TSTP Solution File: LAT005-3 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT005-3 : TPTP v8.1.2. Released v1.0.0.
% 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 : n025.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 06:27:04 EDT 2023
% Result : Unsatisfiable 2.39s 0.79s
% Output : Proof 2.39s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LAT005-3 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.35 % Computer : n025.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Thu Aug 24 09:22:21 EDT 2023
% 0.15/0.35 % CPUTime :
% 2.39/0.79 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 2.39/0.79
% 2.39/0.79 % SZS status Unsatisfiable
% 2.39/0.79
% 2.39/0.80 % SZS output start Proof
% 2.39/0.80 Take the following subset of the input axioms:
% 2.39/0.80 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 2.39/0.80 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 2.39/0.80 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 2.39/0.80 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 2.39/0.80 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 2.39/0.80 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 2.39/0.80 fof(complement_meet, axiom, ![X2, Y2]: (~complement(X2, Y2) | meet(X2, Y2)=n0)).
% 2.39/0.80 fof(complement_of_a_join_b, hypothesis, complement(r1, join(a, b))).
% 2.39/0.80 fof(complement_of_a_meet_b, hypothesis, complement(r2, meet(a, b))).
% 2.39/0.80 fof(modular, axiom, ![X2, Y2, Z2]: (meet(X2, Z2)!=X2 | meet(Z2, join(X2, Y2))=join(X2, meet(Y2, Z2)))).
% 2.39/0.80 fof(prove_lemma, negated_conjecture, r1!=meet(join(r1, meet(r2, b)), join(r1, meet(r2, a)))).
% 2.39/0.80 fof(x_join_0, axiom, ![X2]: join(X2, n0)=X2).
% 2.39/0.80
% 2.39/0.80 Now clausify the problem and encode Horn clauses using encoding 3 of
% 2.39/0.80 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 2.39/0.80 We repeatedly replace C & s=t => u=v by the two clauses:
% 2.39/0.80 fresh(y, y, x1...xn) = u
% 2.39/0.80 C => fresh(s, t, x1...xn) = v
% 2.39/0.80 where fresh is a fresh function symbol and x1..xn are the free
% 2.39/0.80 variables of u and v.
% 2.39/0.80 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 2.39/0.80 input problem has no model of domain size 1).
% 2.39/0.80
% 2.39/0.80 The encoding turns the above axioms into the following unit equations and goals:
% 2.39/0.80
% 2.39/0.80 Axiom 1 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 2.39/0.80 Axiom 2 (commutativity_of_join): join(X, Y) = join(Y, X).
% 2.39/0.80 Axiom 3 (x_join_0): join(X, n0) = X.
% 2.39/0.80 Axiom 4 (absorption1): meet(X, join(X, Y)) = X.
% 2.39/0.80 Axiom 5 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 2.39/0.80 Axiom 6 (absorption2): join(X, meet(X, Y)) = X.
% 2.39/0.80 Axiom 7 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 2.39/0.80 Axiom 8 (complement_of_a_meet_b): complement(r2, meet(a, b)) = true.
% 2.39/0.80 Axiom 9 (complement_of_a_join_b): complement(r1, join(a, b)) = true.
% 2.39/0.80 Axiom 10 (complement_meet): fresh5(X, X, Y, Z) = n0.
% 2.39/0.80 Axiom 11 (modular): fresh(X, X, Y, Z, W) = join(Y, meet(W, Z)).
% 2.39/0.80 Axiom 12 (complement_meet): fresh5(complement(X, Y), true, X, Y) = meet(X, Y).
% 2.39/0.80 Axiom 13 (modular): fresh(meet(X, Y), X, X, Y, Z) = meet(Y, join(X, Z)).
% 2.39/0.80
% 2.39/0.80 Lemma 14: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 2.39/0.81 Proof:
% 2.39/0.81 meet(Y, meet(Z, X))
% 2.39/0.81 = { by axiom 1 (commutativity_of_meet) R->L }
% 2.39/0.81 meet(meet(Z, X), Y)
% 2.39/0.81 = { by axiom 1 (commutativity_of_meet) }
% 2.39/0.81 meet(meet(X, Z), Y)
% 2.39/0.81 = { by axiom 5 (associativity_of_meet) }
% 2.39/0.81 meet(X, meet(Z, Y))
% 2.39/0.81 = { by axiom 1 (commutativity_of_meet) }
% 2.39/0.81 meet(X, meet(Y, Z))
% 2.39/0.81
% 2.39/0.81 Lemma 15: join(X, meet(Y, X)) = X.
% 2.39/0.81 Proof:
% 2.39/0.81 join(X, meet(Y, X))
% 2.39/0.81 = { by axiom 1 (commutativity_of_meet) R->L }
% 2.39/0.81 join(X, meet(X, Y))
% 2.39/0.81 = { by axiom 6 (absorption2) }
% 2.39/0.81 X
% 2.39/0.81
% 2.39/0.81 Lemma 16: join(X, meet(Y, join(X, Z))) = meet(join(X, Z), join(X, Y)).
% 2.39/0.81 Proof:
% 2.39/0.81 join(X, meet(Y, join(X, Z)))
% 2.39/0.81 = { by axiom 11 (modular) R->L }
% 2.39/0.81 fresh(X, X, X, join(X, Z), Y)
% 2.39/0.81 = { by axiom 4 (absorption1) R->L }
% 2.39/0.81 fresh(meet(X, join(X, Z)), X, X, join(X, Z), Y)
% 2.39/0.81 = { by axiom 13 (modular) }
% 2.39/0.81 meet(join(X, Z), join(X, Y))
% 2.39/0.81
% 2.39/0.81 Lemma 17: join(X, join(Y, meet(Z, X))) = join(Y, X).
% 2.39/0.81 Proof:
% 2.39/0.81 join(X, join(Y, meet(Z, X)))
% 2.39/0.81 = { by axiom 2 (commutativity_of_join) R->L }
% 2.39/0.81 join(X, join(meet(Z, X), Y))
% 2.39/0.81 = { by axiom 7 (associativity_of_join) R->L }
% 2.39/0.81 join(join(X, meet(Z, X)), Y)
% 2.39/0.81 = { by lemma 15 }
% 2.39/0.81 join(X, Y)
% 2.39/0.81 = { by axiom 2 (commutativity_of_join) }
% 2.39/0.81 join(Y, X)
% 2.39/0.81
% 2.39/0.81 Lemma 18: meet(join(r1, meet(r2, b)), join(r1, a)) = r1.
% 2.39/0.81 Proof:
% 2.39/0.81 meet(join(r1, meet(r2, b)), join(r1, a))
% 2.39/0.81 = { by axiom 1 (commutativity_of_meet) R->L }
% 2.39/0.81 meet(join(r1, a), join(r1, meet(r2, b)))
% 2.39/0.81 = { by lemma 16 R->L }
% 2.39/0.81 join(r1, meet(meet(r2, b), join(r1, a)))
% 2.39/0.81 = { by axiom 1 (commutativity_of_meet) R->L }
% 2.39/0.81 join(r1, meet(join(r1, a), meet(r2, b)))
% 2.39/0.81 = { by lemma 14 R->L }
% 2.39/0.81 join(r1, meet(r2, meet(b, join(r1, a))))
% 2.39/0.81 = { by axiom 1 (commutativity_of_meet) R->L }
% 2.39/0.81 join(r1, meet(r2, meet(join(r1, a), b)))
% 2.39/0.81 = { by axiom 4 (absorption1) R->L }
% 2.39/0.81 join(r1, meet(r2, meet(join(r1, a), meet(b, join(b, a)))))
% 2.39/0.81 = { by axiom 2 (commutativity_of_join) }
% 2.39/0.81 join(r1, meet(r2, meet(join(r1, a), meet(b, join(a, b)))))
% 2.39/0.81 = { by axiom 1 (commutativity_of_meet) R->L }
% 2.39/0.81 join(r1, meet(r2, meet(join(r1, a), meet(join(a, b), b))))
% 2.39/0.81 = { by axiom 5 (associativity_of_meet) R->L }
% 2.39/0.81 join(r1, meet(r2, meet(meet(join(r1, a), join(a, b)), b)))
% 2.39/0.81 = { by axiom 1 (commutativity_of_meet) }
% 2.39/0.81 join(r1, meet(r2, meet(b, meet(join(r1, a), join(a, b)))))
% 2.39/0.81 = { by axiom 2 (commutativity_of_join) R->L }
% 2.39/0.81 join(r1, meet(r2, meet(b, meet(join(a, r1), join(a, b)))))
% 2.39/0.81 = { by axiom 1 (commutativity_of_meet) R->L }
% 2.39/0.81 join(r1, meet(r2, meet(b, meet(join(a, b), join(a, r1)))))
% 2.39/0.81 = { by lemma 16 R->L }
% 2.39/0.81 join(r1, meet(r2, meet(b, join(a, meet(r1, join(a, b))))))
% 2.39/0.81 = { by axiom 12 (complement_meet) R->L }
% 2.39/0.81 join(r1, meet(r2, meet(b, join(a, fresh5(complement(r1, join(a, b)), true, r1, join(a, b))))))
% 2.39/0.81 = { by axiom 9 (complement_of_a_join_b) }
% 2.39/0.81 join(r1, meet(r2, meet(b, join(a, fresh5(true, true, r1, join(a, b))))))
% 2.39/0.81 = { by axiom 10 (complement_meet) }
% 2.39/0.81 join(r1, meet(r2, meet(b, join(a, n0))))
% 2.39/0.81 = { by axiom 3 (x_join_0) }
% 2.39/0.81 join(r1, meet(r2, meet(b, a)))
% 2.39/0.81 = { by axiom 1 (commutativity_of_meet) }
% 2.39/0.81 join(r1, meet(r2, meet(a, b)))
% 2.39/0.81 = { by axiom 12 (complement_meet) R->L }
% 2.39/0.81 join(r1, fresh5(complement(r2, meet(a, b)), true, r2, meet(a, b)))
% 2.39/0.81 = { by axiom 8 (complement_of_a_meet_b) }
% 2.39/0.81 join(r1, fresh5(true, true, r2, meet(a, b)))
% 2.39/0.81 = { by axiom 10 (complement_meet) }
% 2.39/0.81 join(r1, n0)
% 2.39/0.81 = { by axiom 3 (x_join_0) }
% 2.39/0.81 r1
% 2.39/0.81
% 2.39/0.81 Lemma 19: join(X, meet(join(X, Y), join(X, Z))) = meet(join(X, Y), join(X, Z)).
% 2.39/0.81 Proof:
% 2.39/0.81 join(X, meet(join(X, Y), join(X, Z)))
% 2.39/0.81 = { by axiom 2 (commutativity_of_join) R->L }
% 2.39/0.81 join(meet(join(X, Y), join(X, Z)), X)
% 2.39/0.81 = { by axiom 4 (absorption1) R->L }
% 2.39/0.81 join(meet(join(X, Y), join(X, Z)), meet(X, join(X, Z)))
% 2.39/0.81 = { by axiom 1 (commutativity_of_meet) R->L }
% 2.39/0.81 join(meet(join(X, Y), join(X, Z)), meet(join(X, Z), X))
% 2.39/0.81 = { by axiom 4 (absorption1) R->L }
% 2.39/0.81 join(meet(join(X, Y), join(X, Z)), meet(join(X, Z), meet(X, join(X, Y))))
% 2.39/0.81 = { by lemma 14 R->L }
% 2.39/0.81 join(meet(join(X, Y), join(X, Z)), meet(X, meet(join(X, Y), join(X, Z))))
% 2.39/0.81 = { by lemma 15 }
% 2.39/0.81 meet(join(X, Y), join(X, Z))
% 2.39/0.81
% 2.39/0.81 Lemma 20: join(X, join(Y, meet(join(X, Z), join(X, W)))) = join(Y, meet(join(X, Z), join(X, W))).
% 2.39/0.81 Proof:
% 2.39/0.81 join(X, join(Y, meet(join(X, Z), join(X, W))))
% 2.39/0.81 = { by axiom 2 (commutativity_of_join) R->L }
% 2.39/0.81 join(X, join(meet(join(X, Z), join(X, W)), Y))
% 2.39/0.81 = { by axiom 7 (associativity_of_join) R->L }
% 2.39/0.81 join(join(X, meet(join(X, Z), join(X, W))), Y)
% 2.39/0.81 = { by lemma 19 }
% 2.39/0.81 join(meet(join(X, Z), join(X, W)), Y)
% 2.39/0.81 = { by axiom 2 (commutativity_of_join) }
% 2.39/0.81 join(Y, meet(join(X, Z), join(X, W)))
% 2.39/0.81
% 2.39/0.81 Goal 1 (prove_lemma): r1 = meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))).
% 2.39/0.81 Proof:
% 2.39/0.81 r1
% 2.39/0.81 = { by lemma 18 R->L }
% 2.39/0.81 meet(join(r1, meet(r2, b)), join(r1, a))
% 2.39/0.81 = { by lemma 17 R->L }
% 2.39/0.81 meet(join(r1, meet(r2, b)), join(a, join(r1, meet(r2, a))))
% 2.39/0.81 = { by axiom 6 (absorption2) R->L }
% 2.39/0.81 meet(join(r1, meet(r2, b)), join(a, join(join(r1, meet(r2, a)), meet(join(r1, meet(r2, a)), join(r1, meet(r2, b))))))
% 2.39/0.81 = { by axiom 1 (commutativity_of_meet) R->L }
% 2.39/0.81 meet(join(r1, meet(r2, b)), join(a, join(join(r1, meet(r2, a)), meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))))))
% 2.39/0.81 = { by axiom 7 (associativity_of_join) }
% 2.39/0.81 meet(join(r1, meet(r2, b)), join(a, join(r1, join(meet(r2, a), meet(join(r1, meet(r2, b)), join(r1, meet(r2, a)))))))
% 2.39/0.81 = { by lemma 20 }
% 2.39/0.81 meet(join(r1, meet(r2, b)), join(a, join(meet(r2, a), meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))))))
% 2.39/0.81 = { by axiom 2 (commutativity_of_join) }
% 2.39/0.81 meet(join(r1, meet(r2, b)), join(a, join(meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), meet(r2, a))))
% 2.39/0.81 = { by lemma 17 }
% 2.39/0.81 meet(join(r1, meet(r2, b)), join(meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), a))
% 2.39/0.81 = { by axiom 2 (commutativity_of_join) }
% 2.39/0.81 meet(join(r1, meet(r2, b)), join(a, meet(join(r1, meet(r2, b)), join(r1, meet(r2, a)))))
% 2.39/0.81 = { by lemma 20 R->L }
% 2.39/0.81 meet(join(r1, meet(r2, b)), join(r1, join(a, meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))))))
% 2.39/0.81 = { by axiom 7 (associativity_of_join) R->L }
% 2.39/0.81 meet(join(r1, meet(r2, b)), join(join(r1, a), meet(join(r1, meet(r2, b)), join(r1, meet(r2, a)))))
% 2.39/0.81 = { by axiom 2 (commutativity_of_join) R->L }
% 2.39/0.81 meet(join(r1, meet(r2, b)), join(meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), join(r1, a)))
% 2.39/0.81 = { by axiom 13 (modular) R->L }
% 2.39/0.81 fresh(meet(meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), join(r1, meet(r2, b))), meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), join(r1, meet(r2, b)), join(r1, a))
% 2.39/0.81 = { by lemma 15 R->L }
% 2.39/0.81 fresh(meet(meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), join(join(r1, meet(r2, b)), meet(join(r1, meet(r2, a)), join(r1, meet(r2, b))))), meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), join(r1, meet(r2, b)), join(r1, a))
% 2.39/0.81 = { by axiom 1 (commutativity_of_meet) R->L }
% 2.39/0.81 fresh(meet(meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), join(join(r1, meet(r2, b)), meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))))), meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), join(r1, meet(r2, b)), join(r1, a))
% 2.39/0.81 = { by axiom 2 (commutativity_of_join) }
% 2.39/0.81 fresh(meet(meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), join(meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), join(r1, meet(r2, b)))), meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), join(r1, meet(r2, b)), join(r1, a))
% 2.39/0.81 = { by axiom 4 (absorption1) }
% 2.39/0.81 fresh(meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), join(r1, meet(r2, b)), join(r1, a))
% 2.39/0.81 = { by axiom 11 (modular) }
% 2.39/0.81 join(meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), meet(join(r1, a), join(r1, meet(r2, b))))
% 2.39/0.81 = { by axiom 1 (commutativity_of_meet) }
% 2.39/0.81 join(meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), meet(join(r1, meet(r2, b)), join(r1, a)))
% 2.39/0.81 = { by lemma 18 }
% 2.39/0.81 join(meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))), r1)
% 2.39/0.81 = { by axiom 2 (commutativity_of_join) }
% 2.39/0.81 join(r1, meet(join(r1, meet(r2, b)), join(r1, meet(r2, a))))
% 2.39/0.81 = { by lemma 19 }
% 2.39/0.81 meet(join(r1, meet(r2, b)), join(r1, meet(r2, a)))
% 2.39/0.81 % SZS output end Proof
% 2.39/0.81
% 2.39/0.81 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------