TSTP Solution File: LAT207-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT207-1 : 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 : n010.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 06:27:46 EDT 2023
% Result : Unsatisfiable 41.50s 5.80s
% Output : Proof 43.31s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LAT207-1 : 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
% 0.13/0.35 % Computer : n010.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Thu Aug 24 08:12:20 EDT 2023
% 0.13/0.35 % CPUTime :
% 41.50/5.80 Command-line arguments: --no-flatten-goal
% 41.50/5.80
% 41.50/5.80 % SZS status Unsatisfiable
% 41.50/5.80
% 42.42/5.89 % SZS output start Proof
% 42.42/5.89 Take the following subset of the input axioms:
% 42.42/5.89 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 42.42/5.89 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 42.42/5.90 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 42.42/5.90 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 42.42/5.90 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 42.42/5.90 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 42.42/5.90 fof(complement_join, axiom, ![X2]: join(X2, complement(X2))=one).
% 42.42/5.90 fof(complement_meet, axiom, ![X2]: meet(X2, complement(X2))=zero).
% 42.42/5.90 fof(equation_H61, axiom, ![X2, Y2, Z2]: meet(join(X2, Y2), join(X2, Z2))=join(X2, meet(join(X2, Y2), join(meet(X2, Y2), Z2)))).
% 42.42/5.90 fof(idempotence_of_join, axiom, ![X2]: join(X2, X2)=X2).
% 42.42/5.90 fof(meet_join_complement, axiom, ![X2, Y2]: (meet(X2, Y2)!=zero | (join(X2, Y2)!=one | complement(X2)=Y2))).
% 42.42/5.90 fof(prove_distributivity, negated_conjecture, meet(a, join(b, c))!=join(meet(a, b), meet(a, c))).
% 42.42/5.90
% 42.42/5.90 Now clausify the problem and encode Horn clauses using encoding 3 of
% 42.42/5.90 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 42.42/5.90 We repeatedly replace C & s=t => u=v by the two clauses:
% 42.42/5.90 fresh(y, y, x1...xn) = u
% 42.42/5.90 C => fresh(s, t, x1...xn) = v
% 42.42/5.90 where fresh is a fresh function symbol and x1..xn are the free
% 42.42/5.90 variables of u and v.
% 42.42/5.90 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 42.42/5.90 input problem has no model of domain size 1).
% 42.42/5.90
% 42.42/5.90 The encoding turns the above axioms into the following unit equations and goals:
% 42.42/5.90
% 42.42/5.90 Axiom 1 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 42.42/5.90 Axiom 2 (idempotence_of_join): join(X, X) = X.
% 42.42/5.90 Axiom 3 (commutativity_of_join): join(X, Y) = join(Y, X).
% 42.42/5.90 Axiom 4 (complement_meet): meet(X, complement(X)) = zero.
% 42.42/5.90 Axiom 5 (complement_join): join(X, complement(X)) = one.
% 42.42/5.90 Axiom 6 (meet_join_complement): fresh(X, X, Y, Z) = Z.
% 42.42/5.90 Axiom 7 (meet_join_complement): fresh2(X, X, Y, Z) = complement(Y).
% 42.42/5.90 Axiom 8 (absorption1): meet(X, join(X, Y)) = X.
% 42.42/5.90 Axiom 9 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 42.42/5.90 Axiom 10 (absorption2): join(X, meet(X, Y)) = X.
% 42.42/5.90 Axiom 11 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 42.42/5.90 Axiom 12 (meet_join_complement): fresh2(join(X, Y), one, X, Y) = fresh(meet(X, Y), zero, X, Y).
% 42.42/5.90 Axiom 13 (equation_H61): meet(join(X, Y), join(X, Z)) = join(X, meet(join(X, Y), join(meet(X, Y), Z))).
% 42.42/5.90
% 42.42/5.90 Lemma 14: fresh2(join(X, Y), one, Y, X) = fresh(meet(X, Y), zero, Y, X).
% 42.42/5.90 Proof:
% 42.42/5.90 fresh2(join(X, Y), one, Y, X)
% 42.42/5.90 = { by axiom 3 (commutativity_of_join) R->L }
% 42.42/5.90 fresh2(join(Y, X), one, Y, X)
% 42.42/5.90 = { by axiom 12 (meet_join_complement) }
% 42.42/5.90 fresh(meet(Y, X), zero, Y, X)
% 42.42/5.90 = { by axiom 1 (commutativity_of_meet) }
% 42.42/5.90 fresh(meet(X, Y), zero, Y, X)
% 42.42/5.90
% 42.42/5.90 Lemma 15: complement(complement(X)) = X.
% 42.42/5.90 Proof:
% 42.42/5.90 complement(complement(X))
% 42.42/5.90 = { by axiom 7 (meet_join_complement) R->L }
% 42.42/5.90 fresh2(one, one, complement(X), X)
% 42.42/5.90 = { by axiom 5 (complement_join) R->L }
% 42.42/5.90 fresh2(join(X, complement(X)), one, complement(X), X)
% 42.42/5.90 = { by lemma 14 }
% 42.42/5.90 fresh(meet(X, complement(X)), zero, complement(X), X)
% 42.42/5.90 = { by axiom 4 (complement_meet) }
% 42.42/5.90 fresh(zero, zero, complement(X), X)
% 42.42/5.90 = { by axiom 6 (meet_join_complement) }
% 42.42/5.90 X
% 42.42/5.90
% 42.42/5.90 Lemma 16: join(X, zero) = X.
% 42.42/5.90 Proof:
% 42.42/5.90 join(X, zero)
% 42.42/5.90 = { by axiom 4 (complement_meet) R->L }
% 42.42/5.90 join(X, meet(X, complement(X)))
% 42.42/5.90 = { by axiom 10 (absorption2) }
% 42.42/5.90 X
% 42.42/5.90
% 42.42/5.90 Lemma 17: join(zero, X) = X.
% 42.42/5.90 Proof:
% 42.42/5.90 join(zero, X)
% 42.42/5.90 = { by axiom 3 (commutativity_of_join) R->L }
% 42.42/5.90 join(X, zero)
% 42.42/5.90 = { by lemma 16 }
% 42.42/5.90 X
% 42.42/5.90
% 42.42/5.90 Lemma 18: meet(X, zero) = zero.
% 42.42/5.90 Proof:
% 42.42/5.90 meet(X, zero)
% 42.42/5.90 = { by axiom 1 (commutativity_of_meet) R->L }
% 42.42/5.90 meet(zero, X)
% 42.42/5.90 = { by lemma 17 R->L }
% 42.42/5.90 meet(zero, join(zero, X))
% 42.42/5.90 = { by axiom 8 (absorption1) }
% 42.42/5.90 zero
% 42.42/5.90
% 42.42/5.90 Lemma 19: meet(one, X) = X.
% 42.42/5.90 Proof:
% 42.42/5.90 meet(one, X)
% 42.42/5.90 = { by axiom 1 (commutativity_of_meet) R->L }
% 42.42/5.90 meet(X, one)
% 42.42/5.90 = { by axiom 5 (complement_join) R->L }
% 42.42/5.90 meet(X, join(X, complement(X)))
% 42.42/5.90 = { by axiom 8 (absorption1) }
% 42.42/5.90 X
% 42.42/5.90
% 42.42/5.90 Lemma 20: meet(X, meet(Y, join(Z, meet(X, Y)))) = meet(X, Y).
% 42.42/5.90 Proof:
% 42.42/5.90 meet(X, meet(Y, join(Z, meet(X, Y))))
% 42.42/5.90 = { by axiom 3 (commutativity_of_join) R->L }
% 42.42/5.90 meet(X, meet(Y, join(meet(X, Y), Z)))
% 42.42/5.90 = { by axiom 9 (associativity_of_meet) R->L }
% 42.42/5.90 meet(meet(X, Y), join(meet(X, Y), Z))
% 42.42/5.90 = { by axiom 8 (absorption1) }
% 42.42/5.90 meet(X, Y)
% 42.42/5.90
% 42.42/5.90 Lemma 21: join(X, meet(Y, X)) = X.
% 42.42/5.90 Proof:
% 42.42/5.90 join(X, meet(Y, X))
% 42.42/5.90 = { by axiom 1 (commutativity_of_meet) R->L }
% 42.42/5.90 join(X, meet(X, Y))
% 42.42/5.90 = { by axiom 10 (absorption2) }
% 42.42/5.90 X
% 42.42/5.90
% 42.42/5.90 Lemma 22: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 42.42/5.90 Proof:
% 42.42/5.90 join(X, join(Y, meet(X, Z)))
% 42.42/5.90 = { by axiom 3 (commutativity_of_join) R->L }
% 42.42/5.90 join(X, join(meet(X, Z), Y))
% 42.42/5.90 = { by axiom 11 (associativity_of_join) R->L }
% 42.42/5.90 join(join(X, meet(X, Z)), Y)
% 42.42/5.90 = { by axiom 10 (absorption2) }
% 42.42/5.90 join(X, Y)
% 42.42/5.90
% 42.42/5.90 Lemma 23: meet(X, join(Y, X)) = X.
% 42.42/5.90 Proof:
% 42.42/5.90 meet(X, join(Y, X))
% 42.42/5.90 = { by axiom 3 (commutativity_of_join) R->L }
% 42.42/5.90 meet(X, join(X, Y))
% 42.42/5.90 = { by axiom 8 (absorption1) }
% 42.42/5.90 X
% 42.42/5.90
% 42.42/5.90 Lemma 24: meet(join(X, Y), join(Y, meet(X, Z))) = join(Y, meet(X, Z)).
% 42.42/5.90 Proof:
% 42.42/5.90 meet(join(X, Y), join(Y, meet(X, Z)))
% 42.42/5.90 = { by axiom 1 (commutativity_of_meet) R->L }
% 42.42/5.90 meet(join(Y, meet(X, Z)), join(X, Y))
% 42.42/5.90 = { by lemma 22 R->L }
% 42.42/5.90 meet(join(Y, meet(X, Z)), join(X, join(Y, meet(X, Z))))
% 42.42/5.90 = { by lemma 23 }
% 42.42/5.90 join(Y, meet(X, Z))
% 42.42/5.90
% 42.42/5.90 Lemma 25: meet(X, meet(join(X, Y), Z)) = meet(X, Z).
% 42.42/5.90 Proof:
% 42.42/5.90 meet(X, meet(join(X, Y), Z))
% 42.42/5.90 = { by axiom 9 (associativity_of_meet) R->L }
% 42.42/5.90 meet(meet(X, join(X, Y)), Z)
% 42.42/5.90 = { by axiom 8 (absorption1) }
% 42.42/5.90 meet(X, Z)
% 42.42/5.90
% 42.42/5.90 Lemma 26: meet(X, complement(join(X, Y))) = zero.
% 42.42/5.90 Proof:
% 42.42/5.90 meet(X, complement(join(X, Y)))
% 42.42/5.90 = { by lemma 25 R->L }
% 42.42/5.90 meet(X, meet(join(X, Y), complement(join(X, Y))))
% 42.42/5.90 = { by axiom 4 (complement_meet) }
% 42.42/5.90 meet(X, zero)
% 42.42/5.90 = { by lemma 18 }
% 42.42/5.90 zero
% 42.42/5.90
% 42.42/5.90 Lemma 27: meet(X, meet(Y, complement(X))) = zero.
% 42.42/5.90 Proof:
% 42.42/5.90 meet(X, meet(Y, complement(X)))
% 42.42/5.90 = { by axiom 1 (commutativity_of_meet) R->L }
% 42.42/5.90 meet(X, meet(complement(X), Y))
% 42.42/5.90 = { by axiom 9 (associativity_of_meet) R->L }
% 42.42/5.90 meet(meet(X, complement(X)), Y)
% 42.42/5.90 = { by axiom 4 (complement_meet) }
% 42.42/5.90 meet(zero, Y)
% 42.42/5.90 = { by axiom 1 (commutativity_of_meet) R->L }
% 42.42/5.90 meet(Y, zero)
% 42.42/5.90 = { by lemma 18 }
% 42.42/5.90 zero
% 42.42/5.90
% 42.42/5.90 Lemma 28: join(X, join(Y, complement(join(X, Y)))) = one.
% 42.42/5.90 Proof:
% 42.42/5.90 join(X, join(Y, complement(join(X, Y))))
% 42.42/5.90 = { by axiom 11 (associativity_of_join) R->L }
% 42.42/5.90 join(join(X, Y), complement(join(X, Y)))
% 42.42/5.90 = { by axiom 5 (complement_join) }
% 42.42/5.90 one
% 42.42/5.90
% 42.42/5.90 Lemma 29: join(meet(X, Y), complement(join(complement(X), meet(X, Y)))) = X.
% 42.42/5.90 Proof:
% 42.42/5.90 join(meet(X, Y), complement(join(complement(X), meet(X, Y))))
% 42.42/5.90 = { by axiom 1 (commutativity_of_meet) R->L }
% 42.42/5.90 join(meet(Y, X), complement(join(complement(X), meet(X, Y))))
% 42.42/5.90 = { by axiom 1 (commutativity_of_meet) R->L }
% 42.42/5.90 join(meet(Y, X), complement(join(complement(X), meet(Y, X))))
% 42.42/5.90 = { by lemma 15 R->L }
% 42.42/5.90 join(meet(Y, X), complement(join(complement(X), meet(Y, complement(complement(X))))))
% 42.42/5.90 = { by lemma 15 R->L }
% 42.42/5.90 join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X))))))
% 42.42/5.90 = { by axiom 6 (meet_join_complement) R->L }
% 42.42/5.90 fresh(zero, zero, complement(X), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))
% 42.42/5.90 = { by lemma 27 R->L }
% 42.42/5.90 fresh(meet(complement(X), meet(Y, complement(complement(X)))), zero, complement(X), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))
% 42.42/5.90 = { by lemma 16 R->L }
% 42.42/5.90 fresh(meet(complement(X), join(meet(Y, complement(complement(X))), zero)), zero, complement(X), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))
% 42.42/5.90 = { by axiom 4 (complement_meet) R->L }
% 42.42/5.90 fresh(meet(complement(X), join(meet(Y, complement(complement(X))), meet(join(complement(X), meet(Y, complement(complement(X)))), complement(join(complement(X), meet(Y, complement(complement(X)))))))), zero, complement(X), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))
% 42.42/5.90 = { by axiom 3 (commutativity_of_join) R->L }
% 42.42/5.90 fresh(meet(complement(X), join(meet(Y, complement(complement(X))), meet(join(meet(Y, complement(complement(X))), complement(X)), complement(join(complement(X), meet(Y, complement(complement(X)))))))), zero, complement(X), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))
% 42.42/5.90 = { by lemma 17 R->L }
% 42.42/5.90 fresh(meet(complement(X), join(meet(Y, complement(complement(X))), meet(join(meet(Y, complement(complement(X))), complement(X)), join(zero, complement(join(complement(X), meet(Y, complement(complement(X))))))))), zero, complement(X), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))
% 42.42/5.90 = { by lemma 27 R->L }
% 42.42/5.90 fresh(meet(complement(X), join(meet(Y, complement(complement(X))), meet(join(meet(Y, complement(complement(X))), complement(X)), join(meet(complement(X), meet(Y, complement(complement(X)))), complement(join(complement(X), meet(Y, complement(complement(X))))))))), zero, complement(X), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))
% 42.42/5.90 = { by axiom 1 (commutativity_of_meet) R->L }
% 42.42/5.90 fresh(meet(complement(X), join(meet(Y, complement(complement(X))), meet(join(meet(Y, complement(complement(X))), complement(X)), join(meet(meet(Y, complement(complement(X))), complement(X)), complement(join(complement(X), meet(Y, complement(complement(X))))))))), zero, complement(X), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))
% 42.42/5.90 = { by axiom 13 (equation_H61) R->L }
% 42.42/5.90 fresh(meet(complement(X), meet(join(meet(Y, complement(complement(X))), complement(X)), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))), zero, complement(X), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))
% 42.42/5.90 = { by axiom 3 (commutativity_of_join) }
% 42.42/5.90 fresh(meet(complement(X), meet(join(complement(X), meet(Y, complement(complement(X)))), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))), zero, complement(X), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))
% 42.42/5.90 = { by lemma 25 }
% 42.42/5.90 fresh(meet(complement(X), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X))))))), zero, complement(X), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))
% 42.42/5.90 = { by axiom 12 (meet_join_complement) R->L }
% 42.42/5.90 fresh2(join(complement(X), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X))))))), one, complement(X), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))
% 42.42/5.90 = { by lemma 28 }
% 42.42/5.90 fresh2(one, one, complement(X), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))
% 42.42/5.90 = { by axiom 7 (meet_join_complement) }
% 42.42/5.90 complement(complement(X))
% 42.42/5.90 = { by lemma 15 }
% 42.42/5.90 X
% 42.42/5.90
% 42.42/5.90 Lemma 30: join(X, join(Y, meet(Z, X))) = join(X, Y).
% 42.42/5.90 Proof:
% 42.42/5.90 join(X, join(Y, meet(Z, X)))
% 42.42/5.90 = { by axiom 3 (commutativity_of_join) R->L }
% 42.42/5.90 join(join(Y, meet(Z, X)), X)
% 42.42/5.90 = { by axiom 11 (associativity_of_join) }
% 42.42/5.90 join(Y, join(meet(Z, X), X))
% 42.42/5.90 = { by axiom 3 (commutativity_of_join) }
% 42.42/5.90 join(Y, join(X, meet(Z, X)))
% 42.42/5.90 = { by lemma 21 }
% 42.42/5.90 join(Y, X)
% 42.42/5.90 = { by axiom 3 (commutativity_of_join) }
% 42.42/5.90 join(X, Y)
% 42.42/5.90
% 42.42/5.90 Lemma 31: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 42.42/5.90 Proof:
% 42.42/5.90 meet(Y, meet(X, Z))
% 42.42/5.90 = { by axiom 1 (commutativity_of_meet) R->L }
% 42.42/5.90 meet(meet(X, Z), Y)
% 42.42/5.90 = { by axiom 9 (associativity_of_meet) }
% 42.42/5.90 meet(X, meet(Z, Y))
% 42.42/5.90 = { by axiom 1 (commutativity_of_meet) }
% 42.42/5.90 meet(X, meet(Y, Z))
% 42.42/5.90
% 42.42/5.90 Lemma 32: meet(X, meet(Y, join(Z, X))) = meet(X, Y).
% 42.42/5.90 Proof:
% 42.42/5.90 meet(X, meet(Y, join(Z, X)))
% 42.42/5.90 = { by lemma 31 }
% 42.42/5.90 meet(Y, meet(X, join(Z, X)))
% 42.42/5.90 = { by lemma 23 }
% 42.42/5.90 meet(Y, X)
% 42.42/5.90 = { by axiom 1 (commutativity_of_meet) }
% 42.42/5.90 meet(X, Y)
% 42.42/5.90
% 42.42/5.90 Lemma 33: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 42.42/5.90 Proof:
% 42.42/5.90 join(X, join(Y, meet(Z, join(X, Y))))
% 42.42/5.90 = { by axiom 1 (commutativity_of_meet) R->L }
% 42.42/5.90 join(X, join(Y, meet(join(X, Y), Z)))
% 42.42/5.90 = { by axiom 11 (associativity_of_join) R->L }
% 42.42/5.90 join(join(X, Y), meet(join(X, Y), Z))
% 42.42/5.90 = { by axiom 10 (absorption2) }
% 42.42/5.90 join(X, Y)
% 42.42/5.90
% 42.42/5.90 Lemma 34: meet(join(X, Y), join(X, meet(Z, join(X, Y)))) = join(X, meet(Z, join(X, Y))).
% 42.42/5.90 Proof:
% 42.42/5.90 meet(join(X, Y), join(X, meet(Z, join(X, Y))))
% 42.42/5.90 = { by axiom 3 (commutativity_of_join) R->L }
% 42.42/5.90 meet(join(Y, X), join(X, meet(Z, join(X, Y))))
% 42.42/5.90 = { by axiom 3 (commutativity_of_join) R->L }
% 42.42/5.90 meet(join(Y, X), join(X, meet(Z, join(Y, X))))
% 42.42/5.90 = { by axiom 1 (commutativity_of_meet) R->L }
% 42.42/5.90 meet(join(X, meet(Z, join(Y, X))), join(Y, X))
% 42.42/5.90 = { by lemma 33 R->L }
% 42.42/5.90 meet(join(X, meet(Z, join(Y, X))), join(Y, join(X, meet(Z, join(Y, X)))))
% 42.42/5.90 = { by lemma 23 }
% 42.42/5.90 join(X, meet(Z, join(Y, X)))
% 42.42/5.90 = { by axiom 3 (commutativity_of_join) }
% 42.42/5.90 join(X, meet(Z, join(X, Y)))
% 42.42/5.90
% 42.42/5.90 Lemma 35: join(meet(X, Y), meet(X, Z)) = meet(X, join(Z, meet(X, Y))).
% 42.42/5.90 Proof:
% 42.42/5.90 join(meet(X, Y), meet(X, Z))
% 42.42/5.90 = { by lemma 17 R->L }
% 42.42/5.90 join(meet(X, Y), meet(X, join(zero, Z)))
% 42.42/5.90 = { by lemma 26 R->L }
% 42.42/5.90 join(meet(X, Y), meet(X, join(meet(meet(X, Y), complement(join(meet(X, Y), complement(X)))), Z)))
% 42.42/5.90 = { by axiom 9 (associativity_of_meet) }
% 42.42/5.90 join(meet(X, Y), meet(X, join(meet(X, meet(Y, complement(join(meet(X, Y), complement(X))))), Z)))
% 42.42/5.90 = { by axiom 3 (commutativity_of_join) }
% 42.42/5.90 join(meet(X, Y), meet(X, join(meet(X, meet(Y, complement(join(complement(X), meet(X, Y))))), Z)))
% 42.42/5.90 = { by axiom 9 (associativity_of_meet) R->L }
% 42.42/5.90 join(meet(X, Y), meet(X, join(meet(meet(X, Y), complement(join(complement(X), meet(X, Y)))), Z)))
% 42.42/5.90 = { by lemma 29 R->L }
% 42.42/5.90 join(meet(X, Y), meet(join(meet(X, Y), complement(join(complement(X), meet(X, Y)))), join(meet(meet(X, Y), complement(join(complement(X), meet(X, Y)))), Z)))
% 42.42/5.90 = { by axiom 13 (equation_H61) R->L }
% 42.42/5.90 meet(join(meet(X, Y), complement(join(complement(X), meet(X, Y)))), join(meet(X, Y), Z))
% 42.42/5.90 = { by lemma 29 }
% 42.42/5.90 meet(X, join(meet(X, Y), Z))
% 42.42/5.91 = { by axiom 3 (commutativity_of_join) }
% 42.42/5.91 meet(X, join(Z, meet(X, Y)))
% 42.42/5.91
% 42.42/5.91 Lemma 36: join(meet(X, Y), meet(Y, Z)) = meet(Y, join(Z, meet(X, Y))).
% 42.42/5.91 Proof:
% 42.42/5.91 join(meet(X, Y), meet(Y, Z))
% 42.42/5.91 = { by axiom 1 (commutativity_of_meet) R->L }
% 42.42/5.91 join(meet(Y, X), meet(Y, Z))
% 42.42/5.91 = { by lemma 35 }
% 42.42/5.91 meet(Y, join(Z, meet(Y, X)))
% 42.42/5.91 = { by axiom 1 (commutativity_of_meet) }
% 42.42/5.91 meet(Y, join(Z, meet(X, Y)))
% 42.42/5.91
% 42.42/5.91 Lemma 37: meet(join(X, Y), join(Y, meet(join(Y, X), Z))) = meet(join(X, Y), join(Z, Y)).
% 42.42/5.91 Proof:
% 42.42/5.91 meet(join(X, Y), join(Y, meet(join(Y, X), Z)))
% 42.42/5.91 = { by axiom 3 (commutativity_of_join) R->L }
% 42.42/5.91 meet(join(Y, X), join(Y, meet(join(Y, X), Z)))
% 42.42/5.91 = { by axiom 1 (commutativity_of_meet) R->L }
% 42.42/5.91 meet(join(Y, X), join(Y, meet(Z, join(Y, X))))
% 42.42/5.91 = { by lemma 34 }
% 43.22/5.91 join(Y, meet(Z, join(Y, X)))
% 43.22/5.91 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.22/5.91 join(Y, meet(join(Y, X), Z))
% 43.22/5.91 = { by axiom 8 (absorption1) R->L }
% 43.22/5.91 join(meet(Y, join(Y, X)), meet(join(Y, X), Z))
% 43.22/5.91 = { by lemma 36 }
% 43.22/5.91 meet(join(Y, X), join(Z, meet(Y, join(Y, X))))
% 43.22/5.91 = { by axiom 8 (absorption1) }
% 43.22/5.91 meet(join(Y, X), join(Z, Y))
% 43.22/5.91 = { by axiom 3 (commutativity_of_join) }
% 43.22/5.91 meet(join(Y, X), join(Y, Z))
% 43.22/5.91 = { by axiom 3 (commutativity_of_join) }
% 43.22/5.91 meet(join(X, Y), join(Y, Z))
% 43.22/5.91 = { by axiom 3 (commutativity_of_join) }
% 43.22/5.91 meet(join(X, Y), join(Z, Y))
% 43.22/5.91
% 43.22/5.91 Lemma 38: meet(join(X, Y), join(X, complement(join(X, Y)))) = X.
% 43.22/5.91 Proof:
% 43.22/5.91 meet(join(X, Y), join(X, complement(join(X, Y))))
% 43.22/5.91 = { by axiom 3 (commutativity_of_join) R->L }
% 43.22/5.91 meet(join(Y, X), join(X, complement(join(X, Y))))
% 43.22/5.91 = { by axiom 3 (commutativity_of_join) R->L }
% 43.22/5.91 meet(join(Y, X), join(complement(join(X, Y)), X))
% 43.22/5.91 = { by lemma 37 R->L }
% 43.22/5.91 meet(join(Y, X), join(X, meet(join(X, Y), complement(join(X, Y)))))
% 43.22/5.91 = { by axiom 4 (complement_meet) }
% 43.22/5.91 meet(join(Y, X), join(X, zero))
% 43.22/5.91 = { by lemma 16 }
% 43.22/5.91 meet(join(Y, X), X)
% 43.22/5.91 = { by axiom 1 (commutativity_of_meet) }
% 43.22/5.91 meet(X, join(Y, X))
% 43.22/5.91 = { by lemma 23 }
% 43.22/5.91 X
% 43.22/5.91
% 43.22/5.91 Lemma 39: meet(X, join(Y, complement(join(X, Y)))) = meet(X, Y).
% 43.22/5.91 Proof:
% 43.22/5.91 meet(X, join(Y, complement(join(X, Y))))
% 43.22/5.91 = { by axiom 3 (commutativity_of_join) R->L }
% 43.22/5.91 meet(X, join(Y, complement(join(Y, X))))
% 43.22/5.91 = { by lemma 32 R->L }
% 43.22/5.91 meet(X, meet(join(Y, complement(join(Y, X))), join(Y, X)))
% 43.22/5.91 = { by axiom 1 (commutativity_of_meet) }
% 43.22/5.91 meet(X, meet(join(Y, X), join(Y, complement(join(Y, X)))))
% 43.22/5.91 = { by lemma 38 }
% 43.22/5.91 meet(X, Y)
% 43.22/5.91
% 43.22/5.91 Lemma 40: join(X, complement(join(X, meet(Y, complement(X))))) = complement(meet(Y, complement(X))).
% 43.22/5.91 Proof:
% 43.22/5.91 join(X, complement(join(X, meet(Y, complement(X)))))
% 43.22/5.91 = { by lemma 15 R->L }
% 43.22/5.91 complement(complement(join(X, complement(join(X, meet(Y, complement(X)))))))
% 43.22/5.91 = { by axiom 3 (commutativity_of_join) R->L }
% 43.22/5.91 complement(complement(join(X, complement(join(meet(Y, complement(X)), X)))))
% 43.22/5.91 = { by axiom 7 (meet_join_complement) R->L }
% 43.22/5.91 complement(fresh2(one, one, join(X, complement(join(meet(Y, complement(X)), X))), meet(Y, complement(X))))
% 43.22/5.91 = { by lemma 28 R->L }
% 43.22/5.91 complement(fresh2(join(meet(Y, complement(X)), join(X, complement(join(meet(Y, complement(X)), X)))), one, join(X, complement(join(meet(Y, complement(X)), X))), meet(Y, complement(X))))
% 43.22/5.91 = { by lemma 14 }
% 43.22/5.91 complement(fresh(meet(meet(Y, complement(X)), join(X, complement(join(meet(Y, complement(X)), X)))), zero, join(X, complement(join(meet(Y, complement(X)), X))), meet(Y, complement(X))))
% 43.22/5.91 = { by lemma 39 }
% 43.22/5.91 complement(fresh(meet(meet(Y, complement(X)), X), zero, join(X, complement(join(meet(Y, complement(X)), X))), meet(Y, complement(X))))
% 43.22/5.91 = { by axiom 1 (commutativity_of_meet) }
% 43.22/5.91 complement(fresh(meet(X, meet(Y, complement(X))), zero, join(X, complement(join(meet(Y, complement(X)), X))), meet(Y, complement(X))))
% 43.22/5.91 = { by axiom 3 (commutativity_of_join) }
% 43.22/5.91 complement(fresh(meet(X, meet(Y, complement(X))), zero, join(X, complement(join(X, meet(Y, complement(X))))), meet(Y, complement(X))))
% 43.22/5.91 = { by lemma 27 }
% 43.22/5.91 complement(fresh(zero, zero, join(X, complement(join(X, meet(Y, complement(X))))), meet(Y, complement(X))))
% 43.22/5.91 = { by axiom 6 (meet_join_complement) }
% 43.22/5.91 complement(meet(Y, complement(X)))
% 43.22/5.91
% 43.22/5.91 Lemma 41: join(X, meet(Y, meet(Z, X))) = X.
% 43.22/5.91 Proof:
% 43.22/5.91 join(X, meet(Y, meet(Z, X)))
% 43.22/5.91 = { by axiom 9 (associativity_of_meet) R->L }
% 43.22/5.91 join(X, meet(meet(Y, Z), X))
% 43.22/5.91 = { by lemma 21 }
% 43.22/5.91 X
% 43.22/5.91
% 43.22/5.91 Lemma 42: join(X, meet(Y, join(Z, X))) = meet(join(Y, X), join(Z, X)).
% 43.22/5.91 Proof:
% 43.22/5.91 join(X, meet(Y, join(Z, X)))
% 43.22/5.91 = { by lemma 38 R->L }
% 43.22/5.91 join(X, meet(Y, join(Z, meet(join(X, W), join(X, complement(join(X, W)))))))
% 43.22/5.91 = { by lemma 38 R->L }
% 43.22/5.91 join(meet(join(X, W), join(X, complement(join(X, W)))), meet(Y, join(Z, meet(join(X, W), join(X, complement(join(X, W)))))))
% 43.22/5.91 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.22/5.91 join(meet(join(X, complement(join(X, W))), join(X, W)), meet(Y, join(Z, meet(join(X, W), join(X, complement(join(X, W)))))))
% 43.22/5.91 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.22/5.91 join(meet(join(X, complement(join(X, W))), join(X, W)), meet(Y, join(Z, meet(join(X, complement(join(X, W))), join(X, W)))))
% 43.22/5.91 = { by axiom 3 (commutativity_of_join) R->L }
% 43.22/5.91 join(meet(join(X, complement(join(X, W))), join(X, W)), meet(Y, join(meet(join(X, complement(join(X, W))), join(X, W)), Z)))
% 43.22/5.91 = { by lemma 34 R->L }
% 43.22/5.91 meet(join(meet(join(X, complement(join(X, W))), join(X, W)), Z), join(meet(join(X, complement(join(X, W))), join(X, W)), meet(Y, join(meet(join(X, complement(join(X, W))), join(X, W)), Z))))
% 43.22/5.91 = { by axiom 3 (commutativity_of_join) }
% 43.22/5.91 meet(join(Z, meet(join(X, complement(join(X, W))), join(X, W))), join(meet(join(X, complement(join(X, W))), join(X, W)), meet(Y, join(meet(join(X, complement(join(X, W))), join(X, W)), Z))))
% 43.22/5.91 = { by axiom 3 (commutativity_of_join) }
% 43.22/5.91 meet(join(Z, meet(join(X, complement(join(X, W))), join(X, W))), join(meet(join(X, complement(join(X, W))), join(X, W)), meet(Y, join(Z, meet(join(X, complement(join(X, W))), join(X, W))))))
% 43.22/5.91 = { by axiom 1 (commutativity_of_meet) }
% 43.22/5.91 meet(join(Z, meet(join(X, complement(join(X, W))), join(X, W))), join(meet(join(X, complement(join(X, W))), join(X, W)), meet(join(Z, meet(join(X, complement(join(X, W))), join(X, W))), Y)))
% 43.22/5.91 = { by lemma 41 R->L }
% 43.22/5.91 meet(join(Z, meet(join(X, complement(join(X, W))), join(join(X, W), meet(join(X, complement(join(X, W))), meet(join(X, complement(join(X, W))), join(X, W)))))), join(meet(join(X, complement(join(X, W))), join(X, W)), meet(join(Z, meet(join(X, complement(join(X, W))), join(X, W))), Y)))
% 43.22/5.91 = { by lemma 35 R->L }
% 43.22/5.91 meet(join(Z, join(meet(join(X, complement(join(X, W))), meet(join(X, complement(join(X, W))), join(X, W))), meet(join(X, complement(join(X, W))), join(X, W)))), join(meet(join(X, complement(join(X, W))), join(X, W)), meet(join(Z, meet(join(X, complement(join(X, W))), join(X, W))), Y)))
% 43.22/5.91 = { by axiom 11 (associativity_of_join) R->L }
% 43.22/5.91 meet(join(join(Z, meet(join(X, complement(join(X, W))), meet(join(X, complement(join(X, W))), join(X, W)))), meet(join(X, complement(join(X, W))), join(X, W))), join(meet(join(X, complement(join(X, W))), join(X, W)), meet(join(Z, meet(join(X, complement(join(X, W))), join(X, W))), Y)))
% 43.22/5.91 = { by axiom 3 (commutativity_of_join) R->L }
% 43.22/5.91 meet(join(join(Z, meet(join(X, complement(join(X, W))), meet(join(X, complement(join(X, W))), join(X, W)))), meet(join(X, complement(join(X, W))), join(X, W))), join(meet(join(X, complement(join(X, W))), join(X, W)), meet(join(meet(join(X, complement(join(X, W))), join(X, W)), Z), Y)))
% 43.22/5.91 = { by lemma 30 R->L }
% 43.22/5.91 meet(join(join(Z, meet(join(X, complement(join(X, W))), meet(join(X, complement(join(X, W))), join(X, W)))), meet(join(X, complement(join(X, W))), join(X, W))), join(meet(join(X, complement(join(X, W))), join(X, W)), meet(join(meet(join(X, complement(join(X, W))), join(X, W)), join(Z, meet(join(X, complement(join(X, W))), meet(join(X, complement(join(X, W))), join(X, W))))), Y)))
% 43.22/5.91 = { by lemma 37 }
% 43.22/5.91 meet(join(join(Z, meet(join(X, complement(join(X, W))), meet(join(X, complement(join(X, W))), join(X, W)))), meet(join(X, complement(join(X, W))), join(X, W))), join(Y, meet(join(X, complement(join(X, W))), join(X, W))))
% 43.22/5.91 = { by axiom 11 (associativity_of_join) }
% 43.22/5.91 meet(join(Z, join(meet(join(X, complement(join(X, W))), meet(join(X, complement(join(X, W))), join(X, W))), meet(join(X, complement(join(X, W))), join(X, W)))), join(Y, meet(join(X, complement(join(X, W))), join(X, W))))
% 43.22/5.91 = { by lemma 35 }
% 43.22/5.91 meet(join(Z, meet(join(X, complement(join(X, W))), join(join(X, W), meet(join(X, complement(join(X, W))), meet(join(X, complement(join(X, W))), join(X, W)))))), join(Y, meet(join(X, complement(join(X, W))), join(X, W))))
% 43.22/5.91 = { by lemma 41 }
% 43.22/5.91 meet(join(Z, meet(join(X, complement(join(X, W))), join(X, W))), join(Y, meet(join(X, complement(join(X, W))), join(X, W))))
% 43.22/5.91 = { by axiom 1 (commutativity_of_meet) }
% 43.22/5.91 meet(join(Y, meet(join(X, complement(join(X, W))), join(X, W))), join(Z, meet(join(X, complement(join(X, W))), join(X, W))))
% 43.22/5.91 = { by axiom 1 (commutativity_of_meet) }
% 43.22/5.91 meet(join(Y, meet(join(X, W), join(X, complement(join(X, W))))), join(Z, meet(join(X, complement(join(X, W))), join(X, W))))
% 43.22/5.91 = { by axiom 1 (commutativity_of_meet) }
% 43.22/5.91 meet(join(Y, meet(join(X, W), join(X, complement(join(X, W))))), join(Z, meet(join(X, W), join(X, complement(join(X, W))))))
% 43.22/5.91 = { by lemma 38 }
% 43.22/5.91 meet(join(Y, X), join(Z, meet(join(X, W), join(X, complement(join(X, W))))))
% 43.22/5.91 = { by lemma 38 }
% 43.22/5.91 meet(join(Y, X), join(Z, X))
% 43.22/5.91
% 43.22/5.91 Lemma 43: complement(join(complement(X), meet(X, Y))) = meet(X, complement(meet(X, Y))).
% 43.22/5.91 Proof:
% 43.22/5.91 complement(join(complement(X), meet(X, Y)))
% 43.22/5.91 = { by lemma 16 R->L }
% 43.22/5.91 join(complement(join(complement(X), meet(X, Y))), zero)
% 43.22/5.91 = { by axiom 4 (complement_meet) R->L }
% 43.22/5.91 join(complement(join(complement(X), meet(X, Y))), meet(join(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), complement(complement(join(complement(X), meet(X, Y)))))), complement(join(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), complement(complement(join(complement(X), meet(X, Y)))))))))
% 43.22/5.91 = { by lemma 17 R->L }
% 43.22/5.91 join(complement(join(complement(X), meet(X, Y))), meet(join(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), complement(complement(join(complement(X), meet(X, Y)))))), join(zero, complement(join(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), complement(complement(join(complement(X), meet(X, Y))))))))))
% 43.22/5.91 = { by lemma 27 R->L }
% 43.22/5.91 join(complement(join(complement(X), meet(X, Y))), meet(join(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), complement(complement(join(complement(X), meet(X, Y)))))), join(meet(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), complement(complement(join(complement(X), meet(X, Y)))))), complement(join(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), complement(complement(join(complement(X), meet(X, Y))))))))))
% 43.22/5.91 = { by axiom 13 (equation_H61) R->L }
% 43.22/5.91 meet(join(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), complement(complement(join(complement(X), meet(X, Y)))))), join(complement(join(complement(X), meet(X, Y))), complement(join(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), complement(complement(join(complement(X), meet(X, Y)))))))))
% 43.22/5.91 = { by lemma 40 }
% 43.22/5.91 meet(join(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), complement(complement(join(complement(X), meet(X, Y)))))), complement(meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), complement(complement(join(complement(X), meet(X, Y)))))))
% 43.22/5.91 = { by axiom 1 (commutativity_of_meet) }
% 43.22/5.91 meet(complement(meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), complement(complement(join(complement(X), meet(X, Y)))))), join(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), complement(complement(join(complement(X), meet(X, Y)))))))
% 43.22/5.92 = { by lemma 15 }
% 43.22/5.92 meet(complement(meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), join(complement(X), meet(X, Y)))), join(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), complement(complement(join(complement(X), meet(X, Y)))))))
% 43.22/5.92 = { by lemma 15 }
% 43.22/5.92 meet(complement(meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), join(complement(X), meet(X, Y)))), join(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), join(complement(X), meet(X, Y)))))
% 43.22/5.92 = { by axiom 1 (commutativity_of_meet) }
% 43.22/5.92 meet(complement(meet(join(complement(X), complement(join(complement(X), meet(X, Y)))), join(complement(X), meet(X, Y)))), join(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), meet(X, Y)), join(complement(X), complement(join(complement(X), meet(X, Y)))))))
% 43.22/5.92 = { by axiom 1 (commutativity_of_meet) }
% 43.22/5.92 meet(complement(meet(join(complement(X), meet(X, Y)), join(complement(X), complement(join(complement(X), meet(X, Y)))))), join(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), meet(X, Y)), join(complement(X), complement(join(complement(X), meet(X, Y)))))))
% 43.22/5.92 = { by lemma 38 }
% 43.22/5.92 meet(complement(complement(X)), join(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), meet(X, Y)), join(complement(X), complement(join(complement(X), meet(X, Y)))))))
% 43.22/5.92 = { by lemma 15 }
% 43.22/5.92 meet(X, join(complement(join(complement(X), meet(X, Y))), meet(join(complement(X), meet(X, Y)), join(complement(X), complement(join(complement(X), meet(X, Y)))))))
% 43.22/5.92 = { by lemma 42 }
% 43.22/5.92 meet(X, meet(join(join(complement(X), meet(X, Y)), complement(join(complement(X), meet(X, Y)))), join(complement(X), complement(join(complement(X), meet(X, Y))))))
% 43.22/5.92 = { by axiom 11 (associativity_of_join) }
% 43.22/5.92 meet(X, meet(join(complement(X), join(meet(X, Y), complement(join(complement(X), meet(X, Y))))), join(complement(X), complement(join(complement(X), meet(X, Y))))))
% 43.22/5.92 = { by lemma 28 }
% 43.22/5.92 meet(X, meet(one, join(complement(X), complement(join(complement(X), meet(X, Y))))))
% 43.22/5.92 = { by lemma 19 }
% 43.22/5.92 meet(X, join(complement(X), complement(join(complement(X), meet(X, Y)))))
% 43.22/5.92 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.22/5.92 meet(X, join(complement(X), complement(join(complement(X), meet(Y, X)))))
% 43.22/5.92 = { by lemma 15 R->L }
% 43.22/5.92 meet(X, join(complement(X), complement(join(complement(X), meet(Y, complement(complement(X)))))))
% 43.22/5.92 = { by lemma 40 }
% 43.22/5.92 meet(X, complement(meet(Y, complement(complement(X)))))
% 43.22/5.92 = { by lemma 15 }
% 43.22/5.92 meet(X, complement(meet(Y, X)))
% 43.22/5.92 = { by axiom 1 (commutativity_of_meet) }
% 43.22/5.92 meet(X, complement(meet(X, Y)))
% 43.31/5.92
% 43.31/5.92 Lemma 44: join(X, meet(join(X, Y), join(X, Z))) = meet(join(X, Y), join(X, Z)).
% 43.31/5.92 Proof:
% 43.31/5.92 join(X, meet(join(X, Y), join(X, Z)))
% 43.31/5.92 = { by axiom 13 (equation_H61) }
% 43.31/5.92 join(X, join(X, meet(join(X, Y), join(meet(X, Y), Z))))
% 43.31/5.92 = { by axiom 11 (associativity_of_join) R->L }
% 43.31/5.92 join(join(X, X), meet(join(X, Y), join(meet(X, Y), Z)))
% 43.31/5.92 = { by axiom 2 (idempotence_of_join) }
% 43.31/5.92 join(X, meet(join(X, Y), join(meet(X, Y), Z)))
% 43.31/5.92 = { by axiom 13 (equation_H61) R->L }
% 43.31/5.92 meet(join(X, Y), join(X, Z))
% 43.31/5.92
% 43.31/5.92 Lemma 45: meet(complement(X), join(X, Y)) = meet(Y, complement(meet(X, Y))).
% 43.31/5.92 Proof:
% 43.31/5.92 meet(complement(X), join(X, Y))
% 43.31/5.92 = { by lemma 29 R->L }
% 43.31/5.92 meet(complement(X), join(X, join(meet(Y, X), complement(join(complement(Y), meet(Y, X))))))
% 43.31/5.92 = { by axiom 3 (commutativity_of_join) R->L }
% 43.31/5.92 meet(complement(X), join(X, join(complement(join(complement(Y), meet(Y, X))), meet(Y, X))))
% 43.31/5.92 = { by lemma 30 }
% 43.31/5.92 meet(complement(X), join(X, complement(join(complement(Y), meet(Y, X)))))
% 43.31/5.92 = { by lemma 43 }
% 43.31/5.92 meet(complement(X), join(X, meet(Y, complement(meet(Y, X)))))
% 43.31/5.92 = { by axiom 1 (commutativity_of_meet) }
% 43.31/5.92 meet(complement(X), join(X, meet(Y, complement(meet(X, Y)))))
% 43.31/5.92 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.31/5.92 meet(join(X, meet(Y, complement(meet(X, Y)))), complement(X))
% 43.31/5.92 = { by lemma 38 R->L }
% 43.31/5.92 meet(join(X, meet(Y, complement(meet(X, Y)))), complement(meet(join(X, meet(Y, complement(meet(X, Y)))), join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))))))
% 43.31/5.92 = { by lemma 43 R->L }
% 43.31/5.92 complement(join(complement(join(X, meet(Y, complement(meet(X, Y))))), meet(join(X, meet(Y, complement(meet(X, Y)))), join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))))))
% 43.31/5.92 = { by axiom 3 (commutativity_of_join) R->L }
% 43.31/5.92 complement(join(complement(join(X, meet(Y, complement(meet(X, Y))))), meet(join(X, meet(Y, complement(meet(X, Y)))), join(complement(join(X, meet(Y, complement(meet(X, Y))))), X))))
% 43.31/5.92 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.31/5.92 complement(join(complement(join(X, meet(Y, complement(meet(X, Y))))), meet(join(complement(join(X, meet(Y, complement(meet(X, Y))))), X), join(X, meet(Y, complement(meet(X, Y)))))))
% 43.31/5.92 = { by axiom 3 (commutativity_of_join) R->L }
% 43.31/5.92 complement(join(complement(join(X, meet(Y, complement(meet(X, Y))))), meet(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), join(X, meet(Y, complement(meet(X, Y)))))))
% 43.31/5.92 = { by lemma 44 R->L }
% 43.31/5.92 complement(join(complement(join(X, meet(Y, complement(meet(X, Y))))), join(X, meet(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), join(X, meet(Y, complement(meet(X, Y))))))))
% 43.31/5.92 = { by axiom 11 (associativity_of_join) R->L }
% 43.31/5.92 complement(join(join(complement(join(X, meet(Y, complement(meet(X, Y))))), X), meet(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), join(X, meet(Y, complement(meet(X, Y)))))))
% 43.31/5.92 = { by lemma 44 R->L }
% 43.31/5.92 complement(join(join(complement(join(X, meet(Y, complement(meet(X, Y))))), X), join(X, meet(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), join(X, meet(Y, complement(meet(X, Y))))))))
% 43.31/5.92 = { by axiom 3 (commutativity_of_join) }
% 43.31/5.92 complement(join(join(complement(join(X, meet(Y, complement(meet(X, Y))))), X), join(X, meet(join(complement(join(X, meet(Y, complement(meet(X, Y))))), X), join(X, meet(Y, complement(meet(X, Y))))))))
% 43.31/5.92 = { by lemma 22 }
% 43.31/5.92 complement(join(join(complement(join(X, meet(Y, complement(meet(X, Y))))), X), X))
% 43.31/5.92 = { by axiom 11 (associativity_of_join) }
% 43.31/5.92 complement(join(complement(join(X, meet(Y, complement(meet(X, Y))))), join(X, X)))
% 43.31/5.92 = { by axiom 2 (idempotence_of_join) }
% 43.31/5.92 complement(join(complement(join(X, meet(Y, complement(meet(X, Y))))), X))
% 43.31/5.92 = { by axiom 3 (commutativity_of_join) }
% 43.31/5.92 complement(join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))))
% 43.31/5.92 = { by axiom 3 (commutativity_of_join) R->L }
% 43.31/5.92 complement(join(X, complement(join(meet(Y, complement(meet(X, Y))), X))))
% 43.31/5.92 = { by axiom 7 (meet_join_complement) R->L }
% 43.31/5.92 fresh2(one, one, join(X, complement(join(meet(Y, complement(meet(X, Y))), X))), meet(Y, complement(meet(X, Y))))
% 43.31/5.92 = { by lemma 28 R->L }
% 43.31/5.92 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))))
% 43.31/5.92 = { by lemma 14 }
% 43.31/5.92 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))))
% 43.31/5.92 = { by lemma 39 }
% 43.31/5.92 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))))
% 43.31/5.92 = { by axiom 1 (commutativity_of_meet) }
% 43.31/5.92 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))))
% 43.31/5.92 = { by axiom 3 (commutativity_of_join) }
% 43.31/5.92 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))))
% 43.31/5.92 = { by axiom 9 (associativity_of_meet) R->L }
% 43.31/5.92 fresh(meet(meet(X, Y), complement(meet(X, Y))), zero, join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), meet(Y, complement(meet(X, Y))))
% 43.31/5.92 = { by axiom 4 (complement_meet) }
% 43.31/5.92 fresh(zero, zero, join(X, complement(join(X, meet(Y, complement(meet(X, Y)))))), meet(Y, complement(meet(X, Y))))
% 43.31/5.92 = { by axiom 6 (meet_join_complement) }
% 43.31/5.92 meet(Y, complement(meet(X, Y)))
% 43.31/5.92
% 43.31/5.92 Lemma 46: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 43.31/5.92 Proof:
% 43.31/5.92 meet(X, complement(meet(X, Y)))
% 43.31/5.92 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.31/5.92 meet(X, complement(meet(Y, X)))
% 43.31/5.92 = { by axiom 2 (idempotence_of_join) R->L }
% 43.31/5.92 meet(X, complement(join(meet(Y, X), meet(Y, X))))
% 43.31/5.92 = { by lemma 20 R->L }
% 43.31/5.92 meet(X, meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(Y, X), meet(meet(Y, X), Z))), meet(X, complement(join(meet(Y, X), meet(Y, X)))))))
% 43.31/5.92 = { by axiom 3 (commutativity_of_join) }
% 43.31/5.92 meet(X, meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(X, complement(join(meet(Y, X), meet(Y, X)))), meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(Y, X), meet(meet(Y, X), Z))))))
% 43.31/5.92 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.31/5.92 meet(X, meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(X, complement(join(meet(Y, X), meet(Y, X)))), meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X)))))))
% 43.31/5.92 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.31/5.92 meet(X, meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(complement(join(meet(Y, X), meet(Y, X))), X), meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X)))))))
% 43.31/5.92 = { by axiom 3 (commutativity_of_join) R->L }
% 43.31/5.92 meet(X, meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X)))), meet(complement(join(meet(Y, X), meet(Y, X))), X))))
% 43.31/5.92 = { by lemma 21 R->L }
% 43.31/5.92 meet(X, meet(join(complement(join(meet(Y, X), meet(Y, X))), meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X))))), join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X)))), meet(complement(join(meet(Y, X), meet(Y, X))), X))))
% 43.31/5.92 = { by lemma 24 }
% 43.31/5.92 meet(X, join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X)))), meet(complement(join(meet(Y, X), meet(Y, X))), X)))
% 43.31/5.92 = { by axiom 3 (commutativity_of_join) }
% 43.31/5.92 meet(X, join(meet(complement(join(meet(Y, X), meet(Y, X))), X), meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X))))))
% 43.31/5.92 = { by axiom 1 (commutativity_of_meet) }
% 43.31/5.92 meet(X, join(meet(X, complement(join(meet(Y, X), meet(Y, X)))), meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X))))))
% 43.31/5.92 = { by axiom 1 (commutativity_of_meet) }
% 43.31/5.92 meet(X, join(meet(X, complement(join(meet(Y, X), meet(Y, X)))), meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(Y, X), meet(meet(Y, X), Z)))))
% 43.31/5.92 = { by axiom 3 (commutativity_of_join) }
% 43.31/5.92 meet(X, join(meet(complement(join(meet(Y, X), meet(Y, X))), join(meet(Y, X), meet(meet(Y, X), Z))), meet(X, complement(join(meet(Y, X), meet(Y, X))))))
% 43.31/5.92 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.31/5.92 meet(X, join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), meet(Y, X)))), meet(X, complement(join(meet(Y, X), meet(Y, X))))))
% 43.31/5.92 = { by lemma 22 R->L }
% 43.31/5.92 meet(X, join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(meet(Y, X), join(meet(Y, X), meet(meet(Y, X), Z))))), meet(X, complement(join(meet(Y, X), meet(Y, X))))))
% 43.31/5.93 = { by axiom 3 (commutativity_of_join) R->L }
% 43.31/5.93 meet(X, join(meet(join(meet(Y, X), meet(meet(Y, X), Z)), complement(join(join(meet(Y, X), meet(meet(Y, X), Z)), meet(Y, X)))), meet(X, complement(join(meet(Y, X), meet(Y, X))))))
% 43.31/5.93 = { by lemma 26 }
% 43.31/5.93 meet(X, join(zero, meet(X, complement(join(meet(Y, X), meet(Y, X))))))
% 43.31/5.93 = { by lemma 17 }
% 43.31/5.93 meet(X, meet(X, complement(join(meet(Y, X), meet(Y, X)))))
% 43.31/5.93 = { by axiom 2 (idempotence_of_join) }
% 43.31/5.93 meet(X, meet(X, complement(meet(Y, X))))
% 43.31/5.93 = { by lemma 45 R->L }
% 43.31/5.93 meet(X, meet(complement(Y), join(Y, X)))
% 43.31/5.93 = { by lemma 32 }
% 43.31/5.93 meet(X, complement(Y))
% 43.31/5.93
% 43.31/5.93 Lemma 47: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 43.31/5.93 Proof:
% 43.31/5.93 meet(complement(X), join(X, Y))
% 43.31/5.93 = { by lemma 45 }
% 43.31/5.93 meet(Y, complement(meet(X, Y)))
% 43.31/5.93 = { by axiom 1 (commutativity_of_meet) }
% 43.31/5.93 meet(Y, complement(meet(Y, X)))
% 43.31/5.93 = { by lemma 46 }
% 43.31/5.93 meet(Y, complement(X))
% 43.31/5.93
% 43.31/5.93 Lemma 48: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 43.31/5.93 Proof:
% 43.31/5.93 complement(meet(X, complement(Y)))
% 43.31/5.93 = { by lemma 46 R->L }
% 43.31/5.93 complement(meet(X, complement(meet(X, Y))))
% 43.31/5.93 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.31/5.93 complement(meet(X, complement(meet(Y, X))))
% 43.31/5.93 = { by lemma 15 R->L }
% 43.31/5.93 complement(meet(X, complement(meet(Y, complement(complement(X))))))
% 43.31/5.93 = { by lemma 47 R->L }
% 43.31/5.93 complement(meet(X, complement(meet(complement(complement(X)), join(complement(X), Y)))))
% 43.31/5.93 = { by lemma 15 }
% 43.31/5.93 complement(meet(X, complement(meet(X, join(complement(X), Y)))))
% 43.31/5.93 = { by axiom 3 (commutativity_of_join) }
% 43.31/5.93 complement(meet(X, complement(meet(X, join(Y, complement(X))))))
% 43.31/5.93 = { by lemma 46 }
% 43.31/5.93 complement(meet(X, complement(join(Y, complement(X)))))
% 43.31/5.93 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.31/5.93 complement(meet(complement(join(Y, complement(X))), X))
% 43.31/5.93 = { by lemma 29 R->L }
% 43.31/5.93 complement(meet(complement(join(Y, complement(X))), join(meet(X, join(Y, complement(X))), complement(join(complement(X), meet(X, join(Y, complement(X))))))))
% 43.31/5.93 = { by lemma 42 }
% 43.31/5.93 complement(meet(complement(join(Y, complement(X))), join(meet(X, join(Y, complement(X))), complement(meet(join(X, complement(X)), join(Y, complement(X)))))))
% 43.31/5.93 = { by axiom 5 (complement_join) }
% 43.31/5.93 complement(meet(complement(join(Y, complement(X))), join(meet(X, join(Y, complement(X))), complement(meet(one, join(Y, complement(X)))))))
% 43.31/5.93 = { by lemma 19 }
% 43.31/5.93 complement(meet(complement(join(Y, complement(X))), join(meet(X, join(Y, complement(X))), complement(join(Y, complement(X))))))
% 43.31/5.93 = { by axiom 3 (commutativity_of_join) }
% 43.31/5.93 complement(meet(complement(join(Y, complement(X))), join(complement(join(Y, complement(X))), meet(X, join(Y, complement(X))))))
% 43.31/5.93 = { by axiom 8 (absorption1) }
% 43.31/5.93 complement(complement(join(Y, complement(X))))
% 43.31/5.93 = { by lemma 15 }
% 43.31/5.93 join(Y, complement(X))
% 43.31/5.93
% 43.31/5.93 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 43.31/5.93 Proof:
% 43.31/5.93 meet(a, join(b, c))
% 43.31/5.93 = { by lemma 15 R->L }
% 43.31/5.93 meet(a, join(b, complement(complement(c))))
% 43.31/5.93 = { by lemma 48 R->L }
% 43.31/5.93 meet(a, complement(meet(complement(c), complement(b))))
% 43.31/5.93 = { by lemma 46 R->L }
% 43.31/5.93 meet(a, complement(meet(a, meet(complement(c), complement(b)))))
% 43.31/5.93 = { by lemma 31 }
% 43.31/5.93 meet(a, complement(meet(complement(c), meet(a, complement(b)))))
% 43.31/5.93 = { by lemma 47 R->L }
% 43.31/5.93 meet(a, complement(meet(complement(c), meet(complement(b), join(b, a)))))
% 43.31/5.93 = { by lemma 45 }
% 43.31/5.93 meet(a, complement(meet(complement(c), meet(a, complement(meet(b, a))))))
% 43.31/5.93 = { by lemma 31 R->L }
% 43.31/5.93 meet(a, complement(meet(a, meet(complement(c), complement(meet(b, a))))))
% 43.31/5.93 = { by lemma 46 }
% 43.31/5.93 meet(a, complement(meet(complement(c), complement(meet(b, a)))))
% 43.31/5.93 = { by lemma 48 }
% 43.31/5.93 meet(a, join(meet(b, a), complement(complement(c))))
% 43.31/5.93 = { by axiom 3 (commutativity_of_join) R->L }
% 43.31/5.93 meet(a, join(complement(complement(c)), meet(b, a)))
% 43.31/5.93 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.31/5.93 meet(a, join(complement(complement(c)), meet(a, b)))
% 43.31/5.93 = { by lemma 35 R->L }
% 43.31/5.93 join(meet(a, b), meet(a, complement(complement(c))))
% 43.31/5.93 = { by axiom 3 (commutativity_of_join) }
% 43.31/5.93 join(meet(a, complement(complement(c))), meet(a, b))
% 43.31/5.93 = { by lemma 35 }
% 43.31/5.93 meet(a, join(b, meet(a, complement(complement(c)))))
% 43.31/5.93 = { by lemma 15 }
% 43.31/5.93 meet(a, join(b, meet(a, c)))
% 43.31/5.93 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.31/5.93 meet(a, join(b, meet(c, a)))
% 43.31/5.93 = { by lemma 36 R->L }
% 43.31/5.93 join(meet(c, a), meet(a, b))
% 43.31/5.93 = { by lemma 33 R->L }
% 43.31/5.93 join(meet(c, a), join(meet(a, b), meet(a, join(meet(c, a), meet(a, b)))))
% 43.31/5.93 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.31/5.93 join(meet(c, a), join(meet(b, a), meet(a, join(meet(c, a), meet(a, b)))))
% 43.31/5.93 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.31/5.93 join(meet(c, a), join(meet(b, a), meet(a, join(meet(c, a), meet(b, a)))))
% 43.31/5.93 = { by axiom 3 (commutativity_of_join) R->L }
% 43.31/5.93 join(meet(c, a), join(meet(a, join(meet(c, a), meet(b, a))), meet(b, a)))
% 43.31/5.93 = { by lemma 20 R->L }
% 43.31/5.93 join(meet(c, a), join(meet(a, join(meet(c, a), meet(b, a))), meet(b, meet(a, join(meet(c, a), meet(b, a))))))
% 43.31/5.93 = { by lemma 21 }
% 43.31/5.93 join(meet(c, a), meet(a, join(meet(c, a), meet(b, a))))
% 43.31/5.93 = { by axiom 1 (commutativity_of_meet) }
% 43.31/5.93 join(meet(c, a), meet(a, join(meet(c, a), meet(a, b))))
% 43.31/5.93 = { by axiom 3 (commutativity_of_join) }
% 43.31/5.93 join(meet(c, a), meet(a, join(meet(a, b), meet(c, a))))
% 43.31/5.93 = { by lemma 36 }
% 43.31/5.93 meet(a, join(join(meet(a, b), meet(c, a)), meet(c, a)))
% 43.31/5.93 = { by axiom 11 (associativity_of_join) }
% 43.31/5.93 meet(a, join(meet(a, b), join(meet(c, a), meet(c, a))))
% 43.31/5.93 = { by lemma 35 }
% 43.31/5.93 meet(a, join(meet(a, b), meet(c, join(a, meet(c, a)))))
% 43.31/5.93 = { by lemma 21 }
% 43.31/5.93 meet(a, join(meet(a, b), meet(c, a)))
% 43.31/5.93 = { by axiom 1 (commutativity_of_meet) R->L }
% 43.31/5.93 meet(a, join(meet(a, b), meet(a, c)))
% 43.31/5.93 = { by axiom 10 (absorption2) R->L }
% 43.31/5.93 meet(join(a, meet(a, b)), join(meet(a, b), meet(a, c)))
% 43.31/5.93 = { by lemma 24 }
% 43.31/5.93 join(meet(a, b), meet(a, c))
% 43.31/5.93 % SZS output end Proof
% 43.31/5.93
% 43.31/5.93 RESULT: Unsatisfiable (the axioms are contradictory).
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