1.14 = { by lemma 33 R->L }
% 5.84/1.14 join(X, join(meet(composition(sk1, sk2), sk3), meet(join(complement(X), complement(Y)), join(complement(X), complement(Y)))))
% 5.84/1.14 = { by lemma 27 }
% 5.84/1.14 join(X, join(complement(X), complement(Y)))
% 5.84/1.14 = { by axiom 1 (maddux1_join_commutativity_1) }
% 5.84/1.14 join(X, join(complement(Y), complement(X)))
% 5.84/1.14 = { by lemma 15 }
% 5.84/1.14 join(complement(Y), top)
% 5.84/1.14 = { by lemma 22 }
% 5.84/1.14 top
% 5.84/1.14
% 5.84/1.14 Lemma 38: join(X, meet(composition(sk1, sk2), sk3)) = X.
% 5.84/1.14 Proof:
% 5.84/1.14 join(X, meet(composition(sk1, sk2), sk3))
% 5.84/1.14 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.14 join(meet(composition(sk1, sk2), sk3), X)
% 5.84/1.14 = { by lemma 28 R->L }
% 5.84/1.14 join(meet(composition(sk1, sk2), sk3), join(meet(composition(sk1, sk2), sk3), complement(complement(X))))
% 5.84/1.14 = { by lemma 29 }
% 5.84/1.14 join(meet(composition(sk1, sk2), sk3), complement(complement(X)))
% 5.84/1.14 = { by lemma 26 }
% 5.84/1.14 X
% 5.84/1.14
% 5.84/1.14 Goal 1 (goals_18): meet(sk2, composition(converse(sk1), sk3)) = zero.
% 5.84/1.14 Proof:
% 5.84/1.14 meet(sk2, composition(converse(sk1), sk3))
% 5.84/1.14 = { by lemma 25 R->L }
% 5.84/1.14 join(meet(meet(sk2, composition(converse(sk1), sk3)), complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), composition(converse(sk1), sk3)))), complement(join(complement(meet(sk2, composition(converse(sk1), sk3))), complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), composition(converse(sk1), sk3))))))
% 5.84/1.14 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.14 join(meet(meet(sk2, composition(converse(sk1), sk3)), complement(join(composition(converse(sk1), sk3), composition(converse(sk1), complement(composition(sk1, sk2)))))), complement(join(complement(meet(sk2, composition(converse(sk1), sk3))), complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), composition(converse(sk1), sk3))))))
% 5.84/1.14 = { by lemma 34 R->L }
% 5.84/1.14 join(complement(join(join(composition(converse(sk1), sk3), composition(converse(sk1), complement(composition(sk1, sk2)))), complement(meet(sk2, composition(converse(sk1), sk3))))), complement(join(complement(meet(sk2, composition(converse(sk1), sk3))), complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), composition(converse(sk1), sk3))))))
% 5.84/1.14 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 5.84/1.14 join(complement(join(composition(converse(sk1), sk3), join(composition(converse(sk1), complement(composition(sk1, sk2))), complement(meet(sk2, composition(converse(sk1), sk3)))))), complement(join(complement(meet(sk2, composition(converse(sk1), sk3))), complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), composition(converse(sk1), sk3))))))
% 5.84/1.14 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.14 join(complement(join(join(composition(converse(sk1), complement(composition(sk1, sk2))), complement(meet(sk2, composition(converse(sk1), sk3)))), composition(converse(sk1), sk3))), complement(join(complement(meet(sk2, composition(converse(sk1), sk3))), complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), composition(converse(sk1), sk3))))))
% 5.84/1.14 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 5.84/1.14 join(complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), join(complement(meet(sk2, composition(converse(sk1), sk3))), composition(converse(sk1), sk3)))), complement(join(complement(meet(sk2, composition(converse(sk1), sk3))), complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), composition(converse(sk1), sk3))))))
% 5.84/1.14 = { by axiom 1 (maddux1_join_commutativity_1) }
% 5.84/1.14 join(complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), join(composition(converse(sk1), sk3), complement(meet(sk2, composition(converse(sk1), sk3)))))), complement(join(complement(meet(sk2, composition(converse(sk1), sk3))), complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), composition(converse(sk1), sk3))))))
% 5.84/1.14 = { by lemma 23 R->L }
% 5.84/1.14 join(complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), join(composition(converse(sk1), sk3), complement(meet(composition(converse(sk1), sk3), sk2))))), complement(join(complement(meet(sk2, composition(converse(sk1), sk3))), complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), composition(converse(sk1), sk3))))))
% 5.84/1.14 = { by lemma 37 }
% 5.84/1.14 join(complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), top)), complement(join(complement(meet(sk2, composition(converse(sk1), sk3))), complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), composition(converse(sk1), sk3))))))
% 5.84/1.14 = { by lemma 22 }
% 5.84/1.14 join(complement(top), complement(join(complement(meet(sk2, composition(converse(sk1), sk3))), complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), composition(converse(sk1), sk3))))))
% 5.84/1.14 = { by lemma 21 R->L }
% 5.84/1.14 join(meet(composition(sk1, sk2), sk3), complement(join(complement(meet(sk2, composition(converse(sk1), sk3))), complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), composition(converse(sk1), sk3))))))
% 5.84/1.14 = { by lemma 29 }
% 5.84/1.14 complement(join(complement(meet(sk2, composition(converse(sk1), sk3))), complement(join(composition(converse(sk1), complement(composition(sk1, sk2))), composition(converse(sk1), sk3)))))
% 5.84/1.14 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), join(composition(converse(sk1), complement(composition(sk1, sk2))), composition(converse(sk1), sk3)))
% 5.84/1.14 = { by axiom 1 (maddux1_join_commutativity_1) }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), join(composition(converse(sk1), sk3), composition(converse(sk1), complement(composition(sk1, sk2)))))
% 5.84/1.14 = { by lemma 16 R->L }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), join(composition(converse(sk1), sk3), converse(composition(converse(complement(composition(sk1, sk2))), sk1))))
% 5.84/1.14 = { by lemma 36 R->L }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), converse(join(composition(converse(complement(composition(sk1, sk2))), sk1), converse(composition(converse(sk1), sk3)))))
% 5.84/1.14 = { by axiom 1 (maddux1_join_commutativity_1) }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), converse(join(converse(composition(converse(sk1), sk3)), composition(converse(complement(composition(sk1, sk2))), sk1))))
% 5.84/1.14 = { by axiom 2 (converse_idempotence_8) R->L }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), converse(join(converse(composition(converse(sk1), sk3)), composition(converse(complement(composition(sk1, sk2))), converse(converse(sk1))))))
% 5.84/1.14 = { by axiom 9 (converse_multiplicativity_10) }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), converse(join(composition(converse(sk3), converse(converse(sk1))), composition(converse(complement(composition(sk1, sk2))), converse(converse(sk1))))))
% 5.84/1.14 = { by axiom 12 (composition_distributivity_7) R->L }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), converse(composition(join(converse(sk3), converse(complement(composition(sk1, sk2)))), converse(converse(sk1)))))
% 5.84/1.14 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), converse(composition(join(converse(complement(composition(sk1, sk2))), converse(sk3)), converse(converse(sk1)))))
% 5.84/1.14 = { by lemma 36 R->L }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), converse(composition(converse(join(sk3, converse(converse(complement(composition(sk1, sk2)))))), converse(converse(sk1)))))
% 5.84/1.14 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), converse(converse(composition(converse(sk1), join(sk3, converse(converse(complement(composition(sk1, sk2)))))))))
% 5.84/1.14 = { by lemma 16 }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), converse(composition(converse(join(sk3, converse(converse(complement(composition(sk1, sk2)))))), sk1)))
% 5.84/1.14 = { by lemma 36 }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), converse(composition(join(converse(complement(composition(sk1, sk2))), converse(sk3)), sk1)))
% 5.84/1.14 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), converse(composition(join(converse(sk3), converse(complement(composition(sk1, sk2)))), sk1)))
% 5.84/1.14 = { by axiom 9 (converse_multiplicativity_10) }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), composition(converse(sk1), converse(join(converse(sk3), converse(complement(composition(sk1, sk2)))))))
% 5.84/1.14 = { by lemma 36 }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), composition(converse(sk1), join(complement(composition(sk1, sk2)), converse(converse(sk3)))))
% 5.84/1.14 = { by axiom 2 (converse_idempotence_8) }
% 5.84/1.14 meet(meet(sk2, composition(converse(sk1), sk3)), composition(converse(sk1), join(complement(composition(sk1, sk2)), sk3)))
% 5.84/1.15 = { by axiom 1 (maddux1_join_commutativity_1) }
% 5.84/1.15 meet(meet(sk2, composition(converse(sk1), sk3)), composition(converse(sk1), join(sk3, complement(composition(sk1, sk2)))))
% 5.84/1.15 = { by lemma 32 R->L }
% 5.84/1.15 meet(meet(sk2, composition(converse(sk1), sk3)), composition(converse(sk1), meet(join(sk3, complement(composition(sk1, sk2))), top)))
% 5.84/1.15 = { by lemma 24 R->L }
% 5.84/1.15 meet(meet(sk2, composition(converse(sk1), sk3)), composition(converse(sk1), complement(join(meet(composition(sk1, sk2), sk3), complement(join(sk3, complement(composition(sk1, sk2))))))))
% 5.84/1.15 = { by lemma 38 R->L }
% 5.84/1.15 meet(meet(sk2, composition(converse(sk1), sk3)), composition(converse(sk1), complement(join(meet(composition(sk1, sk2), sk3), join(complement(join(sk3, complement(composition(sk1, sk2)))), meet(composition(sk1, sk2), sk3))))))
% 5.84/1.15 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.15 meet(meet(sk2, composition(converse(sk1), sk3)), composition(converse(sk1), complement(join(meet(composition(sk1, sk2), sk3), join(meet(composition(sk1, sk2), sk3), complement(join(sk3, complement(composition(sk1, sk2)))))))))
% 5.84/1.15 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.15 meet(meet(sk2, composition(converse(sk1), sk3)), composition(converse(sk1), complement(join(meet(composition(sk1, sk2), sk3), join(meet(composition(sk1, sk2), sk3), complement(join(complement(composition(sk1, sk2)), sk3)))))))
% 5.84/1.15 = { by lemma 25 }
% 5.84/1.15 meet(meet(sk2, composition(converse(sk1), sk3)), composition(converse(sk1), complement(join(meet(composition(sk1, sk2), sk3), composition(sk1, sk2)))))
% 5.84/1.15 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 5.84/1.15 meet(meet(sk2, composition(converse(sk1), sk3)), composition(converse(sk1), complement(join(composition(sk1, sk2), meet(composition(sk1, sk2), sk3)))))
% 5.84/1.15 = { by lemma 38 }
% 5.84/1.15 meet(meet(sk2, composition(converse(sk1), sk3)), composition(converse(sk1), complement(composition(sk1, sk2))))
% 5.84/1.15 = { by lemma 23 }
% 5.84/1.15 meet(composition(converse(sk1), complement(composition(sk1, sk2))), meet(sk2, composition(converse(sk1), sk3)))
% 5.84/1.15 = { by lemma 32 R->L }
% 5.84/1.15 meet(composition(converse(sk1), complement(composition(sk1, sk2))), meet(sk2, meet(composition(converse(sk1), sk3), top)))
% 5.84/1.15 = { by lemma 24 R->L }
% 5.84/1.15 meet(composition(converse(sk1), complement(composition(sk1, sk2))), meet(sk2, complement(join(meet(composition(sk1, sk2), sk3), complement(composition(converse(sk1), sk3))))))
% 5.84/1.15 = { by lemma 35 R->L }
% 5.84/1.15 meet(composition(converse(sk1), complement(composition(sk1, sk2))), complement(join(complement(sk2), join(meet(composition(sk1, sk2), sk3), complement(composition(converse(sk1), sk3))))))
% 5.84/1.15 = { by lemma 35 R->L }
% 5.84/1.15 complement(join(complement(composition(converse(sk1), complement(composition(sk1, sk2)))), join(complement(sk2), join(meet(composition(sk1, sk2), sk3), complement(composition(converse(sk1), sk3))))))
% 5.84/1.15 = { by axiom 7 (maddux2_join_associativity_2) }
% 5.84/1.15 complement(join(join(complement(composition(converse(sk1), complement(composition(sk1, sk2)))), complement(sk2)), join(meet(composition(sk1, sk2), sk3), complement(composition(converse(sk1), sk3)))))
% 5.84/1.15 = { by lemma 32 R->L }
% 5.84/1.15 complement(join(join(complement(composition(converse(sk1), complement(composition(sk1, sk2)))), complement(sk2)), meet(join(meet(composition(sk1, sk2), sk3), complement(composition(converse(sk1), sk3))), top)))
% 5.84/1.15 = { by lemma 24 R->L }
% 5.84/1.15 complement(join(join(complement(composition(converse(sk1), complement(composition(sk1, sk2)))), complement(sk2)), complement(join(meet(composition(sk1, sk2), sk3), complement(join(meet(composition(sk1, sk2), sk3), complement(composition(converse(sk1), sk3))))))))
% 5.84/1.15 = { by lemma 34 }
% 5.84/1.15 meet(join(meet(composition(sk1, sk2), sk3), complement(join(meet(composition(sk1, sk2), sk3), complement(composition(converse(sk1), sk3))))), complement(join(complement(composition(converse(sk1), complement(composition(sk1, sk2)))), complement(sk2))))
% 5.84/1.15 = { by lemma 29 }
% 5.84/1.15 meet(complement(join(meet(composition(sk1, sk2), sk3), complement(composition(converse(sk1), sk3)))), complement(join(complement(composition(converse(sk1), complement(composition(sk1, sk2)))), complement(sk2))))
% 5.84/1.15 = { by lemma 23 R->L }
% 5.84/1.15 meet(complement(join(complement(composition(converse(sk1), complement(composition(sk1, sk2)))), complement(sk2))), complement(join(meet(composition(sk1, sk2), sk3), complement(composition(converse(sk1), sk3)))))
% 5.84/1.15 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 5.84/1.15 meet(meet(composition(converse(sk1), complement(composition(sk1, sk2))), sk2), complement(join(meet(composition(sk1, sk2), sk3), complement(composition(converse(sk1), sk3)))))
% 5.84/1.15 = { by lemma 23 R->L }
% 5.84/1.15 meet(complement(join(meet(composition(sk1, sk2), sk3), complement(composition(converse(sk1), sk3)))), meet(composition(converse(sk1), complement(composition(sk1, sk2))), sk2))
% 5.84/1.15 = { by lemma 23 R->L }
% 5.84/1.15 meet(complement(join(meet(composition(sk1, sk2), sk3), complement(composition(converse(sk1), sk3)))), meet(sk2, composition(converse(sk1), complement(composition(sk1, sk2)))))
% 5.84/1.15 = { by lemma 24 }
% 5.84/1.15 meet(meet(composition(converse(sk1), sk3), top), meet(sk2, composition(converse(sk1), complement(composition(sk1, sk2)))))
% 5.84/1.15 = { by lemma 32 }
% 5.84/1.15 meet(composition(converse(sk1), sk3), meet(sk2, composition(converse(sk1), complement(composition(sk1, sk2)))))
% 5.84/1.15 = { by lemma 23 }
% 5.84/1.15 meet(composition(converse(sk1), sk3), meet(composition(converse(sk1), complement(composition(sk1, sk2))), sk2))
% 5.84/1.15 = { by lemma 26 R->L }
% 5.84/1.15 meet(composition(converse(sk1), sk3), meet(composition(converse(sk1), complement(composition(sk1, sk2))), join(meet(composition(sk1, sk2), sk3), complement(complement(sk2)))))
% 5.84/1.15 = { by lemma 29 }
% 5.84/1.15 meet(composition(converse(sk1), sk3), meet(composition(converse(sk1), complement(composition(sk1, sk2))), complement(complement(sk2))))
% 5.84/1.15 = { by lemma 18 R->L }
% 5.84/1.15 meet(composition(converse(sk1), sk3), meet(composition(converse(sk1), complement(composition(sk1, sk2))), complement(join(complement(sk2), composition(converse(sk1), complement(composition(sk1, sk2)))))))
% 5.84/1.15 = { by lemma 35 }
% 5.84/1.15 meet(composition(converse(sk1), sk3), meet(composition(converse(sk1), complement(composition(sk1, sk2))), meet(sk2, complement(composition(converse(sk1), complement(composition(sk1, sk2)))))))
% 5.84/1.15 = { by lemma 23 }
% 5.84/1.15 meet(composition(converse(sk1), sk3), meet(composition(converse(sk1), complement(composition(sk1, sk2))), meet(complement(composition(converse(sk1), complement(composition(sk1, sk2)))), sk2)))
% 5.84/1.15 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 5.84/1.15 meet(composition(converse(sk1), sk3), complement(join(complement(composition(converse(sk1), complement(composition(sk1, sk2)))), complement(meet(complement(composition(converse(sk1), complement(composition(sk1, sk2)))), sk2)))))
% 5.84/1.15 = { by lemma 37 }
% 5.84/1.15 meet(composition(converse(sk1), sk3), complement(top))
% 5.84/1.15 = { by lemma 21 R->L }
% 5.84/1.15 meet(composition(converse(sk1), sk3), meet(composition(sk1, sk2), sk3))
% 5.84/1.15 = { by lemma 31 }
% 5.84/1.15 meet(composition(sk1, sk2), sk3)
% 5.84/1.15 = { by axiom 8 (goals_17) }
% 5.84/1.15 zero
% 5.84/1.15 % SZS output end Proof
% 5.84/1.15
% 5.84/1.15 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------