0.11/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.11/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.14/0.34	% Computer   : n020.cluster.edu
0.14/0.34	% Model      : x86_64 x86_64
0.14/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.14/0.34	% Memory     : 8042.1875MB
0.14/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.14/0.34	% CPULimit   : 180
0.14/0.34	% DateTime   : Thu Aug 29 09:45:55 EDT 2019
0.14/0.34	% CPUTime    : 
0.14/0.38	% SZS status Unsatisfiable
0.14/0.38	
0.14/0.38	% SZS output start Proof
0.14/0.38	Take the following subset of the input axioms:
0.14/0.38	  fof(b_definition, axiom, ![Y, X, Z]: apply(X, apply(Y, Z))=apply(apply(apply(b, X), Y), Z)).
0.14/0.38	  fof(l_definition, axiom, ![Y, X]: apply(apply(l, X), Y)=apply(X, apply(Y, Y))).
0.14/0.38	  fof(m_definition, axiom, ![X]: apply(X, X)=apply(m, X)).
0.14/0.38	  fof(prove_fixed_point, negated_conjecture, ![Y]: apply(Y, f(Y))!=apply(f(Y), apply(Y, f(Y)))).
0.14/0.38	
0.14/0.38	Now clausify the problem and encode Horn clauses using encoding 3 of
0.14/0.38	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.14/0.38	We repeatedly replace C & s=t => u=v by the two clauses:
0.14/0.38	  fresh(y, y, x1...xn) = u
0.14/0.38	  C => fresh(s, t, x1...xn) = v
0.14/0.38	where fresh is a fresh function symbol and x1..xn are the free
0.14/0.38	variables of u and v.
0.14/0.38	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.14/0.38	input problem has no model of domain size 1).
0.14/0.38	
0.14/0.38	The encoding turns the above axioms into the following unit equations and goals:
0.14/0.38	
0.14/0.38	Axiom 1 (m_definition): apply(X, X) = apply(m, X).
0.14/0.38	Axiom 2 (l_definition): apply(apply(l, X), Y) = apply(X, apply(Y, Y)).
0.14/0.38	Axiom 3 (b_definition): apply(X, apply(Y, Z)) = apply(apply(apply(b, X), Y), Z).
0.14/0.38	
0.14/0.38	Goal 1 (prove_fixed_point): apply(X, f(X)) = apply(f(X), apply(X, f(X))).
0.14/0.38	The goal is true when:
0.14/0.38	  X = apply(apply(b, m), l)
0.14/0.38	
0.14/0.38	Proof:
0.14/0.38	  apply(apply(apply(b, m), l), f(apply(apply(b, m), l)))
0.14/0.38	= { by axiom 3 (b_definition) }
0.14/0.38	  apply(m, apply(l, f(apply(apply(b, m), l))))
0.14/0.38	= { by axiom 1 (m_definition) }
0.14/0.38	  apply(apply(l, f(apply(apply(b, m), l))), apply(l, f(apply(apply(b, m), l))))
0.14/0.38	= { by axiom 2 (l_definition) }
0.14/0.38	  apply(f(apply(apply(b, m), l)), apply(apply(l, f(apply(apply(b, m), l))), apply(l, f(apply(apply(b, m), l)))))
0.14/0.38	= { by axiom 1 (m_definition) }
0.14/0.38	  apply(f(apply(apply(b, m), l)), apply(m, apply(l, f(apply(apply(b, m), l)))))
0.14/0.38	= { by axiom 3 (b_definition) }
0.14/0.38	  apply(f(apply(apply(b, m), l)), apply(apply(apply(b, m), l), f(apply(apply(b, m), l))))
0.14/0.38	% SZS output end Proof
0.14/0.38	
0.14/0.38	RESULT: Unsatisfiable (the axioms are contradictory).
0.14/0.39	EOF