0.12/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.12/0.12	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.12/0.33	% Computer   : n002.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   : 1200
0.12/0.33	% WCLimit    : 120
0.12/0.33	% DateTime   : Tue Jul 13 15:55:33 EDT 2021
0.12/0.33	% CPUTime    : 
7.23/1.27	% SZS status Theorem
7.23/1.27	
7.23/1.27	% SZS output start Proof
7.23/1.27	Take the following subset of the input axioms:
7.23/1.28	  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))))))))).
7.23/1.28	  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))))))))).
7.23/1.28	  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))))))))).
7.23/1.28	  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_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)))) & p(s(tyop_2Emin_2Ebool, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), V1t_2E0))))))).
7.23/1.28	
7.23/1.28	Now clausify the problem and encode Horn clauses using encoding 3 of
7.23/1.28	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
7.23/1.28	We repeatedly replace C & s=t => u=v by the two clauses:
7.23/1.28	  fresh(y, y, x1...xn) = u
7.23/1.28	  C => fresh(s, t, x1...xn) = v
7.23/1.28	where fresh is a fresh function symbol and x1..xn are the free
7.23/1.28	variables of u and v.
7.23/1.28	A predicate p(X) is encoded as p(X)=true (this is sound, because the
7.23/1.28	input problem has no model of domain size 1).
7.23/1.28	
7.23/1.28	The encoding turns the above axioms into the following unit equations and goals:
7.23/1.28	
7.23/1.28	Axiom 1 (thm_2Ereal__topology_2EBOUNDED__SUBSET): fresh39(X, X, Y) = true2.
7.23/1.28	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)))).
7.23/1.28	Axiom 3 (thm_2Ereal__topology_2EBOUNDED__SUBSET): fresh40(p(s(tyop_2Emin_2Ebool, c_2Ereal__topology_2Ebounded__def_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), X)))), true2, Y, X) = fresh39(p(s(tyop_2Emin_2Ebool, c_2Epred__set_2ESUBSET_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), Y), s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal, tyop_2Emin_2Ebool), X)))), true2, Y).
7.23/1.28	Axiom 4 (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.
7.23/1.28	Axiom 5 (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.
7.23/1.28	
7.23/1.28	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, v0x_2e0), s(tyop_2Erealax_2Ereal, v1e_2e0)))))))) = true2.
7.23/1.28	Proof:
7.23/1.28	  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))))))))
7.23/1.28	= { by axiom 2 (thm_2Ereal__topology_2EBOUNDED__SUBSET) R->L }
7.23/1.28	  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, v0x_2e0), s(tyop_2Erealax_2Ereal, 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, v0x_2e0), s(tyop_2Erealax_2Ereal, v1e_2e0)))))
7.23/1.28	= { by axiom 4 (thm_2Ereal__topology_2EBOUNDED__CBALL) R->L }
7.23/1.28	  fresh40(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)))))))), 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, v0x_2e0), s(tyop_2Erealax_2Ereal, 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, v0x_2e0), s(tyop_2Erealax_2Ereal, v1e_2e0)))))
7.23/1.28	= { by axiom 3 (thm_2Ereal__topology_2EBOUNDED__SUBSET) }
7.23/1.28	  fresh39(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)))))))), 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, v0x_2e0), s(tyop_2Erealax_2Ereal, v1e_2e0)))))
7.23/1.28	= { by axiom 5 (thm_2Ereal__topology_2EBALL__SUBSET__CBALL) }
7.23/1.28	  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, v0x_2e0), s(tyop_2Erealax_2Ereal, v1e_2e0)))))
7.23/1.28	= { by axiom 1 (thm_2Ereal__topology_2EBOUNDED__SUBSET) }
7.23/1.28	  true2
7.23/1.28	% SZS output end Proof
7.23/1.28	
7.23/1.28	RESULT: Theorem (the conjecture is true).
7.23/1.28	EOF