0.06/0.11	% 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
0.11/0.33	% Computer   : n013.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 10:26:16 EDT 2019
0.11/0.33	% CPUTime    : 
0.36/0.56	% SZS status Unsatisfiable
0.36/0.56	
0.36/0.56	% SZS output start Proof
0.36/0.56	Take the following subset of the input axioms:
0.36/0.56	  fof(b_definition, axiom, ![X, Y, Z]: apply(apply(apply(b, X), Y), Z)=apply(X, apply(Y, Z))).
0.36/0.56	  fof(prove_strong_fixed_point, negated_conjecture, apply(fixed_pt, apply(strong_fixed_point, fixed_pt))!=apply(strong_fixed_point, fixed_pt)).
0.36/0.56	  fof(strong_fixed_point, axiom, strong_fixed_point=apply(apply(b, apply(w, w)), apply(apply(b, w), apply(apply(b, b), b)))).
0.36/0.56	  fof(w_definition, axiom, ![X, Y]: apply(apply(X, Y), Y)=apply(apply(w, X), Y)).
0.36/0.56	
0.36/0.56	Now clausify the problem and encode Horn clauses using encoding 3 of
0.36/0.56	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.36/0.56	We repeatedly replace C & s=t => u=v by the two clauses:
0.36/0.56	  fresh(y, y, x1...xn) = u
0.36/0.56	  C => fresh(s, t, x1...xn) = v
0.36/0.56	where fresh is a fresh function symbol and x1..xn are the free
0.36/0.56	variables of u and v.
0.36/0.56	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.36/0.56	input problem has no model of domain size 1).
0.36/0.56	
0.36/0.56	The encoding turns the above axioms into the following unit equations and goals:
0.36/0.56	
0.36/0.56	Axiom 1 (strong_fixed_point): strong_fixed_point = apply(apply(b, apply(w, w)), apply(apply(b, w), apply(apply(b, b), b))).
0.36/0.56	Axiom 2 (w_definition): apply(apply(X, Y), Y) = apply(apply(w, X), Y).
0.36/0.56	Axiom 3 (b_definition): apply(apply(apply(b, X), Y), Z) = apply(X, apply(Y, Z)).
0.36/0.56	
0.36/0.56	Lemma 4: apply(apply(w, w), apply(w, apply(b, apply(b, X)))) = apply(strong_fixed_point, X).
0.36/0.56	Proof:
0.36/0.56	  apply(apply(w, w), apply(w, apply(b, apply(b, X))))
0.36/0.56	= { by axiom 3 (b_definition) }
0.36/0.56	  apply(apply(w, w), apply(w, apply(apply(apply(b, b), b), X)))
0.36/0.56	= { by axiom 3 (b_definition) }
0.36/0.56	  apply(apply(w, w), apply(apply(apply(b, w), apply(apply(b, b), b)), X))
0.36/0.56	= { by axiom 3 (b_definition) }
0.36/0.56	  apply(apply(apply(b, apply(w, w)), apply(apply(b, w), apply(apply(b, b), b))), X)
0.36/0.56	= { by axiom 1 (strong_fixed_point) }
0.36/0.56	  apply(strong_fixed_point, X)
0.36/0.56	
0.36/0.56	Goal 1 (prove_strong_fixed_point): apply(fixed_pt, apply(strong_fixed_point, fixed_pt)) = apply(strong_fixed_point, fixed_pt).
0.36/0.56	Proof:
0.36/0.56	  apply(fixed_pt, apply(strong_fixed_point, fixed_pt))
0.36/0.56	= { by lemma 4 }
0.36/0.56	  apply(fixed_pt, apply(apply(w, w), apply(w, apply(b, apply(b, fixed_pt)))))
0.36/0.56	= { by axiom 2 (w_definition) }
0.36/0.56	  apply(fixed_pt, apply(apply(w, apply(w, apply(b, apply(b, fixed_pt)))), apply(w, apply(b, apply(b, fixed_pt)))))
0.36/0.56	= { by axiom 2 (w_definition) }
0.36/0.56	  apply(fixed_pt, apply(apply(apply(w, apply(b, apply(b, fixed_pt))), apply(w, apply(b, apply(b, fixed_pt)))), apply(w, apply(b, apply(b, fixed_pt)))))
0.36/0.56	= { by axiom 3 (b_definition) }
0.36/0.56	  apply(apply(apply(b, fixed_pt), apply(apply(w, apply(b, apply(b, fixed_pt))), apply(w, apply(b, apply(b, fixed_pt))))), apply(w, apply(b, apply(b, fixed_pt))))
0.36/0.56	= { by axiom 3 (b_definition) }
0.36/0.56	  apply(apply(apply(apply(b, apply(b, fixed_pt)), apply(w, apply(b, apply(b, fixed_pt)))), apply(w, apply(b, apply(b, fixed_pt)))), apply(w, apply(b, apply(b, fixed_pt))))
0.36/0.56	= { by axiom 2 (w_definition) }
0.36/0.56	  apply(apply(apply(w, apply(b, apply(b, fixed_pt))), apply(w, apply(b, apply(b, fixed_pt)))), apply(w, apply(b, apply(b, fixed_pt))))
0.36/0.56	= { by axiom 2 (w_definition) }
0.36/0.56	  apply(apply(w, apply(w, apply(b, apply(b, fixed_pt)))), apply(w, apply(b, apply(b, fixed_pt))))
0.36/0.56	= { by axiom 2 (w_definition) }
0.36/0.56	  apply(apply(w, w), apply(w, apply(b, apply(b, fixed_pt))))
0.36/0.56	= { by lemma 4 }
0.36/0.56	  apply(strong_fixed_point, fixed_pt)
0.36/0.56	% SZS output end Proof
0.36/0.56	
0.36/0.56	RESULT: Unsatisfiable (the axioms are contradictory).
0.36/0.56	EOF