TSTP Solution File: REL005-4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL005-4 : 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 : n011.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 13:43:46 EDT 2023
% Result : Unsatisfiable 3.69s 0.86s
% Output : Proof 4.31s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : REL005-4 : TPTP v8.1.2. Released v4.0.0.
% 0.11/0.13 % 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 : n011.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 19:57:53 EDT 2023
% 0.13/0.35 % CPUTime :
% 3.69/0.86 Command-line arguments: --flatten
% 3.69/0.86
% 3.69/0.86 % SZS status Unsatisfiable
% 3.69/0.86
% 3.69/0.91 % SZS output start Proof
% 3.69/0.91 Take the following subset of the input axioms:
% 3.69/0.91 fof(composition_associativity_5, axiom, ![A, B, C]: composition(A, composition(B, C))=composition(composition(A, B), C)).
% 3.69/0.91 fof(composition_identity_6, axiom, ![A2]: composition(A2, one)=A2).
% 3.69/0.91 fof(converse_additivity_9, axiom, ![A2, B2]: converse(join(A2, B2))=join(converse(A2), converse(B2))).
% 3.69/0.91 fof(converse_cancellativity_11, axiom, ![A2, B2]: join(composition(converse(A2), complement(composition(A2, B2))), complement(B2))=complement(B2)).
% 3.69/0.91 fof(converse_idempotence_8, axiom, ![A2]: converse(converse(A2))=A2).
% 3.69/0.91 fof(converse_multiplicativity_10, axiom, ![A2, B2]: converse(composition(A2, B2))=composition(converse(B2), converse(A2))).
% 3.69/0.91 fof(def_top_12, axiom, ![A2]: top=join(A2, complement(A2))).
% 3.69/0.91 fof(def_zero_13, axiom, ![A2]: zero=meet(A2, complement(A2))).
% 3.69/0.91 fof(goals_17, negated_conjecture, join(converse(meet(sk1, sk2)), meet(converse(sk1), converse(sk2)))!=meet(converse(sk1), converse(sk2)) | join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))!=converse(meet(sk1, sk2))).
% 3.69/0.91 fof(maddux1_join_commutativity_1, axiom, ![A2, B2]: join(A2, B2)=join(B2, A2)).
% 3.69/0.91 fof(maddux2_join_associativity_2, axiom, ![A2, B2, C2]: join(A2, join(B2, C2))=join(join(A2, B2), C2)).
% 3.69/0.91 fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A2, B2]: A2=join(complement(join(complement(A2), complement(B2))), complement(join(complement(A2), B2)))).
% 3.69/0.91 fof(maddux4_definiton_of_meet_4, axiom, ![A2, B2]: meet(A2, B2)=complement(join(complement(A2), complement(B2)))).
% 3.69/0.91
% 3.69/0.91 Now clausify the problem and encode Horn clauses using encoding 3 of
% 3.69/0.91 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 3.69/0.91 We repeatedly replace C & s=t => u=v by the two clauses:
% 3.69/0.91 fresh(y, y, x1...xn) = u
% 3.69/0.91 C => fresh(s, t, x1...xn) = v
% 3.69/0.91 where fresh is a fresh function symbol and x1..xn are the free
% 3.69/0.91 variables of u and v.
% 3.69/0.91 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 3.69/0.91 input problem has no model of domain size 1).
% 3.69/0.91
% 3.69/0.91 The encoding turns the above axioms into the following unit equations and goals:
% 3.69/0.91
% 3.69/0.91 Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 3.69/0.91 Axiom 2 (composition_identity_6): composition(X, one) = X.
% 3.69/0.91 Axiom 3 (converse_idempotence_8): converse(converse(X)) = X.
% 3.69/0.91 Axiom 4 (def_top_12): top = join(X, complement(X)).
% 3.69/0.91 Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 3.69/0.91 Axiom 6 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 3.69/0.91 Axiom 7 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 3.69/0.91 Axiom 8 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 3.69/0.91 Axiom 9 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 3.69/0.91 Axiom 10 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 3.69/0.91 Axiom 11 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 3.69/0.91 Axiom 12 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 3.69/0.91
% 3.69/0.91 Lemma 13: complement(top) = zero.
% 3.69/0.91 Proof:
% 3.69/0.91 complement(top)
% 3.69/0.91 = { by axiom 4 (def_top_12) }
% 3.69/0.91 complement(join(complement(X), complement(complement(X))))
% 3.69/0.91 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 3.69/0.91 meet(X, complement(X))
% 3.69/0.91 = { by axiom 5 (def_zero_13) R->L }
% 4.31/0.91 zero
% 4.31/0.91
% 4.31/0.91 Lemma 14: join(X, join(Y, complement(X))) = join(Y, top).
% 4.31/0.91 Proof:
% 4.31/0.91 join(X, join(Y, complement(X)))
% 4.31/0.91 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.91 join(X, join(complement(X), Y))
% 4.31/0.91 = { by axiom 7 (maddux2_join_associativity_2) }
% 4.31/0.91 join(join(X, complement(X)), Y)
% 4.31/0.91 = { by axiom 4 (def_top_12) R->L }
% 4.31/0.91 join(top, Y)
% 4.31/0.91 = { by axiom 1 (maddux1_join_commutativity_1) }
% 4.31/0.91 join(Y, top)
% 4.31/0.91
% 4.31/0.91 Lemma 15: composition(converse(one), X) = X.
% 4.31/0.91 Proof:
% 4.31/0.91 composition(converse(one), X)
% 4.31/0.91 = { by axiom 3 (converse_idempotence_8) R->L }
% 4.31/0.91 composition(converse(one), converse(converse(X)))
% 4.31/0.91 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 4.31/0.91 converse(composition(converse(X), one))
% 4.31/0.91 = { by axiom 2 (composition_identity_6) }
% 4.31/0.91 converse(converse(X))
% 4.31/0.91 = { by axiom 3 (converse_idempotence_8) }
% 4.31/0.92 X
% 4.31/0.92
% 4.31/0.92 Lemma 16: join(complement(X), complement(X)) = complement(X).
% 4.31/0.92 Proof:
% 4.31/0.92 join(complement(X), complement(X))
% 4.31/0.92 = { by lemma 15 R->L }
% 4.31/0.92 join(complement(X), composition(converse(one), complement(X)))
% 4.31/0.92 = { by lemma 15 R->L }
% 4.31/0.92 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 4.31/0.92 = { by axiom 2 (composition_identity_6) R->L }
% 4.31/0.92 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 4.31/0.92 = { by axiom 9 (composition_associativity_5) R->L }
% 4.31/0.92 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 4.31/0.92 = { by lemma 15 }
% 4.31/0.92 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 4.31/0.92 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.92 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 4.31/0.92 = { by axiom 11 (converse_cancellativity_11) }
% 4.31/0.92 complement(X)
% 4.31/0.92
% 4.31/0.92 Lemma 17: join(top, complement(X)) = top.
% 4.31/0.92 Proof:
% 4.31/0.92 join(top, complement(X))
% 4.31/0.92 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.92 join(complement(X), top)
% 4.31/0.92 = { by lemma 14 R->L }
% 4.31/0.92 join(X, join(complement(X), complement(X)))
% 4.31/0.92 = { by lemma 16 }
% 4.31/0.92 join(X, complement(X))
% 4.31/0.92 = { by axiom 4 (def_top_12) R->L }
% 4.31/0.92 top
% 4.31/0.92
% 4.31/0.92 Lemma 18: join(Y, top) = join(X, top).
% 4.31/0.92 Proof:
% 4.31/0.92 join(Y, top)
% 4.31/0.92 = { by lemma 17 R->L }
% 4.31/0.92 join(Y, join(top, complement(Y)))
% 4.31/0.92 = { by lemma 14 }
% 4.31/0.92 join(top, top)
% 4.31/0.92 = { by lemma 14 R->L }
% 4.31/0.92 join(X, join(top, complement(X)))
% 4.31/0.92 = { by lemma 17 }
% 4.31/0.92 join(X, top)
% 4.31/0.92
% 4.31/0.92 Lemma 19: join(X, top) = top.
% 4.31/0.92 Proof:
% 4.31/0.92 join(X, top)
% 4.31/0.92 = { by lemma 18 }
% 4.31/0.92 join(join(zero, zero), top)
% 4.31/0.92 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.92 join(top, join(zero, zero))
% 4.31/0.92 = { by lemma 13 R->L }
% 4.31/0.92 join(top, join(zero, complement(top)))
% 4.31/0.92 = { by lemma 13 R->L }
% 4.31/0.92 join(top, join(complement(top), complement(top)))
% 4.31/0.92 = { by lemma 16 }
% 4.31/0.92 join(top, complement(top))
% 4.31/0.92 = { by axiom 4 (def_top_12) R->L }
% 4.31/0.92 top
% 4.31/0.92
% 4.31/0.92 Lemma 20: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 4.31/0.92 Proof:
% 4.31/0.92 join(meet(X, Y), complement(join(complement(X), Y)))
% 4.31/0.92 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 4.31/0.92 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 4.31/0.92 = { by axiom 12 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 4.31/0.92 X
% 4.31/0.92
% 4.31/0.92 Lemma 21: join(zero, meet(X, X)) = X.
% 4.31/0.92 Proof:
% 4.31/0.92 join(zero, meet(X, X))
% 4.31/0.92 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 4.31/0.92 join(zero, complement(join(complement(X), complement(X))))
% 4.31/0.92 = { by axiom 5 (def_zero_13) }
% 4.31/0.92 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 4.31/0.92 = { by lemma 20 }
% 4.31/0.92 X
% 4.31/0.92
% 4.31/0.92 Lemma 22: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 4.31/0.92 Proof:
% 4.31/0.92 join(zero, join(X, complement(complement(Y))))
% 4.31/0.92 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.92 join(zero, join(complement(complement(Y)), X))
% 4.31/0.92 = { by lemma 16 R->L }
% 4.31/0.92 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 4.31/0.92 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 4.31/0.92 join(zero, join(meet(Y, Y), X))
% 4.31/0.92 = { by axiom 7 (maddux2_join_associativity_2) }
% 4.31/0.92 join(join(zero, meet(Y, Y)), X)
% 4.31/0.92 = { by lemma 21 }
% 4.31/0.92 join(Y, X)
% 4.31/0.92 = { by axiom 1 (maddux1_join_commutativity_1) }
% 4.31/0.92 join(X, Y)
% 4.31/0.92
% 4.31/0.92 Lemma 23: join(zero, complement(complement(X))) = X.
% 4.31/0.92 Proof:
% 4.31/0.92 join(zero, complement(complement(X)))
% 4.31/0.92 = { by axiom 5 (def_zero_13) }
% 4.31/0.92 join(meet(X, complement(X)), complement(complement(X)))
% 4.31/0.92 = { by lemma 16 R->L }
% 4.31/0.92 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 4.31/0.92 = { by lemma 20 }
% 4.31/0.92 X
% 4.31/0.92
% 4.31/0.92 Lemma 24: join(zero, complement(X)) = complement(X).
% 4.31/0.92 Proof:
% 4.31/0.92 join(zero, complement(X))
% 4.31/0.92 = { by lemma 23 R->L }
% 4.31/0.92 join(zero, join(zero, complement(complement(complement(X)))))
% 4.31/0.92 = { by lemma 16 R->L }
% 4.31/0.92 join(zero, join(zero, join(complement(complement(complement(X))), complement(complement(complement(X))))))
% 4.31/0.92 = { by lemma 22 }
% 4.31/0.92 join(zero, join(complement(complement(complement(X))), complement(X)))
% 4.31/0.92 = { by axiom 1 (maddux1_join_commutativity_1) }
% 4.31/0.92 join(zero, join(complement(X), complement(complement(complement(X)))))
% 4.31/0.92 = { by lemma 22 }
% 4.31/0.92 join(complement(X), complement(X))
% 4.31/0.92 = { by lemma 16 }
% 4.31/0.92 complement(X)
% 4.31/0.92
% 4.31/0.92 Lemma 25: join(zero, X) = X.
% 4.31/0.92 Proof:
% 4.31/0.92 join(zero, X)
% 4.31/0.92 = { by lemma 22 R->L }
% 4.31/0.92 join(zero, join(zero, complement(complement(X))))
% 4.31/0.92 = { by lemma 24 }
% 4.31/0.92 join(zero, complement(complement(X)))
% 4.31/0.92 = { by lemma 23 }
% 4.31/0.92 X
% 4.31/0.92
% 4.31/0.92 Lemma 26: complement(complement(X)) = X.
% 4.31/0.92 Proof:
% 4.31/0.92 complement(complement(X))
% 4.31/0.92 = { by lemma 24 R->L }
% 4.31/0.92 join(zero, complement(complement(X)))
% 4.31/0.92 = { by lemma 23 }
% 4.31/0.92 X
% 4.31/0.92
% 4.31/0.92 Lemma 27: meet(Y, X) = meet(X, Y).
% 4.31/0.92 Proof:
% 4.31/0.92 meet(Y, X)
% 4.31/0.92 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 4.31/0.92 complement(join(complement(Y), complement(X)))
% 4.31/0.92 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.92 complement(join(complement(X), complement(Y)))
% 4.31/0.92 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 4.31/0.92 meet(X, Y)
% 4.31/0.92
% 4.31/0.92 Lemma 28: complement(join(zero, complement(X))) = meet(X, top).
% 4.31/0.92 Proof:
% 4.31/0.92 complement(join(zero, complement(X)))
% 4.31/0.92 = { by lemma 13 R->L }
% 4.31/0.92 complement(join(complement(top), complement(X)))
% 4.31/0.92 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 4.31/0.92 meet(top, X)
% 4.31/0.92 = { by lemma 27 R->L }
% 4.31/0.92 meet(X, top)
% 4.31/0.92
% 4.31/0.92 Lemma 29: join(X, join(complement(X), Y)) = top.
% 4.31/0.92 Proof:
% 4.31/0.92 join(X, join(complement(X), Y))
% 4.31/0.92 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.92 join(X, join(Y, complement(X)))
% 4.31/0.92 = { by lemma 14 }
% 4.31/0.92 join(Y, top)
% 4.31/0.92 = { by lemma 18 R->L }
% 4.31/0.92 join(Z, top)
% 4.31/0.92 = { by lemma 19 }
% 4.31/0.92 top
% 4.31/0.92
% 4.31/0.92 Lemma 30: join(X, complement(zero)) = top.
% 4.31/0.92 Proof:
% 4.31/0.92 join(X, complement(zero))
% 4.31/0.92 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.92 join(complement(zero), X)
% 4.31/0.92 = { by lemma 22 R->L }
% 4.31/0.92 join(zero, join(complement(zero), complement(complement(X))))
% 4.31/0.92 = { by lemma 29 }
% 4.31/0.92 top
% 4.31/0.92
% 4.31/0.92 Lemma 31: meet(X, zero) = zero.
% 4.31/0.92 Proof:
% 4.31/0.92 meet(X, zero)
% 4.31/0.92 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 4.31/0.92 complement(join(complement(X), complement(zero)))
% 4.31/0.92 = { by lemma 30 }
% 4.31/0.92 complement(top)
% 4.31/0.92 = { by lemma 13 }
% 4.31/0.92 zero
% 4.31/0.92
% 4.31/0.92 Lemma 32: join(meet(X, Y), meet(X, complement(Y))) = X.
% 4.31/0.92 Proof:
% 4.31/0.92 join(meet(X, Y), meet(X, complement(Y)))
% 4.31/0.92 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.92 join(meet(X, complement(Y)), meet(X, Y))
% 4.31/0.92 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 4.31/0.92 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 4.31/0.92 = { by lemma 20 }
% 4.31/0.92 X
% 4.31/0.92
% 4.31/0.92 Lemma 33: meet(X, top) = X.
% 4.31/0.92 Proof:
% 4.31/0.92 meet(X, top)
% 4.31/0.92 = { by lemma 28 R->L }
% 4.31/0.92 complement(join(zero, complement(X)))
% 4.31/0.92 = { by lemma 24 R->L }
% 4.31/0.92 join(zero, complement(join(zero, complement(X))))
% 4.31/0.92 = { by lemma 28 }
% 4.31/0.92 join(zero, meet(X, top))
% 4.31/0.92 = { by lemma 30 R->L }
% 4.31/0.92 join(zero, meet(X, join(complement(zero), complement(zero))))
% 4.31/0.92 = { by lemma 16 }
% 4.31/0.92 join(zero, meet(X, complement(zero)))
% 4.31/0.92 = { by lemma 31 R->L }
% 4.31/0.92 join(meet(X, zero), meet(X, complement(zero)))
% 4.31/0.92 = { by lemma 32 }
% 4.31/0.92 X
% 4.31/0.92
% 4.31/0.92 Lemma 34: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 4.31/0.92 Proof:
% 4.31/0.92 meet(X, join(complement(Y), complement(Z)))
% 4.31/0.92 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.92 meet(X, join(complement(Z), complement(Y)))
% 4.31/0.92 = { by lemma 27 }
% 4.31/0.92 meet(join(complement(Z), complement(Y)), X)
% 4.31/0.92 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 4.31/0.92 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 4.31/0.92 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 4.31/0.92 complement(join(meet(Z, Y), complement(X)))
% 4.31/0.92 = { by axiom 1 (maddux1_join_commutativity_1) }
% 4.31/0.92 complement(join(complement(X), meet(Z, Y)))
% 4.31/0.92 = { by lemma 27 R->L }
% 4.31/0.92 complement(join(complement(X), meet(Y, Z)))
% 4.31/0.92
% 4.31/0.92 Lemma 35: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 4.31/0.92 Proof:
% 4.31/0.92 complement(join(X, complement(Y)))
% 4.31/0.92 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.92 complement(join(complement(Y), X))
% 4.31/0.92 = { by lemma 33 R->L }
% 4.31/0.92 complement(join(complement(Y), meet(X, top)))
% 4.31/0.92 = { by lemma 27 R->L }
% 4.31/0.92 complement(join(complement(Y), meet(top, X)))
% 4.31/0.92 = { by lemma 34 R->L }
% 4.31/0.92 meet(Y, join(complement(top), complement(X)))
% 4.31/0.92 = { by lemma 13 }
% 4.31/0.92 meet(Y, join(zero, complement(X)))
% 4.31/0.92 = { by lemma 24 }
% 4.31/0.92 meet(Y, complement(X))
% 4.31/0.92
% 4.31/0.92 Lemma 36: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 4.31/0.92 Proof:
% 4.31/0.92 complement(meet(X, complement(Y)))
% 4.31/0.92 = { by lemma 25 R->L }
% 4.31/0.92 complement(join(zero, meet(X, complement(Y))))
% 4.31/0.92 = { by lemma 35 R->L }
% 4.31/0.92 complement(join(zero, complement(join(Y, complement(X)))))
% 4.31/0.92 = { by lemma 28 }
% 4.31/0.92 meet(join(Y, complement(X)), top)
% 4.31/0.92 = { by lemma 33 }
% 4.31/0.92 join(Y, complement(X))
% 4.31/0.92
% 4.31/0.92 Lemma 37: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 4.31/0.92 Proof:
% 4.31/0.92 complement(join(complement(X), Y))
% 4.31/0.92 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.92 complement(join(Y, complement(X)))
% 4.31/0.92 = { by lemma 35 }
% 4.31/0.92 meet(X, complement(Y))
% 4.31/0.92
% 4.31/0.92 Lemma 38: meet(X, join(X, complement(Y))) = X.
% 4.31/0.92 Proof:
% 4.31/0.92 meet(X, join(X, complement(Y)))
% 4.31/0.92 = { by lemma 36 R->L }
% 4.31/0.92 meet(X, complement(meet(Y, complement(X))))
% 4.31/0.92 = { by lemma 37 R->L }
% 4.31/0.92 complement(join(complement(X), meet(Y, complement(X))))
% 4.31/0.92 = { by lemma 24 R->L }
% 4.31/0.92 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by lemma 13 R->L }
% 4.31/0.92 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by lemma 19 R->L }
% 4.31/0.92 join(complement(join(complement(Y), top)), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by lemma 14 R->L }
% 4.31/0.92 join(complement(join(complement(X), join(complement(Y), complement(complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.92 join(complement(join(complement(X), join(complement(complement(X)), complement(Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by lemma 21 R->L }
% 4.31/0.92 join(complement(join(complement(X), join(zero, meet(join(complement(complement(X)), complement(Y)), join(complement(complement(X)), complement(Y)))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by lemma 34 }
% 4.31/0.92 join(complement(join(complement(X), join(zero, complement(join(complement(join(complement(complement(X)), complement(Y))), meet(complement(X), Y)))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by lemma 24 }
% 4.31/0.92 join(complement(join(complement(X), complement(join(complement(join(complement(complement(X)), complement(Y))), meet(complement(X), Y))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 4.31/0.92 join(complement(join(complement(X), complement(join(meet(complement(X), Y), meet(complement(X), Y))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by lemma 27 }
% 4.31/0.92 join(complement(join(complement(X), complement(join(meet(Y, complement(X)), meet(complement(X), Y))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by lemma 27 }
% 4.31/0.92 join(complement(join(complement(X), complement(join(meet(Y, complement(X)), meet(Y, complement(X)))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 4.31/0.92 join(complement(join(complement(X), complement(join(meet(Y, complement(X)), complement(join(complement(Y), complement(complement(X)))))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 4.31/0.92 join(complement(join(complement(X), complement(join(complement(join(complement(Y), complement(complement(X)))), complement(join(complement(Y), complement(complement(X)))))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by lemma 16 }
% 4.31/0.92 join(complement(join(complement(X), complement(complement(join(complement(Y), complement(complement(X))))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 4.31/0.92 join(complement(join(complement(X), complement(meet(Y, complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by lemma 27 R->L }
% 4.31/0.92 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 4.31/0.92 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by lemma 27 R->L }
% 4.31/0.92 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.31/0.92 = { by lemma 20 }
% 4.31/0.92 X
% 4.31/0.92
% 4.31/0.92 Lemma 39: join(X, meet(X, Y)) = X.
% 4.31/0.92 Proof:
% 4.31/0.92 join(X, meet(X, Y))
% 4.31/0.92 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 4.31/0.92 join(X, complement(join(complement(X), complement(Y))))
% 4.31/0.92 = { by lemma 36 R->L }
% 4.31/0.92 complement(meet(join(complement(X), complement(Y)), complement(X)))
% 4.31/0.92 = { by lemma 27 R->L }
% 4.31/0.92 complement(meet(complement(X), join(complement(X), complement(Y))))
% 4.31/0.92 = { by lemma 38 }
% 4.31/0.92 complement(complement(X))
% 4.31/0.92 = { by lemma 26 }
% 4.31/0.92 X
% 4.31/0.92
% 4.31/0.92 Lemma 40: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 4.31/0.92 Proof:
% 4.31/0.92 join(Y, join(X, Z))
% 4.31/0.92 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.92 join(join(X, Z), Y)
% 4.31/0.92 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 4.31/0.92 join(X, join(Z, Y))
% 4.31/0.93 = { by axiom 1 (maddux1_join_commutativity_1) }
% 4.31/0.93 join(X, join(Y, Z))
% 4.31/0.93
% 4.31/0.93 Lemma 41: join(meet(X, Y), X) = X.
% 4.31/0.93 Proof:
% 4.31/0.93 join(meet(X, Y), X)
% 4.31/0.93 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.93 join(X, meet(X, Y))
% 4.31/0.93 = { by lemma 39 }
% 4.31/0.93 X
% 4.31/0.93
% 4.31/0.93 Lemma 42: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 4.31/0.93 Proof:
% 4.31/0.93 meet(Y, meet(Z, X))
% 4.31/0.93 = { by lemma 33 R->L }
% 4.31/0.93 meet(meet(Y, top), meet(Z, X))
% 4.31/0.93 = { by lemma 28 R->L }
% 4.31/0.93 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 4.31/0.93 = { by lemma 27 }
% 4.31/0.93 meet(complement(join(zero, complement(Y))), meet(X, Z))
% 4.31/0.93 = { by lemma 27 }
% 4.31/0.93 meet(meet(X, Z), complement(join(zero, complement(Y))))
% 4.31/0.93 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 4.31/0.93 meet(complement(join(complement(X), complement(Z))), complement(join(zero, complement(Y))))
% 4.31/0.93 = { by lemma 27 }
% 4.31/0.93 meet(complement(join(zero, complement(Y))), complement(join(complement(X), complement(Z))))
% 4.31/0.93 = { by lemma 24 R->L }
% 4.31/0.93 meet(join(zero, complement(join(zero, complement(Y)))), complement(join(complement(X), complement(Z))))
% 4.31/0.93 = { by lemma 35 R->L }
% 4.31/0.93 complement(join(join(complement(X), complement(Z)), complement(join(zero, complement(join(zero, complement(Y)))))))
% 4.31/0.93 = { by lemma 28 }
% 4.31/0.93 complement(join(join(complement(X), complement(Z)), meet(join(zero, complement(Y)), top)))
% 4.31/0.93 = { by lemma 33 }
% 4.31/0.93 complement(join(join(complement(X), complement(Z)), join(zero, complement(Y))))
% 4.31/0.93 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 4.31/0.93 complement(join(complement(X), join(complement(Z), join(zero, complement(Y)))))
% 4.31/0.93 = { by lemma 37 }
% 4.31/0.93 meet(X, complement(join(complement(Z), join(zero, complement(Y)))))
% 4.31/0.93 = { by lemma 37 }
% 4.31/0.93 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 4.31/0.93 = { by lemma 28 }
% 4.31/0.93 meet(X, meet(Z, meet(Y, top)))
% 4.31/0.93 = { by lemma 33 }
% 4.31/0.93 meet(X, meet(Z, Y))
% 4.31/0.93 = { by lemma 27 R->L }
% 4.31/0.93 meet(X, meet(Y, Z))
% 4.31/0.93
% 4.31/0.93 Lemma 43: join(complement(converse(X)), converse(join(X, Y))) = top.
% 4.31/0.93 Proof:
% 4.31/0.93 join(complement(converse(X)), converse(join(X, Y)))
% 4.31/0.93 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.93 join(converse(join(X, Y)), complement(converse(X)))
% 4.31/0.93 = { by axiom 6 (converse_additivity_9) }
% 4.31/0.93 join(join(converse(X), converse(Y)), complement(converse(X)))
% 4.31/0.93 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 4.31/0.93 join(converse(X), join(converse(Y), complement(converse(X))))
% 4.31/0.93 = { by axiom 1 (maddux1_join_commutativity_1) }
% 4.31/0.93 join(converse(X), join(complement(converse(X)), converse(Y)))
% 4.31/0.93 = { by lemma 29 }
% 4.31/0.93 top
% 4.31/0.93
% 4.31/0.93 Lemma 44: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 4.31/0.93 Proof:
% 4.31/0.93 join(meet(X, Y), meet(Y, complement(X)))
% 4.31/0.93 = { by lemma 27 }
% 4.31/0.93 join(meet(Y, X), meet(Y, complement(X)))
% 4.31/0.93 = { by lemma 32 }
% 4.31/0.93 Y
% 4.31/0.93
% 4.31/0.93 Lemma 45: join(meet(converse(X), converse(Y)), converse(meet(X, Y))) = converse(meet(X, Y)).
% 4.31/0.93 Proof:
% 4.31/0.93 join(meet(converse(X), converse(Y)), converse(meet(X, Y)))
% 4.31/0.93 = { by axiom 3 (converse_idempotence_8) R->L }
% 4.31/0.93 join(converse(converse(meet(converse(X), converse(Y)))), converse(meet(X, Y)))
% 4.31/0.93 = { by axiom 6 (converse_additivity_9) R->L }
% 4.31/0.93 converse(join(converse(meet(converse(X), converse(Y))), meet(X, Y)))
% 4.31/0.93 = { by axiom 1 (maddux1_join_commutativity_1) }
% 4.31/0.93 converse(join(meet(X, Y), converse(meet(converse(X), converse(Y)))))
% 4.31/0.93 = { by lemma 44 R->L }
% 4.31/0.93 converse(join(meet(X, Y), join(meet(complement(X), converse(meet(converse(X), converse(Y)))), meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))
% 4.31/0.93 = { by lemma 27 }
% 4.31/0.93 converse(join(meet(X, Y), join(meet(converse(meet(converse(X), converse(Y))), complement(X)), meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))
% 4.31/0.93 = { by lemma 35 R->L }
% 4.31/0.93 converse(join(meet(X, Y), join(complement(join(X, complement(converse(meet(converse(X), converse(Y)))))), meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))
% 4.31/0.93 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.93 converse(join(meet(X, Y), join(complement(join(complement(converse(meet(converse(X), converse(Y)))), X)), meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))
% 4.31/0.93 = { by axiom 3 (converse_idempotence_8) R->L }
% 4.31/0.93 converse(join(meet(X, Y), join(complement(join(complement(converse(meet(converse(X), converse(Y)))), converse(converse(X)))), meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))
% 4.31/0.93 = { by lemma 20 R->L }
% 4.31/0.93 converse(join(meet(X, Y), join(complement(join(complement(converse(meet(converse(X), converse(Y)))), converse(join(meet(converse(X), converse(Y)), complement(join(complement(converse(X)), converse(Y))))))), meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))
% 4.31/0.93 = { by axiom 1 (maddux1_join_commutativity_1) }
% 4.31/0.93 converse(join(meet(X, Y), join(complement(join(complement(converse(meet(converse(X), converse(Y)))), converse(join(meet(converse(X), converse(Y)), complement(join(converse(Y), complement(converse(X)))))))), meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))
% 4.31/0.93 = { by lemma 43 }
% 4.31/0.93 converse(join(meet(X, Y), join(complement(top), meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))
% 4.31/0.93 = { by lemma 13 }
% 4.31/0.93 converse(join(meet(X, Y), join(zero, meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))
% 4.31/0.93 = { by lemma 25 }
% 4.31/0.93 converse(join(meet(X, Y), meet(converse(meet(converse(X), converse(Y))), complement(complement(X)))))
% 4.31/0.93 = { by lemma 26 }
% 4.31/0.93 converse(join(meet(X, Y), meet(converse(meet(converse(X), converse(Y))), X)))
% 4.31/0.93 = { by lemma 27 R->L }
% 4.31/0.93 converse(join(meet(X, Y), meet(X, converse(meet(converse(X), converse(Y))))))
% 4.31/0.93 = { by lemma 44 R->L }
% 4.31/0.93 converse(join(meet(X, Y), meet(X, join(meet(complement(Y), converse(meet(converse(X), converse(Y)))), meet(converse(meet(converse(X), converse(Y))), complement(complement(Y)))))))
% 4.31/0.93 = { by lemma 27 }
% 4.31/0.93 converse(join(meet(X, Y), meet(X, join(meet(converse(meet(converse(X), converse(Y))), complement(Y)), meet(converse(meet(converse(X), converse(Y))), complement(complement(Y)))))))
% 4.31/0.93 = { by lemma 35 R->L }
% 4.31/0.93 converse(join(meet(X, Y), meet(X, join(complement(join(Y, complement(converse(meet(converse(X), converse(Y)))))), meet(converse(meet(converse(X), converse(Y))), complement(complement(Y)))))))
% 4.31/0.93 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.93 converse(join(meet(X, Y), meet(X, join(complement(join(complement(converse(meet(converse(X), converse(Y)))), Y)), meet(converse(meet(converse(X), converse(Y))), complement(complement(Y)))))))
% 4.31/0.93 = { by axiom 3 (converse_idempotence_8) R->L }
% 4.31/0.93 converse(join(meet(X, Y), meet(X, join(complement(join(complement(converse(meet(converse(X), converse(Y)))), converse(converse(Y)))), meet(converse(meet(converse(X), converse(Y))), complement(complement(Y)))))))
% 4.31/0.93 = { by lemma 44 R->L }
% 4.31/0.93 converse(join(meet(X, Y), meet(X, join(complement(join(complement(converse(meet(converse(X), converse(Y)))), converse(join(meet(converse(X), converse(Y)), meet(converse(Y), complement(converse(X))))))), meet(converse(meet(converse(X), converse(Y))), complement(complement(Y)))))))
% 4.31/0.93 = { by lemma 43 }
% 4.31/0.93 converse(join(meet(X, Y), meet(X, join(complement(top), meet(converse(meet(converse(X), converse(Y))), complement(complement(Y)))))))
% 4.31/0.93 = { by lemma 13 }
% 4.31/0.93 converse(join(meet(X, Y), meet(X, join(zero, meet(converse(meet(converse(X), converse(Y))), complement(complement(Y)))))))
% 4.31/0.93 = { by lemma 25 }
% 4.31/0.93 converse(join(meet(X, Y), meet(X, meet(converse(meet(converse(X), converse(Y))), complement(complement(Y))))))
% 4.31/0.93 = { by lemma 26 }
% 4.31/0.93 converse(join(meet(X, Y), meet(X, meet(converse(meet(converse(X), converse(Y))), Y))))
% 4.31/0.93 = { by lemma 27 R->L }
% 4.31/0.93 converse(join(meet(X, Y), meet(X, meet(Y, converse(meet(converse(X), converse(Y)))))))
% 4.31/0.93 = { by lemma 42 }
% 4.31/0.93 converse(join(meet(X, Y), meet(converse(meet(converse(X), converse(Y))), meet(X, Y))))
% 4.31/0.93 = { by lemma 27 R->L }
% 4.31/0.93 converse(join(meet(X, Y), meet(meet(X, Y), converse(meet(converse(X), converse(Y))))))
% 4.31/0.93 = { by lemma 39 }
% 4.31/0.93 converse(meet(X, Y))
% 4.31/0.93
% 4.31/0.93 Goal 1 (goals_17): tuple(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), join(converse(meet(sk1, sk2)), meet(converse(sk1), converse(sk2)))) = tuple(converse(meet(sk1, sk2)), meet(converse(sk1), converse(sk2))).
% 4.31/0.93 Proof:
% 4.31/0.93 tuple(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), join(converse(meet(sk1, sk2)), meet(converse(sk1), converse(sk2))))
% 4.31/0.93 = { by axiom 1 (maddux1_join_commutativity_1) }
% 4.31/0.93 tuple(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))))
% 4.31/0.93 = { by lemma 45 }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))))
% 4.31/0.93 = { by lemma 38 R->L }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), meet(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), join(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), complement(join(zero, complement(converse(sk1)))))))
% 4.31/0.93 = { by lemma 28 }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), meet(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), join(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), meet(converse(sk1), top))))
% 4.31/0.93 = { by lemma 33 }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), meet(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), join(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), converse(sk1))))
% 4.31/0.93 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), meet(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), join(converse(sk1), join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))))))
% 4.31/0.93 = { by lemma 40 }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), meet(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), join(meet(converse(sk1), converse(sk2)), join(converse(sk1), converse(meet(sk1, sk2))))))
% 4.31/0.93 = { by axiom 6 (converse_additivity_9) R->L }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), meet(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), join(meet(converse(sk1), converse(sk2)), converse(join(sk1, meet(sk1, sk2))))))
% 4.31/0.93 = { by axiom 1 (maddux1_join_commutativity_1) }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), meet(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), join(meet(converse(sk1), converse(sk2)), converse(join(meet(sk1, sk2), sk1)))))
% 4.31/0.93 = { by lemma 41 }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), meet(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), join(meet(converse(sk1), converse(sk2)), converse(sk1))))
% 4.31/0.93 = { by lemma 41 }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), meet(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), converse(sk1)))
% 4.31/0.93 = { by lemma 27 }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))))
% 4.31/0.93 = { by lemma 20 R->L }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), join(meet(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), complement(converse(sk2))), complement(join(complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))), complement(converse(sk2)))))))
% 4.31/0.93 = { by lemma 27 }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), join(meet(complement(converse(sk2)), join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))), complement(join(complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))), complement(converse(sk2)))))))
% 4.31/0.93 = { by lemma 45 }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), join(meet(complement(converse(sk2)), converse(meet(sk1, sk2))), complement(join(complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))), complement(converse(sk2)))))))
% 4.31/0.93 = { by lemma 38 R->L }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), join(meet(meet(complement(converse(sk2)), converse(meet(sk1, sk2))), join(meet(complement(converse(sk2)), converse(meet(sk1, sk2))), complement(join(complement(complement(converse(sk2))), converse(meet(sk1, sk2)))))), complement(join(complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))), complement(converse(sk2)))))))
% 4.31/0.93 = { by lemma 20 }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), join(meet(meet(complement(converse(sk2)), converse(meet(sk1, sk2))), complement(converse(sk2))), complement(join(complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))), complement(converse(sk2)))))))
% 4.31/0.93 = { by lemma 27 R->L }
% 4.31/0.93 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), join(meet(complement(converse(sk2)), meet(complement(converse(sk2)), converse(meet(sk1, sk2)))), complement(join(complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))), complement(converse(sk2)))))))
% 4.31/0.94 = { by lemma 27 R->L }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), join(meet(complement(converse(sk2)), meet(converse(meet(sk1, sk2)), complement(converse(sk2)))), complement(join(complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))), complement(converse(sk2)))))))
% 4.31/0.94 = { by lemma 35 R->L }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), join(meet(complement(converse(sk2)), complement(join(converse(sk2), complement(converse(meet(sk1, sk2)))))), complement(join(complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))), complement(converse(sk2)))))))
% 4.31/0.94 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), join(meet(complement(converse(sk2)), complement(join(complement(converse(meet(sk1, sk2))), converse(sk2)))), complement(join(complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))), complement(converse(sk2)))))))
% 4.31/0.94 = { by lemma 44 R->L }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), join(meet(complement(converse(sk2)), complement(join(complement(converse(meet(sk1, sk2))), converse(join(meet(sk1, sk2), meet(sk2, complement(sk1))))))), complement(join(complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))), complement(converse(sk2)))))))
% 4.31/0.94 = { by lemma 43 }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), join(meet(complement(converse(sk2)), complement(top)), complement(join(complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))), complement(converse(sk2)))))))
% 4.31/0.94 = { by lemma 13 }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), join(meet(complement(converse(sk2)), zero), complement(join(complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))), complement(converse(sk2)))))))
% 4.31/0.94 = { by lemma 31 }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), join(zero, complement(join(complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))), complement(converse(sk2)))))))
% 4.31/0.94 = { by lemma 24 }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), complement(join(complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))), complement(converse(sk2))))))
% 4.31/0.94 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), meet(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), converse(sk2))))
% 4.31/0.94 = { by lemma 27 }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), meet(converse(sk2), join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))))))
% 4.31/0.94 = { by lemma 42 }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), meet(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), meet(converse(sk1), converse(sk2))))
% 4.31/0.94 = { by lemma 27 }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), meet(meet(converse(sk1), converse(sk2)), join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))))
% 4.31/0.94 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), complement(join(complement(meet(converse(sk1), converse(sk2))), complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2)))))))
% 4.31/0.94 = { by lemma 24 R->L }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), join(zero, complement(join(complement(meet(converse(sk1), converse(sk2))), complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))))))))
% 4.31/0.94 = { by lemma 13 R->L }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), join(complement(top), complement(join(complement(meet(converse(sk1), converse(sk2))), complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))))))))
% 4.31/0.94 = { by lemma 29 R->L }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), join(complement(join(meet(converse(sk1), converse(sk2)), join(complement(meet(converse(sk1), converse(sk2))), converse(meet(sk1, sk2))))), complement(join(complement(meet(converse(sk1), converse(sk2))), complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))))))))
% 4.31/0.94 = { by lemma 40 R->L }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), join(complement(join(complement(meet(converse(sk1), converse(sk2))), join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))))), complement(join(complement(meet(converse(sk1), converse(sk2))), complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))))))))
% 4.31/0.94 = { by axiom 1 (maddux1_join_commutativity_1) }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), join(complement(join(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))), complement(meet(converse(sk1), converse(sk2))))), complement(join(complement(meet(converse(sk1), converse(sk2))), complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))))))))
% 4.31/0.94 = { by lemma 35 }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), join(meet(meet(converse(sk1), converse(sk2)), complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))))), complement(join(complement(meet(converse(sk1), converse(sk2))), complement(join(meet(converse(sk1), converse(sk2)), converse(meet(sk1, sk2))))))))
% 4.31/0.94 = { by lemma 20 }
% 4.31/0.94 tuple(converse(meet(sk1, sk2)), meet(converse(sk1), converse(sk2)))
% 4.31/0.94 % SZS output end Proof
% 4.31/0.94
% 4.31/0.94 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------