TSTP Solution File: LCL153-1 by Toma---0.4
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% File : Toma---0.4
% Problem : LCL153-1 : TPTP v8.1.2. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n010.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 08:13:58 EDT 2023
% Result : Unsatisfiable 2.28s 2.72s
% Output : CNFRefutation 2.28s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.14 % Problem : LCL153-1 : TPTP v8.1.2. Released v1.0.0.
% 0.08/0.15 % Command : toma --casc %s
% 0.14/0.35 % Computer : n010.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Fri Aug 25 01:57:50 EDT 2023
% 0.14/0.36 % CPUTime :
% 2.28/2.72 % SZS status Unsatisfiable
% 2.28/2.72 % SZS output start Proof
% 2.28/2.72 original problem:
% 2.28/2.72 axioms:
% 2.28/2.72 implies(truth(), X) = X
% 2.28/2.72 implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth()
% 2.28/2.72 implies(implies(X, Y), Y) = implies(implies(Y, X), X)
% 2.28/2.72 implies(implies(not(X), not(Y)), implies(Y, X)) = truth()
% 2.28/2.72 or(X, Y) = implies(not(X), Y)
% 2.28/2.72 or(or(X, Y), Z) = or(X, or(Y, Z))
% 2.28/2.72 or(X, Y) = or(Y, X)
% 2.28/2.72 and(X, Y) = not(or(not(X), not(Y)))
% 2.28/2.72 and(and(X, Y), Z) = and(X, and(Y, Z))
% 2.28/2.72 and(X, Y) = and(Y, X)
% 2.28/2.72 xor(X, Y) = or(and(X, not(Y)), and(not(X), Y))
% 2.28/2.72 xor(X, Y) = xor(Y, X)
% 2.28/2.72 and_star(X, Y) = not(or(not(X), not(Y)))
% 2.28/2.72 and_star(and_star(X, Y), Z) = and_star(X, and_star(Y, Z))
% 2.28/2.72 and_star(X, Y) = and_star(Y, X)
% 2.28/2.72 not(truth()) = falsehood()
% 2.28/2.72 goal:
% 2.28/2.72 not(x()) != xor(x(), truth())
% 2.28/2.72 To show the unsatisfiability of the original goal,
% 2.28/2.72 it suffices to show that not(x()) = xor(x(), truth()) (skolemized goal) is valid under the axioms.
% 2.28/2.72 Here is an equational proof:
% 2.28/2.72 0: implies(truth(), X0) = X0.
% 2.28/2.72 Proof: Axiom.
% 2.28/2.72
% 2.28/2.72 1: implies(implies(X0, X1), implies(implies(X1, X2), implies(X0, X2))) = truth().
% 2.28/2.72 Proof: Axiom.
% 2.28/2.72
% 2.28/2.72 2: implies(implies(X0, X1), X1) = implies(implies(X1, X0), X0).
% 2.28/2.72 Proof: Axiom.
% 2.28/2.72
% 2.28/2.72 3: implies(implies(not(X0), not(X1)), implies(X1, X0)) = truth().
% 2.28/2.72 Proof: Axiom.
% 2.28/2.72
% 2.28/2.72 4: or(X0, X1) = implies(not(X0), X1).
% 2.28/2.72 Proof: Axiom.
% 2.28/2.72
% 2.28/2.72 6: or(X0, X1) = or(X1, X0).
% 2.28/2.72 Proof: Axiom.
% 2.28/2.72
% 2.28/2.72 7: and(X0, X1) = not(or(not(X0), not(X1))).
% 2.28/2.72 Proof: Axiom.
% 2.28/2.72
% 2.28/2.72 10: xor(X0, X1) = or(and(X0, not(X1)), and(not(X0), X1)).
% 2.28/2.72 Proof: Axiom.
% 2.28/2.72
% 2.28/2.72 17: and(X0, X1) = not(implies(not(not(X0)), not(X1))).
% 2.28/2.72 Proof: Rewrite equation 7,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [4].
% 2.28/2.72
% 2.28/2.72 19: xor(X0, X1) = implies(not(not(implies(not(not(X0)), not(not(X1))))), not(implies(not(not(not(X0))), not(X1)))).
% 2.28/2.72 Proof: Rewrite equation 10,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [17,17,4].
% 2.28/2.72
% 2.28/2.72 21: implies(not(X0), X1) = implies(not(X1), X0).
% 2.28/2.72 Proof: Rewrite equation 6,
% 2.28/2.72 lhs with equations [4]
% 2.28/2.72 rhs with equations [4].
% 2.28/2.72
% 2.28/2.72 25: implies(implies(X2, truth()), truth()) = implies(X2, X2).
% 2.28/2.72 Proof: A critical pair between equations 2 and 0.
% 2.28/2.72
% 2.28/2.72 30: truth() = implies(X3, implies(implies(X3, X2), implies(truth(), X2))).
% 2.28/2.72 Proof: A critical pair between equations 1 and 0.
% 2.28/2.72
% 2.28/2.72 32: truth() = implies(implies(not(not(X1)), X2), implies(X1, X2)).
% 2.28/2.72 Proof: A critical pair between equations 3 and 21.
% 2.28/2.72
% 2.28/2.72 34: implies(implies(X3, not(X2)), not(X2)) = implies(implies(not(X3), X2), X3).
% 2.28/2.72 Proof: A critical pair between equations 2 and 21.
% 2.28/2.72
% 2.28/2.72 38: implies(implies(X3, not(X2)), not(X2)) = implies(or(X3, X2), X3).
% 2.28/2.72 Proof: Rewrite equation 34,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [4].
% 2.28/2.72
% 2.28/2.72 40: truth() = implies(or(not(X1), X2), implies(X1, X2)).
% 2.28/2.72 Proof: Rewrite equation 32,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [4].
% 2.28/2.72
% 2.28/2.72 41: truth() = implies(X3, implies(implies(X3, X2), X2)).
% 2.28/2.72 Proof: Rewrite equation 30,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [0].
% 2.28/2.72
% 2.28/2.72 44: xor(X0, X1) = or(not(or(not(X0), not(not(X1)))), not(or(not(not(X0)), not(X1)))).
% 2.28/2.72 Proof: Rewrite equation 19,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [4,4,4].
% 2.28/2.72
% 2.28/2.72 53: or(X0, X1) = or(X1, X0).
% 2.28/2.72 Proof: Rewrite equation 21,
% 2.28/2.72 lhs with equations [4]
% 2.28/2.72 rhs with equations [4].
% 2.28/2.72
% 2.28/2.72 54: truth() = implies(X4, implies(X4, X4)).
% 2.28/2.72 Proof: A critical pair between equations 41 and 25.
% 2.28/2.72
% 2.28/2.72 56: truth() = implies(implies(truth(), X2), X2).
% 2.28/2.72 Proof: A critical pair between equations 41 and 0.
% 2.28/2.72
% 2.28/2.72 58: truth() = implies(or(not(truth()), X3), X3).
% 2.28/2.72 Proof: A critical pair between equations 40 and 0.
% 2.28/2.72
% 2.28/2.72 68: truth() = implies(implies(not(not(truth())), X3), X3).
% 2.28/2.72 Proof: Rewrite equation 58,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [4].
% 2.28/2.72
% 2.28/2.72 69: truth() = implies(X2, X2).
% 2.28/2.72 Proof: Rewrite equation 56,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [0].
% 2.28/2.72
% 2.28/2.72 70: truth() = implies(X4, truth()).
% 2.28/2.72 Proof: Rewrite equation 54,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [69].
% 2.28/2.72
% 2.28/2.72 75: xor(X0, X1) = implies(not(not(implies(not(not(X0)), not(not(X1))))), not(implies(not(not(not(X0))), not(X1)))).
% 2.28/2.72 Proof: Rewrite equation 44,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [4,4,4].
% 2.28/2.72
% 2.28/2.72 81: implies(implies(X3, not(X2)), not(X2)) = implies(implies(not(X3), X2), X3).
% 2.28/2.72 Proof: Rewrite equation 38,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [4].
% 2.28/2.72
% 2.28/2.72 83: implies(not(X0), X1) = implies(not(X1), X0).
% 2.28/2.72 Proof: Rewrite equation 53,
% 2.28/2.72 lhs with equations [4]
% 2.28/2.72 rhs with equations [4].
% 2.28/2.72
% 2.28/2.72 89: truth() = implies(truth(), not(not(truth()))).
% 2.28/2.72 Proof: A critical pair between equations 68 and 69.
% 2.28/2.72
% 2.28/2.72 99: implies(implies(X3, not(truth())), not(truth())) = implies(truth(), X3).
% 2.28/2.72 Proof: A critical pair between equations 81 and 70.
% 2.28/2.72
% 2.28/2.72 100: implies(implies(X3, not(truth())), not(truth())) = X3.
% 2.28/2.72 Proof: Rewrite equation 99,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [0].
% 2.28/2.72
% 2.28/2.72 108: truth() = not(not(truth())).
% 2.28/2.72 Proof: Rewrite equation 89,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [0].
% 2.28/2.72
% 2.28/2.72 115: xor(X0, X1) = or(not(or(not(X0), not(not(X1)))), not(or(not(not(X0)), not(X1)))).
% 2.28/2.72 Proof: Rewrite equation 75,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [4,4,4].
% 2.28/2.72
% 2.28/2.72 124: or(X0, X1) = or(X1, X0).
% 2.28/2.72 Proof: Rewrite equation 83,
% 2.28/2.72 lhs with equations [4]
% 2.28/2.72 rhs with equations [4].
% 2.28/2.72
% 2.28/2.72 143: xor(X0, X1) = implies(not(not(implies(not(not(X0)), not(not(X1))))), not(implies(not(not(not(X0))), not(X1)))).
% 2.28/2.72 Proof: Rewrite equation 115,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [4,4,4].
% 2.28/2.72
% 2.28/2.72 153: implies(not(X0), X1) = implies(not(X1), X0).
% 2.28/2.72 Proof: Rewrite equation 124,
% 2.28/2.72 lhs with equations [4]
% 2.28/2.72 rhs with equations [4].
% 2.28/2.72
% 2.28/2.72 158: implies(not(X0), not(truth())) = implies(truth(), X0).
% 2.28/2.72 Proof: A critical pair between equations 153 and 108.
% 2.28/2.72
% 2.28/2.72 165: not(X4) = implies(implies(not(not(truth())), X4), not(truth())).
% 2.28/2.72 Proof: A critical pair between equations 100 and 153.
% 2.28/2.72
% 2.28/2.72 173: not(X4) = implies(X4, not(truth())).
% 2.28/2.72 Proof: Rewrite equation 165,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [108,0].
% 2.28/2.72
% 2.28/2.72 179: or(X0, not(truth())) = X0.
% 2.28/2.72 Proof: Rewrite equation 158,
% 2.28/2.72 lhs with equations [4]
% 2.28/2.72 rhs with equations [0].
% 2.28/2.72
% 2.28/2.72 189: xor(X0, X1) = or(not(or(not(X0), not(not(X1)))), not(or(not(not(X0)), not(X1)))).
% 2.28/2.72 Proof: Rewrite equation 143,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [4,4,4].
% 2.28/2.72
% 2.28/2.72 219: xor(X0, X1) = implies(not(not(implies(not(not(X0)), not(not(X1))))), not(implies(not(not(not(X0))), not(X1)))).
% 2.28/2.72 Proof: Rewrite equation 189,
% 2.28/2.72 lhs with equations []
% 2.28/2.72 rhs with equations [4,4,4].
% 2.28/2.72
% 2.28/2.72 227: not(not(X0)) = X0.
% 2.28/2.72 Proof: Rewrite equation 179,
% 2.28/2.72 lhs with equations [4,173]
% 2.28/2.72 rhs with equations [].
% 2.28/2.72
% 2.28/2.72 234: not(x()) = xor(x(), truth()).
% 2.28/2.72 Proof: Rewrite lhs with equations []
% 2.28/2.72 rhs with equations [219,227,227,70,227,227,173,227,0].
% 2.28/2.72
% 2.28/2.72 % SZS output end Proof
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