TSTP Solution File: LAT190-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT190-1 : 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 06:27:42 EDT 2023
% Result : Unsatisfiable 67.44s 9.08s
% Output : Proof 69.91s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LAT190-1 : 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
% 0.14/0.35 % Computer : n028.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Thu Aug 24 05:22:34 EDT 2023
% 0.14/0.35 % CPUTime :
% 67.44/9.08 Command-line arguments: --no-flatten-goal
% 67.44/9.08
% 67.44/9.08 % SZS status Unsatisfiable
% 67.44/9.08
% 68.91/9.32 % SZS output start Proof
% 68.91/9.32 Take the following subset of the input axioms:
% 68.91/9.32 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 68.91/9.32 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 68.91/9.32 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 68.91/9.32 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 68.91/9.32 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 68.91/9.32 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 68.91/9.32 fof(complement_join, axiom, ![X2]: join(X2, complement(X2))=one).
% 68.91/9.32 fof(complement_meet, axiom, ![X2]: meet(X2, complement(X2))=zero).
% 68.91/9.32 fof(equation_H18, axiom, ![X2, Y2, Z2]: join(meet(X2, Y2), meet(X2, Z2))=meet(X2, join(meet(X2, Y2), join(meet(X2, Z2), meet(Y2, join(X2, Z2)))))).
% 68.91/9.32 fof(idempotence_of_join, axiom, ![X2]: join(X2, X2)=X2).
% 68.91/9.32 fof(idempotence_of_meet, axiom, ![X2]: meet(X2, X2)=X2).
% 68.91/9.32 fof(meet_join_complement, axiom, ![X2, Y2]: (meet(X2, Y2)!=zero | (join(X2, Y2)!=one | complement(X2)=Y2))).
% 68.91/9.32 fof(prove_distributivity, negated_conjecture, meet(a, join(b, c))!=join(meet(a, b), meet(a, c))).
% 68.91/9.32
% 68.91/9.32 Now clausify the problem and encode Horn clauses using encoding 3 of
% 68.91/9.32 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 68.91/9.32 We repeatedly replace C & s=t => u=v by the two clauses:
% 68.91/9.32 fresh(y, y, x1...xn) = u
% 68.91/9.32 C => fresh(s, t, x1...xn) = v
% 68.91/9.32 where fresh is a fresh function symbol and x1..xn are the free
% 68.91/9.32 variables of u and v.
% 68.91/9.32 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 68.91/9.32 input problem has no model of domain size 1).
% 68.91/9.32
% 68.91/9.32 The encoding turns the above axioms into the following unit equations and goals:
% 68.91/9.32
% 68.91/9.32 Axiom 1 (idempotence_of_join): join(X, X) = X.
% 68.91/9.32 Axiom 2 (commutativity_of_join): join(X, Y) = join(Y, X).
% 68.91/9.32 Axiom 3 (idempotence_of_meet): meet(X, X) = X.
% 68.91/9.32 Axiom 4 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 68.91/9.32 Axiom 5 (complement_join): join(X, complement(X)) = one.
% 68.91/9.32 Axiom 6 (complement_meet): meet(X, complement(X)) = zero.
% 68.91/9.32 Axiom 7 (meet_join_complement): fresh(X, X, Y, Z) = Z.
% 68.91/9.32 Axiom 8 (meet_join_complement): fresh2(X, X, Y, Z) = complement(Y).
% 68.91/9.32 Axiom 9 (absorption2): join(X, meet(X, Y)) = X.
% 68.91/9.32 Axiom 10 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 68.91/9.32 Axiom 11 (absorption1): meet(X, join(X, Y)) = X.
% 68.91/9.32 Axiom 12 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 68.91/9.32 Axiom 13 (meet_join_complement): fresh2(join(X, Y), one, X, Y) = fresh(meet(X, Y), zero, X, Y).
% 68.91/9.32 Axiom 14 (equation_H18): join(meet(X, Y), meet(X, Z)) = meet(X, join(meet(X, Y), join(meet(X, Z), meet(Y, join(X, Z))))).
% 68.91/9.32
% 68.91/9.32 Lemma 15: fresh2(join(X, Y), one, Y, X) = fresh(meet(X, Y), zero, Y, X).
% 68.91/9.32 Proof:
% 68.91/9.32 fresh2(join(X, Y), one, Y, X)
% 68.91/9.32 = { by axiom 2 (commutativity_of_join) R->L }
% 68.91/9.32 fresh2(join(Y, X), one, Y, X)
% 68.91/9.32 = { by axiom 13 (meet_join_complement) }
% 68.91/9.32 fresh(meet(Y, X), zero, Y, X)
% 68.91/9.32 = { by axiom 4 (commutativity_of_meet) }
% 68.91/9.32 fresh(meet(X, Y), zero, Y, X)
% 68.91/9.32
% 68.91/9.32 Lemma 16: complement(complement(X)) = X.
% 68.91/9.32 Proof:
% 68.91/9.32 complement(complement(X))
% 68.91/9.32 = { by axiom 8 (meet_join_complement) R->L }
% 68.91/9.32 fresh2(one, one, complement(X), X)
% 68.91/9.32 = { by axiom 5 (complement_join) R->L }
% 68.91/9.32 fresh2(join(X, complement(X)), one, complement(X), X)
% 68.91/9.32 = { by lemma 15 }
% 68.91/9.32 fresh(meet(X, complement(X)), zero, complement(X), X)
% 68.91/9.32 = { by axiom 6 (complement_meet) }
% 68.91/9.32 fresh(zero, zero, complement(X), X)
% 68.91/9.32 = { by axiom 7 (meet_join_complement) }
% 68.91/9.32 X
% 68.91/9.32
% 68.91/9.32 Lemma 17: meet(X, one) = X.
% 68.91/9.32 Proof:
% 68.91/9.32 meet(X, one)
% 68.91/9.32 = { by axiom 5 (complement_join) R->L }
% 68.91/9.32 meet(X, join(X, complement(X)))
% 68.91/9.32 = { by axiom 11 (absorption1) }
% 68.91/9.32 X
% 68.91/9.32
% 68.91/9.32 Lemma 18: join(X, zero) = X.
% 68.91/9.32 Proof:
% 68.91/9.32 join(X, zero)
% 68.91/9.32 = { by axiom 6 (complement_meet) R->L }
% 68.91/9.32 join(X, meet(X, complement(X)))
% 68.91/9.32 = { by axiom 9 (absorption2) }
% 68.91/9.32 X
% 68.91/9.32
% 68.91/9.32 Lemma 19: join(zero, X) = X.
% 68.91/9.32 Proof:
% 68.91/9.32 join(zero, X)
% 68.91/9.32 = { by axiom 2 (commutativity_of_join) R->L }
% 68.91/9.32 join(X, zero)
% 68.91/9.32 = { by lemma 18 }
% 68.91/9.32 X
% 68.91/9.32
% 68.91/9.32 Lemma 20: meet(X, zero) = zero.
% 68.91/9.32 Proof:
% 68.91/9.32 meet(X, zero)
% 68.91/9.32 = { by axiom 4 (commutativity_of_meet) R->L }
% 68.91/9.32 meet(zero, X)
% 68.91/9.32 = { by lemma 19 R->L }
% 68.91/9.32 join(zero, meet(zero, X))
% 68.91/9.32 = { by axiom 9 (absorption2) }
% 68.91/9.32 zero
% 68.91/9.32
% 68.91/9.32 Lemma 21: meet(one, X) = X.
% 68.91/9.32 Proof:
% 68.91/9.32 meet(one, X)
% 68.91/9.32 = { by axiom 4 (commutativity_of_meet) R->L }
% 68.91/9.32 meet(X, one)
% 68.91/9.32 = { by lemma 17 }
% 68.91/9.32 X
% 68.91/9.32
% 68.91/9.32 Lemma 22: join(X, one) = one.
% 68.91/9.32 Proof:
% 68.91/9.32 join(X, one)
% 68.91/9.32 = { by axiom 2 (commutativity_of_join) R->L }
% 68.91/9.32 join(one, X)
% 68.91/9.32 = { by lemma 21 R->L }
% 68.91/9.32 join(one, meet(one, X))
% 68.91/9.32 = { by axiom 9 (absorption2) }
% 68.91/9.32 one
% 69.74/9.32
% 69.74/9.32 Lemma 23: join(one, X) = one.
% 69.74/9.32 Proof:
% 69.74/9.32 join(one, X)
% 69.74/9.32 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.32 join(X, one)
% 69.74/9.32 = { by lemma 22 }
% 69.74/9.32 one
% 69.74/9.32
% 69.74/9.32 Lemma 24: join(X, meet(Y, X)) = X.
% 69.74/9.32 Proof:
% 69.74/9.32 join(X, meet(Y, X))
% 69.74/9.32 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.74/9.32 join(X, meet(X, Y))
% 69.74/9.32 = { by axiom 9 (absorption2) }
% 69.74/9.32 X
% 69.74/9.32
% 69.74/9.32 Lemma 25: join(X, join(meet(Y, X), Z)) = join(X, Z).
% 69.74/9.32 Proof:
% 69.74/9.32 join(X, join(meet(Y, X), Z))
% 69.74/9.32 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.32 join(X, join(Z, meet(Y, X)))
% 69.74/9.32 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.32 join(join(Z, meet(Y, X)), X)
% 69.74/9.32 = { by axiom 10 (associativity_of_join) }
% 69.74/9.32 join(Z, join(meet(Y, X), X))
% 69.74/9.32 = { by axiom 2 (commutativity_of_join) }
% 69.74/9.32 join(Z, join(X, meet(Y, X)))
% 69.74/9.32 = { by lemma 24 }
% 69.74/9.32 join(Z, X)
% 69.74/9.32 = { by axiom 2 (commutativity_of_join) }
% 69.74/9.32 join(X, Z)
% 69.74/9.32
% 69.74/9.32 Lemma 26: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 69.74/9.32 Proof:
% 69.74/9.32 join(X, join(Y, meet(X, Z)))
% 69.74/9.32 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.32 join(X, join(meet(X, Z), Y))
% 69.74/9.32 = { by axiom 10 (associativity_of_join) R->L }
% 69.74/9.32 join(join(X, meet(X, Z)), Y)
% 69.74/9.32 = { by axiom 9 (absorption2) }
% 69.74/9.32 join(X, Y)
% 69.74/9.32
% 69.74/9.32 Lemma 27: join(X, join(meet(X, Y), Z)) = join(X, Z).
% 69.74/9.32 Proof:
% 69.74/9.32 join(X, join(meet(X, Y), Z))
% 69.74/9.32 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.32 join(X, join(Z, meet(X, Y)))
% 69.74/9.32 = { by lemma 26 }
% 69.74/9.32 join(X, Z)
% 69.74/9.32
% 69.74/9.32 Lemma 28: join(X, complement(meet(X, Y))) = one.
% 69.74/9.32 Proof:
% 69.74/9.32 join(X, complement(meet(X, Y)))
% 69.74/9.32 = { by lemma 27 R->L }
% 69.74/9.32 join(X, join(meet(X, Y), complement(meet(X, Y))))
% 69.74/9.32 = { by axiom 5 (complement_join) }
% 69.74/9.32 join(X, one)
% 69.74/9.32 = { by lemma 22 }
% 69.74/9.32 one
% 69.74/9.32
% 69.74/9.32 Lemma 29: meet(X, meet(Y, complement(meet(X, Y)))) = zero.
% 69.74/9.32 Proof:
% 69.74/9.32 meet(X, meet(Y, complement(meet(X, Y))))
% 69.74/9.32 = { by axiom 12 (associativity_of_meet) R->L }
% 69.74/9.32 meet(meet(X, Y), complement(meet(X, Y)))
% 69.74/9.32 = { by axiom 6 (complement_meet) }
% 69.74/9.32 zero
% 69.74/9.32
% 69.74/9.32 Lemma 30: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 69.74/9.32 Proof:
% 69.74/9.33 meet(X, meet(Y, join(X, Z)))
% 69.74/9.33 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.74/9.33 meet(X, meet(join(X, Z), Y))
% 69.74/9.33 = { by axiom 12 (associativity_of_meet) R->L }
% 69.74/9.33 meet(meet(X, join(X, Z)), Y)
% 69.74/9.33 = { by axiom 11 (absorption1) }
% 69.74/9.33 meet(X, Y)
% 69.74/9.33
% 69.74/9.33 Lemma 31: join(meet(X, Y), meet(Y, join(X, Z))) = meet(Y, join(X, Z)).
% 69.74/9.33 Proof:
% 69.74/9.33 join(meet(X, Y), meet(Y, join(X, Z)))
% 69.74/9.33 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.33 join(meet(Y, join(X, Z)), meet(X, Y))
% 69.74/9.33 = { by lemma 30 R->L }
% 69.74/9.33 join(meet(Y, join(X, Z)), meet(X, meet(Y, join(X, Z))))
% 69.74/9.33 = { by lemma 24 }
% 69.74/9.33 meet(Y, join(X, Z))
% 69.74/9.33
% 69.74/9.33 Lemma 32: meet(X, join(meet(X, Y), meet(Z, join(X, Y)))) = join(meet(X, Z), meet(X, Y)).
% 69.74/9.33 Proof:
% 69.74/9.33 meet(X, join(meet(X, Y), meet(Z, join(X, Y))))
% 69.74/9.33 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.33 meet(X, join(meet(Z, join(X, Y)), meet(X, Y)))
% 69.74/9.33 = { by lemma 31 R->L }
% 69.74/9.33 meet(X, join(join(meet(X, Z), meet(Z, join(X, Y))), meet(X, Y)))
% 69.74/9.33 = { by axiom 10 (associativity_of_join) }
% 69.74/9.33 meet(X, join(meet(X, Z), join(meet(Z, join(X, Y)), meet(X, Y))))
% 69.74/9.33 = { by axiom 2 (commutativity_of_join) }
% 69.74/9.33 meet(X, join(meet(X, Z), join(meet(X, Y), meet(Z, join(X, Y)))))
% 69.74/9.33 = { by axiom 14 (equation_H18) R->L }
% 69.74/9.33 join(meet(X, Z), meet(X, Y))
% 69.74/9.33
% 69.74/9.33 Lemma 33: join(X, join(Y, complement(X))) = one.
% 69.74/9.33 Proof:
% 69.74/9.33 join(X, join(Y, complement(X)))
% 69.74/9.33 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.33 join(X, join(complement(X), Y))
% 69.74/9.33 = { by axiom 10 (associativity_of_join) R->L }
% 69.74/9.33 join(join(X, complement(X)), Y)
% 69.74/9.33 = { by axiom 5 (complement_join) }
% 69.74/9.33 join(one, Y)
% 69.74/9.33 = { by lemma 23 }
% 69.74/9.33 one
% 69.74/9.33
% 69.74/9.33 Lemma 34: meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))) = complement(Y).
% 69.74/9.33 Proof:
% 69.74/9.33 meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))
% 69.74/9.33 = { by axiom 7 (meet_join_complement) R->L }
% 69.74/9.33 fresh(zero, zero, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 69.74/9.33 = { by lemma 29 R->L }
% 69.74/9.33 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))))))
% 69.74/9.33 = { by axiom 13 (meet_join_complement) R->L }
% 69.74/9.33 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))))))
% 69.74/9.33 = { by lemma 27 R->L }
% 69.74/9.33 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))))))
% 69.74/9.33 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.33 fresh2(join(Y, join(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))), meet(Y, join(X, complement(Y))))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 69.74/9.33 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.74/9.33 fresh2(join(Y, join(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))), meet(join(X, complement(Y)), Y))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 69.74/9.33 = { by lemma 32 R->L }
% 69.74/9.33 fresh2(join(Y, meet(join(X, complement(Y)), join(meet(join(X, complement(Y)), Y), meet(complement(meet(Y, join(X, complement(Y)))), join(join(X, complement(Y)), Y))))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 69.74/9.33 = { by axiom 2 (commutativity_of_join) }
% 69.74/9.33 fresh2(join(Y, meet(join(X, complement(Y)), join(meet(join(X, complement(Y)), Y), meet(complement(meet(Y, join(X, complement(Y)))), join(Y, join(X, complement(Y))))))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 69.74/9.33 = { by lemma 33 }
% 69.74/9.33 fresh2(join(Y, meet(join(X, complement(Y)), join(meet(join(X, complement(Y)), Y), meet(complement(meet(Y, join(X, complement(Y)))), one)))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 69.74/9.33 = { by lemma 17 }
% 69.74/9.33 fresh2(join(Y, meet(join(X, complement(Y)), join(meet(join(X, complement(Y)), Y), complement(meet(Y, join(X, complement(Y))))))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 69.74/9.33 = { by axiom 4 (commutativity_of_meet) }
% 69.74/9.33 fresh2(join(Y, meet(join(X, complement(Y)), join(meet(Y, 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))))))
% 69.74/9.33 = { by axiom 5 (complement_join) }
% 69.74/9.33 fresh2(join(Y, meet(join(X, complement(Y)), one)), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 69.74/9.33 = { by lemma 17 }
% 69.74/9.33 fresh2(join(Y, join(X, complement(Y))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 69.74/9.33 = { by lemma 33 }
% 69.74/9.33 fresh2(one, one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 69.74/9.33 = { by axiom 8 (meet_join_complement) }
% 69.74/9.33 complement(Y)
% 69.74/9.33
% 69.74/9.33 Lemma 35: meet(join(X, Y), complement(meet(complement(X), join(X, Y)))) = X.
% 69.74/9.33 Proof:
% 69.74/9.33 meet(join(X, Y), complement(meet(complement(X), join(X, Y))))
% 69.74/9.33 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.33 meet(join(Y, X), complement(meet(complement(X), join(X, Y))))
% 69.74/9.33 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.33 meet(join(Y, X), complement(meet(complement(X), join(Y, X))))
% 69.74/9.33 = { by lemma 16 R->L }
% 69.74/9.33 meet(join(Y, X), complement(meet(complement(X), join(Y, complement(complement(X))))))
% 69.74/9.33 = { by lemma 16 R->L }
% 69.74/9.33 meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))
% 69.74/9.33 = { by lemma 34 }
% 69.74/9.33 complement(complement(X))
% 69.74/9.33 = { by lemma 16 }
% 69.74/9.33 X
% 69.74/9.33
% 69.74/9.33 Lemma 36: meet(join(X, Y), join(X, Z)) = join(X, meet(Z, join(X, Y))).
% 69.74/9.33 Proof:
% 69.74/9.33 meet(join(X, Y), join(X, Z))
% 69.74/9.33 = { by lemma 17 R->L }
% 69.74/9.33 meet(join(X, Y), join(X, meet(Z, one)))
% 69.74/9.33 = { by lemma 28 R->L }
% 69.74/9.33 meet(join(X, Y), join(X, meet(Z, join(join(X, Y), complement(meet(join(X, Y), complement(X)))))))
% 69.74/9.33 = { by axiom 10 (associativity_of_join) }
% 69.74/9.33 meet(join(X, Y), join(X, meet(Z, join(X, join(Y, complement(meet(join(X, Y), complement(X))))))))
% 69.74/9.33 = { by axiom 4 (commutativity_of_meet) }
% 69.74/9.33 meet(join(X, Y), join(X, meet(Z, join(X, join(Y, complement(meet(complement(X), join(X, Y))))))))
% 69.74/9.33 = { by axiom 10 (associativity_of_join) R->L }
% 69.74/9.33 meet(join(X, Y), join(X, meet(Z, join(join(X, Y), complement(meet(complement(X), join(X, Y)))))))
% 69.74/9.33 = { by lemma 35 R->L }
% 69.74/9.33 meet(join(X, Y), join(meet(join(X, Y), complement(meet(complement(X), join(X, Y)))), meet(Z, join(join(X, Y), complement(meet(complement(X), join(X, Y)))))))
% 69.74/9.33 = { by lemma 32 }
% 69.74/9.33 join(meet(join(X, Y), Z), meet(join(X, Y), complement(meet(complement(X), join(X, Y)))))
% 69.74/9.33 = { by lemma 35 }
% 69.74/9.33 join(meet(join(X, Y), Z), X)
% 69.74/9.33 = { by axiom 2 (commutativity_of_join) }
% 69.74/9.33 join(X, meet(join(X, Y), Z))
% 69.74/9.33 = { by axiom 4 (commutativity_of_meet) }
% 69.74/9.33 join(X, meet(Z, join(X, Y)))
% 69.74/9.33
% 69.74/9.33 Lemma 37: meet(join(X, Y), join(Y, Z)) = join(Y, meet(Z, join(X, Y))).
% 69.74/9.33 Proof:
% 69.74/9.33 meet(join(X, Y), join(Y, Z))
% 69.74/9.33 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.33 meet(join(Y, X), join(Y, Z))
% 69.74/9.33 = { by lemma 36 }
% 69.74/9.33 join(Y, meet(Z, join(Y, X)))
% 69.74/9.33 = { by axiom 2 (commutativity_of_join) }
% 69.74/9.33 join(Y, meet(Z, join(X, Y)))
% 69.74/9.33
% 69.74/9.33 Lemma 38: meet(X, join(Y, meet(X, Z))) = join(meet(X, Z), meet(X, Y)).
% 69.74/9.33 Proof:
% 69.74/9.33 meet(X, join(Y, meet(X, Z)))
% 69.74/9.33 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.33 meet(X, join(meet(X, Z), Y))
% 69.74/9.33 = { by axiom 9 (absorption2) R->L }
% 69.74/9.33 meet(join(X, meet(X, Z)), join(meet(X, Z), Y))
% 69.74/9.33 = { by lemma 37 }
% 69.74/9.33 join(meet(X, Z), meet(Y, join(X, meet(X, Z))))
% 69.74/9.33 = { by axiom 9 (absorption2) }
% 69.74/9.33 join(meet(X, Z), meet(Y, X))
% 69.74/9.33 = { by axiom 4 (commutativity_of_meet) }
% 69.74/9.33 join(meet(X, Z), meet(X, Y))
% 69.74/9.33
% 69.74/9.33 Lemma 39: meet(X, meet(Y, join(Z, meet(X, Y)))) = meet(X, Y).
% 69.74/9.33 Proof:
% 69.74/9.33 meet(X, meet(Y, join(Z, meet(X, Y))))
% 69.74/9.33 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.33 meet(X, meet(Y, join(meet(X, Y), Z)))
% 69.74/9.33 = { by axiom 12 (associativity_of_meet) R->L }
% 69.74/9.33 meet(meet(X, Y), join(meet(X, Y), Z))
% 69.74/9.33 = { by axiom 11 (absorption1) }
% 69.74/9.33 meet(X, Y)
% 69.74/9.33
% 69.74/9.33 Lemma 40: join(meet(X, Y), meet(X, join(Z, meet(X, Y)))) = meet(X, join(Z, meet(X, Y))).
% 69.74/9.33 Proof:
% 69.74/9.33 join(meet(X, Y), meet(X, join(Z, meet(X, Y))))
% 69.74/9.33 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.74/9.33 join(meet(Y, X), meet(X, join(Z, meet(X, Y))))
% 69.74/9.33 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.74/9.34 join(meet(Y, X), meet(X, join(Z, meet(Y, X))))
% 69.74/9.34 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.34 join(meet(X, join(Z, meet(Y, X))), meet(Y, X))
% 69.74/9.34 = { by lemma 39 R->L }
% 69.74/9.34 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(Z, meet(Y, X)))))
% 69.74/9.34 = { by lemma 24 }
% 69.74/9.34 meet(X, join(Z, meet(Y, X)))
% 69.74/9.34 = { by axiom 4 (commutativity_of_meet) }
% 69.74/9.34 meet(X, join(Z, meet(X, Y)))
% 69.74/9.34
% 69.74/9.34 Lemma 41: join(meet(X, Y), meet(X, complement(meet(X, Y)))) = X.
% 69.74/9.34 Proof:
% 69.74/9.34 join(meet(X, Y), meet(X, complement(meet(X, Y))))
% 69.74/9.34 = { by lemma 38 R->L }
% 69.74/9.34 meet(X, join(complement(meet(X, Y)), meet(X, Y)))
% 69.74/9.34 = { by lemma 40 R->L }
% 69.74/9.34 join(meet(X, Y), meet(X, join(complement(meet(X, Y)), meet(X, Y))))
% 69.74/9.34 = { by axiom 2 (commutativity_of_join) }
% 69.74/9.34 join(meet(X, Y), meet(X, join(meet(X, Y), complement(meet(X, Y)))))
% 69.74/9.34 = { by axiom 5 (complement_join) }
% 69.74/9.34 join(meet(X, Y), meet(X, one))
% 69.74/9.34 = { by lemma 17 }
% 69.74/9.34 join(meet(X, Y), X)
% 69.74/9.34 = { by axiom 2 (commutativity_of_join) }
% 69.74/9.34 join(X, meet(X, Y))
% 69.74/9.34 = { by axiom 9 (absorption2) }
% 69.74/9.34 X
% 69.74/9.34
% 69.74/9.34 Lemma 42: join(X, meet(Y, complement(meet(X, Y)))) = join(X, Y).
% 69.74/9.34 Proof:
% 69.74/9.34 join(X, meet(Y, complement(meet(X, Y))))
% 69.74/9.34 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.74/9.34 join(X, meet(Y, complement(meet(Y, X))))
% 69.74/9.34 = { by lemma 25 R->L }
% 69.74/9.34 join(X, join(meet(Y, X), meet(Y, complement(meet(Y, X)))))
% 69.74/9.34 = { by lemma 41 }
% 69.74/9.34 join(X, Y)
% 69.74/9.34
% 69.74/9.34 Lemma 43: meet(X, join(Y, X)) = X.
% 69.74/9.34 Proof:
% 69.74/9.34 meet(X, join(Y, X))
% 69.74/9.34 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.34 meet(X, join(X, Y))
% 69.74/9.34 = { by axiom 11 (absorption1) }
% 69.74/9.34 X
% 69.74/9.34
% 69.74/9.34 Lemma 44: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 69.74/9.34 Proof:
% 69.74/9.34 join(X, join(Y, meet(Z, join(X, Y))))
% 69.74/9.34 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.74/9.34 join(X, join(Y, meet(join(X, Y), Z)))
% 69.74/9.34 = { by axiom 10 (associativity_of_join) R->L }
% 69.74/9.34 join(join(X, Y), meet(join(X, Y), Z))
% 69.74/9.34 = { by axiom 9 (absorption2) }
% 69.74/9.34 join(X, Y)
% 69.74/9.34
% 69.74/9.34 Lemma 45: meet(X, join(Y, meet(X, join(Y, Z)))) = meet(X, join(Y, Z)).
% 69.74/9.34 Proof:
% 69.74/9.34 meet(X, join(Y, meet(X, join(Y, Z))))
% 69.74/9.34 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.34 meet(X, join(Y, meet(X, join(Z, Y))))
% 69.74/9.34 = { by lemma 43 R->L }
% 69.74/9.34 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, join(Y, meet(X, join(Z, Y))))))
% 69.74/9.34 = { by lemma 44 }
% 69.74/9.34 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, Y)))
% 69.74/9.34 = { by axiom 4 (commutativity_of_meet) }
% 69.74/9.34 meet(X, meet(join(Z, Y), join(Y, meet(X, join(Z, Y)))))
% 69.74/9.34 = { by axiom 2 (commutativity_of_join) }
% 69.74/9.34 meet(X, meet(join(Z, Y), join(Y, meet(X, join(Y, Z)))))
% 69.74/9.34 = { by axiom 2 (commutativity_of_join) }
% 69.74/9.34 meet(X, meet(join(Y, Z), join(Y, meet(X, join(Y, Z)))))
% 69.74/9.34 = { by lemma 39 }
% 69.74/9.34 meet(X, join(Y, Z))
% 69.74/9.34
% 69.74/9.34 Lemma 46: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
% 69.74/9.34 Proof:
% 69.74/9.34 meet(X, meet(Y, Z))
% 69.74/9.34 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.74/9.34 meet(meet(Y, Z), X)
% 69.74/9.34 = { by axiom 12 (associativity_of_meet) }
% 69.74/9.34 meet(Y, meet(Z, X))
% 69.74/9.34 = { by axiom 4 (commutativity_of_meet) }
% 69.74/9.34 meet(Y, meet(X, Z))
% 69.74/9.34
% 69.74/9.34 Lemma 47: meet(X, meet(join(X, Y), Z)) = meet(X, Z).
% 69.74/9.34 Proof:
% 69.74/9.34 meet(X, meet(join(X, Y), Z))
% 69.74/9.34 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.74/9.34 meet(X, meet(Z, join(X, Y)))
% 69.74/9.34 = { by lemma 30 }
% 69.74/9.34 meet(X, Z)
% 69.74/9.34
% 69.74/9.34 Lemma 48: join(meet(X, join(Y, Z)), meet(X, meet(Y, W))) = meet(X, join(Y, Z)).
% 69.74/9.34 Proof:
% 69.74/9.34 join(meet(X, join(Y, Z)), meet(X, meet(Y, W)))
% 69.74/9.34 = { by lemma 47 R->L }
% 69.74/9.34 join(meet(X, join(Y, Z)), meet(X, meet(Y, meet(join(Y, Z), W))))
% 69.74/9.34 = { by lemma 46 R->L }
% 69.74/9.34 join(meet(X, join(Y, Z)), meet(X, meet(join(Y, Z), meet(Y, W))))
% 69.74/9.34 = { by axiom 12 (associativity_of_meet) R->L }
% 69.74/9.34 join(meet(X, join(Y, Z)), meet(meet(X, join(Y, Z)), meet(Y, W)))
% 69.74/9.34 = { by axiom 9 (absorption2) }
% 69.74/9.34 meet(X, join(Y, Z))
% 69.74/9.34
% 69.74/9.34 Lemma 49: join(meet(X, meet(Y, Z)), meet(Z, join(X, W))) = meet(Z, join(X, W)).
% 69.74/9.34 Proof:
% 69.74/9.34 join(meet(X, meet(Y, Z)), meet(Z, join(X, W)))
% 69.74/9.34 = { by axiom 12 (associativity_of_meet) R->L }
% 69.74/9.34 join(meet(meet(X, Y), Z), meet(Z, join(X, W)))
% 69.74/9.34 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.74/9.34 join(meet(Z, meet(X, Y)), meet(Z, join(X, W)))
% 69.74/9.34 = { by axiom 2 (commutativity_of_join) R->L }
% 69.74/9.34 join(meet(Z, join(X, W)), meet(Z, meet(X, Y)))
% 69.74/9.34 = { by lemma 48 }
% 69.74/9.34 meet(Z, join(X, W))
% 69.74/9.34
% 69.74/9.34 Lemma 50: meet(X, join(Y, meet(Z, join(Y, X)))) = meet(X, join(Z, Y)).
% 69.74/9.34 Proof:
% 69.74/9.34 meet(X, join(Y, meet(Z, join(Y, X))))
% 69.74/9.34 = { by lemma 45 R->L }
% 69.74/9.34 meet(X, join(Y, meet(Z, join(Y, meet(Z, join(Y, X))))))
% 69.74/9.34 = { by lemma 36 R->L }
% 69.74/9.34 meet(X, join(Y, meet(Z, meet(join(Y, X), join(Y, Z)))))
% 69.74/9.34 = { by lemma 31 R->L }
% 69.74/9.34 meet(X, join(Y, meet(Z, join(meet(Y, join(Y, X)), meet(join(Y, X), join(Y, Z))))))
% 69.74/9.34 = { by axiom 11 (absorption1) }
% 69.74/9.34 meet(X, join(Y, meet(Z, join(Y, meet(join(Y, X), join(Y, Z))))))
% 69.74/9.34 = { by axiom 2 (commutativity_of_join) }
% 69.74/9.34 meet(X, join(Y, meet(Z, join(Y, meet(join(Y, X), join(Z, Y))))))
% 69.74/9.34 = { by lemma 43 R->L }
% 69.74/9.34 meet(X, join(Y, meet(Z, join(Y, meet(join(Y, X), join(Z, meet(Y, join(Z, Y))))))))
% 69.74/9.34 = { by lemma 36 R->L }
% 69.74/9.34 meet(X, join(Y, meet(Z, join(Y, meet(join(Y, X), meet(join(Z, Y), join(Z, Y)))))))
% 69.74/9.34 = { by axiom 12 (associativity_of_meet) R->L }
% 69.74/9.34 meet(X, join(Y, meet(Z, join(Y, meet(meet(join(Y, X), join(Z, Y)), join(Z, Y))))))
% 69.74/9.34 = { by lemma 37 R->L }
% 69.74/9.34 meet(X, join(Y, meet(Z, meet(join(Z, Y), join(Y, meet(join(Y, X), join(Z, Y)))))))
% 69.74/9.34 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.74/9.34 meet(X, join(Y, meet(Z, meet(join(Z, Y), join(Y, meet(join(Z, Y), join(Y, X)))))))
% 69.74/9.34 = { by lemma 45 }
% 69.74/9.34 meet(X, join(Y, meet(Z, meet(join(Z, Y), join(Y, X)))))
% 69.74/9.34 = { by lemma 37 }
% 69.91/9.34 meet(X, join(Y, meet(Z, join(Y, meet(X, join(Z, Y))))))
% 69.91/9.34 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.34 meet(X, join(Y, meet(Z, join(Y, meet(X, join(Y, Z))))))
% 69.91/9.34 = { by lemma 36 R->L }
% 69.91/9.34 meet(X, meet(join(Y, meet(X, join(Y, Z))), join(Y, Z)))
% 69.91/9.34 = { by lemma 46 R->L }
% 69.91/9.34 meet(join(Y, meet(X, join(Y, Z))), meet(X, join(Y, Z)))
% 69.91/9.34 = { by lemma 49 R->L }
% 69.91/9.34 meet(join(Y, meet(X, join(Y, Z))), join(meet(Y, meet(W, X)), meet(X, join(Y, Z))))
% 69.91/9.34 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.34 meet(join(Y, meet(X, join(Y, Z))), join(meet(X, join(Y, Z)), meet(Y, meet(W, X))))
% 69.91/9.34 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.34 meet(join(meet(X, join(Y, Z)), meet(Y, meet(W, X))), join(Y, meet(X, join(Y, Z))))
% 69.91/9.34 = { by lemma 26 R->L }
% 69.91/9.34 meet(join(meet(X, join(Y, Z)), meet(Y, meet(W, X))), join(Y, join(meet(X, join(Y, Z)), meet(Y, meet(W, X)))))
% 69.91/9.34 = { by lemma 43 }
% 69.91/9.34 join(meet(X, join(Y, Z)), meet(Y, meet(W, X)))
% 69.91/9.34 = { by axiom 12 (associativity_of_meet) R->L }
% 69.91/9.34 join(meet(X, join(Y, Z)), meet(meet(Y, W), X))
% 69.91/9.34 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.34 join(meet(X, join(Y, Z)), meet(X, meet(Y, W)))
% 69.91/9.34 = { by lemma 48 }
% 69.91/9.34 meet(X, join(Y, Z))
% 69.91/9.34 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.34 meet(X, join(Z, Y))
% 69.91/9.34
% 69.91/9.34 Lemma 51: meet(X, join(Y, complement(join(X, Y)))) = meet(X, Y).
% 69.91/9.34 Proof:
% 69.91/9.34 meet(X, join(Y, complement(join(X, Y))))
% 69.91/9.34 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.34 meet(X, join(Y, complement(join(Y, X))))
% 69.91/9.34 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.34 meet(X, join(complement(join(Y, X)), Y))
% 69.91/9.34 = { by lemma 50 R->L }
% 69.91/9.34 meet(X, join(Y, meet(complement(join(Y, X)), join(Y, X))))
% 69.91/9.34 = { by axiom 4 (commutativity_of_meet) }
% 69.91/9.34 meet(X, join(Y, meet(join(Y, X), complement(join(Y, X)))))
% 69.91/9.34 = { by axiom 6 (complement_meet) }
% 69.91/9.34 meet(X, join(Y, zero))
% 69.91/9.34 = { by lemma 18 }
% 69.91/9.34 meet(X, Y)
% 69.91/9.34
% 69.91/9.34 Lemma 52: join(X, join(Y, complement(join(X, Y)))) = one.
% 69.91/9.34 Proof:
% 69.91/9.34 join(X, join(Y, complement(join(X, Y))))
% 69.91/9.34 = { by axiom 10 (associativity_of_join) R->L }
% 69.91/9.34 join(join(X, Y), complement(join(X, Y)))
% 69.91/9.35 = { by axiom 5 (complement_join) }
% 69.91/9.35 one
% 69.91/9.35
% 69.91/9.35 Lemma 53: fresh(meet(X, Y), zero, join(X, complement(join(X, Y))), Y) = complement(join(X, complement(join(X, Y)))).
% 69.91/9.35 Proof:
% 69.91/9.35 fresh(meet(X, Y), zero, join(X, complement(join(X, Y))), Y)
% 69.91/9.35 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.35 fresh(meet(X, Y), zero, join(X, complement(join(Y, X))), Y)
% 69.91/9.35 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.35 fresh(meet(Y, X), zero, join(X, complement(join(Y, X))), Y)
% 69.91/9.35 = { by lemma 51 R->L }
% 69.91/9.35 fresh(meet(Y, join(X, complement(join(Y, X)))), zero, join(X, complement(join(Y, X))), Y)
% 69.91/9.35 = { by lemma 15 R->L }
% 69.91/9.35 fresh2(join(Y, join(X, complement(join(Y, X)))), one, join(X, complement(join(Y, X))), Y)
% 69.91/9.35 = { by lemma 52 }
% 69.91/9.35 fresh2(one, one, join(X, complement(join(Y, X))), Y)
% 69.91/9.35 = { by axiom 8 (meet_join_complement) }
% 69.91/9.35 complement(join(X, complement(join(Y, X))))
% 69.91/9.35 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.35 complement(join(X, complement(join(X, Y))))
% 69.91/9.35
% 69.91/9.35 Lemma 54: complement(join(X, complement(join(X, Y)))) = meet(Y, complement(meet(X, Y))).
% 69.91/9.35 Proof:
% 69.91/9.35 complement(join(X, complement(join(X, Y))))
% 69.91/9.35 = { by lemma 42 R->L }
% 69.91/9.35 complement(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))))
% 69.91/9.35 = { by lemma 53 R->L }
% 69.91/9.35 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))))
% 69.91/9.35 = { by lemma 42 }
% 69.91/9.35 fresh(meet(X, meet(Y, complement(meet(X, Y)))), zero, join(X, complement(join(X, Y))), meet(Y, complement(meet(X, Y))))
% 69.91/9.35 = { by lemma 29 }
% 69.91/9.35 fresh(zero, zero, join(X, complement(join(X, Y))), meet(Y, complement(meet(X, Y))))
% 69.91/9.35 = { by axiom 7 (meet_join_complement) }
% 69.91/9.35 meet(Y, complement(meet(X, Y)))
% 69.91/9.35
% 69.91/9.35 Lemma 55: fresh(meet(X, Y), zero, X, join(Y, complement(join(X, Y)))) = complement(X).
% 69.91/9.35 Proof:
% 69.91/9.35 fresh(meet(X, Y), zero, X, join(Y, complement(join(X, Y))))
% 69.91/9.35 = { by lemma 51 R->L }
% 69.91/9.35 fresh(meet(X, join(Y, complement(join(X, Y)))), zero, X, join(Y, complement(join(X, Y))))
% 69.91/9.35 = { by axiom 13 (meet_join_complement) R->L }
% 69.91/9.35 fresh2(join(X, join(Y, complement(join(X, Y)))), one, X, join(Y, complement(join(X, Y))))
% 69.91/9.35 = { by lemma 52 }
% 69.91/9.35 fresh2(one, one, X, join(Y, complement(join(X, Y))))
% 69.91/9.35 = { by axiom 8 (meet_join_complement) }
% 69.91/9.35 complement(X)
% 69.91/9.35
% 69.91/9.35 Lemma 56: join(complement(join(X, Y)), meet(Y, complement(meet(X, Y)))) = complement(X).
% 69.91/9.35 Proof:
% 69.91/9.35 join(complement(join(X, Y)), meet(Y, complement(meet(X, Y))))
% 69.91/9.35 = { by lemma 54 R->L }
% 69.91/9.35 join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))
% 69.91/9.35 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.35 join(complement(join(Y, X)), complement(join(X, complement(join(X, Y)))))
% 69.91/9.35 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.35 join(complement(join(Y, X)), complement(join(X, complement(join(Y, X)))))
% 69.91/9.35 = { by axiom 7 (meet_join_complement) R->L }
% 69.91/9.35 fresh(zero, zero, X, join(complement(join(Y, X)), complement(join(X, complement(join(Y, X))))))
% 69.91/9.35 = { by lemma 20 R->L }
% 69.91/9.35 fresh(meet(X, zero), zero, X, join(complement(join(Y, X)), complement(join(X, complement(join(Y, X))))))
% 69.91/9.35 = { by axiom 6 (complement_meet) R->L }
% 69.91/9.35 fresh(meet(X, meet(join(X, Y), complement(join(X, Y)))), zero, X, join(complement(join(Y, X)), complement(join(X, complement(join(Y, X))))))
% 69.91/9.35 = { by lemma 47 }
% 69.91/9.35 fresh(meet(X, complement(join(X, Y))), zero, X, join(complement(join(Y, X)), complement(join(X, complement(join(Y, X))))))
% 69.91/9.35 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.35 fresh(meet(X, complement(join(Y, X))), zero, X, join(complement(join(Y, X)), complement(join(X, complement(join(Y, X))))))
% 69.91/9.35 = { by lemma 55 }
% 69.91/9.35 complement(X)
% 69.91/9.35
% 69.91/9.35 Lemma 57: join(X, complement(join(X, Y))) = join(X, complement(Y)).
% 69.91/9.35 Proof:
% 69.91/9.35 join(X, complement(join(X, Y)))
% 69.91/9.35 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.35 join(X, complement(join(Y, X)))
% 69.91/9.35 = { by lemma 26 R->L }
% 69.91/9.35 join(X, join(complement(join(Y, X)), meet(X, complement(meet(Y, X)))))
% 69.91/9.35 = { by lemma 56 }
% 69.91/9.35 join(X, complement(Y))
% 69.91/9.35
% 69.91/9.35 Lemma 58: meet(X, complement(meet(Y, X))) = complement(join(Y, complement(X))).
% 69.91/9.35 Proof:
% 69.91/9.35 meet(X, complement(meet(Y, X)))
% 69.91/9.35 = { by lemma 54 R->L }
% 69.91/9.35 complement(join(Y, complement(join(Y, X))))
% 69.91/9.35 = { by lemma 57 }
% 69.91/9.35 complement(join(Y, complement(X)))
% 69.91/9.35
% 69.91/9.35 Lemma 59: meet(X, meet(Y, complement(X))) = zero.
% 69.91/9.35 Proof:
% 69.91/9.35 meet(X, meet(Y, complement(X)))
% 69.91/9.35 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.35 meet(X, meet(complement(X), Y))
% 69.91/9.35 = { by axiom 12 (associativity_of_meet) R->L }
% 69.91/9.35 meet(meet(X, complement(X)), Y)
% 69.91/9.35 = { by axiom 6 (complement_meet) }
% 69.91/9.35 meet(zero, Y)
% 69.91/9.35 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.35 meet(Y, zero)
% 69.91/9.35 = { by lemma 20 }
% 69.91/9.35 zero
% 69.91/9.35
% 69.91/9.35 Lemma 60: join(complement(X), complement(meet(X, Y))) = complement(meet(X, Y)).
% 69.91/9.35 Proof:
% 69.91/9.35 join(complement(X), complement(meet(X, Y)))
% 69.91/9.35 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.35 join(complement(meet(X, Y)), complement(X))
% 69.91/9.35 = { by lemma 41 R->L }
% 69.91/9.35 join(complement(meet(X, Y)), complement(join(meet(X, Y), meet(X, complement(meet(X, Y))))))
% 69.91/9.35 = { by lemma 25 R->L }
% 69.91/9.35 join(complement(meet(X, Y)), join(meet(X, complement(meet(X, Y))), complement(join(meet(X, Y), meet(X, complement(meet(X, Y)))))))
% 69.91/9.35 = { by axiom 7 (meet_join_complement) R->L }
% 69.91/9.35 join(complement(meet(X, Y)), fresh(zero, zero, meet(X, Y), join(meet(X, complement(meet(X, Y))), complement(join(meet(X, Y), meet(X, complement(meet(X, Y))))))))
% 69.91/9.35 = { by lemma 59 R->L }
% 69.91/9.35 join(complement(meet(X, Y)), fresh(meet(meet(X, Y), meet(X, complement(meet(X, Y)))), zero, meet(X, Y), join(meet(X, complement(meet(X, Y))), complement(join(meet(X, Y), meet(X, complement(meet(X, Y))))))))
% 69.91/9.35 = { by lemma 55 }
% 69.91/9.35 join(complement(meet(X, Y)), complement(meet(X, Y)))
% 69.91/9.35 = { by axiom 1 (idempotence_of_join) }
% 69.91/9.35 complement(meet(X, Y))
% 69.91/9.35
% 69.91/9.35 Lemma 61: meet(complement(X), complement(join(X, Y))) = complement(join(X, Y)).
% 69.91/9.35 Proof:
% 69.91/9.35 meet(complement(X), complement(join(X, Y)))
% 69.91/9.35 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.35 meet(complement(join(X, Y)), complement(X))
% 69.91/9.35 = { by lemma 56 R->L }
% 69.91/9.35 meet(complement(join(X, Y)), join(complement(join(X, Y)), meet(Y, complement(meet(X, Y)))))
% 69.91/9.35 = { by axiom 11 (absorption1) }
% 69.91/9.35 complement(join(X, Y))
% 69.91/9.35
% 69.91/9.35 Lemma 62: meet(X, meet(Y, meet(Z, complement(meet(X, Y))))) = zero.
% 69.91/9.35 Proof:
% 69.91/9.35 meet(X, meet(Y, meet(Z, complement(meet(X, Y)))))
% 69.91/9.35 = { by axiom 12 (associativity_of_meet) R->L }
% 69.91/9.35 meet(meet(X, Y), meet(Z, complement(meet(X, Y))))
% 69.91/9.35 = { by lemma 59 }
% 69.91/9.35 zero
% 69.91/9.35
% 69.91/9.35 Lemma 63: complement(join(complement(X), meet(X, Y))) = meet(X, complement(meet(X, Y))).
% 69.91/9.35 Proof:
% 69.91/9.35 complement(join(complement(X), meet(X, Y)))
% 69.91/9.35 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.35 complement(join(meet(X, Y), complement(X)))
% 69.91/9.35 = { by lemma 41 R->L }
% 69.91/9.35 complement(join(meet(X, Y), complement(join(meet(X, Y), meet(X, complement(meet(X, Y)))))))
% 69.91/9.35 = { by lemma 53 R->L }
% 69.91/9.35 fresh(meet(meet(X, Y), meet(X, complement(meet(X, Y)))), zero, join(meet(X, Y), complement(join(meet(X, Y), meet(X, complement(meet(X, Y)))))), meet(X, complement(meet(X, Y))))
% 69.91/9.35 = { by lemma 41 }
% 69.91/9.35 fresh(meet(meet(X, Y), meet(X, complement(meet(X, Y)))), zero, join(meet(X, Y), complement(X)), meet(X, complement(meet(X, Y))))
% 69.91/9.35 = { by axiom 12 (associativity_of_meet) }
% 69.91/9.35 fresh(meet(X, meet(Y, meet(X, complement(meet(X, Y))))), zero, join(meet(X, Y), complement(X)), meet(X, complement(meet(X, Y))))
% 69.91/9.35 = { by lemma 62 }
% 69.91/9.35 fresh(zero, zero, join(meet(X, Y), complement(X)), meet(X, complement(meet(X, Y))))
% 69.91/9.35 = { by axiom 7 (meet_join_complement) }
% 69.91/9.35 meet(X, complement(meet(X, Y)))
% 69.91/9.35
% 69.91/9.35 Lemma 64: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 69.91/9.35 Proof:
% 69.91/9.35 complement(join(X, complement(Y)))
% 69.91/9.35 = { by lemma 58 R->L }
% 69.91/9.35 meet(Y, complement(meet(X, Y)))
% 69.91/9.35 = { by lemma 16 R->L }
% 69.91/9.35 meet(Y, complement(meet(X, complement(complement(Y)))))
% 69.91/9.35 = { by lemma 34 R->L }
% 69.91/9.35 meet(Y, complement(meet(X, complement(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))))))
% 69.91/9.35 = { by lemma 60 R->L }
% 69.91/9.35 meet(Y, complement(meet(X, join(complement(join(X, complement(Y))), complement(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))))))
% 69.91/9.35 = { by lemma 34 }
% 69.91/9.35 meet(Y, complement(meet(X, join(complement(join(X, complement(Y))), complement(complement(Y))))))
% 69.91/9.35 = { by lemma 16 }
% 69.91/9.35 meet(Y, complement(meet(X, join(complement(join(X, complement(Y))), Y))))
% 69.91/9.35 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.35 meet(Y, complement(meet(X, join(Y, complement(join(X, complement(Y)))))))
% 69.91/9.35 = { by lemma 57 R->L }
% 69.91/9.35 meet(Y, complement(meet(X, join(Y, complement(join(X, complement(join(X, Y))))))))
% 69.91/9.35 = { by lemma 16 R->L }
% 69.91/9.35 meet(Y, complement(meet(X, join(Y, complement(join(complement(complement(X)), complement(join(X, Y))))))))
% 69.91/9.35 = { by lemma 61 R->L }
% 69.91/9.35 meet(Y, complement(meet(X, join(Y, complement(join(complement(complement(X)), meet(complement(X), complement(join(X, Y)))))))))
% 69.91/9.35 = { by lemma 63 }
% 69.91/9.35 meet(Y, complement(meet(X, join(Y, meet(complement(X), complement(meet(complement(X), complement(join(X, Y)))))))))
% 69.91/9.36 = { by lemma 61 }
% 69.91/9.36 meet(Y, complement(meet(X, join(Y, meet(complement(X), complement(complement(join(X, Y))))))))
% 69.91/9.36 = { by lemma 16 }
% 69.91/9.36 meet(Y, complement(meet(X, join(Y, meet(complement(X), join(X, Y))))))
% 69.91/9.36 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.36 meet(Y, complement(meet(X, join(Y, meet(complement(X), join(Y, X))))))
% 69.91/9.36 = { by lemma 50 }
% 69.91/9.36 meet(Y, complement(meet(X, join(complement(X), Y))))
% 69.91/9.36 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.36 meet(Y, complement(meet(X, join(Y, complement(X)))))
% 69.91/9.36 = { by lemma 47 R->L }
% 69.91/9.36 meet(Y, meet(join(Y, complement(X)), complement(meet(X, join(Y, complement(X))))))
% 69.91/9.36 = { by lemma 34 }
% 69.91/9.36 meet(Y, complement(X))
% 69.91/9.36
% 69.91/9.36 Lemma 65: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 69.91/9.36 Proof:
% 69.91/9.36 meet(X, complement(meet(X, Y)))
% 69.91/9.36 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.36 meet(X, complement(meet(Y, X)))
% 69.91/9.36 = { by lemma 58 }
% 69.91/9.36 complement(join(Y, complement(X)))
% 69.91/9.36 = { by lemma 64 }
% 69.91/9.36 meet(X, complement(Y))
% 69.91/9.36
% 69.91/9.36 Lemma 66: join(complement(X), meet(X, Y)) = join(Y, complement(X)).
% 69.91/9.36 Proof:
% 69.91/9.36 join(complement(X), meet(X, Y))
% 69.91/9.36 = { by lemma 16 R->L }
% 69.91/9.36 complement(complement(join(complement(X), meet(X, Y))))
% 69.91/9.36 = { by lemma 63 }
% 69.91/9.36 complement(meet(X, complement(meet(X, Y))))
% 69.91/9.36 = { by lemma 60 R->L }
% 69.91/9.36 join(complement(X), complement(meet(X, complement(meet(X, Y)))))
% 69.91/9.36 = { by lemma 65 }
% 69.91/9.36 join(complement(X), complement(meet(X, complement(Y))))
% 69.91/9.36 = { by lemma 60 }
% 69.91/9.36 complement(meet(X, complement(Y)))
% 69.91/9.36 = { by lemma 64 R->L }
% 69.91/9.36 complement(complement(join(Y, complement(X))))
% 69.91/9.36 = { by lemma 16 }
% 69.91/9.36 join(Y, complement(X))
% 69.91/9.36
% 69.91/9.36 Lemma 67: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 69.91/9.36 Proof:
% 69.91/9.36 join(complement(X), complement(Y))
% 69.91/9.36 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.36 join(complement(Y), complement(X))
% 69.91/9.36 = { by lemma 57 R->L }
% 69.91/9.36 join(complement(Y), complement(join(complement(Y), X)))
% 69.91/9.36 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.36 join(complement(Y), complement(join(X, complement(Y))))
% 69.91/9.36 = { by lemma 66 R->L }
% 69.91/9.36 join(complement(Y), complement(join(complement(Y), meet(Y, X))))
% 69.91/9.36 = { by lemma 57 }
% 69.91/9.36 join(complement(Y), complement(meet(Y, X)))
% 69.91/9.36 = { by lemma 60 }
% 69.91/9.36 complement(meet(Y, X))
% 69.91/9.36 = { by axiom 4 (commutativity_of_meet) }
% 69.91/9.36 complement(meet(X, Y))
% 69.91/9.36
% 69.91/9.36 Lemma 68: join(X, meet(Y, complement(X))) = join(X, Y).
% 69.91/9.36 Proof:
% 69.91/9.36 join(X, meet(Y, complement(X)))
% 69.91/9.36 = { by lemma 65 R->L }
% 69.91/9.36 join(X, meet(Y, complement(meet(Y, X))))
% 69.91/9.36 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.36 join(X, meet(Y, complement(meet(X, Y))))
% 69.91/9.36 = { by lemma 42 }
% 69.91/9.36 join(X, Y)
% 69.91/9.36
% 69.91/9.36 Lemma 69: complement(meet(X, meet(Y, complement(Z)))) = join(Z, complement(meet(X, Y))).
% 69.91/9.36 Proof:
% 69.91/9.36 complement(meet(X, meet(Y, complement(Z))))
% 69.91/9.36 = { by lemma 46 R->L }
% 69.91/9.36 complement(meet(Y, meet(X, complement(Z))))
% 69.91/9.36 = { by axiom 8 (meet_join_complement) R->L }
% 69.91/9.36 fresh2(one, one, meet(Y, meet(X, complement(Z))), one)
% 69.91/9.36 = { by lemma 22 R->L }
% 69.91/9.36 fresh2(join(meet(Y, meet(X, complement(Z))), one), one, meet(Y, meet(X, complement(Z))), one)
% 69.91/9.36 = { by axiom 13 (meet_join_complement) }
% 69.91/9.36 fresh(meet(meet(Y, meet(X, complement(Z))), one), zero, meet(Y, meet(X, complement(Z))), one)
% 69.91/9.36 = { by lemma 17 }
% 69.91/9.36 fresh(meet(Y, meet(X, complement(Z))), zero, meet(Y, meet(X, complement(Z))), one)
% 69.91/9.36 = { by lemma 46 R->L }
% 69.91/9.36 fresh(meet(X, meet(Y, complement(Z))), zero, meet(Y, meet(X, complement(Z))), one)
% 69.91/9.36 = { by axiom 12 (associativity_of_meet) R->L }
% 69.91/9.36 fresh(meet(meet(X, Y), complement(Z)), zero, meet(Y, meet(X, complement(Z))), one)
% 69.91/9.36 = { by lemma 30 R->L }
% 69.91/9.36 fresh(meet(meet(X, Y), meet(complement(Z), join(meet(X, Y), Y))), zero, meet(Y, meet(X, complement(Z))), one)
% 69.91/9.36 = { by axiom 12 (associativity_of_meet) }
% 69.91/9.36 fresh(meet(X, meet(Y, meet(complement(Z), join(meet(X, Y), Y)))), zero, meet(Y, meet(X, complement(Z))), one)
% 69.91/9.36 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.36 fresh(meet(X, meet(Y, meet(complement(Z), join(Y, meet(X, Y))))), zero, meet(Y, meet(X, complement(Z))), one)
% 69.91/9.36 = { by axiom 4 (commutativity_of_meet) }
% 69.91/9.36 fresh(meet(X, meet(Y, meet(join(Y, meet(X, Y)), complement(Z)))), zero, meet(Y, meet(X, complement(Z))), one)
% 69.91/9.36 = { by lemma 24 }
% 69.91/9.36 fresh(meet(X, meet(Y, meet(Y, complement(Z)))), zero, meet(Y, meet(X, complement(Z))), one)
% 69.91/9.36 = { by lemma 46 }
% 69.91/9.36 fresh(meet(Y, meet(X, meet(Y, complement(Z)))), zero, meet(Y, meet(X, complement(Z))), one)
% 69.91/9.36 = { by lemma 46 }
% 69.91/9.36 fresh(meet(Y, meet(Y, meet(X, complement(Z)))), zero, meet(Y, meet(X, complement(Z))), one)
% 69.91/9.36 = { by axiom 12 (associativity_of_meet) R->L }
% 69.91/9.36 fresh(meet(Y, meet(meet(Y, X), complement(Z))), zero, meet(Y, meet(X, complement(Z))), one)
% 69.91/9.36 = { by axiom 12 (associativity_of_meet) R->L }
% 69.91/9.36 fresh(meet(Y, meet(meet(Y, X), complement(Z))), zero, meet(meet(Y, X), complement(Z)), one)
% 69.91/9.36 = { by lemma 64 R->L }
% 69.91/9.36 fresh(meet(Y, complement(join(Z, complement(meet(Y, X))))), zero, meet(meet(Y, X), complement(Z)), one)
% 69.91/9.36 = { by lemma 64 R->L }
% 69.91/9.36 fresh(meet(Y, complement(join(Z, complement(meet(Y, X))))), zero, complement(join(Z, complement(meet(Y, X)))), one)
% 69.91/9.36 = { by lemma 23 R->L }
% 69.91/9.36 fresh(meet(Y, complement(join(Z, complement(meet(Y, X))))), zero, complement(join(Z, complement(meet(Y, X)))), join(one, Z))
% 69.91/9.36 = { by lemma 28 R->L }
% 69.91/9.36 fresh(meet(Y, complement(join(Z, complement(meet(Y, X))))), zero, complement(join(Z, complement(meet(Y, X)))), join(join(Y, complement(meet(Y, X))), Z))
% 69.91/9.36 = { by axiom 10 (associativity_of_join) }
% 69.91/9.36 fresh(meet(Y, complement(join(Z, complement(meet(Y, X))))), zero, complement(join(Z, complement(meet(Y, X)))), join(Y, join(complement(meet(Y, X)), Z)))
% 69.91/9.36 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.36 fresh(meet(Y, complement(join(Z, complement(meet(Y, X))))), zero, complement(join(Z, complement(meet(Y, X)))), join(Y, join(Z, complement(meet(Y, X)))))
% 69.91/9.36 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.36 fresh(meet(Y, complement(join(Z, complement(meet(Y, X))))), zero, complement(join(Z, complement(meet(Y, X)))), join(join(Z, complement(meet(Y, X))), Y))
% 69.91/9.36 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.36 fresh(meet(complement(join(Z, complement(meet(Y, X)))), Y), zero, complement(join(Z, complement(meet(Y, X)))), join(join(Z, complement(meet(Y, X))), Y))
% 69.91/9.36 = { by lemma 68 R->L }
% 69.91/9.36 fresh(meet(complement(join(Z, complement(meet(Y, X)))), Y), zero, complement(join(Z, complement(meet(Y, X)))), join(join(Z, complement(meet(Y, X))), meet(Y, complement(join(Z, complement(meet(Y, X)))))))
% 69.91/9.36 = { by axiom 4 (commutativity_of_meet) }
% 69.91/9.36 fresh(meet(complement(join(Z, complement(meet(Y, X)))), Y), zero, complement(join(Z, complement(meet(Y, X)))), join(join(Z, complement(meet(Y, X))), meet(complement(join(Z, complement(meet(Y, X)))), Y)))
% 69.91/9.36 = { by lemma 16 R->L }
% 69.91/9.36 fresh(meet(complement(join(Z, complement(meet(Y, X)))), Y), zero, complement(join(Z, complement(meet(Y, X)))), join(complement(complement(join(Z, complement(meet(Y, X))))), meet(complement(join(Z, complement(meet(Y, X)))), Y)))
% 69.91/9.36 = { by lemma 19 R->L }
% 69.91/9.36 fresh(join(zero, meet(complement(join(Z, complement(meet(Y, X)))), Y)), zero, complement(join(Z, complement(meet(Y, X)))), join(complement(complement(join(Z, complement(meet(Y, X))))), meet(complement(join(Z, complement(meet(Y, X)))), Y)))
% 69.91/9.36 = { by axiom 6 (complement_meet) R->L }
% 69.91/9.36 fresh(join(meet(complement(join(Z, complement(meet(Y, X)))), complement(complement(join(Z, complement(meet(Y, X)))))), meet(complement(join(Z, complement(meet(Y, X)))), Y)), zero, complement(join(Z, complement(meet(Y, X)))), join(complement(complement(join(Z, complement(meet(Y, X))))), meet(complement(join(Z, complement(meet(Y, X)))), Y)))
% 69.91/9.36 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.36 fresh(join(meet(complement(join(Z, complement(meet(Y, X)))), Y), meet(complement(join(Z, complement(meet(Y, X)))), complement(complement(join(Z, complement(meet(Y, X))))))), zero, complement(join(Z, complement(meet(Y, X)))), join(complement(complement(join(Z, complement(meet(Y, X))))), meet(complement(join(Z, complement(meet(Y, X)))), Y)))
% 69.91/9.36 = { by lemma 38 R->L }
% 69.91/9.36 fresh(meet(complement(join(Z, complement(meet(Y, X)))), join(complement(complement(join(Z, complement(meet(Y, X))))), meet(complement(join(Z, complement(meet(Y, X)))), Y))), zero, complement(join(Z, complement(meet(Y, X)))), join(complement(complement(join(Z, complement(meet(Y, X))))), meet(complement(join(Z, complement(meet(Y, X)))), Y)))
% 69.91/9.36 = { by axiom 13 (meet_join_complement) R->L }
% 69.91/9.36 fresh2(join(complement(join(Z, complement(meet(Y, X)))), join(complement(complement(join(Z, complement(meet(Y, X))))), meet(complement(join(Z, complement(meet(Y, X)))), Y))), one, complement(join(Z, complement(meet(Y, X)))), join(complement(complement(join(Z, complement(meet(Y, X))))), meet(complement(join(Z, complement(meet(Y, X)))), Y)))
% 69.91/9.36 = { by lemma 26 }
% 69.91/9.36 fresh2(join(complement(join(Z, complement(meet(Y, X)))), complement(complement(join(Z, complement(meet(Y, X)))))), one, complement(join(Z, complement(meet(Y, X)))), join(complement(complement(join(Z, complement(meet(Y, X))))), meet(complement(join(Z, complement(meet(Y, X)))), Y)))
% 69.91/9.36 = { by axiom 5 (complement_join) }
% 69.91/9.36 fresh2(one, one, complement(join(Z, complement(meet(Y, X)))), join(complement(complement(join(Z, complement(meet(Y, X))))), meet(complement(join(Z, complement(meet(Y, X)))), Y)))
% 69.91/9.36 = { by axiom 8 (meet_join_complement) }
% 69.91/9.36 complement(complement(join(Z, complement(meet(Y, X)))))
% 69.91/9.36 = { by lemma 16 }
% 69.91/9.36 join(Z, complement(meet(Y, X)))
% 69.91/9.36 = { by axiom 4 (commutativity_of_meet) }
% 69.91/9.36 join(Z, complement(meet(X, Y)))
% 69.91/9.36
% 69.91/9.36 Lemma 70: complement(meet(X, meet(complement(Y), Z))) = join(Y, complement(meet(X, Z))).
% 69.91/9.36 Proof:
% 69.91/9.36 complement(meet(X, meet(complement(Y), Z)))
% 69.91/9.36 = { by axiom 12 (associativity_of_meet) R->L }
% 69.91/9.36 complement(meet(meet(X, complement(Y)), Z))
% 69.91/9.36 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.36 complement(meet(Z, meet(X, complement(Y))))
% 69.91/9.36 = { by lemma 69 }
% 69.91/9.36 join(Y, complement(meet(Z, X)))
% 69.91/9.36 = { by axiom 4 (commutativity_of_meet) }
% 69.91/9.37 join(Y, complement(meet(X, Z)))
% 69.91/9.37
% 69.91/9.37 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 69.91/9.37 Proof:
% 69.91/9.37 meet(a, join(b, c))
% 69.91/9.37 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.37 meet(a, join(c, b))
% 69.91/9.37 = { by lemma 43 R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, meet(a, join(c, b))))
% 69.91/9.37 = { by lemma 16 R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(complement(meet(a, join(c, b))))))
% 69.91/9.37 = { by lemma 57 R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(c, complement(meet(a, join(c, b)))))))
% 69.91/9.37 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(c, complement(meet(join(c, b), a))))))
% 69.91/9.37 = { by lemma 68 R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(c, complement(meet(join(c, meet(b, complement(c))), a))))))
% 69.91/9.37 = { by lemma 70 R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(complement(meet(join(c, meet(b, complement(c))), meet(complement(c), a))))))
% 69.91/9.37 = { by lemma 67 R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(complement(join(c, meet(b, complement(c)))), complement(meet(complement(c), a))))))
% 69.91/9.37 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(complement(join(c, meet(complement(c), b))), complement(meet(complement(c), a))))))
% 69.91/9.37 = { by lemma 16 R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(complement(join(complement(complement(c)), meet(complement(c), b))), complement(meet(complement(c), a))))))
% 69.91/9.37 = { by lemma 63 }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(meet(complement(c), complement(meet(complement(c), b))), complement(meet(complement(c), a))))))
% 69.91/9.37 = { by axiom 4 (commutativity_of_meet) }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(meet(complement(c), complement(meet(b, complement(c)))), complement(meet(complement(c), a))))))
% 69.91/9.37 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(meet(complement(meet(b, complement(c))), complement(c)), complement(meet(complement(c), a))))))
% 69.91/9.37 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), meet(complement(meet(b, complement(c))), complement(c))))))
% 69.91/9.37 = { by lemma 44 R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), join(meet(complement(meet(b, complement(c))), complement(c)), meet(complement(meet(b, complement(c))), join(complement(meet(complement(c), a)), meet(complement(meet(b, complement(c))), complement(c)))))))))
% 69.91/9.37 = { by lemma 40 }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), meet(complement(meet(b, complement(c))), join(complement(meet(complement(c), a)), meet(complement(meet(b, complement(c))), complement(c))))))))
% 69.91/9.37 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), meet(complement(meet(b, complement(c))), join(meet(complement(meet(b, complement(c))), complement(c)), complement(meet(complement(c), a))))))))
% 69.91/9.37 = { by lemma 69 R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), meet(complement(meet(b, complement(c))), complement(meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c)))))))))))
% 69.91/9.37 = { by lemma 65 R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), meet(complement(meet(b, complement(c))), complement(meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c))))))))))))
% 69.91/9.37 = { by lemma 63 R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), complement(join(complement(complement(meet(b, complement(c)))), meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c))))))))))))
% 69.91/9.37 = { by axiom 8 (meet_join_complement) R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), fresh2(one, one, join(complement(complement(meet(b, complement(c)))), meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c))))))), complement(meet(b, complement(c))))))))
% 69.91/9.37 = { by axiom 5 (complement_join) R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), fresh2(join(complement(meet(b, complement(c))), complement(complement(meet(b, complement(c))))), one, join(complement(complement(meet(b, complement(c)))), meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c))))))), complement(meet(b, complement(c))))))))
% 69.91/9.37 = { by lemma 26 R->L }
% 69.91/9.37 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), fresh2(join(complement(meet(b, complement(c))), join(complement(complement(meet(b, complement(c)))), meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c)))))))), one, join(complement(complement(meet(b, complement(c)))), meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c))))))), complement(meet(b, complement(c))))))))
% 69.91/9.38 = { by lemma 15 }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), fresh(meet(complement(meet(b, complement(c))), join(complement(complement(meet(b, complement(c)))), meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c)))))))), zero, join(complement(complement(meet(b, complement(c)))), meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c))))))), complement(meet(b, complement(c))))))))
% 69.91/9.38 = { by lemma 38 }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), fresh(join(meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c)))))), meet(complement(meet(b, complement(c))), complement(complement(meet(b, complement(c)))))), zero, join(complement(complement(meet(b, complement(c)))), meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c))))))), complement(meet(b, complement(c))))))))
% 69.91/9.38 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), fresh(join(meet(complement(meet(b, complement(c))), complement(complement(meet(b, complement(c))))), meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c))))))), zero, join(complement(complement(meet(b, complement(c)))), meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c))))))), complement(meet(b, complement(c))))))))
% 69.91/9.38 = { by axiom 6 (complement_meet) }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), fresh(join(zero, meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c))))))), zero, join(complement(complement(meet(b, complement(c)))), meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c))))))), complement(meet(b, complement(c))))))))
% 69.91/9.38 = { by lemma 19 }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), fresh(meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c)))))), zero, join(complement(complement(meet(b, complement(c)))), meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c))))))), complement(meet(b, complement(c))))))))
% 69.91/9.38 = { by lemma 66 }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), fresh(meet(complement(meet(b, complement(c))), meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c)))))), zero, join(meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c))))), complement(complement(meet(b, complement(c))))), complement(meet(b, complement(c))))))))
% 69.91/9.38 = { by lemma 62 }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), fresh(zero, zero, join(meet(complement(c), meet(a, complement(meet(complement(meet(b, complement(c))), complement(c))))), complement(complement(meet(b, complement(c))))), complement(meet(b, complement(c))))))))
% 69.91/9.38 = { by axiom 7 (meet_join_complement) }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(complement(c), a)), complement(meet(b, complement(c)))))))
% 69.91/9.38 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(complement(meet(b, complement(c))), complement(meet(complement(c), a))))))
% 69.91/9.38 = { by lemma 67 }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(complement(meet(meet(b, complement(c)), meet(complement(c), a))))))
% 69.91/9.38 = { by lemma 70 }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(c, complement(meet(meet(b, complement(c)), a))))))
% 69.91/9.38 = { by axiom 12 (associativity_of_meet) }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(c, complement(meet(b, meet(complement(c), a)))))))
% 69.91/9.38 = { by lemma 70 }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(c, join(c, complement(meet(b, a)))))))
% 69.91/9.38 = { by axiom 3 (idempotence_of_meet) R->L }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(c, join(c, complement(meet(meet(b, a), meet(b, a))))))))
% 69.91/9.38 = { by lemma 21 R->L }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(c, meet(one, join(c, complement(meet(meet(b, a), meet(b, a)))))))))
% 69.91/9.38 = { by lemma 28 R->L }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(c, meet(join(meet(meet(b, a), join(meet(b, a), X)), complement(meet(meet(meet(b, a), join(meet(b, a), X)), meet(b, a)))), join(c, complement(meet(meet(b, a), meet(b, a)))))))))
% 69.91/9.38 = { by axiom 4 (commutativity_of_meet) }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(c, meet(join(meet(meet(b, a), join(meet(b, a), X)), complement(meet(meet(b, a), meet(meet(b, a), join(meet(b, a), X))))), join(c, complement(meet(meet(b, a), meet(b, a)))))))))
% 69.91/9.38 = { by lemma 30 }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(c, meet(join(meet(meet(b, a), join(meet(b, a), X)), complement(meet(meet(b, a), meet(b, a)))), join(c, complement(meet(meet(b, a), meet(b, a)))))))))
% 69.91/9.38 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(c, meet(join(complement(meet(meet(b, a), meet(b, a))), meet(meet(b, a), join(meet(b, a), X))), join(c, complement(meet(meet(b, a), meet(b, a)))))))))
% 69.91/9.38 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.38 meet(meet(a, join(c, b)), join(c, complement(join(c, meet(join(c, complement(meet(meet(b, a), meet(b, a)))), join(complement(meet(meet(b, a), meet(b, a))), meet(meet(b, a), join(meet(b, a), X))))))))
% 69.91/9.38 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.39 meet(meet(a, join(c, b)), join(c, complement(join(c, meet(join(c, complement(meet(meet(b, a), meet(b, a)))), join(meet(meet(b, a), join(meet(b, a), X)), complement(meet(meet(b, a), meet(b, a)))))))))
% 69.91/9.39 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.39 meet(meet(a, join(c, b)), join(c, complement(join(c, meet(join(complement(meet(meet(b, a), meet(b, a))), c), join(meet(meet(b, a), join(meet(b, a), X)), complement(meet(meet(b, a), meet(b, a)))))))))
% 69.91/9.39 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.39 meet(meet(a, join(c, b)), join(c, complement(join(c, meet(join(meet(meet(b, a), join(meet(b, a), X)), complement(meet(meet(b, a), meet(b, a)))), join(complement(meet(meet(b, a), meet(b, a))), c))))))
% 69.91/9.39 = { by lemma 31 R->L }
% 69.91/9.39 meet(meet(a, join(c, b)), join(c, complement(join(c, join(meet(complement(meet(meet(b, a), meet(b, a))), join(meet(meet(b, a), join(meet(b, a), X)), complement(meet(meet(b, a), meet(b, a))))), meet(join(meet(meet(b, a), join(meet(b, a), X)), complement(meet(meet(b, a), meet(b, a)))), join(complement(meet(meet(b, a), meet(b, a))), c)))))))
% 69.91/9.39 = { by lemma 43 }
% 69.91/9.39 meet(meet(a, join(c, b)), join(c, complement(join(c, join(complement(meet(meet(b, a), meet(b, a))), meet(join(meet(meet(b, a), join(meet(b, a), X)), complement(meet(meet(b, a), meet(b, a)))), join(complement(meet(meet(b, a), meet(b, a))), c)))))))
% 69.91/9.39 = { by axiom 4 (commutativity_of_meet) }
% 69.91/9.39 meet(meet(a, join(c, b)), join(c, complement(join(c, join(complement(meet(meet(b, a), meet(b, a))), meet(join(complement(meet(meet(b, a), meet(b, a))), c), join(meet(meet(b, a), join(meet(b, a), X)), complement(meet(meet(b, a), meet(b, a))))))))))
% 69.91/9.39 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.39 meet(meet(a, join(c, b)), join(c, complement(join(c, join(complement(meet(meet(b, a), meet(b, a))), meet(join(c, complement(meet(meet(b, a), meet(b, a)))), join(meet(meet(b, a), join(meet(b, a), X)), complement(meet(meet(b, a), meet(b, a))))))))))
% 69.91/9.39 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.39 meet(meet(a, join(c, b)), join(c, complement(join(c, join(complement(meet(meet(b, a), meet(b, a))), meet(join(c, complement(meet(meet(b, a), meet(b, a)))), join(complement(meet(meet(b, a), meet(b, a))), meet(meet(b, a), join(meet(b, a), X)))))))))
% 69.91/9.39 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.39 meet(meet(a, join(c, b)), join(c, complement(join(c, join(complement(meet(meet(b, a), meet(b, a))), meet(join(complement(meet(meet(b, a), meet(b, a))), meet(meet(b, a), join(meet(b, a), X))), join(c, complement(meet(meet(b, a), meet(b, a))))))))))
% 69.91/9.39 = { by lemma 44 }
% 69.91/9.39 meet(meet(a, join(c, b)), join(c, complement(join(c, complement(meet(meet(b, a), meet(b, a)))))))
% 69.91/9.39 = { by axiom 3 (idempotence_of_meet) }
% 69.91/9.39 meet(meet(a, join(c, b)), join(c, complement(join(c, complement(meet(b, a))))))
% 69.91/9.39 = { by axiom 4 (commutativity_of_meet) }
% 69.91/9.39 meet(meet(a, join(c, b)), join(c, complement(join(c, complement(meet(a, b))))))
% 69.91/9.39 = { by lemma 57 }
% 69.91/9.39 meet(meet(a, join(c, b)), join(c, complement(complement(meet(a, b)))))
% 69.91/9.39 = { by lemma 16 }
% 69.91/9.39 meet(meet(a, join(c, b)), join(c, meet(a, b)))
% 69.91/9.39 = { by axiom 12 (associativity_of_meet) }
% 69.91/9.39 meet(a, meet(join(c, b), join(c, meet(a, b))))
% 69.91/9.39 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.39 meet(a, meet(join(c, b), join(c, meet(b, a))))
% 69.91/9.39 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.39 meet(a, meet(join(b, c), join(c, meet(b, a))))
% 69.91/9.39 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.39 meet(a, meet(join(c, meet(b, a)), join(b, c)))
% 69.91/9.39 = { by lemma 49 R->L }
% 69.91/9.39 meet(a, join(meet(b, meet(a, join(c, meet(b, a)))), meet(join(c, meet(b, a)), join(b, c))))
% 69.91/9.39 = { by lemma 39 }
% 69.91/9.39 meet(a, join(meet(b, a), meet(join(c, meet(b, a)), join(b, c))))
% 69.91/9.39 = { by axiom 4 (commutativity_of_meet) R->L }
% 69.91/9.39 meet(a, join(meet(b, a), meet(join(b, c), join(c, meet(b, a)))))
% 69.91/9.39 = { by axiom 2 (commutativity_of_join) R->L }
% 69.91/9.39 meet(a, join(meet(b, a), meet(join(b, c), join(meet(b, a), c))))
% 69.91/9.39 = { by lemma 36 R->L }
% 69.91/9.39 meet(a, meet(join(meet(b, a), c), join(meet(b, a), join(b, c))))
% 69.91/9.39 = { by axiom 4 (commutativity_of_meet) }
% 69.91/9.39 meet(a, meet(join(meet(b, a), join(b, c)), join(meet(b, a), c)))
% 69.91/9.39 = { by lemma 36 }
% 69.91/9.39 meet(a, join(meet(b, a), meet(c, join(meet(b, a), join(b, c)))))
% 69.91/9.39 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.39 meet(a, join(meet(b, a), meet(c, join(join(b, c), meet(b, a)))))
% 69.91/9.39 = { by axiom 10 (associativity_of_join) }
% 69.91/9.39 meet(a, join(meet(b, a), meet(c, join(b, join(c, meet(b, a))))))
% 69.91/9.39 = { by lemma 26 }
% 69.91/9.39 meet(a, join(meet(b, a), meet(c, join(b, c))))
% 69.91/9.39 = { by axiom 4 (commutativity_of_meet) }
% 69.91/9.39 meet(a, join(meet(a, b), meet(c, join(b, c))))
% 69.91/9.39 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.39 meet(a, join(meet(a, b), meet(c, join(c, b))))
% 69.91/9.39 = { by axiom 11 (absorption1) }
% 69.91/9.39 meet(a, join(meet(a, b), c))
% 69.91/9.39 = { by axiom 2 (commutativity_of_join) }
% 69.91/9.39 meet(a, join(c, meet(a, b)))
% 69.91/9.39 = { by lemma 38 }
% 69.91/9.39 join(meet(a, b), meet(a, c))
% 69.91/9.39 % SZS output end Proof
% 69.91/9.39
% 69.91/9.39 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------