TSTP Solution File: LCL158-1 by Toma---0.4
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% File : Toma---0.4
% Problem : LCL158-1 : TPTP v8.1.2. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n016.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 1.40s 1.80s
% Output : CNFRefutation 1.40s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.14 % Problem : LCL158-1 : TPTP v8.1.2. Released v1.0.0.
% 0.14/0.14 % Command : toma --casc %s
% 0.15/0.35 % Computer : n016.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.21/0.35 % CPULimit : 300
% 0.21/0.35 % WCLimit : 300
% 0.21/0.35 % DateTime : Thu Aug 24 18:46:09 EDT 2023
% 0.21/0.35 % CPUTime :
% 1.40/1.80 % SZS status Unsatisfiable
% 1.40/1.80 % SZS output start Proof
% 1.40/1.80 original problem:
% 1.40/1.80 axioms:
% 1.40/1.80 implies(truth(), X) = X
% 1.40/1.80 implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth()
% 1.40/1.80 implies(implies(X, Y), Y) = implies(implies(Y, X), X)
% 1.40/1.80 implies(implies(not(X), not(Y)), implies(Y, X)) = truth()
% 1.40/1.80 or(X, Y) = implies(not(X), Y)
% 1.40/1.80 or(or(X, Y), Z) = or(X, or(Y, Z))
% 1.40/1.80 or(X, Y) = or(Y, X)
% 1.40/1.80 and(X, Y) = not(or(not(X), not(Y)))
% 1.40/1.80 and(and(X, Y), Z) = and(X, and(Y, Z))
% 1.40/1.80 and(X, Y) = and(Y, X)
% 1.40/1.80 xor(X, Y) = or(and(X, not(Y)), and(not(X), Y))
% 1.40/1.80 xor(X, Y) = xor(Y, X)
% 1.40/1.80 and_star(X, Y) = not(or(not(X), not(Y)))
% 1.40/1.80 and_star(and_star(X, Y), Z) = and_star(X, and_star(Y, Z))
% 1.40/1.80 and_star(X, Y) = and_star(Y, X)
% 1.40/1.80 not(truth()) = falsehood()
% 1.40/1.80 goal:
% 1.40/1.80 and_star(xor(truth(), x()), x()) != falsehood()
% 1.40/1.80 To show the unsatisfiability of the original goal,
% 1.40/1.80 it suffices to show that and_star(xor(truth(), x()), x()) = falsehood() (skolemized goal) is valid under the axioms.
% 1.40/1.80 Here is an equational proof:
% 1.40/1.80 0: implies(truth(), X0) = X0.
% 1.40/1.80 Proof: Axiom.
% 1.40/1.80
% 1.40/1.80 1: implies(implies(X0, X1), implies(implies(X1, X2), implies(X0, X2))) = truth().
% 1.40/1.80 Proof: Axiom.
% 1.40/1.80
% 1.40/1.80 2: implies(implies(X0, X1), X1) = implies(implies(X1, X0), X0).
% 1.40/1.80 Proof: Axiom.
% 1.40/1.80
% 1.40/1.80 3: implies(implies(not(X0), not(X1)), implies(X1, X0)) = truth().
% 1.40/1.80 Proof: Axiom.
% 1.40/1.80
% 1.40/1.80 4: or(X0, X1) = implies(not(X0), X1).
% 1.40/1.80 Proof: Axiom.
% 1.40/1.80
% 1.40/1.80 6: or(X0, X1) = or(X1, X0).
% 1.40/1.80 Proof: Axiom.
% 1.40/1.80
% 1.40/1.80 7: and(X0, X1) = not(or(not(X0), not(X1))).
% 1.40/1.80 Proof: Axiom.
% 1.40/1.80
% 1.40/1.80 10: xor(X0, X1) = or(and(X0, not(X1)), and(not(X0), X1)).
% 1.40/1.80 Proof: Axiom.
% 1.40/1.80
% 1.40/1.80 12: and_star(X0, X1) = not(or(not(X0), not(X1))).
% 1.40/1.80 Proof: Axiom.
% 1.40/1.80
% 1.40/1.80 15: not(truth()) = falsehood().
% 1.40/1.80 Proof: Axiom.
% 1.40/1.80
% 1.40/1.80 17: and(X0, X1) = not(implies(not(not(X0)), not(X1))).
% 1.40/1.80 Proof: Rewrite equation 7,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [4].
% 1.40/1.80
% 1.40/1.80 19: xor(X0, X1) = implies(not(not(implies(not(not(X0)), not(not(X1))))), not(implies(not(not(not(X0))), not(X1)))).
% 1.40/1.80 Proof: Rewrite equation 10,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [17,17,4].
% 1.40/1.80
% 1.40/1.80 20: and_star(X0, X1) = not(implies(not(not(X0)), not(X1))).
% 1.40/1.80 Proof: Rewrite equation 12,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [4].
% 1.40/1.80
% 1.40/1.80 21: implies(not(X0), X1) = implies(not(X1), X0).
% 1.40/1.80 Proof: Rewrite equation 6,
% 1.40/1.80 lhs with equations [4]
% 1.40/1.80 rhs with equations [4].
% 1.40/1.80
% 1.40/1.80 25: implies(not(X0), truth()) = implies(falsehood(), X0).
% 1.40/1.80 Proof: A critical pair between equations 21 and 15.
% 1.40/1.80
% 1.40/1.80 26: implies(implies(X2, truth()), truth()) = implies(X2, X2).
% 1.40/1.80 Proof: A critical pair between equations 2 and 0.
% 1.40/1.80
% 1.40/1.80 33: truth() = implies(X3, implies(implies(X3, X2), implies(truth(), X2))).
% 1.40/1.80 Proof: A critical pair between equations 1 and 0.
% 1.40/1.80
% 1.40/1.80 35: truth() = implies(implies(not(not(X1)), X2), implies(X1, X2)).
% 1.40/1.80 Proof: A critical pair between equations 3 and 21.
% 1.40/1.80
% 1.40/1.80 38: truth() = implies(or(not(X1), X2), implies(X1, X2)).
% 1.40/1.80 Proof: Rewrite equation 35,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [4].
% 1.40/1.80
% 1.40/1.80 39: truth() = implies(X3, implies(implies(X3, X2), X2)).
% 1.40/1.80 Proof: Rewrite equation 33,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [0].
% 1.40/1.80
% 1.40/1.80 42: and_star(X0, X1) = not(or(not(X0), not(X1))).
% 1.40/1.80 Proof: Rewrite equation 20,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [4].
% 1.40/1.80
% 1.40/1.80 43: xor(X0, X1) = or(not(or(not(X0), not(not(X1)))), not(or(not(not(X0)), not(X1)))).
% 1.40/1.80 Proof: Rewrite equation 19,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [4,4,4].
% 1.40/1.80
% 1.40/1.80 48: or(X0, truth()) = or(truth(), X0).
% 1.40/1.80 Proof: Rewrite equation 25,
% 1.40/1.80 lhs with equations [4]
% 1.40/1.80 rhs with equations [15,4].
% 1.40/1.80
% 1.40/1.80 52: or(X0, X1) = or(X1, X0).
% 1.40/1.80 Proof: Rewrite equation 21,
% 1.40/1.80 lhs with equations [4]
% 1.40/1.80 rhs with equations [4].
% 1.40/1.80
% 1.40/1.80 53: truth() = implies(X4, implies(X4, X4)).
% 1.40/1.80 Proof: A critical pair between equations 39 and 26.
% 1.40/1.80
% 1.40/1.80 55: truth() = implies(implies(truth(), X2), X2).
% 1.40/1.80 Proof: A critical pair between equations 39 and 0.
% 1.40/1.80
% 1.40/1.80 57: truth() = implies(or(not(truth()), X3), X3).
% 1.40/1.80 Proof: A critical pair between equations 38 and 0.
% 1.40/1.80
% 1.40/1.80 66: truth() = implies(implies(not(falsehood()), X3), X3).
% 1.40/1.80 Proof: Rewrite equation 57,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [15,4].
% 1.40/1.80
% 1.40/1.80 68: truth() = implies(X2, X2).
% 1.40/1.80 Proof: Rewrite equation 55,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [0].
% 1.40/1.80
% 1.40/1.80 69: truth() = implies(X4, truth()).
% 1.40/1.80 Proof: Rewrite equation 53,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [68].
% 1.40/1.80
% 1.40/1.80 74: xor(X0, X1) = implies(not(not(implies(not(not(X0)), not(not(X1))))), not(implies(not(not(not(X0))), not(X1)))).
% 1.40/1.80 Proof: Rewrite equation 43,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [4,4,4].
% 1.40/1.80
% 1.40/1.80 75: and_star(X0, X1) = not(implies(not(not(X0)), not(X1))).
% 1.40/1.80 Proof: Rewrite equation 42,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [4].
% 1.40/1.80
% 1.40/1.80 78: truth() = implies(implies(not(not(X1)), X2), implies(X1, X2)).
% 1.40/1.80 Proof: Rewrite equation 38,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [4].
% 1.40/1.80
% 1.40/1.80 80: implies(not(X0), X1) = implies(not(X1), X0).
% 1.40/1.80 Proof: Rewrite equation 52,
% 1.40/1.80 lhs with equations [4]
% 1.40/1.80 rhs with equations [4].
% 1.40/1.80
% 1.40/1.80 84: truth() = implies(falsehood(), X0).
% 1.40/1.80 Proof: Rewrite equation 48,
% 1.40/1.80 lhs with equations [4,69]
% 1.40/1.80 rhs with equations [4,15].
% 1.40/1.80
% 1.40/1.80 85: truth() = implies(truth(), not(falsehood())).
% 1.40/1.80 Proof: A critical pair between equations 66 and 68.
% 1.40/1.80
% 1.40/1.80 91: truth() = implies(truth(), implies(X1, not(not(X1)))).
% 1.40/1.80 Proof: A critical pair between equations 78 and 68.
% 1.40/1.80
% 1.40/1.80 102: truth() = implies(X1, not(not(X1))).
% 1.40/1.80 Proof: Rewrite equation 91,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [0].
% 1.40/1.80
% 1.40/1.80 107: truth() = not(not(truth())).
% 1.40/1.80 Proof: Rewrite equation 85,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [15,0].
% 1.40/1.80
% 1.40/1.80 108: truth() = or(truth(), X0).
% 1.40/1.80 Proof: Rewrite equation 84,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [15,4].
% 1.40/1.80
% 1.40/1.80 112: and_star(X0, X1) = not(or(not(X0), not(X1))).
% 1.40/1.80 Proof: Rewrite equation 75,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [4].
% 1.40/1.80
% 1.40/1.80 113: xor(X0, X1) = or(not(or(not(X0), not(not(X1)))), not(or(not(not(X0)), not(X1)))).
% 1.40/1.80 Proof: Rewrite equation 74,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [4,4,4].
% 1.40/1.80
% 1.40/1.80 121: or(X0, X1) = or(X1, X0).
% 1.40/1.80 Proof: Rewrite equation 80,
% 1.40/1.80 lhs with equations [4]
% 1.40/1.80 rhs with equations [4].
% 1.40/1.80
% 1.40/1.80 141: xor(X0, X1) = implies(not(not(implies(not(not(X0)), not(not(X1))))), not(implies(not(not(not(X0))), not(X1)))).
% 1.40/1.80 Proof: Rewrite equation 113,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [4,4,4].
% 1.40/1.80
% 1.40/1.80 142: and_star(X0, X1) = not(implies(not(not(X0)), not(X1))).
% 1.40/1.80 Proof: Rewrite equation 112,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [4].
% 1.40/1.80
% 1.40/1.80 146: truth() = implies(falsehood(), X0).
% 1.40/1.80 Proof: Rewrite equation 108,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [4,15].
% 1.40/1.80
% 1.40/1.80 147: truth() = not(falsehood()).
% 1.40/1.80 Proof: Rewrite equation 107,
% 1.40/1.80 lhs with equations []
% 1.40/1.80 rhs with equations [15].
% 1.40/1.80
% 1.40/1.80 153: implies(not(X0), X1) = implies(not(X1), X0).
% 1.40/1.80 Proof: Rewrite equation 121,
% 1.40/1.80 lhs with equations [4]
% 1.40/1.80 rhs with equations [4].
% 1.40/1.80
% 1.40/1.80 157: and_star(xor(truth(), x()), x()) = falsehood().
% 1.40/1.80 Proof: Rewrite lhs with equations [141,15,147,0,15,147,15,146,15,153,147,0,142,153,102,15]
% 1.40/1.80 rhs with equations [].
% 1.40/1.80
% 1.40/1.80 % SZS output end Proof
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