TSTP Solution File: LAT193-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT193-1 : 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 : n025.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 : Unsatisfiable 85.86s 11.52s
% Output : Proof 87.03s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LAT193-1 : TPTP v8.1.2. Released v3.1.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n025.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Aug 24 08:10:06 EDT 2023
% 0.13/0.34 % CPUTime :
% 85.86/11.52 Command-line arguments: --no-flatten-goal
% 85.86/11.52
% 85.86/11.52 % SZS status Unsatisfiable
% 85.86/11.52
% 87.03/11.58 % SZS output start Proof
% 87.03/11.58 Take the following subset of the input axioms:
% 87.03/11.59 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 87.03/11.59 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 87.03/11.59 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 87.03/11.59 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 87.03/11.59 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 87.03/11.59 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 87.03/11.59 fof(complement_join, axiom, ![X2]: join(X2, complement(X2))=one).
% 87.03/11.59 fof(complement_meet, axiom, ![X2]: meet(X2, complement(X2))=zero).
% 87.03/11.59 fof(equation_H24, axiom, ![X2, Y2, Z2]: join(meet(X2, Y2), meet(Y2, Z2))=join(meet(X2, Y2), meet(Y2, join(meet(X2, Y2), meet(Z2, join(X2, Y2)))))).
% 87.03/11.59 fof(idempotence_of_join, axiom, ![X2]: join(X2, X2)=X2).
% 87.03/11.59 fof(idempotence_of_meet, axiom, ![X2]: meet(X2, X2)=X2).
% 87.03/11.59 fof(meet_join_complement, axiom, ![X2, Y2]: (meet(X2, Y2)!=zero | (join(X2, Y2)!=one | complement(X2)=Y2))).
% 87.03/11.59 fof(prove_distributivity, negated_conjecture, meet(a, join(b, c))!=join(meet(a, b), meet(a, c))).
% 87.03/11.59
% 87.03/11.59 Now clausify the problem and encode Horn clauses using encoding 3 of
% 87.03/11.59 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 87.03/11.59 We repeatedly replace C & s=t => u=v by the two clauses:
% 87.03/11.59 fresh(y, y, x1...xn) = u
% 87.03/11.59 C => fresh(s, t, x1...xn) = v
% 87.03/11.59 where fresh is a fresh function symbol and x1..xn are the free
% 87.03/11.59 variables of u and v.
% 87.03/11.59 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 87.03/11.59 input problem has no model of domain size 1).
% 87.03/11.59
% 87.03/11.59 The encoding turns the above axioms into the following unit equations and goals:
% 87.03/11.59
% 87.03/11.59 Axiom 1 (idempotence_of_join): join(X, X) = X.
% 87.03/11.59 Axiom 2 (commutativity_of_join): join(X, Y) = join(Y, X).
% 87.03/11.59 Axiom 3 (idempotence_of_meet): meet(X, X) = X.
% 87.03/11.59 Axiom 4 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 87.03/11.59 Axiom 5 (complement_join): join(X, complement(X)) = one.
% 87.03/11.59 Axiom 6 (complement_meet): meet(X, complement(X)) = zero.
% 87.03/11.59 Axiom 7 (meet_join_complement): fresh(X, X, Y, Z) = Z.
% 87.03/11.59 Axiom 8 (meet_join_complement): fresh2(X, X, Y, Z) = complement(Y).
% 87.03/11.59 Axiom 9 (absorption2): join(X, meet(X, Y)) = X.
% 87.03/11.59 Axiom 10 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 87.03/11.59 Axiom 11 (absorption1): meet(X, join(X, Y)) = X.
% 87.03/11.59 Axiom 12 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 87.03/11.59 Axiom 13 (meet_join_complement): fresh2(join(X, Y), one, X, Y) = fresh(meet(X, Y), zero, X, Y).
% 87.03/11.59 Axiom 14 (equation_H24): join(meet(X, Y), meet(Y, Z)) = join(meet(X, Y), meet(Y, join(meet(X, Y), meet(Z, join(X, Y))))).
% 87.03/11.59
% 87.03/11.59 Lemma 15: fresh2(join(X, Y), one, Y, X) = fresh(meet(X, Y), zero, Y, X).
% 87.03/11.59 Proof:
% 87.03/11.59 fresh2(join(X, Y), one, Y, X)
% 87.03/11.59 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.59 fresh2(join(Y, X), one, Y, X)
% 87.03/11.59 = { by axiom 13 (meet_join_complement) }
% 87.03/11.59 fresh(meet(Y, X), zero, Y, X)
% 87.03/11.59 = { by axiom 4 (commutativity_of_meet) }
% 87.03/11.59 fresh(meet(X, Y), zero, Y, X)
% 87.03/11.59
% 87.03/11.59 Lemma 16: complement(complement(X)) = X.
% 87.03/11.59 Proof:
% 87.03/11.59 complement(complement(X))
% 87.03/11.59 = { by axiom 8 (meet_join_complement) R->L }
% 87.03/11.59 fresh2(one, one, complement(X), X)
% 87.03/11.59 = { by axiom 5 (complement_join) R->L }
% 87.03/11.59 fresh2(join(X, complement(X)), one, complement(X), X)
% 87.03/11.59 = { by lemma 15 }
% 87.03/11.59 fresh(meet(X, complement(X)), zero, complement(X), X)
% 87.03/11.59 = { by axiom 6 (complement_meet) }
% 87.03/11.59 fresh(zero, zero, complement(X), X)
% 87.03/11.59 = { by axiom 7 (meet_join_complement) }
% 87.03/11.59 X
% 87.03/11.59
% 87.03/11.59 Lemma 17: meet(X, one) = X.
% 87.03/11.59 Proof:
% 87.03/11.59 meet(X, one)
% 87.03/11.59 = { by axiom 5 (complement_join) R->L }
% 87.03/11.59 meet(X, join(X, complement(X)))
% 87.03/11.59 = { by axiom 11 (absorption1) }
% 87.03/11.59 X
% 87.03/11.59
% 87.03/11.59 Lemma 18: join(X, one) = one.
% 87.03/11.59 Proof:
% 87.03/11.59 join(X, one)
% 87.03/11.59 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.59 join(one, X)
% 87.03/11.59 = { by lemma 17 R->L }
% 87.03/11.59 join(one, meet(X, one))
% 87.03/11.59 = { by axiom 4 (commutativity_of_meet) }
% 87.03/11.59 join(one, meet(one, X))
% 87.03/11.59 = { by axiom 9 (absorption2) }
% 87.03/11.59 one
% 87.03/11.59
% 87.03/11.59 Lemma 19: join(X, zero) = X.
% 87.03/11.59 Proof:
% 87.03/11.59 join(X, zero)
% 87.03/11.59 = { by axiom 6 (complement_meet) R->L }
% 87.03/11.59 join(X, meet(X, complement(X)))
% 87.03/11.59 = { by axiom 9 (absorption2) }
% 87.03/11.59 X
% 87.03/11.59
% 87.03/11.59 Lemma 20: join(X, join(X, Y)) = join(X, Y).
% 87.03/11.59 Proof:
% 87.03/11.59 join(X, join(X, Y))
% 87.03/11.59 = { by axiom 10 (associativity_of_join) R->L }
% 87.03/11.59 join(join(X, X), Y)
% 87.03/11.59 = { by axiom 1 (idempotence_of_join) }
% 87.03/11.59 join(X, Y)
% 87.03/11.59
% 87.03/11.59 Lemma 21: join(X, join(Y, X)) = join(X, Y).
% 87.03/11.59 Proof:
% 87.03/11.59 join(X, join(Y, X))
% 87.03/11.59 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.59 join(X, join(X, Y))
% 87.03/11.59 = { by lemma 20 }
% 87.03/11.59 join(X, Y)
% 87.03/11.59
% 87.03/11.59 Lemma 22: join(X, meet(Y, X)) = X.
% 87.03/11.59 Proof:
% 87.03/11.59 join(X, meet(Y, X))
% 87.03/11.59 = { by axiom 4 (commutativity_of_meet) R->L }
% 87.03/11.59 join(X, meet(X, Y))
% 87.03/11.59 = { by axiom 9 (absorption2) }
% 87.03/11.59 X
% 87.03/11.59
% 87.03/11.59 Lemma 23: meet(X, meet(Y, complement(meet(X, Y)))) = zero.
% 87.03/11.59 Proof:
% 87.03/11.59 meet(X, meet(Y, complement(meet(X, Y))))
% 87.03/11.59 = { by axiom 12 (associativity_of_meet) R->L }
% 87.03/11.59 meet(meet(X, Y), complement(meet(X, Y)))
% 87.03/11.59 = { by axiom 6 (complement_meet) }
% 87.03/11.59 zero
% 87.03/11.59
% 87.03/11.59 Lemma 24: join(X, join(meet(X, Y), Z)) = join(X, Z).
% 87.03/11.59 Proof:
% 87.03/11.59 join(X, join(meet(X, Y), Z))
% 87.03/11.59 = { by axiom 10 (associativity_of_join) R->L }
% 87.03/11.59 join(join(X, meet(X, Y)), Z)
% 87.03/11.59 = { by axiom 9 (absorption2) }
% 87.03/11.59 join(X, Z)
% 87.03/11.59
% 87.03/11.59 Lemma 25: meet(X, join(meet(Y, X), meet(Z, join(Y, X)))) = join(meet(Y, X), meet(X, Z)).
% 87.03/11.59 Proof:
% 87.03/11.59 meet(X, join(meet(Y, X), meet(Z, join(Y, X))))
% 87.03/11.59 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.59 meet(X, join(meet(Z, join(Y, X)), meet(Y, X)))
% 87.03/11.59 = { by lemma 22 R->L }
% 87.03/11.59 join(meet(X, join(meet(Z, join(Y, X)), meet(Y, X))), meet(Y, meet(X, join(meet(Z, join(Y, X)), meet(Y, X)))))
% 87.03/11.59 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.59 join(meet(X, join(meet(Z, join(Y, X)), meet(Y, X))), meet(Y, meet(X, join(meet(Y, X), meet(Z, join(Y, X))))))
% 87.03/11.59 = { by axiom 12 (associativity_of_meet) R->L }
% 87.03/11.59 join(meet(X, join(meet(Z, join(Y, X)), meet(Y, X))), meet(meet(Y, X), join(meet(Y, X), meet(Z, join(Y, X)))))
% 87.03/11.59 = { by axiom 11 (absorption1) }
% 87.03/11.59 join(meet(X, join(meet(Z, join(Y, X)), meet(Y, X))), meet(Y, X))
% 87.03/11.59 = { by axiom 2 (commutativity_of_join) }
% 87.03/11.59 join(meet(Y, X), meet(X, join(meet(Z, join(Y, X)), meet(Y, X))))
% 87.03/11.59 = { by axiom 2 (commutativity_of_join) }
% 87.03/11.59 join(meet(Y, X), meet(X, join(meet(Y, X), meet(Z, join(Y, X)))))
% 87.03/11.59 = { by axiom 14 (equation_H24) R->L }
% 87.03/11.59 join(meet(Y, X), meet(X, Z))
% 87.03/11.59
% 87.03/11.59 Lemma 26: join(X, join(Y, complement(X))) = one.
% 87.03/11.59 Proof:
% 87.03/11.59 join(X, join(Y, complement(X)))
% 87.03/11.59 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.59 join(X, join(complement(X), Y))
% 87.03/11.59 = { by axiom 10 (associativity_of_join) R->L }
% 87.03/11.59 join(join(X, complement(X)), Y)
% 87.03/11.59 = { by axiom 5 (complement_join) }
% 87.03/11.59 join(one, Y)
% 87.03/11.59 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.59 join(Y, one)
% 87.03/11.59 = { by lemma 18 }
% 87.03/11.59 one
% 87.03/11.59
% 87.03/11.59 Lemma 27: join(meet(X, join(Y, complement(X))), meet(join(Y, complement(X)), complement(meet(X, join(Y, complement(X)))))) = join(Y, complement(X)).
% 87.03/11.59 Proof:
% 87.03/11.59 join(meet(X, join(Y, complement(X))), meet(join(Y, complement(X)), complement(meet(X, join(Y, complement(X))))))
% 87.03/11.59 = { by lemma 25 R->L }
% 87.03/11.59 meet(join(Y, complement(X)), join(meet(X, join(Y, complement(X))), meet(complement(meet(X, join(Y, complement(X)))), join(X, join(Y, complement(X))))))
% 87.03/11.59 = { by lemma 26 }
% 87.03/11.59 meet(join(Y, complement(X)), join(meet(X, join(Y, complement(X))), meet(complement(meet(X, join(Y, complement(X)))), one)))
% 87.03/11.59 = { by lemma 17 }
% 87.03/11.59 meet(join(Y, complement(X)), join(meet(X, join(Y, complement(X))), complement(meet(X, join(Y, complement(X))))))
% 87.03/11.59 = { by axiom 5 (complement_join) }
% 87.03/11.59 meet(join(Y, complement(X)), one)
% 87.03/11.59 = { by lemma 17 }
% 87.03/11.59 join(Y, complement(X))
% 87.03/11.59
% 87.03/11.59 Lemma 28: meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))) = complement(Y).
% 87.03/11.59 Proof:
% 87.03/11.59 meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))
% 87.03/11.59 = { by axiom 7 (meet_join_complement) R->L }
% 87.03/11.59 fresh(zero, zero, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 87.03/11.59 = { by lemma 23 R->L }
% 87.03/11.59 fresh(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 87.03/11.59 = { by axiom 13 (meet_join_complement) R->L }
% 87.03/11.59 fresh2(join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 87.03/11.59 = { by lemma 24 R->L }
% 87.03/11.59 fresh2(join(Y, join(meet(Y, join(X, complement(Y))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 87.03/11.59 = { by lemma 27 }
% 87.03/11.59 fresh2(join(Y, join(X, complement(Y))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 87.03/11.59 = { by lemma 26 }
% 87.03/11.59 fresh2(one, one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 87.03/11.59 = { by axiom 8 (meet_join_complement) }
% 87.03/11.59 complement(Y)
% 87.03/11.59
% 87.03/11.59 Lemma 29: join(X, meet(Y, complement(meet(X, Y)))) = join(X, Y).
% 87.03/11.59 Proof:
% 87.03/11.59 join(X, meet(Y, complement(meet(X, Y))))
% 87.03/11.59 = { by axiom 4 (commutativity_of_meet) R->L }
% 87.03/11.59 join(X, meet(Y, complement(meet(Y, X))))
% 87.03/11.59 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.59 join(meet(Y, complement(meet(Y, X))), X)
% 87.03/11.59 = { by lemma 22 R->L }
% 87.03/11.59 join(meet(Y, complement(meet(Y, X))), join(X, meet(Y, X)))
% 87.03/11.59 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.59 join(join(X, meet(Y, X)), meet(Y, complement(meet(Y, X))))
% 87.03/11.59 = { by axiom 10 (associativity_of_join) }
% 87.03/11.59 join(X, join(meet(Y, X), meet(Y, complement(meet(Y, X)))))
% 87.03/11.59 = { by axiom 4 (commutativity_of_meet) R->L }
% 87.03/11.59 join(X, join(meet(Y, X), meet(complement(meet(Y, X)), Y)))
% 87.03/11.59 = { by axiom 9 (absorption2) R->L }
% 87.03/11.59 join(X, join(meet(Y, X), meet(complement(meet(Y, X)), join(Y, meet(Y, X)))))
% 87.03/11.59 = { by lemma 16 R->L }
% 87.03/11.59 join(X, join(meet(Y, X), meet(complement(meet(Y, X)), join(Y, complement(complement(meet(Y, X)))))))
% 87.03/11.59 = { by lemma 16 R->L }
% 87.03/11.59 join(X, join(complement(complement(meet(Y, X))), meet(complement(meet(Y, X)), join(Y, complement(complement(meet(Y, X)))))))
% 87.03/11.59 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.59 join(X, join(meet(complement(meet(Y, X)), join(Y, complement(complement(meet(Y, X))))), complement(complement(meet(Y, X)))))
% 87.03/11.59 = { by lemma 28 R->L }
% 87.03/11.59 join(X, join(meet(complement(meet(Y, X)), join(Y, complement(complement(meet(Y, X))))), meet(join(Y, complement(complement(meet(Y, X)))), complement(meet(complement(meet(Y, X)), join(Y, complement(complement(meet(Y, X)))))))))
% 87.03/11.59 = { by lemma 27 }
% 87.03/11.59 join(X, join(Y, complement(complement(meet(Y, X)))))
% 87.03/11.59 = { by lemma 16 }
% 87.03/11.59 join(X, join(Y, meet(Y, X)))
% 87.03/11.59 = { by axiom 9 (absorption2) }
% 87.03/11.59 join(X, Y)
% 87.03/11.59
% 87.03/11.59 Lemma 30: join(meet(X, Y), meet(Y, join(X, Z))) = meet(Y, join(X, Z)).
% 87.03/11.59 Proof:
% 87.03/11.59 join(meet(X, Y), meet(Y, join(X, Z)))
% 87.03/11.59 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.59 join(meet(Y, join(X, Z)), meet(X, Y))
% 87.03/11.59 = { by axiom 11 (absorption1) R->L }
% 87.03/11.59 join(meet(Y, join(X, Z)), meet(meet(X, join(X, Z)), Y))
% 87.03/11.59 = { by axiom 12 (associativity_of_meet) }
% 87.03/11.59 join(meet(Y, join(X, Z)), meet(X, meet(join(X, Z), Y)))
% 87.03/11.59 = { by axiom 4 (commutativity_of_meet) }
% 87.03/11.59 join(meet(Y, join(X, Z)), meet(X, meet(Y, join(X, Z))))
% 87.03/11.59 = { by lemma 22 }
% 87.03/11.59 meet(Y, join(X, Z))
% 87.03/11.59
% 87.03/11.59 Lemma 31: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
% 87.03/11.59 Proof:
% 87.03/11.59 meet(X, meet(Y, Z))
% 87.03/11.59 = { by axiom 4 (commutativity_of_meet) R->L }
% 87.03/11.59 meet(meet(Y, Z), X)
% 87.03/11.59 = { by axiom 12 (associativity_of_meet) }
% 87.03/11.59 meet(Y, meet(Z, X))
% 87.03/11.59 = { by axiom 4 (commutativity_of_meet) }
% 87.03/11.59 meet(Y, meet(X, Z))
% 87.03/11.59
% 87.03/11.59 Lemma 32: meet(X, join(Y, X)) = X.
% 87.03/11.59 Proof:
% 87.03/11.59 meet(X, join(Y, X))
% 87.03/11.59 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.59 meet(X, join(X, Y))
% 87.03/11.59 = { by axiom 11 (absorption1) }
% 87.03/11.59 X
% 87.03/11.59
% 87.03/11.59 Lemma 33: meet(X, meet(Y, join(Z, X))) = meet(X, Y).
% 87.03/11.59 Proof:
% 87.03/11.59 meet(X, meet(Y, join(Z, X)))
% 87.03/11.59 = { by lemma 31 R->L }
% 87.03/11.59 meet(Y, meet(X, join(Z, X)))
% 87.03/11.59 = { by lemma 32 }
% 87.03/11.59 meet(Y, X)
% 87.03/11.59 = { by axiom 4 (commutativity_of_meet) }
% 87.03/11.59 meet(X, Y)
% 87.03/11.59
% 87.03/11.59 Lemma 34: meet(join(X, Y), complement(meet(complement(X), join(X, Y)))) = X.
% 87.03/11.59 Proof:
% 87.03/11.59 meet(join(X, Y), complement(meet(complement(X), join(X, Y))))
% 87.03/11.59 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.59 meet(join(Y, X), complement(meet(complement(X), join(X, Y))))
% 87.03/11.59 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.59 meet(join(Y, X), complement(meet(complement(X), join(Y, X))))
% 87.03/11.59 = { by lemma 16 R->L }
% 87.03/11.59 meet(join(Y, X), complement(meet(complement(X), join(Y, complement(complement(X))))))
% 87.03/11.60 = { by lemma 16 R->L }
% 87.03/11.60 meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))
% 87.03/11.60 = { by lemma 28 }
% 87.03/11.60 complement(complement(X))
% 87.03/11.60 = { by lemma 16 }
% 87.03/11.60 X
% 87.03/11.60
% 87.03/11.60 Lemma 35: meet(X, meet(X, Y)) = meet(X, Y).
% 87.03/11.60 Proof:
% 87.03/11.60 meet(X, meet(X, Y))
% 87.03/11.60 = { by axiom 12 (associativity_of_meet) R->L }
% 87.03/11.60 meet(meet(X, X), Y)
% 87.03/11.60 = { by axiom 3 (idempotence_of_meet) }
% 87.03/11.60 meet(X, Y)
% 87.03/11.60
% 87.03/11.60 Lemma 36: meet(X, complement(meet(X, complement(meet(X, Y))))) = meet(X, Y).
% 87.03/11.60 Proof:
% 87.03/11.60 meet(X, complement(meet(X, complement(meet(X, Y)))))
% 87.03/11.60 = { by axiom 4 (commutativity_of_meet) R->L }
% 87.03/11.60 meet(X, complement(meet(X, complement(meet(Y, X)))))
% 87.03/11.60 = { by axiom 4 (commutativity_of_meet) R->L }
% 87.03/11.60 meet(X, complement(meet(complement(meet(Y, X)), X)))
% 87.03/11.60 = { by lemma 22 R->L }
% 87.03/11.60 meet(X, complement(meet(complement(meet(Y, X)), join(X, meet(Y, X)))))
% 87.03/11.60 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.60 meet(X, complement(meet(complement(meet(Y, X)), join(meet(Y, X), X))))
% 87.03/11.60 = { by lemma 33 R->L }
% 87.03/11.60 meet(X, meet(complement(meet(complement(meet(Y, X)), join(meet(Y, X), X))), join(meet(Y, X), X)))
% 87.03/11.60 = { by axiom 4 (commutativity_of_meet) }
% 87.03/11.60 meet(X, meet(join(meet(Y, X), X), complement(meet(complement(meet(Y, X)), join(meet(Y, X), X)))))
% 87.03/11.60 = { by lemma 34 }
% 87.03/11.60 meet(X, meet(Y, X))
% 87.03/11.60 = { by axiom 4 (commutativity_of_meet) R->L }
% 87.03/11.60 meet(X, meet(X, Y))
% 87.03/11.60 = { by lemma 35 }
% 87.03/11.60 meet(X, Y)
% 87.03/11.60
% 87.03/11.60 Lemma 37: meet(X, join(Y, meet(X, Z))) = join(meet(Y, X), meet(X, Z)).
% 87.03/11.60 Proof:
% 87.03/11.60 meet(X, join(Y, meet(X, Z)))
% 87.03/11.60 = { by lemma 36 R->L }
% 87.03/11.60 meet(X, join(Y, meet(X, complement(meet(X, complement(meet(X, Z)))))))
% 87.03/11.60 = { by axiom 4 (commutativity_of_meet) R->L }
% 87.03/11.60 meet(X, join(Y, meet(X, complement(meet(complement(meet(X, Z)), X)))))
% 87.03/11.60 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.60 meet(X, join(meet(X, complement(meet(complement(meet(X, Z)), X))), Y))
% 87.03/11.60 = { by lemma 17 R->L }
% 87.03/11.60 meet(X, join(meet(X, complement(meet(complement(meet(X, Z)), X))), meet(Y, one)))
% 87.03/11.60 = { by lemma 18 R->L }
% 87.03/11.60 meet(X, join(meet(X, complement(meet(complement(meet(X, Z)), X))), meet(Y, join(X, one))))
% 87.03/11.60 = { by axiom 5 (complement_join) R->L }
% 87.03/11.60 meet(X, join(meet(X, complement(meet(complement(meet(X, Z)), X))), meet(Y, join(X, join(meet(X, complement(meet(X, Z))), complement(meet(X, complement(meet(X, Z)))))))))
% 87.03/11.60 = { by lemma 24 }
% 87.03/11.60 meet(X, join(meet(X, complement(meet(complement(meet(X, Z)), X))), meet(Y, join(X, complement(meet(X, complement(meet(X, Z))))))))
% 87.03/11.60 = { by axiom 4 (commutativity_of_meet) }
% 87.03/11.60 meet(X, join(meet(X, complement(meet(complement(meet(X, Z)), X))), meet(Y, join(X, complement(meet(complement(meet(X, Z)), X))))))
% 87.03/11.60 = { by axiom 4 (commutativity_of_meet) R->L }
% 87.03/11.60 meet(X, join(meet(complement(meet(complement(meet(X, Z)), X)), X), meet(Y, join(X, complement(meet(complement(meet(X, Z)), X))))))
% 87.03/11.60 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.60 meet(X, join(meet(complement(meet(complement(meet(X, Z)), X)), X), meet(Y, join(complement(meet(complement(meet(X, Z)), X)), X))))
% 87.03/11.60 = { by lemma 25 }
% 87.03/11.60 join(meet(complement(meet(complement(meet(X, Z)), X)), X), meet(X, Y))
% 87.03/11.60 = { by axiom 4 (commutativity_of_meet) }
% 87.03/11.60 join(meet(X, complement(meet(complement(meet(X, Z)), X))), meet(X, Y))
% 87.03/11.60 = { by axiom 2 (commutativity_of_join) }
% 87.03/11.60 join(meet(X, Y), meet(X, complement(meet(complement(meet(X, Z)), X))))
% 87.03/11.60 = { by axiom 4 (commutativity_of_meet) }
% 87.03/11.60 join(meet(Y, X), meet(X, complement(meet(complement(meet(X, Z)), X))))
% 87.03/11.60 = { by axiom 4 (commutativity_of_meet) }
% 87.03/11.60 join(meet(Y, X), meet(X, complement(meet(X, complement(meet(X, Z))))))
% 87.03/11.60 = { by lemma 36 }
% 87.03/11.60 join(meet(Y, X), meet(X, Z))
% 87.03/11.60
% 87.03/11.60 Lemma 38: meet(join(X, Y), join(Y, Z)) = join(Y, meet(X, join(Y, Z))).
% 87.03/11.60 Proof:
% 87.03/11.60 meet(join(X, Y), join(Y, Z))
% 87.03/11.60 = { by axiom 4 (commutativity_of_meet) R->L }
% 87.03/11.60 meet(join(Y, Z), join(X, Y))
% 87.03/11.60 = { by lemma 34 R->L }
% 87.03/11.60 meet(join(Y, Z), join(X, meet(join(Y, Z), complement(meet(complement(Y), join(Y, Z))))))
% 87.03/11.60 = { by lemma 37 }
% 87.03/11.60 join(meet(X, join(Y, Z)), meet(join(Y, Z), complement(meet(complement(Y), join(Y, Z)))))
% 87.03/11.60 = { by lemma 34 }
% 87.03/11.60 join(meet(X, join(Y, Z)), Y)
% 87.03/11.60 = { by axiom 2 (commutativity_of_join) }
% 87.03/11.60 join(Y, meet(X, join(Y, Z)))
% 87.03/11.60
% 87.03/11.60 Lemma 39: complement(join(X, complement(join(X, Y)))) = meet(Y, complement(meet(X, Y))).
% 87.03/11.60 Proof:
% 87.03/11.60 complement(join(X, complement(join(X, Y))))
% 87.03/11.60 = { by lemma 29 R->L }
% 87.03/11.60 complement(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))))
% 87.03/11.60 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.60 complement(join(X, complement(join(meet(Y, complement(meet(X, Y))), X))))
% 87.03/11.60 = { by axiom 8 (meet_join_complement) R->L }
% 87.03/11.60 fresh2(one, one, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.60 = { by axiom 5 (complement_join) R->L }
% 87.03/11.60 fresh2(join(join(meet(Y, complement(meet(X, Y))), X), complement(join(meet(Y, complement(meet(X, Y))), X))), one, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.60 = { by axiom 10 (associativity_of_join) }
% 87.03/11.60 fresh2(join(meet(Y, complement(meet(X, Y))), join(X, complement(join(meet(Y, complement(meet(X, Y))), X)))), one, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.60 = { by lemma 15 }
% 87.03/11.60 fresh(meet(meet(Y, complement(meet(X, Y))), join(X, complement(join(meet(Y, complement(meet(X, Y))), X)))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.60 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.60 fresh(meet(meet(Y, complement(meet(X, Y))), join(X, complement(join(X, meet(Y, complement(meet(X, Y))))))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.60 = { by lemma 30 R->L }
% 87.03/11.60 fresh(join(meet(X, meet(Y, complement(meet(X, Y)))), meet(meet(Y, complement(meet(X, Y))), join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.60 = { by lemma 37 R->L }
% 87.03/11.60 fresh(meet(meet(Y, complement(meet(X, Y))), join(X, meet(meet(Y, complement(meet(X, Y))), join(X, complement(join(X, meet(Y, complement(meet(X, Y))))))))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.60 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.60 fresh(meet(meet(Y, complement(meet(X, Y))), join(X, meet(meet(Y, complement(meet(X, Y))), join(X, complement(join(meet(Y, complement(meet(X, Y))), X)))))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.60 = { by lemma 20 R->L }
% 87.03/11.60 fresh(meet(meet(Y, complement(meet(X, Y))), join(X, join(X, meet(meet(Y, complement(meet(X, Y))), join(X, complement(join(meet(Y, complement(meet(X, Y))), X))))))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.60 = { by lemma 38 R->L }
% 87.03/11.60 fresh(meet(meet(Y, complement(meet(X, Y))), join(X, meet(join(meet(Y, complement(meet(X, Y))), X), join(X, complement(join(meet(Y, complement(meet(X, Y))), X)))))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.60 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.60 fresh(meet(meet(Y, complement(meet(X, Y))), join(X, meet(join(X, meet(Y, complement(meet(X, Y)))), join(X, complement(join(meet(Y, complement(meet(X, Y))), X)))))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.60 = { by lemma 32 R->L }
% 87.03/11.60 fresh(meet(meet(Y, complement(meet(X, Y))), join(meet(X, join(complement(join(meet(Y, complement(meet(X, Y))), X)), X)), meet(join(X, meet(Y, complement(meet(X, Y)))), join(X, complement(join(meet(Y, complement(meet(X, Y))), X)))))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.60 = { by lemma 30 R->L }
% 87.03/11.60 fresh(meet(meet(Y, complement(meet(X, Y))), join(meet(X, join(complement(join(meet(Y, complement(meet(X, Y))), X)), X)), join(meet(X, join(X, meet(Y, complement(meet(X, Y))))), meet(join(X, meet(Y, complement(meet(X, Y)))), join(X, complement(join(meet(Y, complement(meet(X, Y))), X))))))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.60 = { by axiom 11 (absorption1) }
% 87.03/11.60 fresh(meet(meet(Y, complement(meet(X, Y))), join(meet(X, join(complement(join(meet(Y, complement(meet(X, Y))), X)), X)), join(X, meet(join(X, meet(Y, complement(meet(X, Y)))), join(X, complement(join(meet(Y, complement(meet(X, Y))), X))))))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.60 = { by axiom 2 (commutativity_of_join) }
% 87.03/11.60 fresh(meet(meet(Y, complement(meet(X, Y))), join(meet(X, join(complement(join(meet(Y, complement(meet(X, Y))), X)), X)), join(X, meet(join(X, meet(Y, complement(meet(X, Y)))), join(complement(join(meet(Y, complement(meet(X, Y))), X)), X))))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.60 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.60 fresh(meet(meet(Y, complement(meet(X, Y))), join(meet(X, join(complement(join(meet(Y, complement(meet(X, Y))), X)), X)), join(meet(join(X, meet(Y, complement(meet(X, Y)))), join(complement(join(meet(Y, complement(meet(X, Y))), X)), X)), X))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.60 = { by axiom 10 (associativity_of_join) R->L }
% 87.03/11.61 fresh(meet(meet(Y, complement(meet(X, Y))), join(join(meet(X, join(complement(join(meet(Y, complement(meet(X, Y))), X)), X)), meet(join(X, meet(Y, complement(meet(X, Y)))), join(complement(join(meet(Y, complement(meet(X, Y))), X)), X))), X)), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.61 = { by axiom 4 (commutativity_of_meet) R->L }
% 87.03/11.61 fresh(meet(meet(Y, complement(meet(X, Y))), join(join(meet(X, join(complement(join(meet(Y, complement(meet(X, Y))), X)), X)), meet(join(complement(join(meet(Y, complement(meet(X, Y))), X)), X), join(X, meet(Y, complement(meet(X, Y)))))), X)), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.61 = { by lemma 30 }
% 87.03/11.61 fresh(meet(meet(Y, complement(meet(X, Y))), join(meet(join(complement(join(meet(Y, complement(meet(X, Y))), X)), X), join(X, meet(Y, complement(meet(X, Y))))), X)), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.61 = { by axiom 2 (commutativity_of_join) }
% 87.03/11.61 fresh(meet(meet(Y, complement(meet(X, Y))), join(X, meet(join(complement(join(meet(Y, complement(meet(X, Y))), X)), X), join(X, meet(Y, complement(meet(X, Y))))))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.61 = { by lemma 38 }
% 87.03/11.61 fresh(meet(meet(Y, complement(meet(X, Y))), join(X, join(X, meet(complement(join(meet(Y, complement(meet(X, Y))), X)), join(X, meet(Y, complement(meet(X, Y)))))))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.61 = { by lemma 20 }
% 87.03/11.61 fresh(meet(meet(Y, complement(meet(X, Y))), join(X, meet(complement(join(meet(Y, complement(meet(X, Y))), X)), join(X, meet(Y, complement(meet(X, Y))))))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.61 = { by axiom 2 (commutativity_of_join) }
% 87.03/11.61 fresh(meet(meet(Y, complement(meet(X, Y))), join(X, meet(complement(join(meet(Y, complement(meet(X, Y))), X)), join(meet(Y, complement(meet(X, Y))), X)))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.61 = { by axiom 4 (commutativity_of_meet) }
% 87.03/11.61 fresh(meet(meet(Y, complement(meet(X, Y))), join(X, meet(join(meet(Y, complement(meet(X, Y))), X), complement(join(meet(Y, complement(meet(X, Y))), X))))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.61 = { by axiom 6 (complement_meet) }
% 87.03/11.61 fresh(meet(meet(Y, complement(meet(X, Y))), join(X, zero)), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.61 = { by lemma 19 }
% 87.03/11.61 fresh(meet(meet(Y, complement(meet(X, Y))), X), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.61 = { by axiom 4 (commutativity_of_meet) }
% 87.03/11.61 fresh(meet(X, meet(Y, complement(meet(X, Y)))), zero, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 87.03/11.61 = { by axiom 2 (commutativity_of_join) }
% 87.03/11.61 fresh(meet(X, meet(Y, complement(meet(X, Y)))), zero, join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), meet(Y, complement(meet(X, Y))))
% 87.03/11.61 = { by lemma 23 }
% 87.03/11.61 fresh(zero, zero, join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), meet(Y, complement(meet(X, Y))))
% 87.03/11.61 = { by axiom 7 (meet_join_complement) }
% 87.03/11.61 meet(Y, complement(meet(X, Y)))
% 87.03/11.61
% 87.03/11.61 Lemma 40: meet(complement(Y), join(Y, X)) = meet(X, complement(meet(Y, X))).
% 87.03/11.61 Proof:
% 87.03/11.61 meet(complement(Y), join(Y, X))
% 87.03/11.61 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.61 meet(complement(Y), join(X, Y))
% 87.03/11.61 = { by axiom 4 (commutativity_of_meet) R->L }
% 87.03/11.61 meet(join(X, Y), complement(Y))
% 87.03/11.61 = { by lemma 32 R->L }
% 87.03/11.61 meet(join(X, Y), complement(meet(Y, join(X, Y))))
% 87.03/11.61 = { by lemma 39 R->L }
% 87.03/11.61 complement(join(Y, complement(join(Y, join(X, Y)))))
% 87.03/11.61 = { by lemma 21 }
% 87.03/11.61 complement(join(Y, complement(join(Y, X))))
% 87.03/11.61 = { by lemma 39 }
% 87.03/11.61 meet(X, complement(meet(Y, X)))
% 87.03/11.61
% 87.03/11.61 Lemma 41: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 87.03/11.61 Proof:
% 87.03/11.61 meet(X, complement(meet(X, Y)))
% 87.03/11.61 = { by axiom 4 (commutativity_of_meet) R->L }
% 87.03/11.61 meet(X, complement(meet(Y, X)))
% 87.03/11.61 = { by lemma 35 R->L }
% 87.03/11.61 meet(X, meet(X, complement(meet(Y, X))))
% 87.03/11.61 = { by lemma 40 R->L }
% 87.03/11.61 meet(X, meet(complement(Y), join(Y, X)))
% 87.03/11.61 = { by lemma 33 }
% 87.03/11.61 meet(X, complement(Y))
% 87.03/11.61
% 87.03/11.61 Lemma 42: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 87.03/11.61 Proof:
% 87.03/11.61 meet(complement(X), join(X, Y))
% 87.03/11.61 = { by lemma 40 }
% 87.03/11.61 meet(Y, complement(meet(X, Y)))
% 87.03/11.61 = { by axiom 4 (commutativity_of_meet) }
% 87.03/11.61 meet(Y, complement(meet(Y, X)))
% 87.03/11.61 = { by lemma 41 }
% 87.03/11.61 meet(Y, complement(X))
% 87.03/11.61
% 87.03/11.61 Lemma 43: meet(X, complement(meet(Y, X))) = meet(X, complement(Y)).
% 87.03/11.61 Proof:
% 87.03/11.61 meet(X, complement(meet(Y, X)))
% 87.03/11.61 = { by lemma 19 R->L }
% 87.03/11.61 join(meet(X, complement(meet(Y, X))), zero)
% 87.03/11.61 = { by axiom 2 (commutativity_of_join) }
% 87.03/11.61 join(zero, meet(X, complement(meet(Y, X))))
% 87.03/11.61 = { by axiom 6 (complement_meet) R->L }
% 87.03/11.61 join(meet(Y, complement(Y)), meet(X, complement(meet(Y, X))))
% 87.03/11.61 = { by lemma 40 R->L }
% 87.03/11.61 join(meet(Y, complement(Y)), meet(complement(Y), join(Y, X)))
% 87.03/11.61 = { by lemma 30 }
% 87.03/11.61 meet(complement(Y), join(Y, X))
% 87.03/11.61 = { by lemma 42 }
% 87.03/11.61 meet(X, complement(Y))
% 87.03/11.61
% 87.03/11.61 Lemma 44: complement(join(X, complement(join(X, Y)))) = meet(Y, complement(X)).
% 87.03/11.61 Proof:
% 87.03/11.61 complement(join(X, complement(join(X, Y))))
% 87.03/11.61 = { by lemma 21 R->L }
% 87.03/11.61 complement(join(X, complement(join(X, join(Y, X)))))
% 87.03/11.61 = { by lemma 39 }
% 87.03/11.61 meet(join(Y, X), complement(meet(X, join(Y, X))))
% 87.03/11.61 = { by lemma 43 }
% 87.03/11.61 meet(join(Y, X), complement(X))
% 87.03/11.61 = { by axiom 4 (commutativity_of_meet) }
% 87.03/11.61 meet(complement(X), join(Y, X))
% 87.03/11.61 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.61 meet(complement(X), join(X, Y))
% 87.03/11.61 = { by lemma 42 }
% 87.03/11.61 meet(Y, complement(X))
% 87.03/11.61
% 87.03/11.61 Lemma 45: join(X, complement(join(X, Y))) = complement(meet(Y, complement(X))).
% 87.03/11.61 Proof:
% 87.03/11.61 join(X, complement(join(X, Y)))
% 87.03/11.61 = { by lemma 16 R->L }
% 87.03/11.61 complement(complement(join(X, complement(join(X, Y)))))
% 87.03/11.61 = { by lemma 44 }
% 87.03/11.61 complement(meet(Y, complement(X)))
% 87.03/11.61
% 87.03/11.61 Lemma 46: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 87.03/11.61 Proof:
% 87.03/11.61 meet(complement(X), complement(Y))
% 87.03/11.61 = { by axiom 4 (commutativity_of_meet) R->L }
% 87.03/11.61 meet(complement(Y), complement(X))
% 87.03/11.61 = { by lemma 43 R->L }
% 87.03/11.61 meet(complement(Y), complement(meet(X, complement(Y))))
% 87.03/11.61 = { by lemma 45 R->L }
% 87.03/11.61 meet(complement(Y), join(Y, complement(join(Y, X))))
% 87.03/11.61 = { by lemma 40 }
% 87.03/11.61 meet(complement(join(Y, X)), complement(meet(Y, complement(join(Y, X)))))
% 87.03/11.61 = { by lemma 43 }
% 87.03/11.61 meet(complement(join(Y, X)), complement(Y))
% 87.03/11.61 = { by lemma 44 R->L }
% 87.03/11.61 complement(join(Y, complement(join(Y, complement(join(Y, X))))))
% 87.03/11.61 = { by lemma 39 }
% 87.03/11.61 complement(join(Y, meet(X, complement(meet(Y, X)))))
% 87.03/11.61 = { by lemma 29 }
% 87.03/11.61 complement(join(Y, X))
% 87.03/11.61 = { by axiom 2 (commutativity_of_join) }
% 87.03/11.61 complement(join(X, Y))
% 87.03/11.61
% 87.03/11.61 Lemma 47: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 87.03/11.61 Proof:
% 87.03/11.61 complement(meet(X, complement(Y)))
% 87.03/11.61 = { by lemma 45 R->L }
% 87.03/11.61 join(Y, complement(join(Y, X)))
% 87.03/11.61 = { by lemma 16 R->L }
% 87.03/11.61 join(complement(complement(Y)), complement(join(Y, X)))
% 87.03/11.61 = { by lemma 46 R->L }
% 87.03/11.61 join(complement(complement(Y)), meet(complement(Y), complement(X)))
% 87.03/11.61 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.61 join(meet(complement(Y), complement(X)), complement(complement(Y)))
% 87.03/11.61 = { by axiom 9 (absorption2) R->L }
% 87.03/11.61 join(meet(complement(Y), complement(X)), complement(join(complement(Y), meet(complement(Y), complement(X)))))
% 87.03/11.61 = { by axiom 2 (commutativity_of_join) R->L }
% 87.03/11.61 join(meet(complement(Y), complement(X)), complement(join(meet(complement(Y), complement(X)), complement(Y))))
% 87.03/11.61 = { by lemma 45 }
% 87.03/11.61 complement(meet(complement(Y), complement(meet(complement(Y), complement(X)))))
% 87.03/11.61 = { by lemma 41 }
% 87.03/11.61 complement(meet(complement(Y), complement(complement(X))))
% 87.03/11.61 = { by lemma 46 }
% 87.03/11.61 complement(complement(join(Y, complement(X))))
% 87.03/11.61 = { by lemma 16 }
% 87.03/11.61 join(Y, complement(X))
% 87.03/11.61
% 87.03/11.61 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 87.03/11.61 Proof:
% 87.03/11.61 meet(a, join(b, c))
% 87.03/11.61 = { by lemma 16 R->L }
% 87.03/11.61 meet(a, join(b, complement(complement(c))))
% 87.03/11.61 = { by lemma 47 R->L }
% 87.03/11.61 meet(a, complement(meet(complement(c), complement(b))))
% 87.03/11.61 = { by lemma 41 R->L }
% 87.03/11.61 meet(a, complement(meet(a, meet(complement(c), complement(b)))))
% 87.03/11.61 = { by lemma 31 R->L }
% 87.03/11.61 meet(a, complement(meet(complement(c), meet(a, complement(b)))))
% 87.03/11.61 = { by lemma 42 R->L }
% 87.03/11.61 meet(a, complement(meet(complement(c), meet(complement(b), join(b, a)))))
% 87.03/11.61 = { by lemma 40 }
% 87.03/11.61 meet(a, complement(meet(complement(c), meet(a, complement(meet(b, a))))))
% 87.03/11.61 = { by lemma 31 }
% 87.03/11.61 meet(a, complement(meet(a, meet(complement(c), complement(meet(b, a))))))
% 87.03/11.61 = { by lemma 41 }
% 87.03/11.61 meet(a, complement(meet(complement(c), complement(meet(b, a)))))
% 87.03/11.61 = { by lemma 47 }
% 87.03/11.61 meet(a, join(meet(b, a), complement(complement(c))))
% 87.03/11.61 = { by axiom 2 (commutativity_of_join) }
% 87.03/11.61 meet(a, join(complement(complement(c)), meet(b, a)))
% 87.03/11.61 = { by axiom 4 (commutativity_of_meet) R->L }
% 87.03/11.61 meet(a, join(complement(complement(c)), meet(a, b)))
% 87.03/11.61 = { by lemma 37 }
% 87.03/11.61 join(meet(complement(complement(c)), a), meet(a, b))
% 87.03/11.61 = { by axiom 2 (commutativity_of_join) }
% 87.03/11.61 join(meet(a, b), meet(complement(complement(c)), a))
% 87.03/11.61 = { by axiom 4 (commutativity_of_meet) }
% 87.03/11.61 join(meet(a, b), meet(a, complement(complement(c))))
% 87.03/11.61 = { by lemma 16 }
% 87.03/11.61 join(meet(a, b), meet(a, c))
% 87.03/11.61 % SZS output end Proof
% 87.03/11.61
% 87.03/11.61 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------