TSTP Solution File: BOO026-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : BOO026-1 : TPTP v8.1.2. Released v2.2.0.
% 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 : n023.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 18:11:29 EDT 2023
% Result : Unsatisfiable 0.14s 0.52s
% Output : Proof 0.14s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.10 % Problem : BOO026-1 : TPTP v8.1.2. Released v2.2.0.
% 0.00/0.10 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.10/0.29 % Computer : n023.cluster.edu
% 0.10/0.29 % Model : x86_64 x86_64
% 0.10/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.29 % Memory : 8042.1875MB
% 0.10/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.29 % CPULimit : 300
% 0.10/0.29 % WCLimit : 300
% 0.10/0.29 % DateTime : Sun Aug 27 08:35:11 EDT 2023
% 0.10/0.29 % CPUTime :
% 0.14/0.52 Command-line arguments: --ground-connectedness --complete-subsets
% 0.14/0.52
% 0.14/0.52 % SZS status Unsatisfiable
% 0.14/0.52
% 0.14/0.55 % SZS output start Proof
% 0.14/0.55 Axiom 1 (multiplicative_inverse): multiply(X, inverse(X)) = n0.
% 0.14/0.55 Axiom 2 (additive_inverse): add(X, inverse(X)) = n1.
% 0.14/0.55 Axiom 3 (add_multiply_property): add(X, multiply(Y, Z)) = multiply(add(Y, X), add(Z, X)).
% 0.14/0.55 Axiom 4 (multiply_add_property): multiply(X, add(Y, Z)) = add(multiply(Y, X), multiply(Z, X)).
% 0.14/0.55 Axiom 5 (pixley1_dual): multiply(add(X, inverse(X)), multiply(add(X, Y), add(inverse(X), Y))) = Y.
% 0.14/0.55 Axiom 6 (pixley3_dual): multiply(add(X, inverse(Y)), multiply(add(X, X), add(inverse(Y), X))) = X.
% 0.14/0.55 Axiom 7 (pixley2_dual): multiply(add(X, inverse(Y)), multiply(add(X, Y), add(inverse(Y), Y))) = X.
% 0.14/0.56 Axiom 8 (pixley1): add(multiply(X, inverse(X)), add(multiply(X, Y), multiply(inverse(X), Y))) = Y.
% 0.14/0.56
% 0.14/0.56 Lemma 9: add(n0, multiply(X, n1)) = X.
% 0.14/0.56 Proof:
% 0.14/0.56 add(n0, multiply(X, n1))
% 0.14/0.56 = { by axiom 2 (additive_inverse) R->L }
% 0.14/0.56 add(n0, multiply(X, add(Y, inverse(Y))))
% 0.14/0.56 = { by axiom 4 (multiply_add_property) }
% 0.14/0.56 add(n0, add(multiply(Y, X), multiply(inverse(Y), X)))
% 0.14/0.56 = { by axiom 1 (multiplicative_inverse) R->L }
% 0.14/0.56 add(multiply(Y, inverse(Y)), add(multiply(Y, X), multiply(inverse(Y), X)))
% 0.14/0.56 = { by axiom 8 (pixley1) }
% 0.14/0.56 X
% 0.14/0.56
% 0.14/0.56 Lemma 10: add(multiply(X, n1), multiply(n0, Y)) = multiply(X, add(Y, multiply(X, n1))).
% 0.14/0.56 Proof:
% 0.14/0.56 add(multiply(X, n1), multiply(n0, Y))
% 0.14/0.56 = { by axiom 3 (add_multiply_property) }
% 0.14/0.56 multiply(add(n0, multiply(X, n1)), add(Y, multiply(X, n1)))
% 0.14/0.56 = { by lemma 9 }
% 0.14/0.56 multiply(X, add(Y, multiply(X, n1)))
% 0.14/0.56
% 0.14/0.56 Lemma 11: add(multiply(X, n1), multiply(n0, n0)) = multiply(X, X).
% 0.14/0.56 Proof:
% 0.14/0.56 add(multiply(X, n1), multiply(n0, n0))
% 0.14/0.56 = { by lemma 10 }
% 0.14/0.56 multiply(X, add(n0, multiply(X, n1)))
% 0.14/0.56 = { by lemma 9 }
% 0.14/0.56 multiply(X, X)
% 0.14/0.56
% 0.14/0.56 Lemma 12: multiply(multiply(X, X), add(n0, multiply(n0, n0))) = multiply(n0, X).
% 0.14/0.56 Proof:
% 0.14/0.56 multiply(multiply(X, X), add(n0, multiply(n0, n0)))
% 0.14/0.56 = { by lemma 11 R->L }
% 0.14/0.56 multiply(add(multiply(X, n1), multiply(n0, n0)), add(n0, multiply(n0, n0)))
% 0.14/0.56 = { by axiom 3 (add_multiply_property) R->L }
% 0.14/0.56 add(multiply(n0, n0), multiply(multiply(X, n1), n0))
% 0.14/0.56 = { by axiom 4 (multiply_add_property) R->L }
% 0.14/0.56 multiply(n0, add(n0, multiply(X, n1)))
% 0.14/0.56 = { by lemma 9 }
% 0.14/0.56 multiply(n0, X)
% 0.14/0.56
% 0.14/0.56 Lemma 13: multiply(n1, add(X, n0)) = X.
% 0.14/0.56 Proof:
% 0.14/0.56 multiply(n1, add(X, n0))
% 0.14/0.56 = { by axiom 1 (multiplicative_inverse) R->L }
% 0.14/0.56 multiply(n1, add(X, multiply(Y, inverse(Y))))
% 0.14/0.56 = { by axiom 3 (add_multiply_property) }
% 0.14/0.56 multiply(n1, multiply(add(Y, X), add(inverse(Y), X)))
% 0.14/0.56 = { by axiom 2 (additive_inverse) R->L }
% 0.14/0.56 multiply(add(Y, inverse(Y)), multiply(add(Y, X), add(inverse(Y), X)))
% 0.14/0.56 = { by axiom 5 (pixley1_dual) }
% 0.14/0.56 X
% 0.14/0.56
% 0.14/0.56 Lemma 14: multiply(X, add(n1, multiply(n0, n1))) = multiply(n1, X).
% 0.14/0.56 Proof:
% 0.14/0.56 multiply(X, add(n1, multiply(n0, n1)))
% 0.14/0.56 = { by lemma 13 R->L }
% 0.14/0.56 multiply(multiply(n1, add(X, n0)), add(n1, multiply(n0, n1)))
% 0.14/0.56 = { by axiom 4 (multiply_add_property) }
% 0.14/0.56 multiply(add(multiply(X, n1), multiply(n0, n1)), add(n1, multiply(n0, n1)))
% 0.14/0.56 = { by axiom 3 (add_multiply_property) R->L }
% 0.14/0.56 add(multiply(n0, n1), multiply(multiply(X, n1), n1))
% 0.14/0.56 = { by axiom 4 (multiply_add_property) R->L }
% 0.14/0.56 multiply(n1, add(n0, multiply(X, n1)))
% 0.14/0.56 = { by lemma 9 }
% 0.14/0.56 multiply(n1, X)
% 0.14/0.56
% 0.14/0.56 Lemma 15: multiply(n1, n0) = n0.
% 0.14/0.56 Proof:
% 0.14/0.56 multiply(n1, n0)
% 0.14/0.56 = { by lemma 14 R->L }
% 0.14/0.56 multiply(n0, add(n1, multiply(n0, n1)))
% 0.14/0.56 = { by lemma 10 R->L }
% 0.14/0.56 add(multiply(n0, n1), multiply(n0, n1))
% 0.14/0.56 = { by axiom 4 (multiply_add_property) R->L }
% 0.14/0.56 multiply(n1, add(n0, n0))
% 0.14/0.56 = { by lemma 13 }
% 0.14/0.56 n0
% 0.14/0.56
% 0.14/0.56 Lemma 16: multiply(n0, n1) = n0.
% 0.14/0.56 Proof:
% 0.14/0.56 multiply(n0, n1)
% 0.14/0.56 = { by lemma 12 R->L }
% 0.14/0.56 multiply(multiply(n1, n1), add(n0, multiply(n0, n0)))
% 0.14/0.56 = { by lemma 14 R->L }
% 0.14/0.56 multiply(multiply(n1, add(n1, multiply(n0, n1))), add(n0, multiply(n0, n0)))
% 0.14/0.56 = { by lemma 14 R->L }
% 0.14/0.56 multiply(multiply(add(n1, multiply(n0, n1)), add(n1, multiply(n0, n1))), add(n0, multiply(n0, n0)))
% 0.14/0.56 = { by lemma 12 }
% 0.14/0.56 multiply(n0, add(n1, multiply(n0, n1)))
% 0.14/0.56 = { by lemma 14 }
% 0.14/0.56 multiply(n1, n0)
% 0.14/0.56 = { by lemma 15 }
% 0.14/0.56 n0
% 0.14/0.56
% 0.14/0.56 Lemma 17: multiply(n1, add(n0, X)) = X.
% 0.14/0.56 Proof:
% 0.14/0.56 multiply(n1, add(n0, X))
% 0.14/0.56 = { by axiom 4 (multiply_add_property) }
% 0.14/0.56 add(multiply(n0, n1), multiply(X, n1))
% 0.14/0.56 = { by lemma 16 }
% 0.14/0.56 add(n0, multiply(X, n1))
% 0.14/0.56 = { by lemma 9 }
% 0.14/0.56 X
% 0.14/0.56
% 0.14/0.56 Lemma 18: multiply(n1, n1) = inverse(n0).
% 0.14/0.56 Proof:
% 0.14/0.56 multiply(n1, n1)
% 0.14/0.56 = { by axiom 2 (additive_inverse) R->L }
% 0.14/0.56 multiply(n1, add(n0, inverse(n0)))
% 0.14/0.56 = { by lemma 17 }
% 0.14/0.56 inverse(n0)
% 0.14/0.56
% 0.14/0.56 Lemma 19: add(inverse(X), multiply(Y, X)) = multiply(add(Y, inverse(X)), n1).
% 0.14/0.56 Proof:
% 0.14/0.56 add(inverse(X), multiply(Y, X))
% 0.14/0.56 = { by axiom 3 (add_multiply_property) }
% 0.14/0.56 multiply(add(Y, inverse(X)), add(X, inverse(X)))
% 0.14/0.56 = { by axiom 2 (additive_inverse) }
% 0.14/0.56 multiply(add(Y, inverse(X)), n1)
% 0.14/0.56
% 0.14/0.56 Lemma 20: multiply(add(n1, inverse(add(X, n0))), n1) = add(inverse(add(X, n0)), X).
% 0.14/0.56 Proof:
% 0.14/0.56 multiply(add(n1, inverse(add(X, n0))), n1)
% 0.14/0.56 = { by lemma 19 R->L }
% 0.14/0.56 add(inverse(add(X, n0)), multiply(n1, add(X, n0)))
% 0.14/0.56 = { by lemma 13 }
% 0.14/0.56 add(inverse(add(X, n0)), X)
% 0.14/0.56
% 0.14/0.56 Lemma 21: add(inverse(X), multiply(X, Y)) = multiply(n1, add(Y, inverse(X))).
% 0.14/0.56 Proof:
% 0.14/0.56 add(inverse(X), multiply(X, Y))
% 0.14/0.56 = { by axiom 3 (add_multiply_property) }
% 0.14/0.56 multiply(add(X, inverse(X)), add(Y, inverse(X)))
% 0.14/0.56 = { by axiom 2 (additive_inverse) }
% 0.14/0.56 multiply(n1, add(Y, inverse(X)))
% 0.14/0.56
% 0.14/0.56 Lemma 22: add(inverse(X), add(multiply(X, n1), multiply(n0, n0))) = multiply(n1, n1).
% 0.14/0.56 Proof:
% 0.14/0.56 add(inverse(X), add(multiply(X, n1), multiply(n0, n0)))
% 0.14/0.56 = { by lemma 11 }
% 0.14/0.56 add(inverse(X), multiply(X, X))
% 0.14/0.56 = { by lemma 21 }
% 0.14/0.56 multiply(n1, add(X, inverse(X)))
% 0.14/0.56 = { by axiom 2 (additive_inverse) }
% 0.14/0.56 multiply(n1, n1)
% 0.14/0.56
% 0.14/0.56 Lemma 23: add(n1, inverse(add(n1, n0))) = n1.
% 0.14/0.56 Proof:
% 0.14/0.56 add(n1, inverse(add(n1, n0)))
% 0.14/0.56 = { by lemma 9 R->L }
% 0.14/0.56 add(n0, multiply(add(n1, inverse(add(n1, n0))), n1))
% 0.14/0.56 = { by lemma 20 }
% 0.14/0.56 add(n0, add(inverse(add(n1, n0)), n1))
% 0.14/0.56 = { by lemma 9 R->L }
% 0.14/0.56 add(n0, add(inverse(add(n1, n0)), add(n0, multiply(n1, n1))))
% 0.14/0.56 = { by axiom 3 (add_multiply_property) }
% 0.14/0.56 add(n0, add(inverse(add(n1, n0)), multiply(add(n1, n0), add(n1, n0))))
% 0.14/0.56 = { by lemma 11 R->L }
% 0.14/0.56 add(n0, add(inverse(add(n1, n0)), add(multiply(add(n1, n0), n1), multiply(n0, n0))))
% 0.14/0.56 = { by lemma 22 }
% 0.14/0.56 add(n0, multiply(n1, n1))
% 0.14/0.56 = { by lemma 9 }
% 0.14/0.56 n1
% 0.14/0.56
% 0.14/0.56 Lemma 24: add(n0, multiply(Y, inverse(X))) = multiply(inverse(X), add(X, Y)).
% 0.14/0.56 Proof:
% 0.14/0.56 add(n0, multiply(Y, inverse(X)))
% 0.14/0.56 = { by axiom 1 (multiplicative_inverse) R->L }
% 0.14/0.56 add(multiply(X, inverse(X)), multiply(Y, inverse(X)))
% 0.14/0.56 = { by axiom 4 (multiply_add_property) R->L }
% 0.14/0.56 multiply(inverse(X), add(X, Y))
% 0.14/0.56
% 0.14/0.56 Lemma 25: multiply(inverse(X), add(X, X)) = add(n0, n0).
% 0.14/0.56 Proof:
% 0.14/0.56 multiply(inverse(X), add(X, X))
% 0.14/0.56 = { by lemma 24 R->L }
% 0.14/0.56 add(n0, multiply(X, inverse(X)))
% 0.14/0.56 = { by axiom 1 (multiplicative_inverse) }
% 0.14/0.56 add(n0, n0)
% 0.14/0.56
% 0.14/0.56 Lemma 26: multiply(n1, X) = multiply(X, n1).
% 0.14/0.56 Proof:
% 0.14/0.56 multiply(n1, X)
% 0.14/0.56 = { by lemma 9 R->L }
% 0.14/0.56 multiply(n1, add(n0, multiply(X, n1)))
% 0.14/0.56 = { by lemma 17 }
% 0.14/0.56 multiply(X, n1)
% 0.14/0.56
% 0.14/0.56 Lemma 27: add(n0, n0) = n0.
% 0.14/0.56 Proof:
% 0.14/0.56 add(n0, n0)
% 0.14/0.56 = { by lemma 16 R->L }
% 0.14/0.56 add(n0, multiply(n0, n1))
% 0.14/0.56 = { by lemma 9 }
% 0.14/0.56 n0
% 0.14/0.56
% 0.14/0.56 Lemma 28: add(n1, n0) = add(n1, n1).
% 0.14/0.56 Proof:
% 0.14/0.56 add(n1, n0)
% 0.14/0.56 = { by axiom 1 (multiplicative_inverse) R->L }
% 0.14/0.56 add(n1, multiply(n1, inverse(n1)))
% 0.14/0.56 = { by axiom 3 (add_multiply_property) }
% 0.14/0.56 multiply(add(n1, n1), add(inverse(n1), n1))
% 0.14/0.56 = { by axiom 4 (multiply_add_property) }
% 0.14/0.56 add(multiply(inverse(n1), add(n1, n1)), multiply(n1, add(n1, n1)))
% 0.14/0.56 = { by lemma 25 }
% 0.14/0.56 add(add(n0, n0), multiply(n1, add(n1, n1)))
% 0.14/0.56 = { by lemma 26 }
% 0.14/0.56 add(add(n0, n0), multiply(add(n1, n1), n1))
% 0.14/0.56 = { by lemma 27 }
% 0.14/0.56 add(n0, multiply(add(n1, n1), n1))
% 0.14/0.56 = { by lemma 9 }
% 0.14/0.56 add(n1, n1)
% 0.14/0.56
% 0.14/0.56 Lemma 29: add(n1, X) = add(X, n1).
% 0.14/0.56 Proof:
% 0.14/0.56 add(n1, X)
% 0.14/0.56 = { by lemma 13 R->L }
% 0.14/0.56 add(n1, multiply(n1, add(X, n0)))
% 0.14/0.56 = { by axiom 3 (add_multiply_property) }
% 0.14/0.56 multiply(add(n1, n1), add(add(X, n0), n1))
% 0.14/0.56 = { by lemma 28 R->L }
% 0.14/0.56 multiply(add(n1, n0), add(add(X, n0), n1))
% 0.14/0.56 = { by axiom 4 (multiply_add_property) }
% 0.14/0.56 add(multiply(add(X, n0), add(n1, n0)), multiply(n1, add(n1, n0)))
% 0.14/0.56 = { by lemma 13 }
% 0.14/0.56 add(multiply(add(X, n0), add(n1, n0)), n1)
% 0.14/0.56 = { by axiom 3 (add_multiply_property) R->L }
% 0.14/0.56 add(add(n0, multiply(X, n1)), n1)
% 0.14/0.56 = { by lemma 9 }
% 0.14/0.56 add(X, n1)
% 0.14/0.56
% 0.14/0.56 Lemma 30: inverse(n0) = n1.
% 0.14/0.56 Proof:
% 0.14/0.56 inverse(n0)
% 0.14/0.56 = { by lemma 18 R->L }
% 0.14/0.56 multiply(n1, n1)
% 0.14/0.56 = { by lemma 23 R->L }
% 0.14/0.56 multiply(add(n1, inverse(add(n1, n0))), n1)
% 0.14/0.56 = { by lemma 20 }
% 0.14/0.56 add(inverse(add(n1, n0)), n1)
% 0.14/0.56 = { by lemma 29 R->L }
% 0.14/0.56 add(n1, inverse(add(n1, n0)))
% 0.14/0.56 = { by lemma 23 }
% 0.14/0.56 n1
% 0.14/0.56
% 0.14/0.56 Lemma 31: multiply(add(X, inverse(Y)), add(Y, multiply(X, inverse(Y)))) = X.
% 0.14/0.56 Proof:
% 0.14/0.56 multiply(add(X, inverse(Y)), add(Y, multiply(X, inverse(Y))))
% 0.14/0.56 = { by axiom 3 (add_multiply_property) }
% 0.14/0.56 multiply(add(X, inverse(Y)), multiply(add(X, Y), add(inverse(Y), Y)))
% 0.14/0.56 = { by axiom 7 (pixley2_dual) }
% 0.14/0.56 X
% 0.14/0.56
% 0.14/0.56 Lemma 32: add(multiply(X, n1), n0) = X.
% 0.14/0.56 Proof:
% 0.14/0.56 add(multiply(X, n1), n0)
% 0.14/0.56 = { by lemma 16 R->L }
% 0.14/0.56 add(multiply(X, n1), multiply(n0, n1))
% 0.14/0.56 = { by axiom 4 (multiply_add_property) R->L }
% 0.14/0.56 multiply(n1, add(X, n0))
% 0.14/0.56 = { by lemma 13 }
% 0.14/0.56 X
% 0.14/0.56
% 0.14/0.56 Lemma 33: multiply(n0, n0) = n0.
% 0.14/0.56 Proof:
% 0.14/0.56 multiply(n0, n0)
% 0.14/0.56 = { by lemma 11 R->L }
% 0.14/0.56 add(multiply(n0, n1), multiply(n0, n0))
% 0.14/0.57 = { by lemma 16 }
% 0.14/0.57 add(n0, multiply(n0, n0))
% 0.14/0.57 = { by lemma 15 R->L }
% 0.14/0.57 add(multiply(n1, n0), multiply(n0, n0))
% 0.14/0.57 = { by axiom 4 (multiply_add_property) R->L }
% 0.14/0.57 multiply(n0, add(n1, n0))
% 0.14/0.57 = { by lemma 16 R->L }
% 0.14/0.57 multiply(n0, add(n1, multiply(n0, n1)))
% 0.14/0.57 = { by lemma 10 R->L }
% 0.14/0.57 add(multiply(n0, n1), multiply(n0, n1))
% 0.14/0.57 = { by axiom 4 (multiply_add_property) R->L }
% 0.14/0.57 multiply(n1, add(n0, n0))
% 0.14/0.57 = { by lemma 13 }
% 0.14/0.57 n0
% 0.14/0.57
% 0.14/0.57 Lemma 34: add(inverse(X), X) = n1.
% 0.14/0.57 Proof:
% 0.14/0.57 add(inverse(X), X)
% 0.14/0.57 = { by lemma 32 R->L }
% 0.14/0.57 add(inverse(X), add(multiply(X, n1), n0))
% 0.14/0.57 = { by lemma 33 R->L }
% 0.14/0.57 add(inverse(X), add(multiply(X, n1), multiply(n0, n0)))
% 0.14/0.57 = { by lemma 22 }
% 0.14/0.57 multiply(n1, n1)
% 0.14/0.57 = { by lemma 18 }
% 0.14/0.57 inverse(n0)
% 0.14/0.57 = { by lemma 30 }
% 0.14/0.57 n1
% 0.14/0.57
% 0.14/0.57 Lemma 35: inverse(n1) = n0.
% 0.14/0.57 Proof:
% 0.14/0.57 inverse(n1)
% 0.14/0.57 = { by lemma 30 R->L }
% 0.14/0.57 inverse(inverse(n0))
% 0.14/0.57 = { by lemma 31 R->L }
% 0.14/0.57 multiply(add(inverse(inverse(n0)), inverse(n0)), add(n0, multiply(inverse(inverse(n0)), inverse(n0))))
% 0.14/0.57 = { by lemma 24 }
% 0.14/0.57 multiply(add(inverse(inverse(n0)), inverse(n0)), multiply(inverse(n0), add(n0, inverse(inverse(n0)))))
% 0.14/0.57 = { by lemma 34 }
% 0.14/0.57 multiply(n1, multiply(inverse(n0), add(n0, inverse(inverse(n0)))))
% 0.14/0.57 = { by lemma 30 }
% 0.14/0.57 multiply(n1, multiply(n1, add(n0, inverse(inverse(n0)))))
% 0.14/0.57 = { by lemma 30 }
% 0.14/0.57 multiply(n1, multiply(n1, add(n0, inverse(n1))))
% 0.14/0.57 = { by lemma 17 }
% 0.14/0.57 multiply(n1, inverse(n1))
% 0.14/0.57 = { by axiom 1 (multiplicative_inverse) }
% 0.14/0.57 n0
% 0.14/0.57
% 0.14/0.57 Lemma 36: add(X, X) = X.
% 0.14/0.57 Proof:
% 0.14/0.57 add(X, X)
% 0.14/0.57 = { by lemma 9 R->L }
% 0.14/0.57 add(n0, multiply(add(X, X), n1))
% 0.14/0.57 = { by lemma 26 R->L }
% 0.14/0.57 add(n0, multiply(n1, add(X, X)))
% 0.14/0.57 = { by axiom 4 (multiply_add_property) }
% 0.14/0.57 add(n0, add(multiply(X, n1), multiply(X, n1)))
% 0.14/0.57 = { by lemma 13 R->L }
% 0.14/0.57 add(n0, add(multiply(X, n1), multiply(n1, add(multiply(X, n1), n0))))
% 0.14/0.57 = { by lemma 13 R->L }
% 0.14/0.57 add(n0, add(multiply(n1, add(multiply(X, n1), n0)), multiply(n1, add(multiply(X, n1), n0))))
% 0.14/0.57 = { by axiom 4 (multiply_add_property) R->L }
% 0.14/0.57 add(n0, multiply(add(multiply(X, n1), n0), add(n1, n1)))
% 0.14/0.57 = { by lemma 32 }
% 0.14/0.57 add(n0, multiply(X, add(n1, n1)))
% 0.14/0.57 = { by lemma 28 R->L }
% 0.14/0.57 add(n0, multiply(X, add(n1, n0)))
% 0.14/0.57 = { by lemma 30 R->L }
% 0.14/0.57 add(n0, multiply(X, add(inverse(n0), n0)))
% 0.14/0.57 = { by lemma 18 R->L }
% 0.14/0.57 add(n0, multiply(X, add(multiply(n1, n1), n0)))
% 0.14/0.57 = { by lemma 32 }
% 0.14/0.57 add(n0, multiply(X, n1))
% 0.14/0.57 = { by lemma 9 }
% 0.14/0.57 X
% 0.14/0.57
% 0.14/0.57 Lemma 37: multiply(X, X) = X.
% 0.14/0.57 Proof:
% 0.14/0.57 multiply(X, X)
% 0.14/0.57 = { by lemma 9 R->L }
% 0.14/0.57 multiply(X, add(n0, multiply(X, n1)))
% 0.14/0.57 = { by lemma 10 R->L }
% 0.14/0.57 add(multiply(X, n1), multiply(n0, n0))
% 0.14/0.57 = { by lemma 33 }
% 0.14/0.57 add(multiply(X, n1), n0)
% 0.14/0.57 = { by lemma 32 }
% 0.14/0.57 X
% 0.14/0.57
% 0.14/0.57 Lemma 38: add(X, n0) = multiply(X, n1).
% 0.14/0.57 Proof:
% 0.14/0.57 add(X, n0)
% 0.14/0.57 = { by lemma 27 R->L }
% 0.14/0.57 add(X, add(n0, n0))
% 0.14/0.57 = { by lemma 37 R->L }
% 0.14/0.57 add(multiply(X, X), add(n0, n0))
% 0.14/0.57 = { by lemma 36 R->L }
% 0.14/0.57 add(multiply(X, add(X, X)), add(n0, n0))
% 0.14/0.57 = { by lemma 25 R->L }
% 0.14/0.57 add(multiply(X, add(X, X)), multiply(inverse(X), add(X, X)))
% 0.14/0.57 = { by axiom 4 (multiply_add_property) R->L }
% 0.14/0.57 multiply(add(X, X), add(X, inverse(X)))
% 0.14/0.57 = { by axiom 2 (additive_inverse) }
% 0.14/0.57 multiply(add(X, X), n1)
% 0.14/0.57 = { by lemma 36 }
% 0.14/0.57 multiply(X, n1)
% 0.14/0.57
% 0.14/0.57 Lemma 39: add(Y, X) = add(X, Y).
% 0.14/0.57 Proof:
% 0.14/0.57 add(Y, X)
% 0.14/0.57 = { by lemma 37 R->L }
% 0.14/0.57 add(Y, multiply(X, X))
% 0.14/0.57 = { by axiom 3 (add_multiply_property) }
% 0.14/0.57 multiply(add(X, Y), add(X, Y))
% 0.14/0.57 = { by lemma 37 }
% 0.14/0.57 add(X, Y)
% 0.14/0.57
% 0.14/0.57 Lemma 40: multiply(n1, inverse(X)) = inverse(X).
% 0.14/0.57 Proof:
% 0.14/0.57 multiply(n1, inverse(X))
% 0.14/0.57 = { by lemma 26 }
% 0.14/0.57 multiply(inverse(X), n1)
% 0.14/0.57 = { by axiom 2 (additive_inverse) R->L }
% 0.14/0.57 multiply(inverse(X), add(X, inverse(X)))
% 0.14/0.57 = { by lemma 37 R->L }
% 0.14/0.57 multiply(inverse(X), add(X, multiply(inverse(X), inverse(X))))
% 0.14/0.57 = { by lemma 36 R->L }
% 0.14/0.57 multiply(add(inverse(X), inverse(X)), add(X, multiply(inverse(X), inverse(X))))
% 0.14/0.57 = { by lemma 31 }
% 0.14/0.57 inverse(X)
% 0.14/0.57
% 0.14/0.57 Lemma 41: multiply(add(X, inverse(inverse(X))), multiply(n1, add(inverse(inverse(X)), inverse(X)))) = X.
% 0.14/0.57 Proof:
% 0.14/0.57 multiply(add(X, inverse(inverse(X))), multiply(n1, add(inverse(inverse(X)), inverse(X))))
% 0.14/0.57 = { by lemma 21 R->L }
% 0.14/0.57 multiply(add(X, inverse(inverse(X))), add(inverse(X), multiply(X, inverse(inverse(X)))))
% 0.14/0.57 = { by lemma 31 }
% 0.14/0.57 X
% 0.14/0.57
% 0.14/0.57 Lemma 42: inverse(inverse(X)) = X.
% 0.14/0.57 Proof:
% 0.14/0.57 inverse(inverse(X))
% 0.14/0.57 = { by lemma 40 R->L }
% 0.14/0.57 multiply(n1, inverse(inverse(X)))
% 0.14/0.57 = { by lemma 26 }
% 0.14/0.57 multiply(inverse(inverse(X)), n1)
% 0.14/0.57 = { by lemma 38 R->L }
% 0.14/0.57 add(inverse(inverse(X)), n0)
% 0.14/0.57 = { by axiom 1 (multiplicative_inverse) R->L }
% 0.14/0.57 add(inverse(inverse(X)), multiply(X, inverse(X)))
% 0.14/0.57 = { by lemma 19 }
% 0.14/0.57 multiply(add(X, inverse(inverse(X))), n1)
% 0.14/0.57 = { by lemma 30 R->L }
% 0.14/0.57 multiply(add(X, inverse(inverse(X))), inverse(n0))
% 0.14/0.57 = { by lemma 18 R->L }
% 0.14/0.57 multiply(add(X, inverse(inverse(X))), multiply(n1, n1))
% 0.14/0.57 = { by lemma 34 R->L }
% 0.14/0.57 multiply(add(X, inverse(inverse(X))), multiply(n1, add(inverse(inverse(X)), inverse(X))))
% 0.14/0.57 = { by lemma 41 }
% 0.14/0.57 X
% 0.14/0.57
% 0.14/0.57 Lemma 43: multiply(X, n1) = X.
% 0.14/0.57 Proof:
% 0.14/0.57 multiply(X, n1)
% 0.14/0.57 = { by lemma 30 R->L }
% 0.14/0.57 multiply(X, inverse(n0))
% 0.14/0.57 = { by lemma 18 R->L }
% 0.14/0.57 multiply(X, multiply(n1, n1))
% 0.14/0.57 = { by lemma 34 R->L }
% 0.14/0.57 multiply(X, multiply(n1, add(inverse(inverse(X)), inverse(X))))
% 0.14/0.57 = { by lemma 36 R->L }
% 0.14/0.57 multiply(add(X, X), multiply(n1, add(inverse(inverse(X)), inverse(X))))
% 0.14/0.57 = { by lemma 42 R->L }
% 0.14/0.57 multiply(add(X, inverse(inverse(X))), multiply(n1, add(inverse(inverse(X)), inverse(X))))
% 0.14/0.57 = { by lemma 41 }
% 0.14/0.57 X
% 0.14/0.57
% 0.14/0.57 Goal 1 (prove_multiply_add): multiply(add(a, b), b) = b.
% 0.14/0.57 Proof:
% 0.14/0.57 multiply(add(a, b), b)
% 0.14/0.57 = { by lemma 43 R->L }
% 0.14/0.57 multiply(add(a, b), multiply(b, n1))
% 0.14/0.57 = { by lemma 38 R->L }
% 0.14/0.57 multiply(add(a, b), add(b, n0))
% 0.14/0.57 = { by lemma 39 }
% 0.14/0.57 multiply(add(a, b), add(n0, b))
% 0.14/0.57 = { by axiom 3 (add_multiply_property) R->L }
% 0.14/0.57 add(b, multiply(a, n0))
% 0.14/0.57 = { by lemma 35 R->L }
% 0.14/0.57 add(b, multiply(a, inverse(n1)))
% 0.14/0.57 = { by lemma 43 R->L }
% 0.14/0.57 add(b, multiply(multiply(a, inverse(n1)), n1))
% 0.14/0.57 = { by lemma 38 R->L }
% 0.14/0.57 add(b, add(multiply(a, inverse(n1)), n0))
% 0.14/0.57 = { by axiom 1 (multiplicative_inverse) R->L }
% 0.14/0.57 add(b, add(multiply(a, inverse(n1)), multiply(n1, inverse(n1))))
% 0.14/0.57 = { by axiom 4 (multiply_add_property) R->L }
% 0.14/0.58 add(b, multiply(inverse(n1), add(a, n1)))
% 0.14/0.58 = { by lemma 39 }
% 0.14/0.58 add(b, multiply(inverse(n1), add(n1, a)))
% 0.14/0.58 = { by lemma 42 R->L }
% 0.14/0.58 add(b, multiply(inverse(n1), add(n1, inverse(inverse(a)))))
% 0.14/0.58 = { by lemma 29 }
% 0.14/0.58 add(b, multiply(inverse(n1), add(inverse(inverse(a)), n1)))
% 0.14/0.58 = { by lemma 30 R->L }
% 0.14/0.58 add(b, multiply(inverse(n1), add(inverse(inverse(a)), inverse(n0))))
% 0.14/0.58 = { by lemma 18 R->L }
% 0.14/0.58 add(b, multiply(inverse(n1), add(inverse(inverse(a)), multiply(n1, n1))))
% 0.14/0.58 = { by axiom 3 (add_multiply_property) }
% 0.14/0.58 add(b, multiply(inverse(n1), multiply(add(n1, inverse(inverse(a))), add(n1, inverse(inverse(a))))))
% 0.14/0.58 = { by lemma 40 R->L }
% 0.14/0.58 add(b, multiply(inverse(n1), multiply(add(n1, inverse(inverse(a))), add(n1, multiply(n1, inverse(inverse(a)))))))
% 0.14/0.58 = { by axiom 3 (add_multiply_property) }
% 0.14/0.58 add(b, multiply(inverse(n1), multiply(add(n1, inverse(inverse(a))), multiply(add(n1, n1), add(inverse(inverse(a)), n1)))))
% 0.14/0.58 = { by axiom 6 (pixley3_dual) }
% 0.14/0.58 add(b, multiply(inverse(n1), n1))
% 0.14/0.58 = { by lemma 43 }
% 0.14/0.58 add(b, inverse(n1))
% 0.14/0.58 = { by lemma 35 }
% 0.14/0.58 add(b, n0)
% 0.14/0.58 = { by lemma 38 }
% 0.14/0.58 multiply(b, n1)
% 0.14/0.58 = { by lemma 43 }
% 0.14/0.58 b
% 0.14/0.58 % SZS output end Proof
% 0.14/0.58
% 0.14/0.58 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------