0.07/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.07/0.13 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.14/0.34 % Computer : n025.cluster.edu 0.14/0.34 % Model : x86_64 x86_64 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.14/0.34 % Memory : 8042.1875MB 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.14/0.34 % CPULimit : 180 0.14/0.34 % DateTime : Thu Aug 29 12:19:15 EDT 2019 0.14/0.34 % CPUTime : 3.75/3.92 % SZS status Unsatisfiable 3.75/3.92 3.75/3.92 % SZS output start Proof 3.75/3.92 Take the following subset of the input axioms: 3.75/3.92 fof(prove_wal_axioms_1, negated_conjecture, meet(a, a)!=a). 3.75/3.92 fof(single_axiom, axiom, ![B, A, C, D, E, F]: B=join(meet(join(meet(A, B), meet(B, join(A, B))), C), meet(join(meet(A, join(join(meet(B, D), meet(E, B)), B)), meet(join(meet(B, meet(meet(join(B, D), join(E, B)), B)), meet(F, join(B, meet(meet(join(B, D), join(E, B)), B)))), join(A, join(join(meet(B, D), meet(E, B)), B)))), join(join(meet(A, B), meet(B, join(A, B))), C)))). 3.75/3.92 3.75/3.92 Now clausify the problem and encode Horn clauses using encoding 3 of 3.75/3.92 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 3.75/3.92 We repeatedly replace C & s=t => u=v by the two clauses: 3.75/3.92 fresh(y, y, x1...xn) = u 3.75/3.92 C => fresh(s, t, x1...xn) = v 3.75/3.92 where fresh is a fresh function symbol and x1..xn are the free 3.75/3.92 variables of u and v. 3.75/3.92 A predicate p(X) is encoded as p(X)=true (this is sound, because the 3.75/3.92 input problem has no model of domain size 1). 3.75/3.92 3.75/3.92 The encoding turns the above axioms into the following unit equations and goals: 3.75/3.92 3.88/4.03 Axiom 1 (single_axiom): X = join(meet(join(meet(Y, X), meet(X, join(Y, X))), Z), meet(join(meet(Y, join(join(meet(X, W), meet(V, X)), X)), meet(join(meet(X, meet(meet(join(X, W), join(V, X)), X)), meet(U, join(X, meet(meet(join(X, W), join(V, X)), X)))), join(Y, join(join(meet(X, W), meet(V, X)), X)))), join(join(meet(Y, X), meet(X, join(Y, X))), Z))). 3.88/4.03 3.88/4.03 Lemma 2: join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), W), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), V), meet(U, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), V), meet(U, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), W))) = join(meet(X, Y), meet(Y, join(X, Y))). 3.88/4.03 Proof: 3.88/4.03 join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), W), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), V), meet(U, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), V), meet(U, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), W))) 3.88/4.03 = { by axiom 1 (single_axiom) } 3.88/4.03 join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), W), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), V), meet(U, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), meet(meet(join(join(meet(X, Y), meet(Y, join(X, Y))), V), join(U, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(join(meet(X, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(X, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(X, Y), meet(Y, join(X, Y))), meet(meet(join(join(meet(X, Y), meet(Y, join(X, Y))), V), join(U, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), V), meet(U, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), W))) 3.88/4.03 = { by axiom 1 (single_axiom) } 4.16/4.32 join(meet(X, Y), meet(Y, join(X, Y))) 4.16/4.32 4.16/4.32 Lemma 3: join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), W), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), W))) = join(meet(X, Y), meet(Y, join(X, Y))). 4.16/4.32 Proof: 4.16/4.32 join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), W), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), W))) 4.16/4.32 = { by lemma 2 } 4.16/4.32 join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))))))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), W), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), W))) 4.16/4.32 = { by lemma 2 } 4.16/4.32 join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))))))), meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), W), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), W))) 4.16/4.32 = { by lemma 2 } 4.16/4.33 join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))))))), meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))))))))), W), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), W))) 4.16/4.33 = { by lemma 2 } 4.16/4.34 join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))))))), meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))))))))), W), meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), W))) 4.16/4.34 = { by lemma 2 } 4.19/4.34 join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))))))), meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))))))))), W), meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), W))) 4.19/4.34 = { by lemma 2 } 4.19/4.34 join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))))))), meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), 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Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), W))) 4.19/4.35 = { by lemma 2 } 4.19/4.36 join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(join(meet(Z, join(meet(X, Y), 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meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))))))), meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, 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join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))))))))), W), meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))))), ?), meet(?, join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))))))), join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, 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join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), W))) 4.19/4.36 = { by lemma 2 } 4.19/4.37 join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), 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Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), W))) 4.19/4.37 = { by lemma 2 } 4.19/4.38 join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), 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Y)))))))))), ?), meet(?, join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))))))), join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, 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meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))))))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), W))) 4.19/4.39 = { by lemma 2 } 4.19/4.40 join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))))))), meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, 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Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))))))))), W), meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, 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meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), W))) 4.19/4.41 = { by lemma 2 } 4.19/4.41 join(meet(join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), 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Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))))))))), W))) 4.19/4.42 = { by lemma 2 } 4.19/4.42 join(meet(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))), meet(join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Z, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Z, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Z, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y)))))))))) 4.19/4.42 = { by lemma 2 } 4.63/4.81 join(meet(X, Y), meet(Y, join(X, Y))) 4.63/4.81 4.63/4.81 Lemma 4: join(meet(join(meet(join(meet(U, Y), meet(Y, join(U, Y))), Y), meet(Y, join(join(meet(U, Y), meet(Y, join(U, Y))), Y))), T), meet(Y, join(join(meet(join(meet(U, Y), meet(Y, join(U, Y))), Y), meet(Y, join(join(meet(U, Y), meet(Y, join(U, Y))), Y))), T))) = Y. 4.63/4.81 Proof: 4.63/4.81 join(meet(join(meet(join(meet(U, Y), meet(Y, join(U, Y))), Y), meet(Y, join(join(meet(U, Y), meet(Y, join(U, Y))), Y))), T), meet(Y, join(join(meet(join(meet(U, Y), meet(Y, join(U, Y))), Y), meet(Y, join(join(meet(U, Y), meet(Y, join(U, Y))), Y))), T))) 4.63/4.81 = { by axiom 1 (single_axiom) } 4.63/4.81 join(meet(join(meet(join(meet(U, join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))))))), meet(Y, join(U, Y))), Y), meet(Y, join(join(meet(U, Y), meet(Y, join(U, Y))), Y))), T), meet(Y, join(join(meet(join(meet(U, Y), meet(Y, join(U, Y))), Y), meet(Y, join(join(meet(U, Y), meet(Y, join(U, Y))), Y))), T))) 4.63/4.81 = { by axiom 1 (single_axiom) } 4.63/4.81 join(meet(join(meet(join(meet(U, join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))))))), meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))))), join(U, Y))), Y), meet(Y, join(join(meet(U, Y), meet(Y, join(U, Y))), Y))), T), meet(Y, join(join(meet(join(meet(U, Y), meet(Y, join(U, Y))), Y), meet(Y, join(join(meet(U, Y), meet(Y, join(U, Y))), Y))), T))) 4.63/4.81 = { by axiom 1 (single_axiom) } 4.63/4.81 join(meet(join(meet(join(meet(U, join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))))))), meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, 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Y))), T))) 4.63/4.82 = { by axiom 1 (single_axiom) } 4.63/4.82 join(meet(join(meet(join(meet(U, join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))))))), meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, 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join(join(meet(join(meet(U, Y), meet(Y, join(U, Y))), Y), meet(Y, join(join(meet(U, Y), meet(Y, join(U, Y))), Y))), T))) 4.63/4.82 = { by axiom 1 (single_axiom) } 4.63/4.82 join(meet(join(meet(join(meet(U, join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))))))), meet(join(meet(join(meet(?, Y), 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meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))))), join(U, Y))), Y))), T), meet(Y, join(join(meet(join(meet(U, Y), meet(Y, join(U, Y))), Y), meet(Y, join(join(meet(U, Y), meet(Y, join(U, Y))), Y))), T))) 4.63/4.82 = { by axiom 1 (single_axiom) } 4.63/4.83 join(meet(join(meet(join(meet(U, join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), 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meet(Y, join(U, Y))), Y))), T))) 4.72/4.88 = { by axiom 1 (single_axiom) } 4.72/4.89 join(meet(join(meet(join(meet(U, join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))))))), meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), 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Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))))))), meet(Y, join(U, Y))), Y))), T))) 4.72/4.89 = { by axiom 1 (single_axiom) } 4.72/4.90 join(meet(join(meet(join(meet(U, join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), 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join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))))))))), T), meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, 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Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))))))), meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, 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meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))))))), meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))))), join(join(meet(U, join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))))))), meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))))), join(U, join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))))))))), join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))))))))), T))) 4.78/4.92 = { by lemma 3 } 4.78/4.92 join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))))) 4.78/4.92 = { by axiom 1 (single_axiom) } 4.81/4.96 Y 4.81/4.96 4.81/4.96 Lemma 5: join(meet(join(meet(Z, Y), meet(Y, join(Z, Y))), W), meet(join(meet(Z, join(join(meet(Y, V), meet(U, Y)), Y)), meet(Y, join(Z, join(join(meet(Y, V), meet(U, Y)), Y)))), join(join(meet(Z, Y), meet(Y, join(Z, Y))), W))) = Y. 4.81/4.96 Proof: 4.81/4.96 join(meet(join(meet(Z, Y), meet(Y, join(Z, Y))), W), meet(join(meet(Z, join(join(meet(Y, V), meet(U, Y)), Y)), meet(Y, join(Z, join(join(meet(Y, V), meet(U, Y)), Y)))), join(join(meet(Z, Y), meet(Y, join(Z, Y))), W))) 4.81/4.96 = { by lemma 4 } 4.81/4.96 join(meet(join(meet(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))), meet(Y, join(Z, Y))), W), meet(join(meet(Z, join(join(meet(Y, V), meet(U, Y)), Y)), meet(Y, join(Z, join(join(meet(Y, V), meet(U, Y)), Y)))), join(join(meet(Z, Y), meet(Y, join(Z, Y))), W))) 4.81/4.96 = { by lemma 4 } 4.81/4.96 join(meet(join(meet(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))), meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), join(Z, Y))), W), meet(join(meet(Z, join(join(meet(Y, V), meet(U, Y)), Y)), meet(Y, join(Z, join(join(meet(Y, V), meet(U, Y)), Y)))), join(join(meet(Z, Y), meet(Y, join(Z, Y))), W))) 4.81/4.96 = { by lemma 4 } 4.81/4.96 join(meet(join(meet(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))), meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), join(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))))), W), meet(join(meet(Z, join(join(meet(Y, V), meet(U, Y)), Y)), meet(Y, join(Z, join(join(meet(Y, V), meet(U, Y)), Y)))), join(join(meet(Z, Y), meet(Y, join(Z, Y))), W))) 4.81/4.96 = { by lemma 4 } 4.81/4.96 join(meet(join(meet(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))), meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), join(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))))), W), meet(join(meet(Z, join(join(meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), V), meet(U, Y)), Y)), meet(Y, join(Z, join(join(meet(Y, V), meet(U, Y)), Y)))), join(join(meet(Z, Y), meet(Y, join(Z, Y))), W))) 4.81/4.96 = { by lemma 4 } 4.81/4.96 join(meet(join(meet(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))), meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), join(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))))), W), meet(join(meet(Z, join(join(meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), V), meet(U, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), Y)), meet(Y, join(Z, join(join(meet(Y, V), meet(U, Y)), Y)))), join(join(meet(Z, Y), meet(Y, join(Z, Y))), W))) 4.81/4.96 = { by lemma 4 } 4.81/4.96 join(meet(join(meet(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))), meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), join(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))))), W), meet(join(meet(Z, join(join(meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), V), meet(U, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), meet(Y, join(Z, join(join(meet(Y, V), meet(U, Y)), Y)))), join(join(meet(Z, Y), meet(Y, join(Z, Y))), W))) 4.81/4.96 = { by lemma 4 } 4.81/4.96 join(meet(join(meet(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))), meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), join(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))))), W), meet(join(meet(Z, join(join(meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), V), meet(U, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), meet(Y, join(Z, join(join(meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), V), meet(U, Y)), Y)))), join(join(meet(Z, Y), meet(Y, join(Z, Y))), W))) 4.81/4.96 = { by lemma 4 } 4.81/4.96 join(meet(join(meet(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))), meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), join(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))))), W), meet(join(meet(Z, join(join(meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), V), meet(U, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), meet(Y, join(Z, join(join(meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), V), meet(U, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), Y)))), join(join(meet(Z, Y), meet(Y, join(Z, Y))), W))) 4.81/4.96 = { by lemma 4 } 4.81/4.97 join(meet(join(meet(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))), meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), join(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))))), W), meet(join(meet(Z, join(join(meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), V), meet(U, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), meet(Y, join(Z, join(join(meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), V), meet(U, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))))), join(join(meet(Z, Y), meet(Y, join(Z, Y))), W))) 4.81/4.97 = { by lemma 4 } 4.81/4.97 join(meet(join(meet(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))), meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), join(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))))), W), meet(join(meet(Z, join(join(meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), V), meet(U, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), meet(Y, join(Z, join(join(meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), V), meet(U, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))))), join(join(meet(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))), meet(Y, join(Z, Y))), W))) 4.81/4.97 = { by lemma 4 } 4.81/4.97 join(meet(join(meet(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))), meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), join(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))))), W), meet(join(meet(Z, join(join(meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), V), meet(U, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), meet(Y, join(Z, join(join(meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), V), meet(U, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))))), join(join(meet(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))), meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), join(Z, Y))), W))) 4.81/4.97 = { by lemma 4 } 4.81/4.97 join(meet(join(meet(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))), meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), join(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))))), W), meet(join(meet(Z, join(join(meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), V), meet(U, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), meet(Y, join(Z, join(join(meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), V), meet(U, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))), join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))))))), join(join(meet(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))), meet(join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))), join(Z, join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y)))))), W))) 4.81/4.97 = { by lemma 2 } 4.81/4.97 join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y), meet(Y, join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), Y), meet(Y, join(join(meet(?, Y), meet(Y, join(?, Y))), Y))), Y))) 4.81/4.97 = { by lemma 4 } 4.81/4.97 Y 4.81/4.97 4.81/4.97 Lemma 6: join(meet(join(meet(X, Y), meet(Y, join(X, Y))), Z), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(X, Y), meet(Y, join(X, Y))), Z))) = join(meet(X, Y), meet(Y, join(X, Y))). 4.81/4.97 Proof: 4.81/4.97 join(meet(join(meet(X, Y), meet(Y, join(X, Y))), Z), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(X, Y), meet(Y, join(X, Y))), Z))) 4.81/4.97 = { by lemma 3 } 4.81/4.97 join(meet(join(meet(join(meet(join(meet(?, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(?, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(?, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(?, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(join(meet(?, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(?, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(?, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(?, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), Z), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(X, Y), meet(Y, join(X, Y))), Z))) 4.81/4.97 = { by lemma 3 } 4.81/4.97 join(meet(join(meet(join(meet(join(meet(?, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(?, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(?, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(?, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(join(meet(?, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(?, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(?, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(?, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), Z), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(join(meet(join(meet(?, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(?, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(?, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(?, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(join(meet(?, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(?, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(?, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(?, join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), join(meet(X, Y), meet(Y, join(X, Y)))))), Z))) 4.81/4.97 = { by lemma 3 } 4.81/4.98 join(meet(X, Y), meet(Y, join(X, Y))) 4.81/4.98 4.81/4.98 Lemma 7: join(meet(Y, U), meet(Y, join(Y, U))) = Y. 4.81/4.98 Proof: 4.81/4.98 join(meet(Y, U), meet(Y, join(Y, U))) 4.81/4.98 = { by axiom 1 (single_axiom) } 4.81/4.98 join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))))), U), meet(Y, join(Y, U))) 4.81/4.98 = { by axiom 1 (single_axiom) } 4.81/4.98 join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))))), U), meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))))), join(Y, U))) 4.81/4.98 = { by axiom 1 (single_axiom) } 4.81/4.98 join(meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))))), U), meet(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))))), join(join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))))), U))) 4.81/4.98 = { by lemma 6 } 4.81/4.98 join(meet(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))), meet(join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y)))), join(join(meet(?, Y), meet(Y, join(?, Y))), join(meet(?, join(join(meet(Y, ?), meet(?, Y)), Y)), meet(join(meet(Y, meet(meet(join(Y, ?), join(?, Y)), Y)), meet(?, join(Y, meet(meet(join(Y, ?), join(?, Y)), Y)))), join(?, join(join(meet(Y, ?), meet(?, Y)), Y))))))) 4.81/4.98 = { by axiom 1 (single_axiom) } 4.81/4.98 Y 4.81/4.98 4.81/4.98 Lemma 8: join(meet(join(meet(Y, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Y, join(meet(X, Y), meet(Y, join(X, Y)))))), Z), meet(Y, join(join(meet(Y, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Y, join(meet(X, Y), meet(Y, join(X, Y)))))), Z))) = join(meet(X, Y), meet(Y, join(X, Y))). 4.81/4.98 Proof: 4.81/4.98 join(meet(join(meet(Y, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Y, join(meet(X, Y), meet(Y, join(X, Y)))))), Z), meet(Y, join(join(meet(Y, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Y, join(meet(X, Y), meet(Y, join(X, Y)))))), Z))) 4.81/4.98 = { by lemma 7 } 4.81/4.98 join(meet(join(meet(Y, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Y, join(meet(X, Y), meet(Y, join(X, Y)))))), Z), meet(join(meet(Y, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))), meet(Y, join(Y, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), ?), meet(?, join(meet(X, Y), meet(Y, join(X, Y))))), join(meet(X, Y), meet(Y, join(X, Y))))))), join(join(meet(Y, join(meet(X, Y), meet(Y, join(X, Y)))), meet(join(meet(X, Y), meet(Y, join(X, Y))), join(Y, join(meet(X, Y), meet(Y, join(X, Y)))))), Z))) 4.81/4.98 = { by lemma 2 } 4.81/4.98 join(meet(X, Y), meet(Y, join(X, Y))) 4.81/4.98 4.81/4.98 Lemma 9: join(meet(Y, X), meet(X, join(Y, X))) = X. 4.81/4.98 Proof: 4.81/4.98 join(meet(Y, X), meet(X, join(Y, X))) 4.81/4.98 = { by lemma 6 } 4.81/4.98 join(meet(join(meet(Y, X), meet(X, join(Y, X))), ?), meet(join(meet(Y, X), meet(X, join(Y, X))), join(join(meet(Y, X), meet(X, join(Y, X))), ?))) 4.81/4.98 = { by lemma 8 } 4.81/4.98 join(meet(join(meet(join(meet(X, join(meet(Y, X), meet(X, join(Y, X)))), meet(join(meet(Y, X), meet(X, join(Y, X))), join(X, join(meet(Y, X), meet(X, join(Y, X)))))), X), meet(X, join(join(meet(X, join(meet(Y, X), meet(X, join(Y, X)))), meet(join(meet(Y, X), meet(X, join(Y, X))), join(X, join(meet(Y, X), meet(X, join(Y, X)))))), X))), ?), meet(join(meet(Y, X), meet(X, join(Y, X))), join(join(meet(Y, X), meet(X, join(Y, X))), ?))) 4.81/4.98 = { by lemma 8 } 4.81/4.99 join(meet(join(meet(join(meet(X, join(meet(Y, X), meet(X, join(Y, X)))), meet(join(meet(Y, X), meet(X, join(Y, X))), join(X, join(meet(Y, X), meet(X, join(Y, X)))))), X), meet(X, join(join(meet(X, join(meet(Y, X), meet(X, join(Y, X)))), meet(join(meet(Y, X), meet(X, join(Y, X))), join(X, join(meet(Y, X), meet(X, join(Y, X)))))), X))), ?), meet(join(meet(join(meet(X, join(meet(Y, X), meet(X, join(Y, X)))), meet(join(meet(Y, X), meet(X, join(Y, X))), join(X, join(meet(Y, X), meet(X, join(Y, X)))))), join(join(meet(X, ?), meet(?, X)), X)), meet(X, join(join(meet(X, join(meet(Y, X), meet(X, join(Y, X)))), meet(join(meet(Y, X), meet(X, join(Y, X))), join(X, join(meet(Y, X), meet(X, join(Y, X)))))), join(join(meet(X, ?), meet(?, X)), X)))), join(join(meet(Y, X), meet(X, join(Y, X))), ?))) 4.81/4.99 = { by lemma 8 } 4.81/4.99 join(meet(join(meet(join(meet(X, join(meet(Y, X), meet(X, join(Y, X)))), meet(join(meet(Y, X), meet(X, join(Y, X))), join(X, join(meet(Y, X), meet(X, join(Y, X)))))), X), meet(X, join(join(meet(X, join(meet(Y, X), meet(X, join(Y, X)))), meet(join(meet(Y, X), meet(X, join(Y, X))), join(X, join(meet(Y, X), meet(X, join(Y, X)))))), X))), ?), meet(join(meet(join(meet(X, join(meet(Y, X), meet(X, join(Y, X)))), meet(join(meet(Y, X), meet(X, join(Y, X))), join(X, join(meet(Y, X), meet(X, join(Y, X)))))), join(join(meet(X, ?), meet(?, X)), X)), meet(X, join(join(meet(X, join(meet(Y, X), meet(X, join(Y, X)))), meet(join(meet(Y, X), meet(X, join(Y, X))), join(X, join(meet(Y, X), meet(X, join(Y, X)))))), join(join(meet(X, ?), meet(?, X)), X)))), join(join(meet(join(meet(X, join(meet(Y, X), meet(X, join(Y, X)))), meet(join(meet(Y, X), meet(X, join(Y, X))), join(X, join(meet(Y, X), meet(X, join(Y, X)))))), X), meet(X, join(join(meet(X, join(meet(Y, X), meet(X, join(Y, X)))), meet(join(meet(Y, X), meet(X, join(Y, X))), join(X, join(meet(Y, X), meet(X, join(Y, X)))))), X))), ?))) 4.81/4.99 = { by lemma 5 } 4.81/4.99 X 4.81/4.99 4.81/4.99 Lemma 10: join(meet(X, W), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, W))) = X. 4.81/4.99 Proof: 4.81/4.99 join(meet(X, W), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, W))) 4.81/4.99 = { by lemma 9 } 4.81/4.99 join(meet(join(meet(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), X), meet(X, join(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), X))), W), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, W))) 4.81/4.99 = { by lemma 9 } 4.81/4.99 join(meet(join(meet(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), X), meet(X, join(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), X))), W), meet(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), join(X, W))) 4.81/4.99 = { by lemma 6 } 4.81/4.99 join(meet(join(meet(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), X), meet(X, join(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), X))), W), meet(join(meet(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), join(join(meet(X, Y), meet(Z, X)), X)), meet(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), join(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), join(join(meet(X, Y), meet(Z, X)), X)))), join(X, W))) 4.81/4.99 = { by lemma 9 } 4.81/4.99 join(meet(join(meet(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), X), meet(X, join(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), X))), W), meet(join(meet(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), join(join(meet(X, Y), meet(Z, X)), X)), meet(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), join(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), join(join(meet(X, Y), meet(Z, X)), X)))), join(join(meet(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), X), meet(X, join(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), X))), W))) 4.81/4.99 = { by axiom 1 (single_axiom) } 4.81/4.99 X 4.81/4.99 4.81/4.99 Lemma 11: join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))) = Y. 4.81/4.99 Proof: 4.81/4.99 join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))) 4.81/4.99 = { by lemma 5 } 4.81/4.99 join(meet(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y))))), meet(join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))), join(join(meet(X, Y), meet(Y, join(X, Y))), join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y))))))), ?), meet(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))), ?), meet(?, join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))))), join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))))), meet(join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))), join(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))), ?), meet(?, join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))))), join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))))))), join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y))))), meet(join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))), join(join(meet(X, Y), meet(Y, join(X, Y))), join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y))))))), ?))) 4.81/4.99 = { by lemma 5 } 4.81/4.99 join(meet(Y, ?), meet(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))), ?), meet(?, join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))))), join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))))), meet(join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))), join(join(meet(X, Y), meet(Y, join(X, Y))), join(join(meet(join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))), ?), meet(?, join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))))), join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))))))), join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y))))), meet(join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))), join(join(meet(X, Y), meet(Y, join(X, Y))), join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y))))))), ?))) 4.81/4.99 = { by lemma 5 } 4.81/4.99 join(meet(Y, ?), meet(Y, join(join(meet(join(meet(X, Y), meet(Y, join(X, Y))), join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y))))), meet(join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y)))), join(join(meet(X, Y), meet(Y, join(X, Y))), join(meet(X, join(join(meet(Y, Z), meet(W, Y)), Y)), meet(Y, join(X, join(join(meet(Y, Z), meet(W, Y)), Y))))))), ?))) 4.81/4.99 = { by lemma 5 } 4.81/4.99 join(meet(Y, ?), meet(Y, join(Y, ?))) 4.81/4.99 = { by lemma 7 } 4.81/4.99 Y 4.81/4.99 4.81/4.99 Lemma 12: meet(meet(join(X, Y), join(Z, X)), X) = X. 4.81/4.99 Proof: 4.81/4.99 meet(meet(join(X, Y), join(Z, X)), X) 4.81/4.99 = { by lemma 5 } 4.81/4.99 join(meet(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), ?), meet(join(meet(X, join(join(meet(meet(meet(join(X, Y), join(Z, X)), X), ?), meet(?, meet(meet(join(X, Y), join(Z, X)), X))), meet(meet(join(X, Y), join(Z, X)), X))), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, join(join(meet(meet(meet(join(X, Y), join(Z, X)), X), ?), meet(?, meet(meet(join(X, Y), join(Z, X)), X))), meet(meet(join(X, Y), join(Z, X)), X))))), join(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), ?))) 4.81/4.99 = { by lemma 10 } 4.81/4.99 join(meet(X, ?), meet(join(meet(X, join(join(meet(meet(meet(join(X, Y), join(Z, X)), X), ?), meet(?, meet(meet(join(X, Y), join(Z, X)), X))), meet(meet(join(X, Y), join(Z, X)), X))), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, join(join(meet(meet(meet(join(X, Y), join(Z, X)), X), ?), meet(?, meet(meet(join(X, Y), join(Z, X)), X))), meet(meet(join(X, Y), join(Z, X)), X))))), join(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), ?))) 4.81/4.99 = { by lemma 11 } 4.81/4.99 join(meet(X, ?), meet(meet(meet(join(X, Y), join(Z, X)), X), join(join(meet(X, meet(meet(join(X, Y), join(Z, X)), X)), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, meet(meet(join(X, Y), join(Z, X)), X)))), ?))) 4.81/4.99 = { by lemma 10 } 4.81/4.99 join(meet(X, ?), meet(meet(meet(join(X, Y), join(Z, X)), X), join(X, ?))) 4.81/4.99 = { by lemma 10 } 4.81/5.00 X 4.81/5.00 4.81/5.00 Lemma 13: join(meet(X, Z), meet(join(meet(Y, join(join(meet(X, W), meet(V, X)), X)), meet(join(meet(X, X), meet(U, join(X, X))), join(Y, join(join(meet(X, W), meet(V, X)), X)))), join(X, Z))) = X. 4.81/5.00 Proof: 4.81/5.00 join(meet(X, Z), meet(join(meet(Y, join(join(meet(X, W), meet(V, X)), X)), meet(join(meet(X, X), meet(U, join(X, X))), join(Y, join(join(meet(X, W), meet(V, X)), X)))), join(X, Z))) 4.81/5.00 = { by lemma 9 } 4.81/5.00 join(meet(join(meet(Y, X), meet(X, join(Y, X))), Z), meet(join(meet(Y, join(join(meet(X, W), meet(V, X)), X)), meet(join(meet(X, X), meet(U, join(X, X))), join(Y, join(join(meet(X, W), meet(V, X)), X)))), join(X, Z))) 4.81/5.00 = { by lemma 12 } 4.81/5.00 join(meet(join(meet(Y, X), meet(X, join(Y, X))), Z), meet(join(meet(Y, join(join(meet(X, W), meet(V, X)), X)), meet(join(meet(X, meet(meet(join(X, W), join(V, X)), X)), meet(U, join(X, X))), join(Y, join(join(meet(X, W), meet(V, X)), X)))), join(X, Z))) 4.81/5.00 = { by lemma 12 } 4.81/5.00 join(meet(join(meet(Y, X), meet(X, join(Y, X))), Z), meet(join(meet(Y, join(join(meet(X, W), meet(V, X)), X)), meet(join(meet(X, meet(meet(join(X, W), join(V, X)), X)), meet(U, join(X, meet(meet(join(X, W), join(V, X)), X)))), join(Y, join(join(meet(X, W), meet(V, X)), X)))), join(X, Z))) 4.81/5.00 = { by lemma 9 } 4.81/5.00 join(meet(join(meet(Y, X), meet(X, join(Y, X))), Z), meet(join(meet(Y, join(join(meet(X, W), meet(V, X)), X)), meet(join(meet(X, meet(meet(join(X, W), join(V, X)), X)), meet(U, join(X, meet(meet(join(X, W), join(V, X)), X)))), join(Y, join(join(meet(X, W), meet(V, X)), X)))), join(join(meet(Y, X), meet(X, join(Y, X))), Z))) 4.81/5.00 = { by axiom 1 (single_axiom) } 4.81/5.00 X 4.81/5.00 4.81/5.00 Goal 1 (prove_wal_axioms_1): meet(a, a) = a. 4.81/5.00 Proof: 4.81/5.00 meet(a, a) 4.81/5.00 = { by lemma 12 } 4.81/5.00 meet(meet(meet(join(a, ?), join(?, a)), a), a) 4.81/5.00 = { by axiom 1 (single_axiom) } 4.81/5.00 meet(meet(join(meet(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a), meet(join(meet(?, join(join(meet(meet(join(a, ?), join(?, a)), ?), meet(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))), meet(join(meet(meet(join(a, ?), join(?, a)), meet(meet(join(meet(join(a, ?), join(?, a)), ?), join(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))), meet(?, join(meet(join(a, ?), join(?, a)), meet(meet(join(meet(join(a, ?), join(?, a)), ?), join(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))))), join(?, join(join(meet(meet(join(a, ?), join(?, a)), ?), meet(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))))), join(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a))), a), a) 4.81/5.00 = { by lemma 11 } 4.81/5.01 meet(meet(join(meet(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a), meet(join(meet(?, join(join(meet(meet(join(a, ?), join(?, a)), ?), meet(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))), meet(join(meet(meet(join(a, ?), join(?, a)), meet(meet(join(meet(join(a, ?), join(?, a)), ?), join(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))), meet(?, join(meet(join(a, ?), join(?, a)), meet(meet(join(meet(join(a, ?), join(?, a)), ?), join(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))))), join(?, join(join(meet(meet(join(a, ?), join(?, a)), ?), meet(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))))), join(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a))), join(meet(join(join(meet(a, ?), meet(?, a)), a), join(join(meet(a, ?), meet(?, a)), a)), meet(a, join(join(join(meet(a, ?), meet(?, a)), a), join(join(meet(a, ?), meet(?, a)), a))))), a) 4.81/5.01 = { by lemma 13 } 4.81/5.01 meet(meet(join(meet(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a), meet(join(meet(?, join(join(meet(meet(join(a, ?), join(?, a)), ?), meet(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))), meet(join(meet(meet(join(a, ?), join(?, a)), meet(meet(join(meet(join(a, ?), join(?, a)), ?), join(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))), meet(?, join(meet(join(a, ?), join(?, a)), meet(meet(join(meet(join(a, ?), join(?, a)), ?), join(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))))), join(?, join(join(meet(meet(join(a, ?), join(?, a)), ?), meet(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))))), join(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a))), join(meet(join(join(meet(a, ?), meet(?, a)), a), join(join(meet(a, ?), meet(?, a)), a)), meet(join(meet(a, join(join(meet(join(join(meet(a, ?), meet(?, a)), a), ?), meet(?, join(join(meet(a, ?), meet(?, a)), a))), join(join(meet(a, ?), meet(?, a)), a))), meet(join(meet(join(join(meet(a, ?), meet(?, a)), a), join(join(meet(a, ?), meet(?, a)), a)), meet(join(meet(a, a), meet(?, join(a, a))), join(join(join(meet(a, ?), meet(?, a)), a), join(join(meet(a, ?), meet(?, a)), a)))), join(a, join(join(meet(join(join(meet(a, ?), meet(?, a)), a), ?), meet(?, join(join(meet(a, ?), meet(?, a)), a))), join(join(meet(a, ?), meet(?, a)), a))))), join(join(join(meet(a, ?), meet(?, a)), a), join(join(meet(a, ?), meet(?, a)), a))))), a) 4.81/5.01 = { by lemma 13 } 4.81/5.01 meet(meet(join(meet(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a), meet(join(meet(?, join(join(meet(meet(join(a, ?), join(?, a)), ?), meet(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))), meet(join(meet(meet(join(a, ?), join(?, a)), meet(meet(join(meet(join(a, ?), join(?, a)), ?), join(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))), meet(?, join(meet(join(a, ?), join(?, a)), meet(meet(join(meet(join(a, ?), join(?, a)), ?), join(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))))), join(?, join(join(meet(meet(join(a, ?), join(?, a)), ?), meet(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))))), join(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a))), join(join(meet(a, ?), meet(?, a)), a)), a) 4.81/5.01 = { by lemma 12 } 4.81/5.01 meet(meet(join(meet(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a), meet(join(meet(?, join(join(meet(meet(join(a, ?), join(?, a)), ?), meet(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))), meet(join(meet(meet(join(a, ?), join(?, a)), meet(meet(join(meet(join(a, ?), join(?, a)), ?), join(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))), meet(?, join(meet(join(a, ?), join(?, a)), meet(meet(join(meet(join(a, ?), join(?, a)), ?), join(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))))), join(?, join(join(meet(meet(join(a, ?), join(?, a)), ?), meet(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))))), join(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a))), join(join(meet(a, ?), meet(?, a)), meet(meet(join(a, ?), join(?, a)), a))), a) 4.81/5.01 = { by lemma 9 } 4.81/5.01 meet(meet(join(meet(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a), meet(join(meet(?, join(join(meet(meet(join(a, ?), join(?, a)), ?), meet(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))), meet(join(meet(meet(join(a, ?), join(?, a)), meet(meet(join(meet(join(a, ?), join(?, a)), ?), join(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))), meet(?, join(meet(join(a, ?), join(?, a)), meet(meet(join(meet(join(a, ?), join(?, a)), ?), join(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))))), join(?, join(join(meet(meet(join(a, ?), join(?, a)), ?), meet(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))))), join(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a))), join(join(meet(a, ?), meet(?, a)), meet(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a))), a) 4.81/5.01 = { by lemma 12 } 4.81/5.01 meet(meet(join(meet(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a), meet(join(meet(?, join(join(meet(meet(join(a, ?), join(?, a)), ?), meet(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))), meet(join(meet(meet(join(a, ?), join(?, a)), meet(meet(join(meet(join(a, ?), join(?, a)), ?), join(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))), meet(?, join(meet(join(a, ?), join(?, a)), meet(meet(join(meet(join(a, ?), join(?, a)), ?), join(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))))), join(?, join(join(meet(meet(join(a, ?), join(?, a)), ?), meet(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))))), join(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a))), join(join(meet(a, ?), meet(?, a)), meet(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a))), meet(meet(join(a, ?), join(?, a)), a)) 4.81/5.01 = { by lemma 9 } 4.81/5.01 meet(meet(join(meet(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a), meet(join(meet(?, join(join(meet(meet(join(a, ?), join(?, a)), ?), meet(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))), meet(join(meet(meet(join(a, ?), join(?, a)), meet(meet(join(meet(join(a, ?), join(?, a)), ?), join(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))), meet(?, join(meet(join(a, ?), join(?, a)), meet(meet(join(meet(join(a, ?), join(?, a)), ?), join(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))))), join(?, join(join(meet(meet(join(a, ?), join(?, a)), ?), meet(?, meet(join(a, ?), join(?, a)))), meet(join(a, ?), join(?, a)))))), join(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a))), join(join(meet(a, ?), meet(?, a)), meet(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a))), meet(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a)) 4.81/5.01 = { by lemma 12 } 4.81/5.01 meet(join(meet(?, meet(join(a, ?), join(?, a))), meet(meet(join(a, ?), join(?, a)), join(?, meet(join(a, ?), join(?, a))))), a) 4.81/5.01 = { by lemma 9 } 4.81/5.01 meet(meet(join(a, ?), join(?, a)), a) 4.81/5.01 = { by lemma 12 } 4.81/5.01 a 4.81/5.01 % SZS output end Proof 4.81/5.01 4.81/5.01 RESULT: Unsatisfiable (the axioms are contradictory). 4.81/5.01 EOF