0.03/0.12	% 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   : n017.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 12:39:19 EDT 2019
0.12/0.33	% CPUTime    : 
1.66/1.84	% SZS status Unsatisfiable
1.66/1.84	
1.66/1.84	% SZS output start Proof
1.66/1.84	Take the following subset of the input axioms:
1.66/1.84	  fof(b_definition, axiom, ![X, Y, Z]: apply(X, apply(Y, Z))=apply(apply(apply(b, X), Y), Z)).
1.66/1.84	  fof(prove_f_combinator, negated_conjecture, ![X]: apply(apply(apply(X, f(X)), g(X)), h(X))!=apply(apply(h(X), g(X)), f(X))).
1.66/1.84	  fof(t_definition, axiom, ![X, Y]: apply(Y, X)=apply(apply(t, X), Y)).
1.66/1.84	
1.66/1.84	Now clausify the problem and encode Horn clauses using encoding 3 of
1.66/1.84	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
1.66/1.84	We repeatedly replace C & s=t => u=v by the two clauses:
1.66/1.84	  fresh(y, y, x1...xn) = u
1.66/1.84	  C => fresh(s, t, x1...xn) = v
1.66/1.84	where fresh is a fresh function symbol and x1..xn are the free
1.66/1.84	variables of u and v.
1.66/1.84	A predicate p(X) is encoded as p(X)=true (this is sound, because the
1.66/1.84	input problem has no model of domain size 1).
1.66/1.84	
1.66/1.84	The encoding turns the above axioms into the following unit equations and goals:
1.66/1.84	
1.66/1.84	Axiom 1 (t_definition): apply(X, Y) = apply(apply(t, Y), X).
1.66/1.84	Axiom 2 (b_definition): apply(X, apply(Y, Z)) = apply(apply(apply(b, X), Y), Z).
1.66/1.84	
1.66/1.84	Goal 1 (prove_f_combinator): apply(apply(apply(X, f(X)), g(X)), h(X)) = apply(apply(h(X), g(X)), f(X)).
1.66/1.84	The goal is true when:
1.66/1.84	  X = apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t)))
1.66/1.84	
1.66/1.84	Proof:
1.66/1.84	  apply(apply(apply(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t))), f(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t))))), g(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t))))), h(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t)))))
1.66/1.84	= { by axiom 2 (b_definition) }
1.66/1.84	  apply(apply(apply(apply(t, t), apply(apply(apply(b, b), apply(apply(b, b), t)), f(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t)))))), g(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t))))), h(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t)))))
1.66/1.84	= { by axiom 1 (t_definition) }
1.66/1.84	  apply(apply(apply(apply(apply(apply(b, b), apply(apply(b, b), t)), f(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t))))), t), g(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t))))), h(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t)))))
1.66/1.84	= { by axiom 2 (b_definition) }
1.66/1.84	  apply(apply(apply(apply(b, apply(apply(apply(b, b), t), f(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t)))))), t), g(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t))))), h(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t)))))
1.66/1.84	= { by axiom 2 (b_definition) }
1.66/1.84	  apply(apply(apply(apply(apply(b, b), t), f(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t))))), apply(t, g(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t)))))), h(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t)))))
1.66/1.84	= { by axiom 2 (b_definition) }
1.66/1.84	  apply(apply(apply(b, apply(t, f(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t)))))), apply(t, g(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t)))))), h(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t)))))
1.66/1.84	= { by axiom 2 (b_definition) }
1.66/1.84	  apply(apply(t, f(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t))))), apply(apply(t, g(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t))))), h(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t))))))
1.66/1.84	= { by axiom 1 (t_definition) }
1.66/1.84	  apply(apply(t, f(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t))))), apply(h(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t)))), g(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t))))))
1.66/1.84	= { by axiom 1 (t_definition) }
1.66/1.84	  apply(apply(h(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t)))), g(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t))))), f(apply(apply(b, apply(t, t)), apply(apply(b, b), apply(apply(b, b), t)))))
1.66/1.84	% SZS output end Proof
1.66/1.84	
1.66/1.84	RESULT: Unsatisfiable (the axioms are contradictory).
1.66/1.85	EOF