TSTP Solution File: LAT202-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT202-1 : 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 : n029.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 06:27:44 EDT 2023
% Result : Unsatisfiable 92.70s 12.31s
% Output : Proof 93.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LAT202-1 : 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
% 0.14/0.34 % Computer : n029.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Thu Aug 24 08:51:08 EDT 2023
% 0.14/0.35 % CPUTime :
% 92.70/12.31 Command-line arguments: --no-flatten-goal
% 92.70/12.31
% 92.70/12.31 % SZS status Unsatisfiable
% 92.70/12.31
% 93.22/12.40 % SZS output start Proof
% 93.22/12.40 Take the following subset of the input axioms:
% 93.22/12.40 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 93.22/12.40 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 93.22/12.40 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 93.22/12.40 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 93.22/12.40 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 93.22/12.40 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 93.22/12.40 fof(complement_join, axiom, ![X2]: join(X2, complement(X2))=one).
% 93.22/12.40 fof(complement_meet, axiom, ![X2]: meet(X2, complement(X2))=zero).
% 93.22/12.40 fof(equation_H55, axiom, ![X2, Y2, Z2]: join(X2, meet(Y2, join(X2, Z2)))=join(X2, meet(Y2, join(Z2, meet(X2, join(Z2, Y2)))))).
% 93.22/12.40 fof(idempotence_of_join, axiom, ![X2]: join(X2, X2)=X2).
% 93.22/12.40 fof(meet_join_complement, axiom, ![X2, Y2]: (meet(X2, Y2)!=zero | (join(X2, Y2)!=one | complement(X2)=Y2))).
% 93.22/12.40 fof(prove_distributivity, negated_conjecture, meet(a, join(b, c))!=join(meet(a, b), meet(a, c))).
% 93.22/12.40
% 93.22/12.40 Now clausify the problem and encode Horn clauses using encoding 3 of
% 93.22/12.40 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 93.22/12.40 We repeatedly replace C & s=t => u=v by the two clauses:
% 93.22/12.40 fresh(y, y, x1...xn) = u
% 93.22/12.40 C => fresh(s, t, x1...xn) = v
% 93.22/12.40 where fresh is a fresh function symbol and x1..xn are the free
% 93.22/12.40 variables of u and v.
% 93.22/12.40 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 93.22/12.40 input problem has no model of domain size 1).
% 93.22/12.40
% 93.22/12.40 The encoding turns the above axioms into the following unit equations and goals:
% 93.22/12.40
% 93.22/12.40 Axiom 1 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 93.22/12.40 Axiom 2 (idempotence_of_join): join(X, X) = X.
% 93.22/12.40 Axiom 3 (commutativity_of_join): join(X, Y) = join(Y, X).
% 93.22/12.40 Axiom 4 (complement_meet): meet(X, complement(X)) = zero.
% 93.22/12.40 Axiom 5 (complement_join): join(X, complement(X)) = one.
% 93.22/12.40 Axiom 6 (meet_join_complement): fresh(X, X, Y, Z) = Z.
% 93.22/12.40 Axiom 7 (meet_join_complement): fresh2(X, X, Y, Z) = complement(Y).
% 93.22/12.40 Axiom 8 (absorption1): meet(X, join(X, Y)) = X.
% 93.22/12.40 Axiom 9 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 93.22/12.40 Axiom 10 (absorption2): join(X, meet(X, Y)) = X.
% 93.22/12.40 Axiom 11 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 93.22/12.40 Axiom 12 (meet_join_complement): fresh2(join(X, Y), one, X, Y) = fresh(meet(X, Y), zero, X, Y).
% 93.22/12.40 Axiom 13 (equation_H55): join(X, meet(Y, join(X, Z))) = join(X, meet(Y, join(Z, meet(X, join(Z, Y))))).
% 93.22/12.40
% 93.22/12.40 Lemma 14: fresh2(join(X, Y), one, Y, X) = fresh(meet(X, Y), zero, Y, X).
% 93.22/12.40 Proof:
% 93.22/12.40 fresh2(join(X, Y), one, Y, X)
% 93.22/12.40 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.40 fresh2(join(Y, X), one, Y, X)
% 93.22/12.40 = { by axiom 12 (meet_join_complement) }
% 93.22/12.40 fresh(meet(Y, X), zero, Y, X)
% 93.22/12.40 = { by axiom 1 (commutativity_of_meet) }
% 93.22/12.40 fresh(meet(X, Y), zero, Y, X)
% 93.22/12.40
% 93.22/12.40 Lemma 15: complement(complement(X)) = X.
% 93.22/12.40 Proof:
% 93.22/12.40 complement(complement(X))
% 93.22/12.40 = { by axiom 7 (meet_join_complement) R->L }
% 93.22/12.40 fresh2(one, one, complement(X), X)
% 93.22/12.40 = { by axiom 5 (complement_join) R->L }
% 93.22/12.40 fresh2(join(X, complement(X)), one, complement(X), X)
% 93.22/12.40 = { by lemma 14 }
% 93.22/12.40 fresh(meet(X, complement(X)), zero, complement(X), X)
% 93.22/12.40 = { by axiom 4 (complement_meet) }
% 93.22/12.40 fresh(zero, zero, complement(X), X)
% 93.22/12.40 = { by axiom 6 (meet_join_complement) }
% 93.22/12.40 X
% 93.22/12.40
% 93.22/12.40 Lemma 16: join(X, zero) = X.
% 93.22/12.40 Proof:
% 93.22/12.40 join(X, zero)
% 93.22/12.40 = { by axiom 4 (complement_meet) R->L }
% 93.22/12.40 join(X, meet(X, complement(X)))
% 93.22/12.40 = { by axiom 10 (absorption2) }
% 93.22/12.40 X
% 93.22/12.40
% 93.22/12.40 Lemma 17: join(zero, X) = X.
% 93.22/12.40 Proof:
% 93.22/12.40 join(zero, X)
% 93.22/12.40 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.40 join(X, zero)
% 93.22/12.40 = { by lemma 16 }
% 93.22/12.40 X
% 93.22/12.40
% 93.22/12.40 Lemma 18: meet(X, zero) = zero.
% 93.22/12.40 Proof:
% 93.22/12.40 meet(X, zero)
% 93.22/12.40 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.40 meet(zero, X)
% 93.22/12.40 = { by lemma 17 R->L }
% 93.22/12.40 meet(zero, join(zero, X))
% 93.22/12.40 = { by axiom 8 (absorption1) }
% 93.22/12.40 zero
% 93.22/12.40
% 93.22/12.40 Lemma 19: meet(one, X) = X.
% 93.22/12.40 Proof:
% 93.22/12.40 meet(one, X)
% 93.22/12.40 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.40 meet(X, one)
% 93.22/12.40 = { by axiom 5 (complement_join) R->L }
% 93.22/12.40 meet(X, join(X, complement(X)))
% 93.22/12.40 = { by axiom 8 (absorption1) }
% 93.22/12.40 X
% 93.22/12.40
% 93.22/12.40 Lemma 20: meet(X, join(Y, X)) = X.
% 93.22/12.40 Proof:
% 93.22/12.40 meet(X, join(Y, X))
% 93.22/12.40 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.40 meet(X, join(X, Y))
% 93.22/12.40 = { by axiom 8 (absorption1) }
% 93.22/12.40 X
% 93.22/12.40
% 93.22/12.40 Lemma 21: join(X, meet(Y, X)) = X.
% 93.22/12.40 Proof:
% 93.22/12.40 join(X, meet(Y, X))
% 93.22/12.40 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.40 join(X, meet(X, Y))
% 93.22/12.40 = { by axiom 10 (absorption2) }
% 93.22/12.40 X
% 93.22/12.40
% 93.22/12.40 Lemma 22: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 93.22/12.40 Proof:
% 93.22/12.40 join(Y, join(X, Z))
% 93.22/12.40 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.40 join(join(X, Z), Y)
% 93.22/12.40 = { by axiom 11 (associativity_of_join) }
% 93.22/12.40 join(X, join(Z, Y))
% 93.22/12.40 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.40 join(X, join(Y, Z))
% 93.22/12.40
% 93.22/12.40 Lemma 23: meet(X, meet(join(X, Y), Z)) = meet(X, Z).
% 93.22/12.40 Proof:
% 93.22/12.40 meet(X, meet(join(X, Y), Z))
% 93.22/12.40 = { by axiom 9 (associativity_of_meet) R->L }
% 93.22/12.40 meet(meet(X, join(X, Y)), Z)
% 93.22/12.40 = { by axiom 8 (absorption1) }
% 93.22/12.40 meet(X, Z)
% 93.22/12.40
% 93.22/12.40 Lemma 24: meet(X, complement(join(X, Y))) = zero.
% 93.22/12.40 Proof:
% 93.22/12.41 meet(X, complement(join(X, Y)))
% 93.22/12.41 = { by lemma 23 R->L }
% 93.22/12.41 meet(X, meet(join(X, Y), complement(join(X, Y))))
% 93.22/12.41 = { by axiom 4 (complement_meet) }
% 93.22/12.41 meet(X, zero)
% 93.22/12.41 = { by lemma 18 }
% 93.22/12.41 zero
% 93.22/12.41
% 93.22/12.41 Lemma 25: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 93.22/12.41 Proof:
% 93.22/12.41 join(X, join(Y, meet(X, Z)))
% 93.22/12.41 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.41 join(X, join(meet(X, Z), Y))
% 93.22/12.41 = { by axiom 11 (associativity_of_join) R->L }
% 93.22/12.41 join(join(X, meet(X, Z)), Y)
% 93.22/12.41 = { by axiom 10 (absorption2) }
% 93.22/12.41 join(X, Y)
% 93.22/12.41
% 93.22/12.41 Lemma 26: join(X, join(Y, meet(Z, X))) = join(X, Y).
% 93.22/12.41 Proof:
% 93.22/12.41 join(X, join(Y, meet(Z, X)))
% 93.22/12.41 = { by lemma 22 }
% 93.22/12.41 join(Y, join(X, meet(Z, X)))
% 93.22/12.41 = { by lemma 21 }
% 93.22/12.41 join(Y, X)
% 93.22/12.41 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.41 join(X, Y)
% 93.22/12.41
% 93.22/12.41 Lemma 27: meet(join(X, Y), join(Y, meet(Z, X))) = join(Y, meet(X, Z)).
% 93.22/12.41 Proof:
% 93.22/12.41 meet(join(X, Y), join(Y, meet(Z, X)))
% 93.22/12.41 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.41 meet(join(Y, meet(Z, X)), join(X, Y))
% 93.22/12.41 = { by lemma 26 R->L }
% 93.22/12.41 meet(join(Y, meet(Z, X)), join(X, join(Y, meet(Z, X))))
% 93.22/12.41 = { by lemma 20 }
% 93.22/12.41 join(Y, meet(Z, X))
% 93.22/12.41 = { by axiom 1 (commutativity_of_meet) }
% 93.22/12.41 join(Y, meet(X, Z))
% 93.22/12.41
% 93.22/12.41 Lemma 28: meet(X, join(meet(X, Y), complement(join(X, Y)))) = meet(X, join(Y, complement(join(X, Y)))).
% 93.22/12.41 Proof:
% 93.22/12.41 meet(X, join(meet(X, Y), complement(join(X, Y))))
% 93.22/12.41 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.41 meet(X, join(meet(Y, X), complement(join(X, Y))))
% 93.22/12.41 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.41 meet(X, join(complement(join(X, Y)), meet(Y, X)))
% 93.22/12.41 = { by lemma 27 R->L }
% 93.22/12.41 meet(X, meet(join(Y, complement(join(X, Y))), join(complement(join(X, Y)), meet(X, Y))))
% 93.22/12.41 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.41 meet(X, meet(join(Y, complement(join(X, Y))), join(meet(X, Y), complement(join(X, Y)))))
% 93.22/12.41 = { by axiom 9 (associativity_of_meet) R->L }
% 93.22/12.41 meet(meet(X, join(Y, complement(join(X, Y)))), join(meet(X, Y), complement(join(X, Y))))
% 93.22/12.41 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.41 meet(meet(X, join(Y, complement(join(X, Y)))), join(meet(X, Y), complement(join(Y, X))))
% 93.22/12.41 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.41 meet(meet(X, join(Y, complement(join(X, Y)))), join(complement(join(Y, X)), meet(X, Y)))
% 93.22/12.41 = { by lemma 16 R->L }
% 93.22/12.41 meet(meet(X, join(Y, complement(join(X, Y)))), join(complement(join(Y, X)), meet(X, join(Y, zero))))
% 93.22/12.41 = { by axiom 4 (complement_meet) R->L }
% 93.22/12.41 meet(meet(X, join(Y, complement(join(X, Y)))), join(complement(join(Y, X)), meet(X, join(Y, meet(join(Y, X), complement(join(Y, X)))))))
% 93.22/12.41 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.41 meet(meet(X, join(Y, complement(join(X, Y)))), join(complement(join(Y, X)), meet(X, join(Y, meet(complement(join(Y, X)), join(Y, X))))))
% 93.22/12.41 = { by axiom 13 (equation_H55) R->L }
% 93.22/12.41 meet(meet(X, join(Y, complement(join(X, Y)))), join(complement(join(Y, X)), meet(X, join(complement(join(Y, X)), Y))))
% 93.22/12.41 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.41 meet(meet(X, join(Y, complement(join(X, Y)))), join(complement(join(Y, X)), meet(X, join(Y, complement(join(Y, X))))))
% 93.22/12.41 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.41 meet(meet(X, join(Y, complement(join(X, Y)))), join(complement(join(Y, X)), meet(X, join(Y, complement(join(X, Y))))))
% 93.22/12.41 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.41 meet(meet(X, join(Y, complement(join(X, Y)))), join(complement(join(X, Y)), meet(X, join(Y, complement(join(X, Y))))))
% 93.22/12.41 = { by lemma 20 }
% 93.22/12.41 meet(X, join(Y, complement(join(X, Y))))
% 93.22/12.41
% 93.22/12.41 Lemma 29: join(X, join(Y, complement(join(X, Y)))) = one.
% 93.22/12.41 Proof:
% 93.22/12.41 join(X, join(Y, complement(join(X, Y))))
% 93.22/12.41 = { by axiom 11 (associativity_of_join) R->L }
% 93.22/12.41 join(join(X, Y), complement(join(X, Y)))
% 93.22/12.41 = { by axiom 5 (complement_join) }
% 93.22/12.41 one
% 93.22/12.41
% 93.22/12.41 Lemma 30: fresh(meet(X, join(Y, complement(join(X, Y)))), zero, X, join(Y, complement(join(X, Y)))) = complement(X).
% 93.22/12.41 Proof:
% 93.22/12.41 fresh(meet(X, join(Y, complement(join(X, Y)))), zero, X, join(Y, complement(join(X, Y))))
% 93.22/12.41 = { by axiom 12 (meet_join_complement) R->L }
% 93.22/12.41 fresh2(join(X, join(Y, complement(join(X, Y)))), one, X, join(Y, complement(join(X, Y))))
% 93.22/12.41 = { by lemma 29 }
% 93.22/12.41 fresh2(one, one, X, join(Y, complement(join(X, Y))))
% 93.22/12.41 = { by axiom 7 (meet_join_complement) }
% 93.22/12.41 complement(X)
% 93.22/12.41
% 93.22/12.41 Lemma 31: join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))) = complement(Y).
% 93.22/12.41 Proof:
% 93.22/12.41 join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))
% 93.22/12.41 = { by axiom 6 (meet_join_complement) R->L }
% 93.22/12.41 fresh(zero, zero, Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 93.22/12.41 = { by lemma 24 R->L }
% 93.22/12.41 fresh(meet(Y, complement(join(Y, meet(X, complement(Y))))), zero, Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 93.22/12.41 = { by lemma 17 R->L }
% 93.22/12.41 fresh(meet(Y, join(zero, complement(join(Y, meet(X, complement(Y)))))), zero, Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 93.22/12.41 = { by lemma 18 R->L }
% 93.22/12.41 fresh(meet(Y, join(meet(X, zero), complement(join(Y, meet(X, complement(Y)))))), zero, Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 93.22/12.41 = { by axiom 1 (commutativity_of_meet) }
% 93.22/12.41 fresh(meet(Y, join(meet(zero, X), complement(join(Y, meet(X, complement(Y)))))), zero, Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 93.22/12.41 = { by axiom 4 (complement_meet) R->L }
% 93.22/12.41 fresh(meet(Y, join(meet(meet(Y, complement(Y)), X), complement(join(Y, meet(X, complement(Y)))))), zero, Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 93.22/12.41 = { by axiom 9 (associativity_of_meet) }
% 93.22/12.41 fresh(meet(Y, join(meet(Y, meet(complement(Y), X)), complement(join(Y, meet(X, complement(Y)))))), zero, Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 93.22/12.41 = { by axiom 1 (commutativity_of_meet) }
% 93.22/12.41 fresh(meet(Y, join(meet(Y, meet(X, complement(Y))), complement(join(Y, meet(X, complement(Y)))))), zero, Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 93.22/12.41 = { by lemma 28 }
% 93.22/12.41 fresh(meet(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), zero, Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 93.22/12.41 = { by lemma 30 }
% 93.22/12.41 complement(Y)
% 93.22/12.41
% 93.22/12.41 Lemma 32: join(X, complement(join(Y, meet(X, complement(Y))))) = join(X, complement(Y)).
% 93.22/12.41 Proof:
% 93.22/12.41 join(X, complement(join(Y, meet(X, complement(Y)))))
% 93.22/12.41 = { by lemma 25 R->L }
% 93.22/12.41 join(X, join(complement(join(Y, meet(X, complement(Y)))), meet(X, complement(Y))))
% 93.22/12.41 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.41 join(X, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 93.22/12.41 = { by lemma 31 }
% 93.22/12.41 join(X, complement(Y))
% 93.22/12.41
% 93.22/12.41 Lemma 33: meet(X, meet(Y, join(Z, X))) = meet(X, Y).
% 93.22/12.41 Proof:
% 93.22/12.41 meet(X, meet(Y, join(Z, X)))
% 93.22/12.41 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.41 meet(meet(Y, join(Z, X)), X)
% 93.22/12.41 = { by axiom 9 (associativity_of_meet) }
% 93.22/12.41 meet(Y, meet(join(Z, X), X))
% 93.22/12.41 = { by axiom 1 (commutativity_of_meet) }
% 93.22/12.41 meet(Y, meet(X, join(Z, X)))
% 93.22/12.41 = { by lemma 20 }
% 93.22/12.41 meet(Y, X)
% 93.22/12.41 = { by axiom 1 (commutativity_of_meet) }
% 93.22/12.41 meet(X, Y)
% 93.22/12.41
% 93.22/12.41 Lemma 34: meet(X, join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))) = zero.
% 93.22/12.41 Proof:
% 93.22/12.41 meet(X, join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))
% 93.22/12.41 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.42 meet(X, join(complement(join(Y, X)), complement(join(X, complement(join(X, Y))))))
% 93.22/12.42 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.42 meet(X, join(complement(join(Y, X)), complement(join(X, complement(join(Y, X))))))
% 93.22/12.42 = { by lemma 28 R->L }
% 93.22/12.42 meet(X, join(meet(X, complement(join(Y, X))), complement(join(X, complement(join(Y, X))))))
% 93.22/12.42 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.42 meet(X, join(meet(X, complement(join(X, Y))), complement(join(X, complement(join(Y, X))))))
% 93.22/12.42 = { by lemma 24 }
% 93.22/12.42 meet(X, join(zero, complement(join(X, complement(join(Y, X))))))
% 93.22/12.42 = { by lemma 17 }
% 93.22/12.42 meet(X, complement(join(X, complement(join(Y, X)))))
% 93.22/12.42 = { by lemma 24 }
% 93.22/12.42 zero
% 93.22/12.42
% 93.22/12.42 Lemma 35: join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))) = complement(X).
% 93.22/12.42 Proof:
% 93.22/12.42 join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))
% 93.22/12.42 = { by axiom 6 (meet_join_complement) R->L }
% 93.22/12.42 fresh(zero, zero, X, join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))
% 93.22/12.42 = { by lemma 34 R->L }
% 93.22/12.42 fresh(meet(X, join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), zero, X, join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))
% 93.22/12.42 = { by lemma 30 }
% 93.22/12.42 complement(X)
% 93.22/12.42
% 93.22/12.42 Lemma 36: meet(complement(X), complement(join(X, Y))) = complement(join(X, Y)).
% 93.22/12.42 Proof:
% 93.22/12.42 meet(complement(X), complement(join(X, Y)))
% 93.22/12.42 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.42 meet(complement(join(X, Y)), complement(X))
% 93.22/12.42 = { by lemma 35 R->L }
% 93.22/12.42 meet(complement(join(X, Y)), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))
% 93.22/12.42 = { by axiom 8 (absorption1) }
% 93.22/12.42 complement(join(X, Y))
% 93.22/12.42
% 93.22/12.42 Lemma 37: meet(X, complement(join(Y, complement(X)))) = complement(join(Y, complement(X))).
% 93.22/12.42 Proof:
% 93.22/12.42 meet(X, complement(join(Y, complement(X))))
% 93.22/12.42 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.42 meet(X, complement(join(complement(X), Y)))
% 93.22/12.42 = { by lemma 15 R->L }
% 93.22/12.42 meet(complement(complement(X)), complement(join(complement(X), Y)))
% 93.22/12.42 = { by lemma 36 }
% 93.22/12.42 complement(join(complement(X), Y))
% 93.22/12.42 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.42 complement(join(Y, complement(X)))
% 93.22/12.42
% 93.22/12.42 Lemma 38: join(X, complement(join(X, meet(Y, complement(X))))) = complement(meet(Y, complement(X))).
% 93.22/12.42 Proof:
% 93.22/12.42 join(X, complement(join(X, meet(Y, complement(X)))))
% 93.22/12.42 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.42 join(X, complement(join(meet(Y, complement(X)), X)))
% 93.22/12.42 = { by lemma 15 R->L }
% 93.22/12.42 join(X, complement(join(meet(Y, complement(X)), complement(complement(X)))))
% 93.22/12.42 = { by lemma 31 R->L }
% 93.22/12.42 join(X, complement(join(meet(Y, complement(X)), complement(join(meet(Y, complement(X)), complement(join(X, meet(Y, complement(X)))))))))
% 93.22/12.42 = { by lemma 15 R->L }
% 93.22/12.42 join(complement(complement(X)), complement(join(meet(Y, complement(X)), complement(join(meet(Y, complement(X)), complement(join(X, meet(Y, complement(X)))))))))
% 93.22/12.42 = { by lemma 31 R->L }
% 93.22/12.42 join(complement(join(meet(Y, complement(X)), complement(join(X, meet(Y, complement(X)))))), complement(join(meet(Y, complement(X)), complement(join(meet(Y, complement(X)), complement(join(X, meet(Y, complement(X)))))))))
% 93.22/12.42 = { by lemma 35 }
% 93.22/12.42 complement(meet(Y, complement(X)))
% 93.22/12.42
% 93.22/12.42 Lemma 39: meet(X, meet(Y, join(Z, meet(X, Y)))) = meet(X, Y).
% 93.22/12.42 Proof:
% 93.22/12.42 meet(X, meet(Y, join(Z, meet(X, Y))))
% 93.22/12.42 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.42 meet(X, meet(Y, join(meet(X, Y), Z)))
% 93.22/12.42 = { by axiom 9 (associativity_of_meet) R->L }
% 93.22/12.42 meet(meet(X, Y), join(meet(X, Y), Z))
% 93.22/12.42 = { by axiom 8 (absorption1) }
% 93.22/12.42 meet(X, Y)
% 93.22/12.42
% 93.22/12.42 Lemma 40: meet(X, meet(Y, join(Z, join(W, join(V, meet(X, Y)))))) = meet(X, Y).
% 93.22/12.42 Proof:
% 93.22/12.42 meet(X, meet(Y, join(Z, join(W, join(V, meet(X, Y))))))
% 93.22/12.42 = { by axiom 11 (associativity_of_join) R->L }
% 93.22/12.42 meet(X, meet(Y, join(join(Z, W), join(V, meet(X, Y)))))
% 93.22/12.42 = { by axiom 11 (associativity_of_join) R->L }
% 93.22/12.42 meet(X, meet(Y, join(join(join(Z, W), V), meet(X, Y))))
% 93.22/12.42 = { by lemma 39 }
% 93.22/12.42 meet(X, Y)
% 93.22/12.42
% 93.22/12.42 Lemma 41: join(meet(X, Y), complement(join(complement(X), meet(X, Y)))) = X.
% 93.22/12.42 Proof:
% 93.22/12.42 join(meet(X, Y), complement(join(complement(X), meet(X, Y))))
% 93.22/12.42 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.42 join(meet(Y, X), complement(join(complement(X), meet(X, Y))))
% 93.22/12.42 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.42 join(meet(Y, X), complement(join(complement(X), meet(Y, X))))
% 93.22/12.42 = { by lemma 15 R->L }
% 93.22/12.42 join(meet(Y, X), complement(join(complement(X), meet(Y, complement(complement(X))))))
% 93.22/12.42 = { by lemma 15 R->L }
% 93.22/12.42 join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X))))))
% 93.22/12.42 = { by lemma 31 }
% 93.22/12.42 complement(complement(X))
% 93.22/12.42 = { by lemma 15 }
% 93.22/12.42 X
% 93.22/12.42
% 93.22/12.42 Lemma 42: join(complement(X), complement(join(complement(X), meet(X, Y)))) = complement(meet(X, Y)).
% 93.22/12.42 Proof:
% 93.22/12.42 join(complement(X), complement(join(complement(X), meet(X, Y))))
% 93.22/12.42 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.42 join(complement(X), complement(join(meet(X, Y), complement(X))))
% 93.22/12.42 = { by lemma 41 R->L }
% 93.22/12.42 join(complement(X), complement(join(meet(X, Y), complement(join(meet(X, Y), complement(join(complement(X), meet(X, Y))))))))
% 93.22/12.42 = { by lemma 41 R->L }
% 93.22/12.42 join(complement(join(meet(X, Y), complement(join(complement(X), meet(X, Y))))), complement(join(meet(X, Y), complement(join(meet(X, Y), complement(join(complement(X), meet(X, Y))))))))
% 93.22/12.42 = { by lemma 35 }
% 93.22/12.42 complement(meet(X, Y))
% 93.22/12.42
% 93.22/12.42 Lemma 43: join(meet(X, Y), meet(X, join(Z, meet(X, Y)))) = meet(X, join(Z, meet(X, Y))).
% 93.22/12.42 Proof:
% 93.22/12.42 join(meet(X, Y), meet(X, join(Z, meet(X, Y))))
% 93.22/12.42 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.42 join(meet(Y, X), meet(X, join(Z, meet(X, Y))))
% 93.22/12.42 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.42 join(meet(Y, X), meet(X, join(Z, meet(Y, X))))
% 93.22/12.42 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.42 join(meet(X, join(Z, meet(Y, X))), meet(Y, X))
% 93.22/12.42 = { by lemma 39 R->L }
% 93.22/12.42 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(Z, meet(Y, X)))))
% 93.22/12.42 = { by lemma 21 }
% 93.22/12.42 meet(X, join(Z, meet(Y, X)))
% 93.22/12.42 = { by axiom 1 (commutativity_of_meet) }
% 93.22/12.42 meet(X, join(Z, meet(X, Y)))
% 93.22/12.42
% 93.22/12.42 Lemma 44: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 93.22/12.42 Proof:
% 93.22/12.42 join(X, join(Y, meet(Z, join(X, Y))))
% 93.22/12.42 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.42 join(X, join(Y, meet(join(X, Y), Z)))
% 93.22/12.42 = { by axiom 11 (associativity_of_join) R->L }
% 93.22/12.42 join(join(X, Y), meet(join(X, Y), Z))
% 93.22/12.42 = { by axiom 10 (absorption2) }
% 93.22/12.42 join(X, Y)
% 93.22/12.42
% 93.22/12.42 Lemma 45: meet(X, join(complement(X), meet(X, Y))) = meet(X, Y).
% 93.22/12.42 Proof:
% 93.22/12.42 meet(X, join(complement(X), meet(X, Y)))
% 93.22/12.42 = { by lemma 15 R->L }
% 93.22/12.42 complement(complement(meet(X, join(complement(X), meet(X, Y)))))
% 93.22/12.42 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.42 complement(complement(meet(X, join(complement(X), meet(Y, X)))))
% 93.22/12.42 = { by lemma 42 R->L }
% 93.22/12.42 complement(join(complement(X), complement(join(complement(X), meet(X, join(complement(X), meet(Y, X)))))))
% 93.22/12.42 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.42 complement(join(complement(X), complement(join(complement(X), meet(X, join(complement(X), meet(X, Y)))))))
% 93.22/12.42 = { by lemma 43 R->L }
% 93.22/12.42 complement(join(complement(X), complement(join(complement(X), join(meet(X, Y), meet(X, join(complement(X), meet(X, Y))))))))
% 93.22/12.42 = { by lemma 44 }
% 93.22/12.42 complement(join(complement(X), complement(join(complement(X), meet(X, Y)))))
% 93.22/12.42 = { by lemma 42 }
% 93.22/12.42 complement(complement(meet(X, Y)))
% 93.22/12.42 = { by lemma 15 }
% 93.22/12.42 meet(X, Y)
% 93.22/12.42
% 93.22/12.42 Lemma 46: meet(X, join(Y, complement(join(X, Y)))) = meet(X, Y).
% 93.22/12.42 Proof:
% 93.22/12.42 meet(X, join(Y, complement(join(X, Y))))
% 93.22/12.42 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.42 meet(X, join(Y, complement(join(Y, X))))
% 93.22/12.42 = { by lemma 33 R->L }
% 93.22/12.42 meet(X, meet(join(Y, complement(join(Y, X))), join(Y, X)))
% 93.22/12.42 = { by axiom 1 (commutativity_of_meet) }
% 93.22/12.42 meet(X, meet(join(Y, X), join(Y, complement(join(Y, X)))))
% 93.22/12.42 = { by lemma 15 R->L }
% 93.22/12.42 meet(X, complement(complement(meet(join(Y, X), join(Y, complement(join(Y, X)))))))
% 93.22/12.42 = { by lemma 15 R->L }
% 93.22/12.42 meet(X, complement(complement(meet(join(Y, X), complement(complement(join(Y, complement(join(Y, X)))))))))
% 93.22/12.42 = { by lemma 37 R->L }
% 93.22/12.42 meet(X, complement(complement(meet(join(Y, X), complement(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))))))))
% 93.22/12.42 = { by lemma 38 R->L }
% 93.22/12.42 meet(X, complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), meet(join(Y, X), complement(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))))))))))
% 93.22/12.43 = { by lemma 40 R->L }
% 93.22/12.43 meet(X, complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), meet(join(Y, X), complement(meet(join(Y, X), meet(complement(join(Y, complement(join(Y, X)))), join(Z, join(W, join(V, meet(join(Y, X), complement(join(Y, complement(join(Y, X)))))))))))))))))
% 93.22/12.43 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.43 meet(X, complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), meet(join(Y, X), complement(meet(meet(complement(join(Y, complement(join(Y, X)))), join(Z, join(W, join(V, meet(join(Y, X), complement(join(Y, complement(join(Y, X))))))))), join(Y, X)))))))))
% 93.22/12.43 = { by lemma 15 R->L }
% 93.22/12.43 meet(X, complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), meet(join(Y, X), complement(meet(meet(complement(join(Y, complement(join(Y, X)))), join(Z, join(W, join(V, meet(join(Y, X), complement(join(Y, complement(join(Y, X))))))))), complement(complement(join(Y, X)))))))))))
% 93.22/12.43 = { by lemma 15 R->L }
% 93.22/12.43 meet(X, complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), meet(complement(complement(join(Y, X))), complement(meet(meet(complement(join(Y, complement(join(Y, X)))), join(Z, join(W, join(V, meet(join(Y, X), complement(join(Y, complement(join(Y, X))))))))), complement(complement(join(Y, X)))))))))))
% 93.22/12.43 = { by lemma 38 R->L }
% 93.22/12.43 meet(X, complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), meet(complement(complement(join(Y, X))), join(complement(join(Y, X)), complement(join(complement(join(Y, X)), meet(meet(complement(join(Y, complement(join(Y, X)))), join(Z, join(W, join(V, meet(join(Y, X), complement(join(Y, complement(join(Y, X))))))))), complement(complement(join(Y, X)))))))))))))
% 93.22/12.43 = { by lemma 15 R->L }
% 93.22/12.43 meet(X, complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), meet(complement(complement(join(Y, X))), join(complement(complement(complement(join(Y, X)))), complement(join(complement(join(Y, X)), meet(meet(complement(join(Y, complement(join(Y, X)))), join(Z, join(W, join(V, meet(join(Y, X), complement(join(Y, complement(join(Y, X))))))))), complement(complement(join(Y, X)))))))))))))
% 93.22/12.43 = { by lemma 36 R->L }
% 93.22/12.43 meet(X, complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), meet(complement(complement(join(Y, X))), join(complement(complement(complement(join(Y, X)))), meet(complement(complement(join(Y, X))), complement(join(complement(join(Y, X)), meet(meet(complement(join(Y, complement(join(Y, X)))), join(Z, join(W, join(V, meet(join(Y, X), complement(join(Y, complement(join(Y, X))))))))), complement(complement(join(Y, X))))))))))))))
% 93.22/12.43 = { by lemma 45 }
% 93.22/12.43 meet(X, complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), meet(complement(complement(join(Y, X))), complement(join(complement(join(Y, X)), meet(meet(complement(join(Y, complement(join(Y, X)))), join(Z, join(W, join(V, meet(join(Y, X), complement(join(Y, complement(join(Y, X))))))))), complement(complement(join(Y, X))))))))))))
% 93.22/12.43 = { by lemma 36 }
% 93.22/12.43 meet(X, complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(complement(join(Y, X)), meet(meet(complement(join(Y, complement(join(Y, X)))), join(Z, join(W, join(V, meet(join(Y, X), complement(join(Y, complement(join(Y, X))))))))), complement(complement(join(Y, X)))))))))))
% 93.22/12.43 = { by lemma 15 }
% 93.22/12.43 meet(X, complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(complement(join(Y, X)), meet(meet(complement(join(Y, complement(join(Y, X)))), join(Z, join(W, join(V, meet(join(Y, X), complement(join(Y, complement(join(Y, X))))))))), join(Y, X)))))))))
% 93.22/12.43 = { by axiom 1 (commutativity_of_meet) }
% 93.22/12.43 meet(X, complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(complement(join(Y, X)), meet(join(Y, X), meet(complement(join(Y, complement(join(Y, X)))), join(Z, join(W, join(V, meet(join(Y, X), complement(join(Y, complement(join(Y, X)))))))))))))))))
% 93.22/12.43 = { by lemma 40 R->L }
% 93.22/12.43 meet(X, complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(meet(join(Y, X), meet(complement(join(Y, complement(join(Y, X)))), join(Z, join(W, join(V, meet(join(Y, X), complement(join(Y, complement(join(Y, X)))))))))), complement(join(complement(join(Y, X)), meet(join(Y, X), meet(complement(join(Y, complement(join(Y, X)))), join(Z, join(W, join(V, meet(join(Y, X), complement(join(Y, complement(join(Y, X)))))))))))))))))
% 93.22/12.43 = { by lemma 41 }
% 93.22/12.43 meet(X, complement(join(meet(join(Y, X), complement(join(Y, complement(join(Y, X))))), complement(join(Y, X)))))
% 93.22/12.43 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.43 meet(X, complement(join(complement(join(Y, X)), meet(join(Y, X), complement(join(Y, complement(join(Y, X))))))))
% 93.22/12.43 = { by lemma 37 }
% 93.22/12.43 meet(X, complement(join(complement(join(Y, X)), complement(join(Y, complement(join(Y, X)))))))
% 93.22/12.43 = { by axiom 6 (meet_join_complement) R->L }
% 93.22/12.43 meet(X, complement(fresh(zero, zero, Y, join(complement(join(Y, X)), complement(join(Y, complement(join(Y, X))))))))
% 93.22/12.43 = { by lemma 34 R->L }
% 93.22/12.43 meet(X, complement(fresh(meet(Y, join(complement(join(Y, X)), complement(join(Y, complement(join(Y, X)))))), zero, Y, join(complement(join(Y, X)), complement(join(Y, complement(join(Y, X))))))))
% 93.22/12.43 = { by lemma 30 }
% 93.22/12.43 meet(X, complement(complement(Y)))
% 93.22/12.43 = { by lemma 15 }
% 93.22/12.43 meet(X, Y)
% 93.22/12.43
% 93.22/12.43 Lemma 47: fresh(meet(X, Y), zero, join(X, complement(join(X, Y))), Y) = complement(join(X, complement(join(X, Y)))).
% 93.22/12.43 Proof:
% 93.22/12.43 fresh(meet(X, Y), zero, join(X, complement(join(X, Y))), Y)
% 93.22/12.43 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.43 fresh(meet(Y, X), zero, join(X, complement(join(X, Y))), Y)
% 93.22/12.43 = { by lemma 46 R->L }
% 93.22/12.43 fresh(meet(Y, join(X, complement(join(Y, X)))), zero, join(X, complement(join(X, Y))), Y)
% 93.22/12.43 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.43 fresh(meet(Y, join(X, complement(join(X, Y)))), zero, join(X, complement(join(X, Y))), Y)
% 93.22/12.43 = { by lemma 14 R->L }
% 93.22/12.43 fresh2(join(Y, join(X, complement(join(X, Y)))), one, join(X, complement(join(X, Y))), Y)
% 93.22/12.43 = { by lemma 22 R->L }
% 93.22/12.43 fresh2(join(X, join(Y, complement(join(X, Y)))), one, join(X, complement(join(X, Y))), Y)
% 93.22/12.43 = { by lemma 29 }
% 93.22/12.43 fresh2(one, one, join(X, complement(join(X, Y))), Y)
% 93.22/12.43 = { by axiom 7 (meet_join_complement) }
% 93.22/12.43 complement(join(X, complement(join(X, Y))))
% 93.22/12.43
% 93.22/12.43 Lemma 48: complement(join(X, complement(join(X, complement(Y))))) = complement(join(Y, meet(X, complement(Y)))).
% 93.22/12.43 Proof:
% 93.22/12.43 complement(join(X, complement(join(X, complement(Y)))))
% 93.22/12.43 = { by lemma 32 R->L }
% 93.22/12.43 complement(join(X, complement(join(X, complement(join(Y, meet(X, complement(Y))))))))
% 93.22/12.43 = { by lemma 47 R->L }
% 93.22/12.43 fresh(meet(X, complement(join(Y, meet(X, complement(Y))))), zero, join(X, complement(join(X, complement(join(Y, meet(X, complement(Y))))))), complement(join(Y, meet(X, complement(Y)))))
% 93.22/12.43 = { by lemma 32 }
% 93.22/12.43 fresh(meet(X, complement(join(Y, meet(X, complement(Y))))), zero, join(X, complement(join(X, complement(Y)))), complement(join(Y, meet(X, complement(Y)))))
% 93.22/12.43 = { by lemma 36 R->L }
% 93.22/12.43 fresh(meet(X, meet(complement(Y), complement(join(Y, meet(X, complement(Y)))))), zero, join(X, complement(join(X, complement(Y)))), complement(join(Y, meet(X, complement(Y)))))
% 93.22/12.43 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.43 fresh(meet(X, meet(complement(Y), complement(join(meet(X, complement(Y)), Y)))), zero, join(X, complement(join(X, complement(Y)))), complement(join(Y, meet(X, complement(Y)))))
% 93.22/12.43 = { by axiom 9 (associativity_of_meet) R->L }
% 93.22/12.43 fresh(meet(meet(X, complement(Y)), complement(join(meet(X, complement(Y)), Y))), zero, join(X, complement(join(X, complement(Y)))), complement(join(Y, meet(X, complement(Y)))))
% 93.22/12.43 = { by lemma 24 }
% 93.22/12.43 fresh(zero, zero, join(X, complement(join(X, complement(Y)))), complement(join(Y, meet(X, complement(Y)))))
% 93.22/12.43 = { by axiom 6 (meet_join_complement) }
% 93.22/12.43 complement(join(Y, meet(X, complement(Y))))
% 93.22/12.43
% 93.22/12.43 Lemma 49: join(complement(Y), meet(Y, X)) = join(X, complement(join(X, Y))).
% 93.22/12.43 Proof:
% 93.22/12.43 join(complement(Y), meet(Y, X))
% 93.22/12.43 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.43 join(complement(Y), meet(X, Y))
% 93.22/12.43 = { by lemma 15 R->L }
% 93.22/12.43 join(complement(Y), meet(X, complement(complement(Y))))
% 93.22/12.43 = { by lemma 15 R->L }
% 93.22/12.43 complement(complement(join(complement(Y), meet(X, complement(complement(Y))))))
% 93.22/12.43 = { by lemma 48 R->L }
% 93.22/12.43 complement(complement(join(X, complement(join(X, complement(complement(Y)))))))
% 93.22/12.43 = { by lemma 15 }
% 93.22/12.43 join(X, complement(join(X, complement(complement(Y)))))
% 93.22/12.43 = { by lemma 15 }
% 93.22/12.43 join(X, complement(join(X, Y)))
% 93.22/12.43
% 93.22/12.43 Lemma 50: join(X, complement(join(X, Y))) = join(X, complement(Y)).
% 93.22/12.43 Proof:
% 93.22/12.43 join(X, complement(join(X, Y)))
% 93.22/12.43 = { by axiom 2 (idempotence_of_join) R->L }
% 93.22/12.43 join(join(X, X), complement(join(X, Y)))
% 93.22/12.43 = { by axiom 11 (associativity_of_join) }
% 93.22/12.43 join(X, join(X, complement(join(X, Y))))
% 93.22/12.43 = { by lemma 49 R->L }
% 93.22/12.43 join(X, join(complement(Y), meet(Y, X)))
% 93.22/12.43 = { by lemma 26 }
% 93.22/12.43 join(X, complement(Y))
% 93.22/12.43
% 93.22/12.43 Lemma 51: join(X, meet(Y, complement(X))) = join(X, Y).
% 93.22/12.43 Proof:
% 93.22/12.43 join(X, meet(Y, complement(X)))
% 93.22/12.43 = { by lemma 15 R->L }
% 93.22/12.43 complement(complement(join(X, meet(Y, complement(X)))))
% 93.22/12.43 = { by lemma 48 R->L }
% 93.22/12.43 complement(complement(join(Y, complement(join(Y, complement(X))))))
% 93.22/12.43 = { by lemma 15 }
% 93.22/12.43 join(Y, complement(join(Y, complement(X))))
% 93.22/12.43 = { by lemma 50 }
% 93.22/12.43 join(Y, complement(complement(X)))
% 93.22/12.43 = { by lemma 15 }
% 93.22/12.43 join(Y, X)
% 93.22/12.43 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.43 join(X, Y)
% 93.22/12.43
% 93.22/12.43 Lemma 52: meet(join(X, Y), join(X, meet(join(X, Y), Z))) = join(X, meet(Z, join(X, Y))).
% 93.22/12.43 Proof:
% 93.22/12.43 meet(join(X, Y), join(X, meet(join(X, Y), Z)))
% 93.22/12.43 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.43 meet(join(X, Y), join(X, meet(Z, join(X, Y))))
% 93.22/12.43 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.43 meet(join(Y, X), join(X, meet(Z, join(X, Y))))
% 93.22/12.43 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.43 meet(join(Y, X), join(X, meet(Z, join(Y, X))))
% 93.22/12.43 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.43 meet(join(X, meet(Z, join(Y, X))), join(Y, X))
% 93.22/12.43 = { by lemma 44 R->L }
% 93.22/12.43 meet(join(X, meet(Z, join(Y, X))), join(Y, join(X, meet(Z, join(Y, X)))))
% 93.22/12.43 = { by lemma 20 }
% 93.22/12.43 join(X, meet(Z, join(Y, X)))
% 93.22/12.43 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.43 join(X, meet(Z, join(X, Y)))
% 93.22/12.43
% 93.22/12.43 Lemma 53: join(X, meet(Y, meet(Z, complement(X)))) = join(X, meet(Y, Z)).
% 93.22/12.43 Proof:
% 93.22/12.43 join(X, meet(Y, meet(Z, complement(X))))
% 93.22/12.43 = { by axiom 9 (associativity_of_meet) R->L }
% 93.22/12.43 join(X, meet(meet(Y, Z), complement(X)))
% 93.22/12.43 = { by lemma 51 }
% 93.22/12.44 join(X, meet(Y, Z))
% 93.22/12.44
% 93.22/12.44 Lemma 54: join(X, meet(Y, join(X, Z))) = join(X, meet(Y, Z)).
% 93.22/12.44 Proof:
% 93.22/12.44 join(X, meet(Y, join(X, Z)))
% 93.22/12.44 = { by lemma 52 R->L }
% 93.22/12.44 meet(join(X, Z), join(X, meet(join(X, Z), Y)))
% 93.22/12.44 = { by lemma 53 R->L }
% 93.22/12.44 meet(join(X, Z), join(X, meet(join(X, Z), meet(Y, complement(X)))))
% 93.22/12.44 = { by lemma 52 }
% 93.22/12.44 join(X, meet(meet(Y, complement(X)), join(X, Z)))
% 93.22/12.44 = { by axiom 9 (associativity_of_meet) }
% 93.22/12.44 join(X, meet(Y, meet(complement(X), join(X, Z))))
% 93.22/12.44 = { by lemma 21 R->L }
% 93.22/12.44 join(X, meet(Y, join(meet(complement(X), join(X, Z)), meet(Z, meet(complement(X), join(X, Z))))))
% 93.22/12.44 = { by lemma 33 }
% 93.22/12.44 join(X, meet(Y, join(meet(complement(X), join(X, Z)), meet(Z, complement(X)))))
% 93.22/12.44 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.44 join(X, meet(Y, join(meet(Z, complement(X)), meet(complement(X), join(X, Z)))))
% 93.22/12.44 = { by lemma 51 R->L }
% 93.22/12.44 join(X, meet(Y, join(meet(Z, complement(X)), meet(complement(X), join(X, meet(Z, complement(X)))))))
% 93.22/12.44 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.44 join(X, meet(Y, join(meet(Z, complement(X)), meet(complement(X), join(X, meet(complement(X), Z))))))
% 93.22/12.44 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.44 join(X, meet(Y, join(meet(complement(X), Z), meet(complement(X), join(X, meet(complement(X), Z))))))
% 93.22/12.44 = { by lemma 43 }
% 93.22/12.44 join(X, meet(Y, meet(complement(X), join(X, meet(complement(X), Z)))))
% 93.22/12.44 = { by lemma 15 R->L }
% 93.22/12.44 join(X, meet(Y, meet(complement(X), join(complement(complement(X)), meet(complement(X), Z)))))
% 93.22/12.44 = { by lemma 45 }
% 93.22/12.44 join(X, meet(Y, meet(complement(X), Z)))
% 93.22/12.44 = { by axiom 1 (commutativity_of_meet) }
% 93.22/12.44 join(X, meet(Y, meet(Z, complement(X))))
% 93.22/12.44 = { by lemma 53 }
% 93.22/12.44 join(X, meet(Y, Z))
% 93.22/12.44
% 93.22/12.44 Lemma 55: meet(join(X, Y), join(Y, meet(X, Z))) = join(Y, meet(X, Z)).
% 93.22/12.44 Proof:
% 93.22/12.44 meet(join(X, Y), join(Y, meet(X, Z)))
% 93.22/12.44 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.44 meet(join(Y, meet(X, Z)), join(X, Y))
% 93.22/12.44 = { by lemma 25 R->L }
% 93.22/12.44 meet(join(Y, meet(X, Z)), join(X, join(Y, meet(X, Z))))
% 93.22/12.44 = { by lemma 20 }
% 93.22/12.44 join(Y, meet(X, Z))
% 93.22/12.44
% 93.22/12.44 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 93.22/12.44 Proof:
% 93.22/12.44 meet(a, join(b, c))
% 93.22/12.44 = { by lemma 20 R->L }
% 93.22/12.44 meet(meet(a, join(b, c)), join(b, meet(a, join(b, c))))
% 93.22/12.44 = { by lemma 54 }
% 93.22/12.44 meet(meet(a, join(b, c)), join(b, meet(a, c)))
% 93.22/12.44 = { by axiom 9 (associativity_of_meet) }
% 93.22/12.44 meet(a, meet(join(b, c), join(b, meet(a, c))))
% 93.22/12.44 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.44 meet(a, meet(join(c, b), join(b, meet(a, c))))
% 93.22/12.44 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.44 meet(a, meet(join(b, meet(a, c)), join(c, b)))
% 93.22/12.44 = { by lemma 55 R->L }
% 93.22/12.44 meet(a, meet(meet(join(a, b), join(b, meet(a, c))), join(c, b)))
% 93.22/12.44 = { by axiom 9 (associativity_of_meet) }
% 93.22/12.44 meet(a, meet(join(a, b), meet(join(b, meet(a, c)), join(c, b))))
% 93.22/12.44 = { by axiom 1 (commutativity_of_meet) }
% 93.22/12.44 meet(a, meet(join(a, b), meet(join(c, b), join(b, meet(a, c)))))
% 93.22/12.44 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.44 meet(a, meet(join(b, a), meet(join(c, b), join(b, meet(a, c)))))
% 93.22/12.44 = { by lemma 27 }
% 93.22/12.44 meet(a, meet(join(b, a), join(b, meet(c, a))))
% 93.22/12.44 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.44 meet(a, meet(join(a, b), join(b, meet(c, a))))
% 93.22/12.44 = { by axiom 1 (commutativity_of_meet) }
% 93.22/12.44 meet(a, meet(join(a, b), join(b, meet(a, c))))
% 93.22/12.44 = { by lemma 55 }
% 93.22/12.44 meet(a, join(b, meet(a, c)))
% 93.22/12.44 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.44 meet(a, join(b, meet(c, a)))
% 93.22/12.44 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.44 meet(a, join(meet(c, a), b))
% 93.22/12.44 = { by lemma 21 R->L }
% 93.22/12.44 meet(join(a, meet(c, a)), join(meet(c, a), b))
% 93.22/12.44 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.44 meet(join(a, meet(c, a)), join(b, meet(c, a)))
% 93.22/12.44 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.44 meet(join(meet(c, a), a), join(b, meet(c, a)))
% 93.22/12.44 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.44 meet(join(b, meet(c, a)), join(meet(c, a), a))
% 93.22/12.44 = { by lemma 15 R->L }
% 93.22/12.44 complement(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))))
% 93.22/12.44 = { by lemma 19 R->L }
% 93.22/12.44 meet(one, complement(complement(meet(join(b, meet(c, a)), join(meet(c, a), a)))))
% 93.22/12.44 = { by lemma 29 R->L }
% 93.22/12.44 meet(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a))))), complement(complement(meet(join(b, meet(c, a)), join(meet(c, a), a)))))
% 93.22/12.44 = { by lemma 15 R->L }
% 93.22/12.44 complement(complement(meet(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a))))), complement(complement(meet(join(b, meet(c, a)), join(meet(c, a), a)))))))
% 93.22/12.44 = { by lemma 38 R->L }
% 93.22/12.44 complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a))))), complement(complement(meet(join(b, meet(c, a)), join(meet(c, a), a)))))))))
% 93.22/12.44 = { by lemma 51 }
% 93.22/12.44 complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)))))))))
% 93.22/12.44 = { by lemma 50 }
% 93.22/12.44 complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a))))))))
% 93.22/12.44 = { by lemma 47 R->L }
% 93.22/12.44 fresh(meet(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a))))), zero, join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a))))))), join(meet(c, a), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)))))
% 93.22/12.44 = { by lemma 29 }
% 93.22/12.44 fresh(meet(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a))))), zero, join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), complement(one)), join(meet(c, a), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)))))
% 93.22/12.44 = { by lemma 46 }
% 93.22/12.44 fresh(meet(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)), zero, join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), complement(one)), join(meet(c, a), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)))))
% 93.22/12.44 = { by lemma 19 R->L }
% 93.22/12.44 fresh(meet(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)), zero, join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(one, complement(one))), join(meet(c, a), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)))))
% 93.22/12.44 = { by axiom 4 (complement_meet) }
% 93.22/12.44 fresh(meet(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)), zero, join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), zero), join(meet(c, a), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)))))
% 93.22/12.44 = { by lemma 16 }
% 93.22/12.44 fresh(meet(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)), zero, complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), complement(join(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)))))
% 93.22/12.44 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.44 fresh(meet(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)), zero, complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), complement(join(meet(c, a), complement(meet(join(b, meet(c, a)), join(meet(c, a), a)))))))
% 93.22/12.44 = { by lemma 49 R->L }
% 93.22/12.44 fresh(meet(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)), zero, complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(complement(complement(meet(join(b, meet(c, a)), join(meet(c, a), a)))), meet(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a))))
% 93.22/12.44 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.44 fresh(meet(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)), zero, complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(complement(complement(meet(join(b, meet(c, a)), join(meet(c, a), a)))), meet(meet(c, a), complement(meet(join(b, meet(c, a)), join(meet(c, a), a))))))
% 93.22/12.44 = { by lemma 15 R->L }
% 93.22/12.44 fresh(meet(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)), zero, complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(complement(complement(meet(join(b, meet(c, a)), join(meet(c, a), a)))), meet(meet(c, a), complement(complement(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))))))))
% 93.22/12.44 = { by lemma 51 }
% 93.22/12.44 fresh(meet(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)), zero, complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(complement(complement(meet(join(b, meet(c, a)), join(meet(c, a), a)))), meet(c, a)))
% 93.22/12.44 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.45 fresh(meet(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)), zero, complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), complement(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))))))
% 93.22/12.45 = { by lemma 15 }
% 93.22/12.45 fresh(meet(complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), meet(c, a)), zero, complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), meet(join(b, meet(c, a)), join(meet(c, a), a))))
% 93.22/12.45 = { by axiom 1 (commutativity_of_meet) }
% 93.22/12.45 fresh(meet(meet(c, a), complement(meet(join(b, meet(c, a)), join(meet(c, a), a)))), zero, complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), meet(join(b, meet(c, a)), join(meet(c, a), a))))
% 93.22/12.45 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.45 fresh(meet(meet(c, a), complement(meet(join(meet(c, a), b), join(meet(c, a), a)))), zero, complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), meet(join(b, meet(c, a)), join(meet(c, a), a))))
% 93.22/12.45 = { by lemma 23 R->L }
% 93.22/12.45 fresh(meet(meet(c, a), meet(join(meet(c, a), b), complement(meet(join(meet(c, a), b), join(meet(c, a), a))))), zero, complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), meet(join(b, meet(c, a)), join(meet(c, a), a))))
% 93.22/12.45 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.45 fresh(meet(meet(c, a), meet(join(meet(c, a), b), complement(meet(join(meet(c, a), a), join(meet(c, a), b))))), zero, complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), meet(join(b, meet(c, a)), join(meet(c, a), a))))
% 93.22/12.45 = { by lemma 23 R->L }
% 93.22/12.45 fresh(meet(meet(c, a), meet(join(meet(c, a), a), meet(join(meet(c, a), b), complement(meet(join(meet(c, a), a), join(meet(c, a), b)))))), zero, complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), meet(join(b, meet(c, a)), join(meet(c, a), a))))
% 93.22/12.45 = { by axiom 9 (associativity_of_meet) R->L }
% 93.22/12.45 fresh(meet(meet(c, a), meet(meet(join(meet(c, a), a), join(meet(c, a), b)), complement(meet(join(meet(c, a), a), join(meet(c, a), b))))), zero, complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), meet(join(b, meet(c, a)), join(meet(c, a), a))))
% 93.22/12.45 = { by axiom 4 (complement_meet) }
% 93.22/12.45 fresh(meet(meet(c, a), zero), zero, complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), meet(join(b, meet(c, a)), join(meet(c, a), a))))
% 93.22/12.45 = { by lemma 18 }
% 93.22/12.45 fresh(zero, zero, complement(meet(join(b, meet(c, a)), join(meet(c, a), a))), join(meet(c, a), meet(join(b, meet(c, a)), join(meet(c, a), a))))
% 93.22/12.45 = { by axiom 6 (meet_join_complement) }
% 93.22/12.45 join(meet(c, a), meet(join(b, meet(c, a)), join(meet(c, a), a)))
% 93.22/12.45 = { by lemma 54 }
% 93.22/12.45 join(meet(c, a), meet(join(b, meet(c, a)), a))
% 93.22/12.45 = { by axiom 1 (commutativity_of_meet) }
% 93.22/12.45 join(meet(c, a), meet(a, join(b, meet(c, a))))
% 93.22/12.45 = { by axiom 3 (commutativity_of_join) R->L }
% 93.22/12.45 join(meet(c, a), meet(a, join(meet(c, a), b)))
% 93.22/12.45 = { by lemma 54 }
% 93.22/12.45 join(meet(c, a), meet(a, b))
% 93.22/12.45 = { by axiom 3 (commutativity_of_join) }
% 93.22/12.45 join(meet(a, b), meet(c, a))
% 93.22/12.45 = { by axiom 1 (commutativity_of_meet) }
% 93.22/12.45 join(meet(b, a), meet(c, a))
% 93.22/12.45 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.45 join(meet(b, a), meet(a, c))
% 93.22/12.45 = { by axiom 1 (commutativity_of_meet) R->L }
% 93.22/12.45 join(meet(a, b), meet(a, c))
% 93.22/12.45 % SZS output end Proof
% 93.22/12.45
% 93.22/12.45 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------