TSTP Solution File: LAT209-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT209-1 : 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 : n028.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 06:27:46 EDT 2023
% Result : Unsatisfiable 163.23s 21.41s
% Output : Proof 163.61s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LAT209-1 : 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
% 0.14/0.36 % Computer : n028.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Thu Aug 24 05:14:19 EDT 2023
% 0.14/0.36 % CPUTime :
% 163.23/21.41 Command-line arguments: --no-flatten-goal
% 163.23/21.41
% 163.23/21.41 % SZS status Unsatisfiable
% 163.23/21.41
% 163.61/21.46 % SZS output start Proof
% 163.61/21.46 Take the following subset of the input axioms:
% 163.61/21.46 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 163.61/21.46 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 163.61/21.46 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 163.61/21.46 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 163.61/21.46 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 163.61/21.46 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 163.61/21.46 fof(complement_join, axiom, ![X2]: join(X2, complement(X2))=one).
% 163.61/21.46 fof(complement_meet, axiom, ![X2]: meet(X2, complement(X2))=zero).
% 163.61/21.46 fof(equation_H64, axiom, ![X2, Y2, Z2]: meet(X2, join(Y2, Z2))=meet(X2, join(Y2, meet(X2, join(Z2, meet(X2, join(Y2, meet(X2, Z2)))))))).
% 163.61/21.46 fof(idempotence_of_join, axiom, ![X2]: join(X2, X2)=X2).
% 163.61/21.46 fof(idempotence_of_meet, axiom, ![X2]: meet(X2, X2)=X2).
% 163.61/21.46 fof(meet_join_complement, axiom, ![X2, Y2]: (meet(X2, Y2)!=zero | (join(X2, Y2)!=one | complement(X2)=Y2))).
% 163.61/21.46 fof(prove_distributivity, negated_conjecture, meet(a, join(b, c))!=join(meet(a, b), meet(a, c))).
% 163.61/21.46
% 163.61/21.47 Now clausify the problem and encode Horn clauses using encoding 3 of
% 163.61/21.47 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 163.61/21.47 We repeatedly replace C & s=t => u=v by the two clauses:
% 163.61/21.47 fresh(y, y, x1...xn) = u
% 163.61/21.47 C => fresh(s, t, x1...xn) = v
% 163.61/21.47 where fresh is a fresh function symbol and x1..xn are the free
% 163.61/21.47 variables of u and v.
% 163.61/21.47 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 163.61/21.47 input problem has no model of domain size 1).
% 163.61/21.47
% 163.61/21.47 The encoding turns the above axioms into the following unit equations and goals:
% 163.61/21.47
% 163.61/21.47 Axiom 1 (idempotence_of_join): join(X, X) = X.
% 163.61/21.47 Axiom 2 (commutativity_of_join): join(X, Y) = join(Y, X).
% 163.61/21.47 Axiom 3 (idempotence_of_meet): meet(X, X) = X.
% 163.61/21.47 Axiom 4 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 163.61/21.47 Axiom 5 (complement_join): join(X, complement(X)) = one.
% 163.61/21.47 Axiom 6 (complement_meet): meet(X, complement(X)) = zero.
% 163.61/21.47 Axiom 7 (meet_join_complement): fresh(X, X, Y, Z) = Z.
% 163.61/21.47 Axiom 8 (meet_join_complement): fresh2(X, X, Y, Z) = complement(Y).
% 163.61/21.47 Axiom 9 (absorption2): join(X, meet(X, Y)) = X.
% 163.61/21.47 Axiom 10 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 163.61/21.47 Axiom 11 (absorption1): meet(X, join(X, Y)) = X.
% 163.61/21.47 Axiom 12 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 163.61/21.47 Axiom 13 (meet_join_complement): fresh2(join(X, Y), one, X, Y) = fresh(meet(X, Y), zero, X, Y).
% 163.61/21.47 Axiom 14 (equation_H64): meet(X, join(Y, Z)) = meet(X, join(Y, meet(X, join(Z, meet(X, join(Y, meet(X, Z))))))).
% 163.61/21.47
% 163.61/21.47 Lemma 15: fresh2(join(X, Y), one, Y, X) = fresh(meet(X, Y), zero, Y, X).
% 163.61/21.47 Proof:
% 163.61/21.47 fresh2(join(X, Y), one, Y, X)
% 163.61/21.47 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.47 fresh2(join(Y, X), one, Y, X)
% 163.61/21.47 = { by axiom 13 (meet_join_complement) }
% 163.61/21.47 fresh(meet(Y, X), zero, Y, X)
% 163.61/21.47 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.47 fresh(meet(X, Y), zero, Y, X)
% 163.61/21.47
% 163.61/21.47 Lemma 16: complement(complement(X)) = X.
% 163.61/21.47 Proof:
% 163.61/21.47 complement(complement(X))
% 163.61/21.47 = { by axiom 8 (meet_join_complement) R->L }
% 163.61/21.47 fresh2(one, one, complement(X), X)
% 163.61/21.47 = { by axiom 5 (complement_join) R->L }
% 163.61/21.47 fresh2(join(X, complement(X)), one, complement(X), X)
% 163.61/21.47 = { by lemma 15 }
% 163.61/21.47 fresh(meet(X, complement(X)), zero, complement(X), X)
% 163.61/21.47 = { by axiom 6 (complement_meet) }
% 163.61/21.47 fresh(zero, zero, complement(X), X)
% 163.61/21.47 = { by axiom 7 (meet_join_complement) }
% 163.61/21.47 X
% 163.61/21.47
% 163.61/21.47 Lemma 17: meet(X, one) = X.
% 163.61/21.47 Proof:
% 163.61/21.47 meet(X, one)
% 163.61/21.47 = { by axiom 5 (complement_join) R->L }
% 163.61/21.47 meet(X, join(X, complement(X)))
% 163.61/21.47 = { by axiom 11 (absorption1) }
% 163.61/21.47 X
% 163.61/21.47
% 163.61/21.47 Lemma 18: join(X, zero) = X.
% 163.61/21.47 Proof:
% 163.61/21.47 join(X, zero)
% 163.61/21.47 = { by axiom 6 (complement_meet) R->L }
% 163.61/21.47 join(X, meet(X, complement(X)))
% 163.61/21.47 = { by axiom 9 (absorption2) }
% 163.61/21.47 X
% 163.61/21.47
% 163.61/21.47 Lemma 19: join(zero, X) = X.
% 163.61/21.47 Proof:
% 163.61/21.47 join(zero, X)
% 163.61/21.47 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.47 join(X, zero)
% 163.61/21.47 = { by lemma 18 }
% 163.61/21.47 X
% 163.61/21.47
% 163.61/21.47 Lemma 20: meet(X, zero) = zero.
% 163.61/21.47 Proof:
% 163.61/21.47 meet(X, zero)
% 163.61/21.47 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.47 meet(zero, X)
% 163.61/21.47 = { by lemma 19 R->L }
% 163.61/21.47 join(zero, meet(zero, X))
% 163.61/21.47 = { by axiom 9 (absorption2) }
% 163.61/21.47 zero
% 163.61/21.47
% 163.61/21.47 Lemma 21: join(X, one) = one.
% 163.61/21.47 Proof:
% 163.61/21.47 join(X, one)
% 163.61/21.47 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.47 join(one, X)
% 163.61/21.47 = { by lemma 17 R->L }
% 163.61/21.47 join(one, meet(X, one))
% 163.61/21.47 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.47 join(one, meet(one, X))
% 163.61/21.47 = { by axiom 9 (absorption2) }
% 163.61/21.47 one
% 163.61/21.47
% 163.61/21.47 Lemma 22: meet(X, meet(Y, complement(X))) = zero.
% 163.61/21.47 Proof:
% 163.61/21.47 meet(X, meet(Y, complement(X)))
% 163.61/21.47 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.47 meet(X, meet(complement(X), Y))
% 163.61/21.47 = { by axiom 12 (associativity_of_meet) R->L }
% 163.61/21.47 meet(meet(X, complement(X)), Y)
% 163.61/21.47 = { by axiom 6 (complement_meet) }
% 163.61/21.47 meet(zero, Y)
% 163.61/21.47 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.47 meet(Y, zero)
% 163.61/21.47 = { by lemma 20 }
% 163.61/21.47 zero
% 163.61/21.47
% 163.61/21.47 Lemma 23: meet(X, meet(Y, complement(meet(X, Y)))) = zero.
% 163.61/21.47 Proof:
% 163.61/21.47 meet(X, meet(Y, complement(meet(X, Y))))
% 163.61/21.47 = { by axiom 12 (associativity_of_meet) R->L }
% 163.61/21.47 meet(meet(X, Y), complement(meet(X, Y)))
% 163.61/21.47 = { by axiom 6 (complement_meet) }
% 163.61/21.47 zero
% 163.61/21.47
% 163.61/21.47 Lemma 24: meet(X, join(Y, meet(X, join(complement(X), meet(X, Y))))) = meet(X, join(Y, complement(X))).
% 163.61/21.47 Proof:
% 163.61/21.47 meet(X, join(Y, meet(X, join(complement(X), meet(X, Y)))))
% 163.61/21.47 = { by lemma 18 R->L }
% 163.61/21.47 meet(X, join(Y, meet(X, join(complement(X), meet(X, join(Y, zero))))))
% 163.61/21.47 = { by axiom 6 (complement_meet) R->L }
% 163.61/21.47 meet(X, join(Y, meet(X, join(complement(X), meet(X, join(Y, meet(X, complement(X))))))))
% 163.61/21.47 = { by axiom 14 (equation_H64) R->L }
% 163.61/21.47 meet(X, join(Y, complement(X)))
% 163.61/21.47
% 163.61/21.47 Lemma 25: join(X, join(Y, complement(X))) = one.
% 163.61/21.47 Proof:
% 163.61/21.47 join(X, join(Y, complement(X)))
% 163.61/21.47 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.47 join(X, join(complement(X), Y))
% 163.61/21.47 = { by axiom 10 (associativity_of_join) R->L }
% 163.61/21.47 join(join(X, complement(X)), Y)
% 163.61/21.47 = { by axiom 5 (complement_join) }
% 163.61/21.47 join(one, Y)
% 163.61/21.47 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.47 join(Y, one)
% 163.61/21.47 = { by lemma 21 }
% 163.61/21.47 one
% 163.61/21.47
% 163.61/21.47 Lemma 26: fresh(meet(X, join(Y, complement(X))), zero, X, join(Y, complement(X))) = complement(X).
% 163.61/21.47 Proof:
% 163.61/21.47 fresh(meet(X, join(Y, complement(X))), zero, X, join(Y, complement(X)))
% 163.61/21.47 = { by axiom 13 (meet_join_complement) R->L }
% 163.61/21.47 fresh2(join(X, join(Y, complement(X))), one, X, join(Y, complement(X)))
% 163.61/21.47 = { by lemma 25 }
% 163.61/21.47 fresh2(one, one, X, join(Y, complement(X)))
% 163.61/21.47 = { by axiom 8 (meet_join_complement) }
% 163.61/21.47 complement(X)
% 163.61/21.47
% 163.61/21.47 Lemma 27: fresh(meet(X, join(complement(X), Y)), zero, X, join(Y, complement(X))) = complement(X).
% 163.61/21.47 Proof:
% 163.61/21.47 fresh(meet(X, join(complement(X), Y)), zero, X, join(Y, complement(X)))
% 163.61/21.47 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.47 fresh(meet(X, join(Y, complement(X))), zero, X, join(Y, complement(X)))
% 163.61/21.47 = { by lemma 26 }
% 163.61/21.47 complement(X)
% 163.61/21.47
% 163.61/21.47 Lemma 28: join(complement(X), meet(Y, complement(meet(X, Y)))) = complement(X).
% 163.61/21.47 Proof:
% 163.61/21.47 join(complement(X), meet(Y, complement(meet(X, Y))))
% 163.61/21.47 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.47 join(meet(Y, complement(meet(X, Y))), complement(X))
% 163.61/21.47 = { by axiom 7 (meet_join_complement) R->L }
% 163.61/21.47 fresh(zero, zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 163.61/21.47 = { by lemma 23 R->L }
% 163.61/21.47 fresh(meet(X, meet(Y, complement(meet(X, Y)))), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 163.61/21.47 = { by lemma 18 R->L }
% 163.61/21.47 fresh(meet(X, join(meet(Y, complement(meet(X, Y))), zero)), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 163.61/21.47 = { by axiom 6 (complement_meet) R->L }
% 163.61/21.47 fresh(meet(X, join(meet(Y, complement(meet(X, Y))), meet(X, complement(X)))), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 163.61/21.47 = { by lemma 18 R->L }
% 163.61/21.47 fresh(meet(X, join(meet(Y, complement(meet(X, Y))), meet(X, join(complement(X), zero)))), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 163.61/21.47 = { by lemma 23 R->L }
% 163.61/21.47 fresh(meet(X, join(meet(Y, complement(meet(X, Y))), meet(X, join(complement(X), meet(X, meet(Y, complement(meet(X, Y)))))))), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 163.61/21.47 = { by lemma 24 }
% 163.61/21.47 fresh(meet(X, join(meet(Y, complement(meet(X, Y))), complement(X))), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 163.61/21.47 = { by axiom 2 (commutativity_of_join) }
% 163.61/21.47 fresh(meet(X, join(complement(X), meet(Y, complement(meet(X, Y))))), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 163.61/21.47 = { by lemma 27 }
% 163.61/21.47 complement(X)
% 163.61/21.47
% 163.61/21.47 Lemma 29: join(X, complement(meet(Y, complement(X)))) = complement(meet(Y, complement(X))).
% 163.61/21.47 Proof:
% 163.61/21.47 join(X, complement(meet(Y, complement(X))))
% 163.61/21.47 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.47 join(complement(meet(Y, complement(X))), X)
% 163.61/21.47 = { by lemma 17 R->L }
% 163.61/21.47 join(complement(meet(Y, complement(X))), meet(X, one))
% 163.61/21.47 = { by axiom 5 (complement_join) R->L }
% 163.61/21.47 join(complement(meet(Y, complement(X))), meet(X, join(zero, complement(zero))))
% 163.61/21.47 = { by lemma 19 }
% 163.61/21.47 join(complement(meet(Y, complement(X))), meet(X, complement(zero)))
% 163.61/21.47 = { by lemma 22 R->L }
% 163.61/21.47 join(complement(meet(Y, complement(X))), meet(X, complement(meet(X, meet(Y, complement(X))))))
% 163.61/21.47 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.47 join(complement(meet(Y, complement(X))), meet(X, complement(meet(meet(Y, complement(X)), X))))
% 163.61/21.47 = { by lemma 28 }
% 163.61/21.47 complement(meet(Y, complement(X)))
% 163.61/21.47
% 163.61/21.47 Lemma 30: join(complement(X), complement(meet(X, Y))) = complement(meet(X, Y)).
% 163.61/21.47 Proof:
% 163.61/21.47 join(complement(X), complement(meet(X, Y)))
% 163.61/21.47 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.47 join(complement(X), complement(meet(Y, X)))
% 163.61/21.47 = { by lemma 16 R->L }
% 163.61/21.47 join(complement(X), complement(meet(Y, complement(complement(X)))))
% 163.61/21.47 = { by lemma 29 }
% 163.61/21.47 complement(meet(Y, complement(complement(X))))
% 163.61/21.47 = { by lemma 16 }
% 163.61/21.47 complement(meet(Y, X))
% 163.61/21.47 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.47 complement(meet(X, Y))
% 163.61/21.47
% 163.61/21.47 Lemma 31: meet(X, join(Y, X)) = X.
% 163.61/21.47 Proof:
% 163.61/21.47 meet(X, join(Y, X))
% 163.61/21.47 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.47 meet(X, join(X, Y))
% 163.61/21.47 = { by axiom 11 (absorption1) }
% 163.61/21.47 X
% 163.61/21.47
% 163.61/21.47 Lemma 32: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 163.61/21.47 Proof:
% 163.61/21.47 meet(X, complement(meet(X, Y)))
% 163.61/21.47 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.47 meet(X, complement(meet(Y, X)))
% 163.61/21.47 = { by lemma 31 R->L }
% 163.61/21.47 meet(meet(X, complement(meet(Y, X))), join(complement(Y), meet(X, complement(meet(Y, X)))))
% 163.61/21.47 = { by lemma 28 }
% 163.61/21.47 meet(meet(X, complement(meet(Y, X))), complement(Y))
% 163.61/21.47 = { by axiom 12 (associativity_of_meet) }
% 163.61/21.47 meet(X, meet(complement(meet(Y, X)), complement(Y)))
% 163.61/21.47 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.47 meet(X, meet(complement(Y), complement(meet(Y, X))))
% 163.61/21.47 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.47 meet(X, meet(complement(Y), complement(meet(X, Y))))
% 163.61/21.47 = { by lemma 16 R->L }
% 163.61/21.47 meet(X, meet(complement(Y), complement(meet(X, complement(complement(Y))))))
% 163.61/21.47 = { by lemma 29 R->L }
% 163.61/21.47 meet(X, meet(complement(Y), join(complement(Y), complement(meet(X, complement(complement(Y)))))))
% 163.61/21.47 = { by axiom 11 (absorption1) }
% 163.61/21.47 meet(X, complement(Y))
% 163.61/21.47
% 163.61/21.47 Lemma 33: meet(X, complement(join(X, Y))) = zero.
% 163.61/21.47 Proof:
% 163.61/21.47 meet(X, complement(join(X, Y)))
% 163.61/21.47 = { by axiom 11 (absorption1) R->L }
% 163.61/21.47 meet(meet(X, join(X, Y)), complement(join(X, Y)))
% 163.61/21.47 = { by axiom 12 (associativity_of_meet) }
% 163.61/21.47 meet(X, meet(join(X, Y), complement(join(X, Y))))
% 163.61/21.47 = { by axiom 6 (complement_meet) }
% 163.61/21.47 meet(X, zero)
% 163.61/21.47 = { by lemma 20 }
% 163.61/21.47 zero
% 163.61/21.47
% 163.61/21.47 Lemma 34: meet(X, complement(join(Y, X))) = zero.
% 163.61/21.47 Proof:
% 163.61/21.47 meet(X, complement(join(Y, X)))
% 163.61/21.47 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.47 meet(X, complement(join(X, Y)))
% 163.61/21.47 = { by lemma 33 }
% 163.61/21.47 zero
% 163.61/21.47
% 163.61/21.47 Lemma 35: join(complement(X), complement(join(X, Y))) = complement(X).
% 163.61/21.47 Proof:
% 163.61/21.47 join(complement(X), complement(join(X, Y)))
% 163.61/21.47 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.47 join(complement(join(X, Y)), complement(X))
% 163.61/21.47 = { by axiom 7 (meet_join_complement) R->L }
% 163.61/21.47 fresh(zero, zero, X, join(complement(join(X, Y)), complement(X)))
% 163.61/21.47 = { by lemma 34 R->L }
% 163.61/21.47 fresh(meet(X, complement(join(Y, X))), zero, X, join(complement(join(X, Y)), complement(X)))
% 163.61/21.47 = { by lemma 18 R->L }
% 163.61/21.47 fresh(meet(X, join(complement(join(Y, X)), zero)), zero, X, join(complement(join(X, Y)), complement(X)))
% 163.61/21.47 = { by axiom 6 (complement_meet) R->L }
% 163.61/21.47 fresh(meet(X, join(complement(join(Y, X)), meet(X, complement(X)))), zero, X, join(complement(join(X, Y)), complement(X)))
% 163.61/21.47 = { by lemma 18 R->L }
% 163.61/21.47 fresh(meet(X, join(complement(join(Y, X)), meet(X, join(complement(X), zero)))), zero, X, join(complement(join(X, Y)), complement(X)))
% 163.61/21.47 = { by lemma 34 R->L }
% 163.61/21.47 fresh(meet(X, join(complement(join(Y, X)), meet(X, join(complement(X), meet(X, complement(join(Y, X))))))), zero, X, join(complement(join(X, Y)), complement(X)))
% 163.61/21.47 = { by lemma 24 }
% 163.61/21.47 fresh(meet(X, join(complement(join(Y, X)), complement(X))), zero, X, join(complement(join(X, Y)), complement(X)))
% 163.61/21.47 = { by axiom 2 (commutativity_of_join) }
% 163.61/21.47 fresh(meet(X, join(complement(X), complement(join(Y, X)))), zero, X, join(complement(join(X, Y)), complement(X)))
% 163.61/21.47 = { by axiom 2 (commutativity_of_join) }
% 163.61/21.47 fresh(meet(X, join(complement(X), complement(join(X, Y)))), zero, X, join(complement(join(X, Y)), complement(X)))
% 163.61/21.47 = { by lemma 27 }
% 163.61/21.47 complement(X)
% 163.61/21.47
% 163.61/21.47 Lemma 36: join(X, meet(complement(X), Y)) = join(X, Y).
% 163.61/21.47 Proof:
% 163.61/21.47 join(X, meet(complement(X), Y))
% 163.61/21.47 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.47 join(X, meet(Y, complement(X)))
% 163.61/21.47 = { by axiom 9 (absorption2) R->L }
% 163.61/21.47 join(join(X, meet(Y, complement(X))), meet(join(X, meet(Y, complement(X))), Y))
% 163.61/21.48 = { by axiom 10 (associativity_of_join) }
% 163.61/21.48 join(X, join(meet(Y, complement(X)), meet(join(X, meet(Y, complement(X))), Y)))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.48 join(X, join(meet(Y, complement(X)), meet(Y, join(X, meet(Y, complement(X))))))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.48 join(X, join(meet(complement(X), Y), meet(Y, join(X, meet(Y, complement(X))))))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.48 join(X, join(meet(complement(X), Y), meet(Y, join(X, meet(complement(X), Y)))))
% 163.61/21.48 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.48 join(X, join(meet(Y, join(X, meet(complement(X), Y))), meet(complement(X), Y)))
% 163.61/21.48 = { by axiom 11 (absorption1) R->L }
% 163.61/21.48 join(X, join(meet(Y, join(X, meet(complement(X), Y))), meet(meet(complement(X), Y), join(meet(complement(X), Y), X))))
% 163.61/21.48 = { by axiom 12 (associativity_of_meet) }
% 163.61/21.48 join(X, join(meet(Y, join(X, meet(complement(X), Y))), meet(complement(X), meet(Y, join(meet(complement(X), Y), X)))))
% 163.61/21.48 = { by axiom 2 (commutativity_of_join) }
% 163.61/21.48 join(X, join(meet(Y, join(X, meet(complement(X), Y))), meet(complement(X), meet(Y, join(X, meet(complement(X), Y))))))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.48 join(X, join(meet(Y, join(X, meet(complement(X), Y))), meet(meet(Y, join(X, meet(complement(X), Y))), complement(X))))
% 163.61/21.48 = { by axiom 9 (absorption2) }
% 163.61/21.48 join(X, meet(Y, join(X, meet(complement(X), Y))))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.48 join(X, meet(Y, join(X, meet(Y, complement(X)))))
% 163.61/21.48 = { by lemma 16 R->L }
% 163.61/21.48 join(X, complement(complement(meet(Y, join(X, meet(Y, complement(X)))))))
% 163.61/21.48 = { by lemma 30 R->L }
% 163.61/21.48 join(X, complement(join(complement(Y), complement(meet(Y, join(X, meet(Y, complement(X))))))))
% 163.61/21.48 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.48 join(X, complement(join(complement(meet(Y, join(X, meet(Y, complement(X))))), complement(Y))))
% 163.61/21.48 = { by axiom 8 (meet_join_complement) R->L }
% 163.61/21.48 join(X, fresh2(one, one, join(complement(meet(Y, join(X, meet(Y, complement(X))))), complement(Y)), Y))
% 163.61/21.48 = { by lemma 25 R->L }
% 163.61/21.48 join(X, fresh2(join(Y, join(complement(meet(Y, join(X, meet(Y, complement(X))))), complement(Y))), one, join(complement(meet(Y, join(X, meet(Y, complement(X))))), complement(Y)), Y))
% 163.61/21.48 = { by lemma 15 }
% 163.61/21.48 join(X, fresh(meet(Y, join(complement(meet(Y, join(X, meet(Y, complement(X))))), complement(Y))), zero, join(complement(meet(Y, join(X, meet(Y, complement(X))))), complement(Y)), Y))
% 163.61/21.48 = { by axiom 2 (commutativity_of_join) }
% 163.61/21.48 join(X, fresh(meet(Y, join(complement(Y), complement(meet(Y, join(X, meet(Y, complement(X))))))), zero, join(complement(meet(Y, join(X, meet(Y, complement(X))))), complement(Y)), Y))
% 163.61/21.48 = { by axiom 2 (commutativity_of_join) }
% 163.61/21.48 join(X, fresh(meet(Y, join(complement(Y), complement(meet(Y, join(X, meet(Y, complement(X))))))), zero, join(complement(Y), complement(meet(Y, join(X, meet(Y, complement(X)))))), Y))
% 163.61/21.48 = { by lemma 30 }
% 163.61/21.48 join(X, fresh(meet(Y, complement(meet(Y, join(X, meet(Y, complement(X)))))), zero, join(complement(Y), complement(meet(Y, join(X, meet(Y, complement(X)))))), Y))
% 163.61/21.48 = { by lemma 32 }
% 163.61/21.48 join(X, fresh(meet(Y, complement(join(X, meet(Y, complement(X))))), zero, join(complement(Y), complement(meet(Y, join(X, meet(Y, complement(X)))))), Y))
% 163.61/21.48 = { by lemma 30 }
% 163.61/21.48 join(X, fresh(meet(Y, complement(join(X, meet(Y, complement(X))))), zero, complement(meet(Y, join(X, meet(Y, complement(X))))), Y))
% 163.61/21.48 = { by lemma 31 R->L }
% 163.61/21.48 join(X, fresh(meet(Y, meet(complement(join(X, meet(Y, complement(X)))), join(complement(X), complement(join(X, meet(Y, complement(X))))))), zero, complement(meet(Y, join(X, meet(Y, complement(X))))), Y))
% 163.61/21.48 = { by lemma 35 }
% 163.61/21.48 join(X, fresh(meet(Y, meet(complement(join(X, meet(Y, complement(X)))), complement(X))), zero, complement(meet(Y, join(X, meet(Y, complement(X))))), Y))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.48 join(X, fresh(meet(Y, meet(complement(X), complement(join(X, meet(Y, complement(X)))))), zero, complement(meet(Y, join(X, meet(Y, complement(X))))), Y))
% 163.61/21.48 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.48 join(X, fresh(meet(Y, meet(complement(X), complement(join(meet(Y, complement(X)), X)))), zero, complement(meet(Y, join(X, meet(Y, complement(X))))), Y))
% 163.61/21.48 = { by axiom 12 (associativity_of_meet) R->L }
% 163.61/21.48 join(X, fresh(meet(meet(Y, complement(X)), complement(join(meet(Y, complement(X)), X))), zero, complement(meet(Y, join(X, meet(Y, complement(X))))), Y))
% 163.61/21.48 = { by lemma 33 }
% 163.61/21.48 join(X, fresh(zero, zero, complement(meet(Y, join(X, meet(Y, complement(X))))), Y))
% 163.61/21.48 = { by axiom 7 (meet_join_complement) }
% 163.61/21.48 join(X, Y)
% 163.61/21.48
% 163.61/21.48 Lemma 37: meet(X, complement(meet(Y, X))) = meet(X, complement(Y)).
% 163.61/21.48 Proof:
% 163.61/21.48 meet(X, complement(meet(Y, X)))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.48 meet(X, complement(meet(X, Y)))
% 163.61/21.48 = { by lemma 32 }
% 163.61/21.48 meet(X, complement(Y))
% 163.61/21.48
% 163.61/21.48 Lemma 38: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 163.61/21.48 Proof:
% 163.61/21.48 complement(meet(X, complement(Y)))
% 163.61/21.48 = { by lemma 29 R->L }
% 163.61/21.48 join(Y, complement(meet(X, complement(Y))))
% 163.61/21.48 = { by lemma 36 R->L }
% 163.61/21.48 join(Y, meet(complement(Y), complement(meet(X, complement(Y)))))
% 163.61/21.48 = { by lemma 37 }
% 163.61/21.48 join(Y, meet(complement(Y), complement(X)))
% 163.61/21.48 = { by lemma 36 }
% 163.61/21.48 join(Y, complement(X))
% 163.61/21.48
% 163.61/21.48 Lemma 39: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 163.61/21.48 Proof:
% 163.61/21.48 complement(join(X, complement(Y)))
% 163.61/21.48 = { by lemma 16 R->L }
% 163.61/21.48 complement(join(complement(complement(X)), complement(Y)))
% 163.61/21.48 = { by axiom 1 (idempotence_of_join) R->L }
% 163.61/21.48 complement(join(join(complement(complement(X)), complement(complement(X))), complement(Y)))
% 163.61/21.48 = { by axiom 10 (associativity_of_join) }
% 163.61/21.48 complement(join(complement(complement(X)), join(complement(complement(X)), complement(Y))))
% 163.61/21.48 = { by axiom 2 (commutativity_of_join) }
% 163.61/21.48 complement(join(complement(complement(X)), join(complement(Y), complement(complement(X)))))
% 163.61/21.48 = { by axiom 10 (associativity_of_join) R->L }
% 163.61/21.48 complement(join(join(complement(complement(X)), complement(Y)), complement(complement(X))))
% 163.61/21.48 = { by lemma 38 R->L }
% 163.61/21.48 complement(join(complement(meet(Y, complement(complement(complement(X))))), complement(complement(X))))
% 163.61/21.48 = { by lemma 31 R->L }
% 163.61/21.48 meet(complement(join(complement(meet(Y, complement(complement(complement(X))))), complement(complement(X)))), join(complement(X), complement(join(complement(meet(Y, complement(complement(complement(X))))), complement(complement(X))))))
% 163.61/21.48 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.48 meet(complement(join(complement(meet(Y, complement(complement(complement(X))))), complement(complement(X)))), join(complement(X), complement(join(complement(complement(X)), complement(meet(Y, complement(complement(complement(X)))))))))
% 163.61/21.48 = { by lemma 16 R->L }
% 163.61/21.48 meet(complement(join(complement(meet(Y, complement(complement(complement(X))))), complement(complement(X)))), join(complement(complement(complement(X))), complement(join(complement(complement(X)), complement(meet(Y, complement(complement(complement(X)))))))))
% 163.61/21.48 = { by lemma 35 }
% 163.61/21.48 meet(complement(join(complement(meet(Y, complement(complement(complement(X))))), complement(complement(X)))), complement(complement(complement(X))))
% 163.61/21.48 = { by lemma 16 }
% 163.61/21.48 meet(complement(join(complement(meet(Y, complement(complement(complement(X))))), complement(complement(X)))), complement(X))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.48 meet(complement(X), complement(join(complement(meet(Y, complement(complement(complement(X))))), complement(complement(X)))))
% 163.61/21.48 = { by axiom 2 (commutativity_of_join) }
% 163.61/21.48 meet(complement(X), complement(join(complement(complement(X)), complement(meet(Y, complement(complement(complement(X))))))))
% 163.61/21.48 = { by lemma 29 }
% 163.61/21.48 meet(complement(X), complement(complement(meet(Y, complement(complement(complement(X)))))))
% 163.61/21.48 = { by lemma 16 }
% 163.61/21.48 meet(complement(X), meet(Y, complement(complement(complement(X)))))
% 163.61/21.48 = { by lemma 16 }
% 163.61/21.48 meet(complement(X), meet(Y, complement(X)))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.48 meet(complement(X), meet(complement(X), Y))
% 163.61/21.48 = { by axiom 12 (associativity_of_meet) R->L }
% 163.61/21.48 meet(meet(complement(X), complement(X)), Y)
% 163.61/21.48 = { by axiom 3 (idempotence_of_meet) }
% 163.61/21.48 meet(complement(X), Y)
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.48 meet(Y, complement(X))
% 163.61/21.48
% 163.61/21.48 Lemma 40: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
% 163.61/21.48 Proof:
% 163.61/21.48 meet(X, meet(Y, Z))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.48 meet(meet(Y, Z), X)
% 163.61/21.48 = { by axiom 12 (associativity_of_meet) }
% 163.61/21.48 meet(Y, meet(Z, X))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.48 meet(Y, meet(X, Z))
% 163.61/21.48
% 163.61/21.48 Lemma 41: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 163.61/21.48 Proof:
% 163.61/21.48 meet(complement(X), complement(Y))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.48 meet(complement(Y), complement(X))
% 163.61/21.48 = { by lemma 39 R->L }
% 163.61/21.48 complement(join(X, complement(complement(Y))))
% 163.61/21.48 = { by lemma 16 }
% 163.61/21.48 complement(join(X, Y))
% 163.61/21.48
% 163.61/21.48 Lemma 42: join(X, complement(meet(Y, meet(Z, complement(X))))) = join(X, complement(meet(Y, Z))).
% 163.61/21.48 Proof:
% 163.61/21.48 join(X, complement(meet(Y, meet(Z, complement(X)))))
% 163.61/21.48 = { by lemma 40 R->L }
% 163.61/21.48 join(X, complement(meet(Z, meet(Y, complement(X)))))
% 163.61/21.48 = { by axiom 12 (associativity_of_meet) R->L }
% 163.61/21.48 join(X, complement(meet(meet(Z, Y), complement(X))))
% 163.61/21.48 = { by lemma 29 }
% 163.61/21.48 complement(meet(meet(Z, Y), complement(X)))
% 163.61/21.48 = { by lemma 38 }
% 163.61/21.48 join(X, complement(meet(Z, Y)))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.48 join(X, complement(meet(Y, Z)))
% 163.61/21.48
% 163.61/21.48 Lemma 43: meet(X, join(Y, complement(meet(X, Z)))) = meet(X, join(Y, complement(Z))).
% 163.61/21.48 Proof:
% 163.61/21.48 meet(X, join(Y, complement(meet(X, Z))))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.48 meet(X, join(Y, complement(meet(Z, X))))
% 163.61/21.48 = { by lemma 42 R->L }
% 163.61/21.48 meet(X, join(Y, complement(meet(Z, meet(X, complement(Y))))))
% 163.61/21.48 = { by axiom 7 (meet_join_complement) R->L }
% 163.61/21.48 meet(X, fresh(zero, zero, meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, meet(X, complement(Y)))))))
% 163.61/21.48 = { by lemma 22 R->L }
% 163.61/21.48 meet(X, fresh(meet(meet(Z, X), meet(complement(Y), complement(meet(Z, X)))), zero, meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, meet(X, complement(Y)))))))
% 163.61/21.48 = { by axiom 12 (associativity_of_meet) }
% 163.61/21.48 meet(X, fresh(meet(Z, meet(X, meet(complement(Y), complement(meet(Z, X))))), zero, meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, meet(X, complement(Y)))))))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.48 meet(X, fresh(meet(Z, meet(X, meet(complement(meet(Z, X)), complement(Y)))), zero, meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, meet(X, complement(Y)))))))
% 163.61/21.48 = { by lemma 39 R->L }
% 163.61/21.48 meet(X, fresh(meet(Z, meet(X, complement(join(Y, complement(complement(meet(Z, X))))))), zero, meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, meet(X, complement(Y)))))))
% 163.61/21.48 = { by lemma 41 R->L }
% 163.61/21.48 meet(X, fresh(meet(Z, meet(X, meet(complement(Y), complement(complement(complement(meet(Z, X))))))), zero, meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, meet(X, complement(Y)))))))
% 163.61/21.48 = { by lemma 37 R->L }
% 163.61/21.48 meet(X, fresh(meet(Z, meet(X, meet(complement(Y), complement(meet(complement(complement(meet(Z, X))), complement(Y)))))), zero, meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, meet(X, complement(Y)))))))
% 163.61/21.48 = { by lemma 41 }
% 163.61/21.48 meet(X, fresh(meet(Z, meet(X, meet(complement(Y), complement(complement(join(complement(meet(Z, X)), Y)))))), zero, meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, meet(X, complement(Y)))))))
% 163.61/21.48 = { by lemma 41 }
% 163.61/21.48 meet(X, fresh(meet(Z, meet(X, complement(join(Y, complement(join(complement(meet(Z, X)), Y)))))), zero, meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, meet(X, complement(Y)))))))
% 163.61/21.48 = { by lemma 39 }
% 163.61/21.48 meet(X, fresh(meet(Z, meet(X, meet(join(complement(meet(Z, X)), Y), complement(Y)))), zero, meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, meet(X, complement(Y)))))))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.48 meet(X, fresh(meet(Z, meet(X, meet(complement(Y), join(complement(meet(Z, X)), Y)))), zero, meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, meet(X, complement(Y)))))))
% 163.61/21.48 = { by axiom 2 (commutativity_of_join) }
% 163.61/21.48 meet(X, fresh(meet(Z, meet(X, meet(complement(Y), join(Y, complement(meet(Z, X)))))), zero, meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, meet(X, complement(Y)))))))
% 163.61/21.48 = { by axiom 12 (associativity_of_meet) R->L }
% 163.61/21.48 meet(X, fresh(meet(Z, meet(meet(X, complement(Y)), join(Y, complement(meet(Z, X))))), zero, meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, meet(X, complement(Y)))))))
% 163.61/21.48 = { by axiom 12 (associativity_of_meet) R->L }
% 163.61/21.48 meet(X, fresh(meet(meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, X)))), zero, meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, meet(X, complement(Y)))))))
% 163.61/21.48 = { by lemma 42 R->L }
% 163.61/21.48 meet(X, fresh(meet(meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, meet(X, complement(Y)))))), zero, meet(Z, meet(X, complement(Y))), join(Y, complement(meet(Z, meet(X, complement(Y)))))))
% 163.61/21.48 = { by lemma 26 }
% 163.61/21.48 meet(X, complement(meet(Z, meet(X, complement(Y)))))
% 163.61/21.48 = { by lemma 40 R->L }
% 163.61/21.48 meet(X, complement(meet(X, meet(Z, complement(Y)))))
% 163.61/21.48 = { by lemma 32 }
% 163.61/21.48 meet(X, complement(meet(Z, complement(Y))))
% 163.61/21.48 = { by lemma 38 }
% 163.61/21.48 meet(X, join(Y, complement(Z)))
% 163.61/21.48
% 163.61/21.48 Lemma 44: meet(X, join(Y, meet(X, Z))) = meet(X, join(Y, Z)).
% 163.61/21.48 Proof:
% 163.61/21.48 meet(X, join(Y, meet(X, Z)))
% 163.61/21.48 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.48 meet(X, join(Y, meet(Z, X)))
% 163.61/21.48 = { by lemma 16 R->L }
% 163.61/21.48 meet(X, join(Y, complement(complement(meet(Z, X)))))
% 163.61/21.48 = { by lemma 43 R->L }
% 163.61/21.48 meet(X, join(Y, complement(meet(X, complement(meet(Z, X))))))
% 163.61/21.48 = { by lemma 37 }
% 163.61/21.48 meet(X, join(Y, complement(meet(X, complement(Z)))))
% 163.61/21.48 = { by lemma 43 }
% 163.61/21.49 meet(X, join(Y, complement(complement(Z))))
% 163.61/21.49 = { by lemma 16 }
% 163.61/21.49 meet(X, join(Y, Z))
% 163.61/21.49
% 163.61/21.49 Lemma 45: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 163.61/21.49 Proof:
% 163.61/21.49 join(X, join(Y, meet(X, Z)))
% 163.61/21.49 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.49 join(X, join(meet(X, Z), Y))
% 163.61/21.49 = { by axiom 10 (associativity_of_join) R->L }
% 163.61/21.49 join(join(X, meet(X, Z)), Y)
% 163.61/21.49 = { by axiom 9 (absorption2) }
% 163.61/21.49 join(X, Y)
% 163.61/21.49
% 163.61/21.49 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 163.61/21.49 Proof:
% 163.61/21.49 meet(a, join(b, c))
% 163.61/21.49 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.49 meet(a, join(c, b))
% 163.61/21.49 = { by lemma 16 R->L }
% 163.61/21.49 complement(complement(meet(a, join(c, b))))
% 163.61/21.49 = { by lemma 44 R->L }
% 163.61/21.49 complement(complement(meet(a, join(c, meet(a, b)))))
% 163.61/21.49 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.49 complement(complement(meet(a, join(c, meet(b, a)))))
% 163.61/21.49 = { by axiom 2 (commutativity_of_join) R->L }
% 163.61/21.49 complement(complement(meet(a, join(meet(b, a), c))))
% 163.61/21.49 = { by lemma 44 R->L }
% 163.61/21.49 complement(complement(meet(a, join(meet(b, a), meet(a, c)))))
% 163.61/21.49 = { by lemma 16 R->L }
% 163.61/21.49 complement(complement(meet(a, join(meet(b, a), meet(complement(complement(a)), c)))))
% 163.61/21.49 = { by lemma 16 R->L }
% 163.61/21.49 complement(complement(meet(a, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)))))
% 163.61/21.49 = { by axiom 7 (meet_join_complement) R->L }
% 163.61/21.49 complement(fresh(zero, zero, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(a, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c))))))
% 163.61/21.49 = { by lemma 22 R->L }
% 163.61/21.49 complement(fresh(meet(complement(a), meet(join(meet(complement(complement(a)), b), meet(complement(complement(a)), c)), complement(complement(a)))), zero, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(a, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c))))))
% 163.61/21.49 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.49 complement(fresh(meet(complement(a), meet(complement(complement(a)), join(meet(complement(complement(a)), b), meet(complement(complement(a)), c)))), zero, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(a, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c))))))
% 163.61/21.49 = { by axiom 9 (absorption2) R->L }
% 163.61/21.49 complement(fresh(meet(complement(a), meet(join(complement(complement(a)), meet(complement(complement(a)), b)), join(meet(complement(complement(a)), b), meet(complement(complement(a)), c)))), zero, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(a, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c))))))
% 163.61/21.49 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.49 complement(fresh(meet(complement(a), meet(join(meet(complement(complement(a)), b), meet(complement(complement(a)), c)), join(complement(complement(a)), meet(complement(complement(a)), b)))), zero, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(a, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c))))))
% 163.61/21.49 = { by lemma 45 R->L }
% 163.61/21.49 complement(fresh(meet(complement(a), meet(join(meet(complement(complement(a)), b), meet(complement(complement(a)), c)), join(complement(complement(a)), join(meet(complement(complement(a)), b), meet(complement(complement(a)), c))))), zero, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(a, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c))))))
% 163.61/21.49 = { by lemma 31 }
% 163.61/21.49 complement(fresh(meet(complement(a), join(meet(complement(complement(a)), b), meet(complement(complement(a)), c))), zero, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(a, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c))))))
% 163.61/21.49 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.49 complement(fresh(meet(complement(a), join(meet(b, complement(complement(a))), meet(complement(complement(a)), c))), zero, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(a, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c))))))
% 163.61/21.49 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.49 complement(fresh(meet(complement(a), join(meet(b, complement(complement(a))), meet(complement(complement(a)), c))), zero, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), a))))
% 163.61/21.49 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.49 complement(fresh(meet(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(a)), zero, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), a))))
% 163.61/21.49 = { by lemma 32 R->L }
% 163.61/21.49 complement(fresh(meet(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), a))), zero, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), a))))
% 163.61/21.49 = { by axiom 13 (meet_join_complement) R->L }
% 163.61/21.49 complement(fresh2(join(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), a))), one, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), a))))
% 163.61/21.49 = { by lemma 45 R->L }
% 163.61/21.49 complement(fresh2(join(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), join(complement(meet(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), a)), meet(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), a))), one, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), a))))
% 163.61/21.49 = { by axiom 2 (commutativity_of_join) }
% 163.61/21.49 complement(fresh2(join(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), join(meet(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), a), complement(meet(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), a)))), one, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), a))))
% 163.61/21.49 = { by axiom 5 (complement_join) }
% 163.61/21.49 complement(fresh2(join(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), one), one, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), a))))
% 163.61/21.49 = { by lemma 21 }
% 163.61/21.49 complement(fresh2(one, one, join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), complement(meet(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c)), a))))
% 163.61/21.49 = { by axiom 8 (meet_join_complement) }
% 163.61/21.49 complement(complement(join(meet(b, complement(complement(a))), meet(complement(complement(a)), c))))
% 163.61/21.49 = { by lemma 16 }
% 163.61/21.49 complement(complement(join(meet(b, complement(complement(a))), meet(a, c))))
% 163.61/21.49 = { by lemma 16 }
% 163.61/21.49 complement(complement(join(meet(b, a), meet(a, c))))
% 163.61/21.49 = { by axiom 2 (commutativity_of_join) }
% 163.61/21.49 complement(complement(join(meet(a, c), meet(b, a))))
% 163.61/21.49 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.49 complement(complement(join(meet(c, a), meet(b, a))))
% 163.61/21.49 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.49 complement(complement(join(meet(c, a), meet(a, b))))
% 163.61/21.49 = { by lemma 16 }
% 163.61/21.49 join(meet(c, a), meet(a, b))
% 163.61/21.49 = { by axiom 2 (commutativity_of_join) }
% 163.61/21.49 join(meet(a, b), meet(c, a))
% 163.61/21.49 = { by axiom 4 (commutativity_of_meet) }
% 163.61/21.49 join(meet(b, a), meet(c, a))
% 163.61/21.49 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.49 join(meet(b, a), meet(a, c))
% 163.61/21.49 = { by axiom 4 (commutativity_of_meet) R->L }
% 163.61/21.49 join(meet(a, b), meet(a, c))
% 163.61/21.49 % SZS output end Proof
% 163.61/21.49
% 163.61/21.49 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------