TSTP Solution File: LAT064-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT064-1 : TPTP v8.1.2. Released v2.5.0.
% 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 : n008.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 06:27:19 EDT 2023
% Result : Unsatisfiable 242.67s 31.58s
% Output : Proof 243.57s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LAT064-1 : TPTP v8.1.2. Released v2.5.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.17/0.35 % Computer : n008.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Thu Aug 24 04:23:32 EDT 2023
% 0.17/0.35 % CPUTime :
% 242.67/31.58 Command-line arguments: --no-flatten-goal
% 242.67/31.58
% 242.67/31.58 % SZS status Unsatisfiable
% 242.67/31.58
% 243.24/31.64 % SZS output start Proof
% 243.24/31.64 Take the following subset of the input axioms:
% 243.24/31.64 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 243.24/31.64 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 243.24/31.64 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 243.24/31.64 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 243.24/31.64 fof(bottom, axiom, ![A]: meet(A, complement(A))=n0).
% 243.24/31.64 fof(c94_6, axiom, ![B, C, A3]: meet(A3, join(B, meet(C, join(A3, meet(B, C)))))=meet(A3, join(B, meet(A3, C)))).
% 243.24/31.64 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 243.24/31.64 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 243.24/31.64 fof(complements_are_unique, axiom, ![A2, B2]: (join(A2, B2)!=n1 | (meet(A2, B2)!=n0 | complement(A2)=B2))).
% 243.24/31.65 fof(idempotence_of_join, axiom, ![X2]: join(X2, X2)=X2).
% 243.24/31.65 fof(idempotence_of_meet, axiom, ![X2]: meet(X2, X2)=X2).
% 243.24/31.65 fof(prove_distributivity, negated_conjecture, meet(a, join(b, c))!=join(meet(a, b), meet(a, c))).
% 243.24/31.65 fof(top, axiom, ![A3]: join(A3, complement(A3))=n1).
% 243.24/31.65
% 243.24/31.65 Now clausify the problem and encode Horn clauses using encoding 3 of
% 243.24/31.65 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 243.24/31.65 We repeatedly replace C & s=t => u=v by the two clauses:
% 243.24/31.65 fresh(y, y, x1...xn) = u
% 243.24/31.65 C => fresh(s, t, x1...xn) = v
% 243.24/31.65 where fresh is a fresh function symbol and x1..xn are the free
% 243.24/31.65 variables of u and v.
% 243.24/31.65 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 243.24/31.65 input problem has no model of domain size 1).
% 243.24/31.65
% 243.24/31.65 The encoding turns the above axioms into the following unit equations and goals:
% 243.24/31.65
% 243.24/31.65 Axiom 1 (idempotence_of_join): join(X, X) = X.
% 243.24/31.65 Axiom 2 (commutativity_of_join): join(X, Y) = join(Y, X).
% 243.24/31.65 Axiom 3 (idempotence_of_meet): meet(X, X) = X.
% 243.24/31.65 Axiom 4 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 243.24/31.65 Axiom 5 (top): join(X, complement(X)) = n1.
% 243.24/31.65 Axiom 6 (bottom): meet(X, complement(X)) = n0.
% 243.24/31.65 Axiom 7 (complements_are_unique): fresh(X, X, Y, Z) = Z.
% 243.24/31.65 Axiom 8 (complements_are_unique): fresh2(X, X, Y, Z) = complement(Y).
% 243.24/31.65 Axiom 9 (absorption2): join(X, meet(X, Y)) = X.
% 243.24/31.65 Axiom 10 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 243.24/31.65 Axiom 11 (absorption1): meet(X, join(X, Y)) = X.
% 243.24/31.65 Axiom 12 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 243.24/31.65 Axiom 13 (complements_are_unique): fresh2(join(X, Y), n1, X, Y) = fresh(meet(X, Y), n0, X, Y).
% 243.24/31.65 Axiom 14 (c94_6): meet(X, join(Y, meet(Z, join(X, meet(Y, Z))))) = meet(X, join(Y, meet(X, Z))).
% 243.24/31.65
% 243.24/31.65 Lemma 15: fresh2(join(X, Y), n1, Y, X) = fresh(meet(X, Y), n0, Y, X).
% 243.24/31.65 Proof:
% 243.24/31.65 fresh2(join(X, Y), n1, Y, X)
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.65 fresh2(join(Y, X), n1, Y, X)
% 243.24/31.65 = { by axiom 13 (complements_are_unique) }
% 243.24/31.65 fresh(meet(Y, X), n0, Y, X)
% 243.24/31.65 = { by axiom 4 (commutativity_of_meet) }
% 243.24/31.65 fresh(meet(X, Y), n0, Y, X)
% 243.24/31.65
% 243.24/31.65 Lemma 16: complement(complement(X)) = X.
% 243.24/31.65 Proof:
% 243.24/31.65 complement(complement(X))
% 243.24/31.65 = { by axiom 8 (complements_are_unique) R->L }
% 243.24/31.65 fresh2(n1, n1, complement(X), X)
% 243.24/31.65 = { by axiom 5 (top) R->L }
% 243.24/31.65 fresh2(join(X, complement(X)), n1, complement(X), X)
% 243.24/31.65 = { by lemma 15 }
% 243.24/31.65 fresh(meet(X, complement(X)), n0, complement(X), X)
% 243.24/31.65 = { by axiom 6 (bottom) }
% 243.24/31.65 fresh(n0, n0, complement(X), X)
% 243.24/31.65 = { by axiom 7 (complements_are_unique) }
% 243.24/31.65 X
% 243.24/31.65
% 243.24/31.65 Lemma 17: meet(X, n1) = X.
% 243.24/31.65 Proof:
% 243.24/31.65 meet(X, n1)
% 243.24/31.65 = { by axiom 5 (top) R->L }
% 243.24/31.65 meet(X, join(X, complement(X)))
% 243.24/31.65 = { by axiom 11 (absorption1) }
% 243.24/31.65 X
% 243.24/31.65
% 243.24/31.65 Lemma 18: meet(n1, X) = X.
% 243.24/31.65 Proof:
% 243.24/31.65 meet(n1, X)
% 243.24/31.65 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.65 meet(X, n1)
% 243.24/31.65 = { by lemma 17 }
% 243.24/31.65 X
% 243.24/31.65
% 243.24/31.65 Lemma 19: join(X, n1) = n1.
% 243.24/31.65 Proof:
% 243.24/31.65 join(X, n1)
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.65 join(n1, X)
% 243.24/31.65 = { by lemma 18 R->L }
% 243.24/31.65 join(n1, meet(n1, X))
% 243.24/31.65 = { by axiom 9 (absorption2) }
% 243.24/31.65 n1
% 243.24/31.65
% 243.24/31.65 Lemma 20: join(n0, X) = X.
% 243.24/31.65 Proof:
% 243.24/31.65 join(n0, X)
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.65 join(X, n0)
% 243.24/31.65 = { by axiom 6 (bottom) R->L }
% 243.24/31.65 join(X, meet(X, complement(X)))
% 243.24/31.65 = { by axiom 9 (absorption2) }
% 243.24/31.65 X
% 243.24/31.65
% 243.24/31.65 Lemma 21: meet(X, meet(Y, join(Z, meet(X, Y)))) = meet(X, Y).
% 243.24/31.65 Proof:
% 243.24/31.65 meet(X, meet(Y, join(Z, meet(X, Y))))
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.65 meet(X, meet(Y, join(meet(X, Y), Z)))
% 243.24/31.65 = { by axiom 12 (associativity_of_meet) R->L }
% 243.24/31.65 meet(meet(X, Y), join(meet(X, Y), Z))
% 243.24/31.65 = { by axiom 11 (absorption1) }
% 243.24/31.65 meet(X, Y)
% 243.24/31.65
% 243.24/31.65 Lemma 22: join(X, meet(Y, X)) = X.
% 243.24/31.65 Proof:
% 243.24/31.65 join(X, meet(Y, X))
% 243.24/31.65 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.65 join(X, meet(X, Y))
% 243.24/31.65 = { by axiom 9 (absorption2) }
% 243.24/31.65 X
% 243.24/31.65
% 243.24/31.65 Lemma 23: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 243.24/31.65 Proof:
% 243.24/31.65 join(X, join(Y, meet(X, Z)))
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.65 join(X, join(meet(X, Z), Y))
% 243.24/31.65 = { by axiom 10 (associativity_of_join) R->L }
% 243.24/31.65 join(join(X, meet(X, Z)), Y)
% 243.24/31.65 = { by axiom 9 (absorption2) }
% 243.24/31.65 join(X, Y)
% 243.24/31.65
% 243.24/31.65 Lemma 24: meet(X, join(Y, X)) = X.
% 243.24/31.65 Proof:
% 243.24/31.65 meet(X, join(Y, X))
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.65 meet(X, join(X, Y))
% 243.24/31.65 = { by axiom 11 (absorption1) }
% 243.24/31.65 X
% 243.24/31.65
% 243.24/31.65 Lemma 25: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 243.24/31.65 Proof:
% 243.24/31.65 meet(X, meet(Y, join(X, Z)))
% 243.24/31.65 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.65 meet(X, meet(join(X, Z), Y))
% 243.24/31.65 = { by axiom 12 (associativity_of_meet) R->L }
% 243.24/31.65 meet(meet(X, join(X, Z)), Y)
% 243.24/31.65 = { by axiom 11 (absorption1) }
% 243.24/31.65 meet(X, Y)
% 243.24/31.65
% 243.24/31.65 Lemma 26: meet(X, meet(Y, complement(meet(X, Y)))) = n0.
% 243.24/31.65 Proof:
% 243.24/31.65 meet(X, meet(Y, complement(meet(X, Y))))
% 243.24/31.65 = { by axiom 12 (associativity_of_meet) R->L }
% 243.24/31.65 meet(meet(X, Y), complement(meet(X, Y)))
% 243.24/31.65 = { by axiom 6 (bottom) }
% 243.24/31.65 n0
% 243.24/31.65
% 243.24/31.65 Lemma 27: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
% 243.24/31.65 Proof:
% 243.24/31.65 meet(X, meet(Y, Z))
% 243.24/31.65 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.65 meet(meet(Y, Z), X)
% 243.24/31.65 = { by axiom 12 (associativity_of_meet) }
% 243.24/31.65 meet(Y, meet(Z, X))
% 243.24/31.65 = { by axiom 4 (commutativity_of_meet) }
% 243.24/31.65 meet(Y, meet(X, Z))
% 243.24/31.65
% 243.24/31.65 Lemma 28: meet(X, meet(Y, join(Z, X))) = meet(X, Y).
% 243.24/31.65 Proof:
% 243.24/31.65 meet(X, meet(Y, join(Z, X)))
% 243.24/31.65 = { by lemma 27 R->L }
% 243.24/31.65 meet(Y, meet(X, join(Z, X)))
% 243.24/31.65 = { by lemma 24 }
% 243.24/31.65 meet(Y, X)
% 243.24/31.65 = { by axiom 4 (commutativity_of_meet) }
% 243.24/31.65 meet(X, Y)
% 243.24/31.65
% 243.24/31.65 Lemma 29: join(X, meet(join(X, Y), complement(meet(X, Y)))) = join(X, meet(Y, complement(meet(X, Y)))).
% 243.24/31.65 Proof:
% 243.24/31.65 join(X, meet(join(X, Y), complement(meet(X, Y))))
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.65 join(X, meet(join(Y, X), complement(meet(X, Y))))
% 243.24/31.65 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.65 join(X, meet(complement(meet(X, Y)), join(Y, X)))
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.65 join(X, meet(complement(meet(X, Y)), join(X, Y)))
% 243.24/31.65 = { by lemma 22 R->L }
% 243.24/31.65 join(X, join(meet(complement(meet(X, Y)), join(X, Y)), meet(Y, meet(complement(meet(X, Y)), join(X, Y)))))
% 243.24/31.65 = { by lemma 28 }
% 243.24/31.65 join(X, join(meet(complement(meet(X, Y)), join(X, Y)), meet(Y, complement(meet(X, Y)))))
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) }
% 243.24/31.65 join(X, join(meet(Y, complement(meet(X, Y))), meet(complement(meet(X, Y)), join(X, Y))))
% 243.24/31.65 = { by axiom 4 (commutativity_of_meet) }
% 243.24/31.65 join(X, join(meet(Y, complement(meet(X, Y))), meet(join(X, Y), complement(meet(X, Y)))))
% 243.24/31.65 = { by axiom 10 (associativity_of_join) R->L }
% 243.24/31.65 join(join(X, meet(Y, complement(meet(X, Y)))), meet(join(X, Y), complement(meet(X, Y))))
% 243.24/31.65 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.65 join(join(X, meet(Y, complement(meet(X, Y)))), meet(complement(meet(X, Y)), join(X, Y)))
% 243.24/31.65 = { by lemma 17 R->L }
% 243.24/31.65 join(join(X, meet(Y, complement(meet(X, Y)))), meet(complement(meet(X, Y)), join(X, meet(Y, n1))))
% 243.24/31.65 = { by axiom 5 (top) R->L }
% 243.24/31.65 join(join(X, meet(Y, complement(meet(X, Y)))), meet(complement(meet(X, Y)), join(X, meet(Y, join(meet(X, Y), complement(meet(X, Y)))))))
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.65 join(join(X, meet(Y, complement(meet(X, Y)))), meet(complement(meet(X, Y)), join(X, meet(Y, join(complement(meet(X, Y)), meet(X, Y))))))
% 243.24/31.65 = { by axiom 14 (c94_6) }
% 243.24/31.65 join(join(X, meet(Y, complement(meet(X, Y)))), meet(complement(meet(X, Y)), join(X, meet(complement(meet(X, Y)), Y))))
% 243.24/31.65 = { by axiom 4 (commutativity_of_meet) }
% 243.24/31.65 join(join(X, meet(Y, complement(meet(X, Y)))), meet(complement(meet(X, Y)), join(X, meet(Y, complement(meet(X, Y))))))
% 243.24/31.65 = { by lemma 22 }
% 243.24/31.65 join(X, meet(Y, complement(meet(X, Y))))
% 243.24/31.65
% 243.24/31.65 Lemma 30: join(meet(X, Y), meet(X, join(Y, Z))) = meet(X, join(Y, Z)).
% 243.24/31.65 Proof:
% 243.24/31.65 join(meet(X, Y), meet(X, join(Y, Z)))
% 243.24/31.65 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.65 join(meet(Y, X), meet(X, join(Y, Z)))
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.65 join(meet(X, join(Y, Z)), meet(Y, X))
% 243.24/31.65 = { by lemma 25 R->L }
% 243.24/31.65 join(meet(X, join(Y, Z)), meet(Y, meet(X, join(Y, Z))))
% 243.24/31.65 = { by lemma 22 }
% 243.24/31.65 meet(X, join(Y, Z))
% 243.24/31.65
% 243.24/31.65 Lemma 31: join(X, join(meet(X, Y), Z)) = join(X, Z).
% 243.24/31.65 Proof:
% 243.24/31.65 join(X, join(meet(X, Y), Z))
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.65 join(X, join(Z, meet(X, Y)))
% 243.24/31.65 = { by lemma 23 }
% 243.24/31.65 join(X, Z)
% 243.24/31.65
% 243.24/31.65 Lemma 32: join(X, join(Y, complement(join(X, Y)))) = n1.
% 243.24/31.65 Proof:
% 243.24/31.65 join(X, join(Y, complement(join(X, Y))))
% 243.24/31.65 = { by axiom 10 (associativity_of_join) R->L }
% 243.24/31.65 join(join(X, Y), complement(join(X, Y)))
% 243.24/31.65 = { by axiom 5 (top) }
% 243.24/31.65 n1
% 243.24/31.65
% 243.24/31.65 Lemma 33: meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))) = complement(Y).
% 243.24/31.65 Proof:
% 243.24/31.65 meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))
% 243.24/31.65 = { by axiom 7 (complements_are_unique) R->L }
% 243.24/31.65 fresh(n0, n0, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 243.24/31.65 = { by lemma 26 R->L }
% 243.24/31.65 fresh(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), n0, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 243.24/31.65 = { by axiom 13 (complements_are_unique) R->L }
% 243.24/31.65 fresh2(join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), n1, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 243.24/31.65 = { by lemma 29 R->L }
% 243.24/31.65 fresh2(join(Y, meet(join(Y, join(X, complement(Y))), complement(meet(Y, join(X, complement(Y)))))), n1, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.65 fresh2(join(Y, meet(join(Y, join(complement(Y), X)), complement(meet(Y, join(X, complement(Y)))))), n1, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 243.24/31.65 = { by axiom 10 (associativity_of_join) R->L }
% 243.24/31.65 fresh2(join(Y, meet(join(join(Y, complement(Y)), X), complement(meet(Y, join(X, complement(Y)))))), n1, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 243.24/31.65 = { by axiom 5 (top) }
% 243.24/31.65 fresh2(join(Y, meet(join(n1, X), complement(meet(Y, join(X, complement(Y)))))), n1, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.65 fresh2(join(Y, meet(join(X, n1), complement(meet(Y, join(X, complement(Y)))))), n1, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 243.24/31.65 = { by lemma 19 }
% 243.24/31.65 fresh2(join(Y, meet(n1, complement(meet(Y, join(X, complement(Y)))))), n1, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 243.24/31.65 = { by lemma 18 }
% 243.24/31.65 fresh2(join(Y, complement(meet(Y, join(X, complement(Y))))), n1, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 243.24/31.65 = { by lemma 30 R->L }
% 243.24/31.65 fresh2(join(Y, complement(join(meet(Y, X), meet(Y, join(X, complement(Y)))))), n1, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 243.24/31.65 = { by lemma 31 R->L }
% 243.24/31.65 fresh2(join(Y, join(meet(Y, X), complement(join(meet(Y, X), meet(Y, join(X, complement(Y))))))), n1, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.65 fresh2(join(Y, join(meet(Y, X), complement(join(meet(Y, join(X, complement(Y))), meet(Y, X))))), n1, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 243.24/31.65 = { by lemma 31 R->L }
% 243.24/31.65 fresh2(join(Y, join(meet(Y, join(X, complement(Y))), join(meet(Y, X), complement(join(meet(Y, join(X, complement(Y))), meet(Y, X)))))), n1, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 243.24/31.65 = { by lemma 32 }
% 243.24/31.65 fresh2(join(Y, n1), n1, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 243.24/31.65 = { by lemma 19 }
% 243.24/31.65 fresh2(n1, n1, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 243.24/31.65 = { by axiom 8 (complements_are_unique) }
% 243.24/31.65 complement(Y)
% 243.24/31.65
% 243.24/31.65 Lemma 34: meet(join(X, Y), complement(meet(complement(X), join(X, Y)))) = X.
% 243.24/31.65 Proof:
% 243.24/31.65 meet(join(X, Y), complement(meet(complement(X), join(X, Y))))
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.65 meet(join(Y, X), complement(meet(complement(X), join(X, Y))))
% 243.24/31.65 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.65 meet(join(Y, X), complement(meet(complement(X), join(Y, X))))
% 243.24/31.65 = { by lemma 16 R->L }
% 243.24/31.65 meet(join(Y, X), complement(meet(complement(X), join(Y, complement(complement(X))))))
% 243.24/31.65 = { by lemma 16 R->L }
% 243.24/31.65 meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))
% 243.24/31.65 = { by lemma 33 }
% 243.24/31.65 complement(complement(X))
% 243.24/31.65 = { by lemma 16 }
% 243.24/31.65 X
% 243.24/31.65
% 243.24/31.65 Lemma 35: join(X, complement(meet(X, Y))) = n1.
% 243.24/31.65 Proof:
% 243.24/31.65 join(X, complement(meet(X, Y)))
% 243.24/31.65 = { by lemma 31 R->L }
% 243.24/31.65 join(X, join(meet(X, Y), complement(meet(X, Y))))
% 243.24/31.65 = { by axiom 5 (top) }
% 243.24/31.65 join(X, n1)
% 243.24/31.65 = { by lemma 19 }
% 243.24/31.65 n1
% 243.24/31.65
% 243.24/31.65 Lemma 36: meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))) = complement(X).
% 243.24/31.65 Proof:
% 243.24/31.65 meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))
% 243.24/31.66 = { by axiom 7 (complements_are_unique) R->L }
% 243.24/31.66 fresh(n0, n0, X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 243.24/31.66 = { by lemma 26 R->L }
% 243.24/31.66 fresh(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), n0, X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 243.24/31.66 = { by axiom 13 (complements_are_unique) R->L }
% 243.24/31.66 fresh2(join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), n1, X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 243.24/31.66 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.66 fresh2(join(X, meet(complement(meet(Y, X)), complement(meet(X, complement(meet(X, Y)))))), n1, X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 243.24/31.66 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.66 fresh2(join(X, meet(complement(meet(Y, X)), complement(meet(X, complement(meet(Y, X)))))), n1, X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 243.24/31.66 = { by lemma 29 R->L }
% 243.24/31.66 fresh2(join(X, meet(join(X, complement(meet(Y, X))), complement(meet(X, complement(meet(Y, X)))))), n1, X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 243.24/31.66 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.66 fresh2(join(X, meet(join(X, complement(meet(X, Y))), complement(meet(X, complement(meet(Y, X)))))), n1, X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 243.24/31.66 = { by lemma 35 }
% 243.24/31.66 fresh2(join(X, meet(n1, complement(meet(X, complement(meet(Y, X)))))), n1, X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 243.24/31.66 = { by lemma 18 }
% 243.24/31.66 fresh2(join(X, complement(meet(X, complement(meet(Y, X))))), n1, X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 243.24/31.66 = { by lemma 35 }
% 243.24/31.66 fresh2(n1, n1, X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 243.24/31.66 = { by axiom 8 (complements_are_unique) }
% 243.24/31.66 complement(X)
% 243.24/31.66
% 243.24/31.66 Lemma 37: meet(complement(X), complement(meet(complement(X), join(X, Y)))) = complement(join(X, Y)).
% 243.24/31.66 Proof:
% 243.24/31.66 meet(complement(X), complement(meet(complement(X), join(X, Y))))
% 243.24/31.66 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.66 meet(complement(X), complement(meet(join(X, Y), complement(X))))
% 243.24/31.66 = { by lemma 34 R->L }
% 243.24/31.66 meet(complement(X), complement(meet(join(X, Y), complement(meet(join(X, Y), complement(meet(complement(X), join(X, Y))))))))
% 243.24/31.66 = { by lemma 34 R->L }
% 243.24/31.66 meet(complement(meet(join(X, Y), complement(meet(complement(X), join(X, Y))))), complement(meet(join(X, Y), complement(meet(join(X, Y), complement(meet(complement(X), join(X, Y))))))))
% 243.24/31.66 = { by lemma 36 }
% 243.24/31.66 complement(join(X, Y))
% 243.24/31.66
% 243.24/31.66 Lemma 38: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 243.24/31.66 Proof:
% 243.24/31.66 join(X, join(Y, meet(Z, join(X, Y))))
% 243.24/31.66 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.66 join(X, join(Y, meet(join(X, Y), Z)))
% 243.24/31.66 = { by axiom 10 (associativity_of_join) R->L }
% 243.24/31.66 join(join(X, Y), meet(join(X, Y), Z))
% 243.24/31.66 = { by axiom 9 (absorption2) }
% 243.24/31.66 join(X, Y)
% 243.24/31.66
% 243.24/31.66 Lemma 39: join(X, meet(Y, complement(meet(X, Y)))) = join(X, Y).
% 243.24/31.66 Proof:
% 243.24/31.66 join(X, meet(Y, complement(meet(X, Y))))
% 243.24/31.66 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.66 join(X, meet(Y, complement(meet(Y, X))))
% 243.24/31.66 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.66 join(meet(Y, complement(meet(Y, X))), X)
% 243.24/31.66 = { by lemma 22 R->L }
% 243.24/31.66 join(meet(Y, complement(meet(Y, X))), join(X, meet(Y, X)))
% 243.24/31.66 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.66 join(meet(Y, complement(meet(Y, X))), join(meet(Y, X), X))
% 243.24/31.66 = { by axiom 10 (associativity_of_join) R->L }
% 243.24/31.66 join(join(meet(Y, complement(meet(Y, X))), meet(Y, X)), X)
% 243.24/31.66 = { by axiom 2 (commutativity_of_join) }
% 243.24/31.66 join(X, join(meet(Y, complement(meet(Y, X))), meet(Y, X)))
% 243.24/31.66 = { by axiom 2 (commutativity_of_join) }
% 243.24/31.66 join(X, join(meet(Y, X), meet(Y, complement(meet(Y, X)))))
% 243.24/31.66 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.66 join(X, join(meet(Y, X), meet(complement(meet(Y, X)), Y)))
% 243.24/31.66 = { by axiom 9 (absorption2) R->L }
% 243.24/31.66 join(X, join(meet(Y, X), meet(complement(meet(Y, X)), join(Y, meet(Y, X)))))
% 243.24/31.66 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.66 join(X, join(meet(Y, X), meet(complement(meet(Y, X)), join(meet(Y, X), Y))))
% 243.24/31.66 = { by lemma 16 R->L }
% 243.24/31.66 join(X, complement(complement(join(meet(Y, X), meet(complement(meet(Y, X)), join(meet(Y, X), Y))))))
% 243.24/31.66 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.66 join(X, complement(complement(join(meet(Y, X), meet(complement(meet(Y, X)), join(Y, meet(Y, X)))))))
% 243.24/31.66 = { by lemma 37 R->L }
% 243.24/31.66 join(X, complement(meet(complement(meet(Y, X)), complement(meet(complement(meet(Y, X)), join(meet(Y, X), meet(complement(meet(Y, X)), join(Y, meet(Y, X)))))))))
% 243.24/31.66 = { by lemma 24 R->L }
% 243.24/31.66 join(X, complement(meet(complement(meet(Y, X)), complement(meet(complement(meet(Y, X)), meet(join(meet(Y, X), meet(complement(meet(Y, X)), join(Y, meet(Y, X)))), join(Y, join(meet(Y, X), meet(complement(meet(Y, X)), join(Y, meet(Y, X)))))))))))
% 243.24/31.66 = { by lemma 38 }
% 243.24/31.66 join(X, complement(meet(complement(meet(Y, X)), complement(meet(complement(meet(Y, X)), meet(join(meet(Y, X), meet(complement(meet(Y, X)), join(Y, meet(Y, X)))), join(Y, meet(Y, X))))))))
% 243.24/31.66 = { by axiom 4 (commutativity_of_meet) }
% 243.24/31.66 join(X, complement(meet(complement(meet(Y, X)), complement(meet(complement(meet(Y, X)), meet(join(Y, meet(Y, X)), join(meet(Y, X), meet(complement(meet(Y, X)), join(Y, meet(Y, X))))))))))
% 243.24/31.66 = { by axiom 2 (commutativity_of_join) }
% 243.24/31.66 join(X, complement(meet(complement(meet(Y, X)), complement(meet(complement(meet(Y, X)), meet(join(Y, meet(Y, X)), join(meet(Y, X), meet(complement(meet(Y, X)), join(meet(Y, X), Y)))))))))
% 243.24/31.66 = { by axiom 2 (commutativity_of_join) }
% 243.24/31.66 join(X, complement(meet(complement(meet(Y, X)), complement(meet(complement(meet(Y, X)), meet(join(meet(Y, X), Y), join(meet(Y, X), meet(complement(meet(Y, X)), join(meet(Y, X), Y)))))))))
% 243.24/31.66 = { by lemma 21 }
% 243.24/31.66 join(X, complement(meet(complement(meet(Y, X)), complement(meet(complement(meet(Y, X)), join(meet(Y, X), Y))))))
% 243.24/31.66 = { by lemma 37 }
% 243.24/31.66 join(X, complement(complement(join(meet(Y, X), Y))))
% 243.24/31.66 = { by lemma 16 }
% 243.24/31.66 join(X, join(meet(Y, X), Y))
% 243.24/31.66 = { by axiom 2 (commutativity_of_join) }
% 243.24/31.66 join(X, join(Y, meet(Y, X)))
% 243.24/31.66 = { by axiom 9 (absorption2) }
% 243.24/31.66 join(X, Y)
% 243.24/31.66
% 243.24/31.66 Lemma 40: join(meet(X, Y), meet(Y, complement(meet(X, Y)))) = Y.
% 243.24/31.66 Proof:
% 243.24/31.66 join(meet(X, Y), meet(Y, complement(meet(X, Y))))
% 243.24/31.66 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.66 join(meet(X, Y), meet(Y, complement(meet(Y, X))))
% 243.24/31.66 = { by axiom 3 (idempotence_of_meet) R->L }
% 243.24/31.66 join(meet(X, Y), meet(Y, complement(meet(meet(Y, Y), X))))
% 243.24/31.66 = { by axiom 12 (associativity_of_meet) }
% 243.24/31.66 join(meet(X, Y), meet(Y, complement(meet(Y, meet(Y, X)))))
% 243.24/31.66 = { by axiom 4 (commutativity_of_meet) }
% 243.24/31.66 join(meet(X, Y), meet(Y, complement(meet(Y, meet(X, Y)))))
% 243.24/31.66 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.66 join(meet(Y, X), meet(Y, complement(meet(Y, meet(X, Y)))))
% 243.24/31.66 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.66 join(meet(Y, X), meet(Y, complement(meet(Y, meet(Y, X)))))
% 243.24/31.66 = { by axiom 9 (absorption2) R->L }
% 243.24/31.66 join(meet(Y, X), meet(join(Y, meet(Y, X)), complement(meet(Y, meet(Y, X)))))
% 243.24/31.66 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.66 join(meet(Y, X), meet(join(Y, meet(Y, X)), complement(meet(meet(Y, X), Y))))
% 243.24/31.66 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.66 join(meet(Y, X), meet(join(meet(Y, X), Y), complement(meet(meet(Y, X), Y))))
% 243.24/31.66 = { by lemma 29 }
% 243.24/31.66 join(meet(Y, X), meet(Y, complement(meet(meet(Y, X), Y))))
% 243.24/31.66 = { by lemma 39 }
% 243.24/31.66 join(meet(Y, X), Y)
% 243.24/31.66 = { by axiom 2 (commutativity_of_join) }
% 243.24/31.66 join(Y, meet(Y, X))
% 243.24/31.66 = { by axiom 9 (absorption2) }
% 243.24/31.66 Y
% 243.24/31.66
% 243.24/31.66 Lemma 41: meet(join(X, Y), join(X, complement(join(X, Y)))) = X.
% 243.24/31.66 Proof:
% 243.24/31.66 meet(join(X, Y), join(X, complement(join(X, Y))))
% 243.24/31.66 = { by lemma 40 R->L }
% 243.24/31.66 meet(join(X, Y), join(X, complement(join(X, join(meet(Z, Y), meet(Y, complement(meet(Z, Y))))))))
% 243.24/31.66 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.66 meet(join(X, Y), join(complement(join(X, join(meet(Z, Y), meet(Y, complement(meet(Z, Y)))))), X))
% 243.24/31.66 = { by lemma 16 R->L }
% 243.24/31.66 meet(join(X, Y), join(complement(join(X, join(meet(Z, Y), meet(Y, complement(meet(Z, Y)))))), complement(complement(X))))
% 243.24/31.66 = { by lemma 36 R->L }
% 243.24/31.66 meet(join(X, Y), join(complement(join(X, join(meet(Z, Y), meet(Y, complement(meet(Z, Y)))))), meet(complement(meet(complement(X), join(join(meet(Z, Y), meet(Y, complement(meet(Z, Y)))), X))), complement(meet(complement(X), complement(meet(complement(X), join(join(meet(Z, Y), meet(Y, complement(meet(Z, Y)))), X))))))))
% 243.24/31.66 = { by lemma 37 R->L }
% 243.24/31.66 meet(join(X, Y), join(meet(complement(X), complement(meet(complement(X), join(X, join(meet(Z, Y), meet(Y, complement(meet(Z, Y)))))))), meet(complement(meet(complement(X), join(join(meet(Z, Y), meet(Y, complement(meet(Z, Y)))), X))), complement(meet(complement(X), complement(meet(complement(X), join(join(meet(Z, Y), meet(Y, complement(meet(Z, Y)))), X))))))))
% 243.24/31.66 = { by axiom 2 (commutativity_of_join) }
% 243.24/31.66 meet(join(X, Y), join(meet(complement(X), complement(meet(complement(X), join(join(meet(Z, Y), meet(Y, complement(meet(Z, Y)))), X)))), meet(complement(meet(complement(X), join(join(meet(Z, Y), meet(Y, complement(meet(Z, Y)))), X))), complement(meet(complement(X), complement(meet(complement(X), join(join(meet(Z, Y), meet(Y, complement(meet(Z, Y)))), X))))))))
% 243.24/31.66 = { by lemma 40 }
% 243.24/31.66 meet(join(X, Y), complement(meet(complement(X), join(join(meet(Z, Y), meet(Y, complement(meet(Z, Y)))), X))))
% 243.24/31.66 = { by axiom 2 (commutativity_of_join) }
% 243.24/31.66 meet(join(X, Y), complement(meet(complement(X), join(X, join(meet(Z, Y), meet(Y, complement(meet(Z, Y))))))))
% 243.24/31.66 = { by lemma 40 R->L }
% 243.24/31.66 meet(join(X, join(meet(Z, Y), meet(Y, complement(meet(Z, Y))))), complement(meet(complement(X), join(X, join(meet(Z, Y), meet(Y, complement(meet(Z, Y))))))))
% 243.24/31.66 = { by lemma 34 }
% 243.24/31.66 X
% 243.24/31.66
% 243.24/31.66 Lemma 42: meet(complement(X), join(X, Y)) = meet(Y, complement(meet(X, Y))).
% 243.24/31.66 Proof:
% 243.24/31.66 meet(complement(X), join(X, Y))
% 243.24/31.66 = { by lemma 39 R->L }
% 243.24/31.66 meet(complement(X), join(X, meet(Y, complement(meet(X, Y)))))
% 243.24/31.66 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.66 meet(join(X, meet(Y, complement(meet(X, Y)))), complement(X))
% 243.24/31.66 = { by lemma 41 R->L }
% 243.24/31.66 meet(join(X, meet(Y, complement(meet(X, Y)))), complement(meet(join(X, meet(Y, complement(meet(X, Y)))), join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))))))
% 243.24/31.66 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.24/31.66 meet(join(X, meet(Y, complement(meet(X, Y)))), complement(meet(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), join(X, meet(Y, complement(meet(X, Y)))))))
% 243.24/31.66 = { by lemma 16 R->L }
% 243.24/31.66 meet(join(X, meet(Y, complement(meet(X, Y)))), complement(meet(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), complement(complement(join(X, meet(Y, complement(meet(X, Y)))))))))
% 243.24/31.66 = { by lemma 33 R->L }
% 243.24/31.66 meet(join(X, meet(Y, complement(meet(X, Y)))), complement(meet(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), complement(meet(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), complement(meet(join(X, meet(Y, complement(meet(X, Y)))), join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))))))))))
% 243.24/31.66 = { by lemma 16 R->L }
% 243.24/31.66 meet(complement(complement(join(X, meet(Y, complement(meet(X, Y)))))), complement(meet(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), complement(meet(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), complement(meet(join(X, meet(Y, complement(meet(X, Y)))), join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))))))))))
% 243.24/31.66 = { by lemma 33 R->L }
% 243.24/31.66 meet(complement(meet(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), complement(meet(join(X, meet(Y, complement(meet(X, Y)))), join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))))))), complement(meet(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), complement(meet(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), complement(meet(join(X, meet(Y, complement(meet(X, Y)))), join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))))))))))
% 243.24/31.66 = { by lemma 36 }
% 243.24/31.66 complement(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))))
% 243.24/31.66 = { by axiom 2 (commutativity_of_join) }
% 243.24/31.66 complement(join(X, complement(join(meet(Y, complement(meet(X, Y))), X))))
% 243.24/31.66 = { by axiom 8 (complements_are_unique) R->L }
% 243.24/31.66 fresh2(n1, n1, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 243.24/31.66 = { by lemma 32 R->L }
% 243.24/31.66 fresh2(join(meet(Y, complement(meet(X, Y))), join(X, complement(join(meet(Y, complement(meet(X, Y))), X)))), n1, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 243.24/31.67 = { by lemma 15 }
% 243.24/31.67 fresh(meet(meet(Y, complement(meet(X, Y))), join(X, complement(join(meet(Y, complement(meet(X, Y))), X)))), n0, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 243.24/31.67 = { by axiom 2 (commutativity_of_join) R->L }
% 243.24/31.67 fresh(meet(meet(Y, complement(meet(X, Y))), join(X, complement(join(X, meet(Y, complement(meet(X, Y))))))), n0, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 243.24/31.67 = { by lemma 28 R->L }
% 243.24/31.67 fresh(meet(meet(Y, complement(meet(X, Y))), meet(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), join(X, meet(Y, complement(meet(X, Y)))))), n0, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 243.24/31.67 = { by axiom 4 (commutativity_of_meet) }
% 243.24/31.67 fresh(meet(meet(Y, complement(meet(X, Y))), meet(join(X, meet(Y, complement(meet(X, Y)))), join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))))), n0, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 243.24/31.67 = { by lemma 41 }
% 243.24/31.67 fresh(meet(meet(Y, complement(meet(X, Y))), X), n0, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 243.24/31.67 = { by axiom 4 (commutativity_of_meet) }
% 243.24/31.67 fresh(meet(X, meet(Y, complement(meet(X, Y)))), n0, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 243.24/31.67 = { by axiom 2 (commutativity_of_join) }
% 243.24/31.67 fresh(meet(X, meet(Y, complement(meet(X, Y)))), n0, join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), meet(Y, complement(meet(X, Y))))
% 243.24/31.67 = { by lemma 26 }
% 243.24/31.67 fresh(n0, n0, join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), meet(Y, complement(meet(X, Y))))
% 243.24/31.67 = { by axiom 7 (complements_are_unique) }
% 243.24/31.67 meet(Y, complement(meet(X, Y)))
% 243.24/31.67
% 243.24/31.67 Lemma 43: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 243.24/31.67 Proof:
% 243.24/31.67 meet(X, complement(meet(X, Y)))
% 243.57/31.67 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.57/31.67 meet(X, complement(meet(Y, X)))
% 243.57/31.67 = { by axiom 1 (idempotence_of_join) R->L }
% 243.57/31.67 meet(X, complement(join(meet(Y, X), meet(Y, X))))
% 243.57/31.67 = { by lemma 21 R->L }
% 243.57/31.67 meet(X, meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(Y, X), meet(meet(Y, X), Z))), meet(X, complement(join(meet(Y, X), meet(Y, X)))))))
% 243.57/31.67 = { by axiom 2 (commutativity_of_join) }
% 243.57/31.67 meet(X, meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(X, complement(join(meet(Y, X), meet(Y, X)))), meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(Y, X), meet(meet(Y, X), Z))))))
% 243.57/31.67 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.57/31.67 meet(X, meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(X, complement(join(meet(Y, X), meet(Y, X)))), meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X)))))))
% 243.57/31.67 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.57/31.67 meet(X, meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(complement(join(meet(Y, X), meet(Y, X))), X), meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X)))))))
% 243.57/31.67 = { by axiom 2 (commutativity_of_join) R->L }
% 243.57/31.67 meet(X, meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X)))), meet(complement(join(meet(Y, X), meet(Y, X))), X))))
% 243.57/31.67 = { by lemma 22 R->L }
% 243.57/31.67 meet(X, meet(join(complement(join(meet(Y, X), meet(Y, X))), meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X))))), join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X)))), meet(complement(join(meet(Y, X), meet(Y, X))), X))))
% 243.57/31.67 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.57/31.67 meet(X, meet(join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X)))), meet(complement(join(meet(Y, X), meet(Y, X))), X)), join(complement(join(meet(Y, X), meet(Y, X))), meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X)))))))
% 243.57/31.67 = { by lemma 23 R->L }
% 243.57/31.67 meet(X, meet(join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X)))), meet(complement(join(meet(Y, X), meet(Y, X))), X)), join(complement(join(meet(Y, X), meet(Y, X))), join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X)))), meet(complement(join(meet(Y, X), meet(Y, X))), X)))))
% 243.57/31.67 = { by lemma 24 }
% 243.57/31.67 meet(X, join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X)))), meet(complement(join(meet(Y, X), meet(Y, X))), X)))
% 243.57/31.67 = { by axiom 2 (commutativity_of_join) }
% 243.57/31.67 meet(X, join(meet(complement(join(meet(Y, X), meet(Y, X))), X), meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X))))))
% 243.57/31.67 = { by axiom 4 (commutativity_of_meet) }
% 243.57/31.67 meet(X, join(meet(X, complement(join(meet(Y, X), meet(Y, X)))), meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X))))))
% 243.57/31.67 = { by axiom 4 (commutativity_of_meet) }
% 243.57/31.67 meet(X, join(meet(X, complement(join(meet(Y, X), meet(Y, X)))), meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(Y, X), meet(meet(Y, X), Z)))))
% 243.57/31.67 = { by axiom 2 (commutativity_of_join) }
% 243.57/31.67 meet(X, join(meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(Y, X), meet(meet(Y, X), Z))), meet(X, complement(join(meet(Y, X), meet(Y, X))))))
% 243.57/31.67 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.57/31.67 meet(X, join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X)))), meet(X, complement(join(meet(Y, X), meet(Y, X))))))
% 243.57/31.67 = { by lemma 23 R->L }
% 243.57/31.67 meet(X, join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), join(meet(Y, X), meet(meet(Y, X), Z))))), meet(X, complement(join(meet(Y, X), meet(Y, X))))))
% 243.57/31.67 = { by axiom 2 (commutativity_of_join) R->L }
% 243.57/31.67 meet(X, join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(join(meet(Y, X), meet(meet(Y, X), Z)), meet(Y, X)))), meet(X, complement(join(meet(Y, X), meet(Y, X))))))
% 243.57/31.67 = { by lemma 25 R->L }
% 243.57/31.67 meet(X, join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), meet(complement(join(join(meet(Y, X), meet(meet(Y, X), Z)), meet(Y, X))), join(join(meet(Y, X), meet(meet(Y, X), Z)), meet(Y, X)))), meet(X, complement(join(meet(Y, X), meet(Y, X))))))
% 243.57/31.67 = { by axiom 4 (commutativity_of_meet) }
% 243.57/31.67 meet(X, join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), meet(join(join(meet(Y, X), meet(meet(Y, X), Z)), meet(Y, X)), complement(join(join(meet(Y, X), meet(meet(Y, X), Z)), meet(Y, X))))), meet(X, complement(join(meet(Y, X), meet(Y, X))))))
% 243.57/31.67 = { by axiom 6 (bottom) }
% 243.57/31.67 meet(X, join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), n0), meet(X, complement(join(meet(Y, X), meet(Y, X))))))
% 243.57/31.67 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.57/31.67 meet(X, join(meet(n0, join(meet(Y, X), meet(meet(Y, X), Z))), meet(X, complement(join(meet(Y, X), meet(Y, X))))))
% 243.57/31.67 = { by lemma 20 R->L }
% 243.57/31.67 meet(X, join(join(n0, meet(n0, join(meet(Y, X), meet(meet(Y, X), Z)))), meet(X, complement(join(meet(Y, X), meet(Y, X))))))
% 243.57/31.67 = { by axiom 9 (absorption2) }
% 243.57/31.67 meet(X, join(n0, meet(X, complement(join(meet(Y, X), meet(Y, X))))))
% 243.57/31.67 = { by lemma 20 }
% 243.57/31.67 meet(X, meet(X, complement(join(meet(Y, X), meet(Y, X)))))
% 243.57/31.67 = { by axiom 1 (idempotence_of_join) }
% 243.57/31.67 meet(X, meet(X, complement(meet(Y, X))))
% 243.57/31.67 = { by lemma 42 R->L }
% 243.57/31.67 meet(X, meet(complement(Y), join(Y, X)))
% 243.57/31.67 = { by lemma 28 }
% 243.57/31.67 meet(X, complement(Y))
% 243.57/31.67
% 243.57/31.67 Lemma 44: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 243.57/31.67 Proof:
% 243.57/31.67 meet(complement(X), join(X, Y))
% 243.57/31.67 = { by lemma 42 }
% 243.57/31.67 meet(Y, complement(meet(X, Y)))
% 243.57/31.67 = { by axiom 4 (commutativity_of_meet) }
% 243.57/31.67 meet(Y, complement(meet(Y, X)))
% 243.57/31.67 = { by lemma 43 }
% 243.57/31.67 meet(Y, complement(X))
% 243.57/31.67
% 243.57/31.67 Lemma 45: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 243.57/31.67 Proof:
% 243.57/31.67 complement(meet(X, complement(Y)))
% 243.57/31.67 = { by lemma 43 R->L }
% 243.57/31.67 complement(meet(X, complement(meet(X, Y))))
% 243.57/31.67 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.57/31.67 complement(meet(X, complement(meet(Y, X))))
% 243.57/31.67 = { by lemma 16 R->L }
% 243.57/31.67 complement(meet(X, complement(meet(Y, complement(complement(X))))))
% 243.57/31.67 = { by lemma 44 R->L }
% 243.57/31.67 complement(meet(X, complement(meet(complement(complement(X)), join(complement(X), Y)))))
% 243.57/31.67 = { by lemma 16 }
% 243.57/31.67 complement(meet(X, complement(meet(X, join(complement(X), Y)))))
% 243.57/31.67 = { by axiom 2 (commutativity_of_join) }
% 243.57/31.67 complement(meet(X, complement(meet(X, join(Y, complement(X))))))
% 243.57/31.67 = { by lemma 43 }
% 243.57/31.67 complement(meet(X, complement(join(Y, complement(X)))))
% 243.57/31.67 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.57/31.67 complement(meet(complement(join(Y, complement(X))), X))
% 243.57/31.67 = { by lemma 16 R->L }
% 243.57/31.67 complement(meet(complement(join(Y, complement(X))), complement(complement(X))))
% 243.57/31.67 = { by lemma 33 R->L }
% 243.57/31.67 complement(meet(complement(join(Y, complement(X))), complement(meet(join(Y, complement(X)), complement(meet(X, join(Y, complement(X))))))))
% 243.57/31.67 = { by axiom 9 (absorption2) R->L }
% 243.57/31.67 complement(meet(complement(join(Y, complement(X))), join(complement(meet(join(Y, complement(X)), complement(meet(X, join(Y, complement(X)))))), meet(complement(meet(join(Y, complement(X)), complement(meet(X, join(Y, complement(X)))))), complement(meet(join(Y, complement(X)), complement(meet(join(Y, complement(X)), complement(meet(X, join(Y, complement(X))))))))))))
% 243.57/31.67 = { by lemma 36 }
% 243.57/31.67 complement(meet(complement(join(Y, complement(X))), join(complement(meet(join(Y, complement(X)), complement(meet(X, join(Y, complement(X)))))), complement(join(Y, complement(X))))))
% 243.57/31.67 = { by axiom 2 (commutativity_of_join) }
% 243.57/31.67 complement(meet(complement(join(Y, complement(X))), join(complement(join(Y, complement(X))), complement(meet(join(Y, complement(X)), complement(meet(X, join(Y, complement(X)))))))))
% 243.57/31.67 = { by lemma 33 }
% 243.57/31.67 complement(meet(complement(join(Y, complement(X))), join(complement(join(Y, complement(X))), complement(complement(X)))))
% 243.57/31.67 = { by lemma 16 }
% 243.57/31.67 complement(meet(complement(join(Y, complement(X))), join(complement(join(Y, complement(X))), X)))
% 243.57/31.67 = { by axiom 2 (commutativity_of_join) }
% 243.57/31.67 complement(meet(complement(join(Y, complement(X))), join(X, complement(join(Y, complement(X))))))
% 243.57/31.67 = { by lemma 24 }
% 243.57/31.67 complement(complement(join(Y, complement(X))))
% 243.57/31.67 = { by lemma 16 }
% 243.57/31.67 join(Y, complement(X))
% 243.57/31.67
% 243.57/31.67 Lemma 46: meet(X, join(Y, meet(X, Z))) = meet(X, join(Z, Y)).
% 243.57/31.67 Proof:
% 243.57/31.67 meet(X, join(Y, meet(X, Z)))
% 243.57/31.67 = { by lemma 16 R->L }
% 243.57/31.67 meet(X, join(complement(complement(Y)), meet(X, Z)))
% 243.57/31.67 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.57/31.67 meet(X, join(complement(complement(Y)), meet(Z, X)))
% 243.57/31.67 = { by axiom 2 (commutativity_of_join) R->L }
% 243.57/31.67 meet(X, join(meet(Z, X), complement(complement(Y))))
% 243.57/31.67 = { by lemma 45 R->L }
% 243.57/31.67 meet(X, complement(meet(complement(Y), complement(meet(Z, X)))))
% 243.57/31.67 = { by lemma 43 R->L }
% 243.57/31.67 meet(X, complement(meet(X, meet(complement(Y), complement(meet(Z, X))))))
% 243.57/31.67 = { by lemma 27 R->L }
% 243.57/31.67 meet(X, complement(meet(complement(Y), meet(X, complement(meet(Z, X))))))
% 243.57/31.67 = { by lemma 42 R->L }
% 243.57/31.67 meet(X, complement(meet(complement(Y), meet(complement(Z), join(Z, X)))))
% 243.57/31.67 = { by lemma 44 }
% 243.57/31.67 meet(X, complement(meet(complement(Y), meet(X, complement(Z)))))
% 243.57/31.67 = { by lemma 27 }
% 243.57/31.67 meet(X, complement(meet(X, meet(complement(Y), complement(Z)))))
% 243.57/31.67 = { by lemma 43 }
% 243.57/31.67 meet(X, complement(meet(complement(Y), complement(Z))))
% 243.57/31.67 = { by lemma 45 }
% 243.57/31.67 meet(X, join(Z, complement(complement(Y))))
% 243.57/31.67 = { by lemma 16 }
% 243.57/31.68 meet(X, join(Z, Y))
% 243.57/31.68
% 243.57/31.68 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 243.57/31.68 Proof:
% 243.57/31.68 meet(a, join(b, c))
% 243.57/31.68 = { by axiom 2 (commutativity_of_join) R->L }
% 243.57/31.68 meet(a, join(c, b))
% 243.57/31.68 = { by lemma 46 R->L }
% 243.57/31.68 meet(a, join(b, meet(a, c)))
% 243.57/31.68 = { by lemma 30 R->L }
% 243.57/31.68 join(meet(a, b), meet(a, join(b, meet(a, c))))
% 243.57/31.68 = { by lemma 46 R->L }
% 243.57/31.68 join(meet(a, b), meet(a, join(meet(a, c), meet(a, b))))
% 243.57/31.68 = { by axiom 2 (commutativity_of_join) R->L }
% 243.57/31.68 join(meet(a, b), meet(a, join(meet(a, b), meet(a, c))))
% 243.57/31.68 = { by axiom 4 (commutativity_of_meet) R->L }
% 243.57/31.68 join(meet(a, b), meet(a, join(meet(a, b), meet(c, a))))
% 243.57/31.68 = { by lemma 22 R->L }
% 243.57/31.68 join(meet(a, b), join(meet(a, join(meet(a, b), meet(c, a))), meet(c, meet(a, join(meet(a, b), meet(c, a))))))
% 243.57/31.68 = { by lemma 21 }
% 243.57/31.68 join(meet(a, b), join(meet(a, join(meet(a, b), meet(c, a))), meet(c, a)))
% 243.57/31.68 = { by axiom 2 (commutativity_of_join) }
% 243.57/31.68 join(meet(a, b), join(meet(c, a), meet(a, join(meet(a, b), meet(c, a)))))
% 243.57/31.68 = { by axiom 4 (commutativity_of_meet) }
% 243.57/31.68 join(meet(a, b), join(meet(c, a), meet(a, join(meet(a, b), meet(a, c)))))
% 243.57/31.68 = { by axiom 4 (commutativity_of_meet) }
% 243.57/31.68 join(meet(a, b), join(meet(a, c), meet(a, join(meet(a, b), meet(a, c)))))
% 243.57/31.68 = { by lemma 38 }
% 243.57/31.68 join(meet(a, b), meet(a, c))
% 243.57/31.68 % SZS output end Proof
% 243.57/31.68
% 243.57/31.68 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------