TSTP Solution File: REL033-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL033-1 : 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 : n023.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 13:44:17 EDT 2023
% Result : Unsatisfiable 58.07s 7.72s
% Output : Proof 59.09s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : REL033-1 : TPTP v8.1.2. Released v4.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n023.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 23:04:11 EDT 2023
% 0.13/0.34 % CPUTime :
% 58.07/7.72 Command-line arguments: --ground-connectedness --complete-subsets
% 58.07/7.72
% 58.07/7.72 % SZS status Unsatisfiable
% 58.07/7.72
% 59.09/7.78 % SZS output start Proof
% 59.09/7.78 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 59.09/7.78 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 59.09/7.78 Axiom 3 (composition_identity_6): composition(X, one) = X.
% 59.09/7.78 Axiom 4 (goals_14): composition(sk1, top) = sk1.
% 59.09/7.78 Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 59.09/7.78 Axiom 6 (def_top_12): top = join(X, complement(X)).
% 59.09/7.78 Axiom 7 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 59.09/7.78 Axiom 8 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 59.09/7.78 Axiom 9 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 59.09/7.78 Axiom 10 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 59.09/7.78 Axiom 11 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 59.09/7.78 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 59.09/7.78 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 59.09/7.78 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 59.09/7.78
% 59.09/7.78 Lemma 15: complement(top) = zero.
% 59.09/7.78 Proof:
% 59.09/7.78 complement(top)
% 59.09/7.78 = { by axiom 6 (def_top_12) }
% 59.09/7.78 complement(join(complement(X), complement(complement(X))))
% 59.09/7.78 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 59.09/7.78 meet(X, complement(X))
% 59.09/7.78 = { by axiom 5 (def_zero_13) R->L }
% 59.09/7.78 zero
% 59.09/7.78
% 59.09/7.78 Lemma 16: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 59.09/7.78 Proof:
% 59.09/7.78 converse(composition(converse(X), Y))
% 59.09/7.78 = { by axiom 9 (converse_multiplicativity_10) }
% 59.09/7.78 composition(converse(Y), converse(converse(X)))
% 59.09/7.78 = { by axiom 1 (converse_idempotence_8) }
% 59.09/7.78 composition(converse(Y), X)
% 59.09/7.78
% 59.09/7.78 Lemma 17: composition(converse(one), X) = X.
% 59.09/7.78 Proof:
% 59.09/7.78 composition(converse(one), X)
% 59.09/7.78 = { by lemma 16 R->L }
% 59.09/7.78 converse(composition(converse(X), one))
% 59.09/7.78 = { by axiom 3 (composition_identity_6) }
% 59.09/7.78 converse(converse(X))
% 59.09/7.78 = { by axiom 1 (converse_idempotence_8) }
% 59.09/7.78 X
% 59.09/7.78
% 59.09/7.78 Lemma 18: composition(one, X) = X.
% 59.09/7.78 Proof:
% 59.09/7.78 composition(one, X)
% 59.09/7.78 = { by lemma 17 R->L }
% 59.09/7.78 composition(converse(one), composition(one, X))
% 59.09/7.78 = { by axiom 10 (composition_associativity_5) }
% 59.09/7.78 composition(composition(converse(one), one), X)
% 59.09/7.78 = { by axiom 3 (composition_identity_6) }
% 59.09/7.78 composition(converse(one), X)
% 59.09/7.78 = { by lemma 17 }
% 59.09/7.78 X
% 59.09/7.78
% 59.09/7.78 Lemma 19: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 59.09/7.78 Proof:
% 59.09/7.78 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 59.09/7.78 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.78 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 59.09/7.78 = { by axiom 13 (converse_cancellativity_11) }
% 59.09/7.78 complement(X)
% 59.09/7.78
% 59.09/7.78 Lemma 20: join(complement(X), complement(X)) = complement(X).
% 59.09/7.78 Proof:
% 59.09/7.78 join(complement(X), complement(X))
% 59.09/7.78 = { by lemma 17 R->L }
% 59.09/7.78 join(complement(X), composition(converse(one), complement(X)))
% 59.09/7.78 = { by lemma 18 R->L }
% 59.09/7.78 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 59.09/7.78 = { by lemma 19 }
% 59.09/7.78 complement(X)
% 59.09/7.78
% 59.09/7.78 Lemma 21: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 59.09/7.78 Proof:
% 59.09/7.78 join(meet(X, Y), complement(join(complement(X), Y)))
% 59.09/7.78 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 59.09/7.78 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 59.09/7.78 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 59.09/7.78 X
% 59.09/7.78
% 59.09/7.78 Lemma 22: join(zero, meet(X, X)) = X.
% 59.09/7.78 Proof:
% 59.09/7.79 join(zero, meet(X, X))
% 59.09/7.79 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 59.09/7.79 join(zero, complement(join(complement(X), complement(X))))
% 59.09/7.79 = { by axiom 5 (def_zero_13) }
% 59.09/7.79 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 59.09/7.79 = { by lemma 21 }
% 59.09/7.79 X
% 59.09/7.79
% 59.09/7.79 Lemma 23: join(zero, join(X, meet(Y, Y))) = join(X, Y).
% 59.09/7.79 Proof:
% 59.09/7.79 join(zero, join(X, meet(Y, Y)))
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.79 join(zero, join(meet(Y, Y), X))
% 59.09/7.79 = { by axiom 8 (maddux2_join_associativity_2) }
% 59.09/7.79 join(join(zero, meet(Y, Y)), X)
% 59.09/7.79 = { by lemma 22 }
% 59.09/7.79 join(Y, X)
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) }
% 59.09/7.79 join(X, Y)
% 59.09/7.79
% 59.09/7.79 Lemma 24: join(X, join(Y, complement(X))) = join(Y, top).
% 59.09/7.79 Proof:
% 59.09/7.79 join(X, join(Y, complement(X)))
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.79 join(X, join(complement(X), Y))
% 59.09/7.79 = { by axiom 8 (maddux2_join_associativity_2) }
% 59.09/7.79 join(join(X, complement(X)), Y)
% 59.09/7.79 = { by axiom 6 (def_top_12) R->L }
% 59.09/7.79 join(top, Y)
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) }
% 59.09/7.79 join(Y, top)
% 59.09/7.79
% 59.09/7.79 Lemma 25: join(top, complement(X)) = top.
% 59.09/7.79 Proof:
% 59.09/7.79 join(top, complement(X))
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.79 join(complement(X), top)
% 59.09/7.79 = { by lemma 24 R->L }
% 59.09/7.79 join(X, join(complement(X), complement(X)))
% 59.09/7.79 = { by lemma 20 }
% 59.09/7.79 join(X, complement(X))
% 59.09/7.79 = { by axiom 6 (def_top_12) R->L }
% 59.09/7.79 top
% 59.09/7.79
% 59.09/7.79 Lemma 26: join(Y, top) = join(X, top).
% 59.09/7.79 Proof:
% 59.09/7.79 join(Y, top)
% 59.09/7.79 = { by lemma 25 R->L }
% 59.09/7.79 join(Y, join(top, complement(Y)))
% 59.09/7.79 = { by lemma 24 }
% 59.09/7.79 join(top, top)
% 59.09/7.79 = { by lemma 24 R->L }
% 59.09/7.79 join(X, join(top, complement(X)))
% 59.09/7.79 = { by lemma 25 }
% 59.09/7.79 join(X, top)
% 59.09/7.79
% 59.09/7.79 Lemma 27: join(X, top) = top.
% 59.09/7.79 Proof:
% 59.09/7.79 join(X, top)
% 59.09/7.79 = { by lemma 26 }
% 59.09/7.79 join(zero, top)
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.79 join(top, zero)
% 59.09/7.79 = { by lemma 15 R->L }
% 59.09/7.79 join(top, complement(top))
% 59.09/7.79 = { by axiom 6 (def_top_12) R->L }
% 59.09/7.79 top
% 59.09/7.79
% 59.09/7.79 Lemma 28: join(X, join(complement(X), Y)) = top.
% 59.09/7.79 Proof:
% 59.09/7.79 join(X, join(complement(X), Y))
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.79 join(X, join(Y, complement(X)))
% 59.09/7.79 = { by lemma 24 }
% 59.09/7.79 join(Y, top)
% 59.09/7.79 = { by lemma 26 R->L }
% 59.09/7.79 join(Z, top)
% 59.09/7.79 = { by lemma 27 }
% 59.09/7.79 top
% 59.09/7.79
% 59.09/7.79 Lemma 29: join(X, complement(zero)) = top.
% 59.09/7.79 Proof:
% 59.09/7.79 join(X, complement(zero))
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.79 join(complement(zero), X)
% 59.09/7.79 = { by lemma 23 R->L }
% 59.09/7.79 join(zero, join(complement(zero), meet(X, X)))
% 59.09/7.79 = { by lemma 28 }
% 59.09/7.79 top
% 59.09/7.79
% 59.09/7.79 Lemma 30: complement(zero) = top.
% 59.09/7.79 Proof:
% 59.09/7.79 complement(zero)
% 59.09/7.79 = { by lemma 20 R->L }
% 59.09/7.79 join(complement(zero), complement(zero))
% 59.09/7.79 = { by lemma 29 }
% 59.09/7.79 top
% 59.09/7.79
% 59.09/7.79 Lemma 31: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 59.09/7.79 Proof:
% 59.09/7.79 converse(join(X, converse(Y)))
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.79 converse(join(converse(Y), X))
% 59.09/7.79 = { by axiom 7 (converse_additivity_9) }
% 59.09/7.79 join(converse(converse(Y)), converse(X))
% 59.09/7.79 = { by axiom 1 (converse_idempotence_8) }
% 59.09/7.79 join(Y, converse(X))
% 59.09/7.79
% 59.09/7.79 Lemma 32: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 59.09/7.79 Proof:
% 59.09/7.79 converse(join(converse(X), Y))
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.79 converse(join(Y, converse(X)))
% 59.09/7.79 = { by lemma 31 }
% 59.09/7.79 join(X, converse(Y))
% 59.09/7.79
% 59.09/7.79 Lemma 33: join(X, converse(top)) = top.
% 59.09/7.79 Proof:
% 59.09/7.79 join(X, converse(top))
% 59.09/7.79 = { by axiom 6 (def_top_12) }
% 59.09/7.79 join(X, converse(join(converse(complement(X)), complement(converse(complement(X))))))
% 59.09/7.79 = { by lemma 32 }
% 59.09/7.79 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 59.09/7.79 = { by lemma 28 }
% 59.09/7.79 top
% 59.09/7.79
% 59.09/7.79 Lemma 34: converse(top) = top.
% 59.09/7.79 Proof:
% 59.09/7.79 converse(top)
% 59.09/7.79 = { by lemma 27 R->L }
% 59.09/7.79 converse(join(X, top))
% 59.09/7.79 = { by axiom 7 (converse_additivity_9) }
% 59.09/7.79 join(converse(X), converse(top))
% 59.09/7.79 = { by lemma 33 }
% 59.09/7.79 top
% 59.09/7.79
% 59.09/7.79 Lemma 35: complement(complement(X)) = meet(X, X).
% 59.09/7.79 Proof:
% 59.09/7.79 complement(complement(X))
% 59.09/7.79 = { by lemma 20 R->L }
% 59.09/7.79 complement(join(complement(X), complement(X)))
% 59.09/7.79 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 59.09/7.79 meet(X, X)
% 59.09/7.79
% 59.09/7.79 Lemma 36: meet(Y, X) = meet(X, Y).
% 59.09/7.79 Proof:
% 59.09/7.79 meet(Y, X)
% 59.09/7.79 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 59.09/7.79 complement(join(complement(Y), complement(X)))
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.79 complement(join(complement(X), complement(Y)))
% 59.09/7.79 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 59.09/7.79 meet(X, Y)
% 59.09/7.79
% 59.09/7.79 Lemma 37: complement(join(zero, complement(X))) = meet(X, top).
% 59.09/7.79 Proof:
% 59.09/7.79 complement(join(zero, complement(X)))
% 59.09/7.79 = { by lemma 15 R->L }
% 59.09/7.79 complement(join(complement(top), complement(X)))
% 59.09/7.79 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 59.09/7.79 meet(top, X)
% 59.09/7.79 = { by lemma 36 R->L }
% 59.09/7.79 meet(X, top)
% 59.09/7.79
% 59.09/7.79 Lemma 38: join(meet(X, Y), meet(X, complement(Y))) = X.
% 59.09/7.79 Proof:
% 59.09/7.79 join(meet(X, Y), meet(X, complement(Y)))
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.79 join(meet(X, complement(Y)), meet(X, Y))
% 59.09/7.79 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 59.09/7.79 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 59.09/7.79 = { by lemma 21 }
% 59.09/7.79 X
% 59.09/7.79
% 59.09/7.79 Lemma 39: join(zero, meet(X, top)) = X.
% 59.09/7.79 Proof:
% 59.09/7.79 join(zero, meet(X, top))
% 59.09/7.79 = { by lemma 30 R->L }
% 59.09/7.79 join(zero, meet(X, complement(zero)))
% 59.09/7.79 = { by lemma 15 R->L }
% 59.09/7.79 join(complement(top), meet(X, complement(zero)))
% 59.09/7.79 = { by lemma 29 R->L }
% 59.09/7.79 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero)))
% 59.09/7.79 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 59.09/7.79 join(meet(X, zero), meet(X, complement(zero)))
% 59.09/7.79 = { by lemma 38 }
% 59.09/7.79 X
% 59.09/7.79
% 59.09/7.79 Lemma 40: join(zero, complement(X)) = complement(X).
% 59.09/7.79 Proof:
% 59.09/7.79 join(zero, complement(X))
% 59.09/7.79 = { by lemma 22 R->L }
% 59.09/7.79 join(zero, complement(join(zero, meet(X, X))))
% 59.09/7.79 = { by lemma 35 R->L }
% 59.09/7.79 join(zero, complement(join(zero, complement(complement(X)))))
% 59.09/7.79 = { by lemma 37 }
% 59.09/7.79 join(zero, meet(complement(X), top))
% 59.09/7.79 = { by lemma 39 }
% 59.09/7.79 complement(X)
% 59.09/7.79
% 59.09/7.79 Lemma 41: complement(complement(X)) = X.
% 59.09/7.79 Proof:
% 59.09/7.79 complement(complement(X))
% 59.09/7.79 = { by lemma 40 R->L }
% 59.09/7.79 join(zero, complement(complement(X)))
% 59.09/7.79 = { by lemma 35 }
% 59.09/7.79 join(zero, meet(X, X))
% 59.09/7.79 = { by lemma 22 }
% 59.09/7.79 X
% 59.09/7.79
% 59.09/7.79 Lemma 42: join(X, zero) = X.
% 59.09/7.79 Proof:
% 59.09/7.79 join(X, zero)
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.79 join(zero, X)
% 59.09/7.79 = { by lemma 41 R->L }
% 59.09/7.79 join(zero, complement(complement(X)))
% 59.09/7.79 = { by lemma 35 }
% 59.09/7.79 join(zero, meet(X, X))
% 59.09/7.79 = { by lemma 22 }
% 59.09/7.79 X
% 59.09/7.79
% 59.09/7.79 Lemma 43: join(zero, X) = X.
% 59.09/7.79 Proof:
% 59.09/7.79 join(zero, X)
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.79 join(X, zero)
% 59.09/7.79 = { by lemma 42 }
% 59.09/7.79 X
% 59.09/7.79
% 59.09/7.79 Lemma 44: composition(converse(sk1), complement(sk1)) = zero.
% 59.09/7.79 Proof:
% 59.09/7.79 composition(converse(sk1), complement(sk1))
% 59.09/7.79 = { by lemma 43 R->L }
% 59.09/7.79 join(zero, composition(converse(sk1), complement(sk1)))
% 59.09/7.79 = { by lemma 15 R->L }
% 59.09/7.79 join(complement(top), composition(converse(sk1), complement(sk1)))
% 59.09/7.79 = { by axiom 4 (goals_14) R->L }
% 59.09/7.79 join(complement(top), composition(converse(sk1), complement(composition(sk1, top))))
% 59.09/7.79 = { by lemma 19 }
% 59.09/7.79 complement(top)
% 59.09/7.79 = { by lemma 15 }
% 59.09/7.79 zero
% 59.09/7.79
% 59.09/7.79 Lemma 45: join(top, X) = top.
% 59.09/7.79 Proof:
% 59.09/7.79 join(top, X)
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.79 join(X, top)
% 59.09/7.79 = { by lemma 26 R->L }
% 59.09/7.79 join(Y, top)
% 59.09/7.79 = { by lemma 27 }
% 59.09/7.79 top
% 59.09/7.79
% 59.09/7.79 Lemma 46: composition(join(one, Y), X) = join(X, composition(Y, X)).
% 59.09/7.79 Proof:
% 59.09/7.79 composition(join(one, Y), X)
% 59.09/7.79 = { by axiom 12 (composition_distributivity_7) }
% 59.09/7.79 join(composition(one, X), composition(Y, X))
% 59.09/7.79 = { by lemma 18 }
% 59.09/7.79 join(X, composition(Y, X))
% 59.09/7.79
% 59.09/7.79 Lemma 47: join(X, composition(top, X)) = composition(top, X).
% 59.09/7.79 Proof:
% 59.09/7.79 join(X, composition(top, X))
% 59.09/7.79 = { by lemma 34 R->L }
% 59.09/7.79 join(X, composition(converse(top), X))
% 59.09/7.79 = { by lemma 46 R->L }
% 59.09/7.79 composition(join(one, converse(top)), X)
% 59.09/7.79 = { by lemma 33 }
% 59.09/7.79 composition(top, X)
% 59.09/7.79
% 59.09/7.79 Lemma 48: composition(top, zero) = zero.
% 59.09/7.79 Proof:
% 59.09/7.79 composition(top, zero)
% 59.09/7.79 = { by lemma 34 R->L }
% 59.09/7.79 composition(converse(top), zero)
% 59.09/7.79 = { by lemma 43 R->L }
% 59.09/7.79 join(zero, composition(converse(top), zero))
% 59.09/7.79 = { by lemma 15 R->L }
% 59.09/7.79 join(complement(top), composition(converse(top), zero))
% 59.09/7.79 = { by lemma 15 R->L }
% 59.09/7.79 join(complement(top), composition(converse(top), complement(top)))
% 59.09/7.79 = { by lemma 45 R->L }
% 59.09/7.79 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 59.09/7.79 = { by lemma 47 }
% 59.09/7.79 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 59.09/7.79 = { by lemma 19 }
% 59.09/7.79 complement(top)
% 59.09/7.79 = { by lemma 15 }
% 59.09/7.79 zero
% 59.09/7.79
% 59.09/7.79 Lemma 49: composition(converse(complement(sk1)), composition(sk1, X)) = converse(zero).
% 59.09/7.79 Proof:
% 59.09/7.79 composition(converse(complement(sk1)), composition(sk1, X))
% 59.09/7.79 = { by lemma 16 R->L }
% 59.09/7.79 converse(composition(converse(composition(sk1, X)), complement(sk1)))
% 59.09/7.79 = { by axiom 9 (converse_multiplicativity_10) }
% 59.09/7.79 converse(composition(composition(converse(X), converse(sk1)), complement(sk1)))
% 59.09/7.79 = { by axiom 10 (composition_associativity_5) R->L }
% 59.09/7.79 converse(composition(converse(X), composition(converse(sk1), complement(sk1))))
% 59.09/7.79 = { by lemma 44 }
% 59.09/7.79 converse(composition(converse(X), zero))
% 59.09/7.79 = { by lemma 43 R->L }
% 59.09/7.79 converse(join(zero, composition(converse(X), zero)))
% 59.09/7.79 = { by lemma 48 R->L }
% 59.09/7.79 converse(join(composition(top, zero), composition(converse(X), zero)))
% 59.09/7.79 = { by axiom 12 (composition_distributivity_7) R->L }
% 59.09/7.79 converse(composition(join(top, converse(X)), zero))
% 59.09/7.79 = { by lemma 45 }
% 59.09/7.79 converse(composition(top, zero))
% 59.09/7.79 = { by lemma 48 }
% 59.09/7.79 converse(zero)
% 59.09/7.79
% 59.09/7.79 Lemma 50: join(converse(zero), X) = X.
% 59.09/7.79 Proof:
% 59.09/7.79 join(converse(zero), X)
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.79 join(X, converse(zero))
% 59.09/7.79 = { by lemma 49 R->L }
% 59.09/7.79 join(X, composition(converse(complement(sk1)), composition(sk1, composition(Y, X))))
% 59.09/7.79 = { by axiom 10 (composition_associativity_5) }
% 59.09/7.79 join(X, composition(converse(complement(sk1)), composition(composition(sk1, Y), X)))
% 59.09/7.79 = { by axiom 10 (composition_associativity_5) }
% 59.09/7.79 join(X, composition(composition(converse(complement(sk1)), composition(sk1, Y)), X))
% 59.09/7.79 = { by lemma 49 }
% 59.09/7.79 join(X, composition(converse(zero), X))
% 59.09/7.79 = { by lemma 46 R->L }
% 59.09/7.79 composition(join(one, converse(zero)), X)
% 59.09/7.79 = { by lemma 17 R->L }
% 59.09/7.79 composition(join(composition(converse(one), one), converse(zero)), X)
% 59.09/7.79 = { by axiom 3 (composition_identity_6) }
% 59.09/7.79 composition(join(converse(one), converse(zero)), X)
% 59.09/7.79 = { by axiom 7 (converse_additivity_9) R->L }
% 59.09/7.79 composition(converse(join(one, zero)), X)
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) }
% 59.09/7.79 composition(converse(join(zero, one)), X)
% 59.09/7.79 = { by lemma 43 }
% 59.09/7.79 composition(converse(one), X)
% 59.09/7.79 = { by lemma 17 }
% 59.09/7.79 X
% 59.09/7.79
% 59.09/7.79 Lemma 51: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 59.09/7.79 Proof:
% 59.09/7.79 complement(join(complement(X), meet(Y, Z)))
% 59.09/7.79 = { by lemma 36 }
% 59.09/7.79 complement(join(complement(X), meet(Z, Y)))
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.79 complement(join(meet(Z, Y), complement(X)))
% 59.09/7.79 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 59.09/7.79 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 59.09/7.79 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 59.09/7.79 meet(join(complement(Z), complement(Y)), X)
% 59.09/7.79 = { by lemma 36 R->L }
% 59.09/7.79 meet(X, join(complement(Z), complement(Y)))
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) }
% 59.09/7.79 meet(X, join(complement(Y), complement(Z)))
% 59.09/7.79
% 59.09/7.79 Lemma 52: meet(X, top) = X.
% 59.09/7.79 Proof:
% 59.09/7.79 meet(X, top)
% 59.09/7.79 = { by lemma 37 R->L }
% 59.09/7.79 complement(join(zero, complement(X)))
% 59.09/7.79 = { by lemma 40 R->L }
% 59.09/7.79 join(zero, complement(join(zero, complement(X))))
% 59.09/7.79 = { by lemma 37 }
% 59.09/7.79 join(zero, meet(X, top))
% 59.09/7.79 = { by lemma 39 }
% 59.09/7.79 X
% 59.09/7.79
% 59.09/7.79 Lemma 53: meet(top, X) = X.
% 59.09/7.79 Proof:
% 59.09/7.79 meet(top, X)
% 59.09/7.79 = { by lemma 36 }
% 59.09/7.79 meet(X, top)
% 59.09/7.79 = { by lemma 52 }
% 59.09/7.79 X
% 59.09/7.79
% 59.09/7.79 Lemma 54: complement(join(zero, meet(X, Y))) = join(complement(X), complement(Y)).
% 59.09/7.79 Proof:
% 59.09/7.79 complement(join(zero, meet(X, Y)))
% 59.09/7.79 = { by lemma 36 }
% 59.09/7.79 complement(join(zero, meet(Y, X)))
% 59.09/7.79 = { by lemma 15 R->L }
% 59.09/7.79 complement(join(complement(top), meet(Y, X)))
% 59.09/7.79 = { by lemma 51 }
% 59.09/7.79 meet(top, join(complement(Y), complement(X)))
% 59.09/7.79 = { by lemma 53 }
% 59.09/7.79 join(complement(Y), complement(X))
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) }
% 59.09/7.79 join(complement(X), complement(Y))
% 59.09/7.79
% 59.09/7.79 Lemma 55: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 59.09/7.79 Proof:
% 59.09/7.79 join(complement(X), complement(Y))
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.79 join(complement(Y), complement(X))
% 59.09/7.79 = { by lemma 54 R->L }
% 59.09/7.79 complement(join(zero, meet(Y, X)))
% 59.09/7.79 = { by axiom 2 (maddux1_join_commutativity_1) }
% 59.09/7.79 complement(join(meet(Y, X), zero))
% 59.09/7.79 = { by lemma 42 }
% 59.09/7.79 complement(meet(Y, X))
% 59.09/7.79 = { by lemma 36 R->L }
% 59.09/7.79 complement(meet(X, Y))
% 59.09/7.80
% 59.09/7.80 Lemma 56: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 59.09/7.80 Proof:
% 59.09/7.80 complement(meet(X, complement(Y)))
% 59.09/7.80 = { by lemma 36 }
% 59.09/7.80 complement(meet(complement(Y), X))
% 59.09/7.80 = { by lemma 40 R->L }
% 59.09/7.80 complement(meet(join(zero, complement(Y)), X))
% 59.09/7.80 = { by lemma 55 R->L }
% 59.09/7.80 join(complement(join(zero, complement(Y))), complement(X))
% 59.09/7.80 = { by lemma 37 }
% 59.09/7.80 join(meet(Y, top), complement(X))
% 59.09/7.80 = { by lemma 52 }
% 59.09/7.80 join(Y, complement(X))
% 59.09/7.80
% 59.09/7.80 Lemma 57: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 59.09/7.80 Proof:
% 59.09/7.80 complement(meet(complement(X), Y))
% 59.09/7.80 = { by lemma 36 }
% 59.09/7.80 complement(meet(Y, complement(X)))
% 59.09/7.80 = { by lemma 56 }
% 59.09/7.80 join(X, complement(Y))
% 59.09/7.80
% 59.09/7.80 Lemma 58: join(X, complement(meet(X, Y))) = top.
% 59.09/7.80 Proof:
% 59.09/7.80 join(X, complement(meet(X, Y)))
% 59.09/7.80 = { by lemma 36 }
% 59.09/7.80 join(X, complement(meet(Y, X)))
% 59.09/7.80 = { by lemma 55 R->L }
% 59.09/7.80 join(X, join(complement(Y), complement(X)))
% 59.09/7.80 = { by lemma 24 }
% 59.09/7.80 join(complement(Y), top)
% 59.09/7.80 = { by lemma 27 }
% 59.09/7.80 top
% 59.09/7.80
% 59.09/7.80 Lemma 59: meet(X, meet(Y, complement(X))) = zero.
% 59.09/7.80 Proof:
% 59.09/7.80 meet(X, meet(Y, complement(X)))
% 59.09/7.80 = { by lemma 36 }
% 59.09/7.80 meet(X, meet(complement(X), Y))
% 59.09/7.80 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 59.09/7.80 complement(join(complement(X), complement(meet(complement(X), Y))))
% 59.09/7.80 = { by lemma 58 }
% 59.09/7.80 complement(top)
% 59.09/7.80 = { by lemma 15 }
% 59.09/7.80 zero
% 59.09/7.80
% 59.09/7.80 Lemma 60: meet(X, join(X, complement(Y))) = X.
% 59.09/7.80 Proof:
% 59.09/7.80 meet(X, join(X, complement(Y)))
% 59.09/7.80 = { by lemma 56 R->L }
% 59.09/7.80 meet(X, complement(meet(Y, complement(X))))
% 59.09/7.80 = { by lemma 55 R->L }
% 59.09/7.80 meet(X, join(complement(Y), complement(complement(X))))
% 59.09/7.80 = { by lemma 51 R->L }
% 59.09/7.80 complement(join(complement(X), meet(Y, complement(X))))
% 59.09/7.80 = { by lemma 40 R->L }
% 59.09/7.80 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 59.09/7.80 = { by lemma 59 R->L }
% 59.09/7.80 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 59.09/7.80 = { by lemma 21 }
% 59.09/7.80 X
% 59.09/7.80
% 59.09/7.80 Lemma 61: converse(zero) = zero.
% 59.09/7.80 Proof:
% 59.09/7.80 converse(zero)
% 59.09/7.80 = { by lemma 50 R->L }
% 59.09/7.80 join(converse(zero), converse(zero))
% 59.09/7.80 = { by lemma 23 R->L }
% 59.09/7.80 join(zero, join(converse(zero), meet(converse(zero), converse(zero))))
% 59.09/7.80 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 59.09/7.80 join(zero, join(converse(zero), complement(join(complement(converse(zero)), complement(converse(zero))))))
% 59.09/7.80 = { by lemma 57 R->L }
% 59.09/7.80 join(zero, complement(meet(complement(converse(zero)), join(complement(converse(zero)), complement(converse(zero))))))
% 59.09/7.80 = { by lemma 60 }
% 59.09/7.80 join(zero, complement(complement(converse(zero))))
% 59.09/7.80 = { by lemma 41 }
% 59.09/7.80 join(zero, converse(zero))
% 59.09/7.80 = { by axiom 2 (maddux1_join_commutativity_1) }
% 59.09/7.80 join(converse(zero), zero)
% 59.09/7.80 = { by lemma 50 }
% 59.09/7.80 zero
% 59.09/7.80
% 59.09/7.80 Lemma 62: meet(X, X) = X.
% 59.09/7.80 Proof:
% 59.09/7.80 meet(X, X)
% 59.09/7.80 = { by lemma 35 R->L }
% 59.09/7.80 complement(complement(X))
% 59.09/7.80 = { by lemma 41 }
% 59.09/7.80 X
% 59.09/7.80
% 59.09/7.80 Lemma 63: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 59.09/7.80 Proof:
% 59.09/7.80 converse(composition(X, converse(Y)))
% 59.09/7.80 = { by axiom 9 (converse_multiplicativity_10) }
% 59.09/7.80 composition(converse(converse(Y)), converse(X))
% 59.09/7.80 = { by axiom 1 (converse_idempotence_8) }
% 59.09/7.80 composition(Y, converse(X))
% 59.09/7.80
% 59.09/7.80 Lemma 64: composition(complement(sk1), top) = complement(sk1).
% 59.09/7.80 Proof:
% 59.09/7.80 composition(complement(sk1), top)
% 59.09/7.80 = { by lemma 34 R->L }
% 59.09/7.80 composition(complement(sk1), converse(top))
% 59.09/7.80 = { by lemma 63 R->L }
% 59.09/7.80 converse(composition(top, converse(complement(sk1))))
% 59.09/7.80 = { by lemma 47 R->L }
% 59.09/7.80 converse(join(converse(complement(sk1)), composition(top, converse(complement(sk1)))))
% 59.09/7.80 = { by lemma 32 }
% 59.09/7.80 join(complement(sk1), converse(composition(top, converse(complement(sk1)))))
% 59.09/7.80 = { by lemma 63 }
% 59.09/7.80 join(complement(sk1), composition(complement(sk1), converse(top)))
% 59.09/7.80 = { by lemma 34 }
% 59.09/7.80 join(complement(sk1), composition(complement(sk1), top))
% 59.09/7.80 = { by lemma 30 R->L }
% 59.09/7.80 join(complement(sk1), composition(complement(sk1), complement(zero)))
% 59.09/7.80 = { by lemma 61 R->L }
% 59.09/7.80 join(complement(sk1), composition(complement(sk1), complement(converse(zero))))
% 59.09/7.80 = { by axiom 1 (converse_idempotence_8) R->L }
% 59.09/7.80 join(complement(sk1), composition(converse(converse(complement(sk1))), complement(converse(zero))))
% 59.09/7.80 = { by lemma 44 R->L }
% 59.09/7.80 join(complement(sk1), composition(converse(converse(complement(sk1))), complement(converse(composition(converse(sk1), complement(sk1))))))
% 59.09/7.80 = { by lemma 16 }
% 59.09/7.80 join(complement(sk1), composition(converse(converse(complement(sk1))), complement(composition(converse(complement(sk1)), sk1))))
% 59.09/7.80 = { by lemma 19 }
% 59.09/7.80 complement(sk1)
% 59.09/7.80
% 59.09/7.80 Lemma 65: join(meet(X, Y), meet(complement(Y), X)) = X.
% 59.09/7.80 Proof:
% 59.09/7.80 join(meet(X, Y), meet(complement(Y), X))
% 59.09/7.80 = { by lemma 36 }
% 59.09/7.80 join(meet(X, Y), meet(X, complement(Y)))
% 59.09/7.80 = { by lemma 38 }
% 59.09/7.80 X
% 59.09/7.80
% 59.09/7.80 Lemma 66: meet(X, join(Y, complement(X))) = meet(X, Y).
% 59.09/7.80 Proof:
% 59.09/7.80 meet(X, join(Y, complement(X)))
% 59.09/7.80 = { by lemma 57 R->L }
% 59.09/7.80 meet(X, complement(meet(complement(Y), X)))
% 59.09/7.80 = { by lemma 65 R->L }
% 59.09/7.80 join(meet(meet(X, complement(meet(complement(Y), X))), Y), meet(complement(Y), meet(X, complement(meet(complement(Y), X)))))
% 59.09/7.80 = { by lemma 55 R->L }
% 59.09/7.80 join(meet(meet(X, complement(meet(complement(Y), X))), Y), meet(complement(Y), meet(X, join(complement(complement(Y)), complement(X)))))
% 59.09/7.80 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 59.09/7.80 join(meet(meet(X, complement(meet(complement(Y), X))), Y), complement(join(complement(complement(Y)), complement(meet(X, join(complement(complement(Y)), complement(X)))))))
% 59.09/7.80 = { by lemma 55 R->L }
% 59.09/7.80 join(meet(meet(X, complement(meet(complement(Y), X))), Y), complement(join(complement(complement(Y)), join(complement(X), complement(join(complement(complement(Y)), complement(X)))))))
% 59.09/7.80 = { by axiom 8 (maddux2_join_associativity_2) }
% 59.09/7.80 join(meet(meet(X, complement(meet(complement(Y), X))), Y), complement(join(join(complement(complement(Y)), complement(X)), complement(join(complement(complement(Y)), complement(X))))))
% 59.09/7.80 = { by axiom 6 (def_top_12) R->L }
% 59.09/7.80 join(meet(meet(X, complement(meet(complement(Y), X))), Y), complement(top))
% 59.09/7.80 = { by lemma 15 }
% 59.09/7.80 join(meet(meet(X, complement(meet(complement(Y), X))), Y), zero)
% 59.09/7.80 = { by lemma 42 }
% 59.09/7.80 meet(meet(X, complement(meet(complement(Y), X))), Y)
% 59.09/7.80 = { by lemma 36 }
% 59.09/7.80 meet(Y, meet(X, complement(meet(complement(Y), X))))
% 59.09/7.80 = { by lemma 62 R->L }
% 59.09/7.80 meet(meet(Y, meet(X, complement(meet(complement(Y), X)))), meet(Y, meet(X, complement(meet(complement(Y), X)))))
% 59.09/7.80 = { by lemma 35 R->L }
% 59.09/7.80 complement(complement(meet(Y, meet(X, complement(meet(complement(Y), X))))))
% 59.09/7.80 = { by lemma 36 }
% 59.09/7.80 complement(complement(meet(Y, meet(complement(meet(complement(Y), X)), X))))
% 59.09/7.80 = { by lemma 55 R->L }
% 59.09/7.80 complement(join(complement(Y), complement(meet(complement(meet(complement(Y), X)), X))))
% 59.09/7.80 = { by lemma 55 R->L }
% 59.09/7.80 complement(join(complement(Y), join(complement(complement(meet(complement(Y), X))), complement(X))))
% 59.09/7.80 = { by axiom 8 (maddux2_join_associativity_2) }
% 59.09/7.80 complement(join(join(complement(Y), complement(complement(meet(complement(Y), X)))), complement(X)))
% 59.09/7.80 = { by lemma 56 R->L }
% 59.09/7.80 complement(complement(meet(X, complement(join(complement(Y), complement(complement(meet(complement(Y), X))))))))
% 59.09/7.80 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 59.09/7.80 complement(complement(meet(X, meet(Y, complement(meet(complement(Y), X))))))
% 59.09/7.80 = { by lemma 36 R->L }
% 59.09/7.80 complement(complement(meet(X, meet(complement(meet(complement(Y), X)), Y))))
% 59.09/7.80 = { by lemma 41 }
% 59.09/7.80 meet(X, meet(complement(meet(complement(Y), X)), Y))
% 59.09/7.80 = { by lemma 36 R->L }
% 59.09/7.80 meet(X, meet(Y, complement(meet(complement(Y), X))))
% 59.09/7.80 = { by lemma 57 }
% 59.09/7.80 meet(X, meet(Y, join(Y, complement(X))))
% 59.09/7.80 = { by lemma 60 }
% 59.09/7.80 meet(X, Y)
% 59.09/7.80
% 59.09/7.80 Lemma 67: join(meet(Y, X), complement(Y)) = join(X, complement(Y)).
% 59.09/7.80 Proof:
% 59.09/7.80 join(meet(Y, X), complement(Y))
% 59.09/7.80 = { by lemma 56 R->L }
% 59.09/7.80 complement(meet(Y, complement(meet(Y, X))))
% 59.09/7.80 = { by lemma 55 R->L }
% 59.09/7.80 join(complement(Y), complement(complement(meet(Y, X))))
% 59.09/7.80 = { by lemma 54 R->L }
% 59.09/7.80 complement(join(zero, meet(Y, complement(meet(Y, X)))))
% 59.09/7.80 = { by lemma 43 R->L }
% 59.09/7.80 complement(join(zero, meet(Y, complement(join(zero, meet(Y, X))))))
% 59.09/7.80 = { by lemma 54 }
% 59.09/7.80 complement(join(zero, meet(Y, join(complement(Y), complement(X)))))
% 59.09/7.80 = { by axiom 2 (maddux1_join_commutativity_1) }
% 59.09/7.80 complement(join(zero, meet(Y, join(complement(X), complement(Y)))))
% 59.09/7.80 = { by lemma 66 }
% 59.09/7.80 complement(join(zero, meet(Y, complement(X))))
% 59.09/7.80 = { by lemma 54 }
% 59.09/7.80 join(complement(Y), complement(complement(X)))
% 59.09/7.80 = { by lemma 55 }
% 59.09/7.80 complement(meet(Y, complement(X)))
% 59.09/7.80 = { by lemma 56 }
% 59.09/7.80 join(X, complement(Y))
% 59.09/7.80
% 59.09/7.80 Lemma 68: meet(complement(X), join(Y, X)) = meet(Y, complement(X)).
% 59.09/7.80 Proof:
% 59.09/7.80 meet(complement(X), join(Y, X))
% 59.09/7.80 = { by lemma 62 R->L }
% 59.09/7.80 meet(complement(X), join(Y, meet(X, X)))
% 59.09/7.80 = { by lemma 35 R->L }
% 59.09/7.80 meet(complement(X), join(Y, complement(complement(X))))
% 59.09/7.80 = { by lemma 66 }
% 59.09/7.80 meet(complement(X), Y)
% 59.09/7.80 = { by lemma 36 R->L }
% 59.09/7.80 meet(Y, complement(X))
% 59.09/7.80
% 59.09/7.80 Lemma 69: join(complement(sk1), composition(join(X, complement(sk1)), Y)) = join(complement(sk1), composition(X, Y)).
% 59.09/7.80 Proof:
% 59.09/7.80 join(complement(sk1), composition(join(X, complement(sk1)), Y))
% 59.09/7.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.80 join(complement(sk1), composition(join(complement(sk1), X), Y))
% 59.09/7.80 = { by lemma 64 R->L }
% 59.09/7.80 join(composition(complement(sk1), top), composition(join(complement(sk1), X), Y))
% 59.09/7.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.80 join(composition(complement(sk1), top), composition(join(X, complement(sk1)), Y))
% 59.09/7.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.80 join(composition(join(X, complement(sk1)), Y), composition(complement(sk1), top))
% 59.09/7.80 = { by axiom 12 (composition_distributivity_7) }
% 59.09/7.80 join(join(composition(X, Y), composition(complement(sk1), Y)), composition(complement(sk1), top))
% 59.09/7.80 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 59.09/7.80 join(composition(X, Y), join(composition(complement(sk1), Y), composition(complement(sk1), top)))
% 59.09/7.80 = { by axiom 2 (maddux1_join_commutativity_1) }
% 59.09/7.80 join(composition(X, Y), join(composition(complement(sk1), top), composition(complement(sk1), Y)))
% 59.09/7.80 = { by axiom 1 (converse_idempotence_8) R->L }
% 59.09/7.80 join(composition(X, Y), join(composition(complement(sk1), top), composition(complement(sk1), converse(converse(Y)))))
% 59.09/7.80 = { by axiom 1 (converse_idempotence_8) R->L }
% 59.09/7.80 join(composition(X, Y), converse(converse(join(composition(complement(sk1), top), composition(complement(sk1), converse(converse(Y)))))))
% 59.09/7.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.80 join(composition(X, Y), converse(converse(join(composition(complement(sk1), converse(converse(Y))), composition(complement(sk1), top)))))
% 59.09/7.80 = { by axiom 7 (converse_additivity_9) }
% 59.09/7.80 join(composition(X, Y), converse(join(converse(composition(complement(sk1), converse(converse(Y)))), converse(composition(complement(sk1), top)))))
% 59.09/7.80 = { by lemma 63 }
% 59.09/7.80 join(composition(X, Y), converse(join(composition(converse(Y), converse(complement(sk1))), converse(composition(complement(sk1), top)))))
% 59.09/7.80 = { by axiom 2 (maddux1_join_commutativity_1) }
% 59.09/7.80 join(composition(X, Y), converse(join(converse(composition(complement(sk1), top)), composition(converse(Y), converse(complement(sk1))))))
% 59.09/7.80 = { by axiom 9 (converse_multiplicativity_10) }
% 59.09/7.80 join(composition(X, Y), converse(join(composition(converse(top), converse(complement(sk1))), composition(converse(Y), converse(complement(sk1))))))
% 59.09/7.80 = { by axiom 12 (composition_distributivity_7) R->L }
% 59.09/7.80 join(composition(X, Y), converse(composition(join(converse(top), converse(Y)), converse(complement(sk1)))))
% 59.09/7.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.80 join(composition(X, Y), converse(composition(join(converse(Y), converse(top)), converse(complement(sk1)))))
% 59.09/7.80 = { by lemma 31 R->L }
% 59.09/7.80 join(composition(X, Y), converse(composition(converse(join(top, converse(converse(Y)))), converse(complement(sk1)))))
% 59.09/7.80 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 59.09/7.80 join(composition(X, Y), converse(converse(composition(complement(sk1), join(top, converse(converse(Y)))))))
% 59.09/7.80 = { by axiom 1 (converse_idempotence_8) }
% 59.09/7.80 join(composition(X, Y), composition(complement(sk1), join(top, converse(converse(Y)))))
% 59.09/7.80 = { by axiom 1 (converse_idempotence_8) }
% 59.09/7.80 join(composition(X, Y), composition(complement(sk1), join(top, Y)))
% 59.09/7.80 = { by lemma 45 }
% 59.09/7.80 join(composition(X, Y), composition(complement(sk1), top))
% 59.09/7.80 = { by lemma 64 }
% 59.09/7.80 join(composition(X, Y), complement(sk1))
% 59.09/7.80 = { by axiom 2 (maddux1_join_commutativity_1) }
% 59.09/7.80 join(complement(sk1), composition(X, Y))
% 59.09/7.80
% 59.09/7.80 Goal 1 (goals_15): composition(meet(sk1, sk2), sk3) = meet(sk1, composition(sk2, sk3)).
% 59.09/7.80 Proof:
% 59.09/7.80 composition(meet(sk1, sk2), sk3)
% 59.09/7.80 = { by lemma 53 R->L }
% 59.09/7.80 meet(top, composition(meet(sk1, sk2), sk3))
% 59.09/7.80 = { by lemma 36 }
% 59.09/7.80 meet(composition(meet(sk1, sk2), sk3), top)
% 59.09/7.80 = { by lemma 27 R->L }
% 59.09/7.80 meet(composition(meet(sk1, sk2), sk3), join(complement(composition(meet(sk1, sk2), sk3)), top))
% 59.09/7.80 = { by lemma 58 R->L }
% 59.09/7.80 meet(composition(meet(sk1, sk2), sk3), join(complement(composition(meet(sk1, sk2), sk3)), join(sk1, complement(meet(sk1, composition(sk1, sk3))))))
% 59.09/7.80 = { by lemma 36 }
% 59.09/7.80 meet(composition(meet(sk1, sk2), sk3), join(complement(composition(meet(sk1, sk2), sk3)), join(sk1, complement(meet(composition(sk1, sk3), sk1)))))
% 59.09/7.80 = { by lemma 55 R->L }
% 59.09/7.80 meet(composition(meet(sk1, sk2), sk3), join(complement(composition(meet(sk1, sk2), sk3)), join(sk1, join(complement(composition(sk1, sk3)), complement(sk1)))))
% 59.09/7.80 = { by lemma 64 R->L }
% 59.09/7.80 meet(composition(meet(sk1, sk2), sk3), join(complement(composition(meet(sk1, sk2), sk3)), join(sk1, join(complement(composition(sk1, sk3)), composition(complement(sk1), top)))))
% 59.09/7.80 = { by lemma 30 R->L }
% 59.09/7.80 meet(composition(meet(sk1, sk2), sk3), join(complement(composition(meet(sk1, sk2), sk3)), join(sk1, join(complement(composition(sk1, sk3)), composition(complement(sk1), complement(zero))))))
% 59.09/7.81 = { by lemma 61 R->L }
% 59.09/7.81 meet(composition(meet(sk1, sk2), sk3), join(complement(composition(meet(sk1, sk2), sk3)), join(sk1, join(complement(composition(sk1, sk3)), composition(complement(sk1), complement(converse(zero)))))))
% 59.09/7.81 = { by lemma 49 R->L }
% 59.09/7.81 meet(composition(meet(sk1, sk2), sk3), join(complement(composition(meet(sk1, sk2), sk3)), join(sk1, join(complement(composition(sk1, sk3)), composition(complement(sk1), complement(composition(converse(complement(sk1)), composition(sk1, sk3))))))))
% 59.09/7.81 = { by axiom 1 (converse_idempotence_8) R->L }
% 59.09/7.81 meet(composition(meet(sk1, sk2), sk3), join(complement(composition(meet(sk1, sk2), sk3)), join(sk1, join(complement(composition(sk1, sk3)), composition(converse(converse(complement(sk1))), complement(composition(converse(complement(sk1)), composition(sk1, sk3))))))))
% 59.09/7.81 = { by lemma 19 }
% 59.09/7.81 meet(composition(meet(sk1, sk2), sk3), join(complement(composition(meet(sk1, sk2), sk3)), join(sk1, complement(composition(sk1, sk3)))))
% 59.09/7.81 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.81 meet(composition(meet(sk1, sk2), sk3), join(complement(composition(meet(sk1, sk2), sk3)), join(complement(composition(sk1, sk3)), sk1)))
% 59.09/7.81 = { by axiom 8 (maddux2_join_associativity_2) }
% 59.09/7.81 meet(composition(meet(sk1, sk2), sk3), join(join(complement(composition(meet(sk1, sk2), sk3)), complement(composition(sk1, sk3))), sk1))
% 59.09/7.81 = { by lemma 55 }
% 59.09/7.81 meet(composition(meet(sk1, sk2), sk3), join(complement(meet(composition(meet(sk1, sk2), sk3), composition(sk1, sk3))), sk1))
% 59.09/7.81 = { by axiom 2 (maddux1_join_commutativity_1) }
% 59.09/7.81 meet(composition(meet(sk1, sk2), sk3), join(sk1, complement(meet(composition(meet(sk1, sk2), sk3), composition(sk1, sk3)))))
% 59.09/7.81 = { by lemma 38 R->L }
% 59.09/7.81 meet(composition(meet(sk1, sk2), sk3), join(sk1, complement(meet(composition(meet(sk1, sk2), sk3), composition(join(meet(sk1, sk2), meet(sk1, complement(sk2))), sk3)))))
% 59.09/7.81 = { by axiom 12 (composition_distributivity_7) }
% 59.09/7.81 meet(composition(meet(sk1, sk2), sk3), join(sk1, complement(meet(composition(meet(sk1, sk2), sk3), join(composition(meet(sk1, sk2), sk3), composition(meet(sk1, complement(sk2)), sk3))))))
% 59.09/7.81 = { by lemma 62 R->L }
% 59.09/7.81 meet(composition(meet(sk1, sk2), sk3), join(sk1, complement(meet(composition(meet(sk1, sk2), sk3), join(composition(meet(sk1, sk2), sk3), meet(composition(meet(sk1, complement(sk2)), sk3), composition(meet(sk1, complement(sk2)), sk3)))))))
% 59.09/7.81 = { by lemma 35 R->L }
% 59.09/7.81 meet(composition(meet(sk1, sk2), sk3), join(sk1, complement(meet(composition(meet(sk1, sk2), sk3), join(composition(meet(sk1, sk2), sk3), complement(complement(composition(meet(sk1, complement(sk2)), sk3))))))))
% 59.09/7.81 = { by lemma 60 }
% 59.09/7.81 meet(composition(meet(sk1, sk2), sk3), join(sk1, complement(composition(meet(sk1, sk2), sk3))))
% 59.09/7.81 = { by lemma 66 }
% 59.09/7.81 meet(composition(meet(sk1, sk2), sk3), sk1)
% 59.09/7.81 = { by lemma 41 R->L }
% 59.09/7.81 meet(composition(meet(sk1, sk2), sk3), complement(complement(sk1)))
% 59.09/7.81 = { by lemma 68 R->L }
% 59.09/7.81 meet(complement(complement(sk1)), join(composition(meet(sk1, sk2), sk3), complement(sk1)))
% 59.09/7.81 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 59.09/7.81 meet(complement(complement(sk1)), join(complement(sk1), composition(meet(sk1, sk2), sk3)))
% 59.09/7.81 = { by lemma 69 R->L }
% 59.09/7.81 meet(complement(complement(sk1)), join(complement(sk1), composition(join(meet(sk1, sk2), complement(sk1)), sk3)))
% 59.09/7.81 = { by lemma 67 }
% 59.09/7.81 meet(complement(complement(sk1)), join(complement(sk1), composition(join(sk2, complement(sk1)), sk3)))
% 59.09/7.81 = { by lemma 69 }
% 59.09/7.81 meet(complement(complement(sk1)), join(complement(sk1), composition(sk2, sk3)))
% 59.09/7.81 = { by axiom 2 (maddux1_join_commutativity_1) }
% 59.09/7.81 meet(complement(complement(sk1)), join(composition(sk2, sk3), complement(sk1)))
% 59.09/7.81 = { by lemma 67 R->L }
% 59.09/7.81 meet(complement(complement(sk1)), join(meet(sk1, composition(sk2, sk3)), complement(sk1)))
% 59.09/7.81 = { by lemma 68 }
% 59.09/7.81 meet(meet(sk1, composition(sk2, sk3)), complement(complement(sk1)))
% 59.09/7.81 = { by lemma 41 }
% 59.09/7.81 meet(meet(sk1, composition(sk2, sk3)), sk1)
% 59.09/7.81 = { by lemma 36 }
% 59.09/7.81 meet(sk1, meet(sk1, composition(sk2, sk3)))
% 59.09/7.81 = { by lemma 36 }
% 59.09/7.81 meet(sk1, meet(composition(sk2, sk3), sk1))
% 59.09/7.81 = { by lemma 36 }
% 59.09/7.81 meet(meet(composition(sk2, sk3), sk1), sk1)
% 59.09/7.81 = { by lemma 41 R->L }
% 59.09/7.81 meet(meet(composition(sk2, sk3), complement(complement(sk1))), sk1)
% 59.09/7.81 = { by lemma 42 R->L }
% 59.09/7.81 join(meet(meet(composition(sk2, sk3), complement(complement(sk1))), sk1), zero)
% 59.09/7.81 = { by lemma 59 R->L }
% 59.09/7.81 join(meet(meet(composition(sk2, sk3), complement(complement(sk1))), sk1), meet(complement(sk1), meet(composition(sk2, sk3), complement(complement(sk1)))))
% 59.09/7.81 = { by lemma 65 }
% 59.09/7.81 meet(composition(sk2, sk3), complement(complement(sk1)))
% 59.09/7.81 = { by lemma 41 }
% 59.09/7.81 meet(composition(sk2, sk3), sk1)
% 59.09/7.81 = { by lemma 36 R->L }
% 59.09/7.81 meet(sk1, composition(sk2, sk3))
% 59.09/7.81 % SZS output end Proof
% 59.09/7.81
% 59.09/7.81 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------