0.00/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.12/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33	% Computer   : n004.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 11:26:08 EDT 2019
0.12/0.34	% CPUTime    : 
0.55/0.73	% SZS status Unsatisfiable
0.55/0.73	
0.55/0.73	% SZS output start Proof
0.55/0.73	Take the following subset of the input axioms:
0.55/0.73	  fof(b_definition, axiom, ![X, Y, Z]: apply(apply(apply(b, X), Y), Z)=apply(X, apply(Y, Z))).
0.55/0.73	  fof(prove_strong_fixed_point, negated_conjecture, apply(strong_fixed_point, fixed_pt)!=apply(fixed_pt, apply(strong_fixed_point, fixed_pt))).
0.55/0.73	  fof(strong_fixed_point, axiom, strong_fixed_point=apply(apply(b, apply(apply(b, apply(w, w)), apply(apply(b, apply(b, w)), b))), b)).
0.55/0.73	  fof(w_definition, axiom, ![X, Y]: apply(apply(X, Y), Y)=apply(apply(w, X), Y)).
0.55/0.73	
0.55/0.73	Now clausify the problem and encode Horn clauses using encoding 3 of
0.55/0.73	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.55/0.73	We repeatedly replace C & s=t => u=v by the two clauses:
0.55/0.73	  fresh(y, y, x1...xn) = u
0.55/0.73	  C => fresh(s, t, x1...xn) = v
0.55/0.73	where fresh is a fresh function symbol and x1..xn are the free
0.55/0.73	variables of u and v.
0.55/0.73	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.55/0.73	input problem has no model of domain size 1).
0.55/0.73	
0.55/0.73	The encoding turns the above axioms into the following unit equations and goals:
0.55/0.73	
0.55/0.73	Axiom 1 (w_definition): apply(apply(X, Y), Y) = apply(apply(w, X), Y).
0.55/0.73	Axiom 2 (strong_fixed_point): strong_fixed_point = apply(apply(b, apply(apply(b, apply(w, w)), apply(apply(b, apply(b, w)), b))), b).
0.55/0.73	Axiom 3 (b_definition): apply(apply(apply(b, X), Y), Z) = apply(X, apply(Y, Z)).
0.55/0.73	
0.55/0.73	Lemma 4: apply(apply(w, w), apply(apply(b, w), apply(b, apply(b, X)))) = apply(strong_fixed_point, X).
0.55/0.73	Proof:
0.55/0.73	  apply(apply(w, w), apply(apply(b, w), apply(b, apply(b, X))))
0.55/0.73	= { by axiom 3 (b_definition) }
0.55/0.73	  apply(apply(w, w), apply(apply(apply(b, apply(b, w)), b), apply(b, X)))
0.55/0.73	= { by axiom 3 (b_definition) }
0.55/0.73	  apply(apply(apply(b, apply(w, w)), apply(apply(b, apply(b, w)), b)), apply(b, X))
0.55/0.73	= { by axiom 3 (b_definition) }
0.55/0.73	  apply(apply(apply(b, apply(apply(b, apply(w, w)), apply(apply(b, apply(b, w)), b))), b), X)
0.55/0.73	= { by axiom 2 (strong_fixed_point) }
0.55/0.73	  apply(strong_fixed_point, X)
0.55/0.73	
0.55/0.73	Goal 1 (prove_strong_fixed_point): apply(strong_fixed_point, fixed_pt) = apply(fixed_pt, apply(strong_fixed_point, fixed_pt)).
0.55/0.73	Proof:
0.55/0.73	  apply(strong_fixed_point, fixed_pt)
0.55/0.73	= { by lemma 4 }
0.55/0.73	  apply(apply(w, w), apply(apply(b, w), apply(b, apply(b, fixed_pt))))
0.55/0.73	= { by axiom 1 (w_definition) }
0.55/0.73	  apply(apply(w, apply(apply(b, w), apply(b, apply(b, fixed_pt)))), apply(apply(b, w), apply(b, apply(b, fixed_pt))))
0.55/0.73	= { by axiom 1 (w_definition) }
0.55/0.73	  apply(apply(apply(apply(b, w), apply(b, apply(b, fixed_pt))), apply(apply(b, w), apply(b, apply(b, fixed_pt)))), apply(apply(b, w), apply(b, apply(b, fixed_pt))))
0.55/0.73	= { by axiom 3 (b_definition) }
0.55/0.73	  apply(apply(w, apply(apply(b, apply(b, fixed_pt)), apply(apply(b, w), apply(b, apply(b, fixed_pt))))), apply(apply(b, w), apply(b, apply(b, fixed_pt))))
0.55/0.73	= { by axiom 1 (w_definition) }
0.55/0.73	  apply(apply(apply(apply(b, apply(b, fixed_pt)), apply(apply(b, w), apply(b, apply(b, fixed_pt)))), apply(apply(b, w), apply(b, apply(b, fixed_pt)))), apply(apply(b, w), apply(b, apply(b, fixed_pt))))
0.55/0.73	= { by axiom 3 (b_definition) }
0.55/0.73	  apply(apply(apply(b, fixed_pt), apply(apply(apply(b, w), apply(b, apply(b, fixed_pt))), apply(apply(b, w), apply(b, apply(b, fixed_pt))))), apply(apply(b, w), apply(b, apply(b, fixed_pt))))
0.55/0.73	= { by axiom 3 (b_definition) }
0.55/0.73	  apply(fixed_pt, apply(apply(apply(apply(b, w), apply(b, apply(b, fixed_pt))), apply(apply(b, w), apply(b, apply(b, fixed_pt)))), apply(apply(b, w), apply(b, apply(b, fixed_pt)))))
0.55/0.73	= { by axiom 1 (w_definition) }
0.55/0.73	  apply(fixed_pt, apply(apply(w, apply(apply(b, w), apply(b, apply(b, fixed_pt)))), apply(apply(b, w), apply(b, apply(b, fixed_pt)))))
0.55/0.73	= { by axiom 1 (w_definition) }
0.55/0.73	  apply(fixed_pt, apply(apply(w, w), apply(apply(b, w), apply(b, apply(b, fixed_pt)))))
0.55/0.73	= { by lemma 4 }
0.55/0.73	  apply(fixed_pt, apply(strong_fixed_point, fixed_pt))
0.55/0.73	% SZS output end Proof
0.55/0.73	
0.55/0.73	RESULT: Unsatisfiable (the axioms are contradictory).
0.55/0.73	EOF