TSTP Solution File: REL017-2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL017-2 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n012.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 13:43:57 EDT 2023
% Result : Unsatisfiable 104.24s 13.70s
% Output : Proof 104.95s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL017-2 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n012.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 22:31:11 EDT 2023
% 0.13/0.34 % CPUTime :
% 104.24/13.70 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 104.24/13.70
% 104.24/13.70 % SZS status Unsatisfiable
% 104.24/13.70
% 104.95/13.78 % SZS output start Proof
% 104.95/13.78 Take the following subset of the input axioms:
% 104.95/13.78 fof(composition_associativity_5, axiom, ![A, B, C]: composition(A, composition(B, C))=composition(composition(A, B), C)).
% 104.95/13.78 fof(composition_distributivity_7, axiom, ![A2, B2, C2]: composition(join(A2, B2), C2)=join(composition(A2, C2), composition(B2, C2))).
% 104.95/13.78 fof(composition_identity_6, axiom, ![A2]: composition(A2, one)=A2).
% 104.95/13.78 fof(converse_additivity_9, axiom, ![A2, B2]: converse(join(A2, B2))=join(converse(A2), converse(B2))).
% 104.95/13.78 fof(converse_cancellativity_11, axiom, ![A2, B2]: join(composition(converse(A2), complement(composition(A2, B2))), complement(B2))=complement(B2)).
% 104.95/13.78 fof(converse_idempotence_8, axiom, ![A2]: converse(converse(A2))=A2).
% 104.95/13.78 fof(converse_multiplicativity_10, axiom, ![A2, B2]: converse(composition(A2, B2))=composition(converse(B2), converse(A2))).
% 104.95/13.78 fof(def_top_12, axiom, ![A2]: top=join(A2, complement(A2))).
% 104.95/13.78 fof(def_zero_13, axiom, ![A2]: zero=meet(A2, complement(A2))).
% 104.95/13.78 fof(goals_14, negated_conjecture, join(join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), complement(composition(sk1, meet(sk2, complement(sk3))))), composition(sk1, sk3))!=join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)) | join(join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2))), composition(sk1, sk3))!=join(complement(composition(sk1, sk2)), composition(sk1, sk3))).
% 104.95/13.78 fof(maddux1_join_commutativity_1, axiom, ![A2, B2]: join(A2, B2)=join(B2, A2)).
% 104.95/13.78 fof(maddux2_join_associativity_2, axiom, ![A2, B2, C2]: join(A2, join(B2, C2))=join(join(A2, B2), C2)).
% 104.95/13.78 fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A2, B2]: A2=join(complement(join(complement(A2), complement(B2))), complement(join(complement(A2), B2)))).
% 104.95/13.78 fof(maddux4_definiton_of_meet_4, axiom, ![A2, B2]: meet(A2, B2)=complement(join(complement(A2), complement(B2)))).
% 104.95/13.78
% 104.95/13.78 Now clausify the problem and encode Horn clauses using encoding 3 of
% 104.95/13.78 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 104.95/13.78 We repeatedly replace C & s=t => u=v by the two clauses:
% 104.95/13.78 fresh(y, y, x1...xn) = u
% 104.95/13.78 C => fresh(s, t, x1...xn) = v
% 104.95/13.78 where fresh is a fresh function symbol and x1..xn are the free
% 104.95/13.78 variables of u and v.
% 104.95/13.78 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 104.95/13.78 input problem has no model of domain size 1).
% 104.95/13.78
% 104.95/13.78 The encoding turns the above axioms into the following unit equations and goals:
% 104.95/13.78
% 104.95/13.78 Axiom 1 (composition_identity_6): composition(X, one) = X.
% 104.95/13.78 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 104.95/13.78 Axiom 3 (converse_idempotence_8): converse(converse(X)) = X.
% 104.95/13.78 Axiom 4 (def_top_12): top = join(X, complement(X)).
% 104.95/13.78 Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 104.95/13.78 Axiom 6 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 104.95/13.78 Axiom 7 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 104.95/13.78 Axiom 8 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 104.95/13.78 Axiom 9 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 104.95/13.78 Axiom 10 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 104.95/13.78 Axiom 11 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 104.95/13.78 Axiom 12 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 104.95/13.78 Axiom 13 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 104.95/13.78
% 104.95/13.78 Lemma 14: complement(top) = zero.
% 104.95/13.78 Proof:
% 104.95/13.78 complement(top)
% 104.95/13.78 = { by axiom 4 (def_top_12) }
% 104.95/13.78 complement(join(complement(X), complement(complement(X))))
% 104.95/13.78 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 104.95/13.78 meet(X, complement(X))
% 104.95/13.78 = { by axiom 5 (def_zero_13) R->L }
% 104.95/13.78 zero
% 104.95/13.78
% 104.95/13.78 Lemma 15: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 104.95/13.78 Proof:
% 104.95/13.78 converse(composition(converse(X), Y))
% 104.95/13.78 = { by axiom 6 (converse_multiplicativity_10) }
% 104.95/13.78 composition(converse(Y), converse(converse(X)))
% 104.95/13.78 = { by axiom 3 (converse_idempotence_8) }
% 104.95/13.78 composition(converse(Y), X)
% 104.95/13.78
% 104.95/13.78 Lemma 16: composition(converse(one), X) = X.
% 104.95/13.78 Proof:
% 104.95/13.78 composition(converse(one), X)
% 104.95/13.78 = { by lemma 15 R->L }
% 104.95/13.78 converse(composition(converse(X), one))
% 104.95/13.78 = { by axiom 1 (composition_identity_6) }
% 104.95/13.78 converse(converse(X))
% 104.95/13.78 = { by axiom 3 (converse_idempotence_8) }
% 104.95/13.78 X
% 104.95/13.78
% 104.95/13.79 Lemma 17: join(complement(X), complement(X)) = complement(X).
% 104.95/13.79 Proof:
% 104.95/13.79 join(complement(X), complement(X))
% 104.95/13.79 = { by lemma 16 R->L }
% 104.95/13.79 join(complement(X), complement(composition(converse(one), X)))
% 104.95/13.79 = { by axiom 1 (composition_identity_6) R->L }
% 104.95/13.79 join(complement(X), complement(composition(composition(converse(one), one), X)))
% 104.95/13.79 = { by axiom 7 (composition_associativity_5) R->L }
% 104.95/13.79 join(complement(X), complement(composition(converse(one), composition(one, X))))
% 104.95/13.79 = { by lemma 16 }
% 104.95/13.79 join(complement(X), complement(composition(one, X)))
% 104.95/13.79 = { by axiom 3 (converse_idempotence_8) R->L }
% 104.95/13.79 join(complement(X), complement(composition(converse(converse(one)), X)))
% 104.95/13.79 = { by lemma 16 R->L }
% 104.95/13.79 join(complement(X), composition(converse(one), complement(composition(converse(converse(one)), X))))
% 104.95/13.79 = { by axiom 3 (converse_idempotence_8) R->L }
% 104.95/13.79 join(complement(X), composition(converse(converse(converse(one))), complement(composition(converse(converse(one)), X))))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.79 join(composition(converse(converse(converse(one))), complement(composition(converse(converse(one)), X))), complement(X))
% 104.95/13.79 = { by axiom 12 (converse_cancellativity_11) }
% 104.95/13.79 complement(X)
% 104.95/13.79
% 104.95/13.79 Lemma 18: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 104.95/13.79 Proof:
% 104.95/13.79 join(meet(X, Y), complement(join(complement(X), Y)))
% 104.95/13.79 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 104.95/13.79 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 104.95/13.79 = { by axiom 13 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 104.95/13.79 X
% 104.95/13.79
% 104.95/13.79 Lemma 19: join(X, join(Y, complement(X))) = join(Y, top).
% 104.95/13.79 Proof:
% 104.95/13.79 join(X, join(Y, complement(X)))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.79 join(X, join(complement(X), Y))
% 104.95/13.79 = { by axiom 9 (maddux2_join_associativity_2) }
% 104.95/13.79 join(join(X, complement(X)), Y)
% 104.95/13.79 = { by axiom 4 (def_top_12) R->L }
% 104.95/13.79 join(top, Y)
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.79 join(Y, top)
% 104.95/13.79
% 104.95/13.79 Lemma 20: join(X, join(complement(X), Y)) = join(Y, top).
% 104.95/13.79 Proof:
% 104.95/13.79 join(X, join(complement(X), Y))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.79 join(X, join(Y, complement(X)))
% 104.95/13.79 = { by lemma 19 }
% 104.95/13.79 join(Y, top)
% 104.95/13.79
% 104.95/13.79 Lemma 21: join(top, complement(X)) = top.
% 104.95/13.79 Proof:
% 104.95/13.79 join(top, complement(X))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.79 join(complement(X), top)
% 104.95/13.79 = { by lemma 19 R->L }
% 104.95/13.79 join(complement(X), join(complement(X), complement(complement(X))))
% 104.95/13.79 = { by lemma 17 R->L }
% 104.95/13.79 join(complement(X), join(complement(X), complement(join(complement(X), complement(X)))))
% 104.95/13.79 = { by axiom 9 (maddux2_join_associativity_2) }
% 104.95/13.79 join(join(complement(X), complement(X)), complement(join(complement(X), complement(X))))
% 104.95/13.79 = { by axiom 4 (def_top_12) R->L }
% 104.95/13.79 top
% 104.95/13.79
% 104.95/13.79 Lemma 22: join(X, top) = top.
% 104.95/13.79 Proof:
% 104.95/13.79 join(X, top)
% 104.95/13.79 = { by axiom 4 (def_top_12) }
% 104.95/13.79 join(X, join(complement(X), complement(complement(X))))
% 104.95/13.79 = { by lemma 20 }
% 104.95/13.79 join(complement(complement(X)), top)
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.79 join(top, complement(complement(X)))
% 104.95/13.79 = { by lemma 21 }
% 104.95/13.79 top
% 104.95/13.79
% 104.95/13.79 Lemma 23: complement(zero) = top.
% 104.95/13.79 Proof:
% 104.95/13.79 complement(zero)
% 104.95/13.79 = { by lemma 17 R->L }
% 104.95/13.79 join(complement(zero), complement(zero))
% 104.95/13.79 = { by axiom 5 (def_zero_13) }
% 104.95/13.79 join(complement(zero), complement(meet(complement(zero), complement(complement(zero)))))
% 104.95/13.79 = { by lemma 18 R->L }
% 104.95/13.79 join(join(meet(complement(zero), complement(complement(zero))), complement(join(complement(complement(zero)), complement(complement(zero))))), complement(meet(complement(zero), complement(complement(zero)))))
% 104.95/13.79 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 104.95/13.79 join(meet(complement(zero), complement(complement(zero))), join(complement(join(complement(complement(zero)), complement(complement(zero)))), complement(meet(complement(zero), complement(complement(zero))))))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.79 join(meet(complement(zero), complement(complement(zero))), join(complement(meet(complement(zero), complement(complement(zero)))), complement(join(complement(complement(zero)), complement(complement(zero))))))
% 104.95/13.79 = { by lemma 20 }
% 104.95/13.79 join(complement(join(complement(complement(zero)), complement(complement(zero)))), top)
% 104.95/13.79 = { by lemma 22 }
% 104.95/13.79 top
% 104.95/13.79
% 104.95/13.79 Lemma 24: meet(Y, X) = meet(X, Y).
% 104.95/13.79 Proof:
% 104.95/13.79 meet(Y, X)
% 104.95/13.79 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 104.95/13.79 complement(join(complement(Y), complement(X)))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.79 complement(join(complement(X), complement(Y)))
% 104.95/13.79 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 104.95/13.79 meet(X, Y)
% 104.95/13.79
% 104.95/13.79 Lemma 25: complement(complement(X)) = X.
% 104.95/13.79 Proof:
% 104.95/13.79 complement(complement(X))
% 104.95/13.79 = { by lemma 18 R->L }
% 104.95/13.79 join(meet(complement(complement(X)), complement(X)), complement(join(complement(complement(complement(X))), complement(X))))
% 104.95/13.79 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 104.95/13.79 join(meet(complement(complement(X)), complement(X)), meet(complement(complement(X)), X))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.79 join(meet(complement(complement(X)), X), meet(complement(complement(X)), complement(X)))
% 104.95/13.79 = { by lemma 24 R->L }
% 104.95/13.79 join(meet(complement(complement(X)), X), meet(complement(X), complement(complement(X))))
% 104.95/13.79 = { by axiom 5 (def_zero_13) R->L }
% 104.95/13.79 join(meet(complement(complement(X)), X), zero)
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.79 join(zero, meet(complement(complement(X)), X))
% 104.95/13.79 = { by lemma 24 R->L }
% 104.95/13.79 join(zero, meet(X, complement(complement(X))))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.79 join(meet(X, complement(complement(X))), zero)
% 104.95/13.79 = { by lemma 14 R->L }
% 104.95/13.79 join(meet(X, complement(complement(X))), complement(top))
% 104.95/13.79 = { by axiom 4 (def_top_12) }
% 104.95/13.79 join(meet(X, complement(complement(X))), complement(join(complement(X), complement(complement(X)))))
% 104.95/13.79 = { by lemma 18 }
% 104.95/13.79 X
% 104.95/13.79
% 104.95/13.79 Lemma 26: join(zero, meet(X, X)) = X.
% 104.95/13.79 Proof:
% 104.95/13.79 join(zero, meet(X, X))
% 104.95/13.79 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 104.95/13.79 join(zero, complement(join(complement(X), complement(X))))
% 104.95/13.79 = { by axiom 5 (def_zero_13) }
% 104.95/13.79 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 104.95/13.79 = { by lemma 18 }
% 104.95/13.79 X
% 104.95/13.79
% 104.95/13.79 Lemma 27: join(zero, complement(complement(X))) = X.
% 104.95/13.79 Proof:
% 104.95/13.79 join(zero, complement(complement(X)))
% 104.95/13.79 = { by axiom 5 (def_zero_13) }
% 104.95/13.79 join(meet(X, complement(X)), complement(complement(X)))
% 104.95/13.79 = { by lemma 17 R->L }
% 104.95/13.79 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 104.95/13.79 = { by lemma 18 }
% 104.95/13.79 X
% 104.95/13.79
% 104.95/13.79 Lemma 28: join(X, X) = X.
% 104.95/13.79 Proof:
% 104.95/13.79 join(X, X)
% 104.95/13.79 = { by lemma 25 R->L }
% 104.95/13.79 join(X, complement(complement(X)))
% 104.95/13.79 = { by lemma 26 R->L }
% 104.95/13.79 join(join(zero, meet(X, X)), complement(complement(X)))
% 104.95/13.79 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 104.95/13.79 join(zero, join(meet(X, X), complement(complement(X))))
% 104.95/13.79 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 104.95/13.79 join(zero, join(complement(join(complement(X), complement(X))), complement(complement(X))))
% 104.95/13.79 = { by lemma 17 }
% 104.95/13.79 join(zero, join(complement(complement(X)), complement(complement(X))))
% 104.95/13.79 = { by lemma 17 }
% 104.95/13.79 join(zero, complement(complement(X)))
% 104.95/13.79 = { by lemma 27 }
% 104.95/13.79 X
% 104.95/13.79
% 104.95/13.79 Lemma 29: complement(join(zero, complement(X))) = meet(X, top).
% 104.95/13.79 Proof:
% 104.95/13.79 complement(join(zero, complement(X)))
% 104.95/13.79 = { by lemma 14 R->L }
% 104.95/13.79 complement(join(complement(top), complement(X)))
% 104.95/13.79 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 104.95/13.79 meet(top, X)
% 104.95/13.79 = { by lemma 24 R->L }
% 104.95/13.79 meet(X, top)
% 104.95/13.79
% 104.95/13.79 Lemma 30: join(meet(X, Y), complement(join(complement(Y), X))) = Y.
% 104.95/13.79 Proof:
% 104.95/13.79 join(meet(X, Y), complement(join(complement(Y), X)))
% 104.95/13.79 = { by lemma 24 }
% 104.95/13.79 join(meet(Y, X), complement(join(complement(Y), X)))
% 104.95/13.79 = { by lemma 18 }
% 104.95/13.79 Y
% 104.95/13.79
% 104.95/13.79 Lemma 31: join(zero, complement(X)) = complement(X).
% 104.95/13.79 Proof:
% 104.95/13.79 join(zero, complement(X))
% 104.95/13.79 = { by lemma 28 R->L }
% 104.95/13.79 join(zero, complement(join(X, X)))
% 104.95/13.79 = { by lemma 25 R->L }
% 104.95/13.79 join(zero, complement(join(complement(complement(X)), X)))
% 104.95/13.79 = { by axiom 5 (def_zero_13) }
% 104.95/13.79 join(meet(X, complement(X)), complement(join(complement(complement(X)), X)))
% 104.95/13.79 = { by lemma 30 }
% 104.95/13.79 complement(X)
% 104.95/13.79
% 104.95/13.79 Lemma 32: meet(X, top) = X.
% 104.95/13.79 Proof:
% 104.95/13.79 meet(X, top)
% 104.95/13.79 = { by lemma 29 R->L }
% 104.95/13.79 complement(join(zero, complement(X)))
% 104.95/13.79 = { by lemma 31 }
% 104.95/13.79 complement(complement(X))
% 104.95/13.79 = { by lemma 25 }
% 104.95/13.79 X
% 104.95/13.79
% 104.95/13.79 Lemma 33: meet(top, X) = X.
% 104.95/13.79 Proof:
% 104.95/13.79 meet(top, X)
% 104.95/13.79 = { by lemma 24 }
% 104.95/13.79 meet(X, top)
% 104.95/13.79 = { by lemma 32 }
% 104.95/13.79 X
% 104.95/13.79
% 104.95/13.79 Lemma 34: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 104.95/13.79 Proof:
% 104.95/13.79 join(Y, join(X, Z))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.79 join(join(X, Z), Y)
% 104.95/13.79 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 104.95/13.79 join(X, join(Z, Y))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.79 join(X, join(Y, Z))
% 104.95/13.79
% 104.95/13.79 Lemma 35: join(join(X, Y), Z) = join(join(Z, Y), X).
% 104.95/13.79 Proof:
% 104.95/13.79 join(join(X, Y), Z)
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.79 join(Z, join(X, Y))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.79 join(Z, join(Y, X))
% 104.95/13.79 = { by axiom 9 (maddux2_join_associativity_2) }
% 104.95/13.79 join(join(Z, Y), X)
% 104.95/13.79
% 104.95/13.79 Lemma 36: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 104.95/13.79 Proof:
% 104.95/13.79 meet(X, join(complement(Y), complement(Z)))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.79 meet(X, join(complement(Z), complement(Y)))
% 104.95/13.79 = { by lemma 24 }
% 104.95/13.79 meet(join(complement(Z), complement(Y)), X)
% 104.95/13.79 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 104.95/13.79 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 104.95/13.79 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 104.95/13.79 complement(join(meet(Z, Y), complement(X)))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.79 complement(join(complement(X), meet(Z, Y)))
% 104.95/13.79 = { by lemma 24 R->L }
% 104.95/13.79 complement(join(complement(X), meet(Y, Z)))
% 104.95/13.79
% 104.95/13.79 Lemma 37: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 104.95/13.79 Proof:
% 104.95/13.79 complement(join(X, complement(Y)))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.79 complement(join(complement(Y), X))
% 104.95/13.79 = { by lemma 33 R->L }
% 104.95/13.79 complement(join(complement(Y), meet(top, X)))
% 104.95/13.79 = { by lemma 36 R->L }
% 104.95/13.79 meet(Y, join(complement(top), complement(X)))
% 104.95/13.79 = { by lemma 14 }
% 104.95/13.79 meet(Y, join(zero, complement(X)))
% 104.95/13.79 = { by lemma 31 }
% 104.95/13.79 meet(Y, complement(X))
% 104.95/13.79
% 104.95/13.79 Lemma 38: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 104.95/13.79 Proof:
% 104.95/13.79 complement(join(complement(X), Y))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.79 complement(join(Y, complement(X)))
% 104.95/13.79 = { by lemma 37 }
% 104.95/13.79 meet(X, complement(Y))
% 104.95/13.79
% 104.95/13.79 Lemma 39: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 104.95/13.79 Proof:
% 104.95/13.79 complement(meet(X, complement(Y)))
% 104.95/13.79 = { by lemma 24 }
% 104.95/13.79 complement(meet(complement(Y), X))
% 104.95/13.79 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 104.95/13.79 complement(complement(join(complement(complement(Y)), complement(X))))
% 104.95/13.79 = { by lemma 31 R->L }
% 104.95/13.79 complement(join(zero, complement(join(complement(complement(Y)), complement(X)))))
% 104.95/13.79 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 104.95/13.79 complement(join(zero, meet(complement(Y), X)))
% 104.95/13.79 = { by lemma 24 R->L }
% 104.95/13.79 complement(join(zero, meet(X, complement(Y))))
% 104.95/13.79 = { by lemma 37 R->L }
% 104.95/13.79 complement(join(zero, complement(join(Y, complement(X)))))
% 104.95/13.79 = { by lemma 29 }
% 104.95/13.79 meet(join(Y, complement(X)), top)
% 104.95/13.79 = { by lemma 32 }
% 104.95/13.79 join(Y, complement(X))
% 104.95/13.79
% 104.95/13.79 Lemma 40: meet(X, join(X, join(Y, Z))) = X.
% 104.95/13.79 Proof:
% 104.95/13.79 meet(X, join(X, join(Y, Z)))
% 104.95/13.79 = { by lemma 34 }
% 104.95/13.79 meet(X, join(Y, join(X, Z)))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.79 meet(X, join(Y, join(Z, X)))
% 104.95/13.79 = { by lemma 27 R->L }
% 104.95/13.79 join(zero, complement(complement(meet(X, join(Y, join(Z, X))))))
% 104.95/13.79 = { by lemma 25 }
% 104.95/13.79 join(zero, meet(X, join(Y, join(Z, X))))
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.79 join(meet(X, join(Y, join(Z, X))), zero)
% 104.95/13.79 = { by lemma 14 R->L }
% 104.95/13.79 join(meet(X, join(Y, join(Z, X))), complement(top))
% 104.95/13.79 = { by lemma 25 R->L }
% 104.95/13.79 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(top))
% 104.95/13.79 = { by lemma 22 R->L }
% 104.95/13.79 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(join(Y, top)))
% 104.95/13.79 = { by lemma 22 R->L }
% 104.95/13.79 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(join(Y, join(Z, top))))
% 104.95/13.79 = { by axiom 9 (maddux2_join_associativity_2) }
% 104.95/13.79 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(join(join(Y, Z), top)))
% 104.95/13.79 = { by lemma 19 R->L }
% 104.95/13.79 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(join(complement(X), join(join(Y, Z), complement(complement(X))))))
% 104.95/13.79 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 104.95/13.79 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(join(complement(X), join(Y, join(Z, complement(complement(X)))))))
% 104.95/13.79 = { by lemma 18 }
% 104.95/13.79 X
% 104.95/13.79
% 104.95/13.79 Lemma 41: meet(X, join(X, Y)) = X.
% 104.95/13.79 Proof:
% 104.95/13.79 meet(X, join(X, Y))
% 104.95/13.79 = { by lemma 18 R->L }
% 104.95/13.79 meet(X, join(X, join(meet(Y, Z), complement(join(complement(Y), Z)))))
% 104.95/13.79 = { by lemma 40 }
% 104.95/13.79 X
% 104.95/13.79
% 104.95/13.79 Lemma 42: join(complement(X), join(X, Y)) = top.
% 104.95/13.79 Proof:
% 104.95/13.79 join(complement(X), join(X, Y))
% 104.95/13.79 = { by axiom 9 (maddux2_join_associativity_2) }
% 104.95/13.79 join(join(complement(X), X), Y)
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.79 join(join(X, complement(X)), Y)
% 104.95/13.79 = { by axiom 4 (def_top_12) R->L }
% 104.95/13.79 join(top, Y)
% 104.95/13.79 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.79 join(Y, top)
% 104.95/13.79 = { by lemma 22 }
% 104.95/13.79 top
% 104.95/13.79
% 104.95/13.79 Lemma 43: join(join(X, Y), join(Z, W)) = join(W, join(join(X, Y), Z)).
% 104.95/13.79 Proof:
% 104.95/13.79 join(join(X, Y), join(Z, W))
% 104.95/13.80 = { by axiom 9 (maddux2_join_associativity_2) }
% 104.95/13.80 join(join(join(X, Y), Z), W)
% 104.95/13.80 = { by lemma 35 }
% 104.95/13.80 join(join(join(Z, Y), X), W)
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.80 join(W, join(join(Z, Y), X))
% 104.95/13.80 = { by lemma 35 }
% 104.95/13.80 join(W, join(join(X, Y), Z))
% 104.95/13.80
% 104.95/13.80 Lemma 44: join(composition(X, Y), composition(X, Z)) = composition(X, join(Z, Y)).
% 104.95/13.80 Proof:
% 104.95/13.80 join(composition(X, Y), composition(X, Z))
% 104.95/13.80 = { by axiom 3 (converse_idempotence_8) R->L }
% 104.95/13.80 join(composition(X, Y), composition(converse(converse(X)), Z))
% 104.95/13.80 = { by axiom 3 (converse_idempotence_8) R->L }
% 104.95/13.80 join(converse(converse(composition(X, Y))), composition(converse(converse(X)), Z))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.80 join(composition(converse(converse(X)), Z), converse(converse(composition(X, Y))))
% 104.95/13.80 = { by lemma 15 R->L }
% 104.95/13.80 join(converse(composition(converse(Z), converse(X))), converse(converse(composition(X, Y))))
% 104.95/13.80 = { by axiom 8 (converse_additivity_9) R->L }
% 104.95/13.80 converse(join(composition(converse(Z), converse(X)), converse(composition(X, Y))))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.80 converse(join(converse(composition(X, Y)), composition(converse(Z), converse(X))))
% 104.95/13.80 = { by axiom 6 (converse_multiplicativity_10) }
% 104.95/13.80 converse(join(composition(converse(Y), converse(X)), composition(converse(Z), converse(X))))
% 104.95/13.80 = { by axiom 11 (composition_distributivity_7) R->L }
% 104.95/13.80 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.80 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 104.95/13.80 = { by axiom 6 (converse_multiplicativity_10) }
% 104.95/13.80 composition(converse(converse(X)), converse(join(converse(Z), converse(Y))))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.80 composition(converse(converse(X)), converse(join(converse(Y), converse(Z))))
% 104.95/13.80 = { by axiom 8 (converse_additivity_9) }
% 104.95/13.80 composition(converse(converse(X)), join(converse(converse(Y)), converse(converse(Z))))
% 104.95/13.80 = { by axiom 3 (converse_idempotence_8) }
% 104.95/13.80 composition(converse(converse(X)), join(Y, converse(converse(Z))))
% 104.95/13.80 = { by axiom 3 (converse_idempotence_8) }
% 104.95/13.80 composition(X, join(Y, converse(converse(Z))))
% 104.95/13.80 = { by axiom 3 (converse_idempotence_8) }
% 104.95/13.80 composition(X, join(Y, Z))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.80 composition(X, join(Z, Y))
% 104.95/13.80
% 104.95/13.80 Lemma 45: join(join(join(X, W), Y), Z) = join(X, join(join(Y, Z), W)).
% 104.95/13.80 Proof:
% 104.95/13.80 join(join(join(X, W), Y), Z)
% 104.95/13.80 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 104.95/13.80 join(join(X, W), join(Y, Z))
% 104.95/13.80 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 104.95/13.80 join(X, join(W, join(Y, Z)))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.80 join(X, join(join(Y, Z), W))
% 104.95/13.80
% 104.95/13.80 Lemma 46: join(join(join(Z, W), X), Y) = join(join(X, Y), join(Z, W)).
% 104.95/13.80 Proof:
% 104.95/13.80 join(join(join(Z, W), X), Y)
% 104.95/13.80 = { by lemma 45 }
% 104.95/13.80 join(Z, join(join(X, Y), W))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.80 join(join(join(X, Y), W), Z)
% 104.95/13.80 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 104.95/13.80 join(join(X, Y), join(W, Z))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.80 join(join(X, Y), join(Z, W))
% 104.95/13.80
% 104.95/13.80 Lemma 47: join(join(join(X, Y), Z), W) = join(join(join(Z, W), X), Y).
% 104.95/13.80 Proof:
% 104.95/13.80 join(join(join(X, Y), Z), W)
% 104.95/13.80 = { by lemma 46 }
% 104.95/13.80 join(join(Z, W), join(X, Y))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.80 join(join(X, Y), join(Z, W))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.80 join(join(X, Y), join(W, Z))
% 104.95/13.80 = { by axiom 9 (maddux2_join_associativity_2) }
% 104.95/13.80 join(join(join(X, Y), W), Z)
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.80 join(Z, join(join(X, Y), W))
% 104.95/13.80 = { by lemma 45 R->L }
% 104.95/13.80 join(join(join(Z, W), X), Y)
% 104.95/13.80
% 104.95/13.80 Lemma 48: meet(complement(meet(X, Y)), join(X, complement(Y))) = complement(Y).
% 104.95/13.80 Proof:
% 104.95/13.80 meet(complement(meet(X, Y)), join(X, complement(Y)))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.80 meet(complement(meet(X, Y)), join(complement(Y), X))
% 104.95/13.80 = { by lemma 24 }
% 104.95/13.80 meet(complement(meet(Y, X)), join(complement(Y), X))
% 104.95/13.80 = { by lemma 24 }
% 104.95/13.80 meet(join(complement(Y), X), complement(meet(Y, X)))
% 104.95/13.80 = { by lemma 37 R->L }
% 104.95/13.80 complement(join(meet(Y, X), complement(join(complement(Y), X))))
% 104.95/13.80 = { by lemma 18 }
% 104.95/13.80 complement(Y)
% 104.95/13.80
% 104.95/13.80 Lemma 49: join(join(join(X, Y), complement(Z)), Y) = join(join(X, Y), complement(Z)).
% 104.95/13.80 Proof:
% 104.95/13.80 join(join(join(X, Y), complement(Z)), Y)
% 104.95/13.80 = { by lemma 41 R->L }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), join(join(join(join(X, Y), complement(Z)), Y), X))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), join(X, join(join(join(X, Y), complement(Z)), Y)))
% 104.95/13.80 = { by lemma 47 R->L }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), join(X, join(join(join(complement(Z), Y), X), Y)))
% 104.95/13.80 = { by lemma 45 }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), join(X, join(complement(Z), join(join(X, Y), Y))))
% 104.95/13.80 = { by lemma 34 R->L }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), join(complement(Z), join(X, join(join(X, Y), Y))))
% 104.95/13.80 = { by lemma 34 R->L }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), join(complement(Z), join(join(X, Y), join(X, Y))))
% 104.95/13.80 = { by lemma 34 R->L }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), join(join(X, Y), join(complement(Z), join(X, Y))))
% 104.95/13.80 = { by lemma 43 }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), join(join(X, Y), join(join(X, Y), complement(Z))))
% 104.95/13.80 = { by lemma 25 R->L }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), join(join(X, Y), complement(complement(join(join(X, Y), complement(Z))))))
% 104.95/13.80 = { by lemma 33 R->L }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), meet(top, join(join(X, Y), complement(complement(join(join(X, Y), complement(Z)))))))
% 104.95/13.80 = { by lemma 23 R->L }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), meet(complement(zero), join(join(X, Y), complement(complement(join(join(X, Y), complement(Z)))))))
% 104.95/13.80 = { by lemma 14 R->L }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), meet(complement(complement(top)), join(join(X, Y), complement(complement(join(join(X, Y), complement(Z)))))))
% 104.95/13.80 = { by lemma 21 R->L }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), meet(complement(complement(join(top, complement(Z)))), join(join(X, Y), complement(complement(join(join(X, Y), complement(Z)))))))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), meet(complement(complement(join(complement(Z), top))), join(join(X, Y), complement(complement(join(join(X, Y), complement(Z)))))))
% 104.95/13.80 = { by lemma 19 R->L }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), meet(complement(complement(join(join(X, Y), join(complement(Z), complement(join(X, Y)))))), join(join(X, Y), complement(complement(join(join(X, Y), complement(Z)))))))
% 104.95/13.80 = { by lemma 43 }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), meet(complement(complement(join(complement(join(X, Y)), join(join(X, Y), complement(Z))))), join(join(X, Y), complement(complement(join(join(X, Y), complement(Z)))))))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), meet(complement(complement(join(join(join(X, Y), complement(Z)), complement(join(X, Y))))), join(join(X, Y), complement(complement(join(join(X, Y), complement(Z)))))))
% 104.95/13.80 = { by lemma 37 }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), meet(complement(meet(join(X, Y), complement(join(join(X, Y), complement(Z))))), join(join(X, Y), complement(complement(join(join(X, Y), complement(Z)))))))
% 104.95/13.80 = { by lemma 48 }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), complement(complement(join(join(X, Y), complement(Z)))))
% 104.95/13.80 = { by lemma 25 }
% 104.95/13.80 meet(join(join(join(X, Y), complement(Z)), Y), join(join(X, Y), complement(Z)))
% 104.95/13.80 = { by lemma 24 }
% 104.95/13.80 meet(join(join(X, Y), complement(Z)), join(join(join(X, Y), complement(Z)), Y))
% 104.95/13.80 = { by lemma 25 R->L }
% 104.95/13.80 meet(join(join(X, Y), complement(Z)), complement(complement(join(join(join(X, Y), complement(Z)), Y))))
% 104.95/13.80 = { by lemma 48 R->L }
% 104.95/13.80 meet(join(join(X, Y), complement(Z)), meet(complement(meet(join(X, Y), complement(join(join(join(X, Y), complement(Z)), Y)))), join(join(X, Y), complement(complement(join(join(join(X, Y), complement(Z)), Y))))))
% 104.95/13.80 = { by lemma 37 R->L }
% 104.95/13.80 meet(join(join(X, Y), complement(Z)), meet(complement(complement(join(join(join(join(X, Y), complement(Z)), Y), complement(join(X, Y))))), join(join(X, Y), complement(complement(join(join(join(X, Y), complement(Z)), Y))))))
% 104.95/13.80 = { by lemma 47 R->L }
% 104.95/13.80 meet(join(join(X, Y), complement(Z)), meet(complement(complement(join(join(join(join(complement(Z), Y), X), Y), complement(join(X, Y))))), join(join(X, Y), complement(complement(join(join(join(X, Y), complement(Z)), Y))))))
% 104.95/13.80 = { by lemma 46 }
% 104.95/13.80 meet(join(join(X, Y), complement(Z)), meet(complement(complement(join(join(join(X, Y), join(complement(Z), Y)), complement(join(X, Y))))), join(join(X, Y), complement(complement(join(join(join(X, Y), complement(Z)), Y))))))
% 104.95/13.80 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 104.95/13.80 meet(join(join(X, Y), complement(Z)), meet(complement(complement(join(join(X, Y), join(join(complement(Z), Y), complement(join(X, Y)))))), join(join(X, Y), complement(complement(join(join(join(X, Y), complement(Z)), Y))))))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.80 meet(join(join(X, Y), complement(Z)), meet(complement(complement(join(join(X, Y), join(complement(join(X, Y)), join(complement(Z), Y))))), join(join(X, Y), complement(complement(join(join(join(X, Y), complement(Z)), Y))))))
% 104.95/13.80 = { by lemma 20 }
% 104.95/13.80 meet(join(join(X, Y), complement(Z)), meet(complement(complement(join(join(complement(Z), Y), top))), join(join(X, Y), complement(complement(join(join(join(X, Y), complement(Z)), Y))))))
% 104.95/13.80 = { by lemma 22 }
% 104.95/13.80 meet(join(join(X, Y), complement(Z)), meet(complement(complement(top)), join(join(X, Y), complement(complement(join(join(join(X, Y), complement(Z)), Y))))))
% 104.95/13.80 = { by lemma 14 }
% 104.95/13.80 meet(join(join(X, Y), complement(Z)), meet(complement(zero), join(join(X, Y), complement(complement(join(join(join(X, Y), complement(Z)), Y))))))
% 104.95/13.80 = { by lemma 23 }
% 104.95/13.80 meet(join(join(X, Y), complement(Z)), meet(top, join(join(X, Y), complement(complement(join(join(join(X, Y), complement(Z)), Y))))))
% 104.95/13.80 = { by lemma 33 }
% 104.95/13.80 meet(join(join(X, Y), complement(Z)), join(join(X, Y), complement(complement(join(join(join(X, Y), complement(Z)), Y)))))
% 104.95/13.80 = { by lemma 25 }
% 104.95/13.80 meet(join(join(X, Y), complement(Z)), join(join(X, Y), join(join(join(X, Y), complement(Z)), Y)))
% 104.95/13.80 = { by lemma 34 R->L }
% 104.95/13.80 meet(join(join(X, Y), complement(Z)), join(join(join(X, Y), complement(Z)), join(join(X, Y), Y)))
% 104.95/13.80 = { by lemma 41 }
% 104.95/13.80 join(join(X, Y), complement(Z))
% 104.95/13.80
% 104.95/13.80 Lemma 50: meet(join(X, Y), join(X, complement(Y))) = X.
% 104.95/13.80 Proof:
% 104.95/13.80 meet(join(X, Y), join(X, complement(Y)))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.80 meet(join(Y, X), join(X, complement(Y)))
% 104.95/13.80 = { by lemma 24 }
% 104.95/13.80 meet(join(X, complement(Y)), join(Y, X))
% 104.95/13.80 = { by lemma 32 R->L }
% 104.95/13.80 meet(join(X, complement(Y)), join(Y, meet(X, top)))
% 104.95/13.80 = { by lemma 39 R->L }
% 104.95/13.80 meet(complement(meet(Y, complement(X))), join(Y, meet(X, top)))
% 104.95/13.80 = { by lemma 31 R->L }
% 104.95/13.80 meet(complement(meet(Y, join(zero, complement(X)))), join(Y, meet(X, top)))
% 104.95/13.80 = { by lemma 29 R->L }
% 104.95/13.80 meet(complement(meet(Y, join(zero, complement(X)))), join(Y, complement(join(zero, complement(X)))))
% 104.95/13.80 = { by lemma 48 }
% 104.95/13.80 complement(join(zero, complement(X)))
% 104.95/13.80 = { by lemma 29 }
% 104.95/13.80 meet(X, top)
% 104.95/13.80 = { by lemma 32 }
% 104.95/13.80 X
% 104.95/13.80
% 104.95/13.80 Lemma 51: join(meet(X, Y), complement(join(X, complement(Y)))) = Y.
% 104.95/13.80 Proof:
% 104.95/13.80 join(meet(X, Y), complement(join(X, complement(Y))))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.80 join(meet(X, Y), complement(join(complement(Y), X)))
% 104.95/13.80 = { by lemma 30 }
% 104.95/13.80 Y
% 104.95/13.80
% 104.95/13.80 Lemma 52: join(composition(X, meet(Y, complement(Z))), composition(X, Z)) = join(composition(X, Y), composition(X, Z)).
% 104.95/13.80 Proof:
% 104.95/13.80 join(composition(X, meet(Y, complement(Z))), composition(X, Z))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.80 join(composition(X, Z), composition(X, meet(Y, complement(Z))))
% 104.95/13.80 = { by lemma 44 }
% 104.95/13.80 composition(X, join(meet(Y, complement(Z)), Z))
% 104.95/13.80 = { by lemma 25 R->L }
% 104.95/13.80 composition(X, join(meet(Y, complement(Z)), complement(complement(Z))))
% 104.95/13.80 = { by lemma 39 R->L }
% 104.95/13.80 composition(X, complement(meet(complement(Z), complement(meet(Y, complement(Z))))))
% 104.95/13.80 = { by lemma 24 }
% 104.95/13.80 composition(X, complement(meet(complement(meet(Y, complement(Z))), complement(Z))))
% 104.95/13.80 = { by lemma 40 R->L }
% 104.95/13.80 composition(X, complement(meet(complement(meet(meet(Y, complement(Z)), join(meet(Y, complement(Z)), join(Z, meet(Z, Y))))), complement(Z))))
% 104.95/13.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.80 composition(X, complement(meet(complement(meet(meet(Y, complement(Z)), join(meet(Y, complement(Z)), join(meet(Z, Y), Z)))), complement(Z))))
% 104.95/13.80 = { by axiom 9 (maddux2_join_associativity_2) }
% 104.95/13.80 composition(X, complement(meet(complement(meet(meet(Y, complement(Z)), join(join(meet(Y, complement(Z)), meet(Z, Y)), Z))), complement(Z))))
% 104.95/13.80 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 104.95/13.81 composition(X, complement(meet(complement(meet(meet(Y, complement(Z)), join(join(meet(Y, complement(Z)), complement(join(complement(Z), complement(Y)))), Z))), complement(Z))))
% 104.95/13.81 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 104.95/13.81 composition(X, complement(meet(complement(meet(meet(Y, complement(Z)), join(join(meet(Y, complement(Z)), complement(join(complement(Y), complement(Z)))), Z))), complement(Z))))
% 104.95/13.81 = { by lemma 18 }
% 104.95/13.81 composition(X, complement(meet(complement(meet(meet(Y, complement(Z)), join(Y, Z))), complement(Z))))
% 104.95/13.81 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.81 composition(X, complement(meet(complement(meet(meet(Y, complement(Z)), join(Z, Y))), complement(Z))))
% 104.95/13.81 = { by lemma 24 }
% 104.95/13.81 composition(X, complement(meet(complement(meet(join(Z, Y), meet(Y, complement(Z)))), complement(Z))))
% 104.95/13.81 = { by lemma 28 R->L }
% 104.95/13.81 composition(X, complement(meet(complement(join(meet(join(Z, Y), meet(Y, complement(Z))), meet(join(Z, Y), meet(Y, complement(Z))))), complement(Z))))
% 104.95/13.81 = { by lemma 32 R->L }
% 104.95/13.81 composition(X, complement(meet(complement(join(meet(join(Z, Y), meet(Y, complement(Z))), meet(meet(join(Z, Y), meet(Y, complement(Z))), top))), complement(Z))))
% 104.95/13.81 = { by lemma 29 R->L }
% 104.95/13.81 composition(X, complement(meet(complement(join(meet(join(Z, Y), meet(Y, complement(Z))), complement(join(zero, complement(meet(join(Z, Y), meet(Y, complement(Z)))))))), complement(Z))))
% 104.95/13.81 = { by lemma 37 }
% 104.95/13.81 composition(X, complement(meet(meet(join(zero, complement(meet(join(Z, Y), meet(Y, complement(Z))))), complement(meet(join(Z, Y), meet(Y, complement(Z))))), complement(Z))))
% 104.95/13.81 = { by lemma 31 }
% 104.95/13.81 composition(X, complement(meet(meet(complement(meet(join(Z, Y), meet(Y, complement(Z)))), complement(meet(join(Z, Y), meet(Y, complement(Z))))), complement(Z))))
% 104.95/13.81 = { by lemma 38 R->L }
% 104.95/13.81 composition(X, complement(meet(complement(join(complement(complement(meet(join(Z, Y), meet(Y, complement(Z))))), meet(join(Z, Y), meet(Y, complement(Z))))), complement(Z))))
% 104.95/13.81 = { by lemma 36 R->L }
% 104.95/13.81 composition(X, complement(meet(meet(complement(meet(join(Z, Y), meet(Y, complement(Z)))), join(complement(join(Z, Y)), complement(meet(Y, complement(Z))))), complement(Z))))
% 104.95/13.81 = { by lemma 24 }
% 104.95/13.81 composition(X, complement(meet(meet(join(complement(join(Z, Y)), complement(meet(Y, complement(Z)))), complement(meet(join(Z, Y), meet(Y, complement(Z))))), complement(Z))))
% 104.95/13.81 = { by lemma 38 R->L }
% 104.95/13.81 composition(X, complement(meet(complement(join(complement(join(complement(join(Z, Y)), complement(meet(Y, complement(Z))))), meet(join(Z, Y), meet(Y, complement(Z))))), complement(Z))))
% 104.95/13.81 = { by lemma 31 R->L }
% 104.95/13.81 composition(X, complement(meet(join(zero, complement(join(complement(join(complement(join(Z, Y)), complement(meet(Y, complement(Z))))), meet(join(Z, Y), meet(Y, complement(Z)))))), complement(Z))))
% 104.95/13.81 = { by lemma 36 R->L }
% 104.95/13.81 composition(X, complement(meet(join(zero, meet(join(complement(join(Z, Y)), complement(meet(Y, complement(Z)))), join(complement(join(Z, Y)), complement(meet(Y, complement(Z)))))), complement(Z))))
% 104.95/13.81 = { by lemma 26 }
% 104.95/13.81 composition(X, complement(meet(join(complement(join(Z, Y)), complement(meet(Y, complement(Z)))), complement(Z))))
% 104.95/13.81 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.81 composition(X, complement(meet(join(complement(meet(Y, complement(Z))), complement(join(Z, Y))), complement(Z))))
% 104.95/13.81 = { by lemma 39 }
% 104.95/13.81 composition(X, join(Z, complement(join(complement(meet(Y, complement(Z))), complement(join(Z, Y))))))
% 104.95/13.81 = { by lemma 25 R->L }
% 104.95/13.81 composition(X, join(complement(complement(Z)), complement(join(complement(meet(Y, complement(Z))), complement(join(Z, Y))))))
% 104.95/13.81 = { by lemma 51 R->L }
% 104.95/13.81 composition(X, join(complement(join(meet(Y, complement(Z)), complement(join(Y, complement(complement(Z)))))), complement(join(complement(meet(Y, complement(Z))), complement(join(Z, Y))))))
% 104.95/13.81 = { by lemma 37 }
% 104.95/13.81 composition(X, join(meet(join(Y, complement(complement(Z))), complement(meet(Y, complement(Z)))), complement(join(complement(meet(Y, complement(Z))), complement(join(Z, Y))))))
% 104.95/13.81 = { by lemma 25 }
% 104.95/13.81 composition(X, join(meet(join(Y, Z), complement(meet(Y, complement(Z)))), complement(join(complement(meet(Y, complement(Z))), complement(join(Z, Y))))))
% 104.95/13.81 = { by lemma 24 R->L }
% 104.95/13.81 composition(X, join(meet(complement(meet(Y, complement(Z))), join(Y, Z)), complement(join(complement(meet(Y, complement(Z))), complement(join(Z, Y))))))
% 104.95/13.81 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.81 composition(X, join(meet(complement(meet(Y, complement(Z))), join(Z, Y)), complement(join(complement(meet(Y, complement(Z))), complement(join(Z, Y))))))
% 104.95/13.81 = { by lemma 51 }
% 104.95/13.81 composition(X, join(Z, Y))
% 104.95/13.81 = { by lemma 44 R->L }
% 104.95/13.81 join(composition(X, Y), composition(X, Z))
% 104.95/13.81
% 104.95/13.81 Goal 1 (goals_14): tuple(join(join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2))), composition(sk1, sk3)), join(join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), complement(composition(sk1, meet(sk2, complement(sk3))))), composition(sk1, sk3))) = tuple(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3))).
% 104.95/13.81 Proof:
% 104.95/13.81 tuple(join(join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2))), composition(sk1, sk3)), join(join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), complement(composition(sk1, meet(sk2, complement(sk3))))), composition(sk1, sk3)))
% 104.95/13.81 = { by lemma 47 }
% 104.95/13.81 tuple(join(join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2))), composition(sk1, sk3)), join(join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2))), composition(sk1, sk3)))
% 104.95/13.81 = { by lemma 49 }
% 104.95/13.81 tuple(join(join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2))), composition(sk1, sk3)), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2))))
% 104.95/13.81 = { by lemma 33 R->L }
% 104.95/13.81 tuple(join(join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2))), composition(sk1, sk3)), meet(top, join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2)))))
% 104.95/13.81 = { by lemma 42 R->L }
% 104.95/13.81 tuple(join(join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2))), composition(sk1, sk3)), meet(join(complement(composition(sk1, meet(sk2, complement(sk3)))), join(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2)))))
% 104.95/13.81 = { by lemma 52 }
% 104.95/13.81 tuple(join(join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2))), composition(sk1, sk3)), meet(join(complement(composition(sk1, meet(sk2, complement(sk3)))), join(composition(sk1, sk2), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2)))))
% 104.95/13.81 = { by lemma 34 R->L }
% 104.95/13.81 tuple(join(join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2))), composition(sk1, sk3)), meet(join(composition(sk1, sk2), join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2)))))
% 104.95/13.81 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.81 tuple(join(join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2))), composition(sk1, sk3)), meet(join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), composition(sk1, sk2)), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2)))))
% 104.95/13.81 = { by lemma 50 }
% 104.95/13.81 tuple(join(join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2))), composition(sk1, sk3)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)))
% 104.95/13.81 = { by lemma 49 }
% 104.95/13.81 tuple(join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2))), join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)))
% 104.95/13.81 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 104.95/13.81 tuple(join(complement(composition(sk1, meet(sk2, complement(sk3)))), join(composition(sk1, sk3), complement(composition(sk1, sk2)))), join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)))
% 104.95/13.81 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.81 tuple(join(complement(composition(sk1, meet(sk2, complement(sk3)))), join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)))
% 104.95/13.81 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.81 tuple(join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), complement(composition(sk1, meet(sk2, complement(sk3))))), join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)))
% 104.95/13.81 = { by lemma 33 R->L }
% 104.95/13.81 tuple(meet(top, join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), complement(composition(sk1, meet(sk2, complement(sk3)))))), join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)))
% 104.95/13.81 = { by lemma 42 R->L }
% 104.95/13.81 tuple(meet(join(complement(composition(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, sk3))), join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), complement(composition(sk1, meet(sk2, complement(sk3)))))), join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)))
% 104.95/13.81 = { by lemma 52 R->L }
% 104.95/13.81 tuple(meet(join(complement(composition(sk1, sk2)), join(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3))), join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), complement(composition(sk1, meet(sk2, complement(sk3)))))), join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)))
% 104.95/13.81 = { by lemma 34 R->L }
% 104.95/13.81 tuple(meet(join(composition(sk1, meet(sk2, complement(sk3))), join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), complement(composition(sk1, meet(sk2, complement(sk3)))))), join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)))
% 104.95/13.81 = { by axiom 2 (maddux1_join_commutativity_1) }
% 104.95/13.81 tuple(meet(join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), composition(sk1, meet(sk2, complement(sk3)))), join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), complement(composition(sk1, meet(sk2, complement(sk3)))))), join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)))
% 104.95/13.81 = { by lemma 50 }
% 104.95/13.81 tuple(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)))
% 104.95/13.81 % SZS output end Proof
% 104.95/13.81
% 104.95/13.81 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------