TSTP Solution File: GRP002-2 by Toma---0.4
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%------------------------------------------------------------------------------
% File : Toma---0.4
% Problem : GRP002-2 : TPTP v8.1.2. Bugfixed v1.2.1.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n012.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 01:13:31 EDT 2023
% Result : Unsatisfiable 1.61s 1.94s
% Output : CNFRefutation 1.61s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : GRP002-2 : TPTP v8.1.2. Bugfixed v1.2.1.
% 0.00/0.13 % Command : toma --casc %s
% 0.13/0.35 % Computer : n012.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 20:07:39 EDT 2023
% 0.13/0.35 % CPUTime :
% 1.61/1.94 % SZS status Unsatisfiable
% 1.61/1.94 % SZS output start Proof
% 1.61/1.94 original problem:
% 1.61/1.94 axioms:
% 1.61/1.94 multiply(identity(), X) = X
% 1.61/1.94 multiply(inverse(X), X) = identity()
% 1.61/1.94 multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z))
% 1.61/1.94 multiply(X, identity()) = X
% 1.61/1.94 multiply(X, inverse(X)) = identity()
% 1.61/1.94 multiply(X, multiply(X, X)) = identity()
% 1.61/1.94 multiply(a(), b()) = c()
% 1.61/1.94 multiply(c(), inverse(a())) = d()
% 1.61/1.94 multiply(d(), inverse(b())) = h()
% 1.61/1.94 multiply(h(), b()) = j()
% 1.61/1.94 multiply(j(), inverse(h())) = k()
% 1.61/1.94 goal:
% 1.61/1.94 multiply(k(), inverse(b())) != identity()
% 1.61/1.94 To show the unsatisfiability of the original goal,
% 1.61/1.94 it suffices to show that multiply(k(), inverse(b())) = identity() (skolemized goal) is valid under the axioms.
% 1.61/1.94 Here is an equational proof:
% 1.61/1.94 0: multiply(identity(), X0) = X0.
% 1.61/1.94 Proof: Axiom.
% 1.61/1.94
% 1.61/1.94 1: multiply(inverse(X0), X0) = identity().
% 1.61/1.94 Proof: Axiom.
% 1.61/1.94
% 1.61/1.94 2: multiply(multiply(X0, X1), X2) = multiply(X0, multiply(X1, X2)).
% 1.61/1.94 Proof: Axiom.
% 1.61/1.94
% 1.61/1.94 3: multiply(X0, identity()) = X0.
% 1.61/1.94 Proof: Axiom.
% 1.61/1.94
% 1.61/1.94 4: multiply(X0, inverse(X0)) = identity().
% 1.61/1.94 Proof: Axiom.
% 1.61/1.94
% 1.61/1.94 5: multiply(X0, multiply(X0, X0)) = identity().
% 1.61/1.94 Proof: Axiom.
% 1.61/1.94
% 1.61/1.94 6: multiply(a(), b()) = c().
% 1.61/1.94 Proof: Axiom.
% 1.61/1.94
% 1.61/1.94 7: multiply(c(), inverse(a())) = d().
% 1.61/1.94 Proof: Axiom.
% 1.61/1.94
% 1.61/1.94 8: multiply(d(), inverse(b())) = h().
% 1.61/1.94 Proof: Axiom.
% 1.61/1.94
% 1.61/1.94 9: multiply(h(), b()) = j().
% 1.61/1.94 Proof: Axiom.
% 1.61/1.94
% 1.61/1.94 10: multiply(j(), inverse(h())) = k().
% 1.61/1.94 Proof: Axiom.
% 1.61/1.94
% 1.61/1.94 11: multiply(a(), multiply(b(), inverse(a()))) = d().
% 1.61/1.94 Proof: Rewrite equation 7,
% 1.61/1.94 lhs with equations [6,2]
% 1.61/1.94 rhs with equations [].
% 1.61/1.94
% 1.61/1.94 12: multiply(a(), multiply(b(), multiply(inverse(a()), inverse(b())))) = h().
% 1.61/1.94 Proof: Rewrite equation 8,
% 1.61/1.94 lhs with equations [11,2,2]
% 1.61/1.94 rhs with equations [].
% 1.61/1.94
% 1.61/1.94 13: multiply(a(), multiply(b(), inverse(a()))) = j().
% 1.61/1.94 Proof: Rewrite equation 9,
% 1.61/1.94 lhs with equations [12,2,2,2,1,3]
% 1.61/1.94 rhs with equations [].
% 1.61/1.94
% 1.61/1.94 14: multiply(a(), multiply(b(), multiply(inverse(a()), inverse(multiply(a(), multiply(b(), multiply(inverse(a()), inverse(b())))))))) = k().
% 1.61/1.94 Proof: Rewrite equation 10,
% 1.61/1.94 lhs with equations [13,12,2,2]
% 1.61/1.94 rhs with equations [].
% 1.61/1.94
% 1.61/1.94 19: multiply(X3, multiply(multiply(X3, X3), X2)) = multiply(identity(), X2).
% 1.61/1.94 Proof: A critical pair between equations 2 and 5.
% 1.61/1.94
% 1.61/1.94 21: identity() = multiply(multiply(X3, X4), multiply(X3, multiply(X4, multiply(X3, X4)))).
% 1.61/1.94 Proof: A critical pair between equations 5 and 2.
% 1.61/1.94
% 1.61/1.94 22: identity() = multiply(X3, multiply(X4, multiply(X3, multiply(X4, multiply(X3, X4))))).
% 1.61/1.94 Proof: Rewrite equation 21,
% 1.61/1.94 lhs with equations []
% 1.61/1.94 rhs with equations [2].
% 1.61/1.94
% 1.61/1.94 23: multiply(X3, multiply(X3, multiply(X3, X2))) = X2.
% 1.61/1.94 Proof: Rewrite equation 19,
% 1.61/1.94 lhs with equations [2]
% 1.61/1.94 rhs with equations [0].
% 1.61/1.94
% 1.61/1.94 26: multiply(a(), multiply(b(), multiply(inverse(a()), inverse(h())))) = k().
% 1.61/1.94 Proof: Rewrite equation 14,
% 1.61/1.94 lhs with equations [12]
% 1.61/1.94 rhs with equations [].
% 1.61/1.94
% 1.61/1.94 31: inverse(X4) = multiply(X4, multiply(X4, identity())).
% 1.61/1.94 Proof: A critical pair between equations 23 and 4.
% 1.61/1.94
% 1.61/1.94 40: inverse(X4) = multiply(X4, X4).
% 1.61/1.94 Proof: Rewrite equation 31,
% 1.61/1.94 lhs with equations []
% 1.61/1.94 rhs with equations [3].
% 1.61/1.94
% 1.61/1.94 42: multiply(a(), multiply(b(), multiply(a(), multiply(a(), multiply(b(), b()))))) = h().
% 1.61/1.94 Proof: Rewrite equation 12,
% 1.61/1.94 lhs with equations [40,40,2]
% 1.61/1.94 rhs with equations [].
% 1.61/1.94
% 1.61/1.94 44: multiply(a(), multiply(b(), multiply(b(), multiply(a(), multiply(a(), multiply(b(), multiply(b(), multiply(a(), multiply(b(), multiply(a(), multiply(a(), multiply(b(), b())))))))))))) = k().
% 1.61/1.94 Proof: Rewrite equation 26,
% 1.61/1.94 lhs with equations [40,42,40,2,2,2,2,2,2,23]
% 1.61/1.94 rhs with equations [].
% 1.61/1.94
% 1.61/1.94 48: multiply(X6, multiply(X5, multiply(X6, multiply(X5, X6)))) = multiply(X5, multiply(X5, identity())).
% 1.61/1.94 Proof: A critical pair between equations 23 and 22.
% 1.61/1.94
% 1.61/1.94 49: multiply(X5, multiply(multiply(X6, multiply(X5, multiply(X6, multiply(X5, X6)))), X2)) = multiply(identity(), X2).
% 1.61/1.94 Proof: A critical pair between equations 2 and 22.
% 1.61/1.94
% 1.61/1.94 54: identity() = multiply(X3, multiply(X5, multiply(X6, multiply(X3, multiply(multiply(X5, X6), multiply(X3, multiply(X5, X6))))))).
% 1.61/1.94 Proof: A critical pair between equations 22 and 2.
% 1.61/1.94
% 1.61/1.94 55: identity() = multiply(X3, multiply(X5, multiply(X6, multiply(X3, multiply(X5, multiply(X6, multiply(X3, multiply(X5, X6)))))))).
% 1.61/1.94 Proof: Rewrite equation 54,
% 1.61/1.94 lhs with equations []
% 1.61/1.94 rhs with equations [2].
% 1.61/1.94
% 1.61/1.94 56: multiply(X5, multiply(X6, multiply(X5, multiply(X6, multiply(X5, multiply(X6, X2)))))) = X2.
% 1.61/1.94 Proof: Rewrite equation 49,
% 1.61/1.94 lhs with equations [2,2,2,2]
% 1.61/1.94 rhs with equations [0].
% 1.61/1.94
% 1.61/1.94 57: multiply(a(), multiply(b(), multiply(b(), multiply(a(), multiply(a(), multiply(b(), multiply(b(), multiply(a(), multiply(b(), multiply(a(), multiply(a(), inverse(b())))))))))))) = k().
% 1.61/1.94 Proof: Rewrite equation 44,
% 1.61/1.94 lhs with equations [40]
% 1.61/1.94 rhs with equations [].
% 1.61/1.94
% 1.61/1.94 62: multiply(X6, multiply(X5, multiply(X6, multiply(X5, X6)))) = inverse(X5).
% 1.61/1.94 Proof: Rewrite equation 48,
% 1.61/1.94 lhs with equations []
% 1.61/1.94 rhs with equations [3,40].
% 1.61/1.94
% 1.61/1.94 74: multiply(X7, multiply(multiply(X8, multiply(X7, multiply(X8, X7))), X2)) = multiply(inverse(X8), X2).
% 1.61/1.94 Proof: A critical pair between equations 2 and 62.
% 1.61/1.94
% 1.61/1.94 79: multiply(a(), multiply(b(), multiply(b(), multiply(a(), multiply(a(), multiply(b(), multiply(b(), multiply(a(), multiply(b(), multiply(a(), multiply(a(), multiply(b(), b())))))))))))) = k().
% 1.61/1.94 Proof: Rewrite equation 57,
% 1.61/1.94 lhs with equations [40]
% 1.61/1.94 rhs with equations [].
% 1.61/1.94
% 1.61/1.94 80: multiply(X7, multiply(X8, multiply(X7, multiply(X8, multiply(X7, X2))))) = multiply(X8, multiply(X8, X2)).
% 1.61/1.94 Proof: Rewrite equation 74,
% 1.61/1.94 lhs with equations [2,2,2]
% 1.61/1.94 rhs with equations [40,2].
% 1.61/1.94
% 1.61/1.94 89: multiply(X10, multiply(X9, multiply(X10, multiply(X9, X11)))) = multiply(X9, multiply(X9, multiply(X10, multiply(X10, X11)))).
% 1.61/1.94 Proof: A critical pair between equations 23 and 80.
% 1.61/1.94
% 1.61/1.94 93: multiply(X8, multiply(X9, multiply(X7, multiply(X8, multiply(X9, multiply(X7, multiply(X8, X9))))))) = multiply(X7, multiply(X7, identity())).
% 1.61/1.94 Proof: A critical pair between equations 23 and 55.
% 1.61/1.94
% 1.61/1.94 94: X9 = multiply(X5, multiply(multiply(X7, X8), multiply(X5, multiply(multiply(X7, X8), multiply(X5, multiply(X7, multiply(X8, X9))))))).
% 1.61/1.94 Proof: A critical pair between equations 56 and 2.
% 1.61/1.94
% 1.61/1.94 95: X9 = multiply(X5, multiply(X7, multiply(X8, multiply(X5, multiply(X7, multiply(X8, multiply(X5, multiply(X7, multiply(X8, X9))))))))).
% 1.61/1.94 Proof: Rewrite equation 94,
% 1.61/1.94 lhs with equations []
% 1.61/1.94 rhs with equations [2,2].
% 1.61/1.94
% 1.61/1.94 96: multiply(X8, multiply(X9, multiply(X7, multiply(X8, multiply(X9, multiply(X7, multiply(X8, X9))))))) = inverse(X7).
% 1.61/1.94 Proof: Rewrite equation 93,
% 1.61/1.94 lhs with equations []
% 1.61/1.94 rhs with equations [3,40].
% 1.61/1.94
% 1.61/1.94 99: identity() = multiply(X3, inverse(X3)).
% 1.61/1.94 Proof: Rewrite equation 55,
% 1.61/1.94 lhs with equations []
% 1.61/1.94 rhs with equations [96].
% 1.61/1.94
% 1.61/1.94 100: multiply(a(), multiply(a(), multiply(b(), multiply(a(), multiply(a(), multiply(b(), multiply(a(), multiply(a(), inverse(b()))))))))) = k().
% 1.61/1.94 Proof: Rewrite equation 79,
% 1.61/1.94 lhs with equations [40,89,89,89,23]
% 1.61/1.94 rhs with equations [].
% 1.61/1.94
% 1.61/1.94 113: identity() = multiply(X4, multiply(X5, inverse(multiply(X4, X5)))).
% 1.61/1.94 Proof: A critical pair between equations 99 and 2.
% 1.61/1.94
% 1.61/1.94 122: identity() = multiply(X5, multiply(X5, X5)).
% 1.61/1.94 Proof: Rewrite equation 113,
% 1.61/1.94 lhs with equations []
% 1.61/1.94 rhs with equations [40,2,80].
% 1.61/1.94
% 1.61/1.94 126: b() = k().
% 1.61/1.94 Proof: Rewrite equation 100,
% 1.61/1.94 lhs with equations [40,95]
% 1.61/1.94 rhs with equations [].
% 1.61/1.94
% 1.61/1.94 134: multiply(k(), inverse(b())) = identity().
% 1.61/1.94 Proof: Rewrite lhs with equations [126,40,122]
% 1.61/1.94 rhs with equations [].
% 1.61/1.94
% 1.61/1.94 % SZS output end Proof
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