TSTP Solution File: REL008-4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL008-4 : TPTP v8.1.2. Released v4.0.0.
% 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 : n006.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 13:43:48 EDT 2023
% Result : Unsatisfiable 0.18s 0.78s
% Output : Proof 2.32s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL008-4 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.33 % Computer : n006.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Fri Aug 25 19:08:37 EDT 2023
% 0.13/0.33 % CPUTime :
% 0.18/0.78 Command-line arguments: --no-flatten-goal
% 0.18/0.78
% 0.18/0.78 % SZS status Unsatisfiable
% 0.18/0.78
% 2.32/0.80 % SZS output start Proof
% 2.32/0.80 Take the following subset of the input axioms:
% 2.32/0.80 fof(composition_associativity_5, axiom, ![A, B, C]: composition(A, composition(B, C))=composition(composition(A, B), C)).
% 2.32/0.80 fof(composition_distributivity_7, axiom, ![A2, B2, C2]: composition(join(A2, B2), C2)=join(composition(A2, C2), composition(B2, C2))).
% 2.32/0.80 fof(composition_identity_6, axiom, ![A2]: composition(A2, one)=A2).
% 2.32/0.80 fof(converse_additivity_9, axiom, ![A2, B2]: converse(join(A2, B2))=join(converse(A2), converse(B2))).
% 2.32/0.80 fof(converse_cancellativity_11, axiom, ![A2, B2]: join(composition(converse(A2), complement(composition(A2, B2))), complement(B2))=complement(B2)).
% 2.32/0.80 fof(converse_idempotence_8, axiom, ![A2]: converse(converse(A2))=A2).
% 2.32/0.80 fof(converse_multiplicativity_10, axiom, ![A2, B2]: converse(composition(A2, B2))=composition(converse(B2), converse(A2))).
% 2.32/0.80 fof(def_zero_13, axiom, ![A2]: zero=meet(A2, complement(A2))).
% 2.32/0.80 fof(goals_17, negated_conjecture, join(join(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, join(sk2, sk3)))!=composition(sk1, join(sk2, sk3)) | join(join(composition(sk1, join(sk2, sk3)), composition(sk1, sk2)), composition(sk1, sk3))!=join(composition(sk1, sk2), composition(sk1, sk3))).
% 2.32/0.80 fof(maddux1_join_commutativity_1, axiom, ![A2, B2]: join(A2, B2)=join(B2, A2)).
% 2.32/0.80 fof(maddux2_join_associativity_2, axiom, ![A2, B2, C2]: join(A2, join(B2, C2))=join(join(A2, B2), C2)).
% 2.32/0.80 fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A2, B2]: A2=join(complement(join(complement(A2), complement(B2))), complement(join(complement(A2), B2)))).
% 2.32/0.80 fof(maddux4_definiton_of_meet_4, axiom, ![A2, B2]: meet(A2, B2)=complement(join(complement(A2), complement(B2)))).
% 2.32/0.80
% 2.32/0.80 Now clausify the problem and encode Horn clauses using encoding 3 of
% 2.32/0.80 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 2.32/0.80 We repeatedly replace C & s=t => u=v by the two clauses:
% 2.32/0.80 fresh(y, y, x1...xn) = u
% 2.32/0.80 C => fresh(s, t, x1...xn) = v
% 2.32/0.80 where fresh is a fresh function symbol and x1..xn are the free
% 2.32/0.80 variables of u and v.
% 2.32/0.80 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 2.32/0.80 input problem has no model of domain size 1).
% 2.32/0.80
% 2.32/0.80 The encoding turns the above axioms into the following unit equations and goals:
% 2.32/0.80
% 2.32/0.80 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 2.32/0.80 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 2.32/0.80 Axiom 3 (composition_identity_6): composition(X, one) = X.
% 2.32/0.80 Axiom 4 (def_zero_13): zero = meet(X, complement(X)).
% 2.32/0.80 Axiom 5 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 2.32/0.80 Axiom 6 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 2.32/0.80 Axiom 7 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 2.32/0.80 Axiom 8 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 2.32/0.80 Axiom 9 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 2.32/0.80 Axiom 10 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 2.32/0.80 Axiom 11 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 2.32/0.80 Axiom 12 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 2.32/0.80
% 2.32/0.80 Lemma 13: composition(converse(one), X) = X.
% 2.32/0.80 Proof:
% 2.32/0.80 composition(converse(one), X)
% 2.32/0.80 = { by axiom 1 (converse_idempotence_8) R->L }
% 2.32/0.80 composition(converse(one), converse(converse(X)))
% 2.32/0.80 = { by axiom 7 (converse_multiplicativity_10) R->L }
% 2.32/0.80 converse(composition(converse(X), one))
% 2.32/0.80 = { by axiom 3 (composition_identity_6) }
% 2.32/0.80 converse(converse(X))
% 2.32/0.80 = { by axiom 1 (converse_idempotence_8) }
% 2.32/0.80 X
% 2.32/0.80
% 2.32/0.80 Lemma 14: join(complement(X), complement(X)) = complement(X).
% 2.32/0.80 Proof:
% 2.32/0.80 join(complement(X), complement(X))
% 2.32/0.80 = { by lemma 13 R->L }
% 2.32/0.80 join(complement(X), composition(converse(one), complement(X)))
% 2.32/0.80 = { by lemma 13 R->L }
% 2.32/0.80 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 2.32/0.80 = { by axiom 3 (composition_identity_6) R->L }
% 2.32/0.80 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 2.32/0.80 = { by axiom 8 (composition_associativity_5) R->L }
% 2.32/0.80 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 2.32/0.80 = { by lemma 13 }
% 2.32/0.80 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 2.32/0.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 2.32/0.80 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 2.32/0.80 = { by axiom 11 (converse_cancellativity_11) }
% 2.32/0.80 complement(X)
% 2.32/0.80
% 2.32/0.80 Lemma 15: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 2.32/0.80 Proof:
% 2.32/0.80 join(meet(X, Y), complement(join(complement(X), Y)))
% 2.32/0.80 = { by axiom 9 (maddux4_definiton_of_meet_4) }
% 2.32/0.80 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 2.32/0.80 = { by axiom 12 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 2.32/0.80 X
% 2.32/0.80
% 2.32/0.80 Lemma 16: join(zero, complement(complement(X))) = X.
% 2.32/0.80 Proof:
% 2.32/0.80 join(zero, complement(complement(X)))
% 2.32/0.80 = { by axiom 4 (def_zero_13) }
% 2.32/0.80 join(meet(X, complement(X)), complement(complement(X)))
% 2.32/0.80 = { by lemma 14 R->L }
% 2.32/0.80 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 2.32/0.80 = { by lemma 15 }
% 2.32/0.80 X
% 2.32/0.80
% 2.32/0.80 Lemma 17: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 2.32/0.80 Proof:
% 2.32/0.80 join(zero, join(X, complement(complement(Y))))
% 2.32/0.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 2.32/0.80 join(zero, join(complement(complement(Y)), X))
% 2.32/0.80 = { by lemma 14 R->L }
% 2.32/0.80 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 2.32/0.80 = { by axiom 9 (maddux4_definiton_of_meet_4) R->L }
% 2.32/0.80 join(zero, join(meet(Y, Y), X))
% 2.32/0.80 = { by axiom 6 (maddux2_join_associativity_2) }
% 2.32/0.80 join(join(zero, meet(Y, Y)), X)
% 2.32/0.80 = { by axiom 9 (maddux4_definiton_of_meet_4) }
% 2.32/0.80 join(join(zero, complement(join(complement(Y), complement(Y)))), X)
% 2.32/0.80 = { by axiom 4 (def_zero_13) }
% 2.32/0.80 join(join(meet(Y, complement(Y)), complement(join(complement(Y), complement(Y)))), X)
% 2.32/0.80 = { by lemma 15 }
% 2.32/0.80 join(Y, X)
% 2.32/0.80 = { by axiom 2 (maddux1_join_commutativity_1) }
% 2.32/0.80 join(X, Y)
% 2.32/0.80
% 2.32/0.80 Lemma 18: join(X, zero) = join(X, X).
% 2.32/0.80 Proof:
% 2.32/0.80 join(X, zero)
% 2.32/0.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 2.32/0.80 join(zero, X)
% 2.32/0.80 = { by lemma 16 R->L }
% 2.32/0.80 join(zero, join(zero, complement(complement(X))))
% 2.32/0.80 = { by lemma 14 R->L }
% 2.32/0.80 join(zero, join(zero, join(complement(complement(X)), complement(complement(X)))))
% 2.32/0.80 = { by lemma 17 }
% 2.32/0.80 join(zero, join(complement(complement(X)), X))
% 2.32/0.80 = { by axiom 2 (maddux1_join_commutativity_1) }
% 2.32/0.80 join(zero, join(X, complement(complement(X))))
% 2.32/0.80 = { by lemma 17 }
% 2.32/0.80 join(X, X)
% 2.32/0.80
% 2.32/0.80 Lemma 19: join(X, join(Y, X)) = join(X, Y).
% 2.32/0.80 Proof:
% 2.32/0.80 join(X, join(Y, X))
% 2.32/0.80 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 2.32/0.80 join(X, join(X, Y))
% 2.32/0.80 = { by axiom 6 (maddux2_join_associativity_2) }
% 2.32/0.80 join(join(X, X), Y)
% 2.32/0.80 = { by lemma 18 R->L }
% 2.32/0.80 join(join(X, zero), Y)
% 2.32/0.80 = { by axiom 6 (maddux2_join_associativity_2) R->L }
% 2.32/0.80 join(X, join(zero, Y))
% 2.32/0.80 = { by lemma 17 R->L }
% 2.32/0.80 join(X, join(zero, join(zero, complement(complement(Y)))))
% 2.32/0.81 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 2.32/0.81 join(X, join(zero, join(complement(complement(Y)), zero)))
% 2.32/0.81 = { by lemma 18 }
% 2.32/0.81 join(X, join(zero, join(complement(complement(Y)), complement(complement(Y)))))
% 2.32/0.81 = { by lemma 14 }
% 2.32/0.81 join(X, join(zero, complement(complement(Y))))
% 2.32/0.81 = { by lemma 16 }
% 2.32/0.81 join(X, Y)
% 2.32/0.81
% 2.32/0.81 Lemma 20: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 2.32/0.81 Proof:
% 2.32/0.81 join(Y, join(X, Z))
% 2.32/0.81 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 2.32/0.81 join(join(X, Z), Y)
% 2.32/0.81 = { by axiom 6 (maddux2_join_associativity_2) R->L }
% 2.32/0.81 join(X, join(Z, Y))
% 2.32/0.81 = { by axiom 2 (maddux1_join_commutativity_1) }
% 2.32/0.81 join(X, join(Y, Z))
% 2.32/0.81
% 2.32/0.81 Lemma 21: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 2.32/0.81 Proof:
% 2.32/0.81 join(composition(X, Y), composition(X, Z))
% 2.32/0.81 = { by axiom 1 (converse_idempotence_8) R->L }
% 2.32/0.81 join(composition(X, Y), composition(X, converse(converse(Z))))
% 2.32/0.81 = { by axiom 1 (converse_idempotence_8) R->L }
% 2.32/0.81 converse(converse(join(composition(X, Y), composition(X, converse(converse(Z))))))
% 2.32/0.81 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 2.32/0.81 converse(converse(join(composition(X, converse(converse(Z))), composition(X, Y))))
% 2.32/0.81 = { by axiom 5 (converse_additivity_9) }
% 2.32/0.81 converse(join(converse(composition(X, converse(converse(Z)))), converse(composition(X, Y))))
% 2.32/0.81 = { by axiom 7 (converse_multiplicativity_10) }
% 2.32/0.81 converse(join(composition(converse(converse(converse(Z))), converse(X)), converse(composition(X, Y))))
% 2.32/0.81 = { by axiom 1 (converse_idempotence_8) }
% 2.32/0.81 converse(join(composition(converse(Z), converse(X)), converse(composition(X, Y))))
% 2.32/0.81 = { by axiom 2 (maddux1_join_commutativity_1) }
% 2.32/0.81 converse(join(converse(composition(X, Y)), composition(converse(Z), converse(X))))
% 2.32/0.81 = { by axiom 7 (converse_multiplicativity_10) }
% 2.32/0.81 converse(join(composition(converse(Y), converse(X)), composition(converse(Z), converse(X))))
% 2.32/0.81 = { by axiom 10 (composition_distributivity_7) R->L }
% 2.32/0.81 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 2.32/0.81 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 2.32/0.81 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 2.32/0.81 = { by axiom 1 (converse_idempotence_8) R->L }
% 2.32/0.81 converse(composition(join(converse(converse(converse(Z))), converse(Y)), converse(X)))
% 2.32/0.81 = { by axiom 5 (converse_additivity_9) R->L }
% 2.32/0.81 converse(composition(converse(join(converse(converse(Z)), Y)), converse(X)))
% 2.32/0.81 = { by axiom 2 (maddux1_join_commutativity_1) }
% 2.32/0.81 converse(composition(converse(join(Y, converse(converse(Z)))), converse(X)))
% 2.32/0.81 = { by axiom 7 (converse_multiplicativity_10) R->L }
% 2.32/0.81 converse(converse(composition(X, join(Y, converse(converse(Z))))))
% 2.32/0.81 = { by axiom 1 (converse_idempotence_8) }
% 2.32/0.81 composition(X, join(Y, converse(converse(Z))))
% 2.32/0.81 = { by axiom 1 (converse_idempotence_8) }
% 2.32/0.81 composition(X, join(Y, Z))
% 2.32/0.81
% 2.32/0.81 Goal 1 (goals_17): tuple(join(join(composition(sk1, join(sk2, sk3)), composition(sk1, sk2)), composition(sk1, sk3)), join(join(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, join(sk2, sk3)))) = tuple(join(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, join(sk2, sk3))).
% 2.32/0.81 Proof:
% 2.32/0.81 tuple(join(join(composition(sk1, join(sk2, sk3)), composition(sk1, sk2)), composition(sk1, sk3)), join(join(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, join(sk2, sk3))))
% 2.32/0.81 = { by axiom 2 (maddux1_join_commutativity_1) }
% 2.32/0.81 tuple(join(composition(sk1, sk3), join(composition(sk1, join(sk2, sk3)), composition(sk1, sk2))), join(join(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, join(sk2, sk3))))
% 2.32/0.81 = { by axiom 2 (maddux1_join_commutativity_1) }
% 2.32/0.81 tuple(join(composition(sk1, sk3), join(composition(sk1, sk2), composition(sk1, join(sk2, sk3)))), join(join(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, join(sk2, sk3))))
% 2.32/0.81 = { by axiom 2 (maddux1_join_commutativity_1) }
% 2.32/0.81 tuple(join(composition(sk1, sk3), join(composition(sk1, sk2), composition(sk1, join(sk2, sk3)))), join(composition(sk1, join(sk2, sk3)), join(composition(sk1, sk2), composition(sk1, sk3))))
% 2.32/0.81 = { by lemma 20 R->L }
% 2.32/0.81 tuple(join(composition(sk1, sk2), join(composition(sk1, sk3), composition(sk1, join(sk2, sk3)))), join(composition(sk1, join(sk2, sk3)), join(composition(sk1, sk2), composition(sk1, sk3))))
% 2.32/0.81 = { by lemma 20 R->L }
% 2.32/0.81 tuple(join(composition(sk1, sk2), join(composition(sk1, sk3), composition(sk1, join(sk2, sk3)))), join(composition(sk1, sk2), join(composition(sk1, join(sk2, sk3)), composition(sk1, sk3))))
% 2.32/0.81 = { by axiom 2 (maddux1_join_commutativity_1) }
% 2.32/0.81 tuple(join(composition(sk1, sk2), join(composition(sk1, sk3), composition(sk1, join(sk2, sk3)))), join(composition(sk1, sk2), join(composition(sk1, sk3), composition(sk1, join(sk2, sk3)))))
% 2.32/0.81 = { by lemma 21 }
% 2.32/0.81 tuple(join(composition(sk1, sk2), composition(sk1, join(sk3, join(sk2, sk3)))), join(composition(sk1, sk2), join(composition(sk1, sk3), composition(sk1, join(sk2, sk3)))))
% 2.32/0.81 = { by lemma 21 }
% 2.32/0.81 tuple(join(composition(sk1, sk2), composition(sk1, join(sk3, join(sk2, sk3)))), join(composition(sk1, sk2), composition(sk1, join(sk3, join(sk2, sk3)))))
% 2.32/0.81 = { by lemma 19 }
% 2.32/0.81 tuple(join(composition(sk1, sk2), composition(sk1, join(sk3, sk2))), join(composition(sk1, sk2), composition(sk1, join(sk3, join(sk2, sk3)))))
% 2.32/0.81 = { by lemma 19 }
% 2.32/0.81 tuple(join(composition(sk1, sk2), composition(sk1, join(sk3, sk2))), join(composition(sk1, sk2), composition(sk1, join(sk3, sk2))))
% 2.32/0.81 = { by lemma 21 }
% 2.32/0.81 tuple(composition(sk1, join(sk2, join(sk3, sk2))), join(composition(sk1, sk2), composition(sk1, join(sk3, sk2))))
% 2.32/0.81 = { by lemma 21 }
% 2.32/0.81 tuple(composition(sk1, join(sk2, join(sk3, sk2))), composition(sk1, join(sk2, join(sk3, sk2))))
% 2.32/0.81 = { by lemma 19 }
% 2.32/0.81 tuple(composition(sk1, join(sk2, sk3)), composition(sk1, join(sk2, join(sk3, sk2))))
% 2.32/0.81 = { by lemma 19 }
% 2.32/0.81 tuple(composition(sk1, join(sk2, sk3)), composition(sk1, join(sk2, sk3)))
% 2.32/0.81 = { by lemma 21 R->L }
% 2.32/0.81 tuple(join(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, join(sk2, sk3)))
% 2.32/0.81 % SZS output end Proof
% 2.32/0.81
% 2.32/0.81 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------