TSTP Solution File: LCL156-1 by Toma---0.4
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% File : Toma---0.4
% Problem : LCL156-1 : TPTP v8.1.2. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n001.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 3.72s 4.10s
% Output : CNFRefutation 3.72s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL156-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13 % Command : toma --casc %s
% 0.12/0.33 % Computer : n001.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Aug 24 22:50:12 EDT 2023
% 0.12/0.33 % CPUTime :
% 3.72/4.10 % SZS status Unsatisfiable
% 3.72/4.10 % SZS output start Proof
% 3.72/4.10 original problem:
% 3.72/4.10 axioms:
% 3.72/4.10 implies(truth(), X) = X
% 3.72/4.10 implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth()
% 3.72/4.10 implies(implies(X, Y), Y) = implies(implies(Y, X), X)
% 3.72/4.10 implies(implies(not(X), not(Y)), implies(Y, X)) = truth()
% 3.72/4.10 or(X, Y) = implies(not(X), Y)
% 3.72/4.10 or(or(X, Y), Z) = or(X, or(Y, Z))
% 3.72/4.10 or(X, Y) = or(Y, X)
% 3.72/4.10 and(X, Y) = not(or(not(X), not(Y)))
% 3.72/4.10 and(and(X, Y), Z) = and(X, and(Y, Z))
% 3.72/4.10 and(X, Y) = and(Y, X)
% 3.72/4.10 xor(X, Y) = or(and(X, not(Y)), and(not(X), Y))
% 3.72/4.10 xor(X, Y) = xor(Y, X)
% 3.72/4.10 and_star(X, Y) = not(or(not(X), not(Y)))
% 3.72/4.10 and_star(and_star(X, Y), Z) = and_star(X, and_star(Y, Z))
% 3.72/4.10 and_star(X, Y) = and_star(Y, X)
% 3.72/4.10 not(truth()) = falsehood()
% 3.72/4.10 goal:
% 3.72/4.10 and_star(x(), truth()) != x()
% 3.72/4.10 To show the unsatisfiability of the original goal,
% 3.72/4.10 it suffices to show that and_star(x(), truth()) = x() (skolemized goal) is valid under the axioms.
% 3.72/4.10 Here is an equational proof:
% 3.72/4.10 0: implies(truth(), X0) = X0.
% 3.72/4.10 Proof: Axiom.
% 3.72/4.10
% 3.72/4.10 1: implies(implies(X0, X1), implies(implies(X1, X2), implies(X0, X2))) = truth().
% 3.72/4.10 Proof: Axiom.
% 3.72/4.10
% 3.72/4.10 2: implies(implies(X0, X1), X1) = implies(implies(X1, X0), X0).
% 3.72/4.10 Proof: Axiom.
% 3.72/4.10
% 3.72/4.10 3: implies(implies(not(X0), not(X1)), implies(X1, X0)) = truth().
% 3.72/4.10 Proof: Axiom.
% 3.72/4.10
% 3.72/4.10 4: or(X0, X1) = implies(not(X0), X1).
% 3.72/4.10 Proof: Axiom.
% 3.72/4.10
% 3.72/4.10 6: or(X0, X1) = or(X1, X0).
% 3.72/4.10 Proof: Axiom.
% 3.72/4.10
% 3.72/4.10 12: and_star(X0, X1) = not(or(not(X0), not(X1))).
% 3.72/4.10 Proof: Axiom.
% 3.72/4.10
% 3.72/4.10 20: and_star(X0, X1) = not(implies(not(not(X0)), not(X1))).
% 3.72/4.10 Proof: Rewrite equation 12,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [4].
% 3.72/4.10
% 3.72/4.10 22: implies(not(X0), X1) = implies(not(X1), X0).
% 3.72/4.10 Proof: Rewrite equation 6,
% 3.72/4.10 lhs with equations [4]
% 3.72/4.10 rhs with equations [4].
% 3.72/4.10
% 3.72/4.10 26: implies(implies(X2, truth()), truth()) = implies(X2, X2).
% 3.72/4.10 Proof: A critical pair between equations 2 and 0.
% 3.72/4.10
% 3.72/4.10 31: truth() = implies(X3, implies(implies(X3, X2), implies(truth(), X2))).
% 3.72/4.10 Proof: A critical pair between equations 1 and 0.
% 3.72/4.10
% 3.72/4.10 33: truth() = implies(implies(not(not(X1)), X2), implies(X1, X2)).
% 3.72/4.10 Proof: A critical pair between equations 3 and 22.
% 3.72/4.10
% 3.72/4.10 35: implies(implies(X3, not(X2)), not(X2)) = implies(implies(not(X3), X2), X3).
% 3.72/4.10 Proof: A critical pair between equations 2 and 22.
% 3.72/4.10
% 3.72/4.10 39: implies(implies(X3, not(X2)), not(X2)) = implies(or(X3, X2), X3).
% 3.72/4.10 Proof: Rewrite equation 35,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [4].
% 3.72/4.10
% 3.72/4.10 41: truth() = implies(or(not(X1), X2), implies(X1, X2)).
% 3.72/4.10 Proof: Rewrite equation 33,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [4].
% 3.72/4.10
% 3.72/4.10 42: truth() = implies(X3, implies(implies(X3, X2), X2)).
% 3.72/4.10 Proof: Rewrite equation 31,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [0].
% 3.72/4.10
% 3.72/4.10 44: and_star(X0, X1) = not(or(not(X0), not(X1))).
% 3.72/4.10 Proof: Rewrite equation 20,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [4].
% 3.72/4.10
% 3.72/4.10 54: or(X0, X1) = or(X1, X0).
% 3.72/4.10 Proof: Rewrite equation 22,
% 3.72/4.10 lhs with equations [4]
% 3.72/4.10 rhs with equations [4].
% 3.72/4.10
% 3.72/4.10 55: truth() = implies(X4, implies(X4, X4)).
% 3.72/4.10 Proof: A critical pair between equations 42 and 26.
% 3.72/4.10
% 3.72/4.10 57: truth() = implies(implies(truth(), X2), X2).
% 3.72/4.10 Proof: A critical pair between equations 42 and 0.
% 3.72/4.10
% 3.72/4.10 59: truth() = implies(or(not(truth()), X3), X3).
% 3.72/4.10 Proof: A critical pair between equations 41 and 0.
% 3.72/4.10
% 3.72/4.10 69: truth() = implies(implies(not(not(truth())), X3), X3).
% 3.72/4.10 Proof: Rewrite equation 59,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [4].
% 3.72/4.10
% 3.72/4.10 70: truth() = implies(X2, X2).
% 3.72/4.10 Proof: Rewrite equation 57,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [0].
% 3.72/4.10
% 3.72/4.10 71: truth() = implies(X4, truth()).
% 3.72/4.10 Proof: Rewrite equation 55,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [70].
% 3.72/4.10
% 3.72/4.10 77: and_star(X0, X1) = not(implies(not(not(X0)), not(X1))).
% 3.72/4.10 Proof: Rewrite equation 44,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [4].
% 3.72/4.10
% 3.72/4.10 82: implies(implies(X3, not(X2)), not(X2)) = implies(implies(not(X3), X2), X3).
% 3.72/4.10 Proof: Rewrite equation 39,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [4].
% 3.72/4.10
% 3.72/4.10 84: implies(not(X0), X1) = implies(not(X1), X0).
% 3.72/4.10 Proof: Rewrite equation 54,
% 3.72/4.10 lhs with equations [4]
% 3.72/4.10 rhs with equations [4].
% 3.72/4.10
% 3.72/4.10 90: truth() = implies(truth(), not(not(truth()))).
% 3.72/4.10 Proof: A critical pair between equations 69 and 70.
% 3.72/4.10
% 3.72/4.10 100: implies(implies(X3, not(truth())), not(truth())) = implies(truth(), X3).
% 3.72/4.10 Proof: A critical pair between equations 82 and 71.
% 3.72/4.10
% 3.72/4.10 101: implies(implies(X3, not(truth())), not(truth())) = X3.
% 3.72/4.10 Proof: Rewrite equation 100,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [0].
% 3.72/4.10
% 3.72/4.10 109: truth() = not(not(truth())).
% 3.72/4.10 Proof: Rewrite equation 90,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [0].
% 3.72/4.10
% 3.72/4.10 115: and_star(X0, X1) = not(or(not(X0), not(X1))).
% 3.72/4.10 Proof: Rewrite equation 77,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [4].
% 3.72/4.10
% 3.72/4.10 125: or(X0, X1) = or(X1, X0).
% 3.72/4.10 Proof: Rewrite equation 84,
% 3.72/4.10 lhs with equations [4]
% 3.72/4.10 rhs with equations [4].
% 3.72/4.10
% 3.72/4.10 145: and_star(X0, X1) = not(implies(not(not(X0)), not(X1))).
% 3.72/4.10 Proof: Rewrite equation 115,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [4].
% 3.72/4.10
% 3.72/4.10 155: implies(not(X0), X1) = implies(not(X1), X0).
% 3.72/4.10 Proof: Rewrite equation 125,
% 3.72/4.10 lhs with equations [4]
% 3.72/4.10 rhs with equations [4].
% 3.72/4.10
% 3.72/4.10 160: implies(not(X0), not(truth())) = implies(truth(), X0).
% 3.72/4.10 Proof: A critical pair between equations 155 and 109.
% 3.72/4.10
% 3.72/4.10 167: not(X4) = implies(implies(not(not(truth())), X4), not(truth())).
% 3.72/4.10 Proof: A critical pair between equations 101 and 155.
% 3.72/4.10
% 3.72/4.10 175: not(X4) = implies(X4, not(truth())).
% 3.72/4.10 Proof: Rewrite equation 167,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [109,0].
% 3.72/4.10
% 3.72/4.10 181: or(X0, not(truth())) = X0.
% 3.72/4.10 Proof: Rewrite equation 160,
% 3.72/4.10 lhs with equations [4]
% 3.72/4.10 rhs with equations [0].
% 3.72/4.10
% 3.72/4.10 191: and_star(X0, X1) = not(or(not(X0), not(X1))).
% 3.72/4.10 Proof: Rewrite equation 145,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [4].
% 3.72/4.10
% 3.72/4.10 223: and_star(X0, X1) = not(implies(not(not(X0)), not(X1))).
% 3.72/4.10 Proof: Rewrite equation 191,
% 3.72/4.10 lhs with equations []
% 3.72/4.10 rhs with equations [4].
% 3.72/4.10
% 3.72/4.10 232: not(not(X0)) = X0.
% 3.72/4.10 Proof: Rewrite equation 181,
% 3.72/4.10 lhs with equations [4,175]
% 3.72/4.10 rhs with equations [].
% 3.72/4.10
% 3.72/4.10 240: and_star(x(), truth()) = x().
% 3.72/4.10 Proof: Rewrite lhs with equations [223,232,175,232]
% 3.72/4.10 rhs with equations [].
% 3.72/4.10
% 3.72/4.10 % SZS output end Proof
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