TSTP Solution File: LAT195-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT195-1 : TPTP v8.1.2. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 06:27:43 EDT 2023
% Result : Timeout 295.14s 38.54s
% Output : None
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LAT195-1 : TPTP v8.1.2. Released v3.1.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.35 % Computer : n011.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Thu Aug 24 09:06:38 EDT 2023
% 0.15/0.35 % CPUTime :
% 295.14/38.54 Command-line arguments: --no-flatten-goal
% 295.14/38.54
% 295.14/38.54 % SZS status Unsatisfiable
% 295.14/38.54
% 296.05/38.63 % SZS output start Proof
% 296.05/38.63 Take the following subset of the input axioms:
% 296.05/38.64 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 296.05/38.64 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 296.05/38.64 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 296.05/38.64 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 296.05/38.64 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 296.05/38.64 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 296.05/38.64 fof(complement_join, axiom, ![X2]: join(X2, complement(X2))=one).
% 296.05/38.64 fof(complement_meet, axiom, ![X2]: meet(X2, complement(X2))=zero).
% 296.05/38.64 fof(equation_H34, axiom, ![U, X2, Y2, Z2]: meet(X2, join(Y2, meet(Z2, U)))=meet(X2, join(Y2, meet(Z2, join(Y2, meet(U, join(Y2, Z2))))))).
% 296.05/38.64 fof(meet_join_complement, axiom, ![X2, Y2]: (meet(X2, Y2)!=zero | (join(X2, Y2)!=one | complement(X2)=Y2))).
% 296.05/38.64 fof(prove_distributivity, negated_conjecture, meet(a, join(b, c))!=join(meet(a, b), meet(a, c))).
% 296.05/38.64
% 296.05/38.64 Now clausify the problem and encode Horn clauses using encoding 3 of
% 296.05/38.64 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 296.05/38.64 We repeatedly replace C & s=t => u=v by the two clauses:
% 296.05/38.64 fresh(y, y, x1...xn) = u
% 296.05/38.64 C => fresh(s, t, x1...xn) = v
% 296.05/38.64 where fresh is a fresh function symbol and x1..xn are the free
% 296.05/38.64 variables of u and v.
% 296.05/38.64 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 296.05/38.64 input problem has no model of domain size 1).
% 296.05/38.64
% 296.05/38.64 The encoding turns the above axioms into the following unit equations and goals:
% 296.05/38.64
% 296.05/38.64 Axiom 1 (commutativity_of_join): join(X, Y) = join(Y, X).
% 296.05/38.64 Axiom 2 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 296.05/38.64 Axiom 3 (complement_join): join(X, complement(X)) = one.
% 296.05/38.64 Axiom 4 (complement_meet): meet(X, complement(X)) = zero.
% 296.05/38.64 Axiom 5 (meet_join_complement): fresh(X, X, Y, Z) = Z.
% 296.05/38.64 Axiom 6 (meet_join_complement): fresh2(X, X, Y, Z) = complement(Y).
% 296.05/38.64 Axiom 7 (absorption2): join(X, meet(X, Y)) = X.
% 296.05/38.64 Axiom 8 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 296.05/38.64 Axiom 9 (absorption1): meet(X, join(X, Y)) = X.
% 296.05/38.64 Axiom 10 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 296.05/38.64 Axiom 11 (meet_join_complement): fresh2(join(X, Y), one, X, Y) = fresh(meet(X, Y), zero, X, Y).
% 296.05/38.64 Axiom 12 (equation_H34): meet(X, join(Y, meet(Z, W))) = meet(X, join(Y, meet(Z, join(Y, meet(W, join(Y, Z)))))).
% 296.05/38.64
% 296.05/38.64 Lemma 13: fresh2(join(X, Y), one, Y, X) = fresh(meet(X, Y), zero, Y, X).
% 296.05/38.64 Proof:
% 296.05/38.64 fresh2(join(X, Y), one, Y, X)
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.64 fresh2(join(Y, X), one, Y, X)
% 296.05/38.64 = { by axiom 11 (meet_join_complement) }
% 296.05/38.64 fresh(meet(Y, X), zero, Y, X)
% 296.05/38.64 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.64 fresh(meet(X, Y), zero, Y, X)
% 296.05/38.64
% 296.05/38.64 Lemma 14: complement(complement(X)) = X.
% 296.05/38.64 Proof:
% 296.05/38.64 complement(complement(X))
% 296.05/38.64 = { by axiom 6 (meet_join_complement) R->L }
% 296.05/38.64 fresh2(one, one, complement(X), X)
% 296.05/38.64 = { by axiom 3 (complement_join) R->L }
% 296.05/38.64 fresh2(join(X, complement(X)), one, complement(X), X)
% 296.05/38.64 = { by lemma 13 }
% 296.05/38.64 fresh(meet(X, complement(X)), zero, complement(X), X)
% 296.05/38.64 = { by axiom 4 (complement_meet) }
% 296.05/38.64 fresh(zero, zero, complement(X), X)
% 296.05/38.64 = { by axiom 5 (meet_join_complement) }
% 296.05/38.64 X
% 296.05/38.64
% 296.05/38.64 Lemma 15: meet(X, one) = X.
% 296.05/38.64 Proof:
% 296.05/38.64 meet(X, one)
% 296.05/38.64 = { by axiom 3 (complement_join) R->L }
% 296.05/38.64 meet(X, join(X, complement(X)))
% 296.05/38.64 = { by axiom 9 (absorption1) }
% 296.05/38.64 X
% 296.05/38.64
% 296.05/38.64 Lemma 16: join(X, zero) = X.
% 296.05/38.64 Proof:
% 296.05/38.64 join(X, zero)
% 296.05/38.64 = { by axiom 4 (complement_meet) R->L }
% 296.05/38.64 join(X, meet(X, complement(X)))
% 296.05/38.64 = { by axiom 7 (absorption2) }
% 296.05/38.64 X
% 296.05/38.64
% 296.05/38.64 Lemma 17: join(zero, X) = X.
% 296.05/38.64 Proof:
% 296.05/38.64 join(zero, X)
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.64 join(X, zero)
% 296.05/38.64 = { by lemma 16 }
% 296.05/38.64 X
% 296.05/38.64
% 296.05/38.64 Lemma 18: meet(X, zero) = zero.
% 296.05/38.64 Proof:
% 296.05/38.64 meet(X, zero)
% 296.05/38.64 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.64 meet(zero, X)
% 296.05/38.64 = { by lemma 17 R->L }
% 296.05/38.64 join(zero, meet(zero, X))
% 296.05/38.64 = { by axiom 7 (absorption2) }
% 296.05/38.64 zero
% 296.05/38.64
% 296.05/38.64 Lemma 19: join(X, one) = one.
% 296.05/38.64 Proof:
% 296.05/38.64 join(X, one)
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.64 join(one, X)
% 296.05/38.64 = { by lemma 15 R->L }
% 296.05/38.64 join(one, meet(X, one))
% 296.05/38.64 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.64 join(one, meet(one, X))
% 296.05/38.64 = { by axiom 7 (absorption2) }
% 296.05/38.64 one
% 296.05/38.64
% 296.05/38.64 Lemma 20: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
% 296.05/38.64 Proof:
% 296.05/38.64 meet(X, meet(Y, Z))
% 296.05/38.64 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.64 meet(meet(Y, Z), X)
% 296.05/38.64 = { by axiom 10 (associativity_of_meet) }
% 296.05/38.64 meet(Y, meet(Z, X))
% 296.05/38.64 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.64 meet(Y, meet(X, Z))
% 296.05/38.64
% 296.05/38.64 Lemma 21: meet(X, join(Y, X)) = X.
% 296.05/38.64 Proof:
% 296.05/38.64 meet(X, join(Y, X))
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.64 meet(X, join(X, Y))
% 296.05/38.64 = { by axiom 9 (absorption1) }
% 296.05/38.64 X
% 296.05/38.64
% 296.05/38.64 Lemma 22: join(X, meet(Y, X)) = X.
% 296.05/38.64 Proof:
% 296.05/38.64 join(X, meet(Y, X))
% 296.05/38.64 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.64 join(X, meet(X, Y))
% 296.05/38.64 = { by axiom 7 (absorption2) }
% 296.05/38.64 X
% 296.05/38.64
% 296.05/38.64 Lemma 23: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 296.05/38.64 Proof:
% 296.05/38.64 meet(X, meet(Y, join(X, Z)))
% 296.05/38.64 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.64 meet(X, meet(join(X, Z), Y))
% 296.05/38.64 = { by axiom 10 (associativity_of_meet) R->L }
% 296.05/38.64 meet(meet(X, join(X, Z)), Y)
% 296.05/38.64 = { by axiom 9 (absorption1) }
% 296.05/38.64 meet(X, Y)
% 296.05/38.64
% 296.05/38.64 Lemma 24: meet(X, complement(join(X, Y))) = zero.
% 296.05/38.64 Proof:
% 296.05/38.64 meet(X, complement(join(X, Y)))
% 296.05/38.64 = { by lemma 23 R->L }
% 296.05/38.64 meet(X, meet(complement(join(X, Y)), join(X, Y)))
% 296.05/38.64 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.64 meet(X, meet(join(X, Y), complement(join(X, Y))))
% 296.05/38.64 = { by axiom 4 (complement_meet) }
% 296.05/38.64 meet(X, zero)
% 296.05/38.64 = { by lemma 18 }
% 296.05/38.64 zero
% 296.05/38.64
% 296.05/38.64 Lemma 25: meet(X, meet(Y, complement(X))) = zero.
% 296.05/38.64 Proof:
% 296.05/38.64 meet(X, meet(Y, complement(X)))
% 296.05/38.64 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.64 meet(X, meet(complement(X), Y))
% 296.05/38.64 = { by axiom 10 (associativity_of_meet) R->L }
% 296.05/38.64 meet(meet(X, complement(X)), Y)
% 296.05/38.64 = { by axiom 4 (complement_meet) }
% 296.05/38.64 meet(zero, Y)
% 296.05/38.64 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.64 meet(Y, zero)
% 296.05/38.64 = { by lemma 18 }
% 296.05/38.64 zero
% 296.05/38.64
% 296.05/38.64 Lemma 26: meet(X, meet(complement(X), Y)) = zero.
% 296.05/38.64 Proof:
% 296.05/38.64 meet(X, meet(complement(X), Y))
% 296.05/38.64 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.64 meet(X, meet(Y, complement(X)))
% 296.05/38.64 = { by lemma 25 }
% 296.05/38.64 zero
% 296.05/38.64
% 296.05/38.64 Lemma 27: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 296.05/38.64 Proof:
% 296.05/38.64 join(X, join(Y, meet(X, Z)))
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.64 join(X, join(meet(X, Z), Y))
% 296.05/38.64 = { by axiom 8 (associativity_of_join) R->L }
% 296.05/38.64 join(join(X, meet(X, Z)), Y)
% 296.05/38.64 = { by axiom 7 (absorption2) }
% 296.05/38.64 join(X, Y)
% 296.05/38.64
% 296.05/38.64 Lemma 28: join(X, complement(meet(X, Y))) = one.
% 296.05/38.64 Proof:
% 296.05/38.64 join(X, complement(meet(X, Y)))
% 296.05/38.64 = { by lemma 27 R->L }
% 296.05/38.64 join(X, join(complement(meet(X, Y)), meet(X, Y)))
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.64 join(X, join(meet(X, Y), complement(meet(X, Y))))
% 296.05/38.64 = { by axiom 3 (complement_join) }
% 296.05/38.64 join(X, one)
% 296.05/38.64 = { by lemma 19 }
% 296.05/38.64 one
% 296.05/38.64
% 296.05/38.64 Lemma 29: fresh(meet(X, join(Y, complement(X))), zero, X, join(Y, complement(X))) = complement(X).
% 296.05/38.64 Proof:
% 296.05/38.64 fresh(meet(X, join(Y, complement(X))), zero, X, join(Y, complement(X)))
% 296.05/38.64 = { by axiom 11 (meet_join_complement) R->L }
% 296.05/38.64 fresh2(join(X, join(Y, complement(X))), one, X, join(Y, complement(X)))
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.64 fresh2(join(X, join(complement(X), Y)), one, X, join(Y, complement(X)))
% 296.05/38.64 = { by axiom 8 (associativity_of_join) R->L }
% 296.05/38.64 fresh2(join(join(X, complement(X)), Y), one, X, join(Y, complement(X)))
% 296.05/38.64 = { by axiom 3 (complement_join) }
% 296.05/38.64 fresh2(join(one, Y), one, X, join(Y, complement(X)))
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.64 fresh2(join(Y, one), one, X, join(Y, complement(X)))
% 296.05/38.64 = { by lemma 19 }
% 296.05/38.64 fresh2(one, one, X, join(Y, complement(X)))
% 296.05/38.64 = { by axiom 6 (meet_join_complement) }
% 296.05/38.64 complement(X)
% 296.05/38.64
% 296.05/38.64 Lemma 30: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 296.05/38.64 Proof:
% 296.05/38.64 join(X, join(Y, meet(Z, join(X, Y))))
% 296.05/38.64 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.64 join(X, join(Y, meet(join(X, Y), Z)))
% 296.05/38.64 = { by axiom 8 (associativity_of_join) R->L }
% 296.05/38.64 join(join(X, Y), meet(join(X, Y), Z))
% 296.05/38.64 = { by axiom 7 (absorption2) }
% 296.05/38.64 join(X, Y)
% 296.05/38.64
% 296.05/38.64 Lemma 31: meet(join(X, Y), join(X, meet(Z, join(X, Y)))) = join(X, meet(Z, join(X, Y))).
% 296.05/38.64 Proof:
% 296.05/38.64 meet(join(X, Y), join(X, meet(Z, join(X, Y))))
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.64 meet(join(Y, X), join(X, meet(Z, join(X, Y))))
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.64 meet(join(Y, X), join(X, meet(Z, join(Y, X))))
% 296.05/38.64 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.64 meet(join(X, meet(Z, join(Y, X))), join(Y, X))
% 296.05/38.64 = { by lemma 30 R->L }
% 296.05/38.64 meet(join(X, meet(Z, join(Y, X))), join(Y, join(X, meet(Z, join(Y, X)))))
% 296.05/38.64 = { by lemma 21 }
% 296.05/38.64 join(X, meet(Z, join(Y, X)))
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.64 join(X, meet(Z, join(X, Y)))
% 296.05/38.64
% 296.05/38.64 Lemma 32: meet(X, meet(Y, join(Z, meet(X, Y)))) = meet(X, Y).
% 296.05/38.64 Proof:
% 296.05/38.64 meet(X, meet(Y, join(Z, meet(X, Y))))
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.64 meet(X, meet(Y, join(meet(X, Y), Z)))
% 296.05/38.64 = { by axiom 10 (associativity_of_meet) R->L }
% 296.05/38.64 meet(meet(X, Y), join(meet(X, Y), Z))
% 296.05/38.64 = { by axiom 9 (absorption1) }
% 296.05/38.64 meet(X, Y)
% 296.05/38.64
% 296.05/38.64 Lemma 33: meet(X, join(Y, meet(X, join(Y, Z)))) = meet(X, join(Y, Z)).
% 296.05/38.64 Proof:
% 296.05/38.64 meet(X, join(Y, meet(X, join(Y, Z))))
% 296.05/38.64 = { by lemma 31 R->L }
% 296.05/38.64 meet(X, meet(join(Y, Z), join(Y, meet(X, join(Y, Z)))))
% 296.05/38.64 = { by lemma 32 }
% 296.05/38.64 meet(X, join(Y, Z))
% 296.05/38.64
% 296.05/38.64 Lemma 34: meet(X, join(complement(X), meet(X, Y))) = meet(X, join(Y, complement(X))).
% 296.05/38.64 Proof:
% 296.05/38.64 meet(X, join(complement(X), meet(X, Y)))
% 296.05/38.64 = { by axiom 12 (equation_H34) }
% 296.05/38.64 meet(X, join(complement(X), meet(X, join(complement(X), meet(Y, join(complement(X), X))))))
% 296.05/38.64 = { by lemma 33 }
% 296.05/38.64 meet(X, join(complement(X), meet(Y, join(complement(X), X))))
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.64 meet(X, join(complement(X), meet(Y, join(X, complement(X)))))
% 296.05/38.64 = { by axiom 3 (complement_join) }
% 296.05/38.64 meet(X, join(complement(X), meet(Y, one)))
% 296.05/38.64 = { by lemma 15 }
% 296.05/38.64 meet(X, join(complement(X), Y))
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.64 meet(X, join(Y, complement(X)))
% 296.05/38.64
% 296.05/38.64 Lemma 35: meet(complement(X), join(X, complement(join(Y, complement(X))))) = zero.
% 296.05/38.64 Proof:
% 296.05/38.64 meet(complement(X), join(X, complement(join(Y, complement(X)))))
% 296.05/38.64 = { by lemma 14 R->L }
% 296.05/38.64 meet(complement(X), join(complement(complement(X)), complement(join(Y, complement(X)))))
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.64 meet(complement(X), join(complement(complement(X)), complement(join(complement(X), Y))))
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.64 meet(complement(X), join(complement(join(complement(X), Y)), complement(complement(X))))
% 296.05/38.64 = { by lemma 34 R->L }
% 296.05/38.64 meet(complement(X), join(complement(complement(X)), meet(complement(X), complement(join(complement(X), Y)))))
% 296.05/38.64 = { by lemma 24 }
% 296.05/38.64 meet(complement(X), join(complement(complement(X)), zero))
% 296.05/38.64 = { by lemma 16 }
% 296.05/38.64 meet(complement(X), complement(complement(X)))
% 296.05/38.64 = { by axiom 4 (complement_meet) }
% 296.05/38.64 zero
% 296.05/38.64
% 296.05/38.64 Lemma 36: join(meet(X, Y), meet(X, join(Y, Z))) = meet(X, join(Y, Z)).
% 296.05/38.64 Proof:
% 296.05/38.64 join(meet(X, Y), meet(X, join(Y, Z)))
% 296.05/38.64 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.64 join(meet(Y, X), meet(X, join(Y, Z)))
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.64 join(meet(X, join(Y, Z)), meet(Y, X))
% 296.05/38.64 = { by lemma 23 R->L }
% 296.05/38.64 join(meet(X, join(Y, Z)), meet(Y, meet(X, join(Y, Z))))
% 296.05/38.64 = { by lemma 22 }
% 296.05/38.64 meet(X, join(Y, Z))
% 296.05/38.64
% 296.05/38.64 Lemma 37: join(complement(X), meet(X, join(Y, complement(X)))) = join(complement(X), meet(X, Y)).
% 296.05/38.64 Proof:
% 296.05/38.64 join(complement(X), meet(X, join(Y, complement(X))))
% 296.05/38.64 = { by lemma 36 R->L }
% 296.05/38.64 join(complement(X), join(meet(X, Y), meet(X, join(Y, complement(X)))))
% 296.05/38.64 = { by axiom 8 (associativity_of_join) R->L }
% 296.05/38.64 join(join(complement(X), meet(X, Y)), meet(X, join(Y, complement(X))))
% 296.05/38.64 = { by lemma 34 R->L }
% 296.05/38.64 join(join(complement(X), meet(X, Y)), meet(X, join(complement(X), meet(X, Y))))
% 296.05/38.64 = { by lemma 22 }
% 296.05/38.64 join(complement(X), meet(X, Y))
% 296.05/38.64
% 296.05/38.64 Lemma 38: join(meet(X, Y), meet(Y, join(X, Z))) = meet(Y, join(X, Z)).
% 296.05/38.64 Proof:
% 296.05/38.64 join(meet(X, Y), meet(Y, join(X, Z)))
% 296.05/38.64 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.64 join(meet(Y, X), meet(Y, join(X, Z)))
% 296.05/38.64 = { by lemma 36 }
% 296.05/38.64 meet(Y, join(X, Z))
% 296.05/38.64
% 296.05/38.64 Lemma 39: join(X, meet(Y, complement(X))) = join(X, Y).
% 296.05/38.64 Proof:
% 296.05/38.64 join(X, meet(Y, complement(X)))
% 296.05/38.64 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.64 join(X, meet(complement(X), Y))
% 296.05/38.64 = { by lemma 14 R->L }
% 296.05/38.64 join(complement(complement(X)), meet(complement(X), Y))
% 296.05/38.64 = { by lemma 14 R->L }
% 296.05/38.64 complement(complement(join(complement(complement(X)), meet(complement(X), Y))))
% 296.05/38.64 = { by lemma 29 R->L }
% 296.05/38.64 fresh(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), join(meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))), complement(complement(join(complement(complement(X)), meet(complement(X), Y)))))), zero, complement(join(complement(complement(X)), meet(complement(X), Y))), join(meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))), complement(complement(join(complement(complement(X)), meet(complement(X), Y))))))
% 296.05/38.64 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.64 fresh(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), join(complement(complement(join(complement(complement(X)), meet(complement(X), Y)))), meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))))), zero, complement(join(complement(complement(X)), meet(complement(X), Y))), join(meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))), complement(complement(join(complement(complement(X)), meet(complement(X), Y))))))
% 296.05/38.65 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.65 fresh(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), join(complement(complement(join(complement(complement(X)), meet(complement(X), Y)))), meet(Y, complement(meet(Y, complement(join(complement(complement(X)), meet(complement(X), Y)))))))), zero, complement(join(complement(complement(X)), meet(complement(X), Y))), join(meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))), complement(complement(join(complement(complement(X)), meet(complement(X), Y))))))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.65 fresh(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), join(meet(Y, complement(meet(Y, complement(join(complement(complement(X)), meet(complement(X), Y)))))), complement(complement(join(complement(complement(X)), meet(complement(X), Y)))))), zero, complement(join(complement(complement(X)), meet(complement(X), Y))), join(meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))), complement(complement(join(complement(complement(X)), meet(complement(X), Y))))))
% 296.05/38.65 = { by lemma 34 R->L }
% 296.05/38.65 fresh(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), join(complement(complement(join(complement(complement(X)), meet(complement(X), Y)))), meet(complement(join(complement(complement(X)), meet(complement(X), Y))), meet(Y, complement(meet(Y, complement(join(complement(complement(X)), meet(complement(X), Y))))))))), zero, complement(join(complement(complement(X)), meet(complement(X), Y))), join(meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))), complement(complement(join(complement(complement(X)), meet(complement(X), Y))))))
% 296.05/38.65 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.65 fresh(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), join(complement(complement(join(complement(complement(X)), meet(complement(X), Y)))), meet(meet(Y, complement(meet(Y, complement(join(complement(complement(X)), meet(complement(X), Y)))))), complement(join(complement(complement(X)), meet(complement(X), Y)))))), zero, complement(join(complement(complement(X)), meet(complement(X), Y))), join(meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))), complement(complement(join(complement(complement(X)), meet(complement(X), Y))))))
% 296.05/38.65 = { by axiom 10 (associativity_of_meet) }
% 296.05/38.65 fresh(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), join(complement(complement(join(complement(complement(X)), meet(complement(X), Y)))), meet(Y, meet(complement(meet(Y, complement(join(complement(complement(X)), meet(complement(X), Y))))), complement(join(complement(complement(X)), meet(complement(X), Y))))))), zero, complement(join(complement(complement(X)), meet(complement(X), Y))), join(meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))), complement(complement(join(complement(complement(X)), meet(complement(X), Y))))))
% 296.05/38.65 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.65 fresh(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), join(complement(complement(join(complement(complement(X)), meet(complement(X), Y)))), meet(Y, meet(complement(join(complement(complement(X)), meet(complement(X), Y))), complement(meet(Y, complement(join(complement(complement(X)), meet(complement(X), Y))))))))), zero, complement(join(complement(complement(X)), meet(complement(X), Y))), join(meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))), complement(complement(join(complement(complement(X)), meet(complement(X), Y))))))
% 296.05/38.65 = { by axiom 10 (associativity_of_meet) R->L }
% 296.05/38.65 fresh(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), join(complement(complement(join(complement(complement(X)), meet(complement(X), Y)))), meet(meet(Y, complement(join(complement(complement(X)), meet(complement(X), Y)))), complement(meet(Y, complement(join(complement(complement(X)), meet(complement(X), Y)))))))), zero, complement(join(complement(complement(X)), meet(complement(X), Y))), join(meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))), complement(complement(join(complement(complement(X)), meet(complement(X), Y))))))
% 296.05/38.65 = { by axiom 4 (complement_meet) }
% 296.05/38.65 fresh(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), join(complement(complement(join(complement(complement(X)), meet(complement(X), Y)))), zero)), zero, complement(join(complement(complement(X)), meet(complement(X), Y))), join(meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))), complement(complement(join(complement(complement(X)), meet(complement(X), Y))))))
% 296.05/38.65 = { by lemma 16 }
% 296.05/38.65 fresh(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), complement(complement(join(complement(complement(X)), meet(complement(X), Y))))), zero, complement(join(complement(complement(X)), meet(complement(X), Y))), join(meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))), complement(complement(join(complement(complement(X)), meet(complement(X), Y))))))
% 296.05/38.65 = { by axiom 4 (complement_meet) }
% 296.05/38.65 fresh(zero, zero, complement(join(complement(complement(X)), meet(complement(X), Y))), join(meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))), complement(complement(join(complement(complement(X)), meet(complement(X), Y))))))
% 296.05/38.65 = { by axiom 5 (meet_join_complement) }
% 296.05/38.65 join(meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))), complement(complement(join(complement(complement(X)), meet(complement(X), Y)))))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.65 join(complement(complement(join(complement(complement(X)), meet(complement(X), Y)))), meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))))
% 296.05/38.65 = { by lemma 14 }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(complement(join(complement(complement(X)), meet(complement(X), Y))), Y))))
% 296.05/38.65 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, complement(join(complement(complement(X)), meet(complement(X), Y)))))))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, complement(join(meet(complement(X), Y), complement(complement(X))))))))
% 296.05/38.65 = { by lemma 16 R->L }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, join(complement(join(meet(complement(X), Y), complement(complement(X)))), zero)))))
% 296.05/38.65 = { by lemma 35 R->L }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, join(complement(join(meet(complement(X), Y), complement(complement(X)))), meet(complement(complement(X)), join(complement(X), complement(join(meet(complement(X), Y), complement(complement(X)))))))))))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, join(complement(join(meet(complement(X), Y), complement(complement(X)))), meet(complement(complement(X)), join(complement(join(meet(complement(X), Y), complement(complement(X)))), complement(X))))))))
% 296.05/38.65 = { by lemma 31 R->L }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, meet(join(complement(join(meet(complement(X), Y), complement(complement(X)))), complement(X)), join(complement(join(meet(complement(X), Y), complement(complement(X)))), meet(complement(complement(X)), join(complement(join(meet(complement(X), Y), complement(complement(X)))), complement(X)))))))))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, meet(join(complement(X), complement(join(meet(complement(X), Y), complement(complement(X))))), join(complement(join(meet(complement(X), Y), complement(complement(X)))), meet(complement(complement(X)), join(complement(join(meet(complement(X), Y), complement(complement(X)))), complement(X)))))))))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, meet(join(complement(X), complement(join(meet(complement(X), Y), complement(complement(X))))), join(complement(join(meet(complement(X), Y), complement(complement(X)))), meet(complement(complement(X)), join(complement(X), complement(join(meet(complement(X), Y), complement(complement(X))))))))))))
% 296.05/38.65 = { by lemma 35 }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, meet(join(complement(X), complement(join(meet(complement(X), Y), complement(complement(X))))), join(complement(join(meet(complement(X), Y), complement(complement(X)))), zero))))))
% 296.05/38.65 = { by axiom 5 (meet_join_complement) R->L }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, meet(fresh(zero, zero, complement(complement(X)), join(complement(X), complement(join(meet(complement(X), Y), complement(complement(X)))))), join(complement(join(meet(complement(X), Y), complement(complement(X)))), zero))))))
% 296.05/38.65 = { by lemma 35 R->L }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, meet(fresh(meet(complement(complement(X)), join(complement(X), complement(join(meet(complement(X), Y), complement(complement(X)))))), zero, complement(complement(X)), join(complement(X), complement(join(meet(complement(X), Y), complement(complement(X)))))), join(complement(join(meet(complement(X), Y), complement(complement(X)))), zero))))))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, meet(fresh(meet(complement(complement(X)), join(complement(join(meet(complement(X), Y), complement(complement(X)))), complement(X))), zero, complement(complement(X)), join(complement(X), complement(join(meet(complement(X), Y), complement(complement(X)))))), join(complement(join(meet(complement(X), Y), complement(complement(X)))), zero))))))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, meet(fresh(meet(complement(complement(X)), join(complement(join(meet(complement(X), Y), complement(complement(X)))), complement(X))), zero, complement(complement(X)), join(complement(join(meet(complement(X), Y), complement(complement(X)))), complement(X))), join(complement(join(meet(complement(X), Y), complement(complement(X)))), zero))))))
% 296.05/38.65 = { by lemma 14 R->L }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, meet(fresh(meet(complement(complement(X)), join(complement(join(meet(complement(X), Y), complement(complement(X)))), complement(X))), zero, complement(complement(X)), join(complement(join(meet(complement(X), Y), complement(complement(X)))), complement(complement(complement(X))))), join(complement(join(meet(complement(X), Y), complement(complement(X)))), zero))))))
% 296.05/38.65 = { by lemma 14 R->L }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, meet(fresh(meet(complement(complement(X)), join(complement(join(meet(complement(X), Y), complement(complement(X)))), complement(complement(complement(X))))), zero, complement(complement(X)), join(complement(join(meet(complement(X), Y), complement(complement(X)))), complement(complement(complement(X))))), join(complement(join(meet(complement(X), Y), complement(complement(X)))), zero))))))
% 296.05/38.65 = { by lemma 29 }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, meet(complement(complement(complement(X))), join(complement(join(meet(complement(X), Y), complement(complement(X)))), zero))))))
% 296.05/38.65 = { by lemma 14 }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, meet(complement(X), join(complement(join(meet(complement(X), Y), complement(complement(X)))), zero))))))
% 296.05/38.65 = { by lemma 16 }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, meet(complement(X), complement(join(meet(complement(X), Y), complement(complement(X)))))))))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, meet(complement(X), complement(join(complement(complement(X)), meet(complement(X), Y))))))))
% 296.05/38.65 = { by lemma 37 R->L }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(Y, meet(complement(X), complement(join(complement(complement(X)), meet(complement(X), join(Y, complement(complement(X)))))))))))
% 296.05/38.65 = { by axiom 10 (associativity_of_meet) R->L }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(meet(Y, complement(X)), complement(join(complement(complement(X)), meet(complement(X), join(Y, complement(complement(X))))))))))
% 296.05/38.65 = { by lemma 38 R->L }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(meet(Y, complement(X)), complement(join(complement(complement(X)), join(meet(Y, complement(X)), meet(complement(X), join(Y, complement(complement(X)))))))))))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(meet(Y, complement(X)), complement(join(join(meet(Y, complement(X)), meet(complement(X), join(Y, complement(complement(X))))), complement(complement(X))))))))
% 296.05/38.65 = { by axiom 8 (associativity_of_join) }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(meet(Y, complement(X)), complement(join(meet(Y, complement(X)), join(meet(complement(X), join(Y, complement(complement(X)))), complement(complement(X)))))))))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(meet(meet(Y, complement(X)), complement(join(meet(Y, complement(X)), join(complement(complement(X)), meet(complement(X), join(Y, complement(complement(X)))))))))))
% 296.05/38.65 = { by lemma 24 }
% 296.05/38.65 join(join(complement(complement(X)), meet(complement(X), Y)), meet(Y, complement(zero)))
% 296.05/38.65 = { by axiom 8 (associativity_of_join) }
% 296.05/38.65 join(complement(complement(X)), join(meet(complement(X), Y), meet(Y, complement(zero))))
% 296.05/38.65 = { by lemma 17 R->L }
% 296.05/38.65 join(complement(complement(X)), join(meet(complement(X), Y), meet(Y, join(zero, complement(zero)))))
% 296.05/38.65 = { by axiom 3 (complement_join) }
% 296.05/38.65 join(complement(complement(X)), join(meet(complement(X), Y), meet(Y, one)))
% 296.05/38.65 = { by lemma 15 }
% 296.05/38.65 join(complement(complement(X)), join(meet(complement(X), Y), Y))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.65 join(complement(complement(X)), join(Y, meet(complement(X), Y)))
% 296.05/38.65 = { by lemma 22 }
% 296.05/38.65 join(complement(complement(X)), Y)
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.65 join(Y, complement(complement(X)))
% 296.05/38.65 = { by lemma 14 }
% 296.05/38.65 join(Y, X)
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.65 join(X, Y)
% 296.05/38.65
% 296.05/38.65 Lemma 40: join(X, meet(complement(X), Y)) = join(X, Y).
% 296.05/38.65 Proof:
% 296.05/38.65 join(X, meet(complement(X), Y))
% 296.05/38.65 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.65 join(X, meet(Y, complement(X)))
% 296.05/38.65 = { by lemma 39 }
% 296.05/38.65 join(X, Y)
% 296.05/38.65
% 296.05/38.65 Lemma 41: join(X, meet(complement(X), join(X, Y))) = join(X, meet(Y, complement(X))).
% 296.05/38.65 Proof:
% 296.05/38.65 join(X, meet(complement(X), join(X, Y)))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.65 join(X, meet(complement(X), join(Y, X)))
% 296.05/38.65 = { by lemma 14 R->L }
% 296.05/38.65 join(X, meet(complement(X), join(Y, complement(complement(X)))))
% 296.05/38.65 = { by lemma 14 R->L }
% 296.05/38.65 join(complement(complement(X)), meet(complement(X), join(Y, complement(complement(X)))))
% 296.05/38.65 = { by lemma 37 }
% 296.05/38.65 join(complement(complement(X)), meet(complement(X), Y))
% 296.05/38.65 = { by lemma 14 }
% 296.05/38.65 join(X, meet(complement(X), Y))
% 296.05/38.65 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.65 join(X, meet(Y, complement(X)))
% 296.05/38.65
% 296.05/38.65 Lemma 42: join(X, meet(Y, join(meet(Y, Z), X))) = join(X, meet(Y, Z)).
% 296.05/38.65 Proof:
% 296.05/38.65 join(X, meet(Y, join(meet(Y, Z), X)))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.65 join(X, meet(Y, join(X, meet(Y, Z))))
% 296.05/38.65 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.65 join(X, meet(Y, join(X, meet(Z, Y))))
% 296.05/38.65 = { by lemma 22 R->L }
% 296.05/38.65 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, meet(Y, join(X, meet(Z, Y))))))
% 296.05/38.65 = { by lemma 32 }
% 296.05/38.65 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, Y)))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.65 join(X, join(meet(Z, Y), meet(Y, join(X, meet(Z, Y)))))
% 296.05/38.65 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.65 join(X, join(meet(Z, Y), meet(Y, join(X, meet(Y, Z)))))
% 296.05/38.65 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.65 join(X, join(meet(Y, Z), meet(Y, join(X, meet(Y, Z)))))
% 296.05/38.65 = { by lemma 30 }
% 296.05/38.65 join(X, meet(Y, Z))
% 296.05/38.65
% 296.05/38.65 Lemma 43: join(X, meet(Y, meet(complement(X), Z))) = join(X, meet(Z, Y)).
% 296.05/38.65 Proof:
% 296.05/38.65 join(X, meet(Y, meet(complement(X), Z)))
% 296.05/38.65 = { by axiom 10 (associativity_of_meet) R->L }
% 296.05/38.65 join(X, meet(meet(Y, complement(X)), Z))
% 296.05/38.65 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.65 join(X, meet(Z, meet(Y, complement(X))))
% 296.05/38.65 = { by lemma 39 R->L }
% 296.05/38.65 join(X, meet(meet(Z, meet(Y, complement(X))), complement(X)))
% 296.05/38.65 = { by lemma 41 R->L }
% 296.05/38.65 join(X, meet(complement(X), join(X, meet(Z, meet(Y, complement(X))))))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.65 join(X, meet(complement(X), join(meet(Z, meet(Y, complement(X))), X)))
% 296.05/38.65 = { by axiom 10 (associativity_of_meet) R->L }
% 296.05/38.65 join(X, meet(complement(X), join(meet(meet(Z, Y), complement(X)), X)))
% 296.05/38.65 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.65 join(X, meet(complement(X), join(meet(complement(X), meet(Z, Y)), X)))
% 296.05/38.65 = { by lemma 42 }
% 296.05/38.65 join(X, meet(complement(X), meet(Z, Y)))
% 296.05/38.65 = { by lemma 40 }
% 296.05/38.65 join(X, meet(Z, Y))
% 296.05/38.65
% 296.05/38.65 Lemma 44: meet(join(X, Y), join(Y, meet(X, Z))) = join(Y, meet(X, Z)).
% 296.05/38.65 Proof:
% 296.05/38.65 meet(join(X, Y), join(Y, meet(X, Z)))
% 296.05/38.65 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.65 meet(join(Y, meet(X, Z)), join(X, Y))
% 296.05/38.65 = { by lemma 27 R->L }
% 296.05/38.65 meet(join(Y, meet(X, Z)), join(X, join(Y, meet(X, Z))))
% 296.05/38.65 = { by lemma 21 }
% 296.05/38.65 join(Y, meet(X, Z))
% 296.05/38.65
% 296.05/38.65 Lemma 45: meet(X, join(meet(complement(X), Y), meet(Z, meet(W, complement(X))))) = zero.
% 296.05/38.65 Proof:
% 296.05/38.65 meet(X, join(meet(complement(X), Y), meet(Z, meet(W, complement(X)))))
% 296.05/38.65 = { by axiom 10 (associativity_of_meet) R->L }
% 296.05/38.65 meet(X, join(meet(complement(X), Y), meet(meet(Z, W), complement(X))))
% 296.05/38.65 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.65 meet(X, join(meet(Y, complement(X)), meet(meet(Z, W), complement(X))))
% 296.05/38.65 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.65 meet(X, join(meet(Y, complement(X)), meet(complement(X), meet(Z, W))))
% 296.05/38.65 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.65 meet(X, join(meet(complement(X), Y), meet(complement(X), meet(Z, W))))
% 296.05/38.65 = { by lemma 44 R->L }
% 296.05/38.65 meet(X, meet(join(complement(X), meet(complement(X), Y)), join(meet(complement(X), Y), meet(complement(X), meet(Z, W)))))
% 296.05/38.65 = { by axiom 7 (absorption2) }
% 296.05/38.65 meet(X, meet(complement(X), join(meet(complement(X), Y), meet(complement(X), meet(Z, W)))))
% 296.05/38.65 = { by lemma 26 }
% 296.05/38.65 zero
% 296.05/38.65
% 296.05/38.65 Lemma 46: meet(X, join(Y, complement(join(Y, complement(X))))) = X.
% 296.05/38.65 Proof:
% 296.05/38.65 meet(X, join(Y, complement(join(Y, complement(X)))))
% 296.05/38.65 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.65 meet(X, join(Y, complement(join(complement(X), Y))))
% 296.05/38.65 = { by lemma 14 R->L }
% 296.05/38.65 meet(complement(complement(X)), join(Y, complement(join(complement(X), Y))))
% 296.05/38.65 = { by lemma 16 R->L }
% 296.05/38.65 join(meet(complement(complement(X)), join(Y, complement(join(complement(X), Y)))), zero)
% 296.05/38.65 = { by lemma 25 R->L }
% 296.05/38.65 join(meet(complement(complement(X)), join(Y, complement(join(complement(X), Y)))), meet(complement(X), meet(Z, complement(complement(X)))))
% 296.05/38.65 = { by axiom 5 (meet_join_complement) R->L }
% 296.05/38.66 fresh(zero, zero, complement(X), join(meet(complement(complement(X)), join(Y, complement(join(complement(X), Y)))), meet(complement(X), meet(Z, complement(complement(X))))))
% 296.05/38.66 = { by lemma 45 R->L }
% 296.05/38.66 fresh(meet(complement(X), join(meet(complement(complement(X)), join(Y, complement(join(complement(X), Y)))), meet(complement(X), meet(Z, complement(complement(X)))))), zero, complement(X), join(meet(complement(complement(X)), join(Y, complement(join(complement(X), Y)))), meet(complement(X), meet(Z, complement(complement(X))))))
% 296.05/38.66 = { by axiom 11 (meet_join_complement) R->L }
% 296.05/38.66 fresh2(join(complement(X), join(meet(complement(complement(X)), join(Y, complement(join(complement(X), Y)))), meet(complement(X), meet(Z, complement(complement(X)))))), one, complement(X), join(meet(complement(complement(X)), join(Y, complement(join(complement(X), Y)))), meet(complement(X), meet(Z, complement(complement(X))))))
% 296.05/38.66 = { by lemma 27 }
% 296.05/38.66 fresh2(join(complement(X), meet(complement(complement(X)), join(Y, complement(join(complement(X), Y))))), one, complement(X), join(meet(complement(complement(X)), join(Y, complement(join(complement(X), Y)))), meet(complement(X), meet(Z, complement(complement(X))))))
% 296.05/38.66 = { by lemma 40 }
% 296.05/38.66 fresh2(join(complement(X), join(Y, complement(join(complement(X), Y)))), one, complement(X), join(meet(complement(complement(X)), join(Y, complement(join(complement(X), Y)))), meet(complement(X), meet(Z, complement(complement(X))))))
% 296.05/38.66 = { by lemma 25 }
% 296.05/38.66 fresh2(join(complement(X), join(Y, complement(join(complement(X), Y)))), one, complement(X), join(meet(complement(complement(X)), join(Y, complement(join(complement(X), Y)))), zero))
% 296.05/38.66 = { by lemma 16 }
% 296.05/38.66 fresh2(join(complement(X), join(Y, complement(join(complement(X), Y)))), one, complement(X), meet(complement(complement(X)), join(Y, complement(join(complement(X), Y)))))
% 296.05/38.66 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.66 fresh2(join(complement(X), join(Y, complement(join(complement(X), Y)))), one, complement(X), meet(join(Y, complement(join(complement(X), Y))), complement(complement(X))))
% 296.05/38.66 = { by axiom 8 (associativity_of_join) R->L }
% 296.05/38.66 fresh2(join(join(complement(X), Y), complement(join(complement(X), Y))), one, complement(X), meet(join(Y, complement(join(complement(X), Y))), complement(complement(X))))
% 296.05/38.66 = { by axiom 3 (complement_join) }
% 296.05/38.66 fresh2(one, one, complement(X), meet(join(Y, complement(join(complement(X), Y))), complement(complement(X))))
% 296.05/38.66 = { by axiom 6 (meet_join_complement) }
% 296.05/38.66 complement(complement(X))
% 296.05/38.66 = { by lemma 14 }
% 296.05/38.66 X
% 296.05/38.66
% 296.05/38.66 Lemma 47: meet(X, join(Y, meet(X, Z))) = meet(X, join(Y, Z)).
% 296.05/38.66 Proof:
% 296.05/38.66 meet(X, join(Y, meet(X, Z)))
% 296.05/38.66 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.66 meet(X, join(Y, meet(Z, X)))
% 296.05/38.66 = { by lemma 43 R->L }
% 296.05/38.66 meet(X, join(Y, meet(X, meet(complement(Y), Z))))
% 296.05/38.66 = { by lemma 14 R->L }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(complement(meet(complement(Y), Z))))))
% 296.05/38.66 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(complement(meet(Z, complement(Y)))))))
% 296.05/38.66 = { by lemma 23 R->L }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(complement(meet(Z, meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by lemma 14 R->L }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(complement(meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))))
% 296.05/38.66 = { by axiom 6 (meet_join_complement) R->L }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(one, one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by lemma 28 R->L }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(join(Z, complement(complement(Y))), complement(meet(join(Z, complement(complement(Y))), meet(complement(complement(complement(Y))), join(complement(complement(Y)), meet(Z, complement(complement(complement(Y))))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by lemma 20 }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(join(Z, complement(complement(Y))), complement(meet(complement(complement(complement(Y))), meet(join(Z, complement(complement(Y))), join(complement(complement(Y)), meet(Z, complement(complement(complement(Y))))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by lemma 44 }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(join(Z, complement(complement(Y))), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), meet(Z, complement(complement(complement(Y)))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by axiom 8 (associativity_of_join) }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(complement(Y)), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), meet(Z, complement(complement(complement(Y))))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(complement(Y)), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), meet(complement(complement(complement(Y))), Z)))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by lemma 14 R->L }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(complement(Y)), complement(meet(complement(complement(complement(Y))), join(complement(complement(complement(complement(Y)))), meet(complement(complement(complement(Y))), Z)))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by lemma 34 }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(complement(Y)), complement(meet(complement(complement(complement(Y))), join(Z, complement(complement(complement(complement(Y))))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by lemma 14 }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(complement(Y)), complement(meet(complement(complement(complement(Y))), join(Z, complement(complement(Y))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(complement(Y)), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(complement(Y)), complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))), complement(complement(Y)))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by lemma 14 R->L }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))), complement(complement(complement(complement(Y)))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by axiom 6 (meet_join_complement) R->L }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))), fresh2(one, one, complement(complement(complement(Y))), meet(complement(complement(complement(complement(Y)))), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by axiom 3 (complement_join) R->L }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))), fresh2(join(complement(complement(complement(Y))), complement(complement(complement(complement(Y))))), one, complement(complement(complement(Y))), meet(complement(complement(complement(complement(Y)))), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by lemma 15 R->L }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))), fresh2(join(complement(complement(complement(Y))), meet(complement(complement(complement(complement(Y)))), one)), one, complement(complement(complement(Y))), meet(complement(complement(complement(complement(Y)))), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by lemma 28 R->L }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))), fresh2(join(complement(complement(complement(Y))), meet(complement(complement(complement(complement(Y)))), join(complement(complement(complement(Y))), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z)))))), one, complement(complement(complement(Y))), meet(complement(complement(complement(complement(Y)))), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.66 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.66 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))), fresh2(join(complement(complement(complement(Y))), meet(complement(complement(complement(complement(Y)))), join(complement(complement(complement(Y))), complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y)))))))), one, complement(complement(complement(Y))), meet(complement(complement(complement(complement(Y)))), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by lemma 41 }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))), fresh2(join(complement(complement(complement(Y))), meet(complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))), complement(complement(complement(complement(Y)))))), one, complement(complement(complement(Y))), meet(complement(complement(complement(complement(Y)))), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))), fresh2(join(complement(complement(complement(Y))), meet(complement(complement(complement(complement(Y)))), complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))))), one, complement(complement(complement(Y))), meet(complement(complement(complement(complement(Y)))), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))), fresh2(join(complement(complement(complement(Y))), meet(complement(complement(complement(complement(Y)))), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z))))), one, complement(complement(complement(Y))), meet(complement(complement(complement(complement(Y)))), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by axiom 11 (meet_join_complement) }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))), fresh(meet(complement(complement(complement(Y))), meet(complement(complement(complement(complement(Y)))), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z))))), zero, complement(complement(complement(Y))), meet(complement(complement(complement(complement(Y)))), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by lemma 26 }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))), fresh(zero, zero, complement(complement(complement(Y))), meet(complement(complement(complement(complement(Y)))), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by axiom 5 (meet_join_complement) }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))), meet(complement(complement(complement(complement(Y)))), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z)))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by lemma 14 }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))), meet(complement(complement(Y)), complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z)))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(Z, join(complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y))))), meet(complement(complement(Y)), complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y)))))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by lemma 22 }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(Z, complement(meet(join(complement(complement(Y)), Z), complement(complement(complement(Y)))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(Z, complement(meet(complement(complement(complement(Y))), join(complement(complement(Y)), Z)))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(Z, complement(meet(complement(complement(complement(Y))), join(Z, complement(complement(Y)))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by lemma 14 }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(Z, complement(meet(complement(Y), join(Z, complement(complement(Y)))))), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(complement(meet(complement(Y), join(Z, complement(complement(Y))))), Z), one, meet(Z, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(complement(meet(complement(Y), join(Z, complement(complement(Y))))), Z), one, meet(complement(complement(meet(complement(Y), join(Z, complement(complement(Y)))))), Z), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by lemma 16 R->L }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(complement(meet(complement(Y), join(Z, complement(complement(Y))))), Z), one, join(meet(complement(complement(meet(complement(Y), join(Z, complement(complement(Y)))))), Z), zero), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by lemma 25 R->L }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(complement(meet(complement(Y), join(Z, complement(complement(Y))))), Z), one, join(meet(complement(complement(meet(complement(Y), join(Z, complement(complement(Y)))))), Z), meet(complement(meet(complement(Y), join(Z, complement(complement(Y))))), meet(W, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by lemma 40 R->L }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(complement(meet(complement(Y), join(Z, complement(complement(Y))))), meet(complement(complement(meet(complement(Y), join(Z, complement(complement(Y)))))), Z)), one, join(meet(complement(complement(meet(complement(Y), join(Z, complement(complement(Y)))))), Z), meet(complement(meet(complement(Y), join(Z, complement(complement(Y))))), meet(W, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by lemma 27 R->L }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh2(join(complement(meet(complement(Y), join(Z, complement(complement(Y))))), join(meet(complement(complement(meet(complement(Y), join(Z, complement(complement(Y)))))), Z), meet(complement(meet(complement(Y), join(Z, complement(complement(Y))))), meet(W, complement(complement(meet(complement(Y), join(Z, complement(complement(Y)))))))))), one, join(meet(complement(complement(meet(complement(Y), join(Z, complement(complement(Y)))))), Z), meet(complement(meet(complement(Y), join(Z, complement(complement(Y))))), meet(W, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by lemma 13 }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh(meet(complement(meet(complement(Y), join(Z, complement(complement(Y))))), join(meet(complement(complement(meet(complement(Y), join(Z, complement(complement(Y)))))), Z), meet(complement(meet(complement(Y), join(Z, complement(complement(Y))))), meet(W, complement(complement(meet(complement(Y), join(Z, complement(complement(Y)))))))))), zero, join(meet(complement(complement(meet(complement(Y), join(Z, complement(complement(Y)))))), Z), meet(complement(meet(complement(Y), join(Z, complement(complement(Y))))), meet(W, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by lemma 45 }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(fresh(zero, zero, join(meet(complement(complement(meet(complement(Y), join(Z, complement(complement(Y)))))), Z), meet(complement(meet(complement(Y), join(Z, complement(complement(Y))))), meet(W, complement(complement(meet(complement(Y), join(Z, complement(complement(Y))))))))), complement(meet(complement(Y), join(Z, complement(complement(Y))))))))))
% 296.05/38.67 = { by axiom 5 (meet_join_complement) }
% 296.05/38.67 meet(X, join(Y, meet(X, complement(complement(meet(complement(Y), join(Z, complement(complement(Y)))))))))
% 296.05/38.67 = { by lemma 14 }
% 296.05/38.67 meet(X, join(Y, meet(X, meet(complement(Y), join(Z, complement(complement(Y)))))))
% 296.05/38.67 = { by lemma 43 }
% 296.05/38.67 meet(X, join(Y, meet(join(Z, complement(complement(Y))), X)))
% 296.05/38.67 = { by lemma 14 }
% 296.05/38.67 meet(X, join(Y, meet(join(Z, Y), X)))
% 296.05/38.67 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.67 meet(X, join(Y, meet(X, join(Z, Y))))
% 296.05/38.67 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.67 meet(X, join(Y, meet(X, join(Y, Z))))
% 296.05/38.67 = { by lemma 46 R->L }
% 296.05/38.67 meet(X, join(Y, meet(X, join(Y, meet(Z, join(V, complement(join(V, complement(Z)))))))))
% 296.05/38.67 = { by lemma 33 }
% 296.05/38.67 meet(X, join(Y, meet(Z, join(V, complement(join(V, complement(Z)))))))
% 296.05/38.67 = { by lemma 46 }
% 296.05/38.67 meet(X, join(Y, Z))
% 296.05/38.67
% 296.05/38.67 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 296.05/38.67 Proof:
% 296.05/38.67 meet(a, join(b, c))
% 296.05/38.67 = { by lemma 47 R->L }
% 296.05/38.67 meet(a, join(b, meet(a, c)))
% 296.05/38.67 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.67 meet(a, join(meet(a, c), b))
% 296.05/38.67 = { by lemma 38 R->L }
% 296.05/38.67 join(meet(meet(a, c), a), meet(a, join(meet(a, c), b)))
% 296.05/38.67 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.67 join(meet(meet(a, c), a), meet(a, join(b, meet(a, c))))
% 296.05/38.67 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.67 join(meet(a, meet(a, c)), meet(a, join(b, meet(a, c))))
% 296.05/38.67 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.67 join(meet(a, meet(a, c)), meet(a, join(meet(a, c), b)))
% 296.05/38.67 = { by lemma 22 R->L }
% 296.05/38.67 join(meet(a, meet(a, c)), join(meet(a, join(meet(a, c), b)), meet(b, meet(a, join(meet(a, c), b)))))
% 296.05/38.67 = { by lemma 20 R->L }
% 296.05/38.67 join(meet(a, meet(a, c)), join(meet(a, join(meet(a, c), b)), meet(a, meet(b, join(meet(a, c), b)))))
% 296.05/38.67 = { by lemma 21 }
% 296.05/38.67 join(meet(a, meet(a, c)), join(meet(a, join(meet(a, c), b)), meet(a, b)))
% 296.05/38.67 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.67 join(meet(a, meet(a, c)), join(meet(a, join(meet(a, c), b)), meet(b, a)))
% 296.05/38.67 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.67 join(meet(a, meet(a, c)), join(meet(b, a), meet(a, join(meet(a, c), b))))
% 296.05/38.67 = { by axiom 2 (commutativity_of_meet) R->L }
% 296.05/38.67 join(meet(meet(a, c), a), join(meet(b, a), meet(a, join(meet(a, c), b))))
% 296.05/38.67 = { by axiom 1 (commutativity_of_join) R->L }
% 296.05/38.67 join(meet(meet(a, c), a), join(meet(a, join(meet(a, c), b)), meet(b, a)))
% 296.05/38.67 = { by axiom 8 (associativity_of_join) R->L }
% 296.05/38.67 join(join(meet(meet(a, c), a), meet(a, join(meet(a, c), b))), meet(b, a))
% 296.05/38.67 = { by lemma 38 }
% 296.05/38.67 join(meet(a, join(meet(a, c), b)), meet(b, a))
% 296.05/38.67 = { by axiom 1 (commutativity_of_join) }
% 296.05/38.67 join(meet(b, a), meet(a, join(meet(a, c), b)))
% 296.05/38.67 = { by axiom 2 (commutativity_of_meet) }
% 296.05/38.67 join(meet(a, b), meet(a, join(meet(a, c), b)))
% 296.05/38.67 = { by lemma 47 R->L }
% 296.05/38.67 join(meet(a, b), meet(a, join(meet(a, c), meet(a, b))))
% 296.05/38.67 = { by lemma 42 }
% 296.05/38.67 join(meet(a, b), meet(a, c))
% 296.05/38.67 % SZS output end Proof
% 296.05/38.67
% 296.05/38.67 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------