TSTP Solution File: LCL154-1 by Toma---0.4
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% File : Toma---0.4
% Problem : LCL154-1 : TPTP v8.1.2. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n021.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 3.87s 4.24s
% Output : CNFRefutation 3.87s
% 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 : LCL154-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13 % Command : toma --casc %s
% 0.18/0.34 % Computer : n021.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Thu Aug 24 22:23:40 EDT 2023
% 0.20/0.35 % CPUTime :
% 3.87/4.24 % SZS status Unsatisfiable
% 3.87/4.24 % SZS output start Proof
% 3.87/4.24 original problem:
% 3.87/4.24 axioms:
% 3.87/4.24 implies(truth(), X) = X
% 3.87/4.24 implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth()
% 3.87/4.24 implies(implies(X, Y), Y) = implies(implies(Y, X), X)
% 3.87/4.24 implies(implies(not(X), not(Y)), implies(Y, X)) = truth()
% 3.87/4.24 or(X, Y) = implies(not(X), Y)
% 3.87/4.24 or(or(X, Y), Z) = or(X, or(Y, Z))
% 3.87/4.24 or(X, Y) = or(Y, X)
% 3.87/4.24 and(X, Y) = not(or(not(X), not(Y)))
% 3.87/4.24 and(and(X, Y), Z) = and(X, and(Y, Z))
% 3.87/4.24 and(X, Y) = and(Y, X)
% 3.87/4.24 xor(X, Y) = or(and(X, not(Y)), and(not(X), Y))
% 3.87/4.24 xor(X, Y) = xor(Y, X)
% 3.87/4.24 and_star(X, Y) = not(or(not(X), not(Y)))
% 3.87/4.24 and_star(and_star(X, Y), Z) = and_star(X, and_star(Y, Z))
% 3.87/4.24 and_star(X, Y) = and_star(Y, X)
% 3.87/4.24 not(truth()) = falsehood()
% 3.87/4.24 goal:
% 3.87/4.24 xor(x(), falsehood()) != x()
% 3.87/4.24 To show the unsatisfiability of the original goal,
% 3.87/4.24 it suffices to show that xor(x(), falsehood()) = x() (skolemized goal) is valid under the axioms.
% 3.87/4.24 Here is an equational proof:
% 3.87/4.24 0: implies(truth(), X0) = X0.
% 3.87/4.24 Proof: Axiom.
% 3.87/4.24
% 3.87/4.24 1: implies(implies(X0, X1), implies(implies(X1, X2), implies(X0, X2))) = truth().
% 3.87/4.24 Proof: Axiom.
% 3.87/4.24
% 3.87/4.24 2: implies(implies(X0, X1), X1) = implies(implies(X1, X0), X0).
% 3.87/4.24 Proof: Axiom.
% 3.87/4.24
% 3.87/4.24 3: implies(implies(not(X0), not(X1)), implies(X1, X0)) = truth().
% 3.87/4.24 Proof: Axiom.
% 3.87/4.24
% 3.87/4.24 4: or(X0, X1) = implies(not(X0), X1).
% 3.87/4.24 Proof: Axiom.
% 3.87/4.24
% 3.87/4.24 6: or(X0, X1) = or(X1, X0).
% 3.87/4.24 Proof: Axiom.
% 3.87/4.24
% 3.87/4.24 7: and(X0, X1) = not(or(not(X0), not(X1))).
% 3.87/4.24 Proof: Axiom.
% 3.87/4.24
% 3.87/4.24 10: xor(X0, X1) = or(and(X0, not(X1)), and(not(X0), X1)).
% 3.87/4.24 Proof: Axiom.
% 3.87/4.24
% 3.87/4.24 15: not(truth()) = falsehood().
% 3.87/4.24 Proof: Axiom.
% 3.87/4.24
% 3.87/4.24 17: and(X0, X1) = not(implies(not(not(X0)), not(X1))).
% 3.87/4.24 Proof: Rewrite equation 7,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [4].
% 3.87/4.24
% 3.87/4.24 19: xor(X0, X1) = implies(not(not(implies(not(not(X0)), not(not(X1))))), not(implies(not(not(not(X0))), not(X1)))).
% 3.87/4.24 Proof: Rewrite equation 10,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [17,17,4].
% 3.87/4.24
% 3.87/4.24 22: implies(not(X0), X1) = implies(not(X1), X0).
% 3.87/4.24 Proof: Rewrite equation 6,
% 3.87/4.24 lhs with equations [4]
% 3.87/4.24 rhs with equations [4].
% 3.87/4.24
% 3.87/4.24 26: implies(implies(X2, truth()), truth()) = implies(X2, X2).
% 3.87/4.24 Proof: A critical pair between equations 2 and 0.
% 3.87/4.24
% 3.87/4.24 31: truth() = implies(X3, implies(implies(X3, X2), implies(truth(), X2))).
% 3.87/4.24 Proof: A critical pair between equations 1 and 0.
% 3.87/4.24
% 3.87/4.24 33: truth() = implies(implies(not(not(X1)), X2), implies(X1, X2)).
% 3.87/4.24 Proof: A critical pair between equations 3 and 22.
% 3.87/4.24
% 3.87/4.24 35: implies(implies(X3, not(X2)), not(X2)) = implies(implies(not(X3), X2), X3).
% 3.87/4.24 Proof: A critical pair between equations 2 and 22.
% 3.87/4.24
% 3.87/4.24 39: implies(implies(X3, not(X2)), not(X2)) = implies(or(X3, X2), X3).
% 3.87/4.24 Proof: Rewrite equation 35,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [4].
% 3.87/4.24
% 3.87/4.24 41: truth() = implies(or(not(X1), X2), implies(X1, X2)).
% 3.87/4.24 Proof: Rewrite equation 33,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [4].
% 3.87/4.24
% 3.87/4.24 42: truth() = implies(X3, implies(implies(X3, X2), X2)).
% 3.87/4.24 Proof: Rewrite equation 31,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [0].
% 3.87/4.24
% 3.87/4.24 45: xor(X0, X1) = or(not(or(not(X0), not(not(X1)))), not(or(not(not(X0)), not(X1)))).
% 3.87/4.24 Proof: Rewrite equation 19,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [4,4,4].
% 3.87/4.24
% 3.87/4.24 54: or(X0, X1) = or(X1, X0).
% 3.87/4.24 Proof: Rewrite equation 22,
% 3.87/4.24 lhs with equations [4]
% 3.87/4.24 rhs with equations [4].
% 3.87/4.24
% 3.87/4.24 55: truth() = implies(X4, implies(X4, X4)).
% 3.87/4.24 Proof: A critical pair between equations 42 and 26.
% 3.87/4.24
% 3.87/4.24 57: truth() = implies(implies(truth(), X2), X2).
% 3.87/4.24 Proof: A critical pair between equations 42 and 0.
% 3.87/4.24
% 3.87/4.24 63: truth() = implies(or(falsehood(), X2), implies(truth(), X2)).
% 3.87/4.24 Proof: A critical pair between equations 41 and 15.
% 3.87/4.24
% 3.87/4.24 68: truth() = implies(implies(not(not(truth())), X2), X2).
% 3.87/4.24 Proof: Rewrite equation 63,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [15,4,0].
% 3.87/4.24
% 3.87/4.24 69: truth() = implies(X2, X2).
% 3.87/4.24 Proof: Rewrite equation 57,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [0].
% 3.87/4.24
% 3.87/4.24 70: truth() = implies(X4, truth()).
% 3.87/4.24 Proof: Rewrite equation 55,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [69].
% 3.87/4.24
% 3.87/4.24 75: xor(X0, X1) = implies(not(not(implies(not(not(X0)), not(not(X1))))), not(implies(not(not(not(X0))), not(X1)))).
% 3.87/4.24 Proof: Rewrite equation 45,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [4,4,4].
% 3.87/4.24
% 3.87/4.24 81: implies(implies(X3, not(X2)), not(X2)) = implies(implies(not(X3), X2), X3).
% 3.87/4.24 Proof: Rewrite equation 39,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [4].
% 3.87/4.24
% 3.87/4.24 83: implies(not(X0), X1) = implies(not(X1), X0).
% 3.87/4.24 Proof: Rewrite equation 54,
% 3.87/4.24 lhs with equations [4]
% 3.87/4.24 rhs with equations [4].
% 3.87/4.24
% 3.87/4.24 89: truth() = implies(truth(), not(not(truth()))).
% 3.87/4.24 Proof: A critical pair between equations 68 and 69.
% 3.87/4.24
% 3.87/4.24 98: implies(implies(X3, not(truth())), not(truth())) = implies(truth(), X3).
% 3.87/4.24 Proof: A critical pair between equations 81 and 70.
% 3.87/4.24
% 3.87/4.24 101: implies(implies(X3, falsehood()), falsehood()) = X3.
% 3.87/4.24 Proof: Rewrite equation 98,
% 3.87/4.24 lhs with equations [15,15]
% 3.87/4.24 rhs with equations [0].
% 3.87/4.24
% 3.87/4.24 110: truth() = not(falsehood()).
% 3.87/4.24 Proof: Rewrite equation 89,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [15,0].
% 3.87/4.24
% 3.87/4.24 117: xor(X0, X1) = or(not(or(not(X0), not(not(X1)))), not(or(not(not(X0)), not(X1)))).
% 3.87/4.24 Proof: Rewrite equation 75,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [4,4,4].
% 3.87/4.24
% 3.87/4.24 126: or(X0, X1) = or(X1, X0).
% 3.87/4.24 Proof: Rewrite equation 83,
% 3.87/4.24 lhs with equations [4]
% 3.87/4.24 rhs with equations [4].
% 3.87/4.24
% 3.87/4.24 143: xor(X0, X1) = implies(not(not(implies(not(not(X0)), not(not(X1))))), not(implies(not(not(not(X0))), not(X1)))).
% 3.87/4.24 Proof: Rewrite equation 117,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [4,4,4].
% 3.87/4.24
% 3.87/4.24 152: implies(not(X0), X1) = implies(not(X1), X0).
% 3.87/4.24 Proof: Rewrite equation 126,
% 3.87/4.24 lhs with equations [4]
% 3.87/4.24 rhs with equations [4].
% 3.87/4.24
% 3.87/4.24 158: not(X4) = implies(implies(not(falsehood()), X4), falsehood()).
% 3.87/4.24 Proof: A critical pair between equations 101 and 152.
% 3.87/4.24
% 3.87/4.24 179: not(X4) = implies(or(falsehood(), X4), falsehood()).
% 3.87/4.24 Proof: Rewrite equation 158,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [4].
% 3.87/4.24
% 3.87/4.24 194: xor(X0, X1) = or(not(or(not(X0), not(not(X1)))), not(or(not(not(X0)), not(X1)))).
% 3.87/4.24 Proof: Rewrite equation 143,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [4,4,4].
% 3.87/4.24
% 3.87/4.24 199: or(falsehood(), X0) = X0.
% 3.87/4.24 Proof: Rewrite equation 0,
% 3.87/4.24 lhs with equations [110,4]
% 3.87/4.24 rhs with equations [].
% 3.87/4.24
% 3.87/4.24 201: not(falsehood()) = implies(X4, not(falsehood())).
% 3.87/4.24 Proof: Rewrite equation 70,
% 3.87/4.24 lhs with equations [110]
% 3.87/4.24 rhs with equations [110].
% 3.87/4.24
% 3.87/4.24 209: not(X5) = implies(X5, falsehood()).
% 3.87/4.24 Proof: A critical pair between equations 179 and 199.
% 3.87/4.24
% 3.87/4.24 228: implies(falsehood(), falsehood()) = implies(X4, implies(falsehood(), falsehood())).
% 3.87/4.24 Proof: Rewrite equation 201,
% 3.87/4.24 lhs with equations [209]
% 3.87/4.24 rhs with equations [209].
% 3.87/4.24
% 3.87/4.24 232: or(X0, X1) = implies(implies(X0, falsehood()), X1).
% 3.87/4.24 Proof: Rewrite equation 4,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [209].
% 3.87/4.24
% 3.87/4.24 235: xor(X0, X1) = implies(implies(X0, X1), implies(implies(implies(X0, falsehood()), implies(X1, falsehood())), falsehood())).
% 3.87/4.24 Proof: Rewrite equation 194,
% 3.87/4.24 lhs with equations []
% 3.87/4.24 rhs with equations [209,209,209,101,232,101,209,209,209,101,209,232,209,232,101].
% 3.87/4.24
% 3.87/4.24 260: xor(x(), falsehood()) = x().
% 3.87/4.24 Proof: Rewrite lhs with equations [235,228,101,101]
% 3.87/4.24 rhs with equations [].
% 3.87/4.24
% 3.87/4.24 % SZS output end Proof
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