0.04/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.04/0.13	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.14/0.34	% Computer   : n009.cluster.edu
0.14/0.34	% Model      : x86_64 x86_64
0.14/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.14/0.34	% Memory     : 8042.1875MB
0.14/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.14/0.34	% CPULimit   : 1200
0.14/0.34	% WCLimit    : 120
0.14/0.34	% DateTime   : Tue Jul 13 15:39:42 EDT 2021
0.14/0.34	% CPUTime    : 
38.62/5.33	% SZS status Theorem
38.62/5.33	
38.62/5.34	% SZS output start Proof
38.62/5.34	Take the following subset of the input axioms:
38.62/5.34	  fof(thm_2Einteger_2EINT__DIVIDES__NEG, axiom, ![V1q_2E0, V0p_2E0]: (s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0), s(tyop_2Einteger_2Eint, V1q_2E0)))=s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, V1q_2E0))))) & s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0), s(tyop_2Einteger_2Eint, V1q_2E0)))=s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, V0p_2E0))), s(tyop_2Einteger_2Eint, V1q_2E0))))).
38.62/5.34	  fof(thm_2Einteger_2EINT__DIVIDES__RADD, axiom, ![V1q_2E0, V0p_2E0, V2r_2E0]: (p(s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0), s(tyop_2Einteger_2Eint, V1q_2E0)))) => s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__add_2E2(s(tyop_2Einteger_2Eint, V2r_2E0), s(tyop_2Einteger_2Eint, V1q_2E0)))))=s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0), s(tyop_2Einteger_2Eint, V2r_2E0))))).
38.62/5.34	  fof(thm_2Einteger_2EINT__DIVIDES__RSUB, conjecture, ![V1q_2E0, V0p_2E0, V2r_2E0]: (s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__sub_2E2(s(tyop_2Einteger_2Eint, V2r_2E0), s(tyop_2Einteger_2Eint, V1q_2E0)))))=s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0), s(tyop_2Einteger_2Eint, V2r_2E0))) <= p(s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0), s(tyop_2Einteger_2Eint, V1q_2E0)))))).
38.62/5.34	  fof(thm_2Einteger_2Eint__sub, axiom, ![V0x_2E0, V1y_2E0]: s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__add_2E2(s(tyop_2Einteger_2Eint, V0x_2E0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, V1y_2E0)))))=s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__sub_2E2(s(tyop_2Einteger_2Eint, V0x_2E0), s(tyop_2Einteger_2Eint, V1y_2E0)))).
38.62/5.34	
38.62/5.34	Now clausify the problem and encode Horn clauses using encoding 3 of
38.62/5.34	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
38.62/5.34	We repeatedly replace C & s=t => u=v by the two clauses:
38.62/5.34	  fresh(y, y, x1...xn) = u
38.62/5.34	  C => fresh(s, t, x1...xn) = v
38.62/5.34	where fresh is a fresh function symbol and x1..xn are the free
38.62/5.34	variables of u and v.
38.62/5.34	A predicate p(X) is encoded as p(X)=true (this is sound, because the
38.62/5.34	input problem has no model of domain size 1).
38.62/5.34	
38.62/5.34	The encoding turns the above axioms into the following unit equations and goals:
38.62/5.34	
38.62/5.34	Axiom 1 (thm_2Einteger_2EINT__DIVIDES__RADD): fresh39(X, X, Y, Z, W) = s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, Y), s(tyop_2Einteger_2Eint, W))).
38.62/5.34	Axiom 2 (thm_2Einteger_2EINT__DIVIDES__RSUB): p(s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, v1q_2e0)))) = true2.
38.62/5.34	Axiom 3 (thm_2Einteger_2Eint__sub): s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__add_2E2(s(tyop_2Einteger_2Eint, X), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, Y))))) = s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__sub_2E2(s(tyop_2Einteger_2Eint, X), s(tyop_2Einteger_2Eint, Y))).
38.62/5.34	Axiom 4 (thm_2Einteger_2EINT__DIVIDES__NEG): s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, X), s(tyop_2Einteger_2Eint, Y))) = s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, X), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, Y))))).
38.62/5.34	Axiom 5 (thm_2Einteger_2EINT__DIVIDES__RADD): fresh39(p(s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, X), s(tyop_2Einteger_2Eint, Y)))), true2, X, Y, Z) = s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, X), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__add_2E2(s(tyop_2Einteger_2Eint, Z), s(tyop_2Einteger_2Eint, Y))))).
38.62/5.34	
38.62/5.34	Goal 1 (thm_2Einteger_2EINT__DIVIDES__RSUB_1): s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__sub_2E2(s(tyop_2Einteger_2Eint, v2r_2e0), s(tyop_2Einteger_2Eint, v1q_2e0))))) = s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, v2r_2e0))).
38.62/5.34	Proof:
38.62/5.34	  s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__sub_2E2(s(tyop_2Einteger_2Eint, v2r_2e0), s(tyop_2Einteger_2Eint, v1q_2e0)))))
38.62/5.34	= { by axiom 3 (thm_2Einteger_2Eint__sub) R->L }
38.62/5.34	  s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__add_2E2(s(tyop_2Einteger_2Eint, v2r_2e0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)))))))
38.62/5.34	= { by axiom 5 (thm_2Einteger_2EINT__DIVIDES__RADD) R->L }
38.62/5.34	  fresh39(p(s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)))))), true2, v0p_2e0, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)), v2r_2e0)
38.62/5.34	= { by axiom 4 (thm_2Einteger_2EINT__DIVIDES__NEG) R->L }
38.62/5.34	  fresh39(p(s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, v1q_2e0)))), true2, v0p_2e0, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)), v2r_2e0)
38.62/5.34	= { by axiom 2 (thm_2Einteger_2EINT__DIVIDES__RSUB) }
38.62/5.34	  fresh39(true2, true2, v0p_2e0, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)), v2r_2e0)
38.62/5.34	= { by axiom 1 (thm_2Einteger_2EINT__DIVIDES__RADD) }
38.62/5.34	  s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, v2r_2e0)))
38.62/5.34	% SZS output end Proof
38.62/5.34	
38.62/5.34	RESULT: Theorem (the conjecture is true).
39.53/5.35	EOF