TSTP Solution File: LAT243-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT243-1 : TPTP v8.1.2. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n010.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 06:27:55 EDT 2023
% Result : Unsatisfiable 85.50s 11.29s
% Output : Proof 85.50s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : LAT243-1 : TPTP v8.1.2. Released v3.1.0.
% 0.12/0.13 % 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 : n010.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 05:38:50 EDT 2023
% 0.13/0.34 % CPUTime :
% 85.50/11.29 Command-line arguments: --flatten
% 85.50/11.29
% 85.50/11.29 % SZS status Unsatisfiable
% 85.50/11.29
% 85.50/11.31 % SZS output start Proof
% 85.50/11.31 Take the following subset of the input axioms:
% 85.50/11.31 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 85.50/11.31 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 85.50/11.31 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 85.50/11.31 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 85.50/11.31 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 85.50/11.31 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 85.50/11.31 fof(complement_join, axiom, ![X2]: join(X2, complement(X2))=one).
% 85.50/11.31 fof(complement_meet, axiom, ![X2]: meet(X2, complement(X2))=zero).
% 85.50/11.31 fof(equation_H57, axiom, ![X2, Y2, Z2]: meet(X2, join(Y2, meet(X2, join(Y2, Z2))))=meet(X2, join(Y2, meet(join(X2, Y2), join(Z2, meet(X2, Y2)))))).
% 85.50/11.31 fof(idempotence_of_meet, axiom, ![X2]: meet(X2, X2)=X2).
% 85.50/11.31 fof(meet_join_complement, axiom, ![X2, Y2]: (meet(X2, Y2)!=zero | (join(X2, Y2)!=one | complement(X2)=Y2))).
% 85.50/11.31 fof(prove_distributivity, negated_conjecture, join(complement(b), complement(a))!=complement(a)).
% 85.50/11.31 fof(prove_distributivity_hypothesis, hypothesis, meet(b, a)=a).
% 85.50/11.31
% 85.50/11.31 Now clausify the problem and encode Horn clauses using encoding 3 of
% 85.50/11.31 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 85.50/11.31 We repeatedly replace C & s=t => u=v by the two clauses:
% 85.50/11.31 fresh(y, y, x1...xn) = u
% 85.50/11.31 C => fresh(s, t, x1...xn) = v
% 85.50/11.31 where fresh is a fresh function symbol and x1..xn are the free
% 85.50/11.31 variables of u and v.
% 85.50/11.31 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 85.50/11.31 input problem has no model of domain size 1).
% 85.50/11.31
% 85.50/11.31 The encoding turns the above axioms into the following unit equations and goals:
% 85.50/11.31
% 85.50/11.31 Axiom 1 (idempotence_of_meet): meet(X, X) = X.
% 85.50/11.31 Axiom 2 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 85.50/11.31 Axiom 3 (prove_distributivity_hypothesis): meet(b, a) = a.
% 85.50/11.31 Axiom 4 (commutativity_of_join): join(X, Y) = join(Y, X).
% 85.50/11.31 Axiom 5 (complement_meet): meet(X, complement(X)) = zero.
% 85.50/11.31 Axiom 6 (complement_join): join(X, complement(X)) = one.
% 85.50/11.31 Axiom 7 (meet_join_complement): fresh(X, X, Y, Z) = Z.
% 85.50/11.31 Axiom 8 (meet_join_complement): fresh2(X, X, Y, Z) = complement(Y).
% 85.50/11.31 Axiom 9 (absorption1): meet(X, join(X, Y)) = X.
% 85.50/11.31 Axiom 10 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 85.50/11.31 Axiom 11 (absorption2): join(X, meet(X, Y)) = X.
% 85.50/11.31 Axiom 12 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 85.50/11.31 Axiom 13 (meet_join_complement): fresh2(join(X, Y), one, X, Y) = fresh(meet(X, Y), zero, X, Y).
% 85.50/11.31 Axiom 14 (equation_H57): meet(X, join(Y, meet(X, join(Y, Z)))) = meet(X, join(Y, meet(join(X, Y), join(Z, meet(X, Y))))).
% 85.50/11.31
% 85.50/11.31 Lemma 15: join(X, zero) = X.
% 85.50/11.31 Proof:
% 85.50/11.31 join(X, zero)
% 85.50/11.31 = { by axiom 5 (complement_meet) R->L }
% 85.50/11.31 join(X, meet(X, complement(X)))
% 85.50/11.31 = { by axiom 11 (absorption2) }
% 85.50/11.31 X
% 85.50/11.31
% 85.50/11.31 Lemma 16: meet(X, join(Y, X)) = X.
% 85.50/11.31 Proof:
% 85.50/11.31 meet(X, join(Y, X))
% 85.50/11.31 = { by axiom 4 (commutativity_of_join) R->L }
% 85.50/11.31 meet(X, join(X, Y))
% 85.50/11.31 = { by axiom 9 (absorption1) }
% 85.50/11.31 X
% 85.50/11.31
% 85.50/11.31 Lemma 17: meet(join(X, Y), Y) = Y.
% 85.50/11.31 Proof:
% 85.50/11.31 meet(join(X, Y), Y)
% 85.50/11.31 = { by axiom 2 (commutativity_of_meet) R->L }
% 85.50/11.31 meet(Y, join(X, Y))
% 85.50/11.31 = { by lemma 16 }
% 85.50/11.31 Y
% 85.50/11.31
% 85.50/11.31 Lemma 18: meet(meet(X, a), complement(X)) = zero.
% 85.50/11.31 Proof:
% 85.50/11.31 meet(meet(X, a), complement(X))
% 85.50/11.31 = { by axiom 2 (commutativity_of_meet) R->L }
% 85.50/11.31 meet(complement(X), meet(X, a))
% 85.50/11.31 = { by axiom 2 (commutativity_of_meet) }
% 85.50/11.31 meet(complement(X), meet(a, X))
% 85.50/11.31 = { by axiom 3 (prove_distributivity_hypothesis) R->L }
% 85.50/11.31 meet(complement(X), meet(meet(b, a), X))
% 85.50/11.31 = { by axiom 2 (commutativity_of_meet) R->L }
% 85.50/11.31 meet(complement(X), meet(X, meet(b, a)))
% 85.50/11.31 = { by axiom 10 (associativity_of_meet) R->L }
% 85.50/11.31 meet(meet(complement(X), X), meet(b, a))
% 85.50/11.31 = { by axiom 2 (commutativity_of_meet) }
% 85.50/11.31 meet(meet(b, a), meet(complement(X), X))
% 85.50/11.31 = { by axiom 2 (commutativity_of_meet) R->L }
% 85.50/11.31 meet(meet(b, a), meet(X, complement(X)))
% 85.50/11.31 = { by axiom 5 (complement_meet) }
% 85.50/11.31 meet(meet(b, a), zero)
% 85.50/11.31 = { by axiom 2 (commutativity_of_meet) R->L }
% 85.50/11.31 meet(zero, meet(b, a))
% 85.50/11.31 = { by lemma 15 R->L }
% 85.50/11.31 meet(zero, join(meet(b, a), zero))
% 85.50/11.31 = { by axiom 4 (commutativity_of_join) }
% 85.50/11.31 meet(zero, join(zero, meet(b, a)))
% 85.50/11.31 = { by axiom 9 (absorption1) }
% 85.50/11.31 zero
% 85.50/11.31
% 85.50/11.31 Lemma 19: join(complement(b), complement(join(meet(b, a), complement(b)))) = complement(a).
% 85.50/11.31 Proof:
% 85.50/11.31 join(complement(b), complement(join(meet(b, a), complement(b))))
% 85.50/11.31 = { by axiom 7 (meet_join_complement) R->L }
% 85.50/11.31 fresh(zero, zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by lemma 18 R->L }
% 85.50/11.31 fresh(meet(meet(b, a), complement(b)), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by lemma 15 R->L }
% 85.50/11.31 fresh(meet(meet(b, a), join(complement(b), zero)), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by axiom 5 (complement_meet) R->L }
% 85.50/11.31 fresh(meet(meet(b, a), join(complement(b), meet(join(meet(b, a), complement(b)), complement(join(meet(b, a), complement(b)))))), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by lemma 15 R->L }
% 85.50/11.31 fresh(meet(meet(b, a), join(complement(b), meet(join(meet(b, a), complement(b)), join(complement(join(meet(b, a), complement(b))), zero)))), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by lemma 18 R->L }
% 85.50/11.31 fresh(meet(meet(b, a), join(complement(b), meet(join(meet(b, a), complement(b)), join(complement(join(meet(b, a), complement(b))), meet(meet(b, a), complement(b)))))), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by axiom 14 (equation_H57) R->L }
% 85.50/11.31 fresh(meet(meet(b, a), join(complement(b), meet(meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b))))))), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by axiom 4 (commutativity_of_join) R->L }
% 85.50/11.31 fresh(meet(meet(b, a), join(complement(b), meet(meet(b, a), join(complement(join(meet(b, a), complement(b))), complement(b))))), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by lemma 16 R->L }
% 85.50/11.31 fresh(meet(meet(b, a), meet(join(complement(b), meet(meet(b, a), join(complement(join(meet(b, a), complement(b))), complement(b)))), join(complement(join(meet(b, a), complement(b))), join(complement(b), meet(meet(b, a), join(complement(join(meet(b, a), complement(b))), complement(b))))))), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by axiom 2 (commutativity_of_meet) R->L }
% 85.50/11.31 fresh(meet(meet(b, a), meet(join(complement(b), meet(meet(b, a), join(complement(join(meet(b, a), complement(b))), complement(b)))), join(complement(join(meet(b, a), complement(b))), join(complement(b), meet(join(complement(join(meet(b, a), complement(b))), complement(b)), meet(b, a)))))), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by axiom 12 (associativity_of_join) R->L }
% 85.50/11.31 fresh(meet(meet(b, a), meet(join(complement(b), meet(meet(b, a), join(complement(join(meet(b, a), complement(b))), complement(b)))), join(join(complement(join(meet(b, a), complement(b))), complement(b)), meet(join(complement(join(meet(b, a), complement(b))), complement(b)), meet(b, a))))), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by axiom 11 (absorption2) }
% 85.50/11.31 fresh(meet(meet(b, a), meet(join(complement(b), meet(meet(b, a), join(complement(join(meet(b, a), complement(b))), complement(b)))), join(complement(join(meet(b, a), complement(b))), complement(b)))), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by axiom 2 (commutativity_of_meet) }
% 85.50/11.31 fresh(meet(meet(b, a), meet(join(complement(join(meet(b, a), complement(b))), complement(b)), join(complement(b), meet(meet(b, a), join(complement(join(meet(b, a), complement(b))), complement(b)))))), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by axiom 4 (commutativity_of_join) }
% 85.50/11.31 fresh(meet(meet(b, a), meet(join(complement(join(meet(b, a), complement(b))), complement(b)), join(complement(b), meet(meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))))), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by axiom 4 (commutativity_of_join) }
% 85.50/11.31 fresh(meet(meet(b, a), meet(join(complement(b), complement(join(meet(b, a), complement(b)))), join(complement(b), meet(meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))))), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by axiom 4 (commutativity_of_join) R->L }
% 85.50/11.31 fresh(meet(meet(b, a), meet(join(complement(b), complement(join(meet(b, a), complement(b)))), join(meet(meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b))))), complement(b)))), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by axiom 10 (associativity_of_meet) R->L }
% 85.50/11.31 fresh(meet(meet(meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b))))), join(meet(meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b))))), complement(b))), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by axiom 9 (absorption1) }
% 85.50/11.31 fresh(meet(meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b))))), zero, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by axiom 13 (meet_join_complement) R->L }
% 85.50/11.31 fresh2(join(meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b))))), one, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by axiom 12 (associativity_of_join) R->L }
% 85.50/11.31 fresh2(join(join(meet(b, a), complement(b)), complement(join(meet(b, a), complement(b)))), one, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by axiom 6 (complement_join) }
% 85.50/11.31 fresh2(one, one, meet(b, a), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.31 = { by axiom 8 (meet_join_complement) }
% 85.50/11.31 complement(meet(b, a))
% 85.50/11.31 = { by axiom 3 (prove_distributivity_hypothesis) }
% 85.50/11.32 complement(a)
% 85.50/11.32
% 85.50/11.32 Goal 1 (prove_distributivity): join(complement(b), complement(a)) = complement(a).
% 85.50/11.32 Proof:
% 85.50/11.32 join(complement(b), complement(a))
% 85.50/11.32 = { by lemma 17 R->L }
% 85.50/11.32 join(complement(b), meet(join(complement(b), complement(a)), complement(a)))
% 85.50/11.32 = { by axiom 1 (idempotence_of_meet) R->L }
% 85.50/11.32 join(meet(complement(b), complement(b)), meet(join(complement(b), complement(a)), complement(a)))
% 85.50/11.32 = { by lemma 19 R->L }
% 85.50/11.32 join(meet(complement(b), complement(b)), meet(join(complement(b), complement(a)), join(complement(b), complement(join(meet(b, a), complement(b))))))
% 85.50/11.32 = { by axiom 2 (commutativity_of_meet) R->L }
% 85.50/11.32 join(meet(complement(b), complement(b)), meet(join(complement(b), complement(join(meet(b, a), complement(b)))), join(complement(b), complement(a))))
% 85.50/11.32 = { by axiom 4 (commutativity_of_join) R->L }
% 85.50/11.32 join(meet(join(complement(b), complement(join(meet(b, a), complement(b)))), join(complement(b), complement(a))), meet(complement(b), complement(b)))
% 85.50/11.32 = { by axiom 9 (absorption1) R->L }
% 85.50/11.32 join(meet(join(complement(b), complement(join(meet(b, a), complement(b)))), join(complement(b), complement(a))), meet(meet(complement(b), join(complement(b), complement(join(meet(b, a), complement(b))))), complement(b)))
% 85.50/11.32 = { by axiom 10 (associativity_of_meet) }
% 85.50/11.32 join(meet(join(complement(b), complement(join(meet(b, a), complement(b)))), join(complement(b), complement(a))), meet(complement(b), meet(join(complement(b), complement(join(meet(b, a), complement(b)))), complement(b))))
% 85.50/11.32 = { by axiom 2 (commutativity_of_meet) }
% 85.50/11.32 join(meet(join(complement(b), complement(join(meet(b, a), complement(b)))), join(complement(b), complement(a))), meet(complement(b), meet(complement(b), join(complement(b), complement(join(meet(b, a), complement(b)))))))
% 85.50/11.32 = { by axiom 10 (associativity_of_meet) R->L }
% 85.50/11.32 join(meet(join(complement(b), complement(join(meet(b, a), complement(b)))), join(complement(b), complement(a))), meet(meet(complement(b), complement(b)), join(complement(b), complement(join(meet(b, a), complement(b))))))
% 85.50/11.32 = { by axiom 2 (commutativity_of_meet) R->L }
% 85.50/11.32 join(meet(join(complement(b), complement(join(meet(b, a), complement(b)))), join(complement(b), complement(a))), meet(join(complement(b), complement(join(meet(b, a), complement(b)))), meet(complement(b), complement(b))))
% 85.50/11.32 = { by axiom 9 (absorption1) R->L }
% 85.50/11.32 join(meet(join(complement(b), complement(join(meet(b, a), complement(b)))), join(complement(b), complement(a))), meet(join(complement(b), complement(join(meet(b, a), complement(b)))), meet(meet(complement(b), join(complement(b), complement(a))), complement(b))))
% 85.50/11.32 = { by axiom 2 (commutativity_of_meet) }
% 85.50/11.32 join(meet(join(complement(b), complement(join(meet(b, a), complement(b)))), join(complement(b), complement(a))), meet(join(complement(b), complement(join(meet(b, a), complement(b)))), meet(meet(join(complement(b), complement(a)), complement(b)), complement(b))))
% 85.50/11.32 = { by axiom 10 (associativity_of_meet) }
% 85.50/11.32 join(meet(join(complement(b), complement(join(meet(b, a), complement(b)))), join(complement(b), complement(a))), meet(join(complement(b), complement(join(meet(b, a), complement(b)))), meet(join(complement(b), complement(a)), meet(complement(b), complement(b)))))
% 85.50/11.32 = { by axiom 10 (associativity_of_meet) R->L }
% 85.50/11.32 join(meet(join(complement(b), complement(join(meet(b, a), complement(b)))), join(complement(b), complement(a))), meet(meet(join(complement(b), complement(join(meet(b, a), complement(b)))), join(complement(b), complement(a))), meet(complement(b), complement(b))))
% 85.50/11.32 = { by axiom 11 (absorption2) }
% 85.50/11.32 meet(join(complement(b), complement(join(meet(b, a), complement(b)))), join(complement(b), complement(a)))
% 85.50/11.32 = { by axiom 2 (commutativity_of_meet) }
% 85.50/11.32 meet(join(complement(b), complement(a)), join(complement(b), complement(join(meet(b, a), complement(b)))))
% 85.50/11.32 = { by lemma 19 }
% 85.50/11.32 meet(join(complement(b), complement(a)), complement(a))
% 85.50/11.32 = { by lemma 17 }
% 85.50/11.32 complement(a)
% 85.50/11.32 % SZS output end Proof
% 85.50/11.32
% 85.50/11.32 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------