0.11/0.11	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.11/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.11/0.33	% Computer   : n018.cluster.edu
0.11/0.33	% Model      : x86_64 x86_64
0.11/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.11/0.33	% Memory     : 8042.1875MB
0.11/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.11/0.33	% CPULimit   : 180
0.11/0.33	% DateTime   : Thu Aug 29 13:17:36 EDT 2019
0.11/0.33	% CPUTime    : 
3.06/3.25	% SZS status Unsatisfiable
3.06/3.25	
3.06/3.25	% SZS output start Proof
3.06/3.25	Take the following subset of the input axioms:
3.06/3.25	  fof(k_definition, axiom, ![Y, X]: apply(apply(k, X), Y)=X).
3.06/3.25	  fof(prove_fixed_point, negated_conjecture, ![Y]: apply(Y, f(Y))!=apply(f(Y), apply(Y, f(Y)))).
3.06/3.25	  fof(s_definition, axiom, ![Y, X, Z]: apply(apply(X, Z), apply(Y, Z))=apply(apply(apply(s, X), Y), Z)).
3.06/3.25	
3.06/3.25	Now clausify the problem and encode Horn clauses using encoding 3 of
3.06/3.25	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
3.06/3.25	We repeatedly replace C & s=t => u=v by the two clauses:
3.06/3.25	  fresh(y, y, x1...xn) = u
3.06/3.25	  C => fresh(s, t, x1...xn) = v
3.06/3.25	where fresh is a fresh function symbol and x1..xn are the free
3.06/3.25	variables of u and v.
3.06/3.25	A predicate p(X) is encoded as p(X)=true (this is sound, because the
3.06/3.25	input problem has no model of domain size 1).
3.06/3.25	
3.06/3.25	The encoding turns the above axioms into the following unit equations and goals:
3.06/3.25	
3.06/3.25	Axiom 1 (s_definition): apply(apply(X, Y), apply(Z, Y)) = apply(apply(apply(s, X), Z), Y).
3.06/3.25	Axiom 2 (k_definition): apply(apply(k, X), Y) = X.
3.06/3.25	
3.06/3.25	Lemma 3: apply(apply(apply(s, k), Y), X) = X.
3.06/3.25	Proof:
3.06/3.25	  apply(apply(apply(s, k), Y), X)
3.06/3.25	= { by axiom 1 (s_definition) }
3.06/3.25	  apply(apply(k, X), apply(Y, X))
3.06/3.25	= { by axiom 2 (k_definition) }
3.06/3.25	  X
3.06/3.25	
3.06/3.25	Lemma 4: apply(apply(X, Z), Z) = apply(apply(apply(s, X), apply(apply(s, k), ?)), Z).
3.06/3.25	Proof:
3.06/3.25	  apply(apply(X, Z), Z)
3.06/3.25	= { by lemma 3 }
3.06/3.25	  apply(apply(X, Z), apply(apply(apply(s, k), ?), Z))
3.06/3.25	= { by axiom 1 (s_definition) }
3.06/3.25	  apply(apply(apply(s, X), apply(apply(s, k), ?)), Z)
3.06/3.25	
3.06/3.25	Lemma 5: apply(apply(apply(s, apply(s, k)), X), Y) = apply(X, Y).
3.06/3.25	Proof:
3.06/3.25	  apply(apply(apply(s, apply(s, k)), X), Y)
3.06/3.25	= { by axiom 1 (s_definition) }
3.06/3.25	  apply(apply(apply(s, k), Y), apply(X, Y))
3.06/3.25	= { by lemma 3 }
3.06/3.25	  apply(X, Y)
3.06/3.25	
3.06/3.25	Goal 1 (prove_fixed_point): apply(X, f(X)) = apply(f(X), apply(X, f(X))).
3.06/3.25	The goal is true when:
3.06/3.25	  X = apply(apply(s, apply(s, k)), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))))
3.06/3.25	
3.06/3.25	Proof:
3.06/3.25	  apply(apply(apply(s, apply(s, k)), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))))), f(apply(apply(s, apply(s, k)), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))))))
3.06/3.25	= { by lemma 5 }
3.06/3.25	  apply(apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))), f(apply(apply(s, apply(s, k)), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))))))
3.06/3.25	= { by axiom 1 (s_definition) }
3.06/3.25	  apply(apply(apply(apply(s, apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(s, k), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))), f(apply(apply(s, apply(s, k)), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))))))
3.06/3.25	= { by axiom 1 (s_definition) }
3.06/3.25	  apply(apply(apply(apply(s, apply(k, apply(s, apply(s, apply(s, k))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))), apply(apply(apply(s, k), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))))), f(apply(apply(s, apply(s, k)), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))))))
3.06/3.25	= { by lemma 3 }
3.06/3.25	  apply(apply(apply(apply(s, apply(k, apply(s, apply(s, apply(s, k))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))), f(apply(apply(s, apply(s, k)), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))))))
3.06/3.25	= { by axiom 1 (s_definition) }
3.06/3.25	  apply(apply(apply(apply(k, apply(s, apply(s, apply(s, k)))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))))), f(apply(apply(s, apply(s, k)), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))))))
3.06/3.25	= { by axiom 2 (k_definition) }
3.06/3.25	  apply(apply(apply(s, apply(s, apply(s, k))), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))))), f(apply(apply(s, apply(s, k)), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))))))
3.06/3.25	= { by axiom 1 (s_definition) }
3.06/3.25	  apply(apply(apply(s, apply(s, k)), f(apply(apply(s, apply(s, k)), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))))))), apply(apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))), f(apply(apply(s, apply(s, k)), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))))))))
3.06/3.25	= { by lemma 5 }
3.06/3.25	  apply(f(apply(apply(s, apply(s, k)), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))))), apply(apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))), f(apply(apply(s, apply(s, k)), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))))))))
3.06/3.25	= { by lemma 5 }
3.06/3.25	  apply(f(apply(apply(s, apply(s, k)), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k))))))))), apply(apply(apply(s, apply(s, k)), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))))), f(apply(apply(s, apply(s, k)), apply(apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))), apply(apply(apply(s, s), apply(s, k)), apply(s, apply(k, apply(s, apply(s, apply(s, k)))))))))))
3.06/3.25	% SZS output end Proof
3.06/3.25	
3.06/3.25	RESULT: Unsatisfiable (the axioms are contradictory).
3.06/3.25	EOF