TSTP Solution File: LAT001-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT001-1 : 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 : n016.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 06:27:03 EDT 2023
% Result : Unsatisfiable 0.19s 0.89s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LAT001-1 : 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
% 0.17/0.36 % Computer : n016.cluster.edu
% 0.17/0.36 % Model : x86_64 x86_64
% 0.17/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.36 % Memory : 8042.1875MB
% 0.17/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.36 % CPULimit : 300
% 0.17/0.36 % WCLimit : 300
% 0.17/0.36 % DateTime : Thu Aug 24 08:07:09 EDT 2023
% 0.17/0.36 % CPUTime :
% 0.19/0.89 Command-line arguments: --ground-connectedness --complete-subsets
% 0.19/0.89
% 0.19/0.89 % SZS status Unsatisfiable
% 0.19/0.89
% 0.19/0.92 % SZS output start Proof
% 0.19/0.92 Take the following subset of the input axioms:
% 0.19/0.92 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 0.19/0.92 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 0.19/0.92 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 0.19/0.92 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 0.19/0.92 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 0.19/0.92 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 0.19/0.92 fof(complement_join, axiom, ![X2, Y2]: (~complement(X2, Y2) | join(X2, Y2)=n1)).
% 0.19/0.92 fof(complement_meet, axiom, ![X2, Y2]: (~complement(X2, Y2) | meet(X2, Y2)=n0)).
% 0.19/0.92 fof(complement_of_a_join_b, hypothesis, complement(r1, join(a, b))).
% 0.19/0.92 fof(complement_of_a_meet_b, hypothesis, complement(r2, meet(a, b))).
% 0.19/0.92 fof(idempotence_of_meet, axiom, ![X2]: meet(X2, X2)=X2).
% 0.19/0.92 fof(meet_join_complement, axiom, ![X2, Y2]: (meet(X2, Y2)!=n0 | (join(X2, Y2)!=n1 | complement(X2, Y2)))).
% 0.19/0.92 fof(modular, axiom, ![X2, Y2, Z2]: (meet(X2, Z2)!=X2 | meet(Z2, join(X2, Y2))=join(X2, meet(Y2, Z2)))).
% 0.19/0.92 fof(prove_complememt, negated_conjecture, ~complement(a, join(r1, meet(r2, b)))).
% 0.19/0.92 fof(x_join_0, axiom, ![X2]: join(X2, n0)=X2).
% 0.19/0.92 fof(x_join_1, axiom, ![X2]: join(X2, n1)=n1).
% 0.19/0.92 fof(x_meet_1, axiom, ![X2]: meet(X2, n1)=X2).
% 0.19/0.92
% 0.19/0.92 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.92 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.92 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.92 fresh(y, y, x1...xn) = u
% 0.19/0.92 C => fresh(s, t, x1...xn) = v
% 0.19/0.92 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.92 variables of u and v.
% 0.19/0.92 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.92 input problem has no model of domain size 1).
% 0.19/0.92
% 0.19/0.92 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.92
% 0.19/0.92 Axiom 1 (commutativity_of_join): join(X, Y) = join(Y, X).
% 0.19/0.92 Axiom 2 (x_join_0): join(X, n0) = X.
% 0.19/0.92 Axiom 3 (x_join_1): join(X, n1) = n1.
% 0.19/0.92 Axiom 4 (idempotence_of_meet): meet(X, X) = X.
% 0.19/0.92 Axiom 5 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 0.19/0.92 Axiom 6 (x_meet_1): meet(X, n1) = X.
% 0.19/0.92 Axiom 7 (complement_meet): fresh5(X, X, Y, Z) = n0.
% 0.19/0.92 Axiom 8 (complement_join): fresh4(X, X, Y, Z) = n1.
% 0.19/0.92 Axiom 9 (meet_join_complement): fresh3(X, X, Y, Z) = complement(Y, Z).
% 0.19/0.92 Axiom 10 (meet_join_complement): fresh2(X, X, Y, Z) = true.
% 0.19/0.92 Axiom 11 (complement_of_a_join_b): complement(r1, join(a, b)) = true.
% 0.19/0.92 Axiom 12 (complement_of_a_meet_b): complement(r2, meet(a, b)) = true.
% 0.19/0.92 Axiom 13 (absorption2): join(X, meet(X, Y)) = X.
% 0.19/0.92 Axiom 14 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 0.19/0.92 Axiom 15 (absorption1): meet(X, join(X, Y)) = X.
% 0.19/0.92 Axiom 16 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 0.19/0.92 Axiom 17 (modular): fresh(X, X, Y, Z, W) = join(Y, meet(W, Z)).
% 0.19/0.92 Axiom 18 (complement_meet): fresh5(complement(X, Y), true, X, Y) = meet(X, Y).
% 0.19/0.92 Axiom 19 (complement_join): fresh4(complement(X, Y), true, X, Y) = join(X, Y).
% 0.19/0.92 Axiom 20 (meet_join_complement): fresh3(join(X, Y), n1, X, Y) = fresh2(meet(X, Y), n0, X, Y).
% 0.19/0.92 Axiom 21 (modular): fresh(meet(X, Y), X, X, Y, Z) = meet(Y, join(X, Z)).
% 0.19/0.92
% 0.19/0.92 Lemma 22: meet(X, join(Y, X)) = X.
% 0.19/0.92 Proof:
% 0.19/0.92 meet(X, join(Y, X))
% 0.19/0.92 = { by axiom 1 (commutativity_of_join) }
% 0.19/0.92 meet(X, join(X, Y))
% 0.19/0.92 = { by axiom 15 (absorption1) }
% 0.19/0.92 X
% 0.19/0.92
% 0.19/0.92 Lemma 23: join(X, join(Y, Z)) = join(Y, join(X, Z)).
% 0.19/0.92 Proof:
% 0.19/0.92 join(X, join(Y, Z))
% 0.19/0.92 = { by axiom 1 (commutativity_of_join) R->L }
% 0.19/0.92 join(join(Y, Z), X)
% 0.19/0.92 = { by axiom 14 (associativity_of_join) }
% 0.19/0.92 join(Y, join(Z, X))
% 0.19/0.92 = { by axiom 1 (commutativity_of_join) }
% 0.19/0.92 join(Y, join(X, Z))
% 0.19/0.92
% 0.19/0.92 Lemma 24: join(r2, meet(a, b)) = n1.
% 0.19/0.92 Proof:
% 0.19/0.92 join(r2, meet(a, b))
% 0.19/0.92 = { by axiom 19 (complement_join) R->L }
% 0.19/0.92 fresh4(complement(r2, meet(a, b)), true, r2, meet(a, b))
% 0.19/0.92 = { by axiom 12 (complement_of_a_meet_b) }
% 0.19/0.92 fresh4(true, true, r2, meet(a, b))
% 0.19/0.92 = { by axiom 8 (complement_join) }
% 0.19/0.92 n1
% 0.19/0.92
% 0.19/0.92 Lemma 25: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 0.19/0.92 Proof:
% 0.19/0.92 join(X, join(Y, meet(X, Z)))
% 0.19/0.92 = { by axiom 1 (commutativity_of_join) R->L }
% 0.19/0.92 join(X, join(meet(X, Z), Y))
% 0.19/0.92 = { by axiom 14 (associativity_of_join) R->L }
% 0.19/0.92 join(join(X, meet(X, Z)), Y)
% 0.19/0.92 = { by axiom 13 (absorption2) }
% 0.19/0.92 join(X, Y)
% 0.19/0.92
% 0.19/0.92 Lemma 26: meet(X, join(Y, meet(X, Z))) = join(meet(X, Z), meet(X, Y)).
% 0.19/0.92 Proof:
% 0.19/0.92 meet(X, join(Y, meet(X, Z)))
% 0.19/0.92 = { by axiom 1 (commutativity_of_join) R->L }
% 0.19/0.92 meet(X, join(meet(X, Z), Y))
% 0.19/0.92 = { by axiom 21 (modular) R->L }
% 0.19/0.92 fresh(meet(meet(X, Z), X), meet(X, Z), meet(X, Z), X, Y)
% 0.19/0.92 = { by axiom 5 (commutativity_of_meet) }
% 0.19/0.92 fresh(meet(X, meet(X, Z)), meet(X, Z), meet(X, Z), X, Y)
% 0.19/0.92 = { by axiom 16 (associativity_of_meet) R->L }
% 0.19/0.92 fresh(meet(meet(X, X), Z), meet(X, Z), meet(X, Z), X, Y)
% 0.19/0.92 = { by axiom 4 (idempotence_of_meet) }
% 0.19/0.92 fresh(meet(X, Z), meet(X, Z), meet(X, Z), X, Y)
% 0.19/0.92 = { by axiom 17 (modular) }
% 0.19/0.92 join(meet(X, Z), meet(Y, X))
% 0.19/0.92 = { by axiom 5 (commutativity_of_meet) }
% 0.19/0.92 join(meet(X, Z), meet(X, Y))
% 0.19/0.92
% 0.19/0.92 Lemma 27: join(X, meet(Y, join(X, Z))) = join(X, meet(Z, join(X, Y))).
% 0.19/0.92 Proof:
% 0.19/0.92 join(X, meet(Y, join(X, Z)))
% 0.19/0.92 = { by axiom 17 (modular) R->L }
% 0.19/0.92 fresh(X, X, X, join(X, Z), Y)
% 0.19/0.92 = { by axiom 15 (absorption1) R->L }
% 0.19/0.92 fresh(meet(X, join(X, Z)), X, X, join(X, Z), Y)
% 0.19/0.92 = { by axiom 21 (modular) }
% 0.19/0.92 meet(join(X, Z), join(X, Y))
% 0.19/0.92 = { by axiom 1 (commutativity_of_join) }
% 0.19/0.92 meet(join(Z, X), join(X, Y))
% 0.19/0.92 = { by axiom 1 (commutativity_of_join) R->L }
% 0.19/0.92 meet(join(Z, X), join(Y, X))
% 0.19/0.92 = { by axiom 1 (commutativity_of_join) R->L }
% 0.19/0.92 meet(join(X, Z), join(Y, X))
% 0.19/0.92 = { by axiom 5 (commutativity_of_meet) R->L }
% 0.19/0.92 meet(join(Y, X), join(X, Z))
% 0.19/0.92 = { by axiom 21 (modular) R->L }
% 0.19/0.93 fresh(meet(X, join(Y, X)), X, X, join(Y, X), Z)
% 0.19/0.93 = { by lemma 22 }
% 0.19/0.93 fresh(X, X, X, join(Y, X), Z)
% 0.19/0.93 = { by axiom 17 (modular) }
% 0.19/0.93 join(X, meet(Z, join(Y, X)))
% 0.19/0.93 = { by axiom 1 (commutativity_of_join) }
% 0.19/0.93 join(X, meet(Z, join(X, Y)))
% 0.19/0.93
% 0.19/0.93 Lemma 28: join(meet(r2, b), a) = join(a, b).
% 0.19/0.93 Proof:
% 0.19/0.93 join(meet(r2, b), a)
% 0.19/0.93 = { by axiom 1 (commutativity_of_join) R->L }
% 0.19/0.93 join(a, meet(r2, b))
% 0.19/0.93 = { by lemma 25 R->L }
% 0.19/0.93 join(a, join(meet(r2, b), meet(a, r2)))
% 0.19/0.93 = { by axiom 5 (commutativity_of_meet) R->L }
% 0.19/0.93 join(a, join(meet(r2, b), meet(r2, a)))
% 0.19/0.93 = { by axiom 1 (commutativity_of_join) R->L }
% 0.19/0.93 join(a, join(meet(r2, a), meet(r2, b)))
% 0.19/0.93 = { by lemma 26 R->L }
% 0.19/0.93 join(a, meet(r2, join(b, meet(r2, a))))
% 0.19/0.93 = { by axiom 5 (commutativity_of_meet) }
% 0.19/0.93 join(a, meet(r2, join(b, meet(a, r2))))
% 0.19/0.93 = { by axiom 1 (commutativity_of_join) R->L }
% 0.19/0.93 join(a, meet(r2, join(meet(a, r2), b)))
% 0.19/0.93 = { by axiom 13 (absorption2) R->L }
% 0.19/0.93 join(a, meet(r2, join(meet(a, r2), join(b, meet(b, a)))))
% 0.19/0.93 = { by axiom 5 (commutativity_of_meet) }
% 0.19/0.93 join(a, meet(r2, join(meet(a, r2), join(b, meet(a, b)))))
% 0.19/0.93 = { by lemma 23 }
% 0.19/0.93 join(a, meet(r2, join(b, join(meet(a, r2), meet(a, b)))))
% 0.19/0.93 = { by axiom 1 (commutativity_of_join) R->L }
% 0.19/0.93 join(a, meet(r2, join(b, join(meet(a, b), meet(a, r2)))))
% 0.19/0.93 = { by lemma 26 R->L }
% 0.19/0.93 join(a, meet(r2, join(b, meet(a, join(r2, meet(a, b))))))
% 0.19/0.93 = { by lemma 24 }
% 0.19/0.93 join(a, meet(r2, join(b, meet(a, n1))))
% 0.19/0.93 = { by axiom 6 (x_meet_1) }
% 0.19/0.93 join(a, meet(r2, join(b, a)))
% 0.19/0.93 = { by axiom 1 (commutativity_of_join) }
% 0.19/0.93 join(a, meet(r2, join(a, b)))
% 0.19/0.93 = { by lemma 27 }
% 0.19/0.93 join(a, meet(b, join(a, r2)))
% 0.19/0.93 = { by lemma 25 R->L }
% 0.19/0.93 join(a, meet(b, join(a, join(r2, meet(a, b)))))
% 0.19/0.93 = { by lemma 24 }
% 0.19/0.93 join(a, meet(b, join(a, n1)))
% 0.19/0.93 = { by axiom 3 (x_join_1) }
% 0.19/0.93 join(a, meet(b, n1))
% 0.19/0.93 = { by axiom 6 (x_meet_1) }
% 0.19/0.93 join(a, b)
% 0.19/0.93
% 0.19/0.93 Goal 1 (prove_complememt): complement(a, join(r1, meet(r2, b))) = true.
% 0.19/0.93 Proof:
% 0.19/0.93 complement(a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 9 (meet_join_complement) R->L }
% 0.19/0.93 fresh3(n1, n1, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 8 (complement_join) R->L }
% 0.19/0.93 fresh3(fresh4(true, true, r1, join(a, b)), n1, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 11 (complement_of_a_join_b) R->L }
% 0.19/0.93 fresh3(fresh4(complement(r1, join(a, b)), true, r1, join(a, b)), n1, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 19 (complement_join) }
% 0.19/0.93 fresh3(join(r1, join(a, b)), n1, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by lemma 28 R->L }
% 0.19/0.93 fresh3(join(r1, join(meet(r2, b), a)), n1, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by lemma 23 }
% 0.19/0.93 fresh3(join(meet(r2, b), join(r1, a)), n1, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 14 (associativity_of_join) R->L }
% 0.19/0.93 fresh3(join(join(meet(r2, b), r1), a), n1, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 1 (commutativity_of_join) R->L }
% 0.19/0.93 fresh3(join(join(r1, meet(r2, b)), a), n1, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 1 (commutativity_of_join) R->L }
% 0.19/0.93 fresh3(join(a, join(r1, meet(r2, b))), n1, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 20 (meet_join_complement) }
% 0.19/0.93 fresh2(meet(a, join(r1, meet(r2, b))), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 15 (absorption1) R->L }
% 0.19/0.93 fresh2(meet(meet(a, join(a, meet(r2, b))), join(r1, meet(r2, b))), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 16 (associativity_of_meet) }
% 0.19/0.93 fresh2(meet(a, meet(join(a, meet(r2, b)), join(r1, meet(r2, b)))), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 5 (commutativity_of_meet) }
% 0.19/0.93 fresh2(meet(a, meet(join(r1, meet(r2, b)), join(a, meet(r2, b)))), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 1 (commutativity_of_join) R->L }
% 0.19/0.93 fresh2(meet(a, meet(join(r1, meet(r2, b)), join(meet(r2, b), a))), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 21 (modular) R->L }
% 0.19/0.93 fresh2(meet(a, fresh(meet(meet(r2, b), join(r1, meet(r2, b))), meet(r2, b), meet(r2, b), join(r1, meet(r2, b)), a)), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by lemma 22 }
% 0.19/0.93 fresh2(meet(a, fresh(meet(r2, b), meet(r2, b), meet(r2, b), join(r1, meet(r2, b)), a)), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 17 (modular) }
% 0.19/0.93 fresh2(meet(a, join(meet(r2, b), meet(a, join(r1, meet(r2, b))))), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 1 (commutativity_of_join) }
% 0.19/0.93 fresh2(meet(a, join(meet(r2, b), meet(a, join(meet(r2, b), r1)))), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by lemma 27 R->L }
% 0.19/0.93 fresh2(meet(a, join(meet(r2, b), meet(r1, join(meet(r2, b), a)))), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by lemma 28 }
% 0.19/0.93 fresh2(meet(a, join(meet(r2, b), meet(r1, join(a, b)))), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 18 (complement_meet) R->L }
% 0.19/0.93 fresh2(meet(a, join(meet(r2, b), fresh5(complement(r1, join(a, b)), true, r1, join(a, b)))), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 11 (complement_of_a_join_b) }
% 0.19/0.93 fresh2(meet(a, join(meet(r2, b), fresh5(true, true, r1, join(a, b)))), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 7 (complement_meet) }
% 0.19/0.93 fresh2(meet(a, join(meet(r2, b), n0)), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 2 (x_join_0) }
% 0.19/0.93 fresh2(meet(a, meet(r2, b)), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 5 (commutativity_of_meet) R->L }
% 0.19/0.93 fresh2(meet(meet(r2, b), a), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 16 (associativity_of_meet) }
% 0.19/0.93 fresh2(meet(r2, meet(b, a)), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 5 (commutativity_of_meet) }
% 0.19/0.93 fresh2(meet(r2, meet(a, b)), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 18 (complement_meet) R->L }
% 0.19/0.93 fresh2(fresh5(complement(r2, meet(a, b)), true, r2, meet(a, b)), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 12 (complement_of_a_meet_b) }
% 0.19/0.93 fresh2(fresh5(true, true, r2, meet(a, b)), n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 7 (complement_meet) }
% 0.19/0.93 fresh2(n0, n0, a, join(r1, meet(r2, b)))
% 0.19/0.93 = { by axiom 10 (meet_join_complement) }
% 0.19/0.93 true
% 0.19/0.93 % SZS output end Proof
% 0.19/0.93
% 0.19/0.93 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------