TSTP Solution File: LCL112-2 by Toma---0.4
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% File : Toma---0.4
% Problem : LCL112-2 : TPTP v8.1.2. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n007.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 08:13:49 EDT 2023
% Result : Unsatisfiable 1.05s 1.35s
% Output : CNFRefutation 1.05s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : LCL112-2 : TPTP v8.1.2. Released v1.0.0.
% 0.07/0.13 % Command : toma --casc %s
% 0.14/0.35 % Computer : n007.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 : Thu Aug 24 21:10:40 EDT 2023
% 0.14/0.35 % CPUTime :
% 1.05/1.35 % SZS status Unsatisfiable
% 1.05/1.35 % SZS output start Proof
% 1.05/1.35 original problem:
% 1.05/1.35 axioms:
% 1.05/1.35 implies(truth(), X) = X
% 1.05/1.35 implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth()
% 1.05/1.35 implies(implies(X, Y), Y) = implies(implies(Y, X), X)
% 1.05/1.35 implies(implies(not(X), not(Y)), implies(Y, X)) = truth()
% 1.05/1.35 goal:
% 1.05/1.35 implies(x(), not(not(x()))) != truth()
% 1.05/1.35 To show the unsatisfiability of the original goal,
% 1.05/1.35 it suffices to show that implies(x(), not(not(x()))) = truth() (skolemized goal) is valid under the axioms.
% 1.05/1.35 Here is an equational proof:
% 1.05/1.35 0: implies(truth(), X0) = X0.
% 1.05/1.35 Proof: Axiom.
% 1.05/1.35
% 1.05/1.35 1: implies(implies(X0, X1), implies(implies(X1, X2), implies(X0, X2))) = truth().
% 1.05/1.35 Proof: Axiom.
% 1.05/1.35
% 1.05/1.35 2: implies(implies(X0, X1), X1) = implies(implies(X1, X0), X0).
% 1.05/1.35 Proof: Axiom.
% 1.05/1.35
% 1.05/1.35 3: implies(implies(not(X0), not(X1)), implies(X1, X0)) = truth().
% 1.05/1.35 Proof: Axiom.
% 1.05/1.35
% 1.05/1.35 4: implies(implies(X2, truth()), truth()) = implies(X2, X2).
% 1.05/1.35 Proof: A critical pair between equations 2 and 0.
% 1.05/1.35
% 1.05/1.35 6: truth() = implies(implies(not(X2), not(truth())), X2).
% 1.05/1.35 Proof: A critical pair between equations 3 and 0.
% 1.05/1.35
% 1.05/1.35 8: truth() = implies(implies(X0, truth()), implies(X3, implies(X0, X3))).
% 1.05/1.35 Proof: A critical pair between equations 1 and 0.
% 1.05/1.35
% 1.05/1.35 9: truth() = implies(X3, implies(implies(X3, X2), implies(truth(), X2))).
% 1.05/1.35 Proof: A critical pair between equations 1 and 0.
% 1.05/1.35
% 1.05/1.35 10: truth() = implies(implies(implies(implies(X3, X5), X5), X3), truth()).
% 1.05/1.35 Proof: A critical pair between equations 1 and 1.
% 1.05/1.35
% 1.05/1.35 13: implies(implies(X3, implies(X2, X3)), implies(X2, X3)) = implies(implies(implies(X3, X2), X2), X3).
% 1.05/1.35 Proof: A critical pair between equations 2 and 2.
% 1.05/1.35
% 1.05/1.35 16: truth() = implies(implies(implies(X3, implies(X5, X3)), implies(X5, X3)), truth()).
% 1.05/1.35 Proof: Rewrite equation 10,
% 1.05/1.35 lhs with equations []
% 1.05/1.35 rhs with equations [13].
% 1.05/1.35
% 1.05/1.35 17: truth() = implies(X3, implies(implies(X3, X2), X2)).
% 1.05/1.35 Proof: Rewrite equation 9,
% 1.05/1.35 lhs with equations []
% 1.05/1.35 rhs with equations [0].
% 1.05/1.35
% 1.05/1.35 27: truth() = implies(implies(truth(), implies(implies(X7, X7), X7)), truth()).
% 1.05/1.35 Proof: A critical pair between equations 16 and 17.
% 1.05/1.35
% 1.05/1.35 28: truth() = implies(implies(truth(), implies(truth(), implies(X6, truth()))), truth()).
% 1.05/1.35 Proof: A critical pair between equations 16 and 8.
% 1.05/1.35
% 1.05/1.35 31: truth() = implies(X6, X6).
% 1.05/1.35 Proof: Rewrite equation 28,
% 1.05/1.35 lhs with equations []
% 1.05/1.35 rhs with equations [0,0,4].
% 1.05/1.35
% 1.05/1.35 32: truth() = implies(X7, truth()).
% 1.05/1.35 Proof: Rewrite equation 27,
% 1.05/1.35 lhs with equations []
% 1.05/1.35 rhs with equations [31,0,0].
% 1.05/1.35
% 1.05/1.35 33: truth() = implies(X3, implies(X0, X3)).
% 1.05/1.35 Proof: Rewrite equation 8,
% 1.05/1.35 lhs with equations []
% 1.05/1.35 rhs with equations [32,0].
% 1.05/1.35
% 1.05/1.35 38: truth() = implies(truth(), implies(implies(implies(X5, X4), X2), implies(X4, X2))).
% 1.05/1.35 Proof: A critical pair between equations 1 and 33.
% 1.05/1.35
% 1.05/1.35 42: truth() = implies(implies(X0, implies(not(X3), not(truth()))), implies(truth(), implies(X0, X3))).
% 1.05/1.35 Proof: A critical pair between equations 1 and 6.
% 1.05/1.35
% 1.05/1.35 50: truth() = implies(implies(X0, implies(not(X3), not(truth()))), implies(X0, X3)).
% 1.05/1.35 Proof: Rewrite equation 42,
% 1.05/1.35 lhs with equations []
% 1.05/1.35 rhs with equations [0].
% 1.05/1.35
% 1.05/1.35 52: truth() = implies(implies(implies(X5, X4), X2), implies(X4, X2)).
% 1.05/1.35 Proof: Rewrite equation 38,
% 1.05/1.35 lhs with equations []
% 1.05/1.35 rhs with equations [0].
% 1.05/1.35
% 1.05/1.35 63: truth() = implies(truth(), implies(implies(not(not(truth())), not(not(X3))), X3)).
% 1.05/1.35 Proof: A critical pair between equations 50 and 3.
% 1.05/1.35
% 1.05/1.35 69: truth() = implies(implies(not(not(truth())), not(not(X3))), X3).
% 1.05/1.35 Proof: Rewrite equation 63,
% 1.05/1.35 lhs with equations []
% 1.05/1.35 rhs with equations [0].
% 1.05/1.35
% 1.05/1.35 81: truth() = implies(truth(), implies(not(not(X6)), X6)).
% 1.05/1.35 Proof: A critical pair between equations 52 and 69.
% 1.05/1.35
% 1.05/1.35 94: truth() = implies(not(not(X6)), X6).
% 1.05/1.35 Proof: Rewrite equation 81,
% 1.05/1.35 lhs with equations []
% 1.05/1.35 rhs with equations [0].
% 1.05/1.35
% 1.05/1.35 96: truth() = implies(truth(), implies(X1, not(not(X1)))).
% 1.05/1.35 Proof: A critical pair between equations 3 and 94.
% 1.05/1.35
% 1.05/1.35 109: truth() = implies(X1, not(not(X1))).
% 1.05/1.35 Proof: Rewrite equation 96,
% 1.05/1.35 lhs with equations []
% 1.05/1.35 rhs with equations [0].
% 1.05/1.35
% 1.05/1.35 110: implies(x(), not(not(x()))) = truth().
% 1.05/1.35 Proof: Rewrite lhs with equations [109]
% 1.05/1.35 rhs with equations [].
% 1.05/1.35
% 1.05/1.35 % SZS output end Proof
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