TSTP Solution File: LCL159-1 by Toma---0.4
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%------------------------------------------------------------------------------
% File : Toma---0.4
% Problem : LCL159-1 : TPTP v8.1.2. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% 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 : Thu Aug 31 08:13:59 EDT 2023
% Result : Unsatisfiable 5.14s 5.51s
% Output : CNFRefutation 5.14s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.10 % Problem : LCL159-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.10 % Command : toma --casc %s
% 0.08/0.30 % Computer : n023.cluster.edu
% 0.08/0.30 % Model : x86_64 x86_64
% 0.08/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.30 % Memory : 8042.1875MB
% 0.08/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.08/0.30 % CPULimit : 300
% 0.08/0.30 % WCLimit : 300
% 0.08/0.30 % DateTime : Thu Aug 24 17:11:57 EDT 2023
% 0.08/0.30 % CPUTime :
% 5.14/5.51 % SZS status Unsatisfiable
% 5.14/5.51 % SZS output start Proof
% 5.14/5.51 original problem:
% 5.14/5.51 axioms:
% 5.14/5.51 implies(truth(), X) = X
% 5.14/5.51 implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth()
% 5.14/5.51 implies(implies(X, Y), Y) = implies(implies(Y, X), X)
% 5.14/5.51 implies(implies(not(X), not(Y)), implies(Y, X)) = truth()
% 5.14/5.51 or(X, Y) = implies(not(X), Y)
% 5.14/5.51 or(or(X, Y), Z) = or(X, or(Y, Z))
% 5.14/5.51 or(X, Y) = or(Y, X)
% 5.14/5.51 and(X, Y) = not(or(not(X), not(Y)))
% 5.14/5.51 and(and(X, Y), Z) = and(X, and(Y, Z))
% 5.14/5.51 and(X, Y) = and(Y, X)
% 5.14/5.51 xor(X, Y) = or(and(X, not(Y)), and(not(X), Y))
% 5.14/5.51 xor(X, Y) = xor(Y, X)
% 5.14/5.51 and_star(X, Y) = not(or(not(X), not(Y)))
% 5.14/5.51 and_star(and_star(X, Y), Z) = and_star(X, and_star(Y, Z))
% 5.14/5.51 and_star(X, Y) = and_star(Y, X)
% 5.14/5.51 not(truth()) = falsehood()
% 5.14/5.51 goal:
% 5.14/5.51 xor(x(), xor(truth(), y())) != xor(xor(x(), truth()), y())
% 5.14/5.51 To show the unsatisfiability of the original goal,
% 5.14/5.51 it suffices to show that xor(x(), xor(truth(), y())) = xor(xor(x(), truth()), y()) (skolemized goal) is valid under the axioms.
% 5.14/5.51 Here is an equational proof:
% 5.14/5.51 0: implies(truth(), X0) = X0.
% 5.14/5.51 Proof: Axiom.
% 5.14/5.51
% 5.14/5.51 1: implies(implies(X0, X1), implies(implies(X1, X2), implies(X0, X2))) = truth().
% 5.14/5.51 Proof: Axiom.
% 5.14/5.51
% 5.14/5.51 2: implies(implies(X0, X1), X1) = implies(implies(X1, X0), X0).
% 5.14/5.51 Proof: Axiom.
% 5.14/5.51
% 5.14/5.51 3: implies(implies(not(X0), not(X1)), implies(X1, X0)) = truth().
% 5.14/5.51 Proof: Axiom.
% 5.14/5.51
% 5.14/5.51 4: or(X0, X1) = implies(not(X0), X1).
% 5.14/5.51 Proof: Axiom.
% 5.14/5.51
% 5.14/5.51 5: or(or(X0, X1), X2) = or(X0, or(X1, X2)).
% 5.14/5.51 Proof: Axiom.
% 5.14/5.51
% 5.14/5.51 6: or(X0, X1) = or(X1, X0).
% 5.14/5.51 Proof: Axiom.
% 5.14/5.51
% 5.14/5.51 7: and(X0, X1) = not(or(not(X0), not(X1))).
% 5.14/5.51 Proof: Axiom.
% 5.14/5.51
% 5.14/5.51 9: and(X0, X1) = and(X1, X0).
% 5.14/5.51 Proof: Axiom.
% 5.14/5.51
% 5.14/5.51 10: xor(X0, X1) = or(and(X0, not(X1)), and(not(X0), X1)).
% 5.14/5.51 Proof: Axiom.
% 5.14/5.51
% 5.14/5.51 16: implies(not(implies(not(X0), X1)), X2) = implies(not(X0), implies(not(X1), X2)).
% 5.14/5.51 Proof: Rewrite equation 5,
% 5.14/5.51 lhs with equations [4,4]
% 5.14/5.51 rhs with equations [4,4].
% 5.14/5.51
% 5.14/5.51 17: and(X0, X1) = not(implies(not(not(X0)), not(X1))).
% 5.14/5.51 Proof: Rewrite equation 7,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 19: xor(X0, X1) = implies(not(not(implies(not(not(X0)), not(not(X1))))), not(implies(not(not(not(X0))), not(X1)))).
% 5.14/5.51 Proof: Rewrite equation 10,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [17,17,4].
% 5.14/5.51
% 5.14/5.51 21: implies(not(X0), X1) = implies(not(X1), X0).
% 5.14/5.51 Proof: Rewrite equation 6,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 22: not(implies(not(not(X0)), not(X1))) = not(implies(not(not(X1)), not(X0))).
% 5.14/5.51 Proof: Rewrite equation 9,
% 5.14/5.51 lhs with equations [17]
% 5.14/5.51 rhs with equations [17].
% 5.14/5.51
% 5.14/5.51 25: implies(implies(X2, truth()), truth()) = implies(X2, X2).
% 5.14/5.51 Proof: A critical pair between equations 2 and 0.
% 5.14/5.51
% 5.14/5.51 29: truth() = implies(implies(X0, truth()), implies(X3, implies(X0, X3))).
% 5.14/5.51 Proof: A critical pair between equations 1 and 0.
% 5.14/5.51
% 5.14/5.51 30: truth() = implies(X3, implies(implies(X3, X2), implies(truth(), X2))).
% 5.14/5.51 Proof: A critical pair between equations 1 and 0.
% 5.14/5.51
% 5.14/5.51 32: truth() = implies(implies(not(not(X1)), X2), implies(X1, X2)).
% 5.14/5.51 Proof: A critical pair between equations 3 and 21.
% 5.14/5.51
% 5.14/5.51 34: implies(implies(X3, not(X2)), not(X2)) = implies(implies(not(X3), X2), X3).
% 5.14/5.51 Proof: A critical pair between equations 2 and 21.
% 5.14/5.51
% 5.14/5.51 38: implies(implies(X3, not(X2)), not(X2)) = implies(or(X3, X2), X3).
% 5.14/5.51 Proof: Rewrite equation 34,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 40: truth() = implies(or(not(X1), X2), implies(X1, X2)).
% 5.14/5.51 Proof: Rewrite equation 32,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 41: truth() = implies(X3, implies(implies(X3, X2), X2)).
% 5.14/5.51 Proof: Rewrite equation 30,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [0].
% 5.14/5.51
% 5.14/5.51 44: xor(X0, X1) = or(not(or(not(X0), not(not(X1)))), not(or(not(not(X0)), not(X1)))).
% 5.14/5.51 Proof: Rewrite equation 19,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [4,4,4].
% 5.14/5.51
% 5.14/5.51 47: or(or(X0, X1), X2) = or(X0, or(X1, X2)).
% 5.14/5.51 Proof: Rewrite equation 16,
% 5.14/5.51 lhs with equations [4,4]
% 5.14/5.51 rhs with equations [4,4].
% 5.14/5.51
% 5.14/5.51 52: not(or(not(X0), not(X1))) = not(or(not(X1), not(X0))).
% 5.14/5.51 Proof: Rewrite equation 22,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 53: or(X0, X1) = or(X1, X0).
% 5.14/5.51 Proof: Rewrite equation 21,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 54: truth() = implies(X4, implies(X4, X4)).
% 5.14/5.51 Proof: A critical pair between equations 41 and 25.
% 5.14/5.51
% 5.14/5.51 56: truth() = implies(implies(truth(), X2), X2).
% 5.14/5.51 Proof: A critical pair between equations 41 and 0.
% 5.14/5.51
% 5.14/5.51 58: truth() = implies(or(not(truth()), X3), X3).
% 5.14/5.51 Proof: A critical pair between equations 40 and 0.
% 5.14/5.51
% 5.14/5.51 68: truth() = implies(implies(not(not(truth())), X3), X3).
% 5.14/5.51 Proof: Rewrite equation 58,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 69: truth() = implies(X2, X2).
% 5.14/5.51 Proof: Rewrite equation 56,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [0].
% 5.14/5.51
% 5.14/5.51 70: truth() = implies(X4, truth()).
% 5.14/5.51 Proof: Rewrite equation 54,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [69].
% 5.14/5.51
% 5.14/5.51 72: implies(not(implies(not(X0), X1)), X2) = implies(not(X0), implies(not(X1), X2)).
% 5.14/5.51 Proof: Rewrite equation 47,
% 5.14/5.51 lhs with equations [4,4]
% 5.14/5.51 rhs with equations [4,4].
% 5.14/5.51
% 5.14/5.51 75: xor(X0, X1) = implies(not(not(implies(not(not(X0)), not(not(X1))))), not(implies(not(not(not(X0))), not(X1)))).
% 5.14/5.51 Proof: Rewrite equation 44,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [4,4,4].
% 5.14/5.51
% 5.14/5.51 78: truth() = implies(X3, implies(X0, X3)).
% 5.14/5.51 Proof: Rewrite equation 29,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [70,0].
% 5.14/5.51
% 5.14/5.51 81: implies(implies(X3, not(X2)), not(X2)) = implies(implies(not(X3), X2), X3).
% 5.14/5.51 Proof: Rewrite equation 38,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 83: implies(not(X0), X1) = implies(not(X1), X0).
% 5.14/5.51 Proof: Rewrite equation 53,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 84: not(implies(not(not(X0)), not(X1))) = not(implies(not(not(X1)), not(X0))).
% 5.14/5.51 Proof: Rewrite equation 52,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 88: implies(not(truth()), X1) = truth().
% 5.14/5.51 Proof: A critical pair between equations 83 and 70.
% 5.14/5.51
% 5.14/5.51 89: truth() = implies(truth(), not(not(truth()))).
% 5.14/5.51 Proof: A critical pair between equations 68 and 69.
% 5.14/5.51
% 5.14/5.51 96: truth() = implies(X6, implies(not(X4), implies(not(X5), X6))).
% 5.14/5.51 Proof: A critical pair between equations 78 and 72.
% 5.14/5.51
% 5.14/5.51 99: implies(implies(X3, not(truth())), not(truth())) = implies(truth(), X3).
% 5.14/5.51 Proof: A critical pair between equations 81 and 70.
% 5.14/5.51
% 5.14/5.51 100: implies(implies(X3, not(truth())), not(truth())) = X3.
% 5.14/5.51 Proof: Rewrite equation 99,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [0].
% 5.14/5.51
% 5.14/5.51 103: truth() = implies(X6, or(X4, or(X5, X6))).
% 5.14/5.51 Proof: Rewrite equation 96,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [4,4].
% 5.14/5.51
% 5.14/5.51 108: truth() = not(not(truth())).
% 5.14/5.51 Proof: Rewrite equation 89,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [0].
% 5.14/5.51
% 5.14/5.51 109: or(truth(), X1) = truth().
% 5.14/5.51 Proof: Rewrite equation 88,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [].
% 5.14/5.51
% 5.14/5.51 115: xor(X0, X1) = or(not(or(not(X0), not(not(X1)))), not(or(not(not(X0)), not(X1)))).
% 5.14/5.51 Proof: Rewrite equation 75,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [4,4,4].
% 5.14/5.51
% 5.14/5.51 123: not(or(not(X0), not(X1))) = not(or(not(X1), not(X0))).
% 5.14/5.51 Proof: Rewrite equation 84,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 124: or(X0, X1) = or(X1, X0).
% 5.14/5.51 Proof: Rewrite equation 83,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 144: xor(X0, X1) = implies(not(not(implies(not(not(X0)), not(not(X1))))), not(implies(not(not(not(X0))), not(X1)))).
% 5.14/5.51 Proof: Rewrite equation 115,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [4,4,4].
% 5.14/5.51
% 5.14/5.51 150: implies(not(truth()), X1) = truth().
% 5.14/5.51 Proof: Rewrite equation 109,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [].
% 5.14/5.51
% 5.14/5.51 152: truth() = implies(X6, implies(not(X4), implies(not(X5), X6))).
% 5.14/5.51 Proof: Rewrite equation 103,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [4,4].
% 5.14/5.51
% 5.14/5.51 154: implies(not(X0), X1) = implies(not(X1), X0).
% 5.14/5.51 Proof: Rewrite equation 124,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 155: not(implies(not(not(X0)), not(X1))) = not(implies(not(not(X1)), not(X0))).
% 5.14/5.51 Proof: Rewrite equation 123,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 159: implies(not(X0), not(truth())) = implies(truth(), X0).
% 5.14/5.51 Proof: A critical pair between equations 154 and 108.
% 5.14/5.51
% 5.14/5.51 164: truth() = implies(X6, implies(not(implies(not(X5), X6)), X7)).
% 5.14/5.51 Proof: A critical pair between equations 152 and 154.
% 5.14/5.51
% 5.14/5.51 166: not(X4) = implies(implies(not(not(truth())), X4), not(truth())).
% 5.14/5.51 Proof: A critical pair between equations 100 and 154.
% 5.14/5.51
% 5.14/5.51 174: not(X4) = implies(X4, not(truth())).
% 5.14/5.51 Proof: Rewrite equation 166,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [108,0].
% 5.14/5.51
% 5.14/5.51 176: truth() = implies(X6, or(or(X5, X6), X7)).
% 5.14/5.51 Proof: Rewrite equation 164,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [4,4].
% 5.14/5.51
% 5.14/5.51 180: or(X0, not(truth())) = X0.
% 5.14/5.51 Proof: Rewrite equation 159,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [0].
% 5.14/5.51
% 5.14/5.51 184: or(truth(), X1) = truth().
% 5.14/5.51 Proof: Rewrite equation 150,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [].
% 5.14/5.51
% 5.14/5.51 190: xor(X0, X1) = or(not(or(not(X0), not(not(X1)))), not(or(not(not(X0)), not(X1)))).
% 5.14/5.51 Proof: Rewrite equation 144,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [4,4,4].
% 5.14/5.51
% 5.14/5.51 200: not(or(not(X0), not(X1))) = not(or(not(X1), not(X0))).
% 5.14/5.51 Proof: Rewrite equation 155,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 201: or(X0, X1) = or(X1, X0).
% 5.14/5.51 Proof: Rewrite equation 154,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 202: or(X2, truth()) = truth().
% 5.14/5.51 Proof: A critical pair between equations 201 and 184.
% 5.14/5.51
% 5.14/5.51 207: truth() = or(or(X5, truth()), X7).
% 5.14/5.51 Proof: A critical pair between equations 176 and 0.
% 5.14/5.51
% 5.14/5.51 215: truth() = implies(not(truth()), X7).
% 5.14/5.51 Proof: Rewrite equation 207,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [202,4].
% 5.14/5.51
% 5.14/5.51 222: xor(X0, X1) = implies(not(not(implies(not(not(X0)), not(not(X1))))), not(implies(not(not(not(X0))), not(X1)))).
% 5.14/5.51 Proof: Rewrite equation 190,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [4,4,4].
% 5.14/5.51
% 5.14/5.51 231: not(not(X0)) = X0.
% 5.14/5.51 Proof: Rewrite equation 180,
% 5.14/5.51 lhs with equations [4,174]
% 5.14/5.51 rhs with equations [].
% 5.14/5.51
% 5.14/5.51 234: implies(not(X0), X1) = implies(not(X1), X0).
% 5.14/5.51 Proof: Rewrite equation 201,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 235: not(implies(X0, not(X1))) = not(implies(X1, not(X0))).
% 5.14/5.51 Proof: Rewrite equation 200,
% 5.14/5.51 lhs with equations [4,231]
% 5.14/5.51 rhs with equations [4,231].
% 5.14/5.51
% 5.14/5.51 239: implies(not(X0), not(X2)) = implies(X2, X0).
% 5.14/5.51 Proof: A critical pair between equations 234 and 231.
% 5.14/5.51
% 5.14/5.51 246: implies(X2, not(X3)) = not(not(implies(X3, not(X2)))).
% 5.14/5.51 Proof: A critical pair between equations 231 and 235.
% 5.14/5.51
% 5.14/5.51 265: xor(X0, X1) = implies(or(X0, not(X1)), not(implies(X0, X1))).
% 5.14/5.51 Proof: Rewrite equation 222,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [231,231,231,231,4,246,231].
% 5.14/5.51
% 5.14/5.51 271: truth() = or(truth(), X7).
% 5.14/5.51 Proof: Rewrite equation 215,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [4].
% 5.14/5.51
% 5.14/5.51 272: implies(X2, not(X3)) = implies(X3, not(X2)).
% 5.14/5.51 Proof: Rewrite equation 246,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [231].
% 5.14/5.51
% 5.14/5.51 274: or(X0, not(X2)) = implies(X2, X0).
% 5.14/5.51 Proof: Rewrite equation 239,
% 5.14/5.51 lhs with equations [4]
% 5.14/5.51 rhs with equations [].
% 5.14/5.51
% 5.14/5.51 279: or(X1, X0) = or(X0, X1).
% 5.14/5.51 Proof: Rewrite equation 234,
% 5.14/5.51 lhs with equations [274,231]
% 5.14/5.51 rhs with equations [274,231].
% 5.14/5.51
% 5.14/5.51 283: X4 = or(X4, not(truth())).
% 5.14/5.51 Proof: A critical pair between equations 0 and 274.
% 5.14/5.51
% 5.14/5.51 301: xor(X0, X1) = or(not(or(X1, not(X0))), not(or(X0, not(X1)))).
% 5.14/5.51 Proof: Rewrite equation 265,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [274,274].
% 5.14/5.51
% 5.14/5.51 309: not(X4) = or(not(truth()), not(X4)).
% 5.14/5.51 Proof: Rewrite equation 174,
% 5.14/5.51 lhs with equations []
% 5.14/5.51 rhs with equations [274].
% 5.14/5.51
% 5.14/5.51 319: or(not(X3), not(X2)) = or(not(X2), not(X3)).
% 5.14/5.51 Proof: Rewrite equation 272,
% 5.14/5.51 lhs with equations [274]
% 5.14/5.51 rhs with equations [274].
% 5.14/5.51
% 5.14/5.51 320: xor(x(), xor(truth(), y())) = xor(xor(x(), truth()), y()).
% 5.14/5.51 Proof: Rewrite lhs with equations [301,283,271,283,301,319,231,319]
% 5.14/5.51 rhs with equations [301,271,283,309,301,231,279].
% 5.14/5.51
% 5.14/5.51 % SZS output end Proof
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