TSTP Solution File: REL030-3 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL030-3 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n007.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 13:44:14 EDT 2023
% Result : Unsatisfiable 19.68s 2.90s
% Output : Proof 20.69s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : REL030-3 : TPTP v8.1.2. Released v4.0.0.
% 0.06/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n007.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Aug 25 18:48:41 EDT 2023
% 0.12/0.34 % CPUTime :
% 19.68/2.90 Command-line arguments: --flatten
% 19.68/2.90
% 19.68/2.90 % SZS status Unsatisfiable
% 19.68/2.90
% 20.31/2.97 % SZS output start Proof
% 20.31/2.97 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 20.31/2.97 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 20.31/2.97 Axiom 3 (goals_17): join(sk1, one) = one.
% 20.31/2.97 Axiom 4 (composition_identity_6): composition(X, one) = X.
% 20.31/2.97 Axiom 5 (def_top_12): top = join(X, complement(X)).
% 20.31/2.97 Axiom 6 (def_zero_13): zero = meet(X, complement(X)).
% 20.31/2.97 Axiom 7 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 20.31/2.97 Axiom 8 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 20.31/2.97 Axiom 9 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 20.31/2.97 Axiom 10 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 20.31/2.97 Axiom 11 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 20.31/2.97 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 20.31/2.97 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 20.31/2.97 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 20.31/2.97 Axiom 15 (modular_law_1_15): join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z)) = meet(composition(X, meet(Y, composition(converse(X), Z))), Z).
% 20.31/2.97
% 20.31/2.97 Lemma 16: complement(top) = zero.
% 20.31/2.97 Proof:
% 20.31/2.97 complement(top)
% 20.31/2.97 = { by axiom 5 (def_top_12) }
% 20.31/2.97 complement(join(complement(X), complement(complement(X))))
% 20.31/2.97 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 20.31/2.97 meet(X, complement(X))
% 20.31/2.97 = { by axiom 6 (def_zero_13) R->L }
% 20.31/2.97 zero
% 20.31/2.97
% 20.31/2.97 Lemma 17: join(X, join(Y, complement(X))) = join(Y, top).
% 20.31/2.97 Proof:
% 20.31/2.97 join(X, join(Y, complement(X)))
% 20.31/2.97 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.97 join(X, join(complement(X), Y))
% 20.31/2.97 = { by axiom 8 (maddux2_join_associativity_2) }
% 20.31/2.97 join(join(X, complement(X)), Y)
% 20.31/2.97 = { by axiom 5 (def_top_12) R->L }
% 20.31/2.97 join(top, Y)
% 20.31/2.97 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.31/2.97 join(Y, top)
% 20.31/2.97
% 20.31/2.97 Lemma 18: composition(X, join(sk1, one)) = X.
% 20.31/2.97 Proof:
% 20.31/2.97 composition(X, join(sk1, one))
% 20.31/2.97 = { by axiom 3 (goals_17) }
% 20.31/2.97 composition(X, one)
% 20.31/2.97 = { by axiom 4 (composition_identity_6) }
% 20.31/2.97 X
% 20.31/2.97
% 20.31/2.97 Lemma 19: composition(converse(join(sk1, one)), X) = X.
% 20.31/2.97 Proof:
% 20.31/2.97 composition(converse(join(sk1, one)), X)
% 20.31/2.97 = { by axiom 1 (converse_idempotence_8) R->L }
% 20.31/2.97 composition(converse(join(sk1, one)), converse(converse(X)))
% 20.31/2.97 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 20.31/2.97 converse(composition(converse(X), join(sk1, one)))
% 20.31/2.97 = { by lemma 18 }
% 20.31/2.97 converse(converse(X))
% 20.31/2.97 = { by axiom 1 (converse_idempotence_8) }
% 20.31/2.97 X
% 20.31/2.97
% 20.31/2.97 Lemma 20: composition(join(sk1, one), X) = X.
% 20.31/2.97 Proof:
% 20.31/2.97 composition(join(sk1, one), X)
% 20.31/2.97 = { by lemma 19 R->L }
% 20.31/2.97 composition(converse(join(sk1, one)), composition(join(sk1, one), X))
% 20.31/2.97 = { by axiom 10 (composition_associativity_5) }
% 20.31/2.97 composition(composition(converse(join(sk1, one)), join(sk1, one)), X)
% 20.31/2.97 = { by lemma 18 }
% 20.31/2.97 composition(converse(join(sk1, one)), X)
% 20.31/2.97 = { by lemma 19 }
% 20.31/2.97 X
% 20.31/2.97
% 20.31/2.97 Lemma 21: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 20.31/2.98 Proof:
% 20.31/2.98 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.98 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 20.31/2.98 = { by axiom 13 (converse_cancellativity_11) }
% 20.31/2.98 complement(X)
% 20.31/2.98
% 20.31/2.98 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 20.31/2.98 Proof:
% 20.31/2.98 join(complement(X), complement(X))
% 20.31/2.98 = { by lemma 19 R->L }
% 20.31/2.98 join(complement(X), composition(converse(join(sk1, one)), complement(X)))
% 20.31/2.98 = { by lemma 20 R->L }
% 20.31/2.98 join(complement(X), composition(converse(join(sk1, one)), complement(composition(join(sk1, one), X))))
% 20.31/2.98 = { by lemma 21 }
% 20.31/2.98 complement(X)
% 20.31/2.98
% 20.31/2.98 Lemma 23: join(top, complement(X)) = top.
% 20.31/2.98 Proof:
% 20.31/2.98 join(top, complement(X))
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.98 join(complement(X), top)
% 20.31/2.98 = { by lemma 17 R->L }
% 20.31/2.98 join(X, join(complement(X), complement(X)))
% 20.31/2.98 = { by lemma 22 }
% 20.31/2.98 join(X, complement(X))
% 20.31/2.98 = { by axiom 5 (def_top_12) R->L }
% 20.31/2.98 top
% 20.31/2.98
% 20.31/2.98 Lemma 24: join(Y, top) = join(X, top).
% 20.31/2.98 Proof:
% 20.31/2.98 join(Y, top)
% 20.31/2.98 = { by lemma 23 R->L }
% 20.31/2.98 join(Y, join(top, complement(Y)))
% 20.31/2.98 = { by lemma 17 }
% 20.31/2.98 join(top, top)
% 20.31/2.98 = { by lemma 17 R->L }
% 20.31/2.98 join(X, join(top, complement(X)))
% 20.31/2.98 = { by lemma 23 }
% 20.31/2.98 join(X, top)
% 20.31/2.98
% 20.31/2.98 Lemma 25: join(X, top) = top.
% 20.31/2.98 Proof:
% 20.31/2.98 join(X, top)
% 20.31/2.98 = { by lemma 24 }
% 20.31/2.98 join(complement(Y), top)
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.98 join(top, complement(Y))
% 20.31/2.98 = { by lemma 23 }
% 20.31/2.98 top
% 20.31/2.98
% 20.31/2.98 Lemma 26: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 20.31/2.98 Proof:
% 20.31/2.98 converse(join(X, converse(Y)))
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.98 converse(join(converse(Y), X))
% 20.31/2.98 = { by axiom 7 (converse_additivity_9) }
% 20.31/2.98 join(converse(converse(Y)), converse(X))
% 20.31/2.98 = { by axiom 1 (converse_idempotence_8) }
% 20.31/2.98 join(Y, converse(X))
% 20.31/2.98
% 20.31/2.98 Lemma 27: join(X, join(complement(X), Y)) = top.
% 20.31/2.98 Proof:
% 20.31/2.98 join(X, join(complement(X), Y))
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.98 join(X, join(Y, complement(X)))
% 20.31/2.98 = { by lemma 17 }
% 20.31/2.98 join(Y, top)
% 20.31/2.98 = { by lemma 24 R->L }
% 20.31/2.98 join(Z, top)
% 20.31/2.98 = { by lemma 25 }
% 20.31/2.98 top
% 20.31/2.98
% 20.31/2.98 Lemma 28: join(X, converse(top)) = top.
% 20.31/2.98 Proof:
% 20.31/2.98 join(X, converse(top))
% 20.31/2.98 = { by axiom 5 (def_top_12) }
% 20.31/2.98 join(X, converse(join(converse(complement(X)), complement(converse(complement(X))))))
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.98 join(X, converse(join(complement(converse(complement(X))), converse(complement(X)))))
% 20.31/2.98 = { by lemma 26 }
% 20.31/2.98 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 20.31/2.98 = { by lemma 27 }
% 20.31/2.98 top
% 20.31/2.98
% 20.31/2.98 Lemma 29: converse(top) = top.
% 20.31/2.98 Proof:
% 20.31/2.98 converse(top)
% 20.31/2.98 = { by lemma 25 R->L }
% 20.31/2.98 converse(join(X, top))
% 20.31/2.98 = { by axiom 7 (converse_additivity_9) }
% 20.31/2.98 join(converse(X), converse(top))
% 20.31/2.98 = { by lemma 28 }
% 20.31/2.98 top
% 20.31/2.98
% 20.31/2.98 Lemma 30: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 20.31/2.98 Proof:
% 20.31/2.98 join(meet(X, Y), complement(join(complement(X), Y)))
% 20.31/2.98 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 20.31/2.98 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 20.31/2.98 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 20.31/2.98 X
% 20.31/2.98
% 20.31/2.98 Lemma 31: join(zero, meet(X, X)) = X.
% 20.31/2.98 Proof:
% 20.31/2.98 join(zero, meet(X, X))
% 20.31/2.98 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 20.31/2.98 join(zero, complement(join(complement(X), complement(X))))
% 20.31/2.98 = { by axiom 6 (def_zero_13) }
% 20.31/2.98 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 20.31/2.98 = { by lemma 30 }
% 20.31/2.98 X
% 20.31/2.98
% 20.31/2.98 Lemma 32: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 20.31/2.98 Proof:
% 20.31/2.98 join(zero, join(X, complement(complement(Y))))
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.98 join(zero, join(complement(complement(Y)), X))
% 20.31/2.98 = { by lemma 22 R->L }
% 20.31/2.98 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 20.31/2.98 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 20.31/2.98 join(zero, join(meet(Y, Y), X))
% 20.31/2.98 = { by axiom 8 (maddux2_join_associativity_2) }
% 20.31/2.98 join(join(zero, meet(Y, Y)), X)
% 20.31/2.98 = { by lemma 31 }
% 20.31/2.98 join(Y, X)
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.31/2.98 join(X, Y)
% 20.31/2.98
% 20.31/2.98 Lemma 33: join(zero, complement(complement(X))) = X.
% 20.31/2.98 Proof:
% 20.31/2.98 join(zero, complement(complement(X)))
% 20.31/2.98 = { by axiom 6 (def_zero_13) }
% 20.31/2.98 join(meet(X, complement(X)), complement(complement(X)))
% 20.31/2.98 = { by lemma 22 R->L }
% 20.31/2.98 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 20.31/2.98 = { by lemma 30 }
% 20.31/2.98 X
% 20.31/2.98
% 20.31/2.98 Lemma 34: join(zero, complement(X)) = complement(X).
% 20.31/2.98 Proof:
% 20.31/2.98 join(zero, complement(X))
% 20.31/2.98 = { by lemma 33 R->L }
% 20.31/2.98 join(zero, join(zero, complement(complement(complement(X)))))
% 20.31/2.98 = { by lemma 22 R->L }
% 20.31/2.98 join(zero, join(zero, join(complement(complement(complement(X))), complement(complement(complement(X))))))
% 20.31/2.98 = { by lemma 32 }
% 20.31/2.98 join(zero, join(complement(complement(complement(X))), complement(X)))
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.31/2.98 join(zero, join(complement(X), complement(complement(complement(X)))))
% 20.31/2.98 = { by lemma 32 }
% 20.31/2.98 join(complement(X), complement(X))
% 20.31/2.98 = { by lemma 22 }
% 20.31/2.98 complement(X)
% 20.31/2.98
% 20.31/2.98 Lemma 35: join(X, zero) = X.
% 20.31/2.98 Proof:
% 20.31/2.98 join(X, zero)
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.98 join(zero, X)
% 20.31/2.98 = { by lemma 32 R->L }
% 20.31/2.98 join(zero, join(zero, complement(complement(X))))
% 20.31/2.98 = { by lemma 34 }
% 20.31/2.98 join(zero, complement(complement(X)))
% 20.31/2.98 = { by lemma 33 }
% 20.31/2.98 X
% 20.31/2.98
% 20.31/2.98 Lemma 36: join(top, X) = top.
% 20.31/2.98 Proof:
% 20.31/2.98 join(top, X)
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.98 join(X, top)
% 20.31/2.98 = { by lemma 24 R->L }
% 20.31/2.98 join(Y, top)
% 20.31/2.98 = { by lemma 25 }
% 20.31/2.98 top
% 20.31/2.98
% 20.31/2.98 Lemma 37: join(zero, X) = X.
% 20.31/2.98 Proof:
% 20.31/2.98 join(zero, X)
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.98 join(X, zero)
% 20.31/2.98 = { by lemma 35 }
% 20.31/2.98 X
% 20.31/2.98
% 20.31/2.98 Lemma 38: meet(Y, X) = meet(X, Y).
% 20.31/2.98 Proof:
% 20.31/2.98 meet(Y, X)
% 20.31/2.98 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 20.31/2.98 complement(join(complement(Y), complement(X)))
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.98 complement(join(complement(X), complement(Y)))
% 20.31/2.98 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 20.31/2.98 meet(X, Y)
% 20.31/2.98
% 20.31/2.98 Lemma 39: complement(join(zero, complement(X))) = meet(X, top).
% 20.31/2.98 Proof:
% 20.31/2.98 complement(join(zero, complement(X)))
% 20.31/2.98 = { by lemma 16 R->L }
% 20.31/2.98 complement(join(complement(top), complement(X)))
% 20.31/2.98 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 20.31/2.98 meet(top, X)
% 20.31/2.98 = { by lemma 38 R->L }
% 20.31/2.98 meet(X, top)
% 20.31/2.98
% 20.31/2.98 Lemma 40: join(X, complement(zero)) = top.
% 20.31/2.98 Proof:
% 20.31/2.98 join(X, complement(zero))
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.98 join(complement(zero), X)
% 20.31/2.98 = { by lemma 32 R->L }
% 20.31/2.98 join(zero, join(complement(zero), complement(complement(X))))
% 20.31/2.98 = { by lemma 27 }
% 20.31/2.98 top
% 20.31/2.98
% 20.31/2.98 Lemma 41: meet(X, zero) = zero.
% 20.31/2.98 Proof:
% 20.31/2.98 meet(X, zero)
% 20.31/2.98 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 20.31/2.98 complement(join(complement(X), complement(zero)))
% 20.31/2.98 = { by lemma 40 }
% 20.31/2.98 complement(top)
% 20.31/2.98 = { by lemma 16 }
% 20.31/2.98 zero
% 20.31/2.98
% 20.31/2.98 Lemma 42: join(meet(X, Y), meet(X, complement(Y))) = X.
% 20.31/2.98 Proof:
% 20.31/2.98 join(meet(X, Y), meet(X, complement(Y)))
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.98 join(meet(X, complement(Y)), meet(X, Y))
% 20.31/2.98 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 20.31/2.98 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 20.31/2.98 = { by lemma 30 }
% 20.31/2.98 X
% 20.31/2.98
% 20.31/2.98 Lemma 43: meet(X, top) = X.
% 20.31/2.98 Proof:
% 20.31/2.98 meet(X, top)
% 20.31/2.98 = { by lemma 39 R->L }
% 20.31/2.98 complement(join(zero, complement(X)))
% 20.31/2.98 = { by lemma 34 R->L }
% 20.31/2.98 join(zero, complement(join(zero, complement(X))))
% 20.31/2.98 = { by lemma 39 }
% 20.31/2.98 join(zero, meet(X, top))
% 20.31/2.98 = { by lemma 40 R->L }
% 20.31/2.98 join(zero, meet(X, join(complement(zero), complement(zero))))
% 20.31/2.98 = { by lemma 22 }
% 20.31/2.98 join(zero, meet(X, complement(zero)))
% 20.31/2.98 = { by lemma 41 R->L }
% 20.31/2.98 join(meet(X, zero), meet(X, complement(zero)))
% 20.31/2.98 = { by lemma 42 }
% 20.31/2.98 X
% 20.31/2.98
% 20.31/2.98 Lemma 44: meet(zero, X) = zero.
% 20.31/2.98 Proof:
% 20.31/2.98 meet(zero, X)
% 20.31/2.98 = { by lemma 38 }
% 20.31/2.98 meet(X, zero)
% 20.31/2.98 = { by lemma 41 }
% 20.31/2.98 zero
% 20.31/2.98
% 20.31/2.98 Lemma 45: composition(join(join(sk1, one), Y), X) = join(X, composition(Y, X)).
% 20.31/2.98 Proof:
% 20.31/2.98 composition(join(join(sk1, one), Y), X)
% 20.31/2.98 = { by axiom 12 (composition_distributivity_7) }
% 20.31/2.98 join(composition(join(sk1, one), X), composition(Y, X))
% 20.31/2.98 = { by lemma 20 }
% 20.31/2.98 join(X, composition(Y, X))
% 20.31/2.98
% 20.31/2.98 Lemma 46: composition(top, zero) = zero.
% 20.31/2.98 Proof:
% 20.31/2.98 composition(top, zero)
% 20.31/2.98 = { by lemma 29 R->L }
% 20.31/2.98 composition(converse(top), zero)
% 20.31/2.98 = { by lemma 37 R->L }
% 20.31/2.98 join(zero, composition(converse(top), zero))
% 20.31/2.98 = { by lemma 16 R->L }
% 20.31/2.98 join(complement(top), composition(converse(top), zero))
% 20.31/2.98 = { by lemma 16 R->L }
% 20.31/2.98 join(complement(top), composition(converse(top), complement(top)))
% 20.31/2.98 = { by lemma 36 R->L }
% 20.31/2.98 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 20.31/2.98 = { by lemma 29 R->L }
% 20.31/2.98 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 20.31/2.98 = { by lemma 45 R->L }
% 20.31/2.98 join(complement(top), composition(converse(top), complement(composition(join(join(sk1, one), converse(top)), top))))
% 20.31/2.98 = { by lemma 28 }
% 20.31/2.98 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 20.31/2.98 = { by lemma 21 }
% 20.31/2.98 complement(top)
% 20.31/2.98 = { by lemma 16 }
% 20.31/2.98 zero
% 20.31/2.98
% 20.31/2.98 Lemma 47: composition(X, zero) = zero.
% 20.31/2.98 Proof:
% 20.31/2.98 composition(X, zero)
% 20.31/2.98 = { by lemma 37 R->L }
% 20.31/2.98 join(zero, composition(X, zero))
% 20.31/2.98 = { by lemma 46 R->L }
% 20.31/2.98 join(composition(top, zero), composition(X, zero))
% 20.31/2.98 = { by axiom 12 (composition_distributivity_7) R->L }
% 20.31/2.98 composition(join(top, X), zero)
% 20.31/2.98 = { by lemma 36 }
% 20.31/2.98 composition(top, zero)
% 20.31/2.98 = { by lemma 46 }
% 20.31/2.98 zero
% 20.31/2.98
% 20.31/2.98 Lemma 48: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 20.31/2.98 Proof:
% 20.31/2.98 join(Y, join(X, Z))
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.98 join(join(X, Z), Y)
% 20.31/2.98 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 20.31/2.98 join(X, join(Z, Y))
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.31/2.98 join(X, join(Y, Z))
% 20.31/2.98
% 20.31/2.98 Lemma 49: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 20.31/2.98 Proof:
% 20.31/2.98 meet(X, join(complement(Y), complement(Z)))
% 20.31/2.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.98 meet(X, join(complement(Z), complement(Y)))
% 20.31/2.98 = { by lemma 38 }
% 20.31/2.98 meet(join(complement(Z), complement(Y)), X)
% 20.31/2.98 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 20.31/2.98 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 20.31/2.98 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 20.31/2.98 complement(join(meet(Z, Y), complement(X)))
% 20.31/2.99 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.31/2.99 complement(join(complement(X), meet(Z, Y)))
% 20.31/2.99 = { by lemma 38 R->L }
% 20.31/2.99 complement(join(complement(X), meet(Y, Z)))
% 20.31/2.99
% 20.31/2.99 Lemma 50: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 20.31/2.99 Proof:
% 20.31/2.99 complement(join(X, complement(Y)))
% 20.31/2.99 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.99 complement(join(complement(Y), X))
% 20.31/2.99 = { by lemma 43 R->L }
% 20.31/2.99 complement(join(complement(Y), meet(X, top)))
% 20.31/2.99 = { by lemma 38 R->L }
% 20.31/2.99 complement(join(complement(Y), meet(top, X)))
% 20.31/2.99 = { by lemma 49 R->L }
% 20.31/2.99 meet(Y, join(complement(top), complement(X)))
% 20.31/2.99 = { by lemma 16 }
% 20.31/2.99 meet(Y, join(zero, complement(X)))
% 20.31/2.99 = { by lemma 34 }
% 20.31/2.99 meet(Y, complement(X))
% 20.31/2.99
% 20.31/2.99 Lemma 51: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 20.31/2.99 Proof:
% 20.31/2.99 complement(join(complement(X), Y))
% 20.31/2.99 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/2.99 complement(join(Y, complement(X)))
% 20.31/2.99 = { by lemma 50 }
% 20.31/2.99 meet(X, complement(Y))
% 20.31/2.99
% 20.31/2.99 Lemma 52: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 20.31/2.99 Proof:
% 20.31/2.99 complement(meet(X, complement(Y)))
% 20.31/2.99 = { by lemma 37 R->L }
% 20.31/2.99 complement(join(zero, meet(X, complement(Y))))
% 20.31/2.99 = { by lemma 50 R->L }
% 20.31/2.99 complement(join(zero, complement(join(Y, complement(X)))))
% 20.31/2.99 = { by lemma 39 }
% 20.31/2.99 meet(join(Y, complement(X)), top)
% 20.31/2.99 = { by lemma 43 }
% 20.31/2.99 join(Y, complement(X))
% 20.31/2.99
% 20.31/2.99 Lemma 53: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 20.31/2.99 Proof:
% 20.31/2.99 meet(complement(X), complement(Y))
% 20.31/2.99 = { by lemma 38 }
% 20.31/2.99 meet(complement(Y), complement(X))
% 20.31/2.99 = { by lemma 34 R->L }
% 20.31/2.99 meet(join(zero, complement(Y)), complement(X))
% 20.31/2.99 = { by lemma 50 R->L }
% 20.31/2.99 complement(join(X, complement(join(zero, complement(Y)))))
% 20.31/2.99 = { by lemma 39 }
% 20.31/2.99 complement(join(X, meet(Y, top)))
% 20.31/2.99 = { by lemma 43 }
% 20.31/2.99 complement(join(X, Y))
% 20.31/2.99
% 20.31/2.99 Lemma 54: meet(complement(X), complement(Y)) = complement(join(Y, X)).
% 20.31/2.99 Proof:
% 20.31/2.99 meet(complement(X), complement(Y))
% 20.31/2.99 = { by lemma 53 }
% 20.31/2.99 complement(join(X, Y))
% 20.31/2.99 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.31/2.99 complement(join(Y, X))
% 20.31/2.99
% 20.31/2.99 Lemma 55: join(X, complement(meet(X, Y))) = top.
% 20.31/2.99 Proof:
% 20.31/2.99 join(X, complement(meet(X, Y)))
% 20.31/2.99 = { by lemma 38 }
% 20.31/2.99 join(X, complement(meet(Y, X)))
% 20.31/2.99 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 20.31/2.99 join(X, complement(complement(join(complement(Y), complement(X)))))
% 20.31/2.99 = { by lemma 22 R->L }
% 20.31/2.99 join(X, complement(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))))
% 20.31/2.99 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 20.31/2.99 join(X, complement(join(meet(Y, X), complement(join(complement(Y), complement(X))))))
% 20.31/2.99 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 20.31/2.99 join(X, complement(join(meet(Y, X), meet(Y, X))))
% 20.31/2.99 = { by lemma 38 R->L }
% 20.31/2.99 join(X, complement(join(meet(Y, X), meet(X, Y))))
% 20.31/2.99 = { by lemma 38 R->L }
% 20.31/2.99 join(X, complement(join(meet(X, Y), meet(X, Y))))
% 20.31/2.99 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 20.31/2.99 join(X, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y))))
% 20.31/2.99 = { by lemma 34 R->L }
% 20.31/2.99 join(X, join(zero, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y)))))
% 20.31/2.99 = { by lemma 49 R->L }
% 20.31/2.99 join(X, join(zero, meet(join(complement(X), complement(Y)), join(complement(X), complement(Y)))))
% 20.31/2.99 = { by lemma 31 }
% 20.31/2.99 join(X, join(complement(X), complement(Y)))
% 20.31/2.99 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.31/2.99 join(X, join(complement(Y), complement(X)))
% 20.31/2.99 = { by lemma 17 }
% 20.31/2.99 join(complement(Y), top)
% 20.31/2.99 = { by lemma 25 }
% 20.31/2.99 top
% 20.31/2.99
% 20.31/2.99 Lemma 56: meet(complement(Z), meet(Y, X)) = meet(X, meet(Y, complement(Z))).
% 20.31/2.99 Proof:
% 20.31/2.99 meet(complement(Z), meet(Y, X))
% 20.31/2.99 = { by lemma 38 }
% 20.31/2.99 meet(complement(Z), meet(X, Y))
% 20.31/2.99 = { by lemma 38 }
% 20.31/2.99 meet(meet(X, Y), complement(Z))
% 20.31/2.99 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 20.31/2.99 meet(complement(join(complement(X), complement(Y))), complement(Z))
% 20.31/2.99 = { by lemma 53 }
% 20.31/2.99 complement(join(join(complement(X), complement(Y)), Z))
% 20.31/2.99 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 20.31/2.99 complement(join(complement(X), join(complement(Y), Z)))
% 20.31/2.99 = { by lemma 51 }
% 20.31/2.99 meet(X, complement(join(complement(Y), Z)))
% 20.31/2.99 = { by lemma 51 }
% 20.31/2.99 meet(X, meet(Y, complement(Z)))
% 20.31/2.99
% 20.31/2.99 Lemma 57: composition(converse(X), complement(composition(X, top))) = zero.
% 20.31/2.99 Proof:
% 20.31/2.99 composition(converse(X), complement(composition(X, top)))
% 20.31/2.99 = { by lemma 37 R->L }
% 20.31/2.99 join(zero, composition(converse(X), complement(composition(X, top))))
% 20.31/2.99 = { by lemma 16 R->L }
% 20.31/2.99 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 20.31/2.99 = { by lemma 21 }
% 20.31/2.99 complement(top)
% 20.31/2.99 = { by lemma 16 }
% 20.31/2.99 zero
% 20.31/2.99
% 20.31/2.99 Lemma 58: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 20.31/2.99 Proof:
% 20.31/2.99 join(meet(X, Y), meet(Y, complement(X)))
% 20.31/2.99 = { by lemma 38 }
% 20.31/2.99 join(meet(Y, X), meet(Y, complement(X)))
% 20.31/2.99 = { by lemma 42 }
% 20.31/2.99 Y
% 20.31/2.99
% 20.31/2.99 Goal 1 (goals_18): meet(composition(sk1, sk2), complement(sk3)) = meet(composition(sk1, sk2), complement(composition(sk1, sk3))).
% 20.31/2.99 Proof:
% 20.31/2.99 meet(composition(sk1, sk2), complement(sk3))
% 20.31/2.99 = { by lemma 20 R->L }
% 20.31/2.99 meet(composition(sk1, sk2), complement(composition(join(sk1, one), sk3)))
% 20.31/2.99 = { by axiom 12 (composition_distributivity_7) }
% 20.31/2.99 meet(composition(sk1, sk2), complement(join(composition(sk1, sk3), composition(one, sk3))))
% 20.31/2.99 = { by axiom 3 (goals_17) R->L }
% 20.31/2.99 meet(composition(sk1, sk2), complement(join(composition(sk1, sk3), composition(join(sk1, one), sk3))))
% 20.31/2.99 = { by lemma 20 }
% 20.31/2.99 meet(composition(sk1, sk2), complement(join(composition(sk1, sk3), sk3)))
% 20.31/2.99 = { by lemma 54 R->L }
% 20.31/2.99 meet(composition(sk1, sk2), meet(complement(sk3), complement(composition(sk1, sk3))))
% 20.31/2.99 = { by lemma 38 R->L }
% 20.31/2.99 meet(composition(sk1, sk2), meet(complement(composition(sk1, sk3)), complement(sk3)))
% 20.31/2.99 = { by lemma 51 R->L }
% 20.31/2.99 meet(composition(sk1, sk2), complement(join(complement(complement(composition(sk1, sk3))), sk3)))
% 20.31/2.99 = { by lemma 51 R->L }
% 20.31/2.99 complement(join(complement(composition(sk1, sk2)), join(complement(complement(composition(sk1, sk3))), sk3)))
% 20.31/2.99 = { by axiom 8 (maddux2_join_associativity_2) }
% 20.31/2.99 complement(join(join(complement(composition(sk1, sk2)), complement(complement(composition(sk1, sk3)))), sk3))
% 20.31/2.99 = { by lemma 54 R->L }
% 20.31/2.99 meet(complement(sk3), complement(join(complement(composition(sk1, sk2)), complement(complement(composition(sk1, sk3))))))
% 20.31/2.99 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 20.31/2.99 meet(complement(sk3), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))
% 20.31/2.99 = { by lemma 38 }
% 20.31/2.99 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(sk3))
% 20.31/2.99 = { by lemma 20 R->L }
% 20.31/2.99 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), composition(join(sk1, one), complement(sk3)))
% 20.31/2.99 = { by axiom 3 (goals_17) R->L }
% 20.31/2.99 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), composition(join(sk1, join(sk1, one)), complement(sk3)))
% 20.31/2.99 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.31/2.99 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), composition(join(join(sk1, one), sk1), complement(sk3)))
% 20.31/2.99 = { by lemma 45 }
% 20.31/2.99 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), composition(sk1, complement(sk3))))
% 20.31/2.99 = { by lemma 33 R->L }
% 20.31/2.99 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(zero, complement(complement(composition(sk1, complement(sk3)))))))
% 20.31/2.99 = { by lemma 34 }
% 20.31/2.99 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), complement(complement(composition(sk1, complement(sk3))))))
% 20.31/2.99 = { by lemma 30 R->L }
% 20.31/2.99 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), complement(join(meet(complement(composition(sk1, complement(sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(complement(composition(sk1, complement(sk3)))))), complement(join(complement(complement(composition(sk1, complement(sk3)))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(complement(composition(sk1, complement(sk3)))))))))))
% 20.31/2.99 = { by lemma 38 }
% 20.31/2.99 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), complement(join(meet(complement(composition(sk1, complement(sk3))), meet(complement(complement(composition(sk1, complement(sk3)))), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))), complement(join(complement(complement(composition(sk1, complement(sk3)))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(complement(composition(sk1, complement(sk3)))))))))))
% 20.31/2.99 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 20.31/2.99 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), complement(join(complement(join(complement(complement(composition(sk1, complement(sk3)))), complement(meet(complement(complement(composition(sk1, complement(sk3)))), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))), complement(join(complement(complement(composition(sk1, complement(sk3)))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(complement(composition(sk1, complement(sk3)))))))))))
% 20.31/2.99 = { by lemma 55 }
% 20.31/2.99 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), complement(join(complement(top), complement(join(complement(complement(composition(sk1, complement(sk3)))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(complement(composition(sk1, complement(sk3)))))))))))
% 20.31/2.99 = { by lemma 16 }
% 20.31/2.99 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), complement(join(zero, complement(join(complement(complement(composition(sk1, complement(sk3)))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(complement(composition(sk1, complement(sk3)))))))))))
% 20.31/2.99 = { by lemma 34 }
% 20.31/2.99 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), complement(complement(join(complement(complement(composition(sk1, complement(sk3)))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(complement(composition(sk1, complement(sk3))))))))))
% 20.31/2.99 = { by lemma 51 }
% 20.31/2.99 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), complement(meet(complement(composition(sk1, complement(sk3))), complement(meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(complement(composition(sk1, complement(sk3))))))))))
% 20.31/2.99 = { by lemma 52 }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), complement(meet(complement(composition(sk1, complement(sk3))), join(complement(composition(sk1, complement(sk3))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))))
% 20.31/3.00 = { by lemma 38 }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), complement(meet(join(complement(composition(sk1, complement(sk3))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3))))), complement(composition(sk1, complement(sk3)))))))
% 20.31/3.00 = { by lemma 52 }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), complement(join(complement(composition(sk1, complement(sk3))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))))
% 20.31/3.00 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))
% 20.31/3.00 = { by lemma 35 R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), zero))))
% 20.31/3.00 = { by lemma 44 R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(zero, complement(composition(sk1, top)))))))
% 20.31/3.00 = { by lemma 47 R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, zero), complement(composition(sk1, top)))))))
% 20.31/3.00 = { by lemma 41 R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, meet(sk2, zero)), complement(composition(sk1, top)))))))
% 20.31/3.00 = { by lemma 57 R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, meet(sk2, composition(converse(sk1), complement(composition(sk1, top))))), complement(composition(sk1, top)))))))
% 20.31/3.00 = { by axiom 15 (modular_law_1_15) R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, sk2), complement(composition(sk1, top))), meet(composition(sk1, meet(sk2, composition(converse(sk1), complement(composition(sk1, top))))), complement(composition(sk1, top))))))))
% 20.31/3.00 = { by lemma 57 }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, sk2), complement(composition(sk1, top))), meet(composition(sk1, meet(sk2, zero)), complement(composition(sk1, top))))))))
% 20.31/3.00 = { by lemma 41 }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, sk2), complement(composition(sk1, top))), meet(composition(sk1, zero), complement(composition(sk1, top))))))))
% 20.31/3.00 = { by lemma 47 }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, sk2), complement(composition(sk1, top))), meet(zero, complement(composition(sk1, top))))))))
% 20.31/3.00 = { by lemma 44 }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, sk2), complement(composition(sk1, top))), zero)))))
% 20.31/3.00 = { by lemma 35 }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(composition(sk1, top)))))))
% 20.31/3.00 = { by axiom 5 (def_top_12) }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(composition(sk1, join(sk3, complement(sk3)))))))))
% 20.31/3.00 = { by axiom 1 (converse_idempotence_8) R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(composition(sk1, join(sk3, converse(converse(complement(sk3)))))))))))
% 20.31/3.00 = { by axiom 1 (converse_idempotence_8) R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(converse(converse(composition(sk1, join(sk3, converse(converse(complement(sk3)))))))))))))
% 20.31/3.00 = { by axiom 9 (converse_multiplicativity_10) }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(converse(composition(converse(join(sk3, converse(converse(complement(sk3))))), converse(sk1)))))))))
% 20.31/3.00 = { by lemma 26 }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(converse(composition(join(converse(complement(sk3)), converse(sk3)), converse(sk1)))))))))
% 20.31/3.00 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(converse(composition(join(converse(sk3), converse(complement(sk3))), converse(sk1)))))))))
% 20.31/3.00 = { by axiom 12 (composition_distributivity_7) }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(converse(join(composition(converse(sk3), converse(sk1)), composition(converse(complement(sk3)), converse(sk1))))))))))
% 20.31/3.00 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(converse(join(converse(composition(sk1, sk3)), composition(converse(complement(sk3)), converse(sk1))))))))))
% 20.31/3.00 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(converse(join(composition(converse(complement(sk3)), converse(sk1)), converse(composition(sk1, sk3))))))))))
% 20.31/3.00 = { by axiom 1 (converse_idempotence_8) R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(converse(join(composition(converse(converse(converse(complement(sk3)))), converse(sk1)), converse(composition(sk1, sk3))))))))))
% 20.31/3.00 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(converse(join(converse(composition(sk1, converse(converse(complement(sk3))))), converse(composition(sk1, sk3))))))))))
% 20.31/3.00 = { by axiom 7 (converse_additivity_9) R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(converse(converse(join(composition(sk1, converse(converse(complement(sk3)))), composition(sk1, sk3))))))))))
% 20.31/3.00 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(converse(converse(join(composition(sk1, sk3), composition(sk1, converse(converse(complement(sk3)))))))))))))
% 20.31/3.00 = { by axiom 1 (converse_idempotence_8) }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(join(composition(sk1, sk3), composition(sk1, converse(converse(complement(sk3)))))))))))
% 20.31/3.00 = { by axiom 1 (converse_idempotence_8) }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(join(composition(sk1, sk3), composition(sk1, complement(sk3)))))))))
% 20.31/3.00 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), complement(join(composition(sk1, complement(sk3)), composition(sk1, sk3))))))))
% 20.31/3.00 = { by lemma 54 R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), meet(complement(composition(sk1, sk3)), complement(composition(sk1, complement(sk3)))))))))
% 20.31/3.00 = { by lemma 56 R->L }
% 20.31/3.00 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(complement(composition(sk1, complement(sk3))), meet(complement(composition(sk1, sk3)), composition(sk1, sk2)))))))
% 20.31/3.00 = { by lemma 38 }
% 20.31/3.01 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(complement(composition(sk1, complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))))
% 20.31/3.01 = { by lemma 38 R->L }
% 20.31/3.01 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), join(meet(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(composition(sk1, complement(sk3))))))))
% 20.31/3.01 = { by lemma 58 }
% 20.31/3.01 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))
% 20.31/3.01 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.31/3.01 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), composition(sk1, complement(sk3)))))
% 20.31/3.01 = { by lemma 48 }
% 20.31/3.01 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(sk3), composition(sk1, complement(sk3)))))
% 20.31/3.01 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.31/3.01 meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))))
% 20.31/3.01 = { by lemma 38 }
% 20.31/3.01 meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))
% 20.31/3.01 = { by lemma 35 R->L }
% 20.31/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), zero)
% 20.31/3.01 = { by lemma 16 R->L }
% 20.31/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), complement(top))
% 20.31/3.01 = { by lemma 25 R->L }
% 20.31/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), complement(join(composition(sk1, complement(sk3)), top)))
% 20.31/3.01 = { by lemma 55 R->L }
% 20.31/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), complement(join(composition(sk1, complement(sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), sk3))))))
% 20.31/3.01 = { by lemma 48 }
% 20.31/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), sk3))))))
% 20.31/3.01 = { by axiom 8 (maddux2_join_associativity_2) }
% 20.31/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), complement(join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), composition(sk1, complement(sk3))), complement(meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), sk3)))))
% 20.31/3.01 = { by lemma 50 }
% 20.31/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), composition(sk1, complement(sk3))))))
% 20.31/3.01 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.31/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), sk3), complement(join(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))
% 20.69/3.01 = { by lemma 38 }
% 20.69/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(complement(join(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), sk3)))
% 20.69/3.01 = { by lemma 38 }
% 20.69/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(complement(join(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))), meet(sk3, meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))
% 20.69/3.01 = { by lemma 43 R->L }
% 20.69/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(complement(join(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))), meet(sk3, meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), top))))
% 20.69/3.01 = { by lemma 39 R->L }
% 20.69/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(complement(join(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))), meet(sk3, complement(join(zero, complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))))
% 20.69/3.01 = { by lemma 56 R->L }
% 20.69/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(complement(join(zero, complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))), meet(sk3, complement(join(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))))
% 20.69/3.01 = { by lemma 39 }
% 20.69/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), top), meet(sk3, complement(join(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))))
% 20.69/3.01 = { by lemma 43 }
% 20.69/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(sk3, complement(join(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))))
% 20.69/3.01 = { by lemma 50 R->L }
% 20.69/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(join(join(composition(sk1, complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), complement(sk3)))))
% 20.69/3.01 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 20.69/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(join(composition(sk1, complement(sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(sk3))))))
% 20.69/3.01 = { by lemma 48 R->L }
% 20.69/3.01 join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, complement(sk3)), complement(sk3))))))
% 20.69/3.01 = { by lemma 58 }
% 20.69/3.01 meet(composition(sk1, sk2), complement(composition(sk1, sk3)))
% 20.69/3.01 % SZS output end Proof
% 20.69/3.01
% 20.69/3.01 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------