TSTP Solution File: GRP402-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP402-1 : TPTP v8.1.2. Released v2.5.0.
% 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 01:18:17 EDT 2023
% Result : Unsatisfiable 1.62s 0.58s
% Output : Proof 1.62s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GRP402-1 : TPTP v8.1.2. Released v2.5.0.
% 0.03/0.13 % 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 : n016.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 : Mon Aug 28 21:41:45 EDT 2023
% 0.13/0.34 % CPUTime :
% 1.62/0.58 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 1.62/0.58
% 1.62/0.58 % SZS status Unsatisfiable
% 1.62/0.58
% 1.62/0.60 % SZS output start Proof
% 1.62/0.60 Take the following subset of the input axioms:
% 1.62/0.60 fof(associativity_of_multiply, axiom, ![X, Y, Z]: multiply(multiply(X, Y), Z)=multiply(X, multiply(Y, Z))).
% 1.62/0.60 fof(commutator, axiom, ![A, B]: multiply(A, B)=multiply(B, multiply(A, commutator(A, B)))).
% 1.62/0.60 fof(left_cancellation, axiom, ![C, B2, A3]: (multiply(A3, B2)!=multiply(A3, C) | B2=C)).
% 1.62/0.60 fof(nilpotency, axiom, ![B2, C2, A3]: multiply(commutator(A3, B2), C2)=multiply(C2, commutator(A3, B2))).
% 1.62/0.60 fof(prove_commutator_is_associative, negated_conjecture, commutator(commutator(a, b), c)!=commutator(a, commutator(b, c))).
% 1.62/0.60 fof(right_cancellation, axiom, ![A2, B2, C2]: (multiply(A2, B2)!=multiply(C2, B2) | A2=C2)).
% 1.62/0.60
% 1.62/0.60 Now clausify the problem and encode Horn clauses using encoding 3 of
% 1.62/0.60 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 1.62/0.60 We repeatedly replace C & s=t => u=v by the two clauses:
% 1.62/0.60 fresh(y, y, x1...xn) = u
% 1.62/0.60 C => fresh(s, t, x1...xn) = v
% 1.62/0.60 where fresh is a fresh function symbol and x1..xn are the free
% 1.62/0.60 variables of u and v.
% 1.62/0.60 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 1.62/0.60 input problem has no model of domain size 1).
% 1.62/0.60
% 1.62/0.60 The encoding turns the above axioms into the following unit equations and goals:
% 1.62/0.60
% 1.62/0.60 Axiom 1 (associativity_of_multiply): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 1.62/0.60 Axiom 2 (nilpotency): multiply(commutator(X, Y), Z) = multiply(Z, commutator(X, Y)).
% 1.62/0.60 Axiom 3 (left_cancellation): fresh(X, X, Y, Z) = Z.
% 1.62/0.60 Axiom 4 (right_cancellation): fresh2(X, X, Y, Z) = Z.
% 1.62/0.60 Axiom 5 (commutator): multiply(X, Y) = multiply(Y, multiply(X, commutator(X, Y))).
% 1.62/0.60 Axiom 6 (left_cancellation): fresh(multiply(X, Y), multiply(X, Z), Y, Z) = Y.
% 1.62/0.60 Axiom 7 (right_cancellation): fresh2(multiply(X, Y), multiply(Z, Y), X, Z) = X.
% 1.62/0.60
% 1.62/0.60 Lemma 8: multiply(X, commutator(X, X)) = X.
% 1.62/0.60 Proof:
% 1.62/0.60 multiply(X, commutator(X, X))
% 1.62/0.60 = { by axiom 3 (left_cancellation) R->L }
% 1.62/0.60 fresh(multiply(X, X), multiply(X, X), X, multiply(X, commutator(X, X)))
% 1.62/0.60 = { by axiom 5 (commutator) }
% 1.62/0.60 fresh(multiply(X, X), multiply(X, multiply(X, commutator(X, X))), X, multiply(X, commutator(X, X)))
% 1.62/0.60 = { by axiom 6 (left_cancellation) }
% 1.62/0.60 X
% 1.62/0.60
% 1.62/0.60 Lemma 9: multiply(X, commutator(X, commutator(Y, Z))) = X.
% 1.62/0.60 Proof:
% 1.62/0.60 multiply(X, commutator(X, commutator(Y, Z)))
% 1.62/0.60 = { by axiom 3 (left_cancellation) R->L }
% 1.62/0.60 fresh(multiply(X, commutator(Y, Z)), multiply(X, commutator(Y, Z)), X, multiply(X, commutator(X, commutator(Y, Z))))
% 1.62/0.60 = { by axiom 2 (nilpotency) R->L }
% 1.62/0.60 fresh(multiply(commutator(Y, Z), X), multiply(X, commutator(Y, Z)), X, multiply(X, commutator(X, commutator(Y, Z))))
% 1.62/0.60 = { by axiom 5 (commutator) }
% 1.62/0.60 fresh(multiply(commutator(Y, Z), X), multiply(commutator(Y, Z), multiply(X, commutator(X, commutator(Y, Z)))), X, multiply(X, commutator(X, commutator(Y, Z))))
% 1.62/0.60 = { by axiom 6 (left_cancellation) }
% 1.62/0.60 X
% 1.62/0.60
% 1.62/0.60 Lemma 10: multiply(commutator(X, X), multiply(Y, X)) = multiply(Y, X).
% 1.62/0.60 Proof:
% 1.62/0.60 multiply(commutator(X, X), multiply(Y, X))
% 1.62/0.60 = { by axiom 2 (nilpotency) }
% 1.62/0.60 multiply(multiply(Y, X), commutator(X, X))
% 1.62/0.60 = { by axiom 1 (associativity_of_multiply) }
% 1.62/0.60 multiply(Y, multiply(X, commutator(X, X)))
% 1.62/0.60 = { by lemma 8 }
% 1.62/0.60 multiply(Y, X)
% 1.62/0.60
% 1.62/0.60 Lemma 11: commutator(multiply(X, Y), multiply(X, Y)) = commutator(Y, Y).
% 1.62/0.60 Proof:
% 1.62/0.60 commutator(multiply(X, Y), multiply(X, Y))
% 1.62/0.60 = { by axiom 4 (right_cancellation) R->L }
% 1.62/0.60 fresh2(multiply(X, Y), multiply(X, Y), commutator(Y, Y), commutator(multiply(X, Y), multiply(X, Y)))
% 1.62/0.60 = { by lemma 10 R->L }
% 1.62/0.60 fresh2(multiply(commutator(Y, Y), multiply(X, Y)), multiply(X, Y), commutator(Y, Y), commutator(multiply(X, Y), multiply(X, Y)))
% 1.62/0.60 = { by lemma 8 R->L }
% 1.62/0.60 fresh2(multiply(commutator(Y, Y), multiply(X, Y)), multiply(multiply(X, Y), commutator(multiply(X, Y), multiply(X, Y))), commutator(Y, Y), commutator(multiply(X, Y), multiply(X, Y)))
% 1.62/0.60 = { by axiom 2 (nilpotency) R->L }
% 1.62/0.60 fresh2(multiply(commutator(Y, Y), multiply(X, Y)), multiply(commutator(multiply(X, Y), multiply(X, Y)), multiply(X, Y)), commutator(Y, Y), commutator(multiply(X, Y), multiply(X, Y)))
% 1.62/0.60 = { by axiom 7 (right_cancellation) }
% 1.62/0.60 commutator(Y, Y)
% 1.62/0.60
% 1.62/0.60 Lemma 12: fresh(multiply(X, Y), X, Y, commutator(X, X)) = Y.
% 1.62/0.60 Proof:
% 1.62/0.60 fresh(multiply(X, Y), X, Y, commutator(X, X))
% 1.62/0.60 = { by lemma 8 R->L }
% 1.62/0.60 fresh(multiply(X, Y), multiply(X, commutator(X, X)), Y, commutator(X, X))
% 1.62/0.60 = { by axiom 6 (left_cancellation) }
% 1.62/0.60 Y
% 1.62/0.60
% 1.62/0.60 Goal 1 (prove_commutator_is_associative): commutator(commutator(a, b), c) = commutator(a, commutator(b, c)).
% 1.62/0.60 Proof:
% 1.62/0.60 commutator(commutator(a, b), c)
% 1.62/0.60 = { by lemma 12 R->L }
% 1.62/0.60 fresh(multiply(c, commutator(commutator(a, b), c)), c, commutator(commutator(a, b), c), commutator(c, c))
% 1.62/0.60 = { by axiom 4 (right_cancellation) R->L }
% 1.62/0.60 fresh(fresh2(multiply(c, commutator(a, b)), multiply(c, commutator(a, b)), c, multiply(c, commutator(commutator(a, b), c))), c, commutator(commutator(a, b), c), commutator(c, c))
% 1.62/0.60 = { by axiom 2 (nilpotency) R->L }
% 1.62/0.60 fresh(fresh2(multiply(c, commutator(a, b)), multiply(commutator(a, b), c), c, multiply(c, commutator(commutator(a, b), c))), c, commutator(commutator(a, b), c), commutator(c, c))
% 1.62/0.60 = { by axiom 5 (commutator) }
% 1.62/0.60 fresh(fresh2(multiply(c, commutator(a, b)), multiply(c, multiply(commutator(a, b), commutator(commutator(a, b), c))), c, multiply(c, commutator(commutator(a, b), c))), c, commutator(commutator(a, b), c), commutator(c, c))
% 1.62/0.60 = { by axiom 2 (nilpotency) }
% 1.62/0.60 fresh(fresh2(multiply(c, commutator(a, b)), multiply(c, multiply(commutator(commutator(a, b), c), commutator(a, b))), c, multiply(c, commutator(commutator(a, b), c))), c, commutator(commutator(a, b), c), commutator(c, c))
% 1.62/0.60 = { by axiom 1 (associativity_of_multiply) R->L }
% 1.62/0.60 fresh(fresh2(multiply(c, commutator(a, b)), multiply(multiply(c, commutator(commutator(a, b), c)), commutator(a, b)), c, multiply(c, commutator(commutator(a, b), c))), c, commutator(commutator(a, b), c), commutator(c, c))
% 1.62/0.60 = { by axiom 7 (right_cancellation) }
% 1.62/0.60 fresh(c, c, commutator(commutator(a, b), c), commutator(c, c))
% 1.62/0.60 = { by axiom 3 (left_cancellation) }
% 1.62/0.60 commutator(c, c)
% 1.62/0.60 = { by lemma 11 R->L }
% 1.62/0.60 commutator(multiply(a, c), multiply(a, c))
% 1.62/0.60 = { by axiom 5 (commutator) }
% 1.62/0.60 commutator(multiply(a, c), multiply(c, multiply(a, commutator(a, c))))
% 1.62/0.60 = { by axiom 5 (commutator) }
% 1.62/0.60 commutator(multiply(c, multiply(a, commutator(a, c))), multiply(c, multiply(a, commutator(a, c))))
% 1.62/0.60 = { by lemma 11 }
% 1.62/0.60 commutator(multiply(a, commutator(a, c)), multiply(a, commutator(a, c)))
% 1.62/0.60 = { by axiom 5 (commutator) }
% 1.62/0.60 commutator(multiply(a, commutator(a, c)), multiply(commutator(a, c), multiply(a, commutator(a, commutator(a, c)))))
% 1.62/0.60 = { by axiom 5 (commutator) }
% 1.62/0.60 commutator(multiply(commutator(a, c), multiply(a, commutator(a, commutator(a, c)))), multiply(commutator(a, c), multiply(a, commutator(a, commutator(a, c)))))
% 1.62/0.60 = { by lemma 11 }
% 1.62/0.60 commutator(multiply(a, commutator(a, commutator(a, c))), multiply(a, commutator(a, commutator(a, c))))
% 1.62/0.60 = { by lemma 11 }
% 1.62/0.60 commutator(commutator(a, commutator(a, c)), commutator(a, commutator(a, c)))
% 1.62/0.60 = { by lemma 12 R->L }
% 1.62/0.60 fresh(multiply(a, commutator(commutator(a, commutator(a, c)), commutator(a, commutator(a, c)))), a, commutator(commutator(a, commutator(a, c)), commutator(a, commutator(a, c))), commutator(a, a))
% 1.62/0.60 = { by axiom 2 (nilpotency) R->L }
% 1.62/0.60 fresh(multiply(commutator(commutator(a, commutator(a, c)), commutator(a, commutator(a, c))), a), a, commutator(commutator(a, commutator(a, c)), commutator(a, commutator(a, c))), commutator(a, a))
% 1.62/0.60 = { by lemma 9 R->L }
% 1.62/0.60 fresh(multiply(commutator(commutator(a, commutator(a, c)), commutator(a, commutator(a, c))), multiply(a, commutator(a, commutator(a, c)))), a, commutator(commutator(a, commutator(a, c)), commutator(a, commutator(a, c))), commutator(a, a))
% 1.62/0.60 = { by lemma 10 }
% 1.62/0.60 fresh(multiply(a, commutator(a, commutator(a, c))), a, commutator(commutator(a, commutator(a, c)), commutator(a, commutator(a, c))), commutator(a, a))
% 1.62/0.60 = { by lemma 9 }
% 1.62/0.60 fresh(a, a, commutator(commutator(a, commutator(a, c)), commutator(a, commutator(a, c))), commutator(a, a))
% 1.62/0.60 = { by axiom 3 (left_cancellation) }
% 1.62/0.60 commutator(a, a)
% 1.62/0.60 = { by axiom 3 (left_cancellation) R->L }
% 1.62/0.60 fresh(a, a, commutator(a, commutator(b, c)), commutator(a, a))
% 1.62/0.60 = { by lemma 9 R->L }
% 1.62/0.60 fresh(multiply(a, commutator(a, commutator(b, c))), a, commutator(a, commutator(b, c)), commutator(a, a))
% 1.62/0.60 = { by lemma 12 }
% 1.62/0.60 commutator(a, commutator(b, c))
% 1.62/0.60 % SZS output end Proof
% 1.62/0.60
% 1.62/0.60 RESULT: Unsatisfiable (the axioms are contradictory).
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