0.12/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.12/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.34	% Computer   : n026.cluster.edu
0.13/0.34	% Model      : x86_64 x86_64
0.13/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.13/0.34	% Memory     : 8042.1875MB
0.13/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.13/0.34	% CPULimit   : 180
0.13/0.34	% DateTime   : Thu Aug 29 10:32:35 EDT 2019
0.13/0.35	% CPUTime    : 
0.43/0.61	% SZS status Theorem
0.43/0.61	
0.43/0.61	% SZS output start Proof
0.43/0.61	Take the following subset of the input axioms:
0.43/0.61	  fof('ass(cond(33, 0), 0)', axiom, ![Vd46, Vd47, Vd48]: vplus(Vd46, vplus(Vd47, Vd48))=vplus(vplus(Vd46, Vd47), Vd48)).
0.43/0.61	  fof('ass(cond(52, 0), 0)', axiom, ![Vd68, Vd69]: vplus(vsucc(Vd68), Vd69)=vsucc(vplus(Vd68, Vd69))).
0.43/0.61	  fof('ass(cond(61, 0), 0)', axiom, ![Vd78, Vd79]: vplus(Vd78, Vd79)=vplus(Vd79, Vd78)).
0.43/0.61	  fof('ass(cond(conseq(263), 1), 6)', axiom, ![Vd413]: (vplus(vplus(vmul(vd411, Vd413), Vd413), vsucc(vd411))=vplus(vmul(vsucc(vd411), Vd413), vsucc(vd411)) <= vplus(vmul(vd411, Vd413), Vd413)=vmul(vsucc(vd411), Vd413))).
0.43/0.61	  fof('qu(cond(conseq(axiom(3)), 3), and(holds(definiens(29), 45, 0), holds(definiens(29), 44, 0)))', axiom, ![Vd42, Vd43]: (vsucc(vplus(Vd42, Vd43))=vplus(Vd42, vsucc(Vd43)) & vplus(Vd42, v1)=vsucc(Vd42))).
0.43/0.61	  fof('qu(cond(conseq(axiom(3)), 32), and(holds(definiens(249), 399, 0), holds(definiens(249), 398, 0)))', axiom, ![Vd396, Vd397]: (vmul(Vd396, vsucc(Vd397))=vplus(vmul(Vd396, Vd397), Vd396) & Vd396=vmul(Vd396, v1))).
0.43/0.61	  fof('qu(ind(267), imp(267))', conjecture, ![Vd416]: (vplus(vmul(vd411, vsucc(Vd416)), vsucc(Vd416))=vmul(vsucc(vd411), vsucc(Vd416)) <= vplus(vmul(vd411, Vd416), Vd416)=vmul(vsucc(vd411), Vd416))).
0.43/0.61	
0.43/0.61	Now clausify the problem and encode Horn clauses using encoding 3 of
0.43/0.61	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.43/0.61	We repeatedly replace C & s=t => u=v by the two clauses:
0.43/0.61	  fresh(y, y, x1...xn) = u
0.43/0.61	  C => fresh(s, t, x1...xn) = v
0.43/0.61	where fresh is a fresh function symbol and x1..xn are the free
0.43/0.61	variables of u and v.
0.43/0.61	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.43/0.61	input problem has no model of domain size 1).
0.43/0.61	
0.43/0.61	The encoding turns the above axioms into the following unit equations and goals:
0.43/0.61	
0.43/0.61	Axiom 1 (ass(cond(conseq(263), 1), 0)): fresh30(X, X, Y) = vplus(vmul(vd411, vsucc(Y)), vsucc(Y)).
0.43/0.61	Axiom 2 (ass(cond(conseq(263), 1), 2)): fresh28(X, X, Y) = vplus(vmul(vd411, Y), vsucc(vplus(vd411, Y))).
0.43/0.61	Axiom 3 (ass(cond(conseq(263), 1), 5)): fresh25(X, X, Y) = vplus(vmul(vd411, Y), vplus(Y, vsucc(vd411))).
0.43/0.61	Axiom 4 (ass(cond(conseq(263), 1), 6)): fresh24(X, X, Y) = vplus(vmul(vsucc(vd411), Y), vsucc(vd411)).
0.43/0.61	Axiom 5 (ass(cond(conseq(263), 1), 7)): fresh23(X, X, Y) = vmul(vsucc(vd411), vsucc(Y)).
0.43/0.61	Axiom 6 (ass(cond(33, 0), 0)): vplus(X, vplus(Y, Z)) = vplus(vplus(X, Y), Z).
0.43/0.61	Axiom 7 (qu(cond(conseq(axiom(3)), 3), and(holds(definiens(29), 45, 0), holds(definiens(29), 44, 0)))): vsucc(vplus(X, Y)) = vplus(X, vsucc(Y)).
0.43/0.61	Axiom 8 (qu(cond(conseq(axiom(3)), 32), and(holds(definiens(249), 399, 0), holds(definiens(249), 398, 0)))_1): vmul(X, vsucc(Y)) = vplus(vmul(X, Y), X).
0.43/0.61	Axiom 9 (ass(cond(52, 0), 0)): vplus(vsucc(X), Y) = vsucc(vplus(X, Y)).
0.43/0.61	Axiom 10 (ass(cond(61, 0), 0)): vplus(X, Y) = vplus(Y, X).
0.43/0.61	Axiom 11 (ass(cond(conseq(263), 1), 6)): fresh24(vplus(vmul(vd411, X), X), vmul(vsucc(vd411), X), X) = vplus(vplus(vmul(vd411, X), X), vsucc(vd411)).
0.43/0.61	Axiom 12 (qu(ind(267), imp(267))): vplus(vmul(vd411, sK2_qu(ind(267), imp(267))_Vd416), sK2_qu(ind(267), imp(267))_Vd416) = vmul(vsucc(vd411), sK2_qu(ind(267), imp(267))_Vd416).
0.43/0.61	
0.43/0.61	Lemma 13: vplus(sK2_qu(ind(267), imp(267))_Vd416, vmul(vd411, sK2_qu(ind(267), imp(267))_Vd416)) = vmul(vsucc(vd411), sK2_qu(ind(267), imp(267))_Vd416).
0.43/0.61	Proof:
0.43/0.61	  vplus(sK2_qu(ind(267), imp(267))_Vd416, vmul(vd411, sK2_qu(ind(267), imp(267))_Vd416))
0.43/0.61	= { by axiom 10 (ass(cond(61, 0), 0)) }
0.43/0.61	  vplus(vmul(vd411, sK2_qu(ind(267), imp(267))_Vd416), sK2_qu(ind(267), imp(267))_Vd416)
0.43/0.61	= { by axiom 12 (qu(ind(267), imp(267))) }
0.43/0.61	  vmul(vsucc(vd411), sK2_qu(ind(267), imp(267))_Vd416)
0.43/0.61	
0.43/0.61	Lemma 14: fresh24(?, ?, sK2_qu(ind(267), imp(267))_Vd416) = fresh28(?, ?, sK2_qu(ind(267), imp(267))_Vd416).
0.43/0.61	Proof:
0.43/0.61	  fresh24(?, ?, sK2_qu(ind(267), imp(267))_Vd416)
0.43/0.61	= { by axiom 4 (ass(cond(conseq(263), 1), 6)) }
0.43/0.61	  vplus(vmul(vsucc(vd411), sK2_qu(ind(267), imp(267))_Vd416), vsucc(vd411))
0.43/0.61	= { by axiom 4 (ass(cond(conseq(263), 1), 6)) }
0.43/0.61	  fresh24(vmul(vsucc(vd411), sK2_qu(ind(267), imp(267))_Vd416), vmul(vsucc(vd411), sK2_qu(ind(267), imp(267))_Vd416), sK2_qu(ind(267), imp(267))_Vd416)
0.43/0.61	= { by lemma 13 }
0.43/0.61	  fresh24(vplus(sK2_qu(ind(267), imp(267))_Vd416, vmul(vd411, sK2_qu(ind(267), imp(267))_Vd416)), vmul(vsucc(vd411), sK2_qu(ind(267), imp(267))_Vd416), sK2_qu(ind(267), imp(267))_Vd416)
0.43/0.61	= { by axiom 10 (ass(cond(61, 0), 0)) }
0.43/0.61	  fresh24(vplus(vmul(vd411, sK2_qu(ind(267), imp(267))_Vd416), sK2_qu(ind(267), imp(267))_Vd416), vmul(vsucc(vd411), sK2_qu(ind(267), imp(267))_Vd416), sK2_qu(ind(267), imp(267))_Vd416)
0.43/0.61	= { by axiom 11 (ass(cond(conseq(263), 1), 6)) }
0.43/0.61	  vplus(vplus(vmul(vd411, sK2_qu(ind(267), imp(267))_Vd416), sK2_qu(ind(267), imp(267))_Vd416), vsucc(vd411))
0.43/0.61	= { by axiom 6 (ass(cond(33, 0), 0)) }
0.43/0.61	  vplus(vmul(vd411, sK2_qu(ind(267), imp(267))_Vd416), vplus(sK2_qu(ind(267), imp(267))_Vd416, vsucc(vd411)))
0.43/0.61	= { by axiom 7 (qu(cond(conseq(axiom(3)), 3), and(holds(definiens(29), 45, 0), holds(definiens(29), 44, 0)))) }
0.43/0.61	  vplus(vmul(vd411, sK2_qu(ind(267), imp(267))_Vd416), vsucc(vplus(sK2_qu(ind(267), imp(267))_Vd416, vd411)))
0.43/0.61	= { by axiom 10 (ass(cond(61, 0), 0)) }
0.43/0.61	  vplus(vmul(vd411, sK2_qu(ind(267), imp(267))_Vd416), vsucc(vplus(vd411, sK2_qu(ind(267), imp(267))_Vd416)))
0.43/0.61	= { by axiom 2 (ass(cond(conseq(263), 1), 2)) }
0.43/0.61	  fresh28(?, ?, sK2_qu(ind(267), imp(267))_Vd416)
0.43/0.61	
0.43/0.61	Lemma 15: vplus(vsucc(X), Y) = vplus(X, vsucc(Y)).
0.43/0.61	Proof:
0.43/0.61	  vplus(vsucc(X), Y)
0.43/0.61	= { by axiom 9 (ass(cond(52, 0), 0)) }
0.43/0.61	  vsucc(vplus(X, Y))
0.43/0.61	= { by axiom 7 (qu(cond(conseq(axiom(3)), 3), and(holds(definiens(29), 45, 0), holds(definiens(29), 44, 0)))) }
0.43/0.62	  vplus(X, vsucc(Y))
0.43/0.62	
0.43/0.62	Goal 1 (qu(ind(267), imp(267))_1): vplus(vmul(vd411, vsucc(sK2_qu(ind(267), imp(267))_Vd416)), vsucc(sK2_qu(ind(267), imp(267))_Vd416)) = vmul(vsucc(vd411), vsucc(sK2_qu(ind(267), imp(267))_Vd416)).
0.43/0.62	Proof:
0.43/0.62	  vplus(vmul(vd411, vsucc(sK2_qu(ind(267), imp(267))_Vd416)), vsucc(sK2_qu(ind(267), imp(267))_Vd416))
0.43/0.62	= { by axiom 1 (ass(cond(conseq(263), 1), 0)) }
0.43/0.62	  fresh30(?, ?, sK2_qu(ind(267), imp(267))_Vd416)
0.43/0.62	= { by axiom 1 (ass(cond(conseq(263), 1), 0)) }
0.43/0.62	  vplus(vmul(vd411, vsucc(sK2_qu(ind(267), imp(267))_Vd416)), vsucc(sK2_qu(ind(267), imp(267))_Vd416))
0.43/0.62	= { by axiom 10 (ass(cond(61, 0), 0)) }
0.43/0.62	  vplus(vsucc(sK2_qu(ind(267), imp(267))_Vd416), vmul(vd411, vsucc(sK2_qu(ind(267), imp(267))_Vd416)))
0.43/0.62	= { by lemma 15 }
0.43/0.62	  vplus(sK2_qu(ind(267), imp(267))_Vd416, vsucc(vmul(vd411, vsucc(sK2_qu(ind(267), imp(267))_Vd416))))
0.43/0.62	= { by axiom 7 (qu(cond(conseq(axiom(3)), 3), and(holds(definiens(29), 45, 0), holds(definiens(29), 44, 0)))) }
0.43/0.62	  vsucc(vplus(sK2_qu(ind(267), imp(267))_Vd416, vmul(vd411, vsucc(sK2_qu(ind(267), imp(267))_Vd416))))
0.43/0.62	= { by axiom 10 (ass(cond(61, 0), 0)) }
0.43/0.62	  vsucc(vplus(vmul(vd411, vsucc(sK2_qu(ind(267), imp(267))_Vd416)), sK2_qu(ind(267), imp(267))_Vd416))
0.43/0.62	= { by axiom 8 (qu(cond(conseq(axiom(3)), 32), and(holds(definiens(249), 399, 0), holds(definiens(249), 398, 0)))_1) }
0.43/0.62	  vsucc(vplus(vplus(vmul(vd411, sK2_qu(ind(267), imp(267))_Vd416), vd411), sK2_qu(ind(267), imp(267))_Vd416))
0.43/0.62	= { by axiom 6 (ass(cond(33, 0), 0)) }
0.43/0.62	  vsucc(vplus(vmul(vd411, sK2_qu(ind(267), imp(267))_Vd416), vplus(vd411, sK2_qu(ind(267), imp(267))_Vd416)))
0.43/0.62	= { by axiom 10 (ass(cond(61, 0), 0)) }
0.43/0.62	  vsucc(vplus(vplus(vd411, sK2_qu(ind(267), imp(267))_Vd416), vmul(vd411, sK2_qu(ind(267), imp(267))_Vd416)))
0.43/0.62	= { by axiom 6 (ass(cond(33, 0), 0)) }
0.43/0.62	  vsucc(vplus(vd411, vplus(sK2_qu(ind(267), imp(267))_Vd416, vmul(vd411, sK2_qu(ind(267), imp(267))_Vd416))))
0.43/0.62	= { by lemma 13 }
0.43/0.62	  vsucc(vplus(vd411, vmul(vsucc(vd411), sK2_qu(ind(267), imp(267))_Vd416)))
0.43/0.62	= { by axiom 7 (qu(cond(conseq(axiom(3)), 3), and(holds(definiens(29), 45, 0), holds(definiens(29), 44, 0)))) }
0.43/0.62	  vplus(vd411, vsucc(vmul(vsucc(vd411), sK2_qu(ind(267), imp(267))_Vd416)))
0.43/0.62	= { by lemma 15 }
0.43/0.62	  vplus(vsucc(vd411), vmul(vsucc(vd411), sK2_qu(ind(267), imp(267))_Vd416))
0.43/0.62	= { by axiom 10 (ass(cond(61, 0), 0)) }
0.43/0.62	  vplus(vmul(vsucc(vd411), sK2_qu(ind(267), imp(267))_Vd416), vsucc(vd411))
0.43/0.62	= { by axiom 8 (qu(cond(conseq(axiom(3)), 32), and(holds(definiens(249), 399, 0), holds(definiens(249), 398, 0)))_1) }
0.43/0.62	  vmul(vsucc(vd411), vsucc(sK2_qu(ind(267), imp(267))_Vd416))
0.43/0.62	% SZS output end Proof
0.43/0.62	
0.43/0.62	RESULT: Theorem (the conjecture is true).
0.43/0.62	EOF