TSTP Solution File: REL005-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL005-1 : 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 : n029.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 13:43:44 EDT 2023
% Result : Unsatisfiable 0.20s 0.75s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL005-1 : 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 : n029.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 21:33:54 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.75 Command-line arguments: --no-flatten-goal
% 0.20/0.75
% 0.20/0.75 % SZS status Unsatisfiable
% 0.20/0.75
% 0.20/0.83 % SZS output start Proof
% 0.20/0.83 Axiom 1 (composition_identity_6): composition(X, one) = X.
% 0.20/0.83 Axiom 2 (converse_idempotence_8): converse(converse(X)) = X.
% 0.20/0.83 Axiom 3 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 0.20/0.83 Axiom 4 (def_zero_13): zero = meet(X, complement(X)).
% 0.20/0.83 Axiom 5 (def_top_12): top = join(X, complement(X)).
% 0.20/0.83 Axiom 6 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 0.20/0.83 Axiom 7 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 0.20/0.83 Axiom 8 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 0.20/0.83 Axiom 9 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 0.20/0.83 Axiom 10 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 0.20/0.83 Axiom 11 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 0.20/0.83 Axiom 12 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 0.20/0.83
% 0.20/0.83 Lemma 13: complement(top) = zero.
% 0.20/0.83 Proof:
% 0.20/0.83 complement(top)
% 0.20/0.83 = { by axiom 5 (def_top_12) }
% 0.20/0.83 complement(join(complement(X), complement(complement(X))))
% 0.20/0.83 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 0.20/0.83 meet(X, complement(X))
% 0.20/0.83 = { by axiom 4 (def_zero_13) R->L }
% 0.20/0.83 zero
% 0.20/0.83
% 0.20/0.83 Lemma 14: join(X, join(Y, complement(X))) = join(Y, top).
% 0.20/0.83 Proof:
% 0.20/0.83 join(X, join(Y, complement(X)))
% 0.20/0.83 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 0.20/0.83 join(X, join(complement(X), Y))
% 0.20/0.83 = { by axiom 9 (maddux2_join_associativity_2) }
% 0.20/0.83 join(join(X, complement(X)), Y)
% 0.20/0.83 = { by axiom 5 (def_top_12) R->L }
% 0.20/0.83 join(top, Y)
% 0.20/0.83 = { by axiom 3 (maddux1_join_commutativity_1) }
% 0.20/0.83 join(Y, top)
% 0.20/0.83
% 0.20/0.83 Lemma 15: composition(converse(one), X) = X.
% 0.20/0.83 Proof:
% 0.20/0.83 composition(converse(one), X)
% 0.20/0.83 = { by axiom 2 (converse_idempotence_8) R->L }
% 0.20/0.83 composition(converse(one), converse(converse(X)))
% 0.20/0.83 = { by axiom 6 (converse_multiplicativity_10) R->L }
% 0.20/0.83 converse(composition(converse(X), one))
% 0.20/0.83 = { by axiom 1 (composition_identity_6) }
% 0.20/0.83 converse(converse(X))
% 0.20/0.83 = { by axiom 2 (converse_idempotence_8) }
% 0.20/0.83 X
% 0.20/0.83
% 0.20/0.83 Lemma 16: join(complement(X), complement(X)) = complement(X).
% 0.20/0.83 Proof:
% 0.20/0.83 join(complement(X), complement(X))
% 0.20/0.83 = { by lemma 15 R->L }
% 0.20/0.83 join(complement(X), composition(converse(one), complement(X)))
% 0.20/0.83 = { by lemma 15 R->L }
% 0.20/0.83 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 0.20/0.83 = { by axiom 1 (composition_identity_6) R->L }
% 0.20/0.83 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 0.20/0.83 = { by axiom 7 (composition_associativity_5) R->L }
% 0.20/0.83 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 0.20/0.83 = { by lemma 15 }
% 0.20/0.83 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 0.20/0.83 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 0.20/0.83 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 0.20/0.83 = { by axiom 11 (converse_cancellativity_11) }
% 0.20/0.83 complement(X)
% 0.20/0.83
% 0.20/0.83 Lemma 17: join(top, complement(X)) = top.
% 0.20/0.83 Proof:
% 0.20/0.83 join(top, complement(X))
% 0.20/0.83 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 0.20/0.83 join(complement(X), top)
% 0.20/0.83 = { by lemma 14 R->L }
% 0.20/0.83 join(X, join(complement(X), complement(X)))
% 0.20/0.83 = { by lemma 16 }
% 0.20/0.83 join(X, complement(X))
% 0.20/0.83 = { by axiom 5 (def_top_12) R->L }
% 0.20/0.83 top
% 0.20/0.83
% 0.20/0.83 Lemma 18: join(Y, top) = join(X, top).
% 0.20/0.83 Proof:
% 0.20/0.84 join(Y, top)
% 0.20/0.84 = { by lemma 17 R->L }
% 0.20/0.84 join(Y, join(top, complement(Y)))
% 0.20/0.84 = { by lemma 14 }
% 0.20/0.84 join(top, top)
% 0.20/0.84 = { by lemma 14 R->L }
% 0.20/0.84 join(X, join(top, complement(X)))
% 0.20/0.84 = { by lemma 17 }
% 0.20/0.84 join(X, top)
% 0.20/0.84
% 0.20/0.84 Lemma 19: join(X, top) = top.
% 0.20/0.84 Proof:
% 0.20/0.84 join(X, top)
% 0.20/0.84 = { by lemma 18 }
% 0.20/0.84 join(join(zero, zero), top)
% 0.20/0.84 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 0.20/0.84 join(top, join(zero, zero))
% 0.20/0.84 = { by lemma 13 R->L }
% 0.20/0.84 join(top, join(zero, complement(top)))
% 0.20/0.84 = { by lemma 13 R->L }
% 0.20/0.84 join(top, join(complement(top), complement(top)))
% 0.20/0.84 = { by lemma 16 }
% 0.20/0.84 join(top, complement(top))
% 0.20/0.84 = { by axiom 5 (def_top_12) R->L }
% 0.20/0.84 top
% 0.20/0.84
% 0.20/0.84 Lemma 20: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 0.20/0.84 Proof:
% 0.20/0.84 join(meet(X, Y), complement(join(complement(X), Y)))
% 0.20/0.84 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 0.20/0.84 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 0.20/0.84 = { by axiom 12 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 0.20/0.84 X
% 0.20/0.84
% 0.20/0.84 Lemma 21: join(zero, meet(X, X)) = X.
% 0.20/0.84 Proof:
% 0.20/0.84 join(zero, meet(X, X))
% 0.20/0.84 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 0.20/0.84 join(zero, complement(join(complement(X), complement(X))))
% 0.20/0.84 = { by axiom 4 (def_zero_13) }
% 0.20/0.84 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 0.20/0.84 = { by lemma 20 }
% 0.20/0.84 X
% 0.20/0.84
% 0.20/0.84 Lemma 22: complement(complement(X)) = meet(X, X).
% 0.20/0.84 Proof:
% 0.20/0.84 complement(complement(X))
% 0.20/0.84 = { by lemma 16 R->L }
% 0.20/0.84 complement(join(complement(X), complement(X)))
% 0.20/0.84 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 0.20/0.84 meet(X, X)
% 0.20/0.84
% 0.20/0.84 Lemma 23: meet(Y, X) = meet(X, Y).
% 0.20/0.84 Proof:
% 0.20/0.84 meet(Y, X)
% 0.20/0.84 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 0.20/0.84 complement(join(complement(Y), complement(X)))
% 0.20/0.84 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 0.20/0.84 complement(join(complement(X), complement(Y)))
% 0.20/0.84 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 0.20/0.84 meet(X, Y)
% 0.20/0.84
% 0.20/0.84 Lemma 24: complement(join(zero, complement(X))) = meet(X, top).
% 0.20/0.84 Proof:
% 0.20/0.84 complement(join(zero, complement(X)))
% 0.20/0.84 = { by lemma 13 R->L }
% 0.20/0.84 complement(join(complement(top), complement(X)))
% 0.20/0.84 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 0.20/0.84 meet(top, X)
% 0.20/0.84 = { by lemma 23 R->L }
% 0.20/0.84 meet(X, top)
% 0.20/0.84
% 0.20/0.84 Lemma 25: join(X, join(complement(X), Y)) = top.
% 0.20/0.84 Proof:
% 0.20/0.84 join(X, join(complement(X), Y))
% 0.20/0.84 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 0.20/0.84 join(X, join(Y, complement(X)))
% 0.20/0.84 = { by lemma 14 }
% 0.20/0.84 join(Y, top)
% 0.20/0.84 = { by lemma 18 R->L }
% 0.20/0.84 join(Z, top)
% 0.20/0.84 = { by lemma 19 }
% 0.20/0.84 top
% 0.20/0.84
% 0.20/0.84 Lemma 26: join(X, complement(zero)) = top.
% 0.20/0.84 Proof:
% 0.20/0.84 join(X, complement(zero))
% 0.20/0.84 = { by lemma 21 R->L }
% 0.20/0.84 join(join(zero, meet(X, X)), complement(zero))
% 0.20/0.84 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 0.20/0.84 join(zero, join(meet(X, X), complement(zero)))
% 0.20/0.84 = { by axiom 3 (maddux1_join_commutativity_1) }
% 0.20/0.84 join(zero, join(complement(zero), meet(X, X)))
% 0.20/0.84 = { by lemma 25 }
% 0.20/0.84 top
% 0.20/0.84
% 0.20/0.84 Lemma 27: join(meet(X, Y), meet(X, complement(Y))) = X.
% 0.20/0.84 Proof:
% 0.20/0.84 join(meet(X, Y), meet(X, complement(Y)))
% 0.20/0.84 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 0.20/0.84 join(meet(X, complement(Y)), meet(X, Y))
% 0.20/0.84 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 0.20/0.84 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 0.20/0.84 = { by lemma 20 }
% 0.20/0.84 X
% 0.20/0.84
% 0.20/0.84 Lemma 28: join(zero, meet(X, top)) = X.
% 0.20/0.84 Proof:
% 0.20/0.84 join(zero, meet(X, top))
% 0.20/0.84 = { by lemma 26 R->L }
% 0.20/0.84 join(zero, meet(X, join(complement(zero), complement(zero))))
% 0.20/0.84 = { by lemma 16 }
% 0.20/0.84 join(zero, meet(X, complement(zero)))
% 0.20/0.84 = { by lemma 13 R->L }
% 0.20/0.84 join(complement(top), meet(X, complement(zero)))
% 0.20/0.84 = { by lemma 26 R->L }
% 0.20/0.84 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero)))
% 0.20/0.84 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 0.20/0.84 join(meet(X, zero), meet(X, complement(zero)))
% 0.20/0.84 = { by lemma 27 }
% 0.20/0.84 X
% 0.20/0.84
% 0.20/0.84 Lemma 29: join(zero, complement(X)) = complement(X).
% 0.20/0.84 Proof:
% 0.20/0.84 join(zero, complement(X))
% 0.20/0.84 = { by lemma 21 R->L }
% 0.20/0.84 join(zero, complement(join(zero, meet(X, X))))
% 0.20/0.84 = { by lemma 22 R->L }
% 0.20/0.84 join(zero, complement(join(zero, complement(complement(X)))))
% 0.20/0.84 = { by lemma 24 }
% 0.20/0.84 join(zero, meet(complement(X), top))
% 0.20/0.84 = { by lemma 28 }
% 0.20/0.84 complement(X)
% 0.20/0.84
% 0.20/0.84 Lemma 30: complement(complement(X)) = X.
% 0.20/0.84 Proof:
% 0.20/0.84 complement(complement(X))
% 0.20/0.84 = { by lemma 29 R->L }
% 0.20/0.84 join(zero, complement(complement(X)))
% 0.20/0.84 = { by lemma 22 }
% 0.20/0.84 join(zero, meet(X, X))
% 0.20/0.84 = { by lemma 21 }
% 0.20/0.84 X
% 0.20/0.84
% 0.20/0.84 Lemma 31: join(X, zero) = X.
% 0.20/0.84 Proof:
% 0.20/0.84 join(X, zero)
% 0.20/0.84 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 0.20/0.84 join(zero, X)
% 0.20/0.84 = { by lemma 30 R->L }
% 0.20/0.84 join(zero, complement(complement(X)))
% 0.20/0.84 = { by lemma 22 }
% 0.20/0.84 join(zero, meet(X, X))
% 0.20/0.84 = { by lemma 21 }
% 0.20/0.84 X
% 0.20/0.84
% 0.20/0.84 Lemma 32: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 0.20/0.84 Proof:
% 0.20/0.84 converse(join(converse(X), Y))
% 0.20/0.84 = { by axiom 8 (converse_additivity_9) }
% 0.20/0.84 join(converse(converse(X)), converse(Y))
% 0.20/0.84 = { by axiom 2 (converse_idempotence_8) }
% 0.20/0.84 join(X, converse(Y))
% 0.20/0.84
% 0.20/0.84 Lemma 33: meet(X, converse(complement(converse(complement(X))))) = X.
% 0.20/0.84 Proof:
% 0.20/0.84 meet(X, converse(complement(converse(complement(X)))))
% 0.20/0.84 = { by lemma 16 R->L }
% 0.20/0.84 meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X))))))
% 0.20/0.84 = { by lemma 31 R->L }
% 0.20/0.84 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), zero)
% 0.20/0.84 = { by lemma 13 R->L }
% 0.20/0.84 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(top))
% 0.20/0.84 = { by lemma 25 R->L }
% 0.20/0.84 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(join(converse(Y), join(complement(converse(Y)), converse(complement(converse(complement(converse(Y)))))))))
% 0.20/0.84 = { by lemma 32 R->L }
% 0.20/0.84 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(join(converse(Y), converse(join(converse(complement(converse(Y))), complement(converse(complement(converse(Y)))))))))
% 0.20/0.84 = { by axiom 5 (def_top_12) R->L }
% 0.20/0.84 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(join(converse(Y), converse(top))))
% 0.20/0.84 = { by axiom 8 (converse_additivity_9) R->L }
% 0.20/0.84 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(converse(join(Y, top))))
% 0.20/0.84 = { by lemma 19 }
% 0.20/0.84 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(converse(top)))
% 0.20/0.84 = { by lemma 25 R->L }
% 0.20/0.84 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(converse(join(converse(complement(X)), join(complement(converse(complement(X))), complement(converse(complement(X))))))))
% 0.20/0.84 = { by lemma 32 }
% 0.20/0.84 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(join(complement(X), converse(join(complement(converse(complement(X))), complement(converse(complement(X))))))))
% 0.20/0.84 = { by lemma 20 }
% 0.20/0.84 X
% 0.20/0.84
% 0.20/0.84 Lemma 34: meet(X, top) = X.
% 0.20/0.84 Proof:
% 0.20/0.84 meet(X, top)
% 0.20/0.84 = { by lemma 24 R->L }
% 0.20/0.84 complement(join(zero, complement(X)))
% 0.20/0.84 = { by lemma 29 R->L }
% 0.20/0.84 join(zero, complement(join(zero, complement(X))))
% 0.20/0.84 = { by lemma 24 }
% 0.20/0.84 join(zero, meet(X, top))
% 0.20/0.84 = { by lemma 28 }
% 0.20/0.84 X
% 0.20/0.84
% 0.20/0.84 Lemma 35: meet(top, X) = X.
% 0.20/0.84 Proof:
% 0.20/0.84 meet(top, X)
% 0.20/0.84 = { by lemma 23 }
% 0.20/0.84 meet(X, top)
% 0.20/0.84 = { by lemma 34 }
% 0.20/0.84 X
% 0.20/0.84
% 0.20/0.84 Lemma 36: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 0.20/0.84 Proof:
% 0.20/0.84 complement(join(complement(X), meet(Y, Z)))
% 0.20/0.84 = { by lemma 23 }
% 0.20/0.84 complement(join(complement(X), meet(Z, Y)))
% 0.20/0.84 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 0.20/0.84 complement(join(meet(Z, Y), complement(X)))
% 0.20/0.84 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 0.20/0.84 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 0.20/0.84 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 0.20/0.84 meet(join(complement(Z), complement(Y)), X)
% 0.20/0.84 = { by lemma 23 R->L }
% 0.20/0.84 meet(X, join(complement(Z), complement(Y)))
% 0.20/0.84 = { by axiom 3 (maddux1_join_commutativity_1) }
% 0.20/0.84 meet(X, join(complement(Y), complement(Z)))
% 0.20/0.84
% 0.20/0.84 Lemma 37: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 0.20/0.85 Proof:
% 0.20/0.85 join(complement(X), complement(Y))
% 0.20/0.85 = { by lemma 35 R->L }
% 0.20/0.85 meet(top, join(complement(X), complement(Y)))
% 0.20/0.85 = { by lemma 36 R->L }
% 0.20/0.85 complement(join(complement(top), meet(X, Y)))
% 0.20/0.85 = { by lemma 13 }
% 0.20/0.85 complement(join(zero, meet(X, Y)))
% 0.20/0.85 = { by lemma 23 R->L }
% 0.20/0.85 complement(join(zero, meet(Y, X)))
% 0.20/0.85 = { by axiom 3 (maddux1_join_commutativity_1) }
% 0.20/0.85 complement(join(meet(Y, X), zero))
% 0.20/0.85 = { by lemma 31 }
% 0.20/0.85 complement(meet(Y, X))
% 0.20/0.85 = { by lemma 23 R->L }
% 0.20/0.85 complement(meet(X, Y))
% 0.20/0.85
% 0.20/0.85 Lemma 38: join(complement(converse(X)), converse(join(X, Y))) = top.
% 0.20/0.85 Proof:
% 0.20/0.85 join(complement(converse(X)), converse(join(X, Y)))
% 0.20/0.85 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 0.20/0.85 join(converse(join(X, Y)), complement(converse(X)))
% 0.20/0.85 = { by axiom 8 (converse_additivity_9) }
% 0.20/0.85 join(join(converse(X), converse(Y)), complement(converse(X)))
% 0.20/0.85 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 0.20/0.85 join(converse(X), join(converse(Y), complement(converse(X))))
% 0.20/0.85 = { by axiom 3 (maddux1_join_commutativity_1) }
% 0.20/0.85 join(converse(X), join(complement(converse(X)), converse(Y)))
% 0.20/0.85 = { by lemma 25 }
% 0.20/0.85 top
% 0.20/0.85
% 0.20/0.85 Lemma 39: meet(X, X) = X.
% 0.20/0.85 Proof:
% 0.20/0.85 meet(X, X)
% 0.20/0.85 = { by lemma 22 R->L }
% 0.20/0.85 complement(complement(X))
% 0.20/0.85 = { by lemma 30 }
% 0.20/0.85 X
% 0.20/0.85
% 0.20/0.85 Lemma 40: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 0.20/0.85 Proof:
% 0.20/0.85 complement(meet(X, complement(Y)))
% 0.20/0.85 = { by lemma 23 }
% 0.20/0.85 complement(meet(complement(Y), X))
% 0.20/0.85 = { by lemma 29 R->L }
% 0.20/0.85 complement(meet(join(zero, complement(Y)), X))
% 0.20/0.85 = { by lemma 37 R->L }
% 0.20/0.85 join(complement(join(zero, complement(Y))), complement(X))
% 0.20/0.85 = { by lemma 24 }
% 0.20/0.85 join(meet(Y, top), complement(X))
% 0.20/0.85 = { by lemma 34 }
% 0.20/0.85 join(Y, complement(X))
% 0.20/0.85
% 0.20/0.85 Lemma 41: meet(meet(X, Y), Z) = meet(X, meet(Z, Y)).
% 0.20/0.85 Proof:
% 0.20/0.85 meet(meet(X, Y), Z)
% 0.20/0.85 = { by lemma 23 }
% 0.20/0.85 meet(Z, meet(X, Y))
% 0.20/0.85 = { by lemma 39 R->L }
% 0.20/0.85 meet(meet(Z, meet(X, Y)), meet(Z, meet(X, Y)))
% 0.20/0.85 = { by lemma 22 R->L }
% 0.20/0.85 complement(complement(meet(Z, meet(X, Y))))
% 0.20/0.85 = { by lemma 23 }
% 0.20/0.85 complement(complement(meet(Z, meet(Y, X))))
% 0.20/0.85 = { by lemma 37 R->L }
% 0.20/0.85 complement(join(complement(Z), complement(meet(Y, X))))
% 0.20/0.85 = { by lemma 37 R->L }
% 0.20/0.85 complement(join(complement(Z), join(complement(Y), complement(X))))
% 0.20/0.85 = { by axiom 9 (maddux2_join_associativity_2) }
% 0.20/0.85 complement(join(join(complement(Z), complement(Y)), complement(X)))
% 0.20/0.85 = { by lemma 40 R->L }
% 0.20/0.85 complement(complement(meet(X, complement(join(complement(Z), complement(Y))))))
% 0.20/0.85 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 0.20/0.85 complement(complement(meet(X, meet(Z, Y))))
% 0.20/0.85 = { by lemma 23 R->L }
% 0.20/0.85 complement(complement(meet(X, meet(Y, Z))))
% 0.20/0.85 = { by lemma 30 }
% 0.20/0.85 meet(X, meet(Y, Z))
% 0.20/0.85 = { by lemma 23 R->L }
% 0.20/0.85 meet(X, meet(Z, Y))
% 0.20/0.85
% 0.20/0.85 Lemma 42: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 0.20/0.85 Proof:
% 0.20/0.85 join(meet(X, Y), meet(complement(X), Y))
% 0.20/0.85 = { by lemma 23 }
% 0.20/0.85 join(meet(X, Y), meet(Y, complement(X)))
% 0.20/0.85 = { by lemma 23 }
% 0.20/0.85 join(meet(Y, X), meet(Y, complement(X)))
% 0.20/0.85 = { by lemma 27 }
% 0.20/0.85 Y
% 0.20/0.85
% 0.20/0.85 Lemma 43: complement(converse(X)) = converse(complement(X)).
% 0.20/0.85 Proof:
% 0.20/0.85 complement(converse(X))
% 0.20/0.85 = { by lemma 33 R->L }
% 0.20/0.85 complement(converse(meet(X, converse(complement(converse(complement(X)))))))
% 0.20/0.85 = { by lemma 31 R->L }
% 0.20/0.85 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), zero)))
% 0.20/0.85 = { by axiom 4 (def_zero_13) }
% 0.20/0.85 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), complement(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))))))))
% 0.20/0.85 = { by lemma 23 }
% 0.20/0.85 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), complement(meet(converse(complement(converse(complement(X)))), converse(converse(complement(X)))))))))
% 0.20/0.85 = { by lemma 37 R->L }
% 0.20/0.85 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(complement(converse(complement(X))))), complement(converse(converse(complement(X)))))))))
% 0.20/0.85 = { by lemma 33 R->L }
% 0.20/0.85 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(complement(converse(complement(X))))), complement(converse(meet(converse(complement(X)), converse(complement(converse(complement(converse(complement(X))))))))))))))
% 0.20/0.85 = { by lemma 23 }
% 0.20/0.85 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(complement(converse(complement(X))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))))))))
% 0.20/0.85 = { by axiom 2 (converse_idempotence_8) R->L }
% 0.20/0.85 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(converse(converse(complement(converse(complement(converse(complement(X))))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))))))))
% 0.20/0.85 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 0.20/0.85 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))), converse(converse(complement(converse(complement(converse(complement(X))))))))))))
% 0.20/0.85 = { by lemma 20 R->L }
% 0.20/0.85 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))), converse(join(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))), complement(join(complement(converse(complement(converse(complement(converse(complement(X))))))), converse(complement(X)))))))))))
% 0.20/0.85 = { by lemma 38 }
% 0.20/0.85 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), top))))
% 0.20/0.85 = { by lemma 41 }
% 0.20/0.85 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(converse(converse(complement(X))), meet(top, converse(complement(converse(complement(X)))))))))
% 0.20/0.85 = { by lemma 35 }
% 0.20/0.85 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))))))
% 0.20/0.85 = { by axiom 2 (converse_idempotence_8) }
% 0.20/0.85 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(complement(converse(complement(X))))))))
% 0.20/0.85 = { by lemma 42 }
% 0.20/0.85 complement(converse(converse(complement(converse(complement(X))))))
% 0.20/0.85 = { by axiom 2 (converse_idempotence_8) }
% 0.20/0.85 complement(complement(converse(complement(X))))
% 0.20/0.85 = { by lemma 22 }
% 0.20/0.85 meet(converse(complement(X)), converse(complement(X)))
% 0.20/0.85 = { by lemma 39 }
% 0.20/0.85 converse(complement(X))
% 0.20/0.85
% 0.20/0.85 Lemma 44: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 0.20/0.85 Proof:
% 0.20/0.85 complement(join(X, complement(Y)))
% 0.20/0.85 = { by lemma 40 R->L }
% 0.20/0.85 complement(complement(meet(Y, complement(X))))
% 0.20/0.85 = { by lemma 22 }
% 0.20/0.85 meet(meet(Y, complement(X)), meet(Y, complement(X)))
% 0.20/0.85 = { by lemma 39 }
% 0.20/0.85 meet(Y, complement(X))
% 0.20/0.85
% 0.20/0.85 Lemma 45: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 0.20/0.85 Proof:
% 0.20/0.85 complement(join(complement(X), Y))
% 0.20/0.85 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 0.20/0.85 complement(join(Y, complement(X)))
% 0.20/0.85 = { by lemma 44 }
% 0.20/0.85 meet(X, complement(Y))
% 0.20/0.85
% 0.20/0.85 Lemma 46: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 0.20/0.85 Proof:
% 0.20/0.85 complement(meet(complement(X), Y))
% 0.20/0.85 = { by lemma 23 }
% 0.20/0.85 complement(meet(Y, complement(X)))
% 0.20/0.85 = { by lemma 40 }
% 0.20/0.85 join(X, complement(Y))
% 0.20/0.85
% 0.20/0.85 Lemma 47: meet(X, join(X, complement(Y))) = X.
% 0.20/0.85 Proof:
% 0.20/0.85 meet(X, join(X, complement(Y)))
% 0.20/0.85 = { by lemma 40 R->L }
% 0.20/0.85 meet(X, complement(meet(Y, complement(X))))
% 0.20/0.85 = { by lemma 37 R->L }
% 0.20/0.85 meet(X, join(complement(Y), complement(complement(X))))
% 0.20/0.85 = { by lemma 36 R->L }
% 0.20/0.85 complement(join(complement(X), meet(Y, complement(X))))
% 0.20/0.85 = { by lemma 29 R->L }
% 0.20/0.85 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 0.20/0.85 = { by lemma 13 R->L }
% 0.20/0.85 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 0.20/0.85 = { by lemma 19 R->L }
% 0.20/0.85 join(complement(join(complement(Y), top)), complement(join(complement(X), meet(Y, complement(X)))))
% 0.20/0.85 = { by lemma 14 R->L }
% 0.20/0.85 join(complement(join(complement(X), join(complement(Y), complement(complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 0.20/0.85 = { by lemma 37 }
% 0.20/0.85 join(complement(join(complement(X), complement(meet(Y, complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 0.20/0.85 = { by lemma 23 R->L }
% 0.20/0.85 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 0.20/0.85 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 0.20/0.85 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 0.20/0.85 = { by lemma 23 R->L }
% 0.20/0.85 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 0.20/0.85 = { by lemma 20 }
% 0.20/0.85 X
% 0.20/0.85
% 0.20/0.85 Lemma 48: join(X, join(Y, complement(join(X, Y)))) = top.
% 0.20/0.85 Proof:
% 0.20/0.85 join(X, join(Y, complement(join(X, Y))))
% 0.20/0.85 = { by axiom 9 (maddux2_join_associativity_2) }
% 0.20/0.85 join(join(X, Y), complement(join(X, Y)))
% 0.20/0.85 = { by axiom 5 (def_top_12) R->L }
% 0.20/0.86 top
% 0.20/0.86
% 0.20/0.86 Goal 1 (goals_14): converse(meet(sk1, sk2)) = meet(converse(sk1), converse(sk2)).
% 0.20/0.86 Proof:
% 0.20/0.86 converse(meet(sk1, sk2))
% 0.20/0.86 = { by lemma 23 }
% 0.20/0.86 converse(meet(sk2, sk1))
% 0.20/0.86 = { by lemma 30 R->L }
% 0.20/0.86 converse(meet(sk2, complement(complement(sk1))))
% 0.20/0.86 = { by lemma 44 R->L }
% 0.20/0.86 converse(complement(join(complement(sk1), complement(sk2))))
% 0.20/0.86 = { by lemma 43 R->L }
% 0.20/0.86 complement(converse(join(complement(sk1), complement(sk2))))
% 0.20/0.86 = { by axiom 8 (converse_additivity_9) }
% 0.20/0.86 complement(join(converse(complement(sk1)), converse(complement(sk2))))
% 0.20/0.86 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 0.20/0.86 complement(join(converse(complement(sk2)), converse(complement(sk1))))
% 0.20/0.86 = { by lemma 39 R->L }
% 0.20/0.86 complement(join(converse(complement(sk2)), meet(converse(complement(sk1)), converse(complement(sk1)))))
% 0.20/0.86 = { by lemma 22 R->L }
% 0.20/0.86 complement(join(converse(complement(sk2)), complement(complement(converse(complement(sk1))))))
% 0.20/0.86 = { by lemma 44 }
% 0.20/0.86 meet(complement(converse(complement(sk1))), complement(converse(complement(sk2))))
% 0.20/0.86 = { by lemma 23 R->L }
% 0.20/0.86 meet(complement(converse(complement(sk2))), complement(converse(complement(sk1))))
% 0.20/0.86 = { by lemma 45 R->L }
% 0.20/0.86 complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))
% 0.20/0.86 = { by lemma 20 R->L }
% 0.20/0.86 join(meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(converse(join(complement(sk2), X)), complement(join(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), meet(complement(converse(complement(sk2))), converse(complement(sk1)))))))), complement(join(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(converse(join(complement(sk2), X)), complement(join(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), meet(complement(converse(complement(sk2))), converse(complement(sk1))))))))))
% 0.20/0.86 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 0.20/0.86 join(meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(converse(join(complement(sk2), X)), complement(join(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), meet(complement(converse(complement(sk2))), converse(complement(sk1)))))))), complement(join(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(complement(join(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), meet(complement(converse(complement(sk2))), converse(complement(sk1))))), converse(join(complement(sk2), X)))))))
% 0.20/0.86 = { by axiom 9 (maddux2_join_associativity_2) }
% 0.20/0.86 join(meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(converse(join(complement(sk2), X)), complement(join(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), meet(complement(converse(complement(sk2))), converse(complement(sk1)))))))), complement(join(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), join(join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), complement(join(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), meet(complement(converse(complement(sk2))), converse(complement(sk1)))))), converse(join(complement(sk2), X))))))
% 0.20/0.86 = { by axiom 9 (maddux2_join_associativity_2) }
% 0.20/0.86 join(meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(converse(join(complement(sk2), X)), complement(join(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), meet(complement(converse(complement(sk2))), converse(complement(sk1)))))))), complement(join(join(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), complement(join(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), converse(join(complement(sk2), X)))))
% 0.20/0.86 = { by lemma 48 }
% 0.20/0.86 join(meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(converse(join(complement(sk2), X)), complement(join(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), meet(complement(converse(complement(sk2))), converse(complement(sk1)))))))), complement(join(top, converse(join(complement(sk2), X)))))
% 0.20/0.86 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 0.20/0.86 join(meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(converse(join(complement(sk2), X)), complement(join(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), meet(complement(converse(complement(sk2))), converse(complement(sk1)))))))), complement(join(converse(join(complement(sk2), X)), top)))
% 0.20/0.86 = { by lemma 19 }
% 0.20/0.86 join(meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(converse(join(complement(sk2), X)), complement(join(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), meet(complement(converse(complement(sk2))), converse(complement(sk1)))))))), complement(top))
% 0.20/0.86 = { by lemma 45 }
% 0.20/0.86 join(meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(converse(join(complement(sk2), X)), meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))))))), complement(top))
% 0.20/0.86 = { by lemma 13 }
% 0.20/0.86 join(meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(converse(join(complement(sk2), X)), meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))))))), zero)
% 0.20/0.86 = { by lemma 31 }
% 0.20/0.86 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(converse(join(complement(sk2), X)), meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))))
% 0.20/0.86 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 0.20/0.86 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))), converse(join(complement(sk2), X)))))
% 0.20/0.86 = { by axiom 9 (maddux2_join_associativity_2) }
% 0.20/0.86 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))))), converse(join(complement(sk2), X))))
% 0.20/0.86 = { by axiom 3 (maddux1_join_commutativity_1) }
% 0.20/0.86 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))))
% 0.20/0.86 = { by lemma 23 }
% 0.20/0.86 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))))))
% 0.20/0.86 = { by lemma 47 R->L }
% 0.20/0.86 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), complement(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))))))), meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))))))
% 0.20/0.86 = { by lemma 22 }
% 0.20/0.86 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))))), meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))))))
% 0.20/0.86 = { by lemma 39 }
% 0.20/0.86 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))))), meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))))))
% 0.20/0.86 = { by axiom 3 (maddux1_join_commutativity_1) }
% 0.20/0.86 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), meet(complement(converse(complement(sk2))), converse(complement(sk1))))), meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))))))
% 0.20/0.87 = { by lemma 30 R->L }
% 0.20/0.87 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))))))
% 0.20/0.87 = { by lemma 47 R->L }
% 0.20/0.87 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))))))))))
% 0.20/0.87 = { by lemma 46 R->L }
% 0.20/0.87 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(meet(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))))))))))
% 0.20/0.87 = { by lemma 41 R->L }
% 0.20/0.87 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), meet(meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(meet(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))))))
% 0.20/0.87 = { by lemma 31 R->L }
% 0.20/0.87 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), join(meet(meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(meet(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), zero))))
% 0.20/0.87 = { by lemma 13 R->L }
% 0.20/0.87 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), join(meet(meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(meet(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), complement(top)))))
% 0.20/0.87 = { by lemma 48 R->L }
% 0.20/0.87 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), join(meet(meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(meet(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), complement(join(complement(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))))), join(complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))), complement(join(complement(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))))))))))
% 0.20/0.87 = { by lemma 37 }
% 0.20/0.87 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), join(meet(meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(meet(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), complement(join(complement(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))))), complement(meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), join(complement(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))))))))))
% 0.20/0.87 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 0.20/0.87 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), join(meet(meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(meet(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), meet(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), join(complement(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))))))))
% 0.20/0.87 = { by lemma 37 }
% 0.20/0.87 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), join(meet(meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(meet(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), meet(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(meet(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))))))))
% 0.20/0.87 = { by lemma 23 }
% 0.20/0.88 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), join(meet(meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(meet(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), meet(meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(meet(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))))))))
% 0.20/0.88 = { by lemma 27 }
% 0.20/0.88 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), complement(meet(complement(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1))))), complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))))))
% 0.20/0.88 = { by lemma 46 }
% 0.20/0.88 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(meet(complement(converse(complement(sk2))), converse(complement(sk1))), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))), meet(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1)))), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))))))
% 0.20/0.88 = { by lemma 42 }
% 0.20/0.88 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), complement(complement(meet(complement(converse(complement(sk2))), converse(complement(sk1))))))))
% 0.20/0.88 = { by lemma 30 }
% 0.20/0.88 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), meet(complement(converse(complement(sk2))), converse(complement(sk1))))))
% 0.20/0.88 = { by axiom 3 (maddux1_join_commutativity_1) }
% 0.20/0.88 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), join(meet(complement(converse(complement(sk2))), converse(complement(sk1))), complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))))))
% 0.20/0.88 = { by lemma 20 }
% 0.20/0.88 meet(complement(join(complement(complement(converse(complement(sk2)))), converse(complement(sk1)))), join(converse(join(complement(sk2), X)), complement(converse(complement(sk2)))))
% 0.20/0.88 = { by lemma 45 }
% 0.20/0.88 meet(meet(complement(converse(complement(sk2))), complement(converse(complement(sk1)))), join(converse(join(complement(sk2), X)), complement(converse(complement(sk2)))))
% 0.20/0.88 = { by lemma 41 }
% 0.20/0.88 meet(complement(converse(complement(sk2))), meet(join(converse(join(complement(sk2), X)), complement(converse(complement(sk2)))), complement(converse(complement(sk1)))))
% 0.20/0.88 = { by lemma 23 R->L }
% 0.20/0.88 meet(complement(converse(complement(sk2))), meet(complement(converse(complement(sk1))), join(converse(join(complement(sk2), X)), complement(converse(complement(sk2))))))
% 0.20/0.88 = { by axiom 3 (maddux1_join_commutativity_1) }
% 0.20/0.88 meet(complement(converse(complement(sk2))), meet(complement(converse(complement(sk1))), join(complement(converse(complement(sk2))), converse(join(complement(sk2), X)))))
% 0.20/0.88 = { by lemma 38 }
% 0.20/0.88 meet(complement(converse(complement(sk2))), meet(complement(converse(complement(sk1))), top))
% 0.20/0.88 = { by lemma 43 }
% 0.20/0.88 meet(converse(complement(complement(sk2))), meet(complement(converse(complement(sk1))), top))
% 0.20/0.88 = { by lemma 34 }
% 0.20/0.88 meet(converse(complement(complement(sk2))), complement(converse(complement(sk1))))
% 0.20/0.88 = { by lemma 23 R->L }
% 0.20/0.88 meet(complement(converse(complement(sk1))), converse(complement(complement(sk2))))
% 0.20/0.88 = { by lemma 22 }
% 0.20/0.88 meet(complement(converse(complement(sk1))), converse(meet(sk2, sk2)))
% 0.20/0.88 = { by lemma 39 }
% 0.20/0.88 meet(complement(converse(complement(sk1))), converse(sk2))
% 0.20/0.88 = { by lemma 23 R->L }
% 0.20/0.88 meet(converse(sk2), complement(converse(complement(sk1))))
% 0.20/0.88 = { by lemma 43 }
% 0.20/0.88 meet(converse(sk2), converse(complement(complement(sk1))))
% 0.20/0.88 = { by lemma 22 }
% 0.20/0.88 meet(converse(sk2), converse(meet(sk1, sk1)))
% 0.20/0.88 = { by lemma 39 }
% 0.20/0.88 meet(converse(sk2), converse(sk1))
% 0.20/0.88 = { by lemma 23 R->L }
% 0.20/0.88 meet(converse(sk1), converse(sk2))
% 0.20/0.88 % SZS output end Proof
% 0.20/0.88
% 0.20/0.88 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------