TSTP Solution File: LCL164-1 by Toma---0.4
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% File : Toma---0.4
% Problem : LCL164-1 : 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 08:14:00 EDT 2023
% Result : Unsatisfiable 0.20s 0.83s
% Output : CNFRefutation 0.20s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LCL164-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13 % Command : toma --casc %s
% 0.13/0.35 % Computer : n016.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Thu Aug 24 21:02:09 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.83 % SZS status Unsatisfiable
% 0.20/0.83 % SZS output start Proof
% 0.20/0.83 original problem:
% 0.20/0.83 axioms:
% 0.20/0.83 not(X) = xor(X, truth())
% 0.20/0.83 xor(X, falsehood()) = X
% 0.20/0.83 xor(X, X) = falsehood()
% 0.20/0.83 and_star(X, truth()) = X
% 0.20/0.83 and_star(X, falsehood()) = falsehood()
% 0.20/0.83 and_star(xor(truth(), X), X) = falsehood()
% 0.20/0.83 xor(X, xor(truth(), Y)) = xor(xor(X, truth()), Y)
% 0.20/0.83 and_star(xor(and_star(xor(truth(), X), Y), truth()), Y) = and_star(xor(and_star(xor(truth(), Y), X), truth()), X)
% 0.20/0.83 xor(X, Y) = xor(Y, X)
% 0.20/0.83 and_star(and_star(X, Y), Z) = and_star(X, and_star(Y, Z))
% 0.20/0.83 and_star(X, Y) = and_star(Y, X)
% 0.20/0.83 not(truth()) = falsehood()
% 0.20/0.83 implies(X, Y) = xor(truth(), and_star(X, xor(truth(), Y)))
% 0.20/0.83 goal:
% 0.20/0.83 implies(implies(not(x()), not(y())), implies(y(), x())) != truth()
% 0.20/0.83 To show the unsatisfiability of the original goal,
% 0.20/0.83 it suffices to show that implies(implies(not(x()), not(y())), implies(y(), x())) = truth() (skolemized goal) is valid under the axioms.
% 0.20/0.83 Here is an equational proof:
% 0.20/0.83 0: not(X0) = xor(X0, truth()).
% 0.20/0.83 Proof: Axiom.
% 0.20/0.83
% 0.20/0.83 1: xor(X0, falsehood()) = X0.
% 0.20/0.83 Proof: Axiom.
% 0.20/0.83
% 0.20/0.83 5: and_star(xor(truth(), X0), X0) = falsehood().
% 0.20/0.83 Proof: Axiom.
% 0.20/0.83
% 0.20/0.83 6: xor(X0, xor(truth(), X1)) = xor(xor(X0, truth()), X1).
% 0.20/0.83 Proof: Axiom.
% 0.20/0.83
% 0.20/0.83 8: xor(X0, X1) = xor(X1, X0).
% 0.20/0.83 Proof: Axiom.
% 0.20/0.83
% 0.20/0.83 10: and_star(X0, X1) = and_star(X1, X0).
% 0.20/0.83 Proof: Axiom.
% 0.20/0.83
% 0.20/0.83 11: not(truth()) = falsehood().
% 0.20/0.83 Proof: Axiom.
% 0.20/0.83
% 0.20/0.83 12: implies(X0, X1) = xor(truth(), and_star(X0, xor(truth(), X1))).
% 0.20/0.83 Proof: Axiom.
% 0.20/0.83
% 0.20/0.83 13: and_star(X0, xor(truth(), X0)) = falsehood().
% 0.20/0.83 Proof: Rewrite equation 5,
% 0.20/0.83 lhs with equations [10]
% 0.20/0.83 rhs with equations [].
% 0.20/0.83
% 0.20/0.83 14: xor(X0, xor(truth(), X1)) = xor(not(X0), X1).
% 0.20/0.83 Proof: Rewrite equation 6,
% 0.20/0.83 lhs with equations []
% 0.20/0.83 rhs with equations [0].
% 0.20/0.83
% 0.20/0.83 16: xor(falsehood(), X2) = X2.
% 0.20/0.83 Proof: A critical pair between equations 8 and 1.
% 0.20/0.83
% 0.20/0.83 19: xor(truth(), X2) = not(X2).
% 0.20/0.83 Proof: A critical pair between equations 8 and 0.
% 0.20/0.83
% 0.20/0.83 20: falsehood() = and_star(X3, xor(X3, truth())).
% 0.20/0.83 Proof: A critical pair between equations 13 and 8.
% 0.20/0.83
% 0.20/0.83 25: xor(truth(), xor(truth(), X1)) = xor(falsehood(), X1).
% 0.20/0.83 Proof: A critical pair between equations 14 and 11.
% 0.20/0.83
% 0.20/0.83 28: not(not(X1)) = X1.
% 0.20/0.83 Proof: Rewrite equation 25,
% 0.20/0.83 lhs with equations [19,19]
% 0.20/0.83 rhs with equations [16].
% 0.20/0.83
% 0.20/0.83 29: falsehood() = and_star(X3, not(X3)).
% 0.20/0.83 Proof: Rewrite equation 20,
% 0.20/0.83 lhs with equations []
% 0.20/0.83 rhs with equations [0].
% 0.20/0.83
% 0.20/0.83 30: implies(X0, X1) = not(and_star(X0, not(X1))).
% 0.20/0.83 Proof: Rewrite equation 12,
% 0.20/0.83 lhs with equations []
% 0.20/0.83 rhs with equations [19,19].
% 0.20/0.83
% 0.20/0.83 54: xor(truth(), truth()) = falsehood().
% 0.20/0.83 Proof: Rewrite equation 11,
% 0.20/0.83 lhs with equations [19]
% 0.20/0.83 rhs with equations [].
% 0.20/0.83
% 0.20/0.83 55: implies(X0, X1) = xor(truth(), and_star(X0, xor(truth(), X1))).
% 0.20/0.83 Proof: Rewrite equation 30,
% 0.20/0.83 lhs with equations []
% 0.20/0.83 rhs with equations [19,19].
% 0.20/0.83
% 0.20/0.83 58: xor(truth(), truth()) = and_star(X3, xor(truth(), X3)).
% 0.20/0.83 Proof: Rewrite equation 29,
% 0.20/0.83 lhs with equations [54]
% 0.20/0.83 rhs with equations [19].
% 0.20/0.83
% 0.20/0.83 59: xor(truth(), xor(truth(), X1)) = X1.
% 0.20/0.83 Proof: Rewrite equation 28,
% 0.20/0.83 lhs with equations [19,19]
% 0.20/0.83 rhs with equations [].
% 0.20/0.83
% 0.20/0.83 66: implies(implies(not(x()), not(y())), implies(y(), x())) = truth().
% 0.20/0.83 Proof: Rewrite lhs with equations [19,19,55,59,55,10,55,59,10,58,59]
% 0.20/0.83 rhs with equations [].
% 0.20/0.83
% 0.20/0.83 % SZS output end Proof
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