0.04/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.04/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33	% Computer   : n018.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 10:40:51 EDT 2019
0.12/0.34	% CPUTime    : 
9.64/9.84	% SZS status Unsatisfiable
9.64/9.84	
9.64/9.84	% SZS output start Proof
9.64/9.84	Take the following subset of the input axioms:
9.64/9.84	  fof(b_definition, axiom, ![Y, X, Z]: apply(X, apply(Y, Z))=apply(apply(apply(b, X), Y), Z)).
9.64/9.84	  fof(prove_strong_fixed_point, negated_conjecture, apply(strong_fixed_point, fixed_pt)!=apply(fixed_pt, apply(strong_fixed_point, fixed_pt))).
9.64/9.84	  fof(strong_fixed_point, axiom, strong_fixed_point=apply(apply(b, apply(w1, w1)), apply(apply(b, apply(b, w1)), apply(apply(b, b), b)))).
9.64/9.84	  fof(w1_definition, axiom, ![Y, X]: apply(apply(Y, X), X)=apply(apply(w1, X), Y)).
9.64/9.84	
9.64/9.84	Now clausify the problem and encode Horn clauses using encoding 3 of
9.64/9.84	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
9.64/9.84	We repeatedly replace C & s=t => u=v by the two clauses:
9.64/9.84	  fresh(y, y, x1...xn) = u
9.64/9.84	  C => fresh(s, t, x1...xn) = v
9.64/9.84	where fresh is a fresh function symbol and x1..xn are the free
9.64/9.84	variables of u and v.
9.64/9.84	A predicate p(X) is encoded as p(X)=true (this is sound, because the
9.64/9.84	input problem has no model of domain size 1).
9.64/9.84	
9.64/9.84	The encoding turns the above axioms into the following unit equations and goals:
9.64/9.84	
9.64/9.84	Axiom 1 (b_definition): apply(X, apply(Y, Z)) = apply(apply(apply(b, X), Y), Z).
9.64/9.84	Axiom 2 (w1_definition): apply(apply(X, Y), Y) = apply(apply(w1, Y), X).
9.64/9.84	Axiom 3 (strong_fixed_point): strong_fixed_point = apply(apply(b, apply(w1, w1)), apply(apply(b, apply(b, w1)), apply(apply(b, b), b))).
9.64/9.84	
9.64/9.84	Lemma 4: apply(apply(w1, apply(apply(b, apply(b, X)), w1)), w1) = apply(strong_fixed_point, X).
9.64/9.84	Proof:
9.64/9.84	  apply(apply(w1, apply(apply(b, apply(b, X)), w1)), w1)
9.64/9.84	= { by axiom 1 (b_definition) }
9.64/9.84	  apply(apply(apply(apply(b, w1), apply(b, apply(b, X))), w1), w1)
9.64/9.84	= { by axiom 2 (w1_definition) }
9.64/9.84	  apply(apply(w1, w1), apply(apply(b, w1), apply(b, apply(b, X))))
9.64/9.84	= { by axiom 1 (b_definition) }
9.64/9.84	  apply(apply(w1, w1), apply(apply(b, w1), apply(apply(apply(b, b), b), X)))
9.64/9.84	= { by axiom 1 (b_definition) }
9.64/9.84	  apply(apply(w1, w1), apply(apply(apply(b, apply(b, w1)), apply(apply(b, b), b)), X))
9.64/9.84	= { by axiom 1 (b_definition) }
9.64/9.84	  apply(apply(apply(b, apply(w1, w1)), apply(apply(b, apply(b, w1)), apply(apply(b, b), b))), X)
9.64/9.84	= { by axiom 3 (strong_fixed_point) }
9.64/9.84	  apply(strong_fixed_point, X)
9.64/9.84	
9.64/9.84	Goal 1 (prove_strong_fixed_point): apply(strong_fixed_point, fixed_pt) = apply(fixed_pt, apply(strong_fixed_point, fixed_pt)).
9.64/9.84	Proof:
9.64/9.84	  apply(strong_fixed_point, fixed_pt)
9.64/9.84	= { by lemma 4 }
9.64/9.84	  apply(apply(w1, apply(apply(b, apply(b, fixed_pt)), w1)), w1)
9.64/9.84	= { by axiom 2 (w1_definition) }
9.64/9.84	  apply(apply(w1, apply(apply(b, apply(b, fixed_pt)), w1)), apply(apply(b, apply(b, fixed_pt)), w1))
9.64/9.84	= { by axiom 2 (w1_definition) }
9.64/9.84	  apply(apply(apply(apply(b, apply(b, fixed_pt)), w1), apply(apply(b, apply(b, fixed_pt)), w1)), apply(apply(b, apply(b, fixed_pt)), w1))
9.64/9.84	= { by axiom 1 (b_definition) }
9.64/9.84	  apply(apply(apply(b, fixed_pt), apply(w1, apply(apply(b, apply(b, fixed_pt)), w1))), apply(apply(b, apply(b, fixed_pt)), w1))
9.64/9.84	= { by axiom 1 (b_definition) }
9.64/9.84	  apply(fixed_pt, apply(apply(w1, apply(apply(b, apply(b, fixed_pt)), w1)), apply(apply(b, apply(b, fixed_pt)), w1)))
9.64/9.84	= { by axiom 2 (w1_definition) }
9.64/9.84	  apply(fixed_pt, apply(apply(w1, apply(apply(b, apply(b, fixed_pt)), w1)), w1))
9.64/9.84	= { by lemma 4 }
9.64/9.84	  apply(fixed_pt, apply(strong_fixed_point, fixed_pt))
9.64/9.84	% SZS output end Proof
9.64/9.84	
9.64/9.84	RESULT: Unsatisfiable (the axioms are contradictory).
9.64/9.84	EOF