TSTP Solution File: GRP067-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP067-1 : TPTP v8.1.2. Bugfixed v2.3.0.
% 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 : n031.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 01:16:51 EDT 2023
% Result : Unsatisfiable 0.21s 0.41s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : GRP067-1 : TPTP v8.1.2. Bugfixed v2.3.0.
% 0.13/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n031.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Tue Aug 29 00:16:10 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.21/0.41 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.21/0.41
% 0.21/0.41 % SZS status Unsatisfiable
% 0.21/0.41
% 0.21/0.43 % SZS output start Proof
% 0.21/0.43 Take the following subset of the input axioms:
% 0.21/0.43 fof(identity, axiom, ![X]: identity=divide(X, X)).
% 0.21/0.43 fof(inverse, axiom, ![X2]: inverse(X2)=divide(identity, X2)).
% 0.21/0.43 fof(multiply, axiom, ![Y, X2]: multiply(X2, Y)=divide(X2, divide(identity, Y))).
% 0.21/0.43 fof(prove_these_axioms, negated_conjecture, multiply(inverse(a1), a1)!=identity | (multiply(identity, a2)!=a2 | multiply(multiply(a3, b3), c3)!=multiply(a3, multiply(b3, c3)))).
% 0.21/0.43 fof(single_axiom, axiom, ![Z, X2, Y2]: divide(divide(divide(X2, X2), divide(X2, divide(Y2, divide(divide(identity, X2), Z)))), Z)=Y2).
% 0.21/0.44
% 0.21/0.44 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.44 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.44 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.44 fresh(y, y, x1...xn) = u
% 0.21/0.44 C => fresh(s, t, x1...xn) = v
% 0.21/0.44 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.44 variables of u and v.
% 0.21/0.44 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.44 input problem has no model of domain size 1).
% 0.21/0.44
% 0.21/0.44 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.44
% 0.21/0.44 Axiom 1 (identity): identity = divide(X, X).
% 0.21/0.44 Axiom 2 (inverse): inverse(X) = divide(identity, X).
% 0.21/0.44 Axiom 3 (multiply): multiply(X, Y) = divide(X, divide(identity, Y)).
% 0.21/0.44 Axiom 4 (single_axiom): divide(divide(divide(X, X), divide(X, divide(Y, divide(divide(identity, X), Z)))), Z) = Y.
% 0.21/0.44
% 0.21/0.44 Lemma 5: inverse(identity) = identity.
% 0.21/0.44 Proof:
% 0.21/0.44 inverse(identity)
% 0.21/0.44 = { by axiom 2 (inverse) }
% 0.21/0.44 divide(identity, identity)
% 0.21/0.44 = { by axiom 1 (identity) R->L }
% 0.21/0.44 identity
% 0.21/0.44
% 0.21/0.44 Lemma 6: divide(X, inverse(Y)) = multiply(X, Y).
% 0.21/0.44 Proof:
% 0.21/0.44 divide(X, inverse(Y))
% 0.21/0.44 = { by axiom 2 (inverse) }
% 0.21/0.44 divide(X, divide(identity, Y))
% 0.21/0.44 = { by axiom 3 (multiply) R->L }
% 0.21/0.44 multiply(X, Y)
% 0.21/0.44
% 0.21/0.44 Lemma 7: divide(X, identity) = multiply(X, identity).
% 0.21/0.44 Proof:
% 0.21/0.44 divide(X, identity)
% 0.21/0.44 = { by lemma 5 R->L }
% 0.21/0.44 divide(X, inverse(identity))
% 0.21/0.44 = { by lemma 6 }
% 0.21/0.44 multiply(X, identity)
% 0.21/0.44
% 0.21/0.44 Lemma 8: inverse(inverse(X)) = multiply(identity, X).
% 0.21/0.44 Proof:
% 0.21/0.44 inverse(inverse(X))
% 0.21/0.44 = { by axiom 2 (inverse) }
% 0.21/0.44 divide(identity, inverse(X))
% 0.21/0.44 = { by lemma 6 }
% 0.21/0.44 multiply(identity, X)
% 0.21/0.44
% 0.21/0.44 Lemma 9: divide(inverse(divide(X, divide(Y, divide(inverse(X), Z)))), Z) = Y.
% 0.21/0.44 Proof:
% 0.21/0.44 divide(inverse(divide(X, divide(Y, divide(inverse(X), Z)))), Z)
% 0.21/0.44 = { by axiom 2 (inverse) }
% 0.21/0.44 divide(inverse(divide(X, divide(Y, divide(divide(identity, X), Z)))), Z)
% 0.21/0.44 = { by axiom 2 (inverse) }
% 0.21/0.44 divide(divide(identity, divide(X, divide(Y, divide(divide(identity, X), Z)))), Z)
% 0.21/0.44 = { by axiom 1 (identity) }
% 0.21/0.44 divide(divide(divide(X, X), divide(X, divide(Y, divide(divide(identity, X), Z)))), Z)
% 0.21/0.44 = { by axiom 4 (single_axiom) }
% 0.21/0.44 Y
% 0.21/0.44
% 0.21/0.44 Lemma 10: divide(multiply(identity, multiply(X, Y)), Y) = X.
% 0.21/0.44 Proof:
% 0.21/0.44 divide(multiply(identity, multiply(X, Y)), Y)
% 0.21/0.44 = { by lemma 6 R->L }
% 0.21/0.44 divide(multiply(identity, divide(X, inverse(Y))), Y)
% 0.21/0.44 = { by axiom 2 (inverse) }
% 0.21/0.44 divide(multiply(identity, divide(X, divide(identity, Y))), Y)
% 0.21/0.44 = { by lemma 8 R->L }
% 0.21/0.44 divide(inverse(inverse(divide(X, divide(identity, Y)))), Y)
% 0.21/0.44 = { by axiom 2 (inverse) }
% 0.21/0.44 divide(inverse(divide(identity, divide(X, divide(identity, Y)))), Y)
% 0.21/0.44 = { by lemma 5 R->L }
% 0.21/0.44 divide(inverse(divide(identity, divide(X, divide(inverse(identity), Y)))), Y)
% 0.21/0.44 = { by lemma 9 }
% 0.21/0.44 X
% 0.21/0.44
% 0.21/0.44 Lemma 11: multiply(X, multiply(Y, identity)) = multiply(X, Y).
% 0.21/0.44 Proof:
% 0.21/0.44 multiply(X, multiply(Y, identity))
% 0.21/0.44 = { by lemma 6 R->L }
% 0.21/0.44 divide(X, inverse(multiply(Y, identity)))
% 0.21/0.44 = { by lemma 9 R->L }
% 0.21/0.44 divide(X, divide(inverse(divide(Z, divide(inverse(multiply(Y, identity)), divide(inverse(Z), W)))), W))
% 0.21/0.44 = { by lemma 7 R->L }
% 0.21/0.44 divide(X, divide(inverse(divide(Z, divide(inverse(divide(Y, identity)), divide(inverse(Z), W)))), W))
% 0.21/0.44 = { by axiom 1 (identity) }
% 0.21/0.44 divide(X, divide(inverse(divide(Z, divide(inverse(divide(Y, divide(divide(inverse(Y), divide(inverse(Z), W)), divide(inverse(Y), divide(inverse(Z), W))))), divide(inverse(Z), W)))), W))
% 0.21/0.44 = { by lemma 9 }
% 0.21/0.44 divide(X, divide(inverse(divide(Z, divide(inverse(Y), divide(inverse(Z), W)))), W))
% 0.21/0.44 = { by lemma 9 }
% 0.21/0.44 divide(X, inverse(Y))
% 0.21/0.44 = { by lemma 6 }
% 0.21/0.44 multiply(X, Y)
% 0.21/0.44
% 0.21/0.44 Lemma 12: multiply(multiply(identity, X), identity) = X.
% 0.21/0.44 Proof:
% 0.21/0.44 multiply(multiply(identity, X), identity)
% 0.21/0.44 = { by lemma 11 R->L }
% 0.21/0.44 multiply(multiply(identity, multiply(X, identity)), identity)
% 0.21/0.44 = { by lemma 7 R->L }
% 0.21/0.44 divide(multiply(identity, multiply(X, identity)), identity)
% 0.21/0.44 = { by lemma 10 }
% 0.21/0.44 X
% 0.21/0.44
% 0.21/0.44 Lemma 13: multiply(identity, X) = X.
% 0.21/0.44 Proof:
% 0.21/0.44 multiply(identity, X)
% 0.21/0.44 = { by lemma 10 R->L }
% 0.21/0.44 divide(multiply(identity, multiply(multiply(identity, X), identity)), identity)
% 0.21/0.44 = { by lemma 12 }
% 0.21/0.44 divide(multiply(identity, X), identity)
% 0.21/0.44 = { by lemma 7 }
% 0.21/0.44 multiply(multiply(identity, X), identity)
% 0.21/0.44 = { by lemma 12 }
% 0.21/0.44 X
% 0.21/0.44
% 0.21/0.44 Lemma 14: multiply(X, inverse(Y)) = divide(X, Y).
% 0.21/0.44 Proof:
% 0.21/0.44 multiply(X, inverse(Y))
% 0.21/0.44 = { by lemma 6 R->L }
% 0.21/0.44 divide(X, inverse(inverse(Y)))
% 0.21/0.44 = { by lemma 8 }
% 0.21/0.44 divide(X, multiply(identity, Y))
% 0.21/0.44 = { by lemma 13 }
% 0.21/0.44 divide(X, Y)
% 0.21/0.44
% 0.21/0.44 Lemma 15: divide(multiply(X, Y), Y) = X.
% 0.21/0.44 Proof:
% 0.21/0.44 divide(multiply(X, Y), Y)
% 0.21/0.44 = { by lemma 13 R->L }
% 0.21/0.44 divide(multiply(identity, multiply(X, Y)), Y)
% 0.21/0.44 = { by lemma 10 }
% 0.21/0.44 X
% 0.21/0.44
% 0.21/0.44 Lemma 16: divide(inverse(divide(X, Y)), Z) = multiply(Y, divide(inverse(X), Z)).
% 0.21/0.44 Proof:
% 0.21/0.44 divide(inverse(divide(X, Y)), Z)
% 0.21/0.44 = { by lemma 15 R->L }
% 0.21/0.44 divide(inverse(divide(X, divide(multiply(Y, divide(inverse(X), Z)), divide(inverse(X), Z)))), Z)
% 0.21/0.44 = { by lemma 9 }
% 0.21/0.44 multiply(Y, divide(inverse(X), Z))
% 0.21/0.44
% 0.21/0.44 Lemma 17: inverse(divide(X, Y)) = divide(Y, X).
% 0.21/0.44 Proof:
% 0.21/0.44 inverse(divide(X, Y))
% 0.21/0.44 = { by lemma 12 R->L }
% 0.21/0.44 multiply(multiply(identity, inverse(divide(X, Y))), identity)
% 0.21/0.44 = { by lemma 11 R->L }
% 0.21/0.44 multiply(multiply(identity, multiply(inverse(divide(X, Y)), identity)), identity)
% 0.21/0.44 = { by lemma 12 }
% 0.21/0.44 multiply(inverse(divide(X, Y)), identity)
% 0.21/0.44 = { by lemma 7 R->L }
% 0.21/0.44 divide(inverse(divide(X, Y)), identity)
% 0.21/0.44 = { by lemma 16 }
% 0.21/0.44 multiply(Y, divide(inverse(X), identity))
% 0.21/0.44 = { by lemma 7 }
% 0.21/0.44 multiply(Y, multiply(inverse(X), identity))
% 0.21/0.44 = { by lemma 11 }
% 0.21/0.44 multiply(Y, inverse(X))
% 0.21/0.44 = { by lemma 14 }
% 0.21/0.44 divide(Y, X)
% 0.21/0.44
% 0.21/0.44 Lemma 18: divide(X, multiply(Y, X)) = inverse(Y).
% 0.21/0.44 Proof:
% 0.21/0.44 divide(X, multiply(Y, X))
% 0.21/0.44 = { by lemma 17 R->L }
% 0.21/0.44 inverse(divide(multiply(Y, X), X))
% 0.21/0.44 = { by lemma 15 }
% 0.21/0.44 inverse(Y)
% 0.21/0.44
% 0.21/0.44 Goal 1 (prove_these_axioms): tuple(multiply(inverse(a1), a1), multiply(identity, a2), multiply(multiply(a3, b3), c3)) = tuple(identity, a2, multiply(a3, multiply(b3, c3))).
% 0.21/0.44 Proof:
% 0.21/0.44 tuple(multiply(inverse(a1), a1), multiply(identity, a2), multiply(multiply(a3, b3), c3))
% 0.21/0.44 = { by lemma 13 }
% 0.21/0.44 tuple(multiply(inverse(a1), a1), a2, multiply(multiply(a3, b3), c3))
% 0.21/0.44 = { by lemma 6 R->L }
% 0.21/0.44 tuple(divide(inverse(a1), inverse(a1)), a2, multiply(multiply(a3, b3), c3))
% 0.21/0.44 = { by axiom 1 (identity) R->L }
% 0.21/0.44 tuple(identity, a2, multiply(multiply(a3, b3), c3))
% 0.21/0.44 = { by lemma 10 R->L }
% 0.21/0.44 tuple(identity, a2, divide(multiply(identity, multiply(multiply(multiply(a3, b3), c3), inverse(multiply(b3, c3)))), inverse(multiply(b3, c3))))
% 0.21/0.44 = { by lemma 14 }
% 0.21/0.44 tuple(identity, a2, divide(multiply(identity, divide(multiply(multiply(a3, b3), c3), multiply(b3, c3))), inverse(multiply(b3, c3))))
% 0.21/0.44 = { by lemma 6 }
% 0.21/0.44 tuple(identity, a2, multiply(multiply(identity, divide(multiply(multiply(a3, b3), c3), multiply(b3, c3))), multiply(b3, c3)))
% 0.21/0.44 = { by lemma 13 }
% 0.21/0.44 tuple(identity, a2, multiply(divide(multiply(multiply(a3, b3), c3), multiply(b3, c3)), multiply(b3, c3)))
% 0.21/0.44 = { by lemma 17 R->L }
% 0.21/0.44 tuple(identity, a2, multiply(inverse(divide(multiply(b3, c3), multiply(multiply(a3, b3), c3))), multiply(b3, c3)))
% 0.21/0.44 = { by lemma 13 R->L }
% 0.21/0.44 tuple(identity, a2, multiply(inverse(divide(multiply(identity, multiply(b3, c3)), multiply(multiply(a3, b3), c3))), multiply(b3, c3)))
% 0.21/0.44 = { by lemma 8 R->L }
% 0.21/0.44 tuple(identity, a2, multiply(inverse(divide(inverse(inverse(multiply(b3, c3))), multiply(multiply(a3, b3), c3))), multiply(b3, c3)))
% 0.21/0.44 = { by lemma 6 R->L }
% 0.21/0.44 tuple(identity, a2, multiply(inverse(divide(inverse(inverse(divide(b3, inverse(c3)))), multiply(multiply(a3, b3), c3))), multiply(b3, c3)))
% 0.21/0.44 = { by lemma 17 }
% 0.21/0.44 tuple(identity, a2, multiply(inverse(divide(inverse(divide(inverse(c3), b3)), multiply(multiply(a3, b3), c3))), multiply(b3, c3)))
% 0.21/0.44 = { by lemma 16 }
% 0.21/0.44 tuple(identity, a2, multiply(inverse(multiply(b3, divide(inverse(inverse(c3)), multiply(multiply(a3, b3), c3)))), multiply(b3, c3)))
% 0.21/0.44 = { by lemma 8 }
% 0.21/0.44 tuple(identity, a2, multiply(inverse(multiply(b3, divide(multiply(identity, c3), multiply(multiply(a3, b3), c3)))), multiply(b3, c3)))
% 0.21/0.44 = { by lemma 13 }
% 0.21/0.44 tuple(identity, a2, multiply(inverse(multiply(b3, divide(c3, multiply(multiply(a3, b3), c3)))), multiply(b3, c3)))
% 0.21/0.44 = { by lemma 18 }
% 0.21/0.44 tuple(identity, a2, multiply(inverse(multiply(b3, inverse(multiply(a3, b3)))), multiply(b3, c3)))
% 0.21/0.44 = { by lemma 14 }
% 0.21/0.44 tuple(identity, a2, multiply(inverse(divide(b3, multiply(a3, b3))), multiply(b3, c3)))
% 0.21/0.44 = { by lemma 18 }
% 0.21/0.44 tuple(identity, a2, multiply(inverse(inverse(a3)), multiply(b3, c3)))
% 0.21/0.45 = { by lemma 8 }
% 0.21/0.45 tuple(identity, a2, multiply(multiply(identity, a3), multiply(b3, c3)))
% 0.21/0.45 = { by lemma 13 }
% 0.21/0.45 tuple(identity, a2, multiply(a3, multiply(b3, c3)))
% 0.21/0.45 % SZS output end Proof
% 0.21/0.45
% 0.21/0.45 RESULT: Unsatisfiable (the axioms are contradictory).
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