TSTP Solution File: LAT212-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT212-1 : 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 06:27:47 EDT 2023
% Result : Unsatisfiable 161.71s 21.24s
% Output : Proof 162.40s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LAT212-1 : TPTP v8.1.2. Released v3.1.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n029.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Aug 24 07:03:53 EDT 2023
% 0.13/0.34 % CPUTime :
% 161.71/21.24 Command-line arguments: --no-flatten-goal
% 161.71/21.24
% 161.71/21.24 % SZS status Unsatisfiable
% 161.71/21.24
% 161.71/21.29 % SZS output start Proof
% 161.71/21.29 Take the following subset of the input axioms:
% 161.71/21.29 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 161.71/21.29 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 161.71/21.29 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 161.71/21.29 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 161.71/21.29 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 161.71/21.29 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 161.71/21.29 fof(complement_join, axiom, ![X2]: join(X2, complement(X2))=one).
% 161.71/21.29 fof(complement_meet, axiom, ![X2]: meet(X2, complement(X2))=zero).
% 161.71/21.29 fof(equation_H70, axiom, ![U, X2, Y2, Z2]: meet(X2, join(Y2, meet(Z2, join(Y2, U))))=meet(X2, join(Y2, meet(Z2, join(U, meet(Y2, join(X2, Z2))))))).
% 161.71/21.29 fof(idempotence_of_join, axiom, ![X2]: join(X2, X2)=X2).
% 161.71/21.29 fof(idempotence_of_meet, axiom, ![X2]: meet(X2, X2)=X2).
% 161.71/21.29 fof(meet_join_complement, axiom, ![X2, Y2]: (meet(X2, Y2)!=zero | (join(X2, Y2)!=one | complement(X2)=Y2))).
% 161.71/21.29 fof(prove_distributivity, negated_conjecture, meet(a, join(b, c))!=join(meet(a, b), meet(a, c))).
% 161.71/21.29
% 161.71/21.29 Now clausify the problem and encode Horn clauses using encoding 3 of
% 161.71/21.29 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 161.71/21.29 We repeatedly replace C & s=t => u=v by the two clauses:
% 161.71/21.29 fresh(y, y, x1...xn) = u
% 161.71/21.29 C => fresh(s, t, x1...xn) = v
% 161.71/21.29 where fresh is a fresh function symbol and x1..xn are the free
% 161.71/21.29 variables of u and v.
% 161.71/21.29 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 161.71/21.29 input problem has no model of domain size 1).
% 161.71/21.29
% 161.71/21.29 The encoding turns the above axioms into the following unit equations and goals:
% 161.71/21.29
% 161.71/21.29 Axiom 1 (idempotence_of_join): join(X, X) = X.
% 161.71/21.29 Axiom 2 (commutativity_of_join): join(X, Y) = join(Y, X).
% 161.71/21.29 Axiom 3 (idempotence_of_meet): meet(X, X) = X.
% 161.71/21.29 Axiom 4 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 161.71/21.29 Axiom 5 (complement_join): join(X, complement(X)) = one.
% 161.71/21.29 Axiom 6 (complement_meet): meet(X, complement(X)) = zero.
% 161.71/21.29 Axiom 7 (meet_join_complement): fresh(X, X, Y, Z) = Z.
% 161.71/21.29 Axiom 8 (meet_join_complement): fresh2(X, X, Y, Z) = complement(Y).
% 161.71/21.29 Axiom 9 (absorption2): join(X, meet(X, Y)) = X.
% 161.71/21.29 Axiom 10 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 161.71/21.29 Axiom 11 (absorption1): meet(X, join(X, Y)) = X.
% 161.71/21.29 Axiom 12 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 161.71/21.29 Axiom 13 (meet_join_complement): fresh2(join(X, Y), one, X, Y) = fresh(meet(X, Y), zero, X, Y).
% 161.71/21.29 Axiom 14 (equation_H70): meet(X, join(Y, meet(Z, join(Y, W)))) = meet(X, join(Y, meet(Z, join(W, meet(Y, join(X, Z)))))).
% 161.71/21.29
% 161.71/21.29 Lemma 15: complement(complement(X)) = X.
% 161.71/21.29 Proof:
% 161.71/21.29 complement(complement(X))
% 161.71/21.29 = { by axiom 8 (meet_join_complement) R->L }
% 161.71/21.29 fresh2(one, one, complement(X), X)
% 161.71/21.29 = { by axiom 5 (complement_join) R->L }
% 161.71/21.29 fresh2(join(X, complement(X)), one, complement(X), X)
% 161.71/21.29 = { by axiom 2 (commutativity_of_join) R->L }
% 161.71/21.29 fresh2(join(complement(X), X), one, complement(X), X)
% 161.71/21.29 = { by axiom 13 (meet_join_complement) }
% 161.71/21.29 fresh(meet(complement(X), X), zero, complement(X), X)
% 161.71/21.29 = { by axiom 4 (commutativity_of_meet) }
% 161.71/21.29 fresh(meet(X, complement(X)), zero, complement(X), X)
% 161.71/21.29 = { by axiom 6 (complement_meet) }
% 161.71/21.29 fresh(zero, zero, complement(X), X)
% 161.71/21.29 = { by axiom 7 (meet_join_complement) }
% 161.71/21.29 X
% 161.71/21.29
% 161.71/21.29 Lemma 16: meet(X, one) = X.
% 161.71/21.29 Proof:
% 161.71/21.29 meet(X, one)
% 161.71/21.29 = { by axiom 5 (complement_join) R->L }
% 161.71/21.29 meet(X, join(X, complement(X)))
% 161.71/21.29 = { by axiom 11 (absorption1) }
% 161.71/21.29 X
% 161.71/21.29
% 161.71/21.29 Lemma 17: meet(one, X) = X.
% 161.71/21.29 Proof:
% 161.71/21.29 meet(one, X)
% 161.71/21.29 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.29 meet(X, one)
% 161.71/21.29 = { by lemma 16 }
% 161.71/21.29 X
% 161.71/21.29
% 161.71/21.29 Lemma 18: join(X, one) = one.
% 161.71/21.29 Proof:
% 161.71/21.29 join(X, one)
% 161.71/21.29 = { by axiom 2 (commutativity_of_join) R->L }
% 161.71/21.29 join(one, X)
% 161.71/21.29 = { by lemma 17 R->L }
% 161.71/21.29 join(one, meet(one, X))
% 161.71/21.29 = { by axiom 9 (absorption2) }
% 161.71/21.29 one
% 161.71/21.29
% 161.71/21.29 Lemma 19: join(X, zero) = X.
% 161.71/21.29 Proof:
% 161.71/21.29 join(X, zero)
% 161.71/21.29 = { by axiom 6 (complement_meet) R->L }
% 161.71/21.29 join(X, meet(X, complement(X)))
% 161.71/21.29 = { by axiom 9 (absorption2) }
% 161.71/21.29 X
% 161.71/21.29
% 161.71/21.29 Lemma 20: meet(X, join(Y, X)) = X.
% 161.71/21.29 Proof:
% 161.71/21.29 meet(X, join(Y, X))
% 161.71/21.29 = { by axiom 2 (commutativity_of_join) R->L }
% 161.71/21.29 meet(X, join(X, Y))
% 161.71/21.29 = { by axiom 11 (absorption1) }
% 161.71/21.29 X
% 161.71/21.29
% 161.71/21.29 Lemma 21: meet(X, join(complement(X), meet(X, meet(Y, join(Z, complement(X)))))) = meet(X, join(complement(X), meet(X, meet(Y, Z)))).
% 161.71/21.29 Proof:
% 161.71/21.29 meet(X, join(complement(X), meet(X, meet(Y, join(Z, complement(X))))))
% 161.71/21.29 = { by axiom 2 (commutativity_of_join) R->L }
% 161.71/21.29 meet(X, join(complement(X), meet(X, meet(Y, join(complement(X), Z)))))
% 161.71/21.29 = { by axiom 12 (associativity_of_meet) R->L }
% 161.71/21.29 meet(X, join(complement(X), meet(meet(X, Y), join(complement(X), Z))))
% 161.71/21.29 = { by axiom 14 (equation_H70) }
% 161.71/21.29 meet(X, join(complement(X), meet(meet(X, Y), join(Z, meet(complement(X), join(X, meet(X, Y)))))))
% 161.71/21.29 = { by axiom 9 (absorption2) }
% 161.71/21.29 meet(X, join(complement(X), meet(meet(X, Y), join(Z, meet(complement(X), X)))))
% 161.71/21.29 = { by axiom 12 (associativity_of_meet) }
% 161.71/21.29 meet(X, join(complement(X), meet(X, meet(Y, join(Z, meet(complement(X), X))))))
% 161.71/21.29 = { by axiom 4 (commutativity_of_meet) }
% 161.71/21.29 meet(X, join(complement(X), meet(X, meet(Y, join(Z, meet(X, complement(X)))))))
% 161.71/21.29 = { by axiom 6 (complement_meet) }
% 161.71/21.29 meet(X, join(complement(X), meet(X, meet(Y, join(Z, zero)))))
% 161.71/21.29 = { by lemma 19 }
% 161.71/21.29 meet(X, join(complement(X), meet(X, meet(Y, Z))))
% 161.71/21.29
% 161.71/21.29 Lemma 22: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 161.71/21.29 Proof:
% 161.71/21.29 join(X, join(Y, meet(X, Z)))
% 161.71/21.29 = { by axiom 2 (commutativity_of_join) R->L }
% 161.71/21.29 join(X, join(meet(X, Z), Y))
% 161.71/21.29 = { by axiom 10 (associativity_of_join) R->L }
% 161.71/21.29 join(join(X, meet(X, Z)), Y)
% 161.71/21.29 = { by axiom 9 (absorption2) }
% 161.71/21.29 join(X, Y)
% 161.71/21.29
% 161.71/21.29 Lemma 23: join(X, meet(Y, X)) = X.
% 161.71/21.29 Proof:
% 161.71/21.29 join(X, meet(Y, X))
% 161.71/21.29 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.29 join(X, meet(X, Y))
% 161.71/21.29 = { by axiom 9 (absorption2) }
% 161.71/21.29 X
% 161.71/21.29
% 161.71/21.29 Lemma 24: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 161.71/21.29 Proof:
% 161.71/21.29 join(X, join(Y, meet(Z, join(X, Y))))
% 161.71/21.29 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.29 join(X, join(Y, meet(join(X, Y), Z)))
% 161.71/21.29 = { by axiom 10 (associativity_of_join) R->L }
% 161.71/21.29 join(join(X, Y), meet(join(X, Y), Z))
% 161.71/21.29 = { by axiom 9 (absorption2) }
% 161.71/21.29 join(X, Y)
% 161.71/21.29
% 161.71/21.29 Lemma 25: join(X, meet(Y, join(X, meet(Y, Z)))) = join(X, meet(Y, Z)).
% 161.71/21.29 Proof:
% 161.71/21.29 join(X, meet(Y, join(X, meet(Y, Z))))
% 161.71/21.29 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.29 join(X, meet(Y, join(X, meet(Z, Y))))
% 161.71/21.29 = { by lemma 23 R->L }
% 161.71/21.29 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, meet(Y, join(X, meet(Z, Y))))))
% 161.71/21.29 = { by axiom 2 (commutativity_of_join) R->L }
% 161.71/21.29 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, meet(Y, join(meet(Z, Y), X)))))
% 161.71/21.29 = { by axiom 12 (associativity_of_meet) R->L }
% 161.71/21.29 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(meet(Z, Y), join(meet(Z, Y), X))))
% 161.71/21.29 = { by axiom 11 (absorption1) }
% 161.71/21.29 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, Y)))
% 161.71/21.29 = { by axiom 2 (commutativity_of_join) }
% 161.71/21.29 join(X, join(meet(Z, Y), meet(Y, join(X, meet(Z, Y)))))
% 161.71/21.29 = { by axiom 4 (commutativity_of_meet) }
% 161.71/21.29 join(X, join(meet(Z, Y), meet(Y, join(X, meet(Y, Z)))))
% 161.71/21.29 = { by axiom 4 (commutativity_of_meet) }
% 161.71/21.29 join(X, join(meet(Y, Z), meet(Y, join(X, meet(Y, Z)))))
% 161.71/21.29 = { by lemma 24 }
% 161.71/21.29 join(X, meet(Y, Z))
% 161.71/21.29
% 161.71/21.29 Lemma 26: join(X, meet(complement(X), join(X, Y))) = join(X, meet(Y, complement(X))).
% 161.71/21.29 Proof:
% 161.71/21.29 join(X, meet(complement(X), join(X, Y)))
% 161.71/21.29 = { by lemma 25 R->L }
% 161.71/21.29 join(X, meet(complement(X), join(X, meet(complement(X), join(X, Y)))))
% 161.71/21.29 = { by axiom 14 (equation_H70) }
% 161.71/21.29 join(X, meet(complement(X), join(X, meet(complement(X), join(Y, meet(X, join(complement(X), complement(X))))))))
% 161.71/21.29 = { by lemma 25 }
% 161.71/21.29 join(X, meet(complement(X), join(Y, meet(X, join(complement(X), complement(X))))))
% 161.71/21.29 = { by axiom 1 (idempotence_of_join) }
% 161.71/21.29 join(X, meet(complement(X), join(Y, meet(X, complement(X)))))
% 161.71/21.29 = { by axiom 6 (complement_meet) }
% 161.71/21.29 join(X, meet(complement(X), join(Y, zero)))
% 161.71/21.29 = { by lemma 19 }
% 161.71/21.29 join(X, meet(complement(X), Y))
% 161.71/21.29 = { by axiom 4 (commutativity_of_meet) }
% 161.71/21.29 join(X, meet(Y, complement(X)))
% 161.71/21.29
% 161.71/21.29 Lemma 27: meet(complement(X), join(X, meet(Y, complement(X)))) = meet(complement(X), join(X, Y)).
% 161.71/21.29 Proof:
% 161.71/21.29 meet(complement(X), join(X, meet(Y, complement(X))))
% 161.71/21.29 = { by lemma 20 R->L }
% 161.71/21.29 meet(complement(X), meet(join(X, meet(Y, complement(X))), join(Y, join(X, meet(Y, complement(X))))))
% 161.71/21.30 = { by lemma 22 }
% 161.71/21.30 meet(complement(X), meet(join(X, meet(Y, complement(X))), join(Y, X)))
% 161.71/21.30 = { by axiom 4 (commutativity_of_meet) }
% 161.71/21.30 meet(complement(X), meet(join(Y, X), join(X, meet(Y, complement(X)))))
% 161.71/21.30 = { by axiom 2 (commutativity_of_join) }
% 161.71/21.30 meet(complement(X), meet(join(X, Y), join(X, meet(Y, complement(X)))))
% 161.71/21.30 = { by axiom 12 (associativity_of_meet) R->L }
% 161.71/21.30 meet(meet(complement(X), join(X, Y)), join(X, meet(Y, complement(X))))
% 161.71/21.30 = { by lemma 26 R->L }
% 161.71/21.30 meet(meet(complement(X), join(X, Y)), join(X, meet(complement(X), join(X, Y))))
% 161.71/21.30 = { by lemma 20 }
% 161.71/21.30 meet(complement(X), join(X, Y))
% 161.71/21.30
% 161.71/21.30 Lemma 28: meet(complement(X), join(X, meet(complement(X), Y))) = meet(complement(X), join(X, Y)).
% 161.71/21.30 Proof:
% 161.71/21.30 meet(complement(X), join(X, meet(complement(X), Y)))
% 161.71/21.30 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.30 meet(complement(X), join(X, meet(Y, complement(X))))
% 161.71/21.30 = { by lemma 27 }
% 161.71/21.30 meet(complement(X), join(X, Y))
% 161.71/21.30
% 161.71/21.30 Lemma 29: join(X, complement(meet(X, Y))) = one.
% 161.71/21.30 Proof:
% 161.71/21.30 join(X, complement(meet(X, Y)))
% 161.71/21.30 = { by lemma 22 R->L }
% 161.71/21.30 join(X, join(complement(meet(X, Y)), meet(X, Y)))
% 161.71/21.30 = { by axiom 2 (commutativity_of_join) }
% 161.71/21.30 join(X, join(meet(X, Y), complement(meet(X, Y))))
% 161.71/21.30 = { by axiom 5 (complement_join) }
% 161.71/21.30 join(X, one)
% 161.71/21.30 = { by lemma 18 }
% 161.71/21.30 one
% 161.71/21.30
% 161.71/21.30 Lemma 30: meet(complement(X), complement(meet(X, Y))) = complement(X).
% 161.71/21.30 Proof:
% 161.71/21.30 meet(complement(X), complement(meet(X, Y)))
% 161.71/21.30 = { by axiom 7 (meet_join_complement) R->L }
% 161.71/21.30 fresh(zero, zero, X, meet(complement(X), complement(meet(X, Y))))
% 161.71/21.30 = { by axiom 9 (absorption2) R->L }
% 161.71/21.30 fresh(join(zero, meet(zero, complement(meet(X, Y)))), zero, X, meet(complement(X), complement(meet(X, Y))))
% 161.71/21.30 = { by axiom 2 (commutativity_of_join) R->L }
% 161.71/21.30 fresh(join(meet(zero, complement(meet(X, Y))), zero), zero, X, meet(complement(X), complement(meet(X, Y))))
% 161.71/21.30 = { by lemma 19 }
% 161.71/21.30 fresh(meet(zero, complement(meet(X, Y))), zero, X, meet(complement(X), complement(meet(X, Y))))
% 161.71/21.30 = { by axiom 6 (complement_meet) R->L }
% 161.71/21.30 fresh(meet(meet(X, complement(X)), complement(meet(X, Y))), zero, X, meet(complement(X), complement(meet(X, Y))))
% 161.71/21.30 = { by axiom 12 (associativity_of_meet) }
% 161.71/21.30 fresh(meet(X, meet(complement(X), complement(meet(X, Y)))), zero, X, meet(complement(X), complement(meet(X, Y))))
% 161.71/21.30 = { by axiom 13 (meet_join_complement) R->L }
% 161.71/21.30 fresh2(join(X, meet(complement(X), complement(meet(X, Y)))), one, X, meet(complement(X), complement(meet(X, Y))))
% 161.71/21.30 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.30 fresh2(join(X, meet(complement(meet(X, Y)), complement(X))), one, X, meet(complement(X), complement(meet(X, Y))))
% 161.71/21.30 = { by lemma 26 R->L }
% 161.71/21.30 fresh2(join(X, meet(complement(X), join(X, complement(meet(X, Y))))), one, X, meet(complement(X), complement(meet(X, Y))))
% 161.71/21.30 = { by lemma 29 }
% 161.71/21.30 fresh2(join(X, meet(complement(X), one)), one, X, meet(complement(X), complement(meet(X, Y))))
% 161.71/21.30 = { by lemma 16 }
% 161.71/21.30 fresh2(join(X, complement(X)), one, X, meet(complement(X), complement(meet(X, Y))))
% 161.71/21.30 = { by axiom 5 (complement_join) }
% 161.71/21.30 fresh2(one, one, X, meet(complement(X), complement(meet(X, Y))))
% 161.71/21.30 = { by axiom 8 (meet_join_complement) }
% 161.71/21.30 complement(X)
% 161.71/21.30
% 161.71/21.30 Lemma 31: meet(complement(X), complement(meet(Y, X))) = complement(X).
% 161.71/21.30 Proof:
% 161.71/21.30 meet(complement(X), complement(meet(Y, X)))
% 161.71/21.30 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.30 meet(complement(X), complement(meet(X, Y)))
% 161.71/21.30 = { by lemma 30 }
% 161.71/21.30 complement(X)
% 161.71/21.30
% 161.71/21.30 Lemma 32: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 161.71/21.30 Proof:
% 161.71/21.30 meet(X, complement(meet(X, Y)))
% 161.71/21.30 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.30 meet(X, complement(meet(Y, X)))
% 161.71/21.30 = { by lemma 20 R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), join(complement(Y), meet(X, complement(meet(Y, X)))))
% 161.71/21.30 = { by axiom 2 (commutativity_of_join) R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), join(meet(X, complement(meet(Y, X))), complement(Y)))
% 161.71/21.30 = { by axiom 7 (meet_join_complement) R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(zero, zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by axiom 6 (complement_meet) R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(Y, complement(Y)), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by lemma 19 R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(Y, join(complement(Y), zero)), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by axiom 6 (complement_meet) R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(Y, join(complement(Y), meet(meet(Y, X), complement(meet(Y, X))))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by axiom 12 (associativity_of_meet) }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(Y, join(complement(Y), meet(Y, meet(X, complement(meet(Y, X)))))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by lemma 21 R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(Y, join(complement(Y), meet(Y, meet(X, join(complement(meet(Y, X)), complement(Y)))))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(Y, join(complement(Y), meet(meet(X, join(complement(meet(Y, X)), complement(Y))), Y))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by lemma 15 R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(Y, join(complement(Y), meet(meet(X, join(complement(meet(Y, X)), complement(Y))), complement(complement(Y))))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by lemma 15 R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(complement(complement(Y)), join(complement(Y), meet(meet(X, join(complement(meet(Y, X)), complement(Y))), complement(complement(Y))))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by lemma 27 }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(complement(complement(Y)), join(complement(Y), meet(X, join(complement(meet(Y, X)), complement(Y))))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by lemma 15 R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(complement(complement(Y)), join(complement(Y), meet(X, join(complement(meet(Y, X)), complement(complement(complement(Y))))))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by lemma 28 R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(complement(complement(Y)), join(complement(Y), meet(complement(complement(Y)), meet(X, join(complement(meet(Y, X)), complement(complement(complement(Y)))))))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by lemma 15 R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(complement(complement(Y)), join(complement(complement(complement(Y))), meet(complement(complement(Y)), meet(X, join(complement(meet(Y, X)), complement(complement(complement(Y)))))))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by lemma 21 }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(complement(complement(Y)), join(complement(complement(complement(Y))), meet(complement(complement(Y)), meet(X, complement(meet(Y, X)))))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by lemma 15 }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(complement(complement(Y)), join(complement(Y), meet(complement(complement(Y)), meet(X, complement(meet(Y, X)))))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by lemma 28 }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(complement(complement(Y)), join(complement(Y), meet(X, complement(meet(Y, X))))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by lemma 15 }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(Y, join(complement(Y), meet(X, complement(meet(Y, X))))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by axiom 2 (commutativity_of_join) R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh(meet(Y, join(meet(X, complement(meet(Y, X))), complement(Y))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by axiom 13 (meet_join_complement) R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh2(join(Y, join(meet(X, complement(meet(Y, X))), complement(Y))), one, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by axiom 2 (commutativity_of_join) R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh2(join(Y, join(complement(Y), meet(X, complement(meet(Y, X))))), one, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by axiom 10 (associativity_of_join) R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh2(join(join(Y, complement(Y)), meet(X, complement(meet(Y, X)))), one, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by axiom 5 (complement_join) }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh2(join(one, meet(X, complement(meet(Y, X)))), one, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by axiom 2 (commutativity_of_join) R->L }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh2(join(meet(X, complement(meet(Y, X))), one), one, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by lemma 18 }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), fresh2(one, one, Y, join(meet(X, complement(meet(Y, X))), complement(Y))))
% 161.71/21.30 = { by axiom 8 (meet_join_complement) }
% 161.71/21.30 meet(meet(X, complement(meet(Y, X))), complement(Y))
% 161.71/21.30 = { by axiom 12 (associativity_of_meet) }
% 161.71/21.30 meet(X, meet(complement(meet(Y, X)), complement(Y)))
% 161.71/21.30 = { by axiom 4 (commutativity_of_meet) }
% 161.71/21.30 meet(X, meet(complement(Y), complement(meet(Y, X))))
% 161.71/21.30 = { by axiom 4 (commutativity_of_meet) }
% 161.71/21.30 meet(X, meet(complement(Y), complement(meet(X, Y))))
% 161.71/21.30 = { by lemma 31 }
% 161.71/21.30 meet(X, complement(Y))
% 161.71/21.30
% 161.71/21.30 Lemma 33: meet(X, complement(meet(Y, X))) = meet(X, complement(Y)).
% 161.71/21.30 Proof:
% 161.71/21.30 meet(X, complement(meet(Y, X)))
% 161.71/21.30 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.30 meet(X, complement(meet(X, Y)))
% 161.71/21.30 = { by lemma 32 }
% 161.71/21.30 meet(X, complement(Y))
% 161.71/21.30
% 161.71/21.30 Lemma 34: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 161.71/21.30 Proof:
% 161.71/21.30 meet(X, meet(Y, join(X, Z)))
% 161.71/21.30 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.30 meet(X, meet(join(X, Z), Y))
% 161.71/21.30 = { by axiom 12 (associativity_of_meet) R->L }
% 161.71/21.30 meet(meet(X, join(X, Z)), Y)
% 161.71/21.30 = { by axiom 11 (absorption1) }
% 161.71/21.30 meet(X, Y)
% 161.71/21.30
% 161.71/21.30 Lemma 35: join(meet(X, Y), meet(X, join(Y, Z))) = meet(X, join(Y, Z)).
% 161.71/21.30 Proof:
% 161.71/21.30 join(meet(X, Y), meet(X, join(Y, Z)))
% 161.71/21.30 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.30 join(meet(Y, X), meet(X, join(Y, Z)))
% 161.71/21.30 = { by axiom 2 (commutativity_of_join) R->L }
% 161.71/21.30 join(meet(X, join(Y, Z)), meet(Y, X))
% 161.71/21.30 = { by lemma 34 R->L }
% 161.71/21.30 join(meet(X, join(Y, Z)), meet(Y, meet(X, join(Y, Z))))
% 161.71/21.30 = { by lemma 23 }
% 161.71/21.30 meet(X, join(Y, Z))
% 161.71/21.30
% 161.71/21.30 Lemma 36: join(X, join(X, complement(Y))) = join(X, complement(Y)).
% 161.71/21.30 Proof:
% 161.71/21.30 join(X, join(X, complement(Y)))
% 161.71/21.30 = { by axiom 3 (idempotence_of_meet) R->L }
% 161.71/21.30 join(X, join(X, complement(meet(Y, Y))))
% 161.71/21.30 = { by lemma 17 R->L }
% 161.71/21.30 join(X, meet(one, join(X, complement(meet(Y, Y)))))
% 161.71/21.30 = { by lemma 29 R->L }
% 161.71/21.30 join(X, meet(join(meet(Y, join(Y, Z)), complement(meet(meet(Y, join(Y, Z)), Y))), join(X, complement(meet(Y, Y)))))
% 161.71/21.30 = { by axiom 4 (commutativity_of_meet) }
% 161.71/21.30 join(X, meet(join(meet(Y, join(Y, Z)), complement(meet(Y, meet(Y, join(Y, Z))))), join(X, complement(meet(Y, Y)))))
% 161.71/21.30 = { by lemma 34 }
% 161.71/21.30 join(X, meet(join(meet(Y, join(Y, Z)), complement(meet(Y, Y))), join(X, complement(meet(Y, Y)))))
% 161.71/21.30 = { by axiom 2 (commutativity_of_join) }
% 161.71/21.30 join(X, meet(join(complement(meet(Y, Y)), meet(Y, join(Y, Z))), join(X, complement(meet(Y, Y)))))
% 161.71/21.30 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.30 join(X, meet(join(X, complement(meet(Y, Y))), join(complement(meet(Y, Y)), meet(Y, join(Y, Z)))))
% 161.71/21.30 = { by axiom 2 (commutativity_of_join) R->L }
% 161.71/21.30 join(X, meet(join(X, complement(meet(Y, Y))), join(meet(Y, join(Y, Z)), complement(meet(Y, Y)))))
% 161.71/21.30 = { by axiom 2 (commutativity_of_join) R->L }
% 161.71/21.30 join(X, meet(join(complement(meet(Y, Y)), X), join(meet(Y, join(Y, Z)), complement(meet(Y, Y)))))
% 161.71/21.30 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.30 join(X, meet(join(meet(Y, join(Y, Z)), complement(meet(Y, Y))), join(complement(meet(Y, Y)), X)))
% 161.71/21.30 = { by lemma 35 R->L }
% 161.71/21.30 join(X, join(meet(join(meet(Y, join(Y, Z)), complement(meet(Y, Y))), complement(meet(Y, Y))), meet(join(meet(Y, join(Y, Z)), complement(meet(Y, Y))), join(complement(meet(Y, Y)), X))))
% 161.71/21.30 = { by axiom 4 (commutativity_of_meet) }
% 161.71/21.30 join(X, join(meet(complement(meet(Y, Y)), join(meet(Y, join(Y, Z)), complement(meet(Y, Y)))), meet(join(meet(Y, join(Y, Z)), complement(meet(Y, Y))), join(complement(meet(Y, Y)), X))))
% 161.71/21.30 = { by lemma 20 }
% 161.71/21.30 join(X, join(complement(meet(Y, Y)), meet(join(meet(Y, join(Y, Z)), complement(meet(Y, Y))), join(complement(meet(Y, Y)), X))))
% 161.71/21.30 = { by axiom 4 (commutativity_of_meet) }
% 161.71/21.30 join(X, join(complement(meet(Y, Y)), meet(join(complement(meet(Y, Y)), X), join(meet(Y, join(Y, Z)), complement(meet(Y, Y))))))
% 161.71/21.30 = { by axiom 2 (commutativity_of_join) }
% 161.71/21.30 join(X, join(complement(meet(Y, Y)), meet(join(X, complement(meet(Y, Y))), join(meet(Y, join(Y, Z)), complement(meet(Y, Y))))))
% 161.71/21.31 = { by axiom 2 (commutativity_of_join) }
% 161.71/21.31 join(X, join(complement(meet(Y, Y)), meet(join(X, complement(meet(Y, Y))), join(complement(meet(Y, Y)), meet(Y, join(Y, Z))))))
% 161.71/21.31 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.31 join(X, join(complement(meet(Y, Y)), meet(join(complement(meet(Y, Y)), meet(Y, join(Y, Z))), join(X, complement(meet(Y, Y))))))
% 161.71/21.31 = { by lemma 24 }
% 161.71/21.31 join(X, complement(meet(Y, Y)))
% 161.71/21.31 = { by axiom 3 (idempotence_of_meet) }
% 161.71/21.31 join(X, complement(Y))
% 161.71/21.31
% 161.71/21.31 Lemma 37: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 161.71/21.31 Proof:
% 161.71/21.31 join(Y, join(X, Z))
% 161.71/21.31 = { by axiom 2 (commutativity_of_join) R->L }
% 161.71/21.31 join(join(X, Z), Y)
% 161.71/21.31 = { by axiom 10 (associativity_of_join) }
% 161.71/21.31 join(X, join(Z, Y))
% 161.71/21.31 = { by axiom 2 (commutativity_of_join) }
% 161.71/21.31 join(X, join(Y, Z))
% 161.71/21.31
% 161.71/21.31 Lemma 38: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 161.71/21.31 Proof:
% 161.71/21.31 complement(meet(X, complement(Y)))
% 161.71/21.31 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.31 complement(meet(complement(Y), X))
% 161.71/21.31 = { by lemma 15 R->L }
% 161.71/21.31 complement(meet(complement(Y), complement(complement(X))))
% 161.71/21.31 = { by lemma 32 R->L }
% 161.71/21.31 complement(meet(complement(Y), complement(meet(complement(Y), complement(X)))))
% 161.71/21.31 = { by lemma 33 R->L }
% 161.71/21.31 complement(meet(complement(Y), complement(meet(complement(Y), complement(meet(X, complement(Y)))))))
% 161.71/21.31 = { by lemma 23 R->L }
% 161.71/21.31 complement(meet(complement(Y), complement(meet(complement(Y), join(complement(meet(X, complement(Y))), meet(Y, complement(meet(X, complement(Y)))))))))
% 161.71/21.31 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.31 complement(meet(complement(Y), complement(meet(complement(Y), join(complement(meet(X, complement(Y))), meet(Y, complement(meet(complement(Y), X))))))))
% 161.71/21.31 = { by lemma 15 R->L }
% 161.71/21.31 complement(meet(complement(Y), complement(meet(complement(Y), join(complement(meet(X, complement(Y))), meet(complement(complement(Y)), complement(meet(complement(Y), X))))))))
% 161.71/21.31 = { by lemma 30 }
% 161.71/21.31 complement(meet(complement(Y), complement(meet(complement(Y), join(complement(meet(X, complement(Y))), complement(complement(Y)))))))
% 161.71/21.31 = { by lemma 15 }
% 161.71/21.31 complement(meet(complement(Y), complement(meet(complement(Y), join(complement(meet(X, complement(Y))), Y)))))
% 161.71/21.31 = { by axiom 2 (commutativity_of_join) }
% 161.71/21.31 complement(meet(complement(Y), complement(meet(complement(Y), join(Y, complement(meet(X, complement(Y))))))))
% 161.71/21.31 = { by lemma 28 R->L }
% 161.71/21.31 complement(meet(complement(Y), complement(meet(complement(Y), join(Y, meet(complement(Y), complement(meet(X, complement(Y)))))))))
% 161.71/21.31 = { by lemma 33 }
% 161.71/21.31 complement(meet(complement(Y), complement(meet(complement(Y), join(Y, meet(complement(Y), complement(X)))))))
% 161.71/21.31 = { by lemma 28 }
% 161.71/21.31 complement(meet(complement(Y), complement(meet(complement(Y), join(Y, complement(X))))))
% 161.71/21.31 = { by lemma 32 }
% 161.71/21.31 complement(meet(complement(Y), complement(join(Y, complement(X)))))
% 161.71/21.31 = { by lemma 36 R->L }
% 161.71/21.31 complement(meet(complement(Y), complement(join(Y, join(Y, complement(X))))))
% 161.71/21.31 = { by axiom 2 (commutativity_of_join) R->L }
% 161.71/21.31 complement(meet(complement(Y), complement(join(Y, join(complement(X), Y)))))
% 161.71/21.31 = { by lemma 37 }
% 161.71/21.31 complement(meet(complement(Y), complement(join(complement(X), join(Y, Y)))))
% 161.71/21.31 = { by axiom 4 (commutativity_of_meet) R->L }
% 161.71/21.31 complement(meet(complement(join(complement(X), join(Y, Y))), complement(Y)))
% 161.71/21.31 = { by lemma 20 R->L }
% 161.71/21.31 complement(meet(complement(join(complement(X), join(Y, Y))), complement(meet(Y, join(join(complement(X), Y), Y)))))
% 161.71/21.31 = { by axiom 10 (associativity_of_join) }
% 161.71/21.31 complement(meet(complement(join(complement(X), join(Y, Y))), complement(meet(Y, join(complement(X), join(Y, Y))))))
% 161.71/21.31 = { by lemma 31 }
% 161.71/21.31 complement(complement(join(complement(X), join(Y, Y))))
% 161.71/21.31 = { by lemma 37 R->L }
% 161.71/21.31 complement(complement(join(Y, join(complement(X), Y))))
% 161.71/21.31 = { by axiom 2 (commutativity_of_join) }
% 161.71/21.31 complement(complement(join(Y, join(Y, complement(X)))))
% 161.71/21.31 = { by lemma 36 }
% 161.71/21.31 complement(complement(join(Y, complement(X))))
% 161.71/21.31 = { by lemma 15 }
% 162.40/21.31 join(Y, complement(X))
% 162.40/21.31
% 162.40/21.31 Lemma 39: meet(X, join(Y, meet(X, Z))) = meet(X, join(Z, Y)).
% 162.40/21.31 Proof:
% 162.40/21.31 meet(X, join(Y, meet(X, Z)))
% 162.40/21.31 = { by lemma 15 R->L }
% 162.40/21.31 meet(X, join(complement(complement(Y)), meet(X, Z)))
% 162.40/21.31 = { by axiom 2 (commutativity_of_join) R->L }
% 162.40/21.31 meet(X, join(meet(X, Z), complement(complement(Y))))
% 162.40/21.31 = { by lemma 38 R->L }
% 162.40/21.31 meet(X, complement(meet(complement(Y), complement(meet(X, Z)))))
% 162.40/21.31 = { by lemma 32 R->L }
% 162.40/21.31 meet(X, complement(meet(X, meet(complement(Y), complement(meet(X, Z))))))
% 162.40/21.31 = { by axiom 4 (commutativity_of_meet) R->L }
% 162.40/21.31 meet(X, complement(meet(X, meet(complement(meet(X, Z)), complement(Y)))))
% 162.40/21.31 = { by axiom 12 (associativity_of_meet) R->L }
% 162.40/21.31 meet(X, complement(meet(meet(X, complement(meet(X, Z))), complement(Y))))
% 162.40/21.31 = { by lemma 32 }
% 162.40/21.31 meet(X, complement(meet(meet(X, complement(Z)), complement(Y))))
% 162.40/21.31 = { by axiom 12 (associativity_of_meet) }
% 162.40/21.31 meet(X, complement(meet(X, meet(complement(Z), complement(Y)))))
% 162.40/21.31 = { by axiom 4 (commutativity_of_meet) }
% 162.40/21.31 meet(X, complement(meet(X, meet(complement(Y), complement(Z)))))
% 162.40/21.31 = { by lemma 32 }
% 162.40/21.31 meet(X, complement(meet(complement(Y), complement(Z))))
% 162.40/21.31 = { by lemma 38 }
% 162.40/21.31 meet(X, join(Z, complement(complement(Y))))
% 162.40/21.31 = { by lemma 15 }
% 162.40/21.31 meet(X, join(Z, Y))
% 162.40/21.31
% 162.40/21.31 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 162.40/21.31 Proof:
% 162.40/21.31 meet(a, join(b, c))
% 162.40/21.31 = { by axiom 2 (commutativity_of_join) R->L }
% 162.40/21.31 meet(a, join(c, b))
% 162.40/21.31 = { by lemma 39 R->L }
% 162.40/21.31 meet(a, join(b, meet(a, c)))
% 162.40/21.31 = { by lemma 35 R->L }
% 162.40/21.31 join(meet(a, b), meet(a, join(b, meet(a, c))))
% 162.40/21.31 = { by lemma 39 R->L }
% 162.40/21.31 join(meet(a, b), meet(a, join(meet(a, c), meet(a, b))))
% 162.40/21.31 = { by axiom 2 (commutativity_of_join) R->L }
% 162.40/21.31 join(meet(a, b), meet(a, join(meet(a, b), meet(a, c))))
% 162.40/21.31 = { by lemma 25 }
% 162.40/21.31 join(meet(a, b), meet(a, c))
% 162.40/21.31 % SZS output end Proof
% 162.40/21.31
% 162.40/21.31 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------