TSTP Solution File: BOO008-4 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : BOO008-4 : TPTP v8.1.2. Released v1.1.0.
% 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 : n002.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 : Wed Aug 30 18:11:19 EDT 2023
% Result : Unsatisfiable 0.21s 0.46s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : BOO008-4 : TPTP v8.1.2. Released v1.1.0.
% 0.07/0.14 % 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 : n002.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 : Sun Aug 27 08:07:33 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.21/0.46 Command-line arguments: --flatten
% 0.21/0.46
% 0.21/0.46 % SZS status Unsatisfiable
% 0.21/0.46
% 0.21/0.49 % SZS output start Proof
% 0.21/0.49 Axiom 1 (commutativity_of_multiply): multiply(X, Y) = multiply(Y, X).
% 0.21/0.49 Axiom 2 (multiplicative_id1): multiply(X, multiplicative_identity) = X.
% 0.21/0.49 Axiom 3 (commutativity_of_add): add(X, Y) = add(Y, X).
% 0.21/0.49 Axiom 4 (additive_id1): add(X, additive_identity) = X.
% 0.21/0.49 Axiom 5 (multiplicative_inverse1): multiply(X, inverse(X)) = additive_identity.
% 0.21/0.49 Axiom 6 (additive_inverse1): add(X, inverse(X)) = multiplicative_identity.
% 0.21/0.49 Axiom 7 (distributivity1): add(X, multiply(Y, Z)) = multiply(add(X, Y), add(X, Z)).
% 0.21/0.49 Axiom 8 (distributivity2): multiply(X, add(Y, Z)) = add(multiply(X, Y), multiply(X, Z)).
% 0.21/0.49
% 0.21/0.49 Lemma 9: add(X, multiplicative_identity) = multiplicative_identity.
% 0.21/0.49 Proof:
% 0.21/0.49 add(X, multiplicative_identity)
% 0.21/0.49 = { by axiom 2 (multiplicative_id1) R->L }
% 0.21/0.49 multiply(add(X, multiplicative_identity), multiplicative_identity)
% 0.21/0.49 = { by axiom 6 (additive_inverse1) R->L }
% 0.21/0.49 multiply(add(X, multiplicative_identity), add(X, inverse(X)))
% 0.21/0.49 = { by axiom 7 (distributivity1) R->L }
% 0.21/0.49 add(X, multiply(multiplicative_identity, inverse(X)))
% 0.21/0.49 = { by axiom 1 (commutativity_of_multiply) R->L }
% 0.21/0.49 add(X, multiply(inverse(X), multiplicative_identity))
% 0.21/0.49 = { by axiom 2 (multiplicative_id1) }
% 0.21/0.49 add(X, inverse(X))
% 0.21/0.49 = { by axiom 6 (additive_inverse1) }
% 0.21/0.49 multiplicative_identity
% 0.21/0.49
% 0.21/0.49 Lemma 10: add(additive_identity, X) = X.
% 0.21/0.49 Proof:
% 0.21/0.49 add(additive_identity, X)
% 0.21/0.49 = { by axiom 3 (commutativity_of_add) R->L }
% 0.21/0.49 add(X, additive_identity)
% 0.21/0.49 = { by axiom 4 (additive_id1) }
% 0.21/0.49 X
% 0.21/0.49
% 0.21/0.49 Lemma 11: add(X, multiply(X, Y)) = multiply(X, add(Y, multiplicative_identity)).
% 0.21/0.49 Proof:
% 0.21/0.49 add(X, multiply(X, Y))
% 0.21/0.49 = { by axiom 2 (multiplicative_id1) R->L }
% 0.21/0.49 add(multiply(X, multiplicative_identity), multiply(X, Y))
% 0.21/0.49 = { by axiom 8 (distributivity2) R->L }
% 0.21/0.49 multiply(X, add(multiplicative_identity, Y))
% 0.21/0.49 = { by axiom 3 (commutativity_of_add) }
% 0.21/0.49 multiply(X, add(Y, multiplicative_identity))
% 0.21/0.49
% 0.21/0.49 Lemma 12: multiply(additive_identity, X) = additive_identity.
% 0.21/0.49 Proof:
% 0.21/0.49 multiply(additive_identity, X)
% 0.21/0.49 = { by lemma 10 R->L }
% 0.21/0.49 add(additive_identity, multiply(additive_identity, X))
% 0.21/0.49 = { by lemma 11 }
% 0.21/0.49 multiply(additive_identity, add(X, multiplicative_identity))
% 0.21/0.49 = { by lemma 9 }
% 0.21/0.49 multiply(additive_identity, multiplicative_identity)
% 0.21/0.49 = { by axiom 2 (multiplicative_id1) }
% 0.21/0.49 additive_identity
% 0.21/0.49
% 0.21/0.49 Lemma 13: multiply(X, add(Y, X)) = X.
% 0.21/0.49 Proof:
% 0.21/0.49 multiply(X, add(Y, X))
% 0.21/0.49 = { by axiom 3 (commutativity_of_add) R->L }
% 0.21/0.49 multiply(X, add(X, Y))
% 0.21/0.49 = { by axiom 4 (additive_id1) R->L }
% 0.21/0.49 multiply(add(X, additive_identity), add(X, Y))
% 0.21/0.49 = { by axiom 7 (distributivity1) R->L }
% 0.21/0.49 add(X, multiply(additive_identity, Y))
% 0.21/0.49 = { by lemma 12 }
% 0.21/0.49 add(X, additive_identity)
% 0.21/0.49 = { by axiom 4 (additive_id1) }
% 0.21/0.49 X
% 0.21/0.49
% 0.21/0.49 Lemma 14: multiply(add(X, Z), add(Y, X)) = add(X, multiply(Y, Z)).
% 0.21/0.49 Proof:
% 0.21/0.49 multiply(add(X, Z), add(Y, X))
% 0.21/0.49 = { by axiom 3 (commutativity_of_add) R->L }
% 0.21/0.49 multiply(add(X, Z), add(X, Y))
% 0.21/0.49 = { by axiom 7 (distributivity1) R->L }
% 0.21/0.49 add(X, multiply(Z, Y))
% 0.21/0.49 = { by axiom 1 (commutativity_of_multiply) }
% 0.21/0.49 add(X, multiply(Y, Z))
% 0.21/0.49
% 0.21/0.49 Lemma 15: multiply(add(X, Y), X) = X.
% 0.21/0.49 Proof:
% 0.21/0.49 multiply(add(X, Y), X)
% 0.21/0.49 = { by lemma 10 R->L }
% 0.21/0.49 multiply(add(X, Y), add(additive_identity, X))
% 0.21/0.49 = { by lemma 14 }
% 0.21/0.49 add(X, multiply(additive_identity, Y))
% 0.21/0.49 = { by lemma 12 }
% 0.21/0.49 add(X, additive_identity)
% 0.21/0.49 = { by axiom 4 (additive_id1) }
% 0.21/0.49 X
% 0.21/0.49
% 0.21/0.49 Lemma 16: multiply(add(X, Y), Y) = Y.
% 0.21/0.49 Proof:
% 0.21/0.49 multiply(add(X, Y), Y)
% 0.21/0.49 = { by axiom 1 (commutativity_of_multiply) R->L }
% 0.21/0.49 multiply(Y, add(X, Y))
% 0.21/0.49 = { by lemma 13 }
% 0.21/0.49 Y
% 0.21/0.49
% 0.21/0.49 Lemma 17: multiply(add(X, Y), add(Z, X)) = add(X, multiply(Y, Z)).
% 0.21/0.49 Proof:
% 0.21/0.49 multiply(add(X, Y), add(Z, X))
% 0.21/0.49 = { by lemma 14 }
% 0.21/0.49 add(X, multiply(Z, Y))
% 0.21/0.49 = { by axiom 1 (commutativity_of_multiply) }
% 0.21/0.50 add(X, multiply(Y, Z))
% 0.21/0.50
% 0.21/0.50 Lemma 18: add(X, multiply(Y, multiply(X, Z))) = X.
% 0.21/0.50 Proof:
% 0.21/0.50 add(X, multiply(Y, multiply(X, Z)))
% 0.21/0.50 = { by axiom 1 (commutativity_of_multiply) R->L }
% 0.21/0.50 add(X, multiply(multiply(X, Z), Y))
% 0.21/0.50 = { by lemma 17 R->L }
% 0.21/0.50 multiply(add(X, multiply(X, Z)), add(Y, X))
% 0.21/0.50 = { by lemma 11 }
% 0.21/0.50 multiply(multiply(X, add(Z, multiplicative_identity)), add(Y, X))
% 0.21/0.50 = { by lemma 9 }
% 0.21/0.50 multiply(multiply(X, multiplicative_identity), add(Y, X))
% 0.21/0.50 = { by axiom 2 (multiplicative_id1) }
% 0.21/0.50 multiply(X, add(Y, X))
% 0.21/0.50 = { by lemma 13 }
% 0.21/0.50 X
% 0.21/0.50
% 0.21/0.50 Lemma 19: multiply(add(Z, X), add(Y, X)) = add(X, multiply(Y, Z)).
% 0.21/0.50 Proof:
% 0.21/0.50 multiply(add(Z, X), add(Y, X))
% 0.21/0.50 = { by axiom 3 (commutativity_of_add) R->L }
% 0.21/0.50 multiply(add(Z, X), add(X, Y))
% 0.21/0.50 = { by axiom 3 (commutativity_of_add) }
% 0.21/0.50 multiply(add(X, Z), add(X, Y))
% 0.21/0.50 = { by axiom 7 (distributivity1) R->L }
% 0.21/0.50 add(X, multiply(Z, Y))
% 0.21/0.50 = { by axiom 1 (commutativity_of_multiply) }
% 0.21/0.50 add(X, multiply(Y, Z))
% 0.21/0.50
% 0.21/0.50 Goal 1 (prove_associativity): add(a, add(b, c)) = add(add(a, b), c).
% 0.21/0.50 Proof:
% 0.21/0.50 add(a, add(b, c))
% 0.21/0.50 = { by axiom 3 (commutativity_of_add) }
% 0.21/0.50 add(add(b, c), a)
% 0.21/0.50 = { by lemma 13 R->L }
% 0.21/0.50 add(add(b, c), multiply(a, add(add(add(a, b), c), a)))
% 0.21/0.50 = { by lemma 15 R->L }
% 0.21/0.50 add(add(b, c), multiply(a, add(add(add(a, b), c), multiply(add(a, b), a))))
% 0.21/0.50 = { by axiom 1 (commutativity_of_multiply) R->L }
% 0.21/0.50 add(add(b, c), multiply(a, add(add(add(a, b), c), multiply(a, add(a, b)))))
% 0.21/0.50 = { by axiom 4 (additive_id1) R->L }
% 0.21/0.50 add(add(b, c), multiply(a, add(add(add(a, b), c), multiply(a, add(add(a, b), additive_identity)))))
% 0.21/0.50 = { by axiom 5 (multiplicative_inverse1) R->L }
% 0.21/0.50 add(add(b, c), multiply(a, add(add(add(a, b), c), multiply(a, add(add(a, b), multiply(c, inverse(c)))))))
% 0.21/0.50 = { by lemma 17 R->L }
% 0.21/0.50 add(add(b, c), multiply(a, add(add(add(a, b), c), multiply(a, multiply(add(add(a, b), c), add(inverse(c), add(a, b)))))))
% 0.21/0.50 = { by axiom 3 (commutativity_of_add) }
% 0.21/0.50 add(add(b, c), multiply(a, add(add(add(a, b), c), multiply(a, multiply(add(add(a, b), c), add(add(a, b), inverse(c)))))))
% 0.21/0.50 = { by lemma 18 }
% 0.21/0.50 add(add(b, c), multiply(a, add(add(a, b), c)))
% 0.21/0.50 = { by axiom 1 (commutativity_of_multiply) }
% 0.21/0.50 add(add(b, c), multiply(add(add(a, b), c), a))
% 0.21/0.50 = { by lemma 19 R->L }
% 0.21/0.50 multiply(add(a, add(b, c)), add(add(add(a, b), c), add(b, c)))
% 0.21/0.50 = { by axiom 3 (commutativity_of_add) }
% 0.21/0.50 multiply(add(a, add(b, c)), add(add(add(a, b), c), add(c, b)))
% 0.21/0.50 = { by lemma 16 R->L }
% 0.21/0.50 multiply(add(a, add(b, c)), add(add(add(a, b), c), add(c, multiply(add(a, b), b))))
% 0.21/0.50 = { by axiom 1 (commutativity_of_multiply) R->L }
% 0.21/0.50 multiply(add(a, add(b, c)), add(add(add(a, b), c), add(c, multiply(b, add(a, b)))))
% 0.21/0.50 = { by axiom 7 (distributivity1) }
% 0.21/0.50 multiply(add(a, add(b, c)), add(add(add(a, b), c), multiply(add(c, b), add(c, add(a, b)))))
% 0.21/0.50 = { by axiom 3 (commutativity_of_add) R->L }
% 0.21/0.50 multiply(add(a, add(b, c)), add(add(add(a, b), c), multiply(add(b, c), add(c, add(a, b)))))
% 0.21/0.50 = { by axiom 3 (commutativity_of_add) }
% 0.21/0.50 multiply(add(a, add(b, c)), add(add(add(a, b), c), multiply(add(b, c), add(add(a, b), c))))
% 0.21/0.50 = { by axiom 1 (commutativity_of_multiply) }
% 0.21/0.50 multiply(add(a, add(b, c)), add(add(add(a, b), c), multiply(add(add(a, b), c), add(b, c))))
% 0.21/0.50 = { by lemma 11 }
% 0.21/0.50 multiply(add(a, add(b, c)), multiply(add(add(a, b), c), add(add(b, c), multiplicative_identity)))
% 0.21/0.50 = { by lemma 9 }
% 0.21/0.50 multiply(add(a, add(b, c)), multiply(add(add(a, b), c), multiplicative_identity))
% 0.21/0.50 = { by axiom 2 (multiplicative_id1) }
% 0.21/0.50 multiply(add(a, add(b, c)), add(add(a, b), c))
% 0.21/0.50 = { by axiom 1 (commutativity_of_multiply) }
% 0.21/0.50 multiply(add(add(a, b), c), add(a, add(b, c)))
% 0.21/0.50 = { by axiom 2 (multiplicative_id1) R->L }
% 0.21/0.50 multiply(add(add(a, b), c), multiply(add(a, add(b, c)), multiplicative_identity))
% 0.21/0.50 = { by lemma 9 R->L }
% 0.21/0.50 multiply(add(add(a, b), c), multiply(add(a, add(b, c)), add(add(a, b), multiplicative_identity)))
% 0.21/0.50 = { by lemma 11 R->L }
% 0.21/0.50 multiply(add(add(a, b), c), add(add(a, add(b, c)), multiply(add(a, add(b, c)), add(a, b))))
% 0.21/0.50 = { by axiom 1 (commutativity_of_multiply) R->L }
% 0.21/0.50 multiply(add(add(a, b), c), add(add(a, add(b, c)), multiply(add(a, b), add(a, add(b, c)))))
% 0.21/0.50 = { by axiom 3 (commutativity_of_add) }
% 0.21/0.50 multiply(add(add(a, b), c), add(add(a, add(b, c)), multiply(add(a, b), add(add(b, c), a))))
% 0.21/0.50 = { by lemma 14 }
% 0.21/0.50 multiply(add(add(a, b), c), add(add(a, add(b, c)), add(a, multiply(add(b, c), b))))
% 0.21/0.50 = { by lemma 15 }
% 0.21/0.50 multiply(add(add(a, b), c), add(add(a, add(b, c)), add(a, b)))
% 0.21/0.50 = { by lemma 14 }
% 0.21/0.50 add(add(a, b), multiply(add(a, add(b, c)), c))
% 0.21/0.50 = { by axiom 1 (commutativity_of_multiply) R->L }
% 0.21/0.50 add(add(a, b), multiply(c, add(a, add(b, c))))
% 0.21/0.50 = { by lemma 18 R->L }
% 0.21/0.50 add(add(a, b), multiply(c, add(add(a, add(b, c)), multiply(c, multiply(add(a, add(b, c)), add(add(b, c), inverse(a)))))))
% 0.21/0.50 = { by axiom 3 (commutativity_of_add) R->L }
% 0.21/0.50 add(add(a, b), multiply(c, add(add(a, add(b, c)), multiply(c, multiply(add(a, add(b, c)), add(inverse(a), add(b, c)))))))
% 0.21/0.50 = { by lemma 19 }
% 0.21/0.50 add(add(a, b), multiply(c, add(add(a, add(b, c)), multiply(c, add(add(b, c), multiply(inverse(a), a))))))
% 0.21/0.50 = { by axiom 1 (commutativity_of_multiply) }
% 0.21/0.50 add(add(a, b), multiply(c, add(add(a, add(b, c)), multiply(c, add(add(b, c), multiply(a, inverse(a)))))))
% 0.21/0.50 = { by axiom 5 (multiplicative_inverse1) }
% 0.21/0.50 add(add(a, b), multiply(c, add(add(a, add(b, c)), multiply(c, add(add(b, c), additive_identity)))))
% 0.21/0.50 = { by axiom 4 (additive_id1) }
% 0.21/0.50 add(add(a, b), multiply(c, add(add(a, add(b, c)), multiply(c, add(b, c)))))
% 0.21/0.50 = { by axiom 1 (commutativity_of_multiply) }
% 0.21/0.50 add(add(a, b), multiply(c, add(add(a, add(b, c)), multiply(add(b, c), c))))
% 0.21/0.50 = { by lemma 16 }
% 0.21/0.50 add(add(a, b), multiply(c, add(add(a, add(b, c)), c)))
% 0.21/0.50 = { by lemma 13 }
% 0.21/0.50 add(add(a, b), c)
% 0.21/0.50 % SZS output end Proof
% 0.21/0.50
% 0.21/0.50 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------