0.06/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.06/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33	% Computer   : n012.cluster.edu
0.12/0.33	% Model      : x86_64 x86_64
0.12/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.33	% Memory     : 8042.1875MB
0.12/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.33	% CPULimit   : 180
0.12/0.33	% DateTime   : Thu Aug 29 11:22:42 EDT 2019
0.12/0.33	% CPUTime    : 
0.12/0.36	% SZS status Unsatisfiable
0.12/0.36	
0.12/0.36	% SZS output start Proof
0.12/0.36	Take the following subset of the input axioms:
0.12/0.36	  fof(m_definition, axiom, ![X]: apply(X, X)=apply(m, X)).
0.12/0.36	  fof(prove_fixed_point, negated_conjecture, ![Y]: apply(Y, f(Y))!=apply(f(Y), apply(Y, f(Y)))).
0.12/0.36	  fof(q_definition, axiom, ![Y, X, Z]: apply(Y, apply(X, Z))=apply(apply(apply(q, X), Y), Z)).
0.12/0.36	
0.12/0.36	Now clausify the problem and encode Horn clauses using encoding 3 of
0.12/0.36	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.12/0.36	We repeatedly replace C & s=t => u=v by the two clauses:
0.12/0.36	  fresh(y, y, x1...xn) = u
0.12/0.36	  C => fresh(s, t, x1...xn) = v
0.12/0.36	where fresh is a fresh function symbol and x1..xn are the free
0.12/0.36	variables of u and v.
0.12/0.36	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.12/0.36	input problem has no model of domain size 1).
0.12/0.36	
0.12/0.36	The encoding turns the above axioms into the following unit equations and goals:
0.12/0.36	
0.12/0.36	Axiom 1 (m_definition): apply(X, X) = apply(m, X).
0.12/0.36	Axiom 2 (q_definition): apply(X, apply(Y, Z)) = apply(apply(apply(q, Y), X), Z).
0.12/0.36	
0.12/0.36	Goal 1 (prove_fixed_point): apply(X, f(X)) = apply(f(X), apply(X, f(X))).
0.12/0.36	The goal is true when:
0.12/0.36	  X = apply(apply(q, apply(q, m)), m)
0.12/0.36	
0.12/0.36	Proof:
0.12/0.36	  apply(apply(apply(q, apply(q, m)), m), f(apply(apply(q, apply(q, m)), m)))
0.12/0.36	= { by axiom 2 (q_definition) }
0.12/0.36	  apply(m, apply(apply(q, m), f(apply(apply(q, apply(q, m)), m))))
0.12/0.36	= { by axiom 1 (m_definition) }
0.12/0.36	  apply(apply(apply(q, m), f(apply(apply(q, apply(q, m)), m))), apply(apply(q, m), f(apply(apply(q, apply(q, m)), m))))
0.12/0.36	= { by axiom 2 (q_definition) }
0.12/0.36	  apply(f(apply(apply(q, apply(q, m)), m)), apply(m, apply(apply(q, m), f(apply(apply(q, apply(q, m)), m)))))
0.12/0.36	= { by axiom 2 (q_definition) }
0.12/0.36	  apply(f(apply(apply(q, apply(q, m)), m)), apply(apply(apply(q, apply(q, m)), m), f(apply(apply(q, apply(q, m)), m))))
0.12/0.36	% SZS output end Proof
0.12/0.36	
0.12/0.36	RESULT: Unsatisfiable (the axioms are contradictory).
0.12/0.36	EOF