TSTP Solution File: GRP063-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP063-1 : 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 01:16:50 EDT 2023
% Result : Unsatisfiable 0.13s 0.41s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.13 % Problem : GRP063-1 : TPTP v8.1.2. Bugfixed v2.3.0.
% 0.12/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n031.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Mon Aug 28 20:30:25 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.13/0.41 Command-line arguments: --no-flatten-goal
% 0.13/0.41
% 0.13/0.41 % SZS status Unsatisfiable
% 0.13/0.41
% 0.21/0.44 % SZS output start Proof
% 0.21/0.44 Take the following subset of the input axioms:
% 0.21/0.44 fof(inverse, axiom, ![X, Z]: inverse(X)=divide(divide(Z, Z), X)).
% 0.21/0.44 fof(multiply, axiom, ![Y, X2, Z2]: multiply(X2, Y)=divide(X2, divide(divide(Z2, Z2), Y))).
% 0.21/0.44 fof(prove_these_axioms, negated_conjecture, multiply(inverse(a1), a1)!=multiply(inverse(b1), b1) | (multiply(multiply(inverse(b2), b2), a2)!=a2 | multiply(multiply(a3, b3), c3)!=multiply(a3, multiply(b3, c3)))).
% 0.21/0.44 fof(single_axiom, axiom, ![X2, Y2, Z2]: divide(X2, divide(divide(divide(divide(X2, X2), Y2), Z2), divide(divide(divide(X2, X2), X2), Z2)))=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 (inverse): inverse(X) = divide(divide(Y, Y), X).
% 0.21/0.44 Axiom 2 (multiply): multiply(X, Y) = divide(X, divide(divide(Z, Z), Y)).
% 0.21/0.44 Axiom 3 (single_axiom): divide(X, divide(divide(divide(divide(X, X), Y), Z), divide(divide(divide(X, X), X), Z))) = Y.
% 0.21/0.44
% 0.21/0.44 Lemma 4: 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 1 (inverse) }
% 0.21/0.44 divide(X, divide(divide(Z, Z), Y))
% 0.21/0.44 = { by axiom 2 (multiply) R->L }
% 0.21/0.44 multiply(X, Y)
% 0.21/0.44
% 0.21/0.44 Lemma 5: divide(inverse(divide(X, X)), Y) = inverse(Y).
% 0.21/0.44 Proof:
% 0.21/0.44 divide(inverse(divide(X, X)), Y)
% 0.21/0.44 = { by axiom 1 (inverse) }
% 0.21/0.44 divide(divide(divide(X, X), divide(X, X)), Y)
% 0.21/0.44 = { by axiom 1 (inverse) R->L }
% 0.21/0.44 inverse(Y)
% 0.21/0.44
% 0.21/0.44 Lemma 6: divide(X, divide(divide(inverse(Y), Z), divide(inverse(X), Z))) = Y.
% 0.21/0.44 Proof:
% 0.21/0.44 divide(X, divide(divide(inverse(Y), Z), divide(inverse(X), Z)))
% 0.21/0.44 = { by axiom 1 (inverse) }
% 0.21/0.44 divide(X, divide(divide(inverse(Y), Z), divide(divide(divide(X, X), X), Z)))
% 0.21/0.44 = { by axiom 1 (inverse) }
% 0.21/0.44 divide(X, divide(divide(divide(divide(X, X), Y), Z), divide(divide(divide(X, X), X), Z)))
% 0.21/0.44 = { by axiom 3 (single_axiom) }
% 0.21/0.44 Y
% 0.21/0.44
% 0.21/0.44 Lemma 7: inverse(multiply(divide(inverse(X), Y), Y)) = X.
% 0.21/0.44 Proof:
% 0.21/0.44 inverse(multiply(divide(inverse(X), Y), Y))
% 0.21/0.44 = { by lemma 4 R->L }
% 0.21/0.44 inverse(divide(divide(inverse(X), Y), inverse(Y)))
% 0.21/0.44 = { by axiom 1 (inverse) }
% 0.21/0.44 divide(divide(Z, Z), divide(divide(inverse(X), Y), inverse(Y)))
% 0.21/0.44 = { by lemma 5 R->L }
% 0.21/0.44 divide(divide(Z, Z), divide(divide(inverse(X), Y), divide(inverse(divide(Z, Z)), Y)))
% 0.21/0.44 = { by lemma 6 }
% 0.21/0.44 X
% 0.21/0.44
% 0.21/0.44 Lemma 8: inverse(multiply(inverse(X), X)) = divide(Y, Y).
% 0.21/0.44 Proof:
% 0.21/0.44 inverse(multiply(inverse(X), X))
% 0.21/0.44 = { by lemma 5 R->L }
% 0.21/0.44 inverse(multiply(divide(inverse(divide(Y, Y)), X), X))
% 0.21/0.44 = { by lemma 7 }
% 0.21/0.44 divide(Y, Y)
% 0.21/0.44
% 0.21/0.44 Lemma 9: multiply(inverse(X), X) = divide(Y, Y).
% 0.21/0.44 Proof:
% 0.21/0.44 multiply(inverse(X), X)
% 0.21/0.44 = { by lemma 7 R->L }
% 0.21/0.44 inverse(multiply(divide(inverse(multiply(inverse(X), X)), Z), Z))
% 0.21/0.44 = { by lemma 8 }
% 0.21/0.44 inverse(multiply(divide(divide(W, W), Z), Z))
% 0.21/0.44 = { by axiom 1 (inverse) R->L }
% 0.21/0.44 inverse(multiply(inverse(Z), Z))
% 0.21/0.44 = { by lemma 8 }
% 0.21/0.44 divide(Y, Y)
% 0.21/0.44
% 0.21/0.44 Lemma 10: multiply(X, multiply(inverse(X), Y)) = Y.
% 0.21/0.44 Proof:
% 0.21/0.44 multiply(X, multiply(inverse(X), Y))
% 0.21/0.44 = { by lemma 4 R->L }
% 0.21/0.44 multiply(X, divide(inverse(X), inverse(Y)))
% 0.21/0.44 = { by lemma 4 R->L }
% 0.21/0.44 divide(X, inverse(divide(inverse(X), inverse(Y))))
% 0.21/0.44 = { by axiom 1 (inverse) }
% 0.21/0.44 divide(X, divide(divide(inverse(Y), inverse(Y)), divide(inverse(X), inverse(Y))))
% 0.21/0.44 = { by lemma 6 }
% 0.21/0.44 Y
% 0.21/0.44
% 0.21/0.44 Lemma 11: multiply(X, divide(Y, Y)) = X.
% 0.21/0.44 Proof:
% 0.21/0.44 multiply(X, divide(Y, Y))
% 0.21/0.44 = { by lemma 9 R->L }
% 0.21/0.44 multiply(X, multiply(inverse(X), X))
% 0.21/0.44 = { by lemma 10 }
% 0.21/0.44 X
% 0.21/0.44
% 0.21/0.44 Lemma 12: multiply(inverse(inverse(X)), Y) = multiply(X, Y).
% 0.21/0.44 Proof:
% 0.21/0.44 multiply(inverse(inverse(X)), Y)
% 0.21/0.44 = { by lemma 10 R->L }
% 0.21/0.44 multiply(X, multiply(inverse(X), multiply(inverse(inverse(X)), Y)))
% 0.21/0.44 = { by lemma 10 }
% 0.21/0.44 multiply(X, Y)
% 0.21/0.44
% 0.21/0.44 Lemma 13: inverse(inverse(X)) = X.
% 0.21/0.44 Proof:
% 0.21/0.44 inverse(inverse(X))
% 0.21/0.44 = { by lemma 11 R->L }
% 0.21/0.44 multiply(inverse(inverse(X)), divide(Y, Y))
% 0.21/0.44 = { by lemma 12 }
% 0.21/0.44 multiply(X, divide(Y, Y))
% 0.21/0.44 = { by lemma 11 }
% 0.21/0.44 X
% 0.21/0.44
% 0.21/0.44 Lemma 14: divide(X, divide(Y, Y)) = X.
% 0.21/0.44 Proof:
% 0.21/0.44 divide(X, divide(Y, Y))
% 0.21/0.44 = { by lemma 7 R->L }
% 0.21/0.44 divide(X, inverse(multiply(divide(inverse(divide(Y, Y)), Z), Z)))
% 0.21/0.44 = { by lemma 5 }
% 0.21/0.44 divide(X, inverse(multiply(inverse(Z), Z)))
% 0.21/0.44 = { by lemma 5 R->L }
% 0.21/0.44 divide(X, inverse(multiply(divide(inverse(divide(divide(inverse(X), W), divide(inverse(X), W))), Z), Z)))
% 0.21/0.44 = { by lemma 7 }
% 0.21/0.44 divide(X, divide(divide(inverse(X), W), divide(inverse(X), W)))
% 0.21/0.44 = { by lemma 6 }
% 0.21/0.44 X
% 0.21/0.44
% 0.21/0.44 Lemma 15: multiply(divide(X, X), Y) = inverse(inverse(Y)).
% 0.21/0.44 Proof:
% 0.21/0.44 multiply(divide(X, X), Y)
% 0.21/0.44 = { by lemma 4 R->L }
% 0.21/0.44 divide(divide(X, X), inverse(Y))
% 0.21/0.44 = { by axiom 1 (inverse) R->L }
% 0.21/0.45 inverse(inverse(Y))
% 0.21/0.45
% 0.21/0.45 Lemma 16: multiply(divide(inverse(X), Y), Y) = inverse(multiply(divide(X, Z), Z)).
% 0.21/0.45 Proof:
% 0.21/0.45 multiply(divide(inverse(X), Y), Y)
% 0.21/0.45 = { by lemma 7 R->L }
% 0.21/0.45 inverse(multiply(divide(inverse(multiply(divide(inverse(X), Y), Y)), Z), Z))
% 0.21/0.45 = { by lemma 7 }
% 0.21/0.45 inverse(multiply(divide(X, Z), Z))
% 0.21/0.45
% 0.21/0.45 Lemma 17: multiply(divide(X, Y), Y) = X.
% 0.21/0.45 Proof:
% 0.21/0.45 multiply(divide(X, Y), Y)
% 0.21/0.45 = { by lemma 10 R->L }
% 0.21/0.45 multiply(divide(multiply(inverse(divide(Z, Z)), multiply(inverse(inverse(divide(Z, Z))), X)), Y), Y)
% 0.21/0.45 = { by axiom 1 (inverse) }
% 0.21/0.45 multiply(divide(multiply(divide(divide(Z, Z), divide(Z, Z)), multiply(inverse(inverse(divide(Z, Z))), X)), Y), Y)
% 0.21/0.45 = { by lemma 15 }
% 0.21/0.45 multiply(divide(inverse(inverse(multiply(inverse(inverse(divide(Z, Z))), X))), Y), Y)
% 0.21/0.45 = { by lemma 12 }
% 0.21/0.45 multiply(divide(inverse(inverse(multiply(divide(Z, Z), X))), Y), Y)
% 0.21/0.45 = { by lemma 15 }
% 0.21/0.45 multiply(divide(inverse(inverse(inverse(inverse(X)))), Y), Y)
% 0.21/0.45 = { by lemma 16 }
% 0.21/0.45 inverse(multiply(divide(inverse(inverse(inverse(X))), W), W))
% 0.21/0.45 = { by lemma 7 }
% 0.21/0.45 inverse(inverse(X))
% 0.21/0.45 = { by lemma 13 }
% 0.21/0.45 X
% 0.21/0.45
% 0.21/0.45 Lemma 18: inverse(divide(X, Y)) = divide(Y, X).
% 0.21/0.45 Proof:
% 0.21/0.45 inverse(divide(X, Y))
% 0.21/0.45 = { by lemma 6 R->L }
% 0.21/0.45 divide(Y, divide(divide(inverse(inverse(divide(X, Y))), divide(Z, Z)), divide(inverse(Y), divide(Z, Z))))
% 0.21/0.45 = { by lemma 14 }
% 0.21/0.45 divide(Y, divide(inverse(inverse(divide(X, Y))), divide(inverse(Y), divide(Z, Z))))
% 0.21/0.45 = { by lemma 14 }
% 0.21/0.45 divide(Y, divide(inverse(inverse(divide(X, Y))), inverse(Y)))
% 0.21/0.45 = { by lemma 4 }
% 0.21/0.45 divide(Y, multiply(inverse(inverse(divide(X, Y))), Y))
% 0.21/0.45 = { by lemma 12 }
% 0.21/0.45 divide(Y, multiply(divide(X, Y), Y))
% 0.21/0.45 = { by lemma 17 }
% 0.21/0.45 divide(Y, X)
% 0.21/0.45
% 0.21/0.45 Lemma 19: divide(inverse(X), Y) = inverse(multiply(Y, X)).
% 0.21/0.45 Proof:
% 0.21/0.45 divide(inverse(X), Y)
% 0.21/0.45 = { by lemma 18 R->L }
% 0.21/0.45 inverse(divide(Y, inverse(X)))
% 0.21/0.45 = { by lemma 4 }
% 0.21/0.45 inverse(multiply(Y, X))
% 0.21/0.45
% 0.21/0.45 Lemma 20: multiply(X, inverse(Y)) = divide(X, Y).
% 0.21/0.45 Proof:
% 0.21/0.45 multiply(X, inverse(Y))
% 0.21/0.45 = { by lemma 4 R->L }
% 0.21/0.45 divide(X, inverse(inverse(Y)))
% 0.21/0.45 = { by lemma 13 }
% 0.21/0.45 divide(X, Y)
% 0.21/0.45
% 0.21/0.45 Lemma 21: divide(X, divide(Y, Z)) = multiply(X, divide(Z, Y)).
% 0.21/0.45 Proof:
% 0.21/0.45 divide(X, divide(Y, Z))
% 0.21/0.45 = { by lemma 18 R->L }
% 0.21/0.45 divide(X, inverse(divide(Z, Y)))
% 0.21/0.45 = { by lemma 4 }
% 0.21/0.45 multiply(X, divide(Z, Y))
% 0.21/0.45
% 0.21/0.45 Goal 1 (prove_these_axioms): tuple(multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1)) = tuple(a2, multiply(a3, multiply(b3, c3)), multiply(inverse(b1), b1)).
% 0.21/0.45 Proof:
% 0.21/0.45 tuple(multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1))
% 0.21/0.45 = { by lemma 4 R->L }
% 0.21/0.45 tuple(multiply(divide(inverse(b2), inverse(b2)), a2), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1))
% 0.21/0.45 = { by lemma 15 }
% 0.21/0.45 tuple(inverse(inverse(a2)), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1))
% 0.21/0.45 = { by lemma 9 }
% 0.21/0.45 tuple(inverse(inverse(a2)), multiply(multiply(a3, b3), c3), divide(X, X))
% 0.21/0.45 = { by lemma 13 }
% 0.21/0.45 tuple(a2, multiply(multiply(a3, b3), c3), divide(X, X))
% 0.21/0.45 = { by lemma 4 R->L }
% 0.21/0.45 tuple(a2, divide(multiply(a3, b3), inverse(c3)), divide(X, X))
% 0.21/0.45 = { by lemma 18 R->L }
% 0.21/0.45 tuple(a2, inverse(divide(inverse(c3), multiply(a3, b3))), divide(X, X))
% 0.21/0.45 = { by lemma 20 R->L }
% 0.21/0.45 tuple(a2, inverse(multiply(inverse(c3), inverse(multiply(a3, b3)))), divide(X, X))
% 0.21/0.45 = { by lemma 19 R->L }
% 0.21/0.45 tuple(a2, inverse(multiply(inverse(c3), divide(inverse(b3), a3))), divide(X, X))
% 0.21/0.45 = { by lemma 19 R->L }
% 0.21/0.45 tuple(a2, divide(inverse(divide(inverse(b3), a3)), inverse(c3)), divide(X, X))
% 0.21/0.45 = { by lemma 6 R->L }
% 0.21/0.45 tuple(a2, divide(a3, divide(divide(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3))), inverse(multiply(inverse(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3)))), inverse(c3)))), divide(inverse(a3), inverse(multiply(inverse(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3)))), inverse(c3)))))), divide(X, X))
% 0.21/0.45 = { by lemma 4 }
% 0.21/0.45 tuple(a2, divide(a3, divide(multiply(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3))), multiply(inverse(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3)))), inverse(c3))), divide(inverse(a3), inverse(multiply(inverse(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3)))), inverse(c3)))))), divide(X, X))
% 0.21/0.45 = { by lemma 21 }
% 0.21/0.45 tuple(a2, multiply(a3, divide(divide(inverse(a3), inverse(multiply(inverse(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3)))), inverse(c3)))), multiply(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3))), multiply(inverse(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3)))), inverse(c3))))), divide(X, X))
% 0.21/0.45 = { by lemma 19 }
% 0.21/0.45 tuple(a2, multiply(a3, divide(inverse(multiply(inverse(multiply(inverse(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3)))), inverse(c3))), a3)), multiply(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3))), multiply(inverse(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3)))), inverse(c3))))), divide(X, X))
% 0.21/0.45 = { by lemma 19 }
% 0.21/0.45 tuple(a2, multiply(a3, inverse(multiply(multiply(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3))), multiply(inverse(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3)))), inverse(c3))), multiply(inverse(multiply(inverse(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3)))), inverse(c3))), a3)))), divide(X, X))
% 0.21/0.45 = { by lemma 20 }
% 0.21/0.45 tuple(a2, divide(a3, multiply(multiply(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3))), multiply(inverse(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3)))), inverse(c3))), multiply(inverse(multiply(inverse(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3)))), inverse(c3))), a3))), divide(X, X))
% 0.21/0.45 = { by lemma 10 }
% 0.21/0.45 tuple(a2, divide(a3, multiply(inverse(c3), multiply(inverse(multiply(inverse(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3)))), inverse(c3))), a3))), divide(X, X))
% 0.21/0.45 = { by lemma 19 R->L }
% 0.21/0.45 tuple(a2, divide(a3, multiply(inverse(c3), multiply(divide(inverse(inverse(c3)), inverse(inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3))))), a3))), divide(X, X))
% 0.21/0.45 = { by lemma 4 }
% 0.21/0.45 tuple(a2, divide(a3, multiply(inverse(c3), multiply(multiply(inverse(inverse(c3)), inverse(divide(inverse(divide(inverse(b3), a3)), inverse(c3)))), a3))), divide(X, X))
% 0.21/0.45 = { by lemma 20 }
% 0.21/0.45 tuple(a2, divide(a3, multiply(inverse(c3), multiply(divide(inverse(inverse(c3)), divide(inverse(divide(inverse(b3), a3)), inverse(c3))), a3))), divide(X, X))
% 0.21/0.45 = { by lemma 19 }
% 0.21/0.45 tuple(a2, divide(a3, multiply(inverse(c3), multiply(inverse(multiply(divide(inverse(divide(inverse(b3), a3)), inverse(c3)), inverse(c3))), a3))), divide(X, X))
% 0.21/0.45 = { by lemma 7 }
% 0.21/0.46 tuple(a2, divide(a3, multiply(inverse(c3), multiply(divide(inverse(b3), a3), a3))), divide(X, X))
% 0.21/0.46 = { by lemma 16 }
% 0.21/0.46 tuple(a2, divide(a3, multiply(inverse(c3), inverse(multiply(divide(b3, Y), Y)))), divide(X, X))
% 0.21/0.46 = { by lemma 20 }
% 0.21/0.46 tuple(a2, divide(a3, divide(inverse(c3), multiply(divide(b3, Y), Y))), divide(X, X))
% 0.21/0.46 = { by lemma 21 }
% 0.21/0.46 tuple(a2, multiply(a3, divide(multiply(divide(b3, Y), Y), inverse(c3))), divide(X, X))
% 0.21/0.46 = { by lemma 17 }
% 0.21/0.46 tuple(a2, multiply(a3, divide(b3, inverse(c3))), divide(X, X))
% 0.21/0.46 = { by lemma 4 }
% 0.21/0.46 tuple(a2, multiply(a3, multiply(b3, c3)), divide(X, X))
% 0.21/0.46 = { by lemma 9 R->L }
% 0.21/0.46 tuple(a2, multiply(a3, multiply(b3, c3)), multiply(inverse(b1), b1))
% 0.21/0.46 % SZS output end Proof
% 0.21/0.46
% 0.21/0.46 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------