0.00/0.05	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.00/0.06	% Command    : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn
0.03/0.32	% Computer   : n157.star.cs.uiowa.edu
0.03/0.32	% Model      : x86_64 x86_64
0.03/0.32	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.03/0.32	% Memory     : 32218.625MB
0.03/0.32	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.03/0.32	% CPULimit   : 300
0.03/0.32	% DateTime   : Sat Jul 14 05:11:55 CDT 2018
0.03/0.32	% CPUTime    : 
0.07/0.48	% SZS status Theorem
0.07/0.48	
0.07/0.48	% SZS output start Proof
0.07/0.48	Take the following subset of the input axioms:
0.26/0.52	  fof(goals_15, conjecture,
0.26/0.52	      ![X17]: '==>'('==>'('==>'(X17, '1'), X17), X17)='0').
0.26/0.52	  fof(sos_01, axiom,
0.26/0.52	      ![A, B, C]: '+'(A, '+'(B, C))='+'('+'(A, B), C)).
0.26/0.52	  fof(sos_02, axiom, ![A, B]: '+'(A, B)='+'(B, A)).
0.26/0.52	  fof(sos_03, axiom, ![A]: A='+'(A, '0')).
0.26/0.52	  fof(sos_04, axiom, ![A]: '>='(A, A)).
0.26/0.52	  fof(sos_06, axiom,
0.26/0.52	      ![X3, X4]: (X3=X4 <= ('>='(X4, X3) & '>='(X3, X4)))).
0.26/0.52	  fof(sos_07, axiom,
0.26/0.52	      ![X5, X6, X7]:
0.26/0.52	        ('>='('+'(X5, X6), X7) <=> '>='(X6, '==>'(X5, X7)))).
0.26/0.52	  fof(sos_08, axiom, ![A]: '>='(A, '0')).
0.26/0.52	  fof(sos_10, axiom,
0.26/0.52	      ![X11, X12, X13]:
0.26/0.52	        ('>='(X11, X12) => '>='('==>'(X12, X13), '==>'(X11, X13)))).
0.26/0.52	  fof(sos_11, axiom,
0.26/0.52	      ![X14, X15, X16]:
0.26/0.52	        ('>='('==>'(X16, X14), '==>'(X16, X15)) <= '>='(X14, X15))).
0.26/0.52	  fof(sos_13, axiom,
0.26/0.52	      ![A, B]: '==>'('==>'(A, B), B)='==>'('==>'(B, A), A)).
0.26/0.52	  fof(sos_14, axiom, ![A]: A='+'(A, A)).
0.26/0.52	
0.26/0.52	Now clausify the problem and encode Horn clauses using encoding 3 of
0.26/0.52	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.26/0.52	We repeatedly replace C & s=t => u=v by the two clauses:
0.26/0.52	  $$fresh(y, y, x1...xn) = u
0.26/0.52	  C => $$fresh(s, t, x1...xn) = v
0.26/0.52	where $$fresh is a fresh function symbol and x1..xn are the free
0.26/0.52	variables of u and v.
0.26/0.52	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
0.26/0.52	input problem has no model of domain size 1).
0.26/0.52	
0.26/0.52	The encoding turns the above axioms into the following unit equations and goals:
0.26/0.52	
0.26/0.53	Axiom 3 (sos_06): $$fresh9(X, X, Y, Z) = Y.
0.26/0.53	Axiom 4 (sos_06): $$fresh8(X, X, Y, Z) = Z.
0.26/0.53	Axiom 5 (sos_07): $$fresh5(X, X, Y, Z, W) = $$true.
0.26/0.53	Axiom 6 (sos_07_1): $$fresh4(X, X, Y, Z, W) = $$true.
0.26/0.53	Axiom 8 (sos_10): $$fresh2(X, X, Y, Z, W) = $$true.
0.26/0.53	Axiom 9 (sos_11): $$fresh(X, X, Y, Z, W) = $$true.
0.26/0.53	Axiom 10 (sos_13): (X ==> Y) ==> Y = (Y ==> X) ==> X.
0.26/0.53	Axiom 11 (sos_06): $$fresh9(X >= Y, $$true, Y, X) = $$fresh8(Y >= X, $$true, Y, X).
0.26/0.53	Axiom 12 (sos_04): X >= X = $$true.
0.26/0.53	Axiom 13 (sos_02): X + Y = Y + X.
0.26/0.53	Axiom 14 (sos_07_1): $$fresh4((X + Y) >= Z, $$true, X, Y, Z) = Y >= (X ==> Z).
0.26/0.53	Axiom 15 (sos_07): $$fresh5(X >= (Y ==> Z), $$true, Y, X, Z) = (Y + X) >= Z.
0.26/0.53	Axiom 16 (sos_11): $$fresh(X >= Y, $$true, X, Y, Z) = (Z ==> X) >= (Z ==> Y).
0.26/0.53	Axiom 19 (sos_10): $$fresh2(X >= Y, $$true, X, Y, Z) = (Y ==> Z) >= (X ==> Z).
0.26/0.53	Axiom 20 (sos_01): X + (Y + Z) = (X + Y) + Z.
0.26/0.53	Axiom 21 (sos_08): X >= 0 = $$true.
0.26/0.53	Axiom 22 (sos_03): X = X + 0.
0.26/0.53	Axiom 23 (sos_14): X = X + X.
0.26/0.53	
0.26/0.53	Lemma 24: 0 + X = X.
0.26/0.53	Proof:
0.26/0.53	  0 + X
0.26/0.53	= { by axiom 13 (sos_02) }
0.26/0.53	  X + 0
0.26/0.53	= { by axiom 22 (sos_03) }
0.26/0.53	  X
0.26/0.53	
0.26/0.53	Lemma 25: X >= (Y ==> (X + Y)) = $$true.
0.26/0.53	Proof:
0.26/0.53	  X >= (Y ==> (X + Y))
0.26/0.53	= { by axiom 13 (sos_02) }
0.26/0.53	  X >= (Y ==> (Y + X))
0.26/0.53	= { by axiom 14 (sos_07_1) }
0.26/0.53	  $$fresh4((Y + X) >= (Y + X), $$true, Y, X, Y + X)
0.26/0.53	= { by axiom 12 (sos_04) }
0.26/0.53	  $$fresh4($$true, $$true, Y, X, Y + X)
0.26/0.53	= { by axiom 6 (sos_07_1) }
0.26/0.53	  $$true
0.26/0.53	
0.26/0.53	Lemma 26: (X + (X ==> Y)) >= Y = $$true.
0.26/0.53	Proof:
0.26/0.53	  (X + (X ==> Y)) >= Y
0.26/0.53	= { by axiom 15 (sos_07) }
0.26/0.53	  $$fresh5((X ==> Y) >= (X ==> Y), $$true, X, X ==> Y, Y)
0.26/0.53	= { by axiom 12 (sos_04) }
0.26/0.53	  $$fresh5($$true, $$true, X, X ==> Y, Y)
0.26/0.53	= { by axiom 5 (sos_07) }
0.26/0.53	  $$true
0.26/0.53	
0.26/0.53	Lemma 27: 0 ==> X = X.
0.26/0.53	Proof:
0.26/0.53	  0 ==> X
0.26/0.53	= { by axiom 3 (sos_06) }
0.26/0.53	  $$fresh9($$true, $$true, 0 ==> X, X)
0.26/0.53	= { by lemma 25 }
0.26/0.53	  $$fresh9(X >= (0 ==> (X + 0)), $$true, 0 ==> X, X)
0.26/0.53	= { by axiom 22 (sos_03) }
0.26/0.53	  $$fresh9(X >= (0 ==> X), $$true, 0 ==> X, X)
0.26/0.53	= { by axiom 11 (sos_06) }
0.26/0.53	  $$fresh8((0 ==> X) >= X, $$true, 0 ==> X, X)
0.26/0.53	= { by lemma 24 }
0.26/0.53	  $$fresh8((0 + (0 ==> X)) >= X, $$true, 0 ==> X, X)
0.26/0.53	= { by lemma 26 }
0.26/0.53	  $$fresh8($$true, $$true, 0 ==> X, X)
0.26/0.53	= { by axiom 4 (sos_06) }
0.26/0.53	  X
0.26/0.53	
0.26/0.53	Lemma 28: X >= (Y ==> X) = $$true.
0.26/0.53	Proof:
0.26/0.53	  X >= (Y ==> X)
0.26/0.53	= { by lemma 27 }
0.26/0.53	  (0 ==> X) >= (Y ==> X)
0.26/0.53	= { by axiom 19 (sos_10) }
0.26/0.53	  $$fresh2(Y >= 0, $$true, Y, 0, X)
0.26/0.53	= { by axiom 21 (sos_08) }
0.26/0.53	  $$fresh2($$true, $$true, Y, 0, X)
0.26/0.53	= { by axiom 8 (sos_10) }
0.26/0.53	  $$true
0.26/0.53	
0.26/0.53	Lemma 29: X ==> X = 0.
0.26/0.53	Proof:
0.26/0.53	  X ==> X
0.26/0.53	= { by lemma 24 }
0.26/0.53	  X ==> (0 + X)
0.26/0.53	= { by axiom 3 (sos_06) }
0.26/0.53	  $$fresh9($$true, $$true, X ==> (0 + X), 0)
0.26/0.53	= { by lemma 25 }
0.26/0.53	  $$fresh9(0 >= (X ==> (0 + X)), $$true, X ==> (0 + X), 0)
0.26/0.53	= { by axiom 11 (sos_06) }
0.26/0.53	  $$fresh8((X ==> (0 + X)) >= 0, $$true, X ==> (0 + X), 0)
0.26/0.53	= { by axiom 21 (sos_08) }
0.26/0.53	  $$fresh8($$true, $$true, X ==> (0 + X), 0)
0.26/0.53	= { by axiom 4 (sos_06) }
0.27/0.57	  0
0.27/0.57	
0.27/0.57	Goal 1 (goals_15): ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17 = 0.
0.27/0.57	Proof:
0.27/0.57	  ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by lemma 27 }
0.27/0.57	  (0 ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 3 (sos_06) }
0.27/0.57	  ($$fresh9($$true, $$true, 0, sK1_goals_15_X17 ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 21 (sos_08) }
0.27/0.57	  ($$fresh9((sK1_goals_15_X17 ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) >= 0, $$true, 0, sK1_goals_15_X17 ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 11 (sos_06) }
0.27/0.57	  ($$fresh8(0 >= (sK1_goals_15_X17 ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)), $$true, 0, sK1_goals_15_X17 ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 14 (sos_07_1) }
0.27/0.57	  ($$fresh8($$fresh4((sK1_goals_15_X17 + 0) >= ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17), $$true, sK1_goals_15_X17, 0, (sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17), $$true, 0, sK1_goals_15_X17 ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 22 (sos_03) }
0.27/0.57	  ($$fresh8($$fresh4(sK1_goals_15_X17 >= ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17), $$true, sK1_goals_15_X17, 0, (sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17), $$true, 0, sK1_goals_15_X17 ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> sK1_goals_15_X17
0.27/0.57	= { by lemma 28 }
0.27/0.57	  ($$fresh8($$fresh4($$true, $$true, sK1_goals_15_X17, 0, (sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17), $$true, 0, sK1_goals_15_X17 ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 6 (sos_07_1) }
0.27/0.57	  ($$fresh8($$true, $$true, 0, sK1_goals_15_X17 ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 4 (sos_06) }
0.27/0.57	  ((sK1_goals_15_X17 ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> ((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17)) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 10 (sos_13) }
0.27/0.57	  ((((sK1_goals_15_X17 ==> 1) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 10 (sos_13) }
0.27/0.57	  (((sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1)) ==> (sK1_goals_15_X17 ==> 1)) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 3 (sos_06) }
0.27/0.57	  (($$fresh9($$true, $$true, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), sK1_goals_15_X17 ==> 1) ==> (sK1_goals_15_X17 ==> 1)) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 9 (sos_11) }
0.27/0.57	  (($$fresh9($$fresh($$true, $$true, 1, sK1_goals_15_X17 ==> 1, sK1_goals_15_X17), $$true, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), sK1_goals_15_X17 ==> 1) ==> (sK1_goals_15_X17 ==> 1)) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by lemma 28 }
0.27/0.57	  (($$fresh9($$fresh(1 >= (sK1_goals_15_X17 ==> 1), $$true, 1, sK1_goals_15_X17 ==> 1, sK1_goals_15_X17), $$true, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), sK1_goals_15_X17 ==> 1) ==> (sK1_goals_15_X17 ==> 1)) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 16 (sos_11) }
0.27/0.57	  (($$fresh9((sK1_goals_15_X17 ==> 1) >= (sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1)), $$true, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), sK1_goals_15_X17 ==> 1) ==> (sK1_goals_15_X17 ==> 1)) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 11 (sos_06) }
0.27/0.57	  (($$fresh8((sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1)) >= (sK1_goals_15_X17 ==> 1), $$true, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), sK1_goals_15_X17 ==> 1) ==> (sK1_goals_15_X17 ==> 1)) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 14 (sos_07_1) }
0.27/0.57	  (($$fresh8($$fresh4((sK1_goals_15_X17 + (sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1))) >= 1, $$true, sK1_goals_15_X17, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), 1), $$true, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), sK1_goals_15_X17 ==> 1) ==> (sK1_goals_15_X17 ==> 1)) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 23 (sos_14) }
0.27/0.57	  (($$fresh8($$fresh4(((sK1_goals_15_X17 + sK1_goals_15_X17) + (sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1))) >= 1, $$true, sK1_goals_15_X17, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), 1), $$true, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), sK1_goals_15_X17 ==> 1) ==> (sK1_goals_15_X17 ==> 1)) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 20 (sos_01) }
0.27/0.57	  (($$fresh8($$fresh4((sK1_goals_15_X17 + (sK1_goals_15_X17 + (sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1)))) >= 1, $$true, sK1_goals_15_X17, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), 1), $$true, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), sK1_goals_15_X17 ==> 1) ==> (sK1_goals_15_X17 ==> 1)) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 15 (sos_07) }
0.27/0.57	  (($$fresh8($$fresh4($$fresh5((sK1_goals_15_X17 + (sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1))) >= (sK1_goals_15_X17 ==> 1), $$true, sK1_goals_15_X17, sK1_goals_15_X17 + (sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1)), 1), $$true, sK1_goals_15_X17, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), 1), $$true, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), sK1_goals_15_X17 ==> 1) ==> (sK1_goals_15_X17 ==> 1)) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by lemma 26 }
0.27/0.57	  (($$fresh8($$fresh4($$fresh5($$true, $$true, sK1_goals_15_X17, sK1_goals_15_X17 + (sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1)), 1), $$true, sK1_goals_15_X17, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), 1), $$true, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), sK1_goals_15_X17 ==> 1) ==> (sK1_goals_15_X17 ==> 1)) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 5 (sos_07) }
0.27/0.57	  (($$fresh8($$fresh4($$true, $$true, sK1_goals_15_X17, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), 1), $$true, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), sK1_goals_15_X17 ==> 1) ==> (sK1_goals_15_X17 ==> 1)) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 6 (sos_07_1) }
0.27/0.57	  (($$fresh8($$true, $$true, sK1_goals_15_X17 ==> (sK1_goals_15_X17 ==> 1), sK1_goals_15_X17 ==> 1) ==> (sK1_goals_15_X17 ==> 1)) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by axiom 4 (sos_06) }
0.27/0.57	  (((sK1_goals_15_X17 ==> 1) ==> (sK1_goals_15_X17 ==> 1)) ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by lemma 29 }
0.27/0.57	  (0 ==> sK1_goals_15_X17) ==> sK1_goals_15_X17
0.27/0.57	= { by lemma 27 }
0.27/0.57	  sK1_goals_15_X17 ==> sK1_goals_15_X17
0.27/0.57	= { by lemma 29 }
0.27/0.57	  0
0.27/0.57	% SZS output end Proof
0.27/0.57	
0.27/0.57	RESULT: Theorem (the conjecture is true).
0.32/0.68	EOF