TSTP Solution File: LAT217-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT217-1 : 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 06:27:48 EDT 2023
% Result : Unsatisfiable 246.81s 32.00s
% Output : Proof 247.38s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LAT217-1 : 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
% 0.13/0.36 % Computer : n028.cluster.edu
% 0.13/0.36 % Model : x86_64 x86_64
% 0.13/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.36 % Memory : 8042.1875MB
% 0.13/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.36 % CPULimit : 300
% 0.13/0.36 % WCLimit : 300
% 0.13/0.36 % DateTime : Thu Aug 24 09:44:34 EDT 2023
% 0.13/0.36 % CPUTime :
% 246.81/32.00 Command-line arguments: --no-flatten-goal
% 246.81/32.00
% 246.81/32.00 % SZS status Unsatisfiable
% 246.81/32.00
% 247.38/32.13 % SZS output start Proof
% 247.38/32.13 Take the following subset of the input axioms:
% 247.38/32.13 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 247.38/32.13 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 247.38/32.13 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 247.38/32.13 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 247.38/32.13 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 247.38/32.13 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 247.38/32.13 fof(complement_join, axiom, ![X2]: join(X2, complement(X2))=one).
% 247.38/32.13 fof(complement_meet, axiom, ![X2]: meet(X2, complement(X2))=zero).
% 247.38/32.13 fof(equation_H82, axiom, ![X2, Y2, Z2]: meet(X2, join(meet(Y2, join(X2, Z2)), meet(Z2, join(X2, Y2))))=join(meet(X2, Y2), meet(X2, Z2))).
% 247.38/32.13 fof(meet_join_complement, axiom, ![X2, Y2]: (meet(X2, Y2)!=zero | (join(X2, Y2)!=one | complement(X2)=Y2))).
% 247.38/32.13 fof(prove_distributivity, negated_conjecture, meet(a, join(b, c))!=join(meet(a, b), meet(a, c))).
% 247.38/32.13
% 247.38/32.13 Now clausify the problem and encode Horn clauses using encoding 3 of
% 247.38/32.13 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 247.38/32.13 We repeatedly replace C & s=t => u=v by the two clauses:
% 247.38/32.13 fresh(y, y, x1...xn) = u
% 247.38/32.13 C => fresh(s, t, x1...xn) = v
% 247.38/32.13 where fresh is a fresh function symbol and x1..xn are the free
% 247.38/32.13 variables of u and v.
% 247.38/32.13 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 247.38/32.13 input problem has no model of domain size 1).
% 247.38/32.13
% 247.38/32.13 The encoding turns the above axioms into the following unit equations and goals:
% 247.38/32.13
% 247.38/32.13 Axiom 1 (commutativity_of_join): join(X, Y) = join(Y, X).
% 247.38/32.13 Axiom 2 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 247.38/32.13 Axiom 3 (complement_join): join(X, complement(X)) = one.
% 247.38/32.13 Axiom 4 (complement_meet): meet(X, complement(X)) = zero.
% 247.38/32.13 Axiom 5 (meet_join_complement): fresh(X, X, Y, Z) = Z.
% 247.38/32.13 Axiom 6 (meet_join_complement): fresh2(X, X, Y, Z) = complement(Y).
% 247.38/32.13 Axiom 7 (absorption2): join(X, meet(X, Y)) = X.
% 247.38/32.13 Axiom 8 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 247.38/32.13 Axiom 9 (absorption1): meet(X, join(X, Y)) = X.
% 247.38/32.13 Axiom 10 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 247.38/32.13 Axiom 11 (meet_join_complement): fresh2(join(X, Y), one, X, Y) = fresh(meet(X, Y), zero, X, Y).
% 247.38/32.13 Axiom 12 (equation_H82): meet(X, join(meet(Y, join(X, Z)), meet(Z, join(X, Y)))) = join(meet(X, Y), meet(X, Z)).
% 247.38/32.13
% 247.38/32.13 Lemma 13: meet(X, one) = X.
% 247.38/32.13 Proof:
% 247.38/32.13 meet(X, one)
% 247.38/32.13 = { by axiom 3 (complement_join) R->L }
% 247.38/32.13 meet(X, join(X, complement(X)))
% 247.38/32.13 = { by axiom 9 (absorption1) }
% 247.38/32.13 X
% 247.38/32.13
% 247.38/32.13 Lemma 14: meet(one, X) = X.
% 247.38/32.13 Proof:
% 247.38/32.13 meet(one, X)
% 247.38/32.13 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.13 meet(X, one)
% 247.38/32.13 = { by lemma 13 }
% 247.38/32.13 X
% 247.38/32.13
% 247.38/32.13 Lemma 15: complement(one) = zero.
% 247.38/32.13 Proof:
% 247.38/32.13 complement(one)
% 247.38/32.13 = { by lemma 14 R->L }
% 247.38/32.13 meet(one, complement(one))
% 247.38/32.13 = { by axiom 4 (complement_meet) }
% 247.38/32.13 zero
% 247.38/32.13
% 247.38/32.13 Lemma 16: fresh2(join(X, Y), one, Y, X) = fresh(meet(X, Y), zero, Y, X).
% 247.38/32.13 Proof:
% 247.38/32.13 fresh2(join(X, Y), one, Y, X)
% 247.38/32.13 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.13 fresh2(join(Y, X), one, Y, X)
% 247.38/32.13 = { by axiom 11 (meet_join_complement) }
% 247.38/32.13 fresh(meet(Y, X), zero, Y, X)
% 247.38/32.13 = { by axiom 2 (commutativity_of_meet) }
% 247.38/32.13 fresh(meet(X, Y), zero, Y, X)
% 247.38/32.13
% 247.38/32.13 Lemma 17: complement(complement(X)) = X.
% 247.38/32.13 Proof:
% 247.38/32.13 complement(complement(X))
% 247.38/32.13 = { by axiom 6 (meet_join_complement) R->L }
% 247.38/32.13 fresh2(one, one, complement(X), X)
% 247.38/32.13 = { by axiom 3 (complement_join) R->L }
% 247.38/32.13 fresh2(join(X, complement(X)), one, complement(X), X)
% 247.38/32.13 = { by lemma 16 }
% 247.38/32.13 fresh(meet(X, complement(X)), zero, complement(X), X)
% 247.38/32.13 = { by axiom 4 (complement_meet) }
% 247.38/32.13 fresh(zero, zero, complement(X), X)
% 247.38/32.13 = { by axiom 5 (meet_join_complement) }
% 247.38/32.13 X
% 247.38/32.13
% 247.38/32.13 Lemma 18: join(X, one) = one.
% 247.38/32.13 Proof:
% 247.38/32.13 join(X, one)
% 247.38/32.13 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.13 join(one, X)
% 247.38/32.13 = { by lemma 14 R->L }
% 247.38/32.13 join(one, meet(one, X))
% 247.38/32.13 = { by axiom 7 (absorption2) }
% 247.38/32.13 one
% 247.38/32.13
% 247.38/32.13 Lemma 19: join(X, zero) = X.
% 247.38/32.13 Proof:
% 247.38/32.13 join(X, zero)
% 247.38/32.13 = { by axiom 4 (complement_meet) R->L }
% 247.38/32.13 join(X, meet(X, complement(X)))
% 247.38/32.13 = { by axiom 7 (absorption2) }
% 247.38/32.13 X
% 247.38/32.13
% 247.38/32.13 Lemma 20: meet(X, join(Y, X)) = X.
% 247.38/32.13 Proof:
% 247.38/32.13 meet(X, join(Y, X))
% 247.38/32.13 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.13 meet(X, join(X, Y))
% 247.38/32.13 = { by axiom 9 (absorption1) }
% 247.38/32.13 X
% 247.38/32.13
% 247.38/32.13 Lemma 21: meet(X, meet(Y, join(Z, X))) = meet(X, Y).
% 247.38/32.13 Proof:
% 247.38/32.13 meet(X, meet(Y, join(Z, X)))
% 247.38/32.13 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.13 meet(X, meet(join(Z, X), Y))
% 247.38/32.13 = { by axiom 10 (associativity_of_meet) R->L }
% 247.38/32.13 meet(meet(X, join(Z, X)), Y)
% 247.38/32.13 = { by axiom 2 (commutativity_of_meet) }
% 247.38/32.13 meet(Y, meet(X, join(Z, X)))
% 247.38/32.13 = { by lemma 20 }
% 247.38/32.13 meet(Y, X)
% 247.38/32.13 = { by axiom 2 (commutativity_of_meet) }
% 247.38/32.13 meet(X, Y)
% 247.38/32.13
% 247.38/32.13 Lemma 22: meet(X, join(Y, meet(complement(X), join(Y, X)))) = meet(X, Y).
% 247.38/32.13 Proof:
% 247.38/32.13 meet(X, join(Y, meet(complement(X), join(Y, X))))
% 247.38/32.13 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.13 meet(X, join(Y, meet(complement(X), join(X, Y))))
% 247.38/32.13 = { by lemma 13 R->L }
% 247.38/32.13 meet(X, join(meet(Y, one), meet(complement(X), join(X, Y))))
% 247.38/32.13 = { by axiom 3 (complement_join) R->L }
% 247.38/32.13 meet(X, join(meet(Y, join(X, complement(X))), meet(complement(X), join(X, Y))))
% 247.38/32.13 = { by axiom 12 (equation_H82) }
% 247.38/32.13 join(meet(X, Y), meet(X, complement(X)))
% 247.38/32.13 = { by axiom 4 (complement_meet) }
% 247.38/32.13 join(meet(X, Y), zero)
% 247.38/32.13 = { by lemma 19 }
% 247.38/32.13 meet(X, Y)
% 247.38/32.13
% 247.38/32.13 Lemma 23: join(X, join(Y, complement(X))) = one.
% 247.38/32.13 Proof:
% 247.38/32.14 join(X, join(Y, complement(X)))
% 247.38/32.14 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.14 join(X, join(complement(X), Y))
% 247.38/32.14 = { by axiom 8 (associativity_of_join) R->L }
% 247.38/32.14 join(join(X, complement(X)), Y)
% 247.38/32.14 = { by axiom 3 (complement_join) }
% 247.38/32.14 join(one, Y)
% 247.38/32.14 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.14 join(Y, one)
% 247.38/32.14 = { by lemma 18 }
% 247.38/32.14 one
% 247.38/32.14
% 247.38/32.14 Lemma 24: fresh(meet(complement(X), join(X, Y)), zero, complement(X), join(X, Y)) = X.
% 247.38/32.14 Proof:
% 247.38/32.14 fresh(meet(complement(X), join(X, Y)), zero, complement(X), join(X, Y))
% 247.38/32.14 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.14 fresh(meet(complement(X), join(Y, X)), zero, complement(X), join(X, Y))
% 247.38/32.14 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.14 fresh(meet(complement(X), join(Y, X)), zero, complement(X), join(Y, X))
% 247.38/32.14 = { by lemma 17 R->L }
% 247.38/32.14 fresh(meet(complement(X), join(Y, X)), zero, complement(X), join(Y, complement(complement(X))))
% 247.38/32.14 = { by lemma 17 R->L }
% 247.38/32.14 fresh(meet(complement(X), join(Y, complement(complement(X)))), zero, complement(X), join(Y, complement(complement(X))))
% 247.38/32.14 = { by axiom 11 (meet_join_complement) R->L }
% 247.38/32.14 fresh2(join(complement(X), join(Y, complement(complement(X)))), one, complement(X), join(Y, complement(complement(X))))
% 247.38/32.14 = { by lemma 23 }
% 247.38/32.14 fresh2(one, one, complement(X), join(Y, complement(complement(X))))
% 247.38/32.14 = { by axiom 6 (meet_join_complement) }
% 247.38/32.14 complement(complement(X))
% 247.38/32.14 = { by lemma 17 }
% 247.38/32.14 X
% 247.38/32.14
% 247.38/32.14 Lemma 25: join(X, complement(join(Y, complement(X)))) = X.
% 247.38/32.14 Proof:
% 247.38/32.14 join(X, complement(join(Y, complement(X))))
% 247.38/32.14 = { by axiom 5 (meet_join_complement) R->L }
% 247.38/32.14 fresh(zero, zero, complement(X), join(X, complement(join(Y, complement(X)))))
% 247.38/32.14 = { by axiom 4 (complement_meet) R->L }
% 247.38/32.14 fresh(meet(X, complement(X)), zero, complement(X), join(X, complement(join(Y, complement(X)))))
% 247.38/32.14 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.14 fresh(meet(complement(X), X), zero, complement(X), join(X, complement(join(Y, complement(X)))))
% 247.38/32.14 = { by lemma 21 R->L }
% 247.38/32.14 fresh(meet(complement(X), meet(X, join(Y, complement(X)))), zero, complement(X), join(X, complement(join(Y, complement(X)))))
% 247.38/32.14 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.14 fresh(meet(complement(X), meet(join(Y, complement(X)), X)), zero, complement(X), join(X, complement(join(Y, complement(X)))))
% 247.38/32.14 = { by lemma 22 R->L }
% 247.38/32.14 fresh(meet(complement(X), meet(join(Y, complement(X)), join(X, meet(complement(join(Y, complement(X))), join(X, join(Y, complement(X))))))), zero, complement(X), join(X, complement(join(Y, complement(X)))))
% 247.38/32.14 = { by lemma 23 }
% 247.38/32.14 fresh(meet(complement(X), meet(join(Y, complement(X)), join(X, meet(complement(join(Y, complement(X))), one)))), zero, complement(X), join(X, complement(join(Y, complement(X)))))
% 247.38/32.14 = { by lemma 13 }
% 247.38/32.14 fresh(meet(complement(X), meet(join(Y, complement(X)), join(X, complement(join(Y, complement(X)))))), zero, complement(X), join(X, complement(join(Y, complement(X)))))
% 247.38/32.14 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.14 fresh(meet(complement(X), meet(join(X, complement(join(Y, complement(X)))), join(Y, complement(X)))), zero, complement(X), join(X, complement(join(Y, complement(X)))))
% 247.38/32.14 = { by lemma 21 }
% 247.38/32.14 fresh(meet(complement(X), join(X, complement(join(Y, complement(X))))), zero, complement(X), join(X, complement(join(Y, complement(X)))))
% 247.38/32.14 = { by lemma 24 }
% 247.38/32.14 X
% 247.38/32.14
% 247.38/32.14 Lemma 26: meet(X, complement(join(Y, complement(X)))) = complement(join(Y, complement(X))).
% 247.38/32.14 Proof:
% 247.38/32.14 meet(X, complement(join(Y, complement(X))))
% 247.38/32.14 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.14 meet(complement(join(Y, complement(X))), X)
% 247.38/32.14 = { by lemma 25 R->L }
% 247.38/32.14 meet(complement(join(Y, complement(X))), join(X, complement(join(Y, complement(X)))))
% 247.38/32.14 = { by lemma 20 }
% 247.38/32.14 complement(join(Y, complement(X)))
% 247.38/32.14
% 247.38/32.14 Lemma 27: join(complement(X), complement(join(Y, X))) = complement(X).
% 247.38/32.14 Proof:
% 247.38/32.14 join(complement(X), complement(join(Y, X)))
% 247.38/32.14 = { by lemma 17 R->L }
% 247.38/32.14 join(complement(X), complement(join(Y, complement(complement(X)))))
% 247.38/32.14 = { by lemma 25 }
% 247.38/32.14 complement(X)
% 247.38/32.14
% 247.38/32.14 Lemma 28: join(complement(X), complement(meet(Y, X))) = complement(meet(X, Y)).
% 247.38/32.14 Proof:
% 247.38/32.14 join(complement(X), complement(meet(Y, X)))
% 247.38/32.14 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.14 join(complement(X), complement(meet(X, Y)))
% 247.38/32.14 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.14 join(complement(meet(X, Y)), complement(X))
% 247.38/32.14 = { by axiom 7 (absorption2) R->L }
% 247.38/32.14 join(complement(meet(X, Y)), complement(join(X, meet(X, Y))))
% 247.38/32.14 = { by lemma 27 }
% 247.38/32.14 complement(meet(X, Y))
% 247.38/32.14
% 247.38/32.14 Lemma 29: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 247.38/32.14 Proof:
% 247.38/32.14 join(complement(X), complement(Y))
% 247.38/32.14 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.14 join(complement(Y), complement(X))
% 247.38/32.14 = { by lemma 25 R->L }
% 247.38/32.14 join(join(complement(Y), complement(X)), complement(join(meet(Y, X), complement(join(complement(Y), complement(X))))))
% 247.38/32.14 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.14 join(join(complement(Y), complement(X)), complement(join(meet(X, Y), complement(join(complement(Y), complement(X))))))
% 247.38/32.14 = { by lemma 26 R->L }
% 247.38/32.14 join(join(complement(Y), complement(X)), complement(join(meet(X, Y), meet(X, complement(join(complement(Y), complement(X)))))))
% 247.38/32.14 = { by axiom 1 (commutativity_of_join) }
% 247.38/32.14 join(join(complement(Y), complement(X)), complement(join(meet(X, Y), meet(X, complement(join(complement(X), complement(Y)))))))
% 247.38/32.14 = { by lemma 26 R->L }
% 247.38/32.14 join(join(complement(Y), complement(X)), complement(join(meet(X, Y), meet(X, meet(Y, complement(join(complement(X), complement(Y))))))))
% 247.38/32.14 = { by axiom 10 (associativity_of_meet) R->L }
% 247.38/32.14 join(join(complement(Y), complement(X)), complement(join(meet(X, Y), meet(meet(X, Y), complement(join(complement(X), complement(Y)))))))
% 247.38/32.14 = { by axiom 7 (absorption2) }
% 247.38/32.14 join(join(complement(Y), complement(X)), complement(meet(X, Y)))
% 247.38/32.14 = { by axiom 2 (commutativity_of_meet) }
% 247.38/32.14 join(join(complement(Y), complement(X)), complement(meet(Y, X)))
% 247.38/32.14 = { by axiom 8 (associativity_of_join) }
% 247.38/32.14 join(complement(Y), join(complement(X), complement(meet(Y, X))))
% 247.38/32.14 = { by lemma 28 }
% 247.38/32.14 join(complement(Y), complement(meet(X, Y)))
% 247.38/32.14 = { by lemma 28 }
% 247.38/32.14 complement(meet(Y, X))
% 247.38/32.14 = { by axiom 2 (commutativity_of_meet) }
% 247.38/32.14 complement(meet(X, Y))
% 247.38/32.14
% 247.38/32.14 Lemma 30: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 247.38/32.14 Proof:
% 247.38/32.14 complement(meet(X, complement(Y)))
% 247.38/32.14 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.14 complement(meet(complement(Y), X))
% 247.38/32.14 = { by lemma 29 R->L }
% 247.38/32.14 join(complement(complement(Y)), complement(X))
% 247.38/32.14 = { by lemma 17 }
% 247.38/32.14 join(Y, complement(X))
% 247.38/32.14
% 247.38/32.14 Lemma 31: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 247.38/32.14 Proof:
% 247.38/32.14 complement(meet(complement(X), Y))
% 247.38/32.14 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.14 complement(meet(Y, complement(X)))
% 247.38/32.14 = { by lemma 30 }
% 247.38/32.14 join(X, complement(Y))
% 247.38/32.14
% 247.38/32.14 Lemma 32: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 247.38/32.14 Proof:
% 247.38/32.14 complement(join(X, complement(Y)))
% 247.38/32.14 = { by lemma 30 R->L }
% 247.38/32.14 complement(complement(meet(Y, complement(X))))
% 247.38/32.14 = { by lemma 17 }
% 247.38/32.14 meet(Y, complement(X))
% 247.38/32.14
% 247.38/32.14 Lemma 33: join(X, meet(Y, X)) = X.
% 247.38/32.14 Proof:
% 247.38/32.14 join(X, meet(Y, X))
% 247.38/32.14 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.14 join(X, meet(X, Y))
% 247.38/32.14 = { by axiom 7 (absorption2) }
% 247.38/32.14 X
% 247.38/32.14
% 247.38/32.14 Lemma 34: fresh(meet(X, join(complement(X), Y)), zero, join(complement(X), Y), X) = meet(X, complement(Y)).
% 247.38/32.14 Proof:
% 247.38/32.14 fresh(meet(X, join(complement(X), Y)), zero, join(complement(X), Y), X)
% 247.38/32.14 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.14 fresh(meet(X, join(complement(X), Y)), zero, join(Y, complement(X)), X)
% 247.38/32.14 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.14 fresh(meet(X, join(Y, complement(X))), zero, join(Y, complement(X)), X)
% 247.38/32.14 = { by lemma 16 R->L }
% 247.38/32.14 fresh2(join(X, join(Y, complement(X))), one, join(Y, complement(X)), X)
% 247.38/32.14 = { by lemma 23 }
% 247.38/32.14 fresh2(one, one, join(Y, complement(X)), X)
% 247.38/32.14 = { by axiom 6 (meet_join_complement) }
% 247.38/32.14 complement(join(Y, complement(X)))
% 247.38/32.14 = { by lemma 32 }
% 247.38/32.14 meet(X, complement(Y))
% 247.38/32.14
% 247.38/32.14 Lemma 35: join(meet(X, Y), meet(X, join(Z, meet(X, Y)))) = meet(X, join(Z, meet(X, Y))).
% 247.38/32.14 Proof:
% 247.38/32.14 join(meet(X, Y), meet(X, join(Z, meet(X, Y))))
% 247.38/32.14 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.14 join(meet(Y, X), meet(X, join(Z, meet(X, Y))))
% 247.38/32.14 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.14 join(meet(Y, X), meet(X, join(Z, meet(Y, X))))
% 247.38/32.14 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.14 join(meet(X, join(Z, meet(Y, X))), meet(Y, X))
% 247.38/32.14 = { by axiom 9 (absorption1) R->L }
% 247.38/32.14 join(meet(X, join(Z, meet(Y, X))), meet(meet(Y, X), join(meet(Y, X), Z)))
% 247.38/32.14 = { by axiom 10 (associativity_of_meet) }
% 247.38/32.14 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(meet(Y, X), Z))))
% 247.38/32.14 = { by axiom 1 (commutativity_of_join) }
% 247.38/32.15 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(Z, meet(Y, X)))))
% 247.38/32.15 = { by lemma 33 }
% 247.38/32.15 meet(X, join(Z, meet(Y, X)))
% 247.38/32.15 = { by axiom 2 (commutativity_of_meet) }
% 247.38/32.15 meet(X, join(Z, meet(X, Y)))
% 247.38/32.15
% 247.38/32.15 Lemma 36: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 247.38/32.15 Proof:
% 247.38/32.15 join(X, join(Y, meet(Z, join(X, Y))))
% 247.38/32.15 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.15 join(X, join(Y, meet(join(X, Y), Z)))
% 247.38/32.15 = { by axiom 8 (associativity_of_join) R->L }
% 247.38/32.15 join(join(X, Y), meet(join(X, Y), Z))
% 247.38/32.15 = { by axiom 7 (absorption2) }
% 247.38/32.15 join(X, Y)
% 247.38/32.15
% 247.38/32.15 Lemma 37: join(X, meet(Y, join(meet(Y, Z), X))) = join(X, meet(Y, Z)).
% 247.38/32.15 Proof:
% 247.38/32.15 join(X, meet(Y, join(meet(Y, Z), X)))
% 247.38/32.15 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.15 join(X, meet(Y, join(X, meet(Y, Z))))
% 247.38/32.15 = { by lemma 35 R->L }
% 247.38/32.15 join(X, join(meet(Y, Z), meet(Y, join(X, meet(Y, Z)))))
% 247.38/32.15 = { by lemma 36 }
% 247.38/32.15 join(X, meet(Y, Z))
% 247.38/32.15
% 247.38/32.15 Lemma 38: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 247.38/32.15 Proof:
% 247.38/32.15 meet(X, meet(Y, join(X, Z)))
% 247.38/32.15 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.15 meet(X, meet(join(X, Z), Y))
% 247.38/32.15 = { by axiom 10 (associativity_of_meet) R->L }
% 247.38/32.15 meet(meet(X, join(X, Z)), Y)
% 247.38/32.15 = { by axiom 9 (absorption1) }
% 247.38/32.15 meet(X, Y)
% 247.38/32.15
% 247.38/32.15 Lemma 39: join(X, join(Y, complement(join(X, Y)))) = one.
% 247.38/32.15 Proof:
% 247.38/32.15 join(X, join(Y, complement(join(X, Y))))
% 247.38/32.15 = { by axiom 8 (associativity_of_join) R->L }
% 247.38/32.15 join(join(X, Y), complement(join(X, Y)))
% 247.38/32.15 = { by axiom 3 (complement_join) }
% 247.38/32.15 one
% 247.38/32.15
% 247.38/32.15 Lemma 40: meet(X, join(Y, complement(X))) = meet(X, Y).
% 247.38/32.15 Proof:
% 247.38/32.15 meet(X, join(Y, complement(X)))
% 247.38/32.15 = { by lemma 31 R->L }
% 247.38/32.15 meet(X, complement(meet(complement(Y), X)))
% 247.38/32.15 = { by lemma 34 R->L }
% 247.38/32.15 fresh(meet(X, join(complement(X), meet(complement(Y), X))), zero, join(complement(X), meet(complement(Y), X)), X)
% 247.38/32.15 = { by lemma 37 R->L }
% 247.38/32.15 fresh(meet(X, join(complement(X), meet(complement(Y), X))), zero, join(complement(X), meet(complement(Y), join(meet(complement(Y), X), complement(X)))), X)
% 247.38/32.15 = { by lemma 37 R->L }
% 247.38/32.15 fresh(meet(X, join(complement(X), meet(complement(Y), join(meet(complement(Y), X), complement(X))))), zero, join(complement(X), meet(complement(Y), join(meet(complement(Y), X), complement(X)))), X)
% 247.38/32.15 = { by lemma 34 }
% 247.38/32.15 meet(X, complement(meet(complement(Y), join(meet(complement(Y), X), complement(X)))))
% 247.38/32.15 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.15 meet(X, complement(meet(join(meet(complement(Y), X), complement(X)), complement(Y))))
% 247.38/32.15 = { by lemma 32 R->L }
% 247.38/32.15 meet(X, complement(complement(join(Y, complement(join(meet(complement(Y), X), complement(X)))))))
% 247.38/32.15 = { by axiom 6 (meet_join_complement) R->L }
% 247.38/32.15 meet(X, complement(fresh2(one, one, join(Y, complement(join(meet(complement(Y), X), complement(X)))), join(X, join(meet(complement(Y), X), complement(X))))))
% 247.38/32.15 = { by lemma 18 R->L }
% 247.38/32.15 meet(X, complement(fresh2(join(X, one), one, join(Y, complement(join(meet(complement(Y), X), complement(X)))), join(X, join(meet(complement(Y), X), complement(X))))))
% 247.38/32.15 = { by lemma 23 R->L }
% 247.38/32.15 meet(X, complement(fresh2(join(X, join(join(meet(complement(Y), X), complement(X)), join(Y, complement(join(meet(complement(Y), X), complement(X)))))), one, join(Y, complement(join(meet(complement(Y), X), complement(X)))), join(X, join(meet(complement(Y), X), complement(X))))))
% 247.38/32.15 = { by axiom 8 (associativity_of_join) R->L }
% 247.38/32.15 meet(X, complement(fresh2(join(join(X, join(meet(complement(Y), X), complement(X))), join(Y, complement(join(meet(complement(Y), X), complement(X))))), one, join(Y, complement(join(meet(complement(Y), X), complement(X)))), join(X, join(meet(complement(Y), X), complement(X))))))
% 247.38/32.15 = { by lemma 16 }
% 247.38/32.15 meet(X, complement(fresh(meet(join(X, join(meet(complement(Y), X), complement(X))), join(Y, complement(join(meet(complement(Y), X), complement(X))))), zero, join(Y, complement(join(meet(complement(Y), X), complement(X)))), join(X, join(meet(complement(Y), X), complement(X))))))
% 247.38/32.15 = { by lemma 23 }
% 247.38/32.15 meet(X, complement(fresh(meet(one, join(Y, complement(join(meet(complement(Y), X), complement(X))))), zero, join(Y, complement(join(meet(complement(Y), X), complement(X)))), join(X, join(meet(complement(Y), X), complement(X))))))
% 247.38/32.15 = { by lemma 14 }
% 247.38/32.15 meet(X, complement(fresh(join(Y, complement(join(meet(complement(Y), X), complement(X)))), zero, join(Y, complement(join(meet(complement(Y), X), complement(X)))), join(X, join(meet(complement(Y), X), complement(X))))))
% 247.38/32.15 = { by lemma 32 }
% 247.38/32.15 meet(X, complement(fresh(join(Y, meet(X, complement(meet(complement(Y), X)))), zero, join(Y, complement(join(meet(complement(Y), X), complement(X)))), join(X, join(meet(complement(Y), X), complement(X))))))
% 247.38/32.15 = { by lemma 32 }
% 247.38/32.15 meet(X, complement(fresh(join(Y, meet(X, complement(meet(complement(Y), X)))), zero, join(Y, meet(X, complement(meet(complement(Y), X)))), join(X, join(meet(complement(Y), X), complement(X))))))
% 247.38/32.15 = { by lemma 23 }
% 247.38/32.15 meet(X, complement(fresh(join(Y, meet(X, complement(meet(complement(Y), X)))), zero, join(Y, meet(X, complement(meet(complement(Y), X)))), one)))
% 247.38/32.15 = { by lemma 13 R->L }
% 247.38/32.15 meet(X, complement(fresh(meet(join(Y, meet(X, complement(meet(complement(Y), X)))), one), zero, join(Y, meet(X, complement(meet(complement(Y), X)))), one)))
% 247.38/32.15 = { by axiom 11 (meet_join_complement) R->L }
% 247.38/32.15 meet(X, complement(fresh2(join(join(Y, meet(X, complement(meet(complement(Y), X)))), one), one, join(Y, meet(X, complement(meet(complement(Y), X)))), one)))
% 247.38/32.15 = { by lemma 18 }
% 247.38/32.15 meet(X, complement(fresh2(one, one, join(Y, meet(X, complement(meet(complement(Y), X)))), one)))
% 247.38/32.15 = { by axiom 6 (meet_join_complement) }
% 247.38/32.15 meet(X, complement(complement(join(Y, meet(X, complement(meet(complement(Y), X)))))))
% 247.38/32.15 = { by lemma 17 }
% 247.38/32.15 meet(X, join(Y, meet(X, complement(meet(complement(Y), X)))))
% 247.38/32.15 = { by lemma 31 }
% 247.38/32.15 meet(X, join(Y, meet(X, join(Y, complement(X)))))
% 247.38/32.15 = { by axiom 5 (meet_join_complement) R->L }
% 247.38/32.15 meet(X, fresh(zero, zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.15 = { by axiom 7 (absorption2) R->L }
% 247.38/32.15 meet(X, fresh(join(zero, meet(zero, join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(complement(join(Y, X)), Y))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.15 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.15 meet(X, fresh(join(meet(zero, join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(complement(join(Y, X)), Y)))), zero), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.15 = { by lemma 19 }
% 247.38/32.15 meet(X, fresh(meet(zero, join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(complement(join(Y, X)), Y)))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by axiom 2 (commutativity_of_meet) }
% 247.38/32.16 meet(X, fresh(meet(join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(complement(join(Y, X)), Y))), zero), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by axiom 4 (complement_meet) R->L }
% 247.38/32.16 meet(X, fresh(meet(join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(complement(join(Y, X)), Y))), meet(join(join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(complement(join(Y, X)), Y))), complement(join(Y, X))), complement(join(join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(complement(join(Y, X)), Y))), complement(join(Y, X)))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.16 meet(X, fresh(meet(join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(complement(join(Y, X)), Y))), meet(complement(join(join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(complement(join(Y, X)), Y))), complement(join(Y, X)))), join(join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(complement(join(Y, X)), Y))), complement(join(Y, X))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by lemma 38 }
% 247.38/32.16 meet(X, fresh(meet(join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(complement(join(Y, X)), Y))), complement(join(join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(complement(join(Y, X)), Y))), complement(join(Y, X))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by axiom 1 (commutativity_of_join) }
% 247.38/32.16 meet(X, fresh(meet(join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(complement(join(Y, X)), Y))), complement(join(complement(join(Y, X)), join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(complement(join(Y, X)), Y)))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by lemma 36 }
% 247.38/32.16 meet(X, fresh(meet(join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(complement(join(Y, X)), Y))), complement(join(complement(join(Y, X)), Y))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by axiom 2 (commutativity_of_meet) }
% 247.38/32.16 meet(X, fresh(meet(complement(join(complement(join(Y, X)), Y)), join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(complement(join(Y, X)), Y)))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by axiom 1 (commutativity_of_join) }
% 247.38/32.16 meet(X, fresh(meet(complement(join(complement(join(Y, X)), Y)), join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(Y, complement(join(Y, X)))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by axiom 1 (commutativity_of_join) }
% 247.38/32.16 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(X, meet(complement(Y), join(Y, X))), join(Y, complement(join(Y, X)))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by axiom 2 (commutativity_of_meet) }
% 247.38/32.16 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(Y, complement(join(Y, X))), join(X, meet(complement(Y), join(Y, X)))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.16 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(Y, complement(join(Y, X))), join(X, meet(join(Y, X), complement(Y)))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by lemma 32 R->L }
% 247.38/32.16 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(Y, complement(join(Y, X))), join(X, complement(join(Y, complement(join(Y, X)))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.16 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(Y, complement(join(X, Y))), join(X, complement(join(Y, complement(join(Y, X)))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.16 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(Y, complement(join(X, Y))), join(X, complement(join(Y, complement(join(X, Y)))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by lemma 13 R->L }
% 247.38/32.16 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(Y, complement(join(X, Y))), join(X, meet(complement(join(Y, complement(join(X, Y)))), one))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by lemma 39 R->L }
% 247.38/32.16 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(Y, complement(join(X, Y))), join(X, meet(complement(join(Y, complement(join(X, Y)))), join(X, join(Y, complement(join(X, Y))))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by lemma 22 }
% 247.38/32.16 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(Y, complement(join(X, Y))), X))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by lemma 19 R->L }
% 247.38/32.16 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(Y, complement(join(X, Y))), join(X, zero)))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by lemma 15 R->L }
% 247.38/32.16 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(Y, complement(join(X, Y))), join(X, complement(one))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by lemma 39 R->L }
% 247.38/32.16 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(Y, complement(join(X, Y))), join(X, complement(join(X, join(Y, complement(join(X, Y))))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.16 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(Y, complement(join(X, Y))), join(X, complement(join(join(Y, complement(join(X, Y))), X)))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.16 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(X, complement(join(join(Y, complement(join(X, Y))), X))), join(Y, complement(join(X, Y)))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by lemma 22 R->L }
% 247.38/32.16 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(X, complement(join(join(Y, complement(join(X, Y))), X))), join(join(Y, complement(join(X, Y))), meet(complement(join(X, complement(join(join(Y, complement(join(X, Y))), X)))), join(join(Y, complement(join(X, Y))), join(X, complement(join(join(Y, complement(join(X, Y))), X))))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.16 = { by lemma 39 }
% 247.38/32.16 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(X, complement(join(join(Y, complement(join(X, Y))), X))), join(join(Y, complement(join(X, Y))), meet(complement(join(X, complement(join(join(Y, complement(join(X, Y))), X)))), one))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by lemma 13 }
% 247.38/32.17 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(X, complement(join(join(Y, complement(join(X, Y))), X))), join(join(Y, complement(join(X, Y))), complement(join(X, complement(join(join(Y, complement(join(X, Y))), X)))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by axiom 1 (commutativity_of_join) }
% 247.38/32.17 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(X, complement(join(join(Y, complement(join(X, Y))), X))), join(join(Y, complement(join(X, Y))), complement(join(X, complement(join(X, join(Y, complement(join(X, Y))))))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by axiom 1 (commutativity_of_join) }
% 247.38/32.17 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(X, complement(join(X, join(Y, complement(join(X, Y)))))), join(join(Y, complement(join(X, Y))), complement(join(X, complement(join(X, join(Y, complement(join(X, Y))))))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by lemma 39 }
% 247.38/32.17 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(X, complement(one)), join(join(Y, complement(join(X, Y))), complement(join(X, complement(join(X, join(Y, complement(join(X, Y))))))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by lemma 15 }
% 247.38/32.17 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(join(X, zero), join(join(Y, complement(join(X, Y))), complement(join(X, complement(join(X, join(Y, complement(join(X, Y))))))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by lemma 19 }
% 247.38/32.17 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(X, join(join(Y, complement(join(X, Y))), complement(join(X, complement(join(X, join(Y, complement(join(X, Y))))))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by axiom 8 (associativity_of_join) }
% 247.38/32.17 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(X, join(Y, join(complement(join(X, Y)), complement(join(X, complement(join(X, join(Y, complement(join(X, Y)))))))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by lemma 39 }
% 247.38/32.17 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(X, join(Y, join(complement(join(X, Y)), complement(join(X, complement(one)))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by lemma 15 }
% 247.38/32.17 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(X, join(Y, join(complement(join(X, Y)), complement(join(X, zero))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by lemma 19 }
% 247.38/32.17 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(X, join(Y, join(complement(join(X, Y)), complement(X)))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by axiom 1 (commutativity_of_join) }
% 247.38/32.17 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(X, join(Y, join(complement(X), complement(join(X, Y))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by axiom 1 (commutativity_of_join) }
% 247.38/32.17 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(X, join(Y, join(complement(X), complement(join(Y, X))))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by lemma 27 }
% 247.38/32.17 meet(X, fresh(meet(complement(join(Y, complement(join(Y, X)))), join(Y, meet(X, join(Y, complement(X))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by lemma 32 }
% 247.38/32.17 meet(X, fresh(meet(meet(join(Y, X), complement(Y)), join(Y, meet(X, join(Y, complement(X))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by axiom 10 (associativity_of_meet) }
% 247.38/32.17 meet(X, fresh(meet(join(Y, X), meet(complement(Y), join(Y, meet(X, join(Y, complement(X)))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.17 meet(X, fresh(meet(join(X, Y), meet(complement(Y), join(Y, meet(X, join(Y, complement(X)))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.17 meet(X, fresh(meet(join(X, Y), meet(join(Y, meet(X, join(Y, complement(X)))), complement(Y))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by axiom 10 (associativity_of_meet) R->L }
% 247.38/32.17 meet(X, fresh(meet(meet(join(X, Y), join(Y, meet(X, join(Y, complement(X))))), complement(Y)), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.17 meet(X, fresh(meet(meet(join(Y, meet(X, join(Y, complement(X)))), join(X, Y)), complement(Y)), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by axiom 7 (absorption2) R->L }
% 247.38/32.17 meet(X, fresh(meet(meet(join(Y, meet(X, join(Y, complement(X)))), join(join(X, meet(X, join(Y, complement(X)))), Y)), complement(Y)), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by axiom 8 (associativity_of_join) }
% 247.38/32.17 meet(X, fresh(meet(meet(join(Y, meet(X, join(Y, complement(X)))), join(X, join(meet(X, join(Y, complement(X))), Y))), complement(Y)), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by axiom 1 (commutativity_of_join) }
% 247.38/32.17 meet(X, fresh(meet(meet(join(Y, meet(X, join(Y, complement(X)))), join(X, join(Y, meet(X, join(Y, complement(X)))))), complement(Y)), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by lemma 20 }
% 247.38/32.17 meet(X, fresh(meet(join(Y, meet(X, join(Y, complement(X)))), complement(Y)), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by axiom 2 (commutativity_of_meet) }
% 247.38/32.17 meet(X, fresh(meet(complement(Y), join(Y, meet(X, join(Y, complement(X))))), zero, complement(Y), join(Y, meet(X, join(Y, complement(X))))))
% 247.38/32.17 = { by lemma 24 }
% 247.38/32.17 meet(X, Y)
% 247.38/32.17
% 247.38/32.17 Lemma 41: join(meet(X, Y), meet(X, join(meet(X, Y), Z))) = meet(X, join(Z, meet(X, Y))).
% 247.38/32.17 Proof:
% 247.38/32.17 join(meet(X, Y), meet(X, join(meet(X, Y), Z)))
% 247.38/32.17 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.17 join(meet(X, Y), meet(X, join(Z, meet(X, Y))))
% 247.38/32.17 = { by lemma 35 }
% 247.38/32.17 meet(X, join(Z, meet(X, Y)))
% 247.38/32.17
% 247.38/32.17 Lemma 42: meet(X, join(Y, join(Z, complement(X)))) = meet(X, join(Y, Z)).
% 247.38/32.17 Proof:
% 247.38/32.17 meet(X, join(Y, join(Z, complement(X))))
% 247.38/32.17 = { by axiom 8 (associativity_of_join) R->L }
% 247.38/32.17 meet(X, join(join(Y, Z), complement(X)))
% 247.38/32.17 = { by lemma 40 }
% 247.38/32.17 meet(X, join(Y, Z))
% 247.38/32.17
% 247.38/32.17 Lemma 43: meet(X, join(Y, meet(X, Z))) = meet(X, join(Y, Z)).
% 247.38/32.17 Proof:
% 247.38/32.17 meet(X, join(Y, meet(X, Z)))
% 247.38/32.17 = { by lemma 41 R->L }
% 247.38/32.17 join(meet(X, Z), meet(X, join(meet(X, Z), Y)))
% 247.38/32.17 = { by lemma 42 R->L }
% 247.38/32.17 join(meet(X, Z), meet(X, join(meet(X, Z), join(Y, complement(X)))))
% 247.38/32.17 = { by lemma 41 }
% 247.38/32.17 meet(X, join(join(Y, complement(X)), meet(X, Z)))
% 247.38/32.17 = { by axiom 8 (associativity_of_join) }
% 247.38/32.17 meet(X, join(Y, join(complement(X), meet(X, Z))))
% 247.38/32.17 = { by axiom 1 (commutativity_of_join) R->L }
% 247.38/32.17 meet(X, join(Y, join(meet(X, Z), complement(X))))
% 247.38/32.17 = { by lemma 30 R->L }
% 247.38/32.17 meet(X, join(Y, complement(meet(X, complement(meet(X, Z))))))
% 247.38/32.17 = { by axiom 2 (commutativity_of_meet) R->L }
% 247.38/32.17 meet(X, join(Y, complement(meet(X, complement(meet(Z, X))))))
% 247.38/32.17 = { by lemma 29 R->L }
% 247.38/32.17 meet(X, join(Y, complement(meet(X, join(complement(Z), complement(X))))))
% 247.38/32.17 = { by lemma 40 }
% 247.38/32.17 meet(X, join(Y, complement(meet(X, complement(Z)))))
% 247.38/32.17 = { by lemma 30 }
% 247.38/32.17 meet(X, join(Y, join(Z, complement(X))))
% 247.38/32.17 = { by lemma 42 }
% 247.38/32.17 meet(X, join(Y, Z))
% 247.38/32.17
% 247.38/32.17 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 247.38/32.17 Proof:
% 247.38/32.17 meet(a, join(b, c))
% 247.38/32.17 = { by lemma 43 R->L }
% 247.38/32.17 meet(a, join(b, meet(a, c)))
% 247.38/32.17 = { by lemma 33 R->L }
% 247.38/32.17 join(meet(a, join(b, meet(a, c))), meet(b, meet(a, join(b, meet(a, c)))))
% 247.38/32.17 = { by lemma 38 }
% 247.38/32.17 join(meet(a, join(b, meet(a, c))), meet(b, a))
% 247.38/32.17 = { by axiom 1 (commutativity_of_join) }
% 247.38/32.17 join(meet(b, a), meet(a, join(b, meet(a, c))))
% 247.38/32.17 = { by axiom 2 (commutativity_of_meet) }
% 247.38/32.17 join(meet(a, b), meet(a, join(b, meet(a, c))))
% 247.38/32.17 = { by axiom 1 (commutativity_of_join) }
% 247.38/32.17 join(meet(a, b), meet(a, join(meet(a, c), b)))
% 247.38/32.17 = { by lemma 43 R->L }
% 247.38/32.17 join(meet(a, b), meet(a, join(meet(a, c), meet(a, b))))
% 247.38/32.17 = { by lemma 37 }
% 247.38/32.17 join(meet(a, b), meet(a, c))
% 247.38/32.17 % SZS output end Proof
% 247.38/32.17
% 247.38/32.17 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------