0.00/0.06	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.00/0.07	% Command    : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn
0.03/0.27	% Computer   : n172.star.cs.uiowa.edu
0.03/0.27	% Model      : x86_64 x86_64
0.03/0.27	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.03/0.27	% Memory     : 32218.625MB
0.03/0.27	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.03/0.27	% CPULimit   : 300
0.03/0.27	% DateTime   : Sat Jul 14 04:19:55 CDT 2018
0.03/0.27	% CPUTime    : 
0.07/0.34	% SZS status Theorem
0.07/0.34	
0.07/0.34	% SZS output start Proof
0.07/0.34	Take the following subset of the input axioms:
0.07/0.34	  fof(complement, axiom,
0.07/0.34	      ![X, Z]:
0.07/0.34	        (member(Z, complement(X))
0.07/0.34	           <=> (member(Z, universal_class) & ~member(Z, X)))).
0.07/0.34	  fof(disjoint_defn, axiom,
0.07/0.34	      ![X, Y]:
0.07/0.34	        (![U]: ~(member(U, Y) & member(U, X)) <=> disjoint(X, Y))).
0.07/0.34	  fof(domain_of, axiom,
0.07/0.34	      ![X, Z]:
0.07/0.34	        ((member(Z, universal_class)
0.07/0.34	            & restrict(X, singleton(Z), universal_class)!=null_class)
0.07/0.34	           <=> member(Z, domain_of(X)))).
0.07/0.34	  fof(existence_of_null_class, conjecture,
0.07/0.34	      ?[X]: ![Z]: ~member(Z, X)).
0.07/0.34	  fof(null_class_defn, axiom, ![X]: ~member(X, null_class)).
0.07/0.34	
0.07/0.34	Now clausify the problem and encode Horn clauses using encoding 3 of
0.07/0.34	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.07/0.34	We repeatedly replace C & s=t => u=v by the two clauses:
0.07/0.34	  $$fresh(y, y, x1...xn) = u
0.07/0.34	  C => $$fresh(s, t, x1...xn) = v
0.07/0.34	where $$fresh is a fresh function symbol and x1..xn are the free
0.07/0.34	variables of u and v.
0.07/0.34	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
0.07/0.34	input problem has no model of domain size 1).
0.07/0.34	
0.07/0.34	The encoding turns the above axioms into the following unit equations and goals:
0.07/0.34	
0.07/0.34	Axiom 146 (existence_of_null_class): member(sK1_existence_of_null_class_Z(X), X) = $$true2.
0.07/0.34	
0.07/0.34	Goal 1 (null_class_defn): member(X, null_class) = $$true2.
0.07/0.34	The goal is true when:
0.07/0.34	  X = sK1_existence_of_null_class_Z(null_class)
0.07/0.34	
0.07/0.34	Proof:
0.07/0.34	  member(sK1_existence_of_null_class_Z(null_class), null_class)
0.07/0.34	= { by axiom 146 (existence_of_null_class) }
0.07/0.34	  $$true2
0.07/0.34	% SZS output end Proof
0.07/0.34	
0.07/0.34	RESULT: Theorem (the conjecture is true).
0.07/0.36	EOF