0.12/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.12/0.14	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.14/0.35	% Computer   : n022.cluster.edu
0.14/0.35	% Model      : x86_64 x86_64
0.14/0.35	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.14/0.35	% Memory     : 8042.1875MB
0.14/0.35	% OS         : Linux 3.10.0-693.el7.x86_64
0.14/0.35	% CPULimit   : 180
0.14/0.35	% DateTime   : Thu Aug 29 09:11:43 EDT 2019
0.14/0.35	% CPUTime    : 
0.14/0.37	% SZS status Unsatisfiable
0.14/0.37	
0.14/0.37	% SZS output start Proof
0.14/0.37	Take the following subset of the input axioms:
0.14/0.37	  fof(k_definition, axiom, ![X, Y]: apply(apply(k, X), Y)=X).
0.14/0.37	  fof(prove_u_combinator, negated_conjecture, apply(y, apply(apply(x, x), y))!=apply(apply(apply(apply(s, apply(k, apply(s, apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))), x), y)).
0.14/0.37	  fof(s_definition, axiom, ![X, Y, Z]: apply(apply(apply(s, X), Y), Z)=apply(apply(X, Z), apply(Y, Z))).
0.14/0.37	
0.14/0.37	Now clausify the problem and encode Horn clauses using encoding 3 of
0.14/0.37	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.14/0.37	We repeatedly replace C & s=t => u=v by the two clauses:
0.14/0.37	  fresh(y, y, x1...xn) = u
0.14/0.37	  C => fresh(s, t, x1...xn) = v
0.14/0.37	where fresh is a fresh function symbol and x1..xn are the free
0.14/0.37	variables of u and v.
0.14/0.37	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.14/0.37	input problem has no model of domain size 1).
0.14/0.37	
0.14/0.37	The encoding turns the above axioms into the following unit equations and goals:
0.14/0.37	
0.14/0.37	Axiom 1 (k_definition): apply(apply(k, X), Y) = X.
0.14/0.37	Axiom 2 (s_definition): apply(apply(apply(s, X), Y), Z) = apply(apply(X, Z), apply(Y, Z)).
0.14/0.37	
0.14/0.37	Goal 1 (prove_u_combinator): apply(y, apply(apply(x, x), y)) = apply(apply(apply(apply(s, apply(k, apply(s, apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))), x), y).
0.14/0.37	Proof:
0.14/0.37	  apply(y, apply(apply(x, x), y))
0.14/0.37	= { by axiom 1 (k_definition) }
0.14/0.37	  apply(apply(apply(k, y), apply(k, y)), apply(apply(x, x), y))
0.14/0.37	= { by axiom 2 (s_definition) }
0.14/0.37	  apply(apply(apply(apply(s, k), k), y), apply(apply(x, x), y))
0.14/0.37	= { by axiom 1 (k_definition) }
0.14/0.37	  apply(apply(apply(apply(s, k), k), y), apply(apply(apply(apply(k, x), apply(k, x)), x), y))
0.14/0.37	= { by axiom 2 (s_definition) }
0.14/0.37	  apply(apply(apply(apply(s, k), k), y), apply(apply(apply(apply(apply(s, k), k), x), x), y))
0.14/0.37	= { by axiom 1 (k_definition) }
0.14/0.37	  apply(apply(apply(apply(s, k), k), y), apply(apply(apply(apply(apply(s, k), k), x), apply(apply(k, x), apply(k, x))), y))
0.14/0.37	= { by axiom 2 (s_definition) }
0.14/0.37	  apply(apply(apply(apply(s, k), k), y), apply(apply(apply(apply(apply(s, k), k), x), apply(apply(apply(s, k), k), x)), y))
0.14/0.37	= { by axiom 2 (s_definition) }
0.14/0.37	  apply(apply(apply(apply(s, k), k), y), apply(apply(apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)), x), y))
0.14/0.37	= { by axiom 2 (s_definition) }
0.14/0.37	  apply(apply(apply(s, apply(apply(s, k), k)), apply(apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)), x)), y)
0.14/0.37	= { by axiom 1 (k_definition) }
0.14/0.37	  apply(apply(apply(apply(k, apply(s, apply(apply(s, k), k))), x), apply(apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)), x)), y)
0.14/0.37	= { by axiom 2 (s_definition) }
0.14/0.37	  apply(apply(apply(apply(s, apply(k, apply(s, apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))), x), y)
0.14/0.37	% SZS output end Proof
0.14/0.37	
0.14/0.37	RESULT: Unsatisfiable (the axioms are contradictory).
0.14/0.37	EOF