TSTP Solution File: LAT201-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT201-1 : 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 : n020.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 185.88s 24.15s
% Output : Proof 187.03s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LAT201-1 : TPTP v8.1.2. Released v3.1.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n020.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Thu Aug 24 06:10:30 EDT 2023
% 0.14/0.35 % CPUTime :
% 185.88/24.15 Command-line arguments: --no-flatten-goal
% 185.88/24.15
% 185.88/24.15 % SZS status Unsatisfiable
% 185.88/24.15
% 186.70/24.24 % SZS output start Proof
% 186.70/24.24 Take the following subset of the input axioms:
% 186.70/24.24 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 186.70/24.24 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 186.70/24.24 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 186.70/24.24 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 186.70/24.24 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 186.70/24.24 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 186.70/24.24 fof(complement_join, axiom, ![X2]: join(X2, complement(X2))=one).
% 186.70/24.24 fof(complement_meet, axiom, ![X2]: meet(X2, complement(X2))=zero).
% 186.70/24.24 fof(equation_H51, axiom, ![U, X2, Y2, Z2]: meet(X2, join(Y2, meet(Z2, join(X2, U))))=meet(X2, join(Y2, join(meet(X2, Z2), meet(Z2, U))))).
% 186.70/24.24 fof(meet_join_complement, axiom, ![X2, Y2]: (meet(X2, Y2)!=zero | (join(X2, Y2)!=one | complement(X2)=Y2))).
% 186.70/24.24 fof(prove_distributivity, negated_conjecture, meet(a, join(b, c))!=join(meet(a, b), meet(a, c))).
% 186.70/24.24
% 186.70/24.24 Now clausify the problem and encode Horn clauses using encoding 3 of
% 186.70/24.24 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 186.70/24.24 We repeatedly replace C & s=t => u=v by the two clauses:
% 186.70/24.24 fresh(y, y, x1...xn) = u
% 186.70/24.24 C => fresh(s, t, x1...xn) = v
% 186.70/24.24 where fresh is a fresh function symbol and x1..xn are the free
% 186.70/24.24 variables of u and v.
% 186.70/24.24 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 186.70/24.24 input problem has no model of domain size 1).
% 186.70/24.24
% 186.70/24.24 The encoding turns the above axioms into the following unit equations and goals:
% 186.70/24.24
% 186.70/24.24 Axiom 1 (commutativity_of_join): join(X, Y) = join(Y, X).
% 186.70/24.24 Axiom 2 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 186.70/24.24 Axiom 3 (complement_join): join(X, complement(X)) = one.
% 186.70/24.24 Axiom 4 (complement_meet): meet(X, complement(X)) = zero.
% 186.70/24.24 Axiom 5 (meet_join_complement): fresh(X, X, Y, Z) = Z.
% 186.70/24.24 Axiom 6 (meet_join_complement): fresh2(X, X, Y, Z) = complement(Y).
% 186.70/24.24 Axiom 7 (absorption2): join(X, meet(X, Y)) = X.
% 186.70/24.24 Axiom 8 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 186.70/24.24 Axiom 9 (absorption1): meet(X, join(X, Y)) = X.
% 186.70/24.24 Axiom 10 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 186.70/24.24 Axiom 11 (meet_join_complement): fresh2(join(X, Y), one, X, Y) = fresh(meet(X, Y), zero, X, Y).
% 186.70/24.24 Axiom 12 (equation_H51): meet(X, join(Y, meet(Z, join(X, W)))) = meet(X, join(Y, join(meet(X, Z), meet(Z, W)))).
% 186.70/24.24
% 186.70/24.24 Lemma 13: fresh2(join(X, Y), one, Y, X) = fresh(meet(X, Y), zero, Y, X).
% 186.70/24.24 Proof:
% 186.70/24.24 fresh2(join(X, Y), one, Y, X)
% 186.70/24.24 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.24 fresh2(join(Y, X), one, Y, X)
% 186.70/24.24 = { by axiom 11 (meet_join_complement) }
% 186.70/24.24 fresh(meet(Y, X), zero, Y, X)
% 186.70/24.24 = { by axiom 2 (commutativity_of_meet) }
% 186.70/24.24 fresh(meet(X, Y), zero, Y, X)
% 186.70/24.24
% 186.70/24.24 Lemma 14: complement(complement(X)) = X.
% 186.70/24.24 Proof:
% 186.70/24.24 complement(complement(X))
% 186.70/24.24 = { by axiom 6 (meet_join_complement) R->L }
% 186.70/24.24 fresh2(one, one, complement(X), X)
% 186.70/24.24 = { by axiom 3 (complement_join) R->L }
% 186.70/24.25 fresh2(join(X, complement(X)), one, complement(X), X)
% 186.70/24.25 = { by lemma 13 }
% 186.70/24.25 fresh(meet(X, complement(X)), zero, complement(X), X)
% 186.70/24.25 = { by axiom 4 (complement_meet) }
% 186.70/24.25 fresh(zero, zero, complement(X), X)
% 186.70/24.25 = { by axiom 5 (meet_join_complement) }
% 186.70/24.25 X
% 186.70/24.25
% 186.70/24.25 Lemma 15: meet(X, one) = X.
% 186.70/24.25 Proof:
% 186.70/24.25 meet(X, one)
% 186.70/24.25 = { by axiom 3 (complement_join) R->L }
% 186.70/24.25 meet(X, join(X, complement(X)))
% 186.70/24.25 = { by axiom 9 (absorption1) }
% 186.70/24.25 X
% 186.70/24.25
% 186.70/24.25 Lemma 16: join(X, zero) = X.
% 186.70/24.25 Proof:
% 186.70/24.25 join(X, zero)
% 186.70/24.25 = { by axiom 4 (complement_meet) R->L }
% 186.70/24.25 join(X, meet(X, complement(X)))
% 186.70/24.25 = { by axiom 7 (absorption2) }
% 186.70/24.25 X
% 186.70/24.25
% 186.70/24.25 Lemma 17: join(zero, X) = X.
% 186.70/24.25 Proof:
% 186.70/24.25 join(zero, X)
% 186.70/24.25 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.25 join(X, zero)
% 186.70/24.25 = { by lemma 16 }
% 186.70/24.25 X
% 186.70/24.25
% 186.70/24.25 Lemma 18: meet(X, zero) = zero.
% 186.70/24.25 Proof:
% 186.70/24.25 meet(X, zero)
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.25 meet(zero, X)
% 186.70/24.25 = { by lemma 17 R->L }
% 186.70/24.25 join(zero, meet(zero, X))
% 186.70/24.25 = { by axiom 7 (absorption2) }
% 186.70/24.25 zero
% 186.70/24.25
% 186.70/24.25 Lemma 19: meet(one, X) = X.
% 186.70/24.25 Proof:
% 186.70/24.25 meet(one, X)
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.25 meet(X, one)
% 186.70/24.25 = { by lemma 15 }
% 186.70/24.25 X
% 186.70/24.25
% 186.70/24.25 Lemma 20: join(X, one) = one.
% 186.70/24.25 Proof:
% 186.70/24.25 join(X, one)
% 186.70/24.25 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.25 join(one, X)
% 186.70/24.25 = { by lemma 19 R->L }
% 186.70/24.25 join(one, meet(one, X))
% 186.70/24.25 = { by axiom 7 (absorption2) }
% 186.70/24.25 one
% 186.70/24.25
% 186.70/24.25 Lemma 21: meet(X, join(Y, X)) = X.
% 186.70/24.25 Proof:
% 186.70/24.25 meet(X, join(Y, X))
% 186.70/24.25 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.25 meet(X, join(X, Y))
% 186.70/24.25 = { by axiom 9 (absorption1) }
% 186.70/24.25 X
% 186.70/24.25
% 186.70/24.25 Lemma 22: join(X, meet(Y, X)) = X.
% 186.70/24.25 Proof:
% 186.70/24.25 join(X, meet(Y, X))
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.25 join(X, meet(X, Y))
% 186.70/24.25 = { by axiom 7 (absorption2) }
% 186.70/24.25 X
% 186.70/24.25
% 186.70/24.25 Lemma 23: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 186.70/24.25 Proof:
% 186.70/24.25 meet(X, meet(Y, join(X, Z)))
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.25 meet(X, meet(join(X, Z), Y))
% 186.70/24.25 = { by axiom 10 (associativity_of_meet) R->L }
% 186.70/24.25 meet(meet(X, join(X, Z)), Y)
% 186.70/24.25 = { by axiom 9 (absorption1) }
% 186.70/24.25 meet(X, Y)
% 186.70/24.25
% 186.70/24.25 Lemma 24: meet(X, complement(join(X, Y))) = zero.
% 186.70/24.25 Proof:
% 186.70/24.25 meet(X, complement(join(X, Y)))
% 186.70/24.25 = { by lemma 23 R->L }
% 186.70/24.25 meet(X, meet(complement(join(X, Y)), join(X, Y)))
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) }
% 186.70/24.25 meet(X, meet(join(X, Y), complement(join(X, Y))))
% 186.70/24.25 = { by axiom 4 (complement_meet) }
% 186.70/24.25 meet(X, zero)
% 186.70/24.25 = { by lemma 18 }
% 186.70/24.25 zero
% 186.70/24.25
% 186.70/24.25 Lemma 25: meet(X, meet(Y, complement(X))) = zero.
% 186.70/24.25 Proof:
% 186.70/24.25 meet(X, meet(Y, complement(X)))
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.25 meet(X, meet(complement(X), Y))
% 186.70/24.25 = { by axiom 10 (associativity_of_meet) R->L }
% 186.70/24.25 meet(meet(X, complement(X)), Y)
% 186.70/24.25 = { by axiom 4 (complement_meet) }
% 186.70/24.25 meet(zero, Y)
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.25 meet(Y, zero)
% 186.70/24.25 = { by lemma 18 }
% 186.70/24.25 zero
% 186.70/24.25
% 186.70/24.25 Lemma 26: meet(X, meet(complement(X), Y)) = zero.
% 186.70/24.25 Proof:
% 186.70/24.25 meet(X, meet(complement(X), Y))
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.25 meet(X, meet(Y, complement(X)))
% 186.70/24.25 = { by lemma 25 }
% 186.70/24.25 zero
% 186.70/24.25
% 186.70/24.25 Lemma 27: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 186.70/24.25 Proof:
% 186.70/24.25 join(X, join(Y, meet(X, Z)))
% 186.70/24.25 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.25 join(X, join(meet(X, Z), Y))
% 186.70/24.25 = { by axiom 8 (associativity_of_join) R->L }
% 186.70/24.25 join(join(X, meet(X, Z)), Y)
% 186.70/24.25 = { by axiom 7 (absorption2) }
% 186.70/24.25 join(X, Y)
% 186.70/24.25
% 186.70/24.25 Lemma 28: meet(X, join(meet(X, Y), meet(X, Z))) = join(meet(X, Y), meet(X, Z)).
% 186.70/24.25 Proof:
% 186.70/24.25 meet(X, join(meet(X, Y), meet(X, Z)))
% 186.70/24.25 = { by axiom 7 (absorption2) R->L }
% 186.70/24.25 meet(join(X, meet(X, Y)), join(meet(X, Y), meet(X, Z)))
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.25 meet(join(meet(X, Y), meet(X, Z)), join(X, meet(X, Y)))
% 186.70/24.25 = { by lemma 27 R->L }
% 186.70/24.25 meet(join(meet(X, Y), meet(X, Z)), join(X, join(meet(X, Y), meet(X, Z))))
% 186.70/24.25 = { by lemma 21 }
% 186.70/24.25 join(meet(X, Y), meet(X, Z))
% 186.70/24.25
% 186.70/24.25 Lemma 29: meet(X, join(meet(Y, complement(X)), meet(Z, complement(X)))) = zero.
% 186.70/24.25 Proof:
% 186.70/24.25 meet(X, join(meet(Y, complement(X)), meet(Z, complement(X))))
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.25 meet(X, join(meet(Y, complement(X)), meet(complement(X), Z)))
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.25 meet(X, join(meet(complement(X), Y), meet(complement(X), Z)))
% 186.70/24.25 = { by lemma 28 R->L }
% 186.70/24.25 meet(X, meet(complement(X), join(meet(complement(X), Y), meet(complement(X), Z))))
% 186.70/24.25 = { by lemma 26 }
% 186.70/24.25 zero
% 186.70/24.25
% 186.70/24.25 Lemma 30: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 186.70/24.25 Proof:
% 186.70/24.25 join(X, join(Y, meet(Z, join(X, Y))))
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.25 join(X, join(Y, meet(join(X, Y), Z)))
% 186.70/24.25 = { by axiom 8 (associativity_of_join) R->L }
% 186.70/24.25 join(join(X, Y), meet(join(X, Y), Z))
% 186.70/24.25 = { by axiom 7 (absorption2) }
% 186.70/24.25 join(X, Y)
% 186.70/24.25
% 186.70/24.25 Lemma 31: meet(X, join(Y, meet(Z, join(X, Y)))) = meet(X, join(Y, meet(X, Z))).
% 186.70/24.25 Proof:
% 186.70/24.25 meet(X, join(Y, meet(Z, join(X, Y))))
% 186.70/24.25 = { by lemma 27 R->L }
% 186.70/24.25 meet(X, join(Y, meet(Z, join(X, join(Y, meet(X, Z))))))
% 186.70/24.25 = { by axiom 12 (equation_H51) }
% 186.70/24.25 meet(X, join(Y, join(meet(X, Z), meet(Z, join(Y, meet(X, Z))))))
% 186.70/24.25 = { by lemma 30 }
% 186.70/24.25 meet(X, join(Y, meet(X, Z)))
% 186.70/24.25
% 186.70/24.25 Lemma 32: meet(X, join(complement(X), meet(X, Y))) = meet(X, join(Y, complement(X))).
% 186.70/24.25 Proof:
% 186.70/24.25 meet(X, join(complement(X), meet(X, Y)))
% 186.70/24.25 = { by lemma 31 R->L }
% 186.70/24.25 meet(X, join(complement(X), meet(Y, join(X, complement(X)))))
% 186.70/24.25 = { by axiom 3 (complement_join) }
% 186.70/24.25 meet(X, join(complement(X), meet(Y, one)))
% 186.70/24.25 = { by lemma 15 }
% 186.70/24.25 meet(X, join(complement(X), Y))
% 186.70/24.25 = { by axiom 1 (commutativity_of_join) }
% 186.70/24.25 meet(X, join(Y, complement(X)))
% 186.70/24.25
% 186.70/24.25 Lemma 33: join(X, join(Y, complement(X))) = one.
% 186.70/24.25 Proof:
% 186.70/24.25 join(X, join(Y, complement(X)))
% 186.70/24.25 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.25 join(X, join(complement(X), Y))
% 186.70/24.25 = { by axiom 8 (associativity_of_join) R->L }
% 186.70/24.25 join(join(X, complement(X)), Y)
% 186.70/24.25 = { by axiom 3 (complement_join) }
% 186.70/24.25 join(one, Y)
% 186.70/24.25 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.25 join(Y, one)
% 186.70/24.25 = { by lemma 20 }
% 186.70/24.25 one
% 186.70/24.25
% 186.70/24.25 Lemma 34: fresh(meet(X, join(Y, complement(X))), zero, X, join(Y, complement(X))) = complement(X).
% 186.70/24.25 Proof:
% 186.70/24.25 fresh(meet(X, join(Y, complement(X))), zero, X, join(Y, complement(X)))
% 186.70/24.25 = { by axiom 11 (meet_join_complement) R->L }
% 186.70/24.25 fresh2(join(X, join(Y, complement(X))), one, X, join(Y, complement(X)))
% 186.70/24.25 = { by lemma 33 }
% 186.70/24.25 fresh2(one, one, X, join(Y, complement(X)))
% 186.70/24.25 = { by axiom 6 (meet_join_complement) }
% 186.70/24.25 complement(X)
% 186.70/24.25
% 186.70/24.25 Lemma 35: fresh(meet(X, join(complement(X), Y)), zero, X, join(Y, complement(X))) = complement(X).
% 186.70/24.25 Proof:
% 186.70/24.25 fresh(meet(X, join(complement(X), Y)), zero, X, join(Y, complement(X)))
% 186.70/24.25 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.25 fresh(meet(X, join(Y, complement(X))), zero, X, join(Y, complement(X)))
% 186.70/24.25 = { by lemma 34 }
% 186.70/24.25 complement(X)
% 186.70/24.25
% 186.70/24.25 Lemma 36: join(complement(X), meet(Y, complement(meet(X, Y)))) = complement(X).
% 186.70/24.25 Proof:
% 186.70/24.25 join(complement(X), meet(Y, complement(meet(X, Y))))
% 186.70/24.25 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.25 join(meet(Y, complement(meet(X, Y))), complement(X))
% 186.70/24.25 = { by axiom 5 (meet_join_complement) R->L }
% 186.70/24.25 fresh(zero, zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 186.70/24.25 = { by axiom 4 (complement_meet) R->L }
% 186.70/24.25 fresh(meet(X, complement(X)), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 186.70/24.25 = { by lemma 16 R->L }
% 186.70/24.25 fresh(meet(X, join(complement(X), zero)), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 186.70/24.25 = { by axiom 4 (complement_meet) R->L }
% 186.70/24.25 fresh(meet(X, join(complement(X), meet(meet(Y, X), complement(meet(Y, X))))), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 186.70/24.25 = { by axiom 10 (associativity_of_meet) }
% 186.70/24.25 fresh(meet(X, join(complement(X), meet(Y, meet(X, complement(meet(Y, X)))))), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.25 fresh(meet(X, join(complement(X), meet(Y, meet(complement(meet(Y, X)), X)))), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 186.70/24.25 = { by axiom 10 (associativity_of_meet) R->L }
% 186.70/24.25 fresh(meet(X, join(complement(X), meet(meet(Y, complement(meet(Y, X))), X))), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.25 fresh(meet(X, join(complement(X), meet(X, meet(Y, complement(meet(Y, X)))))), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 186.70/24.25 = { by lemma 32 }
% 186.70/24.25 fresh(meet(X, join(meet(Y, complement(meet(Y, X))), complement(X))), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 186.70/24.25 = { by axiom 1 (commutativity_of_join) }
% 186.70/24.25 fresh(meet(X, join(complement(X), meet(Y, complement(meet(Y, X))))), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) }
% 186.70/24.25 fresh(meet(X, join(complement(X), meet(Y, complement(meet(X, Y))))), zero, X, join(meet(Y, complement(meet(X, Y))), complement(X)))
% 186.70/24.25 = { by lemma 35 }
% 186.70/24.25 complement(X)
% 186.70/24.25
% 186.70/24.25 Lemma 37: join(X, meet(Y, complement(X))) = join(X, Y).
% 186.70/24.25 Proof:
% 186.70/24.25 join(X, meet(Y, complement(X)))
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.25 join(X, meet(complement(X), Y))
% 186.70/24.25 = { by lemma 14 R->L }
% 186.70/24.25 join(complement(complement(X)), meet(complement(X), Y))
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.25 join(complement(complement(X)), meet(Y, complement(X)))
% 186.70/24.25 = { by lemma 14 R->L }
% 186.70/24.25 complement(complement(join(complement(complement(X)), meet(Y, complement(X)))))
% 186.70/24.25 = { by lemma 36 R->L }
% 186.70/24.25 join(complement(complement(join(complement(complement(X)), meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(complement(complement(X)), meet(Y, complement(X)))), Y))))
% 186.70/24.25 = { by lemma 14 }
% 186.70/24.25 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(complement(join(complement(complement(X)), meet(Y, complement(X)))), Y))))
% 186.70/24.25 = { by axiom 2 (commutativity_of_meet) }
% 186.70/24.25 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, complement(join(complement(complement(X)), meet(Y, complement(X))))))))
% 186.70/24.25 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.25 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, complement(join(meet(Y, complement(X)), complement(complement(X))))))))
% 186.70/24.25 = { by lemma 21 R->L }
% 186.70/24.25 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, meet(complement(join(meet(Y, complement(X)), complement(complement(X)))), join(complement(X), complement(join(meet(Y, complement(X)), complement(complement(X))))))))))
% 186.70/24.25 = { by axiom 5 (meet_join_complement) R->L }
% 186.70/24.25 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, meet(complement(join(meet(Y, complement(X)), complement(complement(X)))), fresh(zero, zero, complement(complement(X)), join(complement(X), complement(join(meet(Y, complement(X)), complement(complement(X)))))))))))
% 186.70/24.25 = { by axiom 4 (complement_meet) R->L }
% 186.70/24.25 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, meet(complement(join(meet(Y, complement(X)), complement(complement(X)))), fresh(meet(complement(complement(X)), complement(complement(complement(X)))), zero, complement(complement(X)), join(complement(X), complement(join(meet(Y, complement(X)), complement(complement(X)))))))))))
% 186.70/24.25 = { by lemma 16 R->L }
% 186.70/24.25 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, meet(complement(join(meet(Y, complement(X)), complement(complement(X)))), fresh(meet(complement(complement(X)), join(complement(complement(complement(X))), zero)), zero, complement(complement(X)), join(complement(X), complement(join(meet(Y, complement(X)), complement(complement(X)))))))))))
% 186.70/24.26 = { by lemma 24 R->L }
% 186.70/24.26 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, meet(complement(join(meet(Y, complement(X)), complement(complement(X)))), fresh(meet(complement(complement(X)), join(complement(complement(complement(X))), meet(complement(complement(X)), complement(join(complement(complement(X)), meet(Y, complement(X))))))), zero, complement(complement(X)), join(complement(X), complement(join(meet(Y, complement(X)), complement(complement(X)))))))))))
% 186.70/24.26 = { by lemma 32 }
% 186.70/24.26 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, meet(complement(join(meet(Y, complement(X)), complement(complement(X)))), fresh(meet(complement(complement(X)), join(complement(join(complement(complement(X)), meet(Y, complement(X)))), complement(complement(complement(X))))), zero, complement(complement(X)), join(complement(X), complement(join(meet(Y, complement(X)), complement(complement(X)))))))))))
% 186.70/24.26 = { by axiom 1 (commutativity_of_join) }
% 186.70/24.26 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, meet(complement(join(meet(Y, complement(X)), complement(complement(X)))), fresh(meet(complement(complement(X)), join(complement(complement(complement(X))), complement(join(complement(complement(X)), meet(Y, complement(X)))))), zero, complement(complement(X)), join(complement(X), complement(join(meet(Y, complement(X)), complement(complement(X)))))))))))
% 186.70/24.26 = { by lemma 14 }
% 186.70/24.26 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, meet(complement(join(meet(Y, complement(X)), complement(complement(X)))), fresh(meet(complement(complement(X)), join(complement(X), complement(join(complement(complement(X)), meet(Y, complement(X)))))), zero, complement(complement(X)), join(complement(X), complement(join(meet(Y, complement(X)), complement(complement(X)))))))))))
% 186.70/24.26 = { by axiom 1 (commutativity_of_join) }
% 186.70/24.26 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, meet(complement(join(meet(Y, complement(X)), complement(complement(X)))), fresh(meet(complement(complement(X)), join(complement(X), complement(join(meet(Y, complement(X)), complement(complement(X)))))), zero, complement(complement(X)), join(complement(X), complement(join(meet(Y, complement(X)), complement(complement(X)))))))))))
% 186.70/24.26 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.26 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, meet(complement(join(meet(Y, complement(X)), complement(complement(X)))), fresh(meet(complement(complement(X)), join(complement(join(meet(Y, complement(X)), complement(complement(X)))), complement(X))), zero, complement(complement(X)), join(complement(X), complement(join(meet(Y, complement(X)), complement(complement(X)))))))))))
% 186.70/24.26 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.26 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, meet(complement(join(meet(Y, complement(X)), complement(complement(X)))), fresh(meet(complement(complement(X)), join(complement(join(meet(Y, complement(X)), complement(complement(X)))), complement(X))), zero, complement(complement(X)), join(complement(join(meet(Y, complement(X)), complement(complement(X)))), complement(X))))))))
% 186.70/24.26 = { by lemma 14 R->L }
% 186.70/24.26 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, meet(complement(join(meet(Y, complement(X)), complement(complement(X)))), fresh(meet(complement(complement(X)), join(complement(join(meet(Y, complement(X)), complement(complement(X)))), complement(X))), zero, complement(complement(X)), join(complement(join(meet(Y, complement(X)), complement(complement(X)))), complement(complement(complement(X))))))))))
% 186.70/24.26 = { by lemma 14 R->L }
% 186.70/24.26 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, meet(complement(join(meet(Y, complement(X)), complement(complement(X)))), fresh(meet(complement(complement(X)), join(complement(join(meet(Y, complement(X)), complement(complement(X)))), complement(complement(complement(X))))), zero, complement(complement(X)), join(complement(join(meet(Y, complement(X)), complement(complement(X)))), complement(complement(complement(X))))))))))
% 186.70/24.26 = { by lemma 34 }
% 186.70/24.26 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, meet(complement(join(meet(Y, complement(X)), complement(complement(X)))), complement(complement(complement(X))))))))
% 186.70/24.26 = { by lemma 14 }
% 186.70/24.26 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, meet(complement(join(meet(Y, complement(X)), complement(complement(X)))), complement(X))))))
% 186.70/24.26 = { by axiom 2 (commutativity_of_meet) }
% 186.70/24.26 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(Y, meet(complement(X), complement(join(meet(Y, complement(X)), complement(complement(X)))))))))
% 186.70/24.26 = { by axiom 10 (associativity_of_meet) R->L }
% 186.70/24.26 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(meet(meet(Y, complement(X)), complement(join(meet(Y, complement(X)), complement(complement(X))))))))
% 186.70/24.26 = { by lemma 24 }
% 186.70/24.26 join(join(complement(complement(X)), meet(Y, complement(X))), meet(Y, complement(zero)))
% 186.70/24.26 = { by axiom 8 (associativity_of_join) }
% 186.70/24.26 join(complement(complement(X)), join(meet(Y, complement(X)), meet(Y, complement(zero))))
% 186.70/24.26 = { by lemma 17 R->L }
% 186.70/24.26 join(complement(complement(X)), join(meet(Y, complement(X)), meet(Y, join(zero, complement(zero)))))
% 186.70/24.26 = { by axiom 3 (complement_join) }
% 186.70/24.26 join(complement(complement(X)), join(meet(Y, complement(X)), meet(Y, one)))
% 186.70/24.26 = { by lemma 15 }
% 186.70/24.26 join(complement(complement(X)), join(meet(Y, complement(X)), Y))
% 186.70/24.26 = { by axiom 1 (commutativity_of_join) }
% 186.70/24.26 join(complement(complement(X)), join(Y, meet(Y, complement(X))))
% 186.70/24.26 = { by axiom 7 (absorption2) }
% 186.70/24.26 join(complement(complement(X)), Y)
% 186.70/24.26 = { by axiom 1 (commutativity_of_join) }
% 186.70/24.26 join(Y, complement(complement(X)))
% 186.70/24.26 = { by lemma 14 }
% 186.70/24.26 join(Y, X)
% 186.70/24.26 = { by axiom 1 (commutativity_of_join) }
% 186.70/24.26 join(X, Y)
% 186.70/24.26
% 186.70/24.26 Lemma 38: join(meet(X, Y), meet(X, join(Y, Z))) = meet(X, join(Y, Z)).
% 186.70/24.26 Proof:
% 186.70/24.26 join(meet(X, Y), meet(X, join(Y, Z)))
% 186.70/24.26 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.26 join(meet(Y, X), meet(X, join(Y, Z)))
% 186.70/24.26 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.26 join(meet(X, join(Y, Z)), meet(Y, X))
% 186.70/24.26 = { by lemma 23 R->L }
% 186.70/24.26 join(meet(X, join(Y, Z)), meet(Y, meet(X, join(Y, Z))))
% 186.70/24.26 = { by lemma 22 }
% 186.70/24.26 meet(X, join(Y, Z))
% 186.70/24.26
% 186.70/24.26 Lemma 39: join(X, join(meet(X, Y), Z)) = join(X, Z).
% 186.70/24.26 Proof:
% 186.70/24.26 join(X, join(meet(X, Y), Z))
% 186.70/24.26 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.26 join(X, join(Z, meet(X, Y)))
% 186.70/24.26 = { by lemma 27 }
% 186.70/24.26 join(X, Z)
% 186.70/24.26
% 186.70/24.26 Lemma 40: join(X, complement(meet(X, Y))) = one.
% 186.70/24.26 Proof:
% 186.70/24.26 join(X, complement(meet(X, Y)))
% 186.70/24.26 = { by lemma 39 R->L }
% 186.70/24.26 join(X, join(meet(X, Y), complement(meet(X, Y))))
% 186.70/24.26 = { by axiom 3 (complement_join) }
% 186.70/24.26 join(X, one)
% 186.70/24.26 = { by lemma 20 }
% 186.70/24.26 one
% 186.70/24.26
% 186.70/24.26 Lemma 41: meet(complement(X), complement(meet(X, Y))) = complement(X).
% 186.70/24.26 Proof:
% 186.70/24.26 meet(complement(X), complement(meet(X, Y)))
% 186.70/24.26 = { by axiom 5 (meet_join_complement) R->L }
% 186.70/24.26 fresh(zero, zero, X, meet(complement(X), complement(meet(X, Y))))
% 186.70/24.26 = { by lemma 26 R->L }
% 186.70/24.26 fresh(meet(X, meet(complement(X), complement(meet(X, Y)))), zero, X, meet(complement(X), complement(meet(X, Y))))
% 186.70/24.26 = { by axiom 11 (meet_join_complement) R->L }
% 186.70/24.26 fresh2(join(X, meet(complement(X), complement(meet(X, Y)))), one, X, meet(complement(X), complement(meet(X, Y))))
% 186.70/24.26 = { by lemma 14 R->L }
% 186.70/24.26 fresh2(join(complement(complement(X)), meet(complement(X), complement(meet(X, Y)))), one, X, meet(complement(X), complement(meet(X, Y))))
% 186.70/24.26 = { by lemma 22 R->L }
% 186.70/24.26 fresh2(join(join(complement(complement(X)), meet(complement(X), complement(meet(X, Y)))), meet(complement(X), join(complement(complement(X)), meet(complement(X), complement(meet(X, Y)))))), one, X, meet(complement(X), complement(meet(X, Y))))
% 186.70/24.26 = { by lemma 32 }
% 186.70/24.26 fresh2(join(join(complement(complement(X)), meet(complement(X), complement(meet(X, Y)))), meet(complement(X), join(complement(meet(X, Y)), complement(complement(X))))), one, X, meet(complement(X), complement(meet(X, Y))))
% 186.70/24.26 = { by axiom 8 (associativity_of_join) }
% 186.70/24.26 fresh2(join(complement(complement(X)), join(meet(complement(X), complement(meet(X, Y))), meet(complement(X), join(complement(meet(X, Y)), complement(complement(X)))))), one, X, meet(complement(X), complement(meet(X, Y))))
% 186.70/24.26 = { by lemma 38 }
% 186.70/24.26 fresh2(join(complement(complement(X)), meet(complement(X), join(complement(meet(X, Y)), complement(complement(X))))), one, X, meet(complement(X), complement(meet(X, Y))))
% 186.70/24.26 = { by lemma 14 }
% 186.70/24.26 fresh2(join(X, meet(complement(X), join(complement(meet(X, Y)), complement(complement(X))))), one, X, meet(complement(X), complement(meet(X, Y))))
% 186.70/24.26 = { by lemma 14 }
% 186.70/24.26 fresh2(join(X, meet(complement(X), join(complement(meet(X, Y)), X))), one, X, meet(complement(X), complement(meet(X, Y))))
% 186.70/24.26 = { by axiom 1 (commutativity_of_join) }
% 186.70/24.26 fresh2(join(X, meet(complement(X), join(X, complement(meet(X, Y))))), one, X, meet(complement(X), complement(meet(X, Y))))
% 186.70/24.26 = { by lemma 40 }
% 186.70/24.26 fresh2(join(X, meet(complement(X), one)), one, X, meet(complement(X), complement(meet(X, Y))))
% 186.70/24.26 = { by lemma 15 }
% 186.70/24.26 fresh2(join(X, complement(X)), one, X, meet(complement(X), complement(meet(X, Y))))
% 186.70/24.26 = { by axiom 3 (complement_join) }
% 186.70/24.26 fresh2(one, one, X, meet(complement(X), complement(meet(X, Y))))
% 186.70/24.26 = { by axiom 6 (meet_join_complement) }
% 186.70/24.26 complement(X)
% 186.70/24.26
% 186.70/24.26 Lemma 42: meet(X, join(Y, complement(X))) = meet(X, Y).
% 186.70/24.26 Proof:
% 186.70/24.26 meet(X, join(Y, complement(X)))
% 186.70/24.26 = { by lemma 14 R->L }
% 186.70/24.26 complement(complement(meet(X, join(Y, complement(X)))))
% 186.70/24.26 = { by axiom 5 (meet_join_complement) R->L }
% 186.70/24.26 complement(fresh(zero, zero, join(meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), meet(complement(meet(X, join(Y, complement(X)))), meet(Z, complement(complement(meet(X, join(Y, complement(X)))))))), complement(meet(X, join(Y, complement(X))))))
% 186.70/24.26 = { by lemma 29 R->L }
% 186.70/24.26 complement(fresh(meet(complement(meet(X, join(Y, complement(X)))), join(meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), meet(meet(complement(meet(X, join(Y, complement(X)))), Z), complement(complement(meet(X, join(Y, complement(X)))))))), zero, join(meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), meet(complement(meet(X, join(Y, complement(X)))), meet(Z, complement(complement(meet(X, join(Y, complement(X)))))))), complement(meet(X, join(Y, complement(X))))))
% 186.70/24.26 = { by axiom 10 (associativity_of_meet) }
% 186.70/24.26 complement(fresh(meet(complement(meet(X, join(Y, complement(X)))), join(meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), meet(complement(meet(X, join(Y, complement(X)))), meet(Z, complement(complement(meet(X, join(Y, complement(X))))))))), zero, join(meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), meet(complement(meet(X, join(Y, complement(X)))), meet(Z, complement(complement(meet(X, join(Y, complement(X)))))))), complement(meet(X, join(Y, complement(X))))))
% 186.70/24.26 = { by lemma 13 R->L }
% 186.70/24.26 complement(fresh2(join(complement(meet(X, join(Y, complement(X)))), join(meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), meet(complement(meet(X, join(Y, complement(X)))), meet(Z, complement(complement(meet(X, join(Y, complement(X))))))))), one, join(meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), meet(complement(meet(X, join(Y, complement(X)))), meet(Z, complement(complement(meet(X, join(Y, complement(X)))))))), complement(meet(X, join(Y, complement(X))))))
% 186.70/24.26 = { by lemma 27 }
% 186.70/24.26 complement(fresh2(join(complement(meet(X, join(Y, complement(X)))), meet(Y, complement(complement(meet(X, join(Y, complement(X))))))), one, join(meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), meet(complement(meet(X, join(Y, complement(X)))), meet(Z, complement(complement(meet(X, join(Y, complement(X)))))))), complement(meet(X, join(Y, complement(X))))))
% 186.70/24.26 = { by lemma 37 }
% 186.70/24.26 complement(fresh2(join(complement(meet(X, join(Y, complement(X)))), Y), one, join(meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), meet(complement(meet(X, join(Y, complement(X)))), meet(Z, complement(complement(meet(X, join(Y, complement(X)))))))), complement(meet(X, join(Y, complement(X))))))
% 186.70/24.26 = { by lemma 25 }
% 186.70/24.26 complement(fresh2(join(complement(meet(X, join(Y, complement(X)))), Y), one, join(meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), zero), complement(meet(X, join(Y, complement(X))))))
% 186.70/24.26 = { by lemma 16 }
% 186.70/24.26 complement(fresh2(join(complement(meet(X, join(Y, complement(X)))), Y), one, meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), complement(meet(X, join(Y, complement(X))))))
% 186.70/24.26 = { by axiom 1 (commutativity_of_join) }
% 186.70/24.26 complement(fresh2(join(Y, complement(meet(X, join(Y, complement(X))))), one, meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), complement(meet(X, join(Y, complement(X))))))
% 186.70/24.26 = { by lemma 22 R->L }
% 186.70/24.26 complement(fresh2(join(Y, join(complement(meet(X, join(Y, complement(X)))), meet(complement(X), complement(meet(X, join(Y, complement(X))))))), one, meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), complement(meet(X, join(Y, complement(X))))))
% 186.70/24.26 = { by lemma 41 }
% 186.70/24.26 complement(fresh2(join(Y, join(complement(meet(X, join(Y, complement(X)))), complement(X))), one, meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), complement(meet(X, join(Y, complement(X))))))
% 186.70/24.27 = { by axiom 1 (commutativity_of_join) }
% 186.70/24.27 complement(fresh2(join(Y, join(complement(X), complement(meet(X, join(Y, complement(X)))))), one, meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), complement(meet(X, join(Y, complement(X))))))
% 186.70/24.27 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.27 complement(fresh2(join(Y, join(complement(X), complement(meet(join(Y, complement(X)), X)))), one, meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), complement(meet(X, join(Y, complement(X))))))
% 186.70/24.27 = { by axiom 8 (associativity_of_join) R->L }
% 186.70/24.27 complement(fresh2(join(join(Y, complement(X)), complement(meet(join(Y, complement(X)), X))), one, meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), complement(meet(X, join(Y, complement(X))))))
% 186.70/24.27 = { by lemma 40 }
% 186.70/24.27 complement(fresh2(one, one, meet(Y, complement(complement(meet(X, join(Y, complement(X)))))), complement(meet(X, join(Y, complement(X))))))
% 186.70/24.27 = { by axiom 6 (meet_join_complement) }
% 186.70/24.27 complement(complement(meet(Y, complement(complement(meet(X, join(Y, complement(X))))))))
% 186.70/24.27 = { by lemma 14 }
% 186.70/24.27 complement(complement(meet(Y, meet(X, join(Y, complement(X))))))
% 186.70/24.27 = { by lemma 23 }
% 186.70/24.27 complement(complement(meet(Y, X)))
% 186.70/24.27 = { by axiom 2 (commutativity_of_meet) }
% 186.70/24.27 complement(complement(meet(X, Y)))
% 186.70/24.27 = { by lemma 14 }
% 186.70/24.27 meet(X, Y)
% 186.70/24.27
% 186.70/24.27 Lemma 43: meet(X, join(complement(X), Y)) = meet(X, Y).
% 186.70/24.27 Proof:
% 186.70/24.27 meet(X, join(complement(X), Y))
% 186.70/24.27 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.27 meet(X, join(Y, complement(X)))
% 186.70/24.27 = { by lemma 42 }
% 186.70/24.27 meet(X, Y)
% 186.70/24.27
% 186.70/24.27 Lemma 44: join(X, meet(complement(X), Y)) = join(X, Y).
% 186.70/24.27 Proof:
% 186.70/24.27 join(X, meet(complement(X), Y))
% 186.70/24.27 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.27 join(X, meet(Y, complement(X)))
% 186.70/24.27 = { by lemma 37 }
% 186.70/24.27 join(X, Y)
% 186.70/24.27
% 186.70/24.27 Lemma 45: join(meet(X, Y), meet(X, join(Z, meet(X, Y)))) = meet(X, join(Z, meet(X, Y))).
% 186.70/24.27 Proof:
% 186.70/24.27 join(meet(X, Y), meet(X, join(Z, meet(X, Y))))
% 186.70/24.27 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.27 join(meet(Y, X), meet(X, join(Z, meet(X, Y))))
% 186.70/24.27 = { by axiom 2 (commutativity_of_meet) R->L }
% 186.70/24.27 join(meet(Y, X), meet(X, join(Z, meet(Y, X))))
% 186.70/24.27 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.27 join(meet(X, join(Z, meet(Y, X))), meet(Y, X))
% 186.70/24.27 = { by axiom 9 (absorption1) R->L }
% 186.70/24.27 join(meet(X, join(Z, meet(Y, X))), meet(meet(Y, X), join(meet(Y, X), Z)))
% 186.70/24.27 = { by axiom 10 (associativity_of_meet) }
% 186.70/24.27 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(meet(Y, X), Z))))
% 186.70/24.27 = { by axiom 1 (commutativity_of_join) }
% 186.70/24.27 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(Z, meet(Y, X)))))
% 186.70/24.27 = { by lemma 22 }
% 186.70/24.27 meet(X, join(Z, meet(Y, X)))
% 186.70/24.27 = { by axiom 2 (commutativity_of_meet) }
% 186.70/24.27 meet(X, join(Z, meet(X, Y)))
% 186.70/24.27
% 186.70/24.27 Lemma 46: join(meet(X, Y), meet(X, join(meet(X, Y), Z))) = meet(X, join(Z, meet(X, Y))).
% 186.70/24.27 Proof:
% 186.70/24.27 join(meet(X, Y), meet(X, join(meet(X, Y), Z)))
% 186.70/24.27 = { by axiom 1 (commutativity_of_join) R->L }
% 186.70/24.27 join(meet(X, Y), meet(X, join(Z, meet(X, Y))))
% 186.70/24.27 = { by lemma 45 }
% 186.70/24.27 meet(X, join(Z, meet(X, Y)))
% 186.70/24.27
% 186.70/24.27 Lemma 47: meet(X, join(Y, join(Z, complement(X)))) = meet(X, join(Y, Z)).
% 186.70/24.27 Proof:
% 186.70/24.27 meet(X, join(Y, join(Z, complement(X))))
% 186.70/24.27 = { by lemma 43 R->L }
% 186.70/24.27 meet(X, join(complement(X), join(Y, join(Z, complement(X)))))
% 186.70/24.27 = { by axiom 8 (associativity_of_join) R->L }
% 186.70/24.27 meet(X, join(complement(X), join(join(Y, Z), complement(X))))
% 186.70/24.27 = { by lemma 43 }
% 186.70/24.27 meet(X, join(join(Y, Z), complement(X)))
% 186.70/24.27 = { by lemma 42 }
% 186.70/24.27 meet(X, join(Y, Z))
% 186.70/24.27
% 186.70/24.27 Lemma 48: meet(X, join(Y, meet(X, Z))) = meet(X, join(Y, Z)).
% 186.70/24.27 Proof:
% 186.70/24.27 meet(X, join(Y, meet(X, Z)))
% 186.70/24.27 = { by lemma 46 R->L }
% 186.70/24.27 join(meet(X, Z), meet(X, join(meet(X, Z), Y)))
% 186.70/24.27 = { by lemma 47 R->L }
% 186.70/24.27 join(meet(X, Z), meet(X, join(meet(X, Z), join(Y, complement(X)))))
% 186.70/24.27 = { by lemma 46 }
% 186.70/24.27 meet(X, join(join(Y, complement(X)), meet(X, Z)))
% 186.70/24.27 = { by lemma 31 R->L }
% 186.70/24.27 meet(X, join(join(Y, complement(X)), meet(Z, join(X, join(Y, complement(X))))))
% 186.70/24.27 = { by axiom 2 (commutativity_of_meet) }
% 186.70/24.27 meet(X, join(join(Y, complement(X)), meet(join(X, join(Y, complement(X))), Z)))
% 187.03/24.27 = { by lemma 33 }
% 187.03/24.27 meet(X, join(join(Y, complement(X)), meet(one, Z)))
% 187.03/24.27 = { by axiom 8 (associativity_of_join) }
% 187.03/24.27 meet(X, join(Y, join(complement(X), meet(one, Z))))
% 187.03/24.27 = { by lemma 19 }
% 187.03/24.27 meet(X, join(Y, join(complement(X), Z)))
% 187.03/24.27 = { by axiom 1 (commutativity_of_join) }
% 187.03/24.27 meet(X, join(Y, join(Z, complement(X))))
% 187.03/24.27 = { by lemma 47 }
% 187.03/24.27 meet(X, join(Y, Z))
% 187.03/24.27
% 187.03/24.27 Lemma 49: join(meet(X, Y), meet(Y, join(X, Z))) = meet(Y, join(X, Z)).
% 187.03/24.27 Proof:
% 187.03/24.27 join(meet(X, Y), meet(Y, join(X, Z)))
% 187.03/24.27 = { by axiom 2 (commutativity_of_meet) R->L }
% 187.03/24.27 join(meet(Y, X), meet(Y, join(X, Z)))
% 187.03/24.27 = { by lemma 38 }
% 187.03/24.27 meet(Y, join(X, Z))
% 187.03/24.27
% 187.03/24.27 Lemma 50: join(X, join(Y, meet(Z, meet(W, join(X, meet(Y, V)))))) = join(X, Y).
% 187.03/24.27 Proof:
% 187.03/24.27 join(X, join(Y, meet(Z, meet(W, join(X, meet(Y, V))))))
% 187.03/24.27 = { by axiom 1 (commutativity_of_join) R->L }
% 187.03/24.27 join(X, join(meet(Z, meet(W, join(X, meet(Y, V)))), Y))
% 187.03/24.27 = { by axiom 8 (associativity_of_join) R->L }
% 187.03/24.27 join(join(X, meet(Z, meet(W, join(X, meet(Y, V))))), Y)
% 187.03/24.27 = { by axiom 1 (commutativity_of_join) R->L }
% 187.03/24.27 join(Y, join(X, meet(Z, meet(W, join(X, meet(Y, V))))))
% 187.03/24.27 = { by axiom 1 (commutativity_of_join) R->L }
% 187.03/24.27 join(Y, join(X, meet(Z, meet(W, join(meet(Y, V), X)))))
% 187.03/24.27 = { by lemma 39 R->L }
% 187.03/24.27 join(Y, join(meet(Y, V), join(X, meet(Z, meet(W, join(meet(Y, V), X))))))
% 187.03/24.27 = { by axiom 10 (associativity_of_meet) R->L }
% 187.03/24.27 join(Y, join(meet(Y, V), join(X, meet(meet(Z, W), join(meet(Y, V), X)))))
% 187.03/24.27 = { by lemma 30 }
% 187.03/24.27 join(Y, join(meet(Y, V), X))
% 187.03/24.27 = { by lemma 39 }
% 187.03/24.27 join(Y, X)
% 187.03/24.27 = { by axiom 1 (commutativity_of_join) }
% 187.03/24.27 join(X, Y)
% 187.03/24.27
% 187.03/24.27 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 187.03/24.27 Proof:
% 187.03/24.27 meet(a, join(b, c))
% 187.03/24.27 = { by axiom 1 (commutativity_of_join) R->L }
% 187.03/24.27 meet(a, join(c, b))
% 187.03/24.27 = { by lemma 14 R->L }
% 187.03/24.27 complement(complement(meet(a, join(c, b))))
% 187.03/24.27 = { by axiom 1 (commutativity_of_join) R->L }
% 187.03/24.27 complement(complement(meet(a, join(b, c))))
% 187.03/24.27 = { by lemma 48 R->L }
% 187.03/24.27 complement(complement(meet(a, join(b, meet(a, c)))))
% 187.03/24.27 = { by axiom 1 (commutativity_of_join) R->L }
% 187.03/24.27 complement(complement(meet(a, join(meet(a, c), b))))
% 187.03/24.27 = { by lemma 49 R->L }
% 187.03/24.27 complement(complement(join(meet(meet(a, c), a), meet(a, join(meet(a, c), b)))))
% 187.03/24.27 = { by axiom 1 (commutativity_of_join) R->L }
% 187.03/24.27 complement(complement(join(meet(meet(a, c), a), meet(a, join(b, meet(a, c))))))
% 187.03/24.27 = { by axiom 2 (commutativity_of_meet) R->L }
% 187.03/24.27 complement(complement(join(meet(a, meet(a, c)), meet(a, join(b, meet(a, c))))))
% 187.03/24.27 = { by axiom 1 (commutativity_of_join) R->L }
% 187.03/24.27 complement(complement(join(meet(a, meet(a, c)), meet(a, join(meet(a, c), b)))))
% 187.03/24.27 = { by lemma 22 R->L }
% 187.03/24.27 complement(complement(join(meet(a, meet(a, c)), join(meet(a, join(meet(a, c), b)), meet(b, meet(a, join(meet(a, c), b)))))))
% 187.03/24.27 = { by axiom 2 (commutativity_of_meet) R->L }
% 187.03/24.27 complement(complement(join(meet(a, meet(a, c)), join(meet(a, join(meet(a, c), b)), meet(b, meet(join(meet(a, c), b), a))))))
% 187.03/24.27 = { by axiom 10 (associativity_of_meet) R->L }
% 187.03/24.27 complement(complement(join(meet(a, meet(a, c)), join(meet(a, join(meet(a, c), b)), meet(meet(b, join(meet(a, c), b)), a)))))
% 187.03/24.27 = { by axiom 2 (commutativity_of_meet) }
% 187.03/24.27 complement(complement(join(meet(a, meet(a, c)), join(meet(a, join(meet(a, c), b)), meet(a, meet(b, join(meet(a, c), b)))))))
% 187.03/24.27 = { by lemma 21 }
% 187.03/24.27 complement(complement(join(meet(a, meet(a, c)), join(meet(a, join(meet(a, c), b)), meet(a, b)))))
% 187.03/24.27 = { by axiom 2 (commutativity_of_meet) }
% 187.03/24.27 complement(complement(join(meet(a, meet(a, c)), join(meet(a, join(meet(a, c), b)), meet(b, a)))))
% 187.03/24.27 = { by axiom 1 (commutativity_of_join) }
% 187.03/24.27 complement(complement(join(meet(a, meet(a, c)), join(meet(b, a), meet(a, join(meet(a, c), b))))))
% 187.03/24.27 = { by axiom 2 (commutativity_of_meet) R->L }
% 187.03/24.27 complement(complement(join(meet(meet(a, c), a), join(meet(b, a), meet(a, join(meet(a, c), b))))))
% 187.03/24.27 = { by axiom 1 (commutativity_of_join) R->L }
% 187.03/24.27 complement(complement(join(meet(meet(a, c), a), join(meet(a, join(meet(a, c), b)), meet(b, a)))))
% 187.03/24.27 = { by axiom 8 (associativity_of_join) R->L }
% 187.03/24.27 complement(complement(join(join(meet(meet(a, c), a), meet(a, join(meet(a, c), b))), meet(b, a))))
% 187.03/24.27 = { by lemma 49 }
% 187.03/24.27 complement(complement(join(meet(a, join(meet(a, c), b)), meet(b, a))))
% 187.03/24.27 = { by axiom 1 (commutativity_of_join) }
% 187.03/24.27 complement(complement(join(meet(b, a), meet(a, join(meet(a, c), b)))))
% 187.03/24.27 = { by axiom 2 (commutativity_of_meet) }
% 187.03/24.27 complement(complement(join(meet(a, b), meet(a, join(meet(a, c), b)))))
% 187.03/24.27 = { by lemma 48 R->L }
% 187.03/24.27 complement(complement(join(meet(a, b), meet(a, join(meet(a, c), meet(a, b))))))
% 187.03/24.27 = { by axiom 1 (commutativity_of_join) R->L }
% 187.03/24.27 complement(complement(join(meet(a, b), meet(a, join(meet(a, b), meet(a, c))))))
% 187.03/24.27 = { by lemma 45 R->L }
% 187.03/24.27 complement(complement(join(meet(a, b), join(meet(a, c), meet(a, join(meet(a, b), meet(a, c)))))))
% 187.03/24.27 = { by lemma 30 }
% 187.03/24.27 complement(complement(join(meet(a, b), meet(a, c))))
% 187.03/24.27 = { by lemma 28 R->L }
% 187.03/24.27 complement(complement(meet(a, join(meet(a, b), meet(a, c)))))
% 187.03/24.27 = { by axiom 2 (commutativity_of_meet) }
% 187.03/24.27 complement(complement(meet(a, join(meet(a, b), meet(c, a)))))
% 187.03/24.27 = { by axiom 2 (commutativity_of_meet) }
% 187.03/24.27 complement(complement(meet(a, join(meet(b, a), meet(c, a)))))
% 187.03/24.27 = { by lemma 14 R->L }
% 187.03/24.27 complement(complement(meet(a, complement(complement(join(meet(b, a), meet(c, a)))))))
% 187.03/24.27 = { by lemma 22 R->L }
% 187.03/24.27 complement(join(complement(meet(a, complement(complement(join(meet(b, a), meet(c, a)))))), meet(complement(join(meet(b, a), meet(c, a))), complement(meet(a, complement(complement(join(meet(b, a), meet(c, a)))))))))
% 187.03/24.27 = { by axiom 2 (commutativity_of_meet) R->L }
% 187.03/24.27 complement(join(complement(meet(a, complement(complement(join(meet(b, a), meet(c, a)))))), meet(complement(join(meet(b, a), meet(c, a))), complement(meet(complement(complement(join(meet(b, a), meet(c, a)))), a)))))
% 187.03/24.27 = { by lemma 14 R->L }
% 187.03/24.27 complement(join(complement(meet(a, complement(complement(join(meet(b, a), meet(c, a)))))), meet(complement(complement(complement(join(meet(b, a), meet(c, a))))), complement(meet(complement(complement(join(meet(b, a), meet(c, a)))), a)))))
% 187.03/24.27 = { by lemma 41 }
% 187.03/24.27 complement(join(complement(meet(a, complement(complement(join(meet(b, a), meet(c, a)))))), complement(complement(complement(join(meet(b, a), meet(c, a)))))))
% 187.03/24.27 = { by lemma 14 }
% 187.03/24.27 complement(join(complement(meet(a, complement(complement(join(meet(b, a), meet(c, a)))))), complement(join(meet(b, a), meet(c, a)))))
% 187.03/24.27 = { by axiom 1 (commutativity_of_join) }
% 187.03/24.27 complement(join(complement(join(meet(b, a), meet(c, a))), complement(meet(a, complement(complement(join(meet(b, a), meet(c, a))))))))
% 187.03/24.27 = { by axiom 2 (commutativity_of_meet) }
% 187.03/24.27 complement(join(complement(join(meet(b, a), meet(c, a))), complement(meet(complement(complement(join(meet(b, a), meet(c, a)))), a))))
% 187.03/24.27 = { by lemma 44 R->L }
% 187.03/24.27 complement(join(complement(join(meet(b, a), meet(c, a))), meet(complement(complement(join(meet(b, a), meet(c, a)))), complement(meet(complement(complement(join(meet(b, a), meet(c, a)))), a)))))
% 187.03/24.28 = { by lemma 21 R->L }
% 187.03/24.28 complement(join(complement(join(meet(b, a), meet(c, a))), meet(meet(complement(complement(join(meet(b, a), meet(c, a)))), complement(meet(complement(complement(join(meet(b, a), meet(c, a)))), a))), join(complement(a), meet(complement(complement(join(meet(b, a), meet(c, a)))), complement(meet(complement(complement(join(meet(b, a), meet(c, a)))), a)))))))
% 187.03/24.28 = { by axiom 2 (commutativity_of_meet) R->L }
% 187.03/24.28 complement(join(complement(join(meet(b, a), meet(c, a))), meet(meet(complement(complement(join(meet(b, a), meet(c, a)))), complement(meet(complement(complement(join(meet(b, a), meet(c, a)))), a))), join(complement(a), meet(complement(complement(join(meet(b, a), meet(c, a)))), complement(meet(a, complement(complement(join(meet(b, a), meet(c, a)))))))))))
% 187.03/24.28 = { by lemma 36 }
% 187.03/24.28 complement(join(complement(join(meet(b, a), meet(c, a))), meet(meet(complement(complement(join(meet(b, a), meet(c, a)))), complement(meet(complement(complement(join(meet(b, a), meet(c, a)))), a))), complement(a))))
% 187.03/24.28 = { by axiom 10 (associativity_of_meet) }
% 187.03/24.28 complement(join(complement(join(meet(b, a), meet(c, a))), meet(complement(complement(join(meet(b, a), meet(c, a)))), meet(complement(meet(complement(complement(join(meet(b, a), meet(c, a)))), a)), complement(a)))))
% 187.03/24.28 = { by axiom 2 (commutativity_of_meet) }
% 187.03/24.28 complement(join(complement(join(meet(b, a), meet(c, a))), meet(complement(complement(join(meet(b, a), meet(c, a)))), meet(complement(a), complement(meet(complement(complement(join(meet(b, a), meet(c, a)))), a))))))
% 187.03/24.28 = { by axiom 2 (commutativity_of_meet) R->L }
% 187.03/24.28 complement(join(complement(join(meet(b, a), meet(c, a))), meet(complement(complement(join(meet(b, a), meet(c, a)))), meet(complement(a), complement(meet(a, complement(complement(join(meet(b, a), meet(c, a))))))))))
% 187.03/24.28 = { by lemma 41 }
% 187.03/24.28 complement(join(complement(join(meet(b, a), meet(c, a))), meet(complement(complement(join(meet(b, a), meet(c, a)))), complement(a))))
% 187.03/24.28 = { by lemma 44 }
% 187.03/24.28 complement(join(complement(join(meet(b, a), meet(c, a))), complement(a)))
% 187.03/24.28 = { by axiom 1 (commutativity_of_join) }
% 187.03/24.28 complement(join(complement(a), complement(join(meet(b, a), meet(c, a)))))
% 187.03/24.28 = { by axiom 5 (meet_join_complement) R->L }
% 187.03/24.28 complement(fresh(zero, zero, join(meet(b, a), meet(c, a)), join(complement(a), complement(join(meet(b, a), meet(c, a))))))
% 187.03/24.28 = { by lemma 29 R->L }
% 187.03/24.28 complement(fresh(meet(complement(a), join(meet(b, complement(complement(a))), meet(c, complement(complement(a))))), zero, join(meet(b, a), meet(c, a)), join(complement(a), complement(join(meet(b, a), meet(c, a))))))
% 187.03/24.28 = { by lemma 14 }
% 187.03/24.28 complement(fresh(meet(complement(a), join(meet(b, a), meet(c, complement(complement(a))))), zero, join(meet(b, a), meet(c, a)), join(complement(a), complement(join(meet(b, a), meet(c, a))))))
% 187.03/24.28 = { by lemma 14 }
% 187.03/24.28 complement(fresh(meet(complement(a), join(meet(b, a), meet(c, a))), zero, join(meet(b, a), meet(c, a)), join(complement(a), complement(join(meet(b, a), meet(c, a))))))
% 187.03/24.28 = { by axiom 2 (commutativity_of_meet) R->L }
% 187.03/24.28 complement(fresh(meet(join(meet(b, a), meet(c, a)), complement(a)), zero, join(meet(b, a), meet(c, a)), join(complement(a), complement(join(meet(b, a), meet(c, a))))))
% 187.03/24.28 = { by axiom 1 (commutativity_of_join) R->L }
% 187.03/24.28 complement(fresh(meet(join(meet(b, a), meet(c, a)), complement(a)), zero, join(meet(b, a), meet(c, a)), join(complement(join(meet(b, a), meet(c, a))), complement(a))))
% 187.03/24.28 = { by lemma 50 R->L }
% 187.03/24.28 complement(fresh(meet(join(meet(b, a), meet(c, a)), complement(a)), zero, join(meet(b, a), meet(c, a)), join(complement(join(meet(b, a), meet(c, a))), join(complement(a), meet(X, meet(Y, join(complement(join(meet(b, a), meet(c, a))), meet(complement(a), Z))))))))
% 187.03/24.28 = { by lemma 43 R->L }
% 187.03/24.28 complement(fresh(meet(join(meet(b, a), meet(c, a)), join(complement(join(meet(b, a), meet(c, a))), complement(a))), zero, join(meet(b, a), meet(c, a)), join(complement(join(meet(b, a), meet(c, a))), join(complement(a), meet(X, meet(Y, join(complement(join(meet(b, a), meet(c, a))), meet(complement(a), Z))))))))
% 187.03/24.28 = { by lemma 50 R->L }
% 187.03/24.28 complement(fresh(meet(join(meet(b, a), meet(c, a)), join(complement(join(meet(b, a), meet(c, a))), join(complement(a), meet(X, meet(Y, join(complement(join(meet(b, a), meet(c, a))), meet(complement(a), Z))))))), zero, join(meet(b, a), meet(c, a)), join(complement(join(meet(b, a), meet(c, a))), join(complement(a), meet(X, meet(Y, join(complement(join(meet(b, a), meet(c, a))), meet(complement(a), Z))))))))
% 187.03/24.28 = { by axiom 1 (commutativity_of_join) R->L }
% 187.03/24.28 complement(fresh(meet(join(meet(b, a), meet(c, a)), join(complement(join(meet(b, a), meet(c, a))), join(complement(a), meet(X, meet(Y, join(complement(join(meet(b, a), meet(c, a))), meet(complement(a), Z))))))), zero, join(meet(b, a), meet(c, a)), join(join(complement(a), meet(X, meet(Y, join(complement(join(meet(b, a), meet(c, a))), meet(complement(a), Z))))), complement(join(meet(b, a), meet(c, a))))))
% 187.03/24.28 = { by lemma 35 }
% 187.03/24.28 complement(complement(join(meet(b, a), meet(c, a))))
% 187.03/24.28 = { by axiom 1 (commutativity_of_join) }
% 187.03/24.28 complement(complement(join(meet(c, a), meet(b, a))))
% 187.03/24.28 = { by axiom 2 (commutativity_of_meet) }
% 187.03/24.28 complement(complement(join(meet(c, a), meet(a, b))))
% 187.03/24.28 = { by lemma 14 }
% 187.03/24.28 join(meet(c, a), meet(a, b))
% 187.03/24.28 = { by axiom 1 (commutativity_of_join) }
% 187.03/24.28 join(meet(a, b), meet(c, a))
% 187.03/24.28 = { by axiom 2 (commutativity_of_meet) }
% 187.03/24.28 join(meet(b, a), meet(c, a))
% 187.03/24.28 = { by axiom 2 (commutativity_of_meet) R->L }
% 187.03/24.28 join(meet(b, a), meet(a, c))
% 187.03/24.28 = { by axiom 2 (commutativity_of_meet) R->L }
% 187.03/24.28 join(meet(a, b), meet(a, c))
% 187.03/24.28 % SZS output end Proof
% 187.03/24.28
% 187.03/24.28 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------