0.03/0.11	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.03/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33	% Computer   : n014.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 12:39:24 EDT 2019
0.12/0.33	% CPUTime    : 
0.18/0.48	% SZS status Unsatisfiable
0.18/0.48	
0.18/0.48	% SZS output start Proof
0.18/0.48	Take the following subset of the input axioms:
0.18/0.48	  fof(b_definition, axiom, ![X, Y, Z]: apply(X, apply(Y, Z))=apply(apply(apply(b, X), Y), Z)).
0.18/0.48	  fof(prove_q1_combinator, negated_conjecture, ![X]: apply(f(X), apply(h(X), g(X)))!=apply(apply(apply(X, f(X)), g(X)), h(X))).
0.18/0.48	  fof(t_definition, axiom, ![X, Y]: apply(apply(t, X), Y)=apply(Y, X)).
0.18/0.48	
0.18/0.48	Now clausify the problem and encode Horn clauses using encoding 3 of
0.18/0.48	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.18/0.48	We repeatedly replace C & s=t => u=v by the two clauses:
0.18/0.48	  fresh(y, y, x1...xn) = u
0.18/0.48	  C => fresh(s, t, x1...xn) = v
0.18/0.48	where fresh is a fresh function symbol and x1..xn are the free
0.18/0.48	variables of u and v.
0.18/0.48	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.18/0.48	input problem has no model of domain size 1).
0.18/0.48	
0.18/0.48	The encoding turns the above axioms into the following unit equations and goals:
0.18/0.48	
0.18/0.48	Axiom 1 (t_definition): apply(apply(t, X), Y) = apply(Y, X).
0.18/0.48	Axiom 2 (b_definition): apply(X, apply(Y, Z)) = apply(apply(apply(b, X), Y), Z).
0.18/0.48	
0.18/0.48	Goal 1 (prove_q1_combinator): apply(f(X), apply(h(X), g(X))) = apply(apply(apply(X, f(X)), g(X)), h(X)).
0.18/0.48	The goal is true when:
0.18/0.48	  X = apply(apply(b, apply(t, t)), apply(apply(b, b), b))
0.18/0.48	
0.18/0.48	Proof:
0.18/0.48	  apply(f(apply(apply(b, apply(t, t)), apply(apply(b, b), b))), apply(h(apply(apply(b, apply(t, t)), apply(apply(b, b), b))), g(apply(apply(b, apply(t, t)), apply(apply(b, b), b)))))
0.18/0.48	= { by axiom 1 (t_definition) }
0.18/0.48	  apply(f(apply(apply(b, apply(t, t)), apply(apply(b, b), b))), apply(apply(t, g(apply(apply(b, apply(t, t)), apply(apply(b, b), b)))), h(apply(apply(b, apply(t, t)), apply(apply(b, b), b)))))
0.18/0.48	= { by axiom 2 (b_definition) }
0.18/0.48	  apply(apply(apply(b, f(apply(apply(b, apply(t, t)), apply(apply(b, b), b)))), apply(t, g(apply(apply(b, apply(t, t)), apply(apply(b, b), b))))), h(apply(apply(b, apply(t, t)), apply(apply(b, b), b))))
0.18/0.48	= { by axiom 2 (b_definition) }
0.18/0.48	  apply(apply(apply(apply(b, apply(b, f(apply(apply(b, apply(t, t)), apply(apply(b, b), b))))), t), g(apply(apply(b, apply(t, t)), apply(apply(b, b), b)))), h(apply(apply(b, apply(t, t)), apply(apply(b, b), b))))
0.18/0.48	= { by axiom 2 (b_definition) }
0.18/0.48	  apply(apply(apply(apply(apply(apply(b, b), b), f(apply(apply(b, apply(t, t)), apply(apply(b, b), b)))), t), g(apply(apply(b, apply(t, t)), apply(apply(b, b), b)))), h(apply(apply(b, apply(t, t)), apply(apply(b, b), b))))
0.18/0.48	= { by axiom 1 (t_definition) }
0.18/0.48	  apply(apply(apply(apply(t, t), apply(apply(apply(b, b), b), f(apply(apply(b, apply(t, t)), apply(apply(b, b), b))))), g(apply(apply(b, apply(t, t)), apply(apply(b, b), b)))), h(apply(apply(b, apply(t, t)), apply(apply(b, b), b))))
0.18/0.48	= { by axiom 2 (b_definition) }
0.18/0.48	  apply(apply(apply(apply(apply(b, apply(t, t)), apply(apply(b, b), b)), f(apply(apply(b, apply(t, t)), apply(apply(b, b), b)))), g(apply(apply(b, apply(t, t)), apply(apply(b, b), b)))), h(apply(apply(b, apply(t, t)), apply(apply(b, b), b))))
0.18/0.48	% SZS output end Proof
0.18/0.48	
0.18/0.48	RESULT: Unsatisfiable (the axioms are contradictory).
0.18/0.48	EOF