TSTP Solution File: BOO004-4 by Toma---0.4
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%------------------------------------------------------------------------------
% File : Toma---0.4
% Problem : BOO004-4 : TPTP v8.1.2. Released v1.1.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n027.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 18:10:54 EDT 2023
% Result : Unsatisfiable 1.41s 1.83s
% Output : CNFRefutation 1.41s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : BOO004-4 : TPTP v8.1.2. Released v1.1.0.
% 0.12/0.13 % Command : toma --casc %s
% 0.12/0.34 % Computer : n027.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Sun Aug 27 08:17:06 EDT 2023
% 0.12/0.34 % CPUTime :
% 1.41/1.83 % SZS status Unsatisfiable
% 1.41/1.83 % SZS output start Proof
% 1.41/1.83 original problem:
% 1.41/1.83 axioms:
% 1.41/1.83 add(X, Y) = add(Y, X)
% 1.41/1.83 multiply(X, Y) = multiply(Y, X)
% 1.41/1.83 add(X, multiply(Y, Z)) = multiply(add(X, Y), add(X, Z))
% 1.41/1.83 multiply(X, add(Y, Z)) = add(multiply(X, Y), multiply(X, Z))
% 1.41/1.83 add(X, additive_identity()) = X
% 1.41/1.83 multiply(X, multiplicative_identity()) = X
% 1.41/1.83 add(X, inverse(X)) = multiplicative_identity()
% 1.41/1.83 multiply(X, inverse(X)) = additive_identity()
% 1.41/1.83 goal:
% 1.41/1.83 add(a(), a()) != a()
% 1.41/1.83 To show the unsatisfiability of the original goal,
% 1.41/1.83 it suffices to show that add(a(), a()) = a() (skolemized goal) is valid under the axioms.
% 1.41/1.83 Here is an equational proof:
% 1.41/1.83 0: add(X0, X1) = add(X1, X0).
% 1.41/1.83 Proof: Axiom.
% 1.41/1.83
% 1.41/1.83 1: multiply(X0, X1) = multiply(X1, X0).
% 1.41/1.83 Proof: Axiom.
% 1.41/1.83
% 1.41/1.83 2: add(X0, multiply(X1, X2)) = multiply(add(X0, X1), add(X0, X2)).
% 1.41/1.83 Proof: Axiom.
% 1.41/1.83
% 1.41/1.83 3: multiply(X0, add(X1, X2)) = add(multiply(X0, X1), multiply(X0, X2)).
% 1.41/1.83 Proof: Axiom.
% 1.41/1.83
% 1.41/1.83 4: add(X0, additive_identity()) = X0.
% 1.41/1.83 Proof: Axiom.
% 1.41/1.83
% 1.41/1.83 5: multiply(X0, multiplicative_identity()) = X0.
% 1.41/1.83 Proof: Axiom.
% 1.41/1.83
% 1.41/1.83 6: add(X0, inverse(X0)) = multiplicative_identity().
% 1.41/1.83 Proof: Axiom.
% 1.41/1.83
% 1.41/1.83 8: multiply(X0, add(X1, X2)) = multiply(multiply(add(X0, X0), add(X0, X1)), add(multiply(X0, X1), X2)).
% 1.41/1.83 Proof: Rewrite equation 3,
% 1.41/1.83 lhs with equations []
% 1.41/1.83 rhs with equations [2,0,2].
% 1.41/1.83
% 1.41/1.83 9: add(additive_identity(), X2) = X2.
% 1.41/1.83 Proof: A critical pair between equations 0 and 4.
% 1.41/1.83
% 1.41/1.83 10: multiply(multiplicative_identity(), X2) = X2.
% 1.41/1.83 Proof: A critical pair between equations 1 and 5.
% 1.41/1.83
% 1.41/1.83 13: add(multiply(X4, X5), X3) = multiply(add(X3, X4), add(X3, X5)).
% 1.41/1.83 Proof: A critical pair between equations 0 and 2.
% 1.41/1.83
% 1.41/1.83 18: multiply(X3, add(multiplicative_identity(), X2)) = multiply(multiply(add(X3, X3), add(X3, multiplicative_identity())), add(X3, X2)).
% 1.41/1.83 Proof: A critical pair between equations 8 and 5.
% 1.41/1.83
% 1.41/1.83 23: multiply(X3, add(multiplicative_identity(), X2)) = add(X3, multiply(X3, X2)).
% 1.41/1.83 Proof: Rewrite equation 18,
% 1.41/1.83 lhs with equations []
% 1.41/1.83 rhs with equations [13,5,13,0].
% 1.41/1.83
% 1.41/1.83 27: add(multiply(X4, X5), X3) = add(X3, multiply(X4, X5)).
% 1.41/1.83 Proof: Rewrite equation 13,
% 1.41/1.83 lhs with equations []
% 1.41/1.83 rhs with equations [2].
% 1.41/1.83
% 1.41/1.83 31: inverse(additive_identity()) = multiplicative_identity().
% 1.41/1.83 Proof: A critical pair between equations 9 and 6.
% 1.41/1.83
% 1.41/1.83 35: add(X3, multiply(X6, multiplicative_identity())) = add(X6, X3).
% 1.41/1.83 Proof: A critical pair between equations 27 and 5.
% 1.41/1.83
% 1.41/1.83 37: add(X3, multiply(X3, additive_identity())) = multiply(X3, multiplicative_identity()).
% 1.41/1.83 Proof: A critical pair between equations 23 and 4.
% 1.41/1.83
% 1.41/1.83 40: multiply(add(multiplicative_identity(), X5), X4) = add(X4, multiply(X4, X5)).
% 1.41/1.83 Proof: A critical pair between equations 1 and 23.
% 1.41/1.83
% 1.41/1.83 46: multiply(X0, inverse(additive_identity())) = X0.
% 1.41/1.83 Proof: Rewrite equation 5,
% 1.41/1.83 lhs with equations [31]
% 1.41/1.83 rhs with equations [].
% 1.41/1.83
% 1.41/1.83 47: multiply(inverse(additive_identity()), X2) = X2.
% 1.41/1.83 Proof: Rewrite equation 10,
% 1.41/1.83 lhs with equations [31]
% 1.41/1.83 rhs with equations [].
% 1.41/1.83
% 1.41/1.83 51: multiply(add(inverse(additive_identity()), X5), X4) = multiply(add(X4, X4), add(X4, X5)).
% 1.41/1.83 Proof: Rewrite equation 40,
% 1.41/1.83 lhs with equations [31]
% 1.41/1.83 rhs with equations [2].
% 1.41/1.83
% 1.41/1.83 53: multiply(X3, add(X3, X3)) = X3.
% 1.41/1.83 Proof: Rewrite equation 37,
% 1.41/1.83 lhs with equations [2,4,1]
% 1.41/1.83 rhs with equations [31,46].
% 1.41/1.83
% 1.41/1.83 54: add(X3, X6) = add(X6, X3).
% 1.41/1.83 Proof: Rewrite equation 35,
% 1.41/1.83 lhs with equations [31,46]
% 1.41/1.83 rhs with equations [].
% 1.41/1.83
% 1.41/1.83 62: add(multiply(X4, X5), X3) = multiply(add(X3, X4), add(X3, X5)).
% 1.41/1.83 Proof: Rewrite equation 27,
% 1.41/1.83 lhs with equations []
% 1.41/1.83 rhs with equations [2].
% 1.41/1.83
% 1.41/1.83 68: add(inverse(additive_identity()), inverse(additive_identity())) = inverse(additive_identity()).
% 1.41/1.83 Proof: A critical pair between equations 47 and 53.
% 1.41/1.83
% 1.41/1.83 77: add(multiplicative_identity(), multiplicative_identity()) = multiplicative_identity().
% 1.41/1.83 Proof: Rewrite equation 68,
% 1.41/1.83 lhs with equations [31,31]
% 1.41/1.83 rhs with equations [31].
% 1.41/1.83
% 1.41/1.83 83: multiply(multiplicative_identity(), X2) = X2.
% 1.41/1.83 Proof: Rewrite equation 47,
% 1.41/1.83 lhs with equations [31]
% 1.41/1.84 rhs with equations [].
% 1.41/1.84
% 1.41/1.84 84: multiply(X0, multiplicative_identity()) = X0.
% 1.41/1.84 Proof: Rewrite equation 46,
% 1.41/1.84 lhs with equations [31]
% 1.41/1.84 rhs with equations [].
% 1.41/1.84
% 1.41/1.84 93: multiply(add(multiplicative_identity(), X5), X4) = add(X4, multiply(X4, X5)).
% 1.41/1.84 Proof: Rewrite equation 51,
% 1.41/1.84 lhs with equations [31]
% 1.41/1.84 rhs with equations [62,54].
% 1.41/1.84
% 1.41/1.84 131: multiply(add(multiplicative_identity(), multiplicative_identity()), X6) = add(X6, X6).
% 1.41/1.84 Proof: A critical pair between equations 93 and 84.
% 1.41/1.84
% 1.41/1.84 145: X6 = add(X6, X6).
% 1.41/1.84 Proof: Rewrite equation 131,
% 1.41/1.84 lhs with equations [77,83]
% 1.41/1.84 rhs with equations [].
% 1.41/1.84
% 1.41/1.84 167: add(a(), a()) = a().
% 1.41/1.84 Proof: Rewrite lhs with equations [145]
% 1.41/1.84 rhs with equations [].
% 1.41/1.84
% 1.41/1.84 % SZS output end Proof
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