TSTP Solution File: LDA001-1 by Toma---0.4
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% File : Toma---0.4
% Problem : LDA001-1 : TPTP v8.1.2. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n003.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 09:04:38 EDT 2023
% Result : Unsatisfiable 1.56s 1.91s
% Output : CNFRefutation 1.56s
% 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 : LDA001-1 : TPTP v8.1.2. Released v1.0.0.
% 0.13/0.13 % Command : toma --casc %s
% 0.13/0.34 % Computer : n003.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Sun Aug 27 01:13:39 EDT 2023
% 0.13/0.35 % CPUTime :
% 1.56/1.91 % SZS status Unsatisfiable
% 1.56/1.91 % SZS output start Proof
% 1.56/1.91 original problem:
% 1.56/1.91 axioms:
% 1.56/1.91 f(X, f(Y, Z)) = f(f(X, Y), f(X, Z))
% 1.56/1.91 n2() = f(n1(), n1())
% 1.56/1.91 n3() = f(n2(), n1())
% 1.56/1.91 u() = f(n2(), n2())
% 1.56/1.91 goal:
% 1.56/1.91 f(f(n3(), n2()), u()) != f(f(u(), u()), u())
% 1.56/1.91 To show the unsatisfiability of the original goal,
% 1.56/1.91 it suffices to show that f(f(n3(), n2()), u()) = f(f(u(), u()), u()) (skolemized goal) is valid under the axioms.
% 1.56/1.91 Here is an equational proof:
% 1.56/1.91 0: f(X0, f(X1, X2)) = f(f(X0, X1), f(X0, X2)).
% 1.56/1.91 Proof: Axiom.
% 1.56/1.91
% 1.56/1.91 1: n2() = f(n1(), n1()).
% 1.56/1.91 Proof: Axiom.
% 1.56/1.91
% 1.56/1.91 2: n3() = f(n2(), n1()).
% 1.56/1.91 Proof: Axiom.
% 1.56/1.91
% 1.56/1.91 3: u() = f(n2(), n2()).
% 1.56/1.91 Proof: Axiom.
% 1.56/1.91
% 1.56/1.91 4: n3() = f(f(n1(), n1()), n1()).
% 1.56/1.91 Proof: Rewrite equation 2,
% 1.56/1.91 lhs with equations []
% 1.56/1.91 rhs with equations [1].
% 1.56/1.91
% 1.56/1.91 5: u() = f(n1(), f(n1(), n1())).
% 1.56/1.91 Proof: Rewrite equation 3,
% 1.56/1.91 lhs with equations []
% 1.56/1.91 rhs with equations [1,1,0].
% 1.56/1.91
% 1.56/1.91 6: f(f(X3, X4), f(X1, f(X3, X5))) = f(f(f(X3, X4), X1), f(X3, f(X4, X5))).
% 1.56/1.91 Proof: A critical pair between equations 0 and 0.
% 1.56/1.91
% 1.56/1.91 8: u() = f(n1(), n2()).
% 1.56/1.91 Proof: Rewrite equation 5,
% 1.56/1.91 lhs with equations []
% 1.56/1.91 rhs with equations [1].
% 1.56/1.91
% 1.56/1.91 9: n3() = f(n2(), n1()).
% 1.56/1.91 Proof: Rewrite equation 4,
% 1.56/1.91 lhs with equations []
% 1.56/1.91 rhs with equations [1].
% 1.56/1.91
% 1.56/1.91 17: f(f(f(X6, X7), X6), f(X1, f(f(X6, X7), X8))) = f(f(f(f(X6, X7), X6), X1), f(X6, f(X7, X8))).
% 1.56/1.91 Proof: A critical pair between equations 6 and 0.
% 1.56/1.91
% 1.56/1.91 36: u() = f(n1(), f(n1(), n1())).
% 1.56/1.91 Proof: Rewrite equation 8,
% 1.56/1.91 lhs with equations []
% 1.56/1.91 rhs with equations [1].
% 1.56/1.91
% 1.56/1.91 37: n3() = f(f(n1(), n1()), n1()).
% 1.56/1.91 Proof: Rewrite equation 9,
% 1.56/1.91 lhs with equations []
% 1.56/1.91 rhs with equations [1].
% 1.56/1.91
% 1.56/1.91 38: f(f(n3(), n2()), u()) = f(f(u(), u()), u()).
% 1.56/1.91 Proof: Rewrite lhs with equations [37,1,36,17,0,0]
% 1.56/1.91 rhs with equations [36,36,0,0,36,0,0].
% 1.56/1.91
% 1.56/1.91 % SZS output end Proof
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