TSTP Solution File: GRP002-4 by Toma---0.4
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% File : Toma---0.4
% Problem : GRP002-4 : TPTP v8.1.2. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n024.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 01:13:32 EDT 2023
% Result : Unsatisfiable 11.12s 11.51s
% Output : CNFRefutation 11.12s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : GRP002-4 : TPTP v8.1.2. Released v1.0.0.
% 0.07/0.14 % Command : toma --casc %s
% 0.13/0.35 % Computer : n024.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 22:03:23 EDT 2023
% 0.13/0.35 % CPUTime :
% 11.12/11.51 % SZS status Unsatisfiable
% 11.12/11.51 % SZS output start Proof
% 11.12/11.51 original problem:
% 11.12/11.51 axioms:
% 11.12/11.51 multiply(identity(), X) = X
% 11.12/11.51 multiply(inverse(X), X) = identity()
% 11.12/11.51 multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z))
% 11.12/11.51 multiply(X, identity()) = X
% 11.12/11.51 multiply(X, inverse(X)) = identity()
% 11.12/11.51 commutator(X, Y) = multiply(X, multiply(Y, multiply(inverse(X), inverse(Y))))
% 11.12/11.51 multiply(X, multiply(X, X)) = identity()
% 11.12/11.51 goal:
% 11.12/11.51 commutator(commutator(a(), b()), b()) != identity()
% 11.12/11.51 To show the unsatisfiability of the original goal,
% 11.12/11.51 it suffices to show that commutator(commutator(a(), b()), b()) = identity() (skolemized goal) is valid under the axioms.
% 11.12/11.51 Here is an equational proof:
% 11.12/11.51 0: multiply(identity(), X0) = X0.
% 11.12/11.51 Proof: Axiom.
% 11.12/11.51
% 11.12/11.51 1: multiply(inverse(X0), X0) = identity().
% 11.12/11.51 Proof: Axiom.
% 11.12/11.51
% 11.12/11.51 2: multiply(multiply(X0, X1), X2) = multiply(X0, multiply(X1, X2)).
% 11.12/11.51 Proof: Axiom.
% 11.12/11.51
% 11.12/11.51 3: multiply(X0, identity()) = X0.
% 11.12/11.51 Proof: Axiom.
% 11.12/11.51
% 11.12/11.51 4: multiply(X0, inverse(X0)) = identity().
% 11.12/11.51 Proof: Axiom.
% 11.12/11.51
% 11.12/11.51 5: commutator(X0, X1) = multiply(X0, multiply(X1, multiply(inverse(X0), inverse(X1)))).
% 11.12/11.51 Proof: Axiom.
% 11.12/11.51
% 11.12/11.51 6: multiply(X0, multiply(X0, X0)) = identity().
% 11.12/11.51 Proof: Axiom.
% 11.12/11.51
% 11.12/11.51 8: multiply(inverse(X3), multiply(X3, X2)) = multiply(identity(), X2).
% 11.12/11.51 Proof: A critical pair between equations 2 and 1.
% 11.12/11.51
% 11.12/11.51 9: multiply(X3, multiply(inverse(X3), X2)) = multiply(identity(), X2).
% 11.12/11.51 Proof: A critical pair between equations 2 and 4.
% 11.12/11.51
% 11.12/11.51 10: multiply(X0, multiply(X1, inverse(multiply(X0, X1)))) = identity().
% 11.12/11.51 Proof: A critical pair between equations 2 and 4.
% 11.12/11.51
% 11.12/11.51 11: multiply(X3, multiply(multiply(X3, X3), X2)) = multiply(identity(), X2).
% 11.12/11.51 Proof: A critical pair between equations 2 and 6.
% 11.12/11.51
% 11.12/11.51 15: multiply(X3, multiply(X3, multiply(X3, X2))) = X2.
% 11.12/11.51 Proof: Rewrite equation 11,
% 11.12/11.51 lhs with equations [2]
% 11.12/11.51 rhs with equations [0].
% 11.12/11.51
% 11.12/11.51 16: multiply(X3, multiply(inverse(X3), X2)) = X2.
% 11.12/11.51 Proof: Rewrite equation 9,
% 11.12/11.51 lhs with equations []
% 11.12/11.51 rhs with equations [0].
% 11.12/11.51
% 11.12/11.51 17: multiply(inverse(X3), multiply(X3, X2)) = X2.
% 11.12/11.51 Proof: Rewrite equation 8,
% 11.12/11.51 lhs with equations []
% 11.12/11.51 rhs with equations [0].
% 11.12/11.51
% 11.12/11.51 21: inverse(X4) = multiply(X4, multiply(X4, identity())).
% 11.12/11.51 Proof: A critical pair between equations 15 and 4.
% 11.12/11.51
% 11.12/11.51 28: multiply(X5, inverse(multiply(inverse(X3), X5))) = multiply(X3, identity()).
% 11.12/11.51 Proof: A critical pair between equations 16 and 10.
% 11.12/11.51
% 11.12/11.51 29: multiply(X5, inverse(multiply(X4, X5))) = multiply(inverse(X4), identity()).
% 11.12/11.51 Proof: A critical pair between equations 17 and 10.
% 11.12/11.51
% 11.12/11.51 30: multiply(X5, multiply(X4, multiply(X5, multiply(X4, X5)))) = multiply(X4, X4).
% 11.12/11.51 Proof: Rewrite equation 29,
% 11.12/11.51 lhs with equations [21,2,3,2]
% 11.12/11.51 rhs with equations [21,3,2,3].
% 11.12/11.51
% 11.12/11.51 31: multiply(X5, multiply(X3, multiply(X3, multiply(X5, multiply(X3, multiply(X3, X5)))))) = X3.
% 11.12/11.51 Proof: Rewrite equation 28,
% 11.12/11.51 lhs with equations [21,3,2,21,2,2,3,2,2]
% 11.12/11.51 rhs with equations [3].
% 11.12/11.51
% 11.12/11.51 32: inverse(X4) = multiply(X4, X4).
% 11.12/11.51 Proof: Rewrite equation 21,
% 11.12/11.51 lhs with equations []
% 11.12/11.51 rhs with equations [3].
% 11.12/11.51
% 11.12/11.51 33: commutator(X0, X1) = multiply(X0, multiply(X1, multiply(X0, multiply(X0, multiply(X1, X1))))).
% 11.12/11.51 Proof: Rewrite equation 5,
% 11.12/11.51 lhs with equations []
% 11.12/11.51 rhs with equations [32,32,2].
% 11.12/11.51
% 11.12/11.51 35: multiply(X7, multiply(X6, multiply(X7, X6))) = multiply(X6, multiply(X6, multiply(X7, X7))).
% 11.12/11.51 Proof: A critical pair between equations 15 and 30.
% 11.12/11.51
% 11.12/11.51 38: multiply(X6, multiply(multiply(X7, multiply(X6, multiply(X7, X6))), X2)) = multiply(multiply(X7, X7), X2).
% 11.12/11.51 Proof: A critical pair between equations 2 and 30.
% 11.12/11.51
% 11.12/11.51 39: multiply(X7, multiply(X7, multiply(X6, multiply(X7, multiply(X7, X6))))) = multiply(X6, multiply(X6, X7)).
% 11.12/11.51 Proof: A critical pair between equations 15 and 31.
% 11.12/11.51
% 11.12/11.51 43: multiply(multiply(X6, X7), multiply(X6, X7)) = multiply(X8, multiply(multiply(X6, X7), multiply(X8, multiply(X6, multiply(X7, X8))))).
% 11.12/11.51 Proof: A critical pair between equations 30 and 2.
% 11.12/11.51
% 11.12/11.51 44: multiply(multiply(X6, X7), multiply(X6, X7)) = multiply(X5, multiply(X6, multiply(X7, multiply(X5, multiply(multiply(X6, X7), X5))))).
% 11.12/11.51 Proof: A critical pair between equations 30 and 2.
% 11.12/11.51
% 11.12/11.51 47: multiply(X6, multiply(X7, multiply(X6, X7))) = multiply(X8, multiply(X6, multiply(X7, multiply(X8, multiply(X6, multiply(X7, X8)))))).
% 11.12/11.51 Proof: Rewrite equation 43,
% 11.12/11.51 lhs with equations [2]
% 11.12/11.51 rhs with equations [2].
% 11.12/11.51
% 11.12/11.51 49: multiply(X6, multiply(X7, multiply(X6, multiply(X7, multiply(X6, X2))))) = multiply(inverse(X7), X2).
% 11.12/11.51 Proof: Rewrite equation 38,
% 11.12/11.51 lhs with equations [2,2,2]
% 11.12/11.51 rhs with equations [32].
% 11.12/11.51
% 11.12/11.51 51: multiply(X0, inverse(X0)) = identity().
% 11.12/11.51 Proof: Rewrite equation 6,
% 11.12/11.51 lhs with equations [32]
% 11.12/11.51 rhs with equations [].
% 11.12/11.51
% 11.12/11.51 52: commutator(X0, X1) = multiply(X0, multiply(X1, multiply(X0, multiply(X0, inverse(X1))))).
% 11.12/11.51 Proof: Rewrite equation 33,
% 11.12/11.51 lhs with equations []
% 11.12/11.51 rhs with equations [32].
% 11.12/11.51
% 11.12/11.51 54: inverse(multiply(X6, X7)) = multiply(X6, multiply(X7, multiply(X6, X7))).
% 11.12/11.51 Proof: Rewrite equation 44,
% 11.12/11.51 lhs with equations [32]
% 11.12/11.51 rhs with equations [2,47].
% 11.12/11.51
% 11.12/11.51 55: inverse(multiply(X7, X6)) = multiply(X6, multiply(X6, inverse(X7))).
% 11.12/11.51 Proof: Rewrite equation 35,
% 11.12/11.51 lhs with equations [54]
% 11.12/11.51 rhs with equations [32].
% 11.12/11.51
% 11.12/11.51 63: inverse(X9) = multiply(X8, inverse(multiply(X9, X8))).
% 11.12/11.51 Proof: A critical pair between equations 15 and 55.
% 11.12/11.51
% 11.12/11.51 69: multiply(X9, X9) = multiply(X8, multiply(X9, multiply(X8, multiply(X9, X8)))).
% 11.12/11.51 Proof: Rewrite equation 63,
% 11.12/11.51 lhs with equations [32]
% 11.12/11.51 rhs with equations [54].
% 11.12/11.51
% 11.12/11.51 70: commutator(X0, X1) = multiply(X0, multiply(X1, multiply(X0, multiply(X0, multiply(X1, X1))))).
% 11.12/11.51 Proof: Rewrite equation 52,
% 11.12/11.51 lhs with equations []
% 11.12/11.51 rhs with equations [32].
% 11.12/11.51
% 11.12/11.51 71: multiply(X0, multiply(X0, X0)) = identity().
% 11.12/11.51 Proof: Rewrite equation 51,
% 11.12/11.51 lhs with equations [32]
% 11.12/11.51 rhs with equations [].
% 11.12/11.51
% 11.12/11.51 73: multiply(X6, multiply(X7, multiply(X6, multiply(X7, multiply(X6, X2))))) = multiply(X7, multiply(X7, X2)).
% 11.12/11.51 Proof: Rewrite equation 49,
% 11.12/11.51 lhs with equations []
% 11.12/11.51 rhs with equations [32,2].
% 11.12/11.51
% 11.12/11.51 80: multiply(X11, multiply(X10, multiply(X10, multiply(X11, X10)))) = multiply(X10, multiply(X10, multiply(X11, multiply(X10, X11)))).
% 11.12/11.51 Proof: A critical pair between equations 15 and 47.
% 11.12/11.51
% 11.12/11.51 82: multiply(X9, multiply(X8, multiply(X9, multiply(X8, X10)))) = multiply(X8, multiply(X8, multiply(X9, multiply(X9, X10)))).
% 11.12/11.51 Proof: A critical pair between equations 15 and 73.
% 11.12/11.51
% 11.12/11.51 85: multiply(X8, multiply(multiply(X8, X9), X2)) = multiply(multiply(X9, multiply(X9, multiply(X8, multiply(X9, multiply(X9, X8))))), X2).
% 11.12/11.51 Proof: A critical pair between equations 2 and 39.
% 11.12/11.51
% 11.12/11.51 86: multiply(X8, multiply(X8, multiply(X9, X2))) = multiply(X9, multiply(X9, multiply(X8, multiply(X9, multiply(X9, multiply(X8, X2)))))).
% 11.12/11.51 Proof: Rewrite equation 85,
% 11.12/11.51 lhs with equations [2]
% 11.12/11.51 rhs with equations [2,2,2,2,2].
% 11.12/11.51
% 11.12/11.51 88: multiply(X0, inverse(X0)) = identity().
% 11.12/11.51 Proof: Rewrite equation 71,
% 11.12/11.51 lhs with equations [32]
% 11.12/11.51 rhs with equations [].
% 11.12/11.51
% 11.12/11.51 89: commutator(X0, X1) = multiply(X0, multiply(X1, multiply(X0, multiply(X0, inverse(X1))))).
% 11.12/11.51 Proof: Rewrite equation 70,
% 11.12/11.51 lhs with equations []
% 11.12/11.51 rhs with equations [32].
% 11.12/11.51
% 11.12/11.51 90: inverse(X9) = multiply(X8, multiply(X9, multiply(X8, multiply(X9, X8)))).
% 11.12/11.51 Proof: Rewrite equation 69,
% 11.12/11.51 lhs with equations [32]
% 11.12/11.51 rhs with equations [].
% 11.12/11.51
% 11.12/11.51 98: identity() = multiply(X3, multiply(X4, inverse(multiply(X3, X4)))).
% 11.12/11.51 Proof: A critical pair between equations 88 and 2.
% 11.12/11.51
% 11.12/11.51 109: identity() = multiply(X4, multiply(X4, X4)).
% 11.12/11.51 Proof: Rewrite equation 98,
% 11.12/11.51 lhs with equations []
% 11.12/11.51 rhs with equations [32,2,73].
% 11.12/11.51
% 11.12/11.51 110: commutator(X0, X1) = multiply(X0, multiply(X1, multiply(X0, multiply(X0, multiply(X1, X1))))).
% 11.12/11.51 Proof: Rewrite equation 89,
% 11.12/11.51 lhs with equations []
% 11.12/11.51 rhs with equations [32].
% 11.12/11.51
% 11.12/11.51 113: multiply(X9, X9) = multiply(X8, multiply(X9, multiply(X8, multiply(X9, X8)))).
% 11.12/11.51 Proof: Rewrite equation 90,
% 11.12/11.51 lhs with equations [32]
% 11.12/11.51 rhs with equations [].
% 11.12/11.51
% 11.12/11.51 119: commutator(commutator(a(), b()), b()) = identity().
% 11.12/11.51 Proof: Rewrite lhs with equations [110,110,2,2,2,2,2,109,3,2,2,2,2,2,82,86,15,82,86,109,3,15,15,86,80,15,2,2,2,2,2,86,113,109,3,80,15,113,109]
% 11.12/11.51 rhs with equations [].
% 11.12/11.51
% 11.12/11.51 % SZS output end Proof
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