TSTP Solution File: BOO003-4 by Toma---0.4
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% File : Toma---0.4
% Problem : BOO003-4 : TPTP v8.1.2. Released v1.1.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n032.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 0.97s 1.35s
% Output : CNFRefutation 0.97s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.10 % Problem : BOO003-4 : TPTP v8.1.2. Released v1.1.0.
% 0.00/0.10 % Command : toma --casc %s
% 0.10/0.29 % Computer : n032.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 08:08:29 EDT 2023
% 0.10/0.29 % CPUTime :
% 0.97/1.35 % SZS status Unsatisfiable
% 0.97/1.35 % SZS output start Proof
% 0.97/1.35 original problem:
% 0.97/1.35 axioms:
% 0.97/1.35 add(X, Y) = add(Y, X)
% 0.97/1.35 multiply(X, Y) = multiply(Y, X)
% 0.97/1.35 add(X, multiply(Y, Z)) = multiply(add(X, Y), add(X, Z))
% 0.97/1.35 multiply(X, add(Y, Z)) = add(multiply(X, Y), multiply(X, Z))
% 0.97/1.35 add(X, additive_identity()) = X
% 0.97/1.35 multiply(X, multiplicative_identity()) = X
% 0.97/1.35 add(X, inverse(X)) = multiplicative_identity()
% 0.97/1.35 multiply(X, inverse(X)) = additive_identity()
% 0.97/1.35 goal:
% 0.97/1.35 multiply(a(), a()) != a()
% 0.97/1.35 To show the unsatisfiability of the original goal,
% 0.97/1.35 it suffices to show that multiply(a(), a()) = a() (skolemized goal) is valid under the axioms.
% 0.97/1.35 Here is an equational proof:
% 0.97/1.35 0: add(X0, X1) = add(X1, X0).
% 0.97/1.35 Proof: Axiom.
% 0.97/1.35
% 0.97/1.35 1: multiply(X0, X1) = multiply(X1, X0).
% 0.97/1.35 Proof: Axiom.
% 0.97/1.35
% 0.97/1.35 2: add(X0, multiply(X1, X2)) = multiply(add(X0, X1), add(X0, X2)).
% 0.97/1.35 Proof: Axiom.
% 0.97/1.35
% 0.97/1.35 3: multiply(X0, add(X1, X2)) = add(multiply(X0, X1), multiply(X0, X2)).
% 0.97/1.35 Proof: Axiom.
% 0.97/1.35
% 0.97/1.35 4: add(X0, additive_identity()) = X0.
% 0.97/1.35 Proof: Axiom.
% 0.97/1.35
% 0.97/1.35 5: multiply(X0, multiplicative_identity()) = X0.
% 0.97/1.35 Proof: Axiom.
% 0.97/1.35
% 0.97/1.35 6: add(X0, inverse(X0)) = multiplicative_identity().
% 0.97/1.35 Proof: Axiom.
% 0.97/1.35
% 0.97/1.35 7: multiply(X0, inverse(X0)) = additive_identity().
% 0.97/1.35 Proof: Axiom.
% 0.97/1.35
% 0.97/1.35 8: multiply(X0, add(X1, X2)) = multiply(multiply(add(X0, X0), add(X0, X1)), add(multiply(X0, X1), X2)).
% 0.97/1.35 Proof: Rewrite equation 3,
% 0.97/1.35 lhs with equations []
% 0.97/1.35 rhs with equations [2,0,2].
% 0.97/1.35
% 0.97/1.35 9: add(additive_identity(), X2) = X2.
% 0.97/1.35 Proof: A critical pair between equations 0 and 4.
% 0.97/1.35
% 0.97/1.35 10: multiply(multiplicative_identity(), X2) = X2.
% 0.97/1.35 Proof: A critical pair between equations 1 and 5.
% 0.97/1.35
% 0.97/1.35 13: add(multiply(X4, X5), X3) = multiply(add(X3, X4), add(X3, X5)).
% 0.97/1.35 Proof: A critical pair between equations 0 and 2.
% 0.97/1.35
% 0.97/1.35 14: multiply(add(X0, X3), add(X0, X4)) = add(X0, multiply(X4, X3)).
% 0.97/1.35 Proof: A critical pair between equations 2 and 1.
% 0.97/1.35
% 0.97/1.35 18: multiply(X3, add(multiplicative_identity(), X2)) = multiply(multiply(add(X3, X3), add(X3, multiplicative_identity())), add(X3, X2)).
% 0.97/1.35 Proof: A critical pair between equations 8 and 5.
% 0.97/1.35
% 0.97/1.35 23: multiply(X3, add(multiplicative_identity(), X2)) = add(X3, multiply(X3, X2)).
% 0.97/1.35 Proof: Rewrite equation 18,
% 0.97/1.35 lhs with equations []
% 0.97/1.35 rhs with equations [13,5,13,0].
% 0.97/1.35
% 0.97/1.35 27: add(multiply(X4, X5), X3) = add(X3, multiply(X4, X5)).
% 0.97/1.35 Proof: Rewrite equation 13,
% 0.97/1.35 lhs with equations []
% 0.97/1.35 rhs with equations [2].
% 0.97/1.35
% 0.97/1.35 29: add(X0, multiply(X3, X4)) = add(X0, multiply(X4, X3)).
% 0.97/1.35 Proof: Rewrite equation 14,
% 0.97/1.35 lhs with equations [2]
% 0.97/1.35 rhs with equations [].
% 0.97/1.35
% 0.97/1.35 30: inverse(multiplicative_identity()) = additive_identity().
% 0.97/1.35 Proof: A critical pair between equations 10 and 7.
% 0.97/1.35
% 0.97/1.35 31: inverse(additive_identity()) = multiplicative_identity().
% 0.97/1.35 Proof: A critical pair between equations 9 and 6.
% 0.97/1.35
% 0.97/1.35 37: add(X3, multiply(X3, additive_identity())) = multiply(X3, multiplicative_identity()).
% 0.97/1.35 Proof: A critical pair between equations 23 and 4.
% 0.97/1.35
% 0.97/1.35 39: add(multiply(X7, X8), X6) = add(X6, multiply(X8, X7)).
% 0.97/1.35 Proof: A critical pair between equations 27 and 29.
% 0.97/1.35
% 0.97/1.35 44: inverse(inverse(additive_identity())) = additive_identity().
% 0.97/1.35 Proof: Rewrite equation 30,
% 0.97/1.35 lhs with equations [31]
% 0.97/1.35 rhs with equations [].
% 0.97/1.35
% 0.97/1.35 46: multiply(X0, inverse(additive_identity())) = X0.
% 0.97/1.35 Proof: Rewrite equation 5,
% 0.97/1.35 lhs with equations [31]
% 0.97/1.35 rhs with equations [].
% 0.97/1.35
% 0.97/1.35 52: add(multiply(X7, X8), X6) = multiply(add(X6, X8), add(X6, X7)).
% 0.97/1.35 Proof: Rewrite equation 39,
% 0.97/1.35 lhs with equations []
% 0.97/1.35 rhs with equations [2].
% 0.97/1.35
% 0.97/1.35 53: multiply(X3, add(X3, X3)) = X3.
% 0.97/1.35 Proof: Rewrite equation 37,
% 0.97/1.35 lhs with equations [2,4,1]
% 0.97/1.35 rhs with equations [31,46].
% 0.97/1.35
% 0.97/1.35 63: additive_identity() = multiply(additive_identity(), additive_identity()).
% 0.97/1.35 Proof: A critical pair between equations 53 and 4.
% 0.97/1.35
% 0.97/1.35 71: add(multiply(additive_identity(), X8), X9) = multiply(add(X9, X8), X9).
% 0.97/1.35 Proof: A critical pair between equations 52 and 4.
% 0.97/1.35
% 0.97/1.35 76: add(multiply(additive_identity(), X8), X9) = multiply(X9, add(X9, X8)).
% 0.97/1.35 Proof: Rewrite equation 71,
% 0.97/1.35 lhs with equations []
% 0.97/1.35 rhs with equations [1].
% 0.97/1.35
% 0.97/1.35 87: inverse(multiplicative_identity()) = additive_identity().
% 0.97/1.35 Proof: Rewrite equation 44,
% 0.97/1.35 lhs with equations [31]
% 0.97/1.35 rhs with equations [].
% 0.97/1.35
% 0.97/1.35 103: add(multiply(additive_identity(), additive_identity()), X10) = multiply(X10, X10).
% 0.97/1.35 Proof: A critical pair between equations 76 and 4.
% 0.97/1.35
% 0.97/1.35 116: add(inverse(multiplicative_identity()), X2) = X2.
% 0.97/1.35 Proof: Rewrite equation 9,
% 0.97/1.35 lhs with equations [87]
% 0.97/1.35 rhs with equations [].
% 0.97/1.35
% 0.97/1.35 120: inverse(multiplicative_identity()) = multiply(inverse(multiplicative_identity()), inverse(multiplicative_identity())).
% 0.97/1.35 Proof: Rewrite equation 63,
% 0.97/1.35 lhs with equations [87]
% 0.97/1.35 rhs with equations [87,87].
% 0.97/1.35
% 0.97/1.35 124: X10 = multiply(X10, X10).
% 0.97/1.35 Proof: Rewrite equation 103,
% 0.97/1.35 lhs with equations [87,87,120,116]
% 0.97/1.35 rhs with equations [].
% 0.97/1.35
% 0.97/1.35 130: multiply(a(), a()) = a().
% 0.97/1.35 Proof: Rewrite lhs with equations [124]
% 0.97/1.35 rhs with equations [].
% 0.97/1.35
% 0.97/1.35 % SZS output end Proof
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