TSTP Solution File: REL016-3 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL016-3 : 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 : n017.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 13:43:56 EDT 2023
% Result : Unsatisfiable 12.29s 1.99s
% Output : Proof 13.34s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : REL016-3 : TPTP v8.1.2. Released v4.0.0.
% 0.12/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.33 % Computer : n017.cluster.edu
% 0.14/0.33 % Model : x86_64 x86_64
% 0.14/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.33 % Memory : 8042.1875MB
% 0.14/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.33 % CPULimit : 300
% 0.14/0.33 % WCLimit : 300
% 0.14/0.33 % DateTime : Fri Aug 25 22:02:40 EDT 2023
% 0.14/0.33 % CPUTime :
% 12.29/1.99 Command-line arguments: --flatten
% 12.29/1.99
% 12.29/1.99 % SZS status Unsatisfiable
% 12.29/1.99
% 13.34/2.04 % SZS output start Proof
% 13.34/2.04 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 13.34/2.04 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 13.34/2.04 Axiom 3 (composition_identity_6): composition(X, one) = X.
% 13.34/2.04 Axiom 4 (def_top_12): top = join(X, complement(X)).
% 13.34/2.04 Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 13.34/2.04 Axiom 6 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 13.34/2.04 Axiom 7 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 13.34/2.04 Axiom 8 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 13.34/2.04 Axiom 9 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 13.34/2.04 Axiom 10 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 13.34/2.04 Axiom 11 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 13.34/2.04 Axiom 12 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 13.34/2.04 Axiom 13 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 13.34/2.04
% 13.34/2.04 Lemma 14: complement(top) = zero.
% 13.34/2.04 Proof:
% 13.34/2.04 complement(top)
% 13.34/2.04 = { by axiom 4 (def_top_12) }
% 13.34/2.04 complement(join(complement(X), complement(complement(X))))
% 13.34/2.04 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 13.34/2.04 meet(X, complement(X))
% 13.34/2.04 = { by axiom 5 (def_zero_13) R->L }
% 13.34/2.04 zero
% 13.34/2.04
% 13.34/2.04 Lemma 15: join(X, join(Y, complement(X))) = join(Y, top).
% 13.34/2.04 Proof:
% 13.34/2.04 join(X, join(Y, complement(X)))
% 13.34/2.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.04 join(X, join(complement(X), Y))
% 13.34/2.04 = { by axiom 7 (maddux2_join_associativity_2) }
% 13.34/2.04 join(join(X, complement(X)), Y)
% 13.34/2.04 = { by axiom 4 (def_top_12) R->L }
% 13.34/2.04 join(top, Y)
% 13.34/2.04 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.34/2.04 join(Y, top)
% 13.34/2.04
% 13.34/2.04 Lemma 16: composition(converse(one), X) = X.
% 13.34/2.04 Proof:
% 13.34/2.04 composition(converse(one), X)
% 13.34/2.04 = { by axiom 1 (converse_idempotence_8) R->L }
% 13.34/2.04 composition(converse(one), converse(converse(X)))
% 13.34/2.04 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 13.34/2.04 converse(composition(converse(X), one))
% 13.34/2.04 = { by axiom 3 (composition_identity_6) }
% 13.34/2.04 converse(converse(X))
% 13.34/2.04 = { by axiom 1 (converse_idempotence_8) }
% 13.34/2.05 X
% 13.34/2.05
% 13.34/2.05 Lemma 17: join(complement(X), complement(X)) = complement(X).
% 13.34/2.05 Proof:
% 13.34/2.05 join(complement(X), complement(X))
% 13.34/2.05 = { by lemma 16 R->L }
% 13.34/2.05 join(complement(X), composition(converse(one), complement(X)))
% 13.34/2.05 = { by lemma 16 R->L }
% 13.34/2.05 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 13.34/2.05 = { by axiom 3 (composition_identity_6) R->L }
% 13.34/2.05 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 13.34/2.05 = { by axiom 9 (composition_associativity_5) R->L }
% 13.34/2.05 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 13.34/2.05 = { by lemma 16 }
% 13.34/2.05 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.05 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 13.34/2.05 = { by axiom 12 (converse_cancellativity_11) }
% 13.34/2.05 complement(X)
% 13.34/2.05
% 13.34/2.05 Lemma 18: join(top, complement(X)) = top.
% 13.34/2.05 Proof:
% 13.34/2.05 join(top, complement(X))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.05 join(complement(X), top)
% 13.34/2.05 = { by lemma 15 R->L }
% 13.34/2.05 join(X, join(complement(X), complement(X)))
% 13.34/2.05 = { by lemma 17 }
% 13.34/2.05 join(X, complement(X))
% 13.34/2.05 = { by axiom 4 (def_top_12) R->L }
% 13.34/2.05 top
% 13.34/2.05
% 13.34/2.05 Lemma 19: join(X, top) = top.
% 13.34/2.05 Proof:
% 13.34/2.05 join(X, top)
% 13.34/2.05 = { by lemma 18 R->L }
% 13.34/2.05 join(X, join(top, complement(X)))
% 13.34/2.05 = { by lemma 15 }
% 13.34/2.05 join(top, top)
% 13.34/2.05 = { by lemma 15 R->L }
% 13.34/2.05 join(complement(Y), join(top, complement(complement(Y))))
% 13.34/2.05 = { by lemma 18 }
% 13.34/2.05 join(complement(Y), top)
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.05 join(top, complement(Y))
% 13.34/2.05 = { by lemma 18 }
% 13.34/2.05 top
% 13.34/2.05
% 13.34/2.05 Lemma 20: meet(X, X) = complement(complement(X)).
% 13.34/2.05 Proof:
% 13.34/2.05 meet(X, X)
% 13.34/2.05 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 13.34/2.05 complement(join(complement(X), complement(X)))
% 13.34/2.05 = { by lemma 17 }
% 13.34/2.05 complement(complement(X))
% 13.34/2.05
% 13.34/2.05 Lemma 21: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 13.34/2.05 Proof:
% 13.34/2.05 join(meet(X, Y), complement(join(complement(X), Y)))
% 13.34/2.05 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 13.34/2.05 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 13.34/2.05 = { by axiom 13 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 13.34/2.05 X
% 13.34/2.05
% 13.34/2.05 Lemma 22: join(zero, meet(X, X)) = X.
% 13.34/2.05 Proof:
% 13.34/2.05 join(zero, meet(X, X))
% 13.34/2.05 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 13.34/2.05 join(zero, complement(join(complement(X), complement(X))))
% 13.34/2.05 = { by axiom 5 (def_zero_13) }
% 13.34/2.05 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 13.34/2.05 = { by lemma 21 }
% 13.34/2.05 X
% 13.34/2.05
% 13.34/2.05 Lemma 23: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 13.34/2.05 Proof:
% 13.34/2.05 join(zero, join(X, complement(complement(Y))))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.05 join(zero, join(complement(complement(Y)), X))
% 13.34/2.05 = { by lemma 20 R->L }
% 13.34/2.05 join(zero, join(meet(Y, Y), X))
% 13.34/2.05 = { by axiom 7 (maddux2_join_associativity_2) }
% 13.34/2.05 join(join(zero, meet(Y, Y)), X)
% 13.34/2.05 = { by lemma 22 }
% 13.34/2.05 join(Y, X)
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.34/2.05 join(X, Y)
% 13.34/2.05
% 13.34/2.05 Lemma 24: join(zero, complement(complement(X))) = X.
% 13.34/2.05 Proof:
% 13.34/2.05 join(zero, complement(complement(X)))
% 13.34/2.05 = { by axiom 5 (def_zero_13) }
% 13.34/2.05 join(meet(X, complement(X)), complement(complement(X)))
% 13.34/2.05 = { by lemma 17 R->L }
% 13.34/2.05 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 13.34/2.05 = { by lemma 21 }
% 13.34/2.05 X
% 13.34/2.05
% 13.34/2.05 Lemma 25: join(zero, complement(X)) = complement(X).
% 13.34/2.05 Proof:
% 13.34/2.05 join(zero, complement(X))
% 13.34/2.05 = { by lemma 24 R->L }
% 13.34/2.05 join(zero, join(zero, complement(complement(complement(X)))))
% 13.34/2.05 = { by lemma 17 R->L }
% 13.34/2.05 join(zero, join(zero, join(complement(complement(complement(X))), complement(complement(complement(X))))))
% 13.34/2.05 = { by lemma 23 }
% 13.34/2.05 join(zero, join(complement(complement(complement(X))), complement(X)))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.34/2.05 join(zero, join(complement(X), complement(complement(complement(X)))))
% 13.34/2.05 = { by lemma 23 }
% 13.34/2.05 join(complement(X), complement(X))
% 13.34/2.05 = { by lemma 17 }
% 13.34/2.05 complement(X)
% 13.34/2.05
% 13.34/2.05 Lemma 26: join(X, zero) = X.
% 13.34/2.05 Proof:
% 13.34/2.05 join(X, zero)
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.05 join(zero, X)
% 13.34/2.05 = { by lemma 23 R->L }
% 13.34/2.05 join(zero, join(zero, complement(complement(X))))
% 13.34/2.05 = { by lemma 25 }
% 13.34/2.05 join(zero, complement(complement(X)))
% 13.34/2.05 = { by lemma 24 }
% 13.34/2.05 X
% 13.34/2.05
% 13.34/2.05 Lemma 27: meet(Y, X) = meet(X, Y).
% 13.34/2.05 Proof:
% 13.34/2.05 meet(Y, X)
% 13.34/2.05 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 13.34/2.05 complement(join(complement(Y), complement(X)))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.05 complement(join(complement(X), complement(Y)))
% 13.34/2.05 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 13.34/2.05 meet(X, Y)
% 13.34/2.05
% 13.34/2.05 Lemma 28: complement(join(complement(X), complement(Y))) = meet(Y, X).
% 13.34/2.05 Proof:
% 13.34/2.05 complement(join(complement(X), complement(Y)))
% 13.34/2.05 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 13.34/2.05 meet(X, Y)
% 13.34/2.05 = { by lemma 27 R->L }
% 13.34/2.05 meet(Y, X)
% 13.34/2.05
% 13.34/2.05 Lemma 29: complement(complement(X)) = X.
% 13.34/2.05 Proof:
% 13.34/2.05 complement(complement(X))
% 13.34/2.05 = { by lemma 20 R->L }
% 13.34/2.05 meet(X, X)
% 13.34/2.05 = { by lemma 28 R->L }
% 13.34/2.05 complement(join(complement(X), complement(X)))
% 13.34/2.05 = { by lemma 25 R->L }
% 13.34/2.05 join(zero, complement(join(complement(X), complement(X))))
% 13.34/2.05 = { by lemma 28 }
% 13.34/2.05 join(zero, meet(X, X))
% 13.34/2.05 = { by lemma 22 }
% 13.34/2.05 X
% 13.34/2.05
% 13.34/2.05 Lemma 30: complement(join(zero, complement(X))) = meet(X, top).
% 13.34/2.05 Proof:
% 13.34/2.05 complement(join(zero, complement(X)))
% 13.34/2.05 = { by lemma 14 R->L }
% 13.34/2.05 complement(join(complement(top), complement(X)))
% 13.34/2.05 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 13.34/2.05 meet(top, X)
% 13.34/2.05 = { by lemma 27 R->L }
% 13.34/2.05 meet(X, top)
% 13.34/2.05
% 13.34/2.05 Lemma 31: join(X, complement(zero)) = top.
% 13.34/2.05 Proof:
% 13.34/2.05 join(X, complement(zero))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.05 join(complement(zero), X)
% 13.34/2.05 = { by lemma 23 R->L }
% 13.34/2.05 join(zero, join(complement(zero), complement(complement(X))))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.05 join(zero, join(complement(complement(X)), complement(zero)))
% 13.34/2.05 = { by lemma 15 }
% 13.34/2.05 join(complement(complement(X)), top)
% 13.34/2.05 = { by lemma 19 }
% 13.34/2.05 top
% 13.34/2.05
% 13.34/2.05 Lemma 32: meet(X, top) = X.
% 13.34/2.05 Proof:
% 13.34/2.05 meet(X, top)
% 13.34/2.05 = { by lemma 30 R->L }
% 13.34/2.05 complement(join(zero, complement(X)))
% 13.34/2.05 = { by lemma 25 R->L }
% 13.34/2.05 join(zero, complement(join(zero, complement(X))))
% 13.34/2.05 = { by lemma 30 }
% 13.34/2.05 join(zero, meet(X, top))
% 13.34/2.05 = { by lemma 31 R->L }
% 13.34/2.05 join(zero, meet(X, join(complement(zero), complement(zero))))
% 13.34/2.05 = { by lemma 17 }
% 13.34/2.05 join(zero, meet(X, complement(zero)))
% 13.34/2.05 = { by lemma 14 R->L }
% 13.34/2.05 join(complement(top), meet(X, complement(zero)))
% 13.34/2.05 = { by lemma 31 R->L }
% 13.34/2.05 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero)))
% 13.34/2.05 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 13.34/2.05 join(meet(X, zero), meet(X, complement(zero)))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.05 join(meet(X, complement(zero)), meet(X, zero))
% 13.34/2.05 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 13.34/2.05 join(meet(X, complement(zero)), complement(join(complement(X), complement(zero))))
% 13.34/2.05 = { by lemma 21 }
% 13.34/2.05 X
% 13.34/2.05
% 13.34/2.05 Lemma 33: join(meet(X, complement(Y)), meet(X, Y)) = X.
% 13.34/2.05 Proof:
% 13.34/2.05 join(meet(X, complement(Y)), meet(X, Y))
% 13.34/2.05 = { by lemma 28 R->L }
% 13.34/2.05 join(meet(X, complement(Y)), complement(join(complement(Y), complement(X))))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.05 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 13.34/2.05 = { by lemma 21 }
% 13.34/2.05 X
% 13.34/2.05
% 13.34/2.05 Lemma 34: join(meet(X, complement(Y)), join(Z, meet(X, Y))) = join(Z, X).
% 13.34/2.05 Proof:
% 13.34/2.05 join(meet(X, complement(Y)), join(Z, meet(X, Y)))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.05 join(meet(X, complement(Y)), join(meet(X, Y), Z))
% 13.34/2.05 = { by axiom 7 (maddux2_join_associativity_2) }
% 13.34/2.05 join(join(meet(X, complement(Y)), meet(X, Y)), Z)
% 13.34/2.05 = { by lemma 33 }
% 13.34/2.05 join(X, Z)
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.34/2.05 join(Z, X)
% 13.34/2.05
% 13.34/2.05 Lemma 35: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 13.34/2.05 Proof:
% 13.34/2.05 join(meet(X, Y), meet(X, Y))
% 13.34/2.05 = { by lemma 27 }
% 13.34/2.05 join(meet(Y, X), meet(X, Y))
% 13.34/2.05 = { by lemma 27 }
% 13.34/2.05 join(meet(Y, X), meet(Y, X))
% 13.34/2.05 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 13.34/2.05 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 13.34/2.05 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 13.34/2.05 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 13.34/2.05 = { by lemma 17 }
% 13.34/2.05 complement(join(complement(Y), complement(X)))
% 13.34/2.05 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 13.34/2.05 meet(Y, X)
% 13.34/2.05 = { by lemma 27 R->L }
% 13.34/2.05 meet(X, Y)
% 13.34/2.05
% 13.34/2.05 Lemma 36: join(X, meet(X, Y)) = X.
% 13.34/2.05 Proof:
% 13.34/2.05 join(X, meet(X, Y))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.05 join(meet(X, Y), X)
% 13.34/2.05 = { by lemma 34 R->L }
% 13.34/2.05 join(meet(X, complement(Y)), join(meet(X, Y), meet(X, Y)))
% 13.34/2.05 = { by lemma 35 }
% 13.34/2.05 join(meet(X, complement(Y)), meet(X, Y))
% 13.34/2.05 = { by lemma 33 }
% 13.34/2.05 X
% 13.34/2.05
% 13.34/2.05 Lemma 37: join(X, meet(Y, X)) = X.
% 13.34/2.05 Proof:
% 13.34/2.05 join(X, meet(Y, X))
% 13.34/2.05 = { by lemma 27 }
% 13.34/2.05 join(X, meet(X, Y))
% 13.34/2.05 = { by lemma 36 }
% 13.34/2.05 X
% 13.34/2.05
% 13.34/2.05 Lemma 38: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 13.34/2.05 Proof:
% 13.34/2.05 meet(X, join(complement(Y), complement(Z)))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.05 meet(X, join(complement(Z), complement(Y)))
% 13.34/2.05 = { by lemma 27 }
% 13.34/2.05 meet(join(complement(Z), complement(Y)), X)
% 13.34/2.05 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 13.34/2.05 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 13.34/2.05 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 13.34/2.05 complement(join(meet(Z, Y), complement(X)))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.34/2.05 complement(join(complement(X), meet(Z, Y)))
% 13.34/2.05 = { by lemma 27 R->L }
% 13.34/2.05 complement(join(complement(X), meet(Y, Z)))
% 13.34/2.05
% 13.34/2.05 Lemma 39: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 13.34/2.05 Proof:
% 13.34/2.05 complement(join(X, complement(Y)))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.05 complement(join(complement(Y), X))
% 13.34/2.05 = { by lemma 32 R->L }
% 13.34/2.05 complement(join(complement(Y), meet(X, top)))
% 13.34/2.05 = { by lemma 27 R->L }
% 13.34/2.05 complement(join(complement(Y), meet(top, X)))
% 13.34/2.05 = { by lemma 38 R->L }
% 13.34/2.05 meet(Y, join(complement(top), complement(X)))
% 13.34/2.05 = { by lemma 14 }
% 13.34/2.05 meet(Y, join(zero, complement(X)))
% 13.34/2.05 = { by lemma 25 }
% 13.34/2.05 meet(Y, complement(X))
% 13.34/2.05
% 13.34/2.05 Lemma 40: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 13.34/2.05 Proof:
% 13.34/2.05 complement(join(complement(X), Y))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.05 complement(join(Y, complement(X)))
% 13.34/2.05 = { by lemma 39 }
% 13.34/2.05 meet(X, complement(Y))
% 13.34/2.05
% 13.34/2.05 Lemma 41: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 13.34/2.05 Proof:
% 13.34/2.05 complement(meet(X, complement(Y)))
% 13.34/2.05 = { by lemma 26 R->L }
% 13.34/2.05 complement(join(meet(X, complement(Y)), zero))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.34/2.05 complement(join(zero, meet(X, complement(Y))))
% 13.34/2.05 = { by lemma 39 R->L }
% 13.34/2.05 complement(join(zero, complement(join(Y, complement(X)))))
% 13.34/2.05 = { by lemma 30 }
% 13.34/2.05 meet(join(Y, complement(X)), top)
% 13.34/2.05 = { by lemma 32 }
% 13.34/2.05 join(Y, complement(X))
% 13.34/2.05
% 13.34/2.05 Lemma 42: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 13.34/2.05 Proof:
% 13.34/2.05 meet(Y, meet(Z, X))
% 13.34/2.05 = { by lemma 32 R->L }
% 13.34/2.05 meet(meet(Y, top), meet(Z, X))
% 13.34/2.05 = { by lemma 30 R->L }
% 13.34/2.05 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 13.34/2.05 = { by lemma 27 }
% 13.34/2.05 meet(complement(join(zero, complement(Y))), meet(X, Z))
% 13.34/2.05 = { by lemma 27 }
% 13.34/2.05 meet(meet(X, Z), complement(join(zero, complement(Y))))
% 13.34/2.05 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 13.34/2.05 meet(complement(join(complement(X), complement(Z))), complement(join(zero, complement(Y))))
% 13.34/2.05 = { by lemma 27 }
% 13.34/2.05 meet(complement(join(zero, complement(Y))), complement(join(complement(X), complement(Z))))
% 13.34/2.05 = { by lemma 25 R->L }
% 13.34/2.05 meet(join(zero, complement(join(zero, complement(Y)))), complement(join(complement(X), complement(Z))))
% 13.34/2.05 = { by lemma 39 R->L }
% 13.34/2.05 complement(join(join(complement(X), complement(Z)), complement(join(zero, complement(join(zero, complement(Y)))))))
% 13.34/2.05 = { by lemma 30 }
% 13.34/2.05 complement(join(join(complement(X), complement(Z)), meet(join(zero, complement(Y)), top)))
% 13.34/2.05 = { by lemma 32 }
% 13.34/2.05 complement(join(join(complement(X), complement(Z)), join(zero, complement(Y))))
% 13.34/2.05 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 13.34/2.05 complement(join(complement(X), join(complement(Z), join(zero, complement(Y)))))
% 13.34/2.05 = { by lemma 40 }
% 13.34/2.05 meet(X, complement(join(complement(Z), join(zero, complement(Y)))))
% 13.34/2.05 = { by lemma 40 }
% 13.34/2.05 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 13.34/2.05 = { by lemma 30 }
% 13.34/2.05 meet(X, meet(Z, meet(Y, top)))
% 13.34/2.05 = { by lemma 32 }
% 13.34/2.05 meet(X, meet(Z, Y))
% 13.34/2.05 = { by lemma 27 R->L }
% 13.34/2.05 meet(X, meet(Y, Z))
% 13.34/2.05
% 13.34/2.05 Lemma 43: join(X, complement(meet(X, Y))) = top.
% 13.34/2.05 Proof:
% 13.34/2.05 join(X, complement(meet(X, Y)))
% 13.34/2.05 = { by lemma 35 R->L }
% 13.34/2.05 join(X, complement(join(meet(X, Y), meet(X, Y))))
% 13.34/2.05 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 13.34/2.05 join(X, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y))))
% 13.34/2.05 = { by lemma 25 R->L }
% 13.34/2.05 join(X, join(zero, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y)))))
% 13.34/2.05 = { by lemma 38 R->L }
% 13.34/2.05 join(X, join(zero, meet(join(complement(X), complement(Y)), join(complement(X), complement(Y)))))
% 13.34/2.05 = { by lemma 22 }
% 13.34/2.05 join(X, join(complement(X), complement(Y)))
% 13.34/2.05 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.34/2.05 join(X, join(complement(Y), complement(X)))
% 13.34/2.05 = { by lemma 15 }
% 13.34/2.05 join(complement(Y), top)
% 13.34/2.05 = { by lemma 19 }
% 13.34/2.06 top
% 13.34/2.06
% 13.34/2.06 Lemma 44: join(X, complement(meet(Y, X))) = top.
% 13.34/2.06 Proof:
% 13.34/2.06 join(X, complement(meet(Y, X)))
% 13.34/2.06 = { by lemma 27 R->L }
% 13.34/2.06 join(X, complement(meet(X, Y)))
% 13.34/2.06 = { by lemma 43 }
% 13.34/2.06 top
% 13.34/2.06
% 13.34/2.06 Lemma 45: join(meet(X, complement(Y)), Y) = join(X, Y).
% 13.34/2.06 Proof:
% 13.34/2.06 join(meet(X, complement(Y)), Y)
% 13.34/2.06 = { by lemma 37 R->L }
% 13.34/2.06 join(meet(X, complement(Y)), join(Y, meet(X, Y)))
% 13.34/2.06 = { by lemma 34 }
% 13.34/2.06 join(Y, X)
% 13.34/2.06 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.34/2.06 join(X, Y)
% 13.34/2.06
% 13.34/2.06 Lemma 46: join(composition(X, Z), composition(X, Y)) = composition(X, join(Y, Z)).
% 13.34/2.06 Proof:
% 13.34/2.06 join(composition(X, Z), composition(X, Y))
% 13.34/2.06 = { by axiom 1 (converse_idempotence_8) R->L }
% 13.34/2.06 join(composition(X, Z), composition(X, converse(converse(Y))))
% 13.34/2.06 = { by axiom 1 (converse_idempotence_8) R->L }
% 13.34/2.06 converse(converse(join(composition(X, Z), composition(X, converse(converse(Y))))))
% 13.34/2.06 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.06 converse(converse(join(composition(X, converse(converse(Y))), composition(X, Z))))
% 13.34/2.06 = { by axiom 6 (converse_additivity_9) }
% 13.34/2.06 converse(join(converse(composition(X, converse(converse(Y)))), converse(composition(X, Z))))
% 13.34/2.06 = { by axiom 8 (converse_multiplicativity_10) }
% 13.34/2.06 converse(join(composition(converse(converse(converse(Y))), converse(X)), converse(composition(X, Z))))
% 13.34/2.06 = { by axiom 1 (converse_idempotence_8) }
% 13.34/2.06 converse(join(composition(converse(Y), converse(X)), converse(composition(X, Z))))
% 13.34/2.06 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.34/2.06 converse(join(converse(composition(X, Z)), composition(converse(Y), converse(X))))
% 13.34/2.06 = { by axiom 8 (converse_multiplicativity_10) }
% 13.34/2.06 converse(join(composition(converse(Z), converse(X)), composition(converse(Y), converse(X))))
% 13.34/2.06 = { by axiom 11 (composition_distributivity_7) R->L }
% 13.34/2.06 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 13.34/2.06 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.06 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 13.34/2.06 = { by axiom 1 (converse_idempotence_8) R->L }
% 13.34/2.06 converse(composition(join(converse(converse(converse(Y))), converse(Z)), converse(X)))
% 13.34/2.06 = { by axiom 6 (converse_additivity_9) R->L }
% 13.34/2.06 converse(composition(converse(join(converse(converse(Y)), Z)), converse(X)))
% 13.34/2.06 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.34/2.06 converse(composition(converse(join(Z, converse(converse(Y)))), converse(X)))
% 13.34/2.06 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 13.34/2.06 converse(converse(composition(X, join(Z, converse(converse(Y))))))
% 13.34/2.06 = { by axiom 1 (converse_idempotence_8) }
% 13.34/2.06 composition(X, join(Z, converse(converse(Y))))
% 13.34/2.06 = { by axiom 1 (converse_idempotence_8) }
% 13.34/2.06 composition(X, join(Z, Y))
% 13.34/2.06 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.34/2.06 composition(X, join(Y, Z))
% 13.34/2.06
% 13.34/2.06 Lemma 47: meet(X, join(complement(Y), meet(X, Y))) = X.
% 13.34/2.06 Proof:
% 13.34/2.06 meet(X, join(complement(Y), meet(X, Y)))
% 13.34/2.06 = { by lemma 26 R->L }
% 13.34/2.06 join(meet(X, join(complement(Y), meet(X, Y))), zero)
% 13.34/2.06 = { by lemma 14 R->L }
% 13.34/2.06 join(meet(X, join(complement(Y), meet(X, Y))), complement(top))
% 13.34/2.06 = { by axiom 4 (def_top_12) }
% 13.34/2.06 join(meet(X, join(complement(Y), meet(X, Y))), complement(join(join(complement(X), complement(Y)), complement(join(complement(X), complement(Y))))))
% 13.34/2.06 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 13.34/2.06 join(meet(X, join(complement(Y), meet(X, Y))), complement(join(join(complement(X), complement(Y)), meet(X, Y))))
% 13.34/2.06 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 13.34/2.06 join(meet(X, join(complement(Y), meet(X, Y))), complement(join(complement(X), join(complement(Y), meet(X, Y)))))
% 13.34/2.06 = { by lemma 21 }
% 13.34/2.06 X
% 13.34/2.06
% 13.34/2.06 Goal 1 (goals_17): meet(composition(sk1, sk2), complement(composition(sk1, sk3))) = meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))).
% 13.34/2.06 Proof:
% 13.34/2.06 meet(composition(sk1, sk2), complement(composition(sk1, sk3)))
% 13.34/2.06 = { by lemma 37 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))
% 13.34/2.06 = { by lemma 27 }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))
% 13.34/2.06 = { by lemma 42 }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(complement(composition(sk1, sk3)), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), composition(sk1, meet(sk2, complement(sk3))))))
% 13.34/2.06 = { by lemma 42 }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), meet(complement(composition(sk1, sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))
% 13.34/2.06 = { by lemma 27 }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(composition(sk1, sk3)))))
% 13.34/2.06 = { by lemma 40 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), complement(join(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), composition(sk1, sk3)))))
% 13.34/2.06 = { by lemma 25 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), join(zero, complement(join(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), composition(sk1, sk3))))))
% 13.34/2.06 = { by lemma 14 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), join(complement(top), complement(join(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), composition(sk1, sk3))))))
% 13.34/2.06 = { by lemma 44 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), join(complement(join(complement(composition(sk1, sk3)), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))), complement(join(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), composition(sk1, sk3))))))
% 13.34/2.06 = { by lemma 28 }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), join(meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), composition(sk1, sk3)), complement(join(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), composition(sk1, sk3))))))
% 13.34/2.06 = { by lemma 21 }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))
% 13.34/2.06 = { by lemma 27 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), meet(complement(composition(sk1, sk3)), composition(sk1, sk2))))
% 13.34/2.06 = { by lemma 42 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(complement(composition(sk1, sk3)), meet(composition(sk1, sk2), composition(sk1, meet(sk2, complement(sk3))))))
% 13.34/2.06 = { by lemma 42 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))
% 13.34/2.06 = { by lemma 27 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, sk2)))
% 13.34/2.06 = { by lemma 32 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, meet(sk2, top))))
% 13.34/2.06 = { by lemma 35 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, join(meet(sk2, top), meet(sk2, top)))))
% 13.34/2.06 = { by lemma 32 }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, join(sk2, meet(sk2, top)))))
% 13.34/2.06 = { by lemma 32 }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, join(sk2, sk2))))
% 13.34/2.06 = { by lemma 34 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, join(meet(sk2, complement(sk3)), join(sk2, meet(sk2, sk3))))))
% 13.34/2.06 = { by lemma 36 }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, join(meet(sk2, complement(sk3)), sk2))))
% 13.34/2.06 = { by lemma 46 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), join(composition(sk1, sk2), composition(sk1, meet(sk2, complement(sk3))))))
% 13.34/2.06 = { by lemma 34 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), join(composition(sk1, sk2), meet(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3))))))
% 13.34/2.06 = { by lemma 32 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(join(composition(sk1, sk2), meet(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3))), top))))
% 13.34/2.06 = { by lemma 30 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), complement(join(zero, complement(join(composition(sk1, sk2), meet(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3)))))))))
% 13.34/2.06 = { by lemma 41 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), complement(meet(join(zero, complement(join(composition(sk1, sk2), meet(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3))))), complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))))))))
% 13.34/2.06 = { by lemma 40 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(join(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), meet(join(zero, complement(join(composition(sk1, sk2), meet(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3))))), complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))))))))
% 13.34/2.06 = { by lemma 25 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(zero, complement(join(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), meet(join(zero, complement(join(composition(sk1, sk2), meet(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3))))), complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))))))
% 13.34/2.06 = { by lemma 14 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(top), complement(join(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), meet(join(zero, complement(join(composition(sk1, sk2), meet(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3))))), complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))))))
% 13.34/2.06 = { by lemma 43 R->L }
% 13.34/2.06 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(join(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), complement(meet(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), join(zero, complement(join(composition(sk1, sk2), meet(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3))))))))), complement(join(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), meet(join(zero, complement(join(composition(sk1, sk2), meet(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3))))), complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))))))
% 13.34/2.06 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 13.34/2.07 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), join(zero, complement(join(composition(sk1, sk2), meet(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3))))))), complement(join(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), meet(join(zero, complement(join(composition(sk1, sk2), meet(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3))))), complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))))))
% 13.34/2.07 = { by lemma 27 R->L }
% 13.34/2.07 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(join(zero, complement(join(composition(sk1, sk2), meet(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3))))), complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))), complement(join(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), meet(join(zero, complement(join(composition(sk1, sk2), meet(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3))))), complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))))))
% 13.34/2.07 = { by lemma 21 }
% 13.34/2.07 join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))))
% 13.34/2.07 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))
% 13.34/2.07 = { by lemma 32 R->L }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), top))
% 13.34/2.07 = { by lemma 44 R->L }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, sk3)), complement(meet(composition(sk1, sk2), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, sk3)))))))
% 13.34/2.07 = { by lemma 45 }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, sk3)), complement(meet(composition(sk1, sk2), join(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3)))))))
% 13.34/2.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, sk3)), complement(meet(composition(sk1, sk2), join(composition(sk1, sk3), composition(sk1, meet(sk2, complement(sk3)))))))))
% 13.34/2.07 = { by lemma 46 }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, sk3)), complement(meet(composition(sk1, sk2), composition(sk1, join(meet(sk2, complement(sk3)), sk3)))))))
% 13.34/2.07 = { by lemma 45 }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, sk3)), complement(meet(composition(sk1, sk2), composition(sk1, join(sk2, sk3)))))))
% 13.34/2.07 = { by lemma 46 R->L }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, sk3)), complement(meet(composition(sk1, sk2), join(composition(sk1, sk3), composition(sk1, sk2)))))))
% 13.34/2.07 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, sk3)), complement(meet(composition(sk1, sk2), join(composition(sk1, sk2), composition(sk1, sk3)))))))
% 13.34/2.07 = { by lemma 45 R->L }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, sk3)), complement(meet(composition(sk1, sk2), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), composition(sk1, sk3)))))))
% 13.34/2.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, sk3)), complement(meet(composition(sk1, sk2), join(composition(sk1, sk3), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))))
% 13.34/2.07 = { by lemma 29 R->L }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, sk3)), complement(meet(composition(sk1, sk2), join(complement(complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))))
% 13.34/2.07 = { by lemma 47 }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, sk3)), complement(composition(sk1, sk2)))))
% 13.34/2.07 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), join(composition(sk1, sk3), complement(composition(sk1, sk2))))))
% 13.34/2.07 = { by lemma 41 R->L }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))
% 13.34/2.07 = { by lemma 41 R->L }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))))))))
% 13.34/2.07 = { by lemma 40 R->L }
% 13.34/2.07 join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), complement(join(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))))))))
% 13.34/2.07 = { by lemma 41 R->L }
% 13.34/2.07 complement(meet(join(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))), complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))))))
% 13.34/2.07 = { by lemma 27 R->L }
% 13.34/2.07 complement(meet(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), join(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))))))))
% 13.34/2.07 = { by lemma 27 }
% 13.34/2.07 complement(meet(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), join(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))
% 13.34/2.07 = { by lemma 47 }
% 13.34/2.07 complement(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))
% 13.34/2.07 = { by lemma 29 }
% 13.34/2.07 meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))
% 13.34/2.07 % SZS output end Proof
% 13.34/2.07
% 13.34/2.07 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------