0.03/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.03/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.14/0.35	% Computer   : n004.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   : 960
0.14/0.35	% WCLimit    : 120
0.14/0.35	% DateTime   : Thu Jul  2 07:21:43 EDT 2020
0.14/0.35	% CPUTime    : 
2.33/0.71	% SZS status Theorem
2.33/0.71	
2.33/0.71	% SZS output start Proof
2.33/0.71	Take the following subset of the input axioms:
2.33/0.72	  fof(thm_2Ereal__topology_2EBALL__SUBSET__CBALL, axiom, ![V0x_2E0, V1e_2E0]: p(s(tyop_2Emin_2Ebool, c_2Epred__set_2ESUBSET_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), c_2Ereal__topology_2Eball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, V0x_2E0), s(tyop_2Erealax_2Ereal, V1e_2E0))))), s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), c_2Ereal__topology_2Ecball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, V0x_2E0), s(tyop_2Erealax_2Ereal, V1e_2E0))))))))).
2.33/0.72	  fof(thm_2Ereal__topology_2EBOUNDED__BALL, conjecture, ![V0x_2E0, V1e_2E0]: p(s(tyop_2Emin_2Ebool, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), c_2Ereal__topology_2Eball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, V0x_2E0), s(tyop_2Erealax_2Ereal, V1e_2E0))))))))).
2.33/0.72	  fof(thm_2Ereal__topology_2EBOUNDED__CBALL, axiom, ![V0x_2E0, V1e_2E0]: p(s(tyop_2Emin_2Ebool, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), c_2Ereal__topology_2Ecball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, V0x_2E0), s(tyop_2Erealax_2Ereal, V1e_2E0))))))))).
2.33/0.72	  fof(thm_2Ereal__topology_2EBOUNDED__SUBSET, axiom, ![V0s_2E0, V1t_2E0]: (p(s(tyop_2Emin_2Ebool, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), V0s_2E0)))) <= (p(s(tyop_2Emin_2Ebool, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), V1t_2E0)))) & p(s(tyop_2Emin_2Ebool, c_2Epred__set_2ESUBSET_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), V0s_2E0), s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), V1t_2E0))))))).
2.33/0.72	
2.33/0.72	Now clausify the problem and encode Horn clauses using encoding 3 of
2.33/0.72	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
2.33/0.72	We repeatedly replace C & s=t => u=v by the two clauses:
2.33/0.72	  fresh(y, y, x1...xn) = u
2.33/0.72	  C => fresh(s, t, x1...xn) = v
2.33/0.72	where fresh is a fresh function symbol and x1..xn are the free
2.33/0.72	variables of u and v.
2.33/0.72	A predicate p(X) is encoded as p(X)=true (this is sound, because the
2.33/0.72	input problem has no model of domain size 1).
2.33/0.72	
2.33/0.72	The encoding turns the above axioms into the following unit equations and goals:
2.33/0.72	
2.33/0.72	Axiom 1 (thm_2Ebool_2EIMP__CLAUSES_7): fresh44(X, X, Y) = p(s(tyop_2Emin_2Ebool, Y)).
2.33/0.72	Axiom 2 (thm_2Ereal__topology_2EBOUNDED__SUBSET): fresh40(X, X, Y, Z) = p(s(tyop_2Emin_2Ebool, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), Y)))).
2.33/0.72	Axiom 3 (thm_2Ereal__topology_2EBOUNDED__SUBSET): fresh39(X, X, Y) = true2.
2.33/0.72	Axiom 4 (thm_2Ereal__topology_2EBOUNDED__SUBSET): fresh40(p(s(tyop_2Emin_2Ebool, c_2Epred__set_2ESUBSET_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), X), s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), Y)))), true2, X, Y) = fresh39(p(s(tyop_2Emin_2Ebool, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), Y)))), true2, X).
2.33/0.72	Axiom 5 (thm_2Ereal__topology_2EBOUNDED__CBALL): p(s(tyop_2Emin_2Ebool, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), c_2Ereal__topology_2Ecball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, X), s(tyop_2Erealax_2Ereal, Y)))))))) = true2.
2.33/0.72	Axiom 6 (thm_2Ereal__topology_2EBALL__SUBSET__CBALL): p(s(tyop_2Emin_2Ebool, c_2Epred__set_2ESUBSET_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), c_2Ereal__topology_2Eball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, X), s(tyop_2Erealax_2Ereal, Y))))), s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), c_2Ereal__topology_2Ecball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, X), s(tyop_2Erealax_2Ereal, Y)))))))) = true2.
2.33/0.72	
2.33/0.72	Lemma 7: fresh44(?, ?, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), Y))) = fresh40(?, ?, Y, ?).
2.33/0.72	Proof:
2.33/0.72	  fresh44(?, ?, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), Y)))
2.33/0.72	= { by axiom 1 (thm_2Ebool_2EIMP__CLAUSES_7) }
2.33/0.72	  p(s(tyop_2Emin_2Ebool, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), Y))))
2.33/0.72	= { by axiom 2 (thm_2Ereal__topology_2EBOUNDED__SUBSET) }
2.33/0.72	  fresh40(?, ?, Y, ?)
2.33/0.72	
2.33/0.72	Goal 1 (thm_2Ereal__topology_2EBOUNDED__BALL): p(s(tyop_2Emin_2Ebool, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), c_2Ereal__topology_2Eball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, sK5_thm_2Ereal__topology_2EBOUNDED__BALL_V0x_2E0), s(tyop_2Erealax_2Ereal, sK4_thm_2Ereal__topology_2EBOUNDED__BALL_V1e_2E0)))))))) = true2.
2.33/0.72	Proof:
2.33/0.72	  p(s(tyop_2Emin_2Ebool, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), c_2Ereal__topology_2Eball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, sK5_thm_2Ereal__topology_2EBOUNDED__BALL_V0x_2E0), s(tyop_2Erealax_2Ereal, sK4_thm_2Ereal__topology_2EBOUNDED__BALL_V1e_2E0))))))))
2.33/0.72	= { by axiom 1 (thm_2Ebool_2EIMP__CLAUSES_7) }
2.33/0.72	  fresh44(?, ?, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), c_2Ereal__topology_2Eball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, sK5_thm_2Ereal__topology_2EBOUNDED__BALL_V0x_2E0), s(tyop_2Erealax_2Ereal, sK4_thm_2Ereal__topology_2EBOUNDED__BALL_V1e_2E0)))))))
2.33/0.72	= { by lemma 7 }
2.33/0.72	  fresh40(?, ?, c_2Ereal__topology_2Eball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, sK5_thm_2Ereal__topology_2EBOUNDED__BALL_V0x_2E0), s(tyop_2Erealax_2Ereal, sK4_thm_2Ereal__topology_2EBOUNDED__BALL_V1e_2E0)))), ?)
2.33/0.72	= { by axiom 2 (thm_2Ereal__topology_2EBOUNDED__SUBSET) }
2.33/0.72	  p(s(tyop_2Emin_2Ebool, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), c_2Ereal__topology_2Eball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, sK5_thm_2Ereal__topology_2EBOUNDED__BALL_V0x_2E0), s(tyop_2Erealax_2Ereal, sK4_thm_2Ereal__topology_2EBOUNDED__BALL_V1e_2E0))))))))
2.33/0.72	= { by axiom 2 (thm_2Ereal__topology_2EBOUNDED__SUBSET) }
2.33/0.72	  fresh40(true2, true2, c_2Ereal__topology_2Eball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, sK5_thm_2Ereal__topology_2EBOUNDED__BALL_V0x_2E0), s(tyop_2Erealax_2Ereal, sK4_thm_2Ereal__topology_2EBOUNDED__BALL_V1e_2E0)))), c_2Ereal__topology_2Ecball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, sK5_thm_2Ereal__topology_2EBOUNDED__BALL_V0x_2E0), s(tyop_2Erealax_2Ereal, sK4_thm_2Ereal__topology_2EBOUNDED__BALL_V1e_2E0)))))
2.33/0.72	= { by axiom 6 (thm_2Ereal__topology_2EBALL__SUBSET__CBALL) }
2.33/0.72	  fresh40(p(s(tyop_2Emin_2Ebool, c_2Epred__set_2ESUBSET_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), c_2Ereal__topology_2Eball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, sK5_thm_2Ereal__topology_2EBOUNDED__BALL_V0x_2E0), s(tyop_2Erealax_2Ereal, sK4_thm_2Ereal__topology_2EBOUNDED__BALL_V1e_2E0))))), s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), c_2Ereal__topology_2Ecball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, sK5_thm_2Ereal__topology_2EBOUNDED__BALL_V0x_2E0), s(tyop_2Erealax_2Ereal, sK4_thm_2Ereal__topology_2EBOUNDED__BALL_V1e_2E0)))))))), true2, c_2Ereal__topology_2Eball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, sK5_thm_2Ereal__topology_2EBOUNDED__BALL_V0x_2E0), s(tyop_2Erealax_2Ereal, sK4_thm_2Ereal__topology_2EBOUNDED__BALL_V1e_2E0)))), c_2Ereal__topology_2Ecball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, sK5_thm_2Ereal__topology_2EBOUNDED__BALL_V0x_2E0), s(tyop_2Erealax_2Ereal, sK4_thm_2Ereal__topology_2EBOUNDED__BALL_V1e_2E0)))))
2.33/0.72	= { by axiom 4 (thm_2Ereal__topology_2EBOUNDED__SUBSET) }
2.33/0.72	  fresh39(p(s(tyop_2Emin_2Ebool, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), c_2Ereal__topology_2Ecball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, sK5_thm_2Ereal__topology_2EBOUNDED__BALL_V0x_2E0), s(tyop_2Erealax_2Ereal, sK4_thm_2Ereal__topology_2EBOUNDED__BALL_V1e_2E0)))))))), true2, c_2Ereal__topology_2Eball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, sK5_thm_2Ereal__topology_2EBOUNDED__BALL_V0x_2E0), s(tyop_2Erealax_2Ereal, sK4_thm_2Ereal__topology_2EBOUNDED__BALL_V1e_2E0)))))
2.33/0.72	= { by axiom 5 (thm_2Ereal__topology_2EBOUNDED__CBALL) }
2.33/0.72	  fresh39(true2, true2, c_2Ereal__topology_2Eball_2E1(s(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal, tyop_2Erealax_2Ereal), c_2Epair_2E_2C_2E2(s(tyop_2Erealax_2Ereal, sK5_thm_2Ereal__topology_2EBOUNDED__BALL_V0x_2E0), s(tyop_2Erealax_2Ereal, sK4_thm_2Ereal__topology_2EBOUNDED__BALL_V1e_2E0)))))
2.33/0.72	= { by axiom 3 (thm_2Ereal__topology_2EBOUNDED__SUBSET) }
2.33/0.72	  true2
2.33/0.72	% SZS output end Proof
2.33/0.72	
2.33/0.72	RESULT: Theorem (the conjecture is true).
2.33/0.73	EOF