0.00/0.03	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.00/0.04	% Command    : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn
0.03/0.23	% Computer   : n135.star.cs.uiowa.edu
0.03/0.23	% Model      : x86_64 x86_64
0.03/0.23	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.03/0.23	% Memory     : 32218.625MB
0.03/0.23	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.03/0.23	% CPULimit   : 300
0.03/0.23	% DateTime   : Sat Jul 14 05:53:09 CDT 2018
0.03/0.23	% CPUTime    : 
5.25/5.47	% SZS status Theorem
5.25/5.47	
5.32/5.54	% SZS output start Proof
5.32/5.54	Take the following subset of the input axioms:
5.32/5.54	  fof(goals_15, conjecture,
5.32/5.54	      ![X17, X18]:
5.32/5.54	        '==>'('==>'(X17, X18), X18)='==>'('==>'(X18, X17), X17)).
5.32/5.54	  fof(sos_01, axiom,
5.32/5.54	      ![A, B, C]: '+'('+'(A, B), C)='+'(A, '+'(B, C))).
5.32/5.54	  fof(sos_02, axiom, ![A, B]: '+'(A, B)='+'(B, A)).
5.32/5.54	  fof(sos_03, axiom, ![A]: A='+'(A, '0')).
5.32/5.54	  fof(sos_04, axiom, ![A]: '>='(A, A)).
5.32/5.54	  fof(sos_06, axiom,
5.32/5.54	      ![X3, X4]: (('>='(X4, X3) & '>='(X3, X4)) => X4=X3)).
5.32/5.54	  fof(sos_07, axiom,
5.32/5.54	      ![X5, X6, X7]:
5.32/5.54	        ('>='(X6, '==>'(X5, X7)) <=> '>='('+'(X5, X6), X7))).
5.32/5.54	  fof(sos_08, axiom, ![A]: '>='(A, '0')).
5.32/5.54	  fof(sos_09, axiom,
5.32/5.54	      ![X8, X9, X10]:
5.32/5.54	        ('>='(X8, X9) => '>='('+'(X8, X10), '+'(X9, X10)))).
5.32/5.54	  fof(sos_12, axiom,
5.32/5.54	      ![A, B]: '+'(B, '==>'(B, A))='+'(A, '==>'(A, B))).
5.32/5.54	  fof(sos_13, axiom, ![A]: '==>'('==>'(A, '1'), '1')=A).
5.32/5.54	  fof(sos_14, axiom, ![A]: '1'='+'(A, '1')).
5.32/5.54	
5.32/5.54	Now clausify the problem and encode Horn clauses using encoding 3 of
5.32/5.54	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
5.32/5.54	We repeatedly replace C & s=t => u=v by the two clauses:
5.32/5.54	  $$fresh(y, y, x1...xn) = u
5.32/5.54	  C => $$fresh(s, t, x1...xn) = v
5.32/5.54	where $$fresh is a fresh function symbol and x1..xn are the free
5.32/5.54	variables of u and v.
5.32/5.54	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
5.32/5.54	input problem has no model of domain size 1).
5.32/5.54	
5.32/5.54	The encoding turns the above axioms into the following unit equations and goals:
5.32/5.54	
5.32/5.54	Axiom 3 (sos_06): $$fresh(X, X, Y, Z) = Y.
5.32/5.54	Axiom 4 (sos_06): $$fresh2(X, X, Y, Z) = Z.
5.32/5.54	Axiom 6 (sos_07_1): $$fresh4(X, X, Y, Z, W) = $$true.
5.32/5.54	Axiom 7 (sos_09): $$fresh8(X, X, Y, Z, W) = $$true.
5.32/5.54	Axiom 10 (sos_01): (X + Y) + Z = X + (Y + Z).
5.32/5.54	Axiom 11 (sos_04): X >= X = $$true.
5.32/5.54	Axiom 12 (sos_13): (X ==> 1) ==> 1 = X.
5.32/5.54	Axiom 13 (sos_08): X >= 0 = $$true.
5.32/5.54	Axiom 14 (sos_09): $$fresh8(X >= Y, $$true, X, Y, Z) = (X + Z) >= (Y + Z).
5.32/5.54	Axiom 17 (sos_14): 1 = X + 1.
5.32/5.54	Axiom 18 (sos_02): X + Y = Y + X.
5.32/5.54	Axiom 19 (sos_03): X = X + 0.
5.32/5.54	Axiom 20 (sos_06): $$fresh2(X >= Y, $$true, Y, X) = $$fresh(Y >= X, $$true, Y, X).
5.32/5.54	Axiom 21 (sos_07_1): $$fresh4((X + Y) >= Z, $$true, X, Y, Z) = Y >= (X ==> Z).
5.32/5.54	Axiom 24 (sos_12): X + (X ==> Y) = Y + (Y ==> X).
5.32/5.54	
5.32/5.54	Lemma 25: 0 + X = X.
5.32/5.54	Proof:
5.32/5.54	  0 + X
5.32/5.54	= { by axiom 18 (sos_02) }
5.32/5.54	  X + 0
5.32/5.54	= { by axiom 19 (sos_03) }
5.32/5.54	  X
5.32/5.54	
5.32/5.54	Lemma 26: (X + Y) >= X = $$true.
5.32/5.54	Proof:
5.32/5.54	  (X + Y) >= X
5.32/5.54	= { by axiom 18 (sos_02) }
5.32/5.54	  (Y + X) >= X
5.32/5.54	= { by lemma 25 }
5.32/5.54	  (Y + X) >= (0 + X)
5.32/5.54	= { by axiom 14 (sos_09) }
5.32/5.54	  $$fresh8(Y >= 0, $$true, Y, 0, X)
5.32/5.54	= { by axiom 13 (sos_08) }
5.32/5.54	  $$fresh8($$true, $$true, Y, 0, X)
5.32/5.54	= { by axiom 7 (sos_09) }
5.32/5.54	  $$true
5.32/5.54	
5.32/5.54	Lemma 27: X + (Y + Z) = Y + (X + Z).
5.32/5.54	Proof:
5.32/5.54	  X + (Y + Z)
5.32/5.54	= { by axiom 18 (sos_02) }
5.32/5.54	  (Y + Z) + X
5.32/5.54	= { by axiom 10 (sos_01) }
5.32/5.54	  Y + (Z + X)
5.32/5.54	= { by axiom 18 (sos_02) }
5.32/5.54	  Y + (X + Z)
5.32/5.54	
5.32/5.54	Lemma 28: X >= (Y ==> (X + Y)) = $$true.
5.32/5.54	Proof:
5.32/5.54	  X >= (Y ==> (X + Y))
5.32/5.54	= { by axiom 18 (sos_02) }
5.32/5.54	  X >= (Y ==> (Y + X))
5.32/5.54	= { by axiom 21 (sos_07_1) }
5.32/5.54	  $$fresh4((Y + X) >= (Y + X), $$true, Y, X, Y + X)
5.32/5.54	= { by axiom 11 (sos_04) }
5.32/5.54	  $$fresh4($$true, $$true, Y, X, Y + X)
5.32/5.54	= { by axiom 6 (sos_07_1) }
5.32/5.54	  $$true
5.32/5.54	
5.32/5.54	Lemma 29: X + (Z + (X ==> Y)) = Y + ((Y ==> X) + Z).
5.32/5.54	Proof:
5.32/5.54	  X + (Z + (X ==> Y))
5.32/5.54	= { by axiom 18 (sos_02) }
5.32/5.54	  X + ((X ==> Y) + Z)
5.32/5.54	= { by axiom 10 (sos_01) }
5.32/5.54	  (X + (X ==> Y)) + Z
5.32/5.54	= { by axiom 24 (sos_12) }
5.32/5.54	  (Y + (Y ==> X)) + Z
5.32/5.54	= { by axiom 10 (sos_01) }
5.32/5.54	  Y + ((Y ==> X) + Z)
5.32/5.54	
5.32/5.54	Lemma 30: X + ((X ==> 1) + Y) = 1.
5.32/5.54	Proof:
5.32/5.54	  X + ((X ==> 1) + Y)
5.32/5.54	= { by lemma 29 }
5.32/5.54	  1 + (Y + (1 ==> X))
5.32/5.54	= { by axiom 18 (sos_02) }
5.32/5.54	  (Y + (1 ==> X)) + 1
5.32/5.54	= { by axiom 17 (sos_14) }
5.32/5.54	  1
5.32/5.54	
5.32/5.54	Lemma 31: Y + (Z + (Z ==> X)) = X + (Y + (X ==> Z)).
5.32/5.54	Proof:
5.32/5.54	  Y + (Z + (Z ==> X))
5.32/5.54	= { by axiom 24 (sos_12) }
5.32/5.54	  Y + (X + (X ==> Z))
5.32/5.54	= { by lemma 27 }
5.32/5.54	  X + (Y + (X ==> Z))
5.32/5.54	
5.32/5.54	Lemma 32: X ==> (Y ==> (X + Y)) = 0.
5.32/5.54	Proof:
5.32/5.54	  X ==> (Y ==> (X + Y))
5.32/5.54	= { by axiom 3 (sos_06) }
5.32/5.54	  $$fresh($$true, $$true, X ==> (Y ==> (X + Y)), 0)
5.32/5.54	= { by axiom 13 (sos_08) }
5.32/5.54	  $$fresh((X ==> (Y ==> (X + Y))) >= 0, $$true, X ==> (Y ==> (X + Y)), 0)
5.32/5.54	= { by axiom 20 (sos_06) }
5.32/5.54	  $$fresh2(0 >= (X ==> (Y ==> (X + Y))), $$true, X ==> (Y ==> (X + Y)), 0)
5.32/5.54	= { by axiom 21 (sos_07_1) }
5.32/5.54	  $$fresh2($$fresh4((X + 0) >= (Y ==> (X + Y)), $$true, X, 0, Y ==> (X + Y)), $$true, X ==> (Y ==> (X + Y)), 0)
5.32/5.54	= { by axiom 19 (sos_03) }
5.32/5.54	  $$fresh2($$fresh4(X >= (Y ==> (X + Y)), $$true, X, 0, Y ==> (X + Y)), $$true, X ==> (Y ==> (X + Y)), 0)
5.32/5.54	= { by lemma 28 }
5.32/5.54	  $$fresh2($$fresh4($$true, $$true, X, 0, Y ==> (X + Y)), $$true, X ==> (Y ==> (X + Y)), 0)
5.32/5.54	= { by axiom 6 (sos_07_1) }
5.32/5.54	  $$fresh2($$true, $$true, X ==> (Y ==> (X + Y)), 0)
5.32/5.54	= { by axiom 4 (sos_06) }
5.32/5.55	  0
5.32/5.55	
5.32/5.55	Lemma 33: (X ==> (Y ==> 1)) >= (Y ==> (X ==> 1)) = $$true.
5.32/5.55	Proof:
5.32/5.55	  (X ==> (Y ==> 1)) >= (Y ==> (X ==> 1))
5.32/5.55	= { by lemma 30 }
5.32/5.55	  (X ==> (Y ==> 1)) >= (Y ==> (X ==> (Y + ((Y ==> 1) + ((Y ==> 1) ==> X)))))
5.32/5.55	= { by axiom 24 (sos_12) }
5.32/5.55	  (X ==> (Y ==> 1)) >= (Y ==> (X ==> (Y + (X + (X ==> (Y ==> 1))))))
5.32/5.55	= { by axiom 10 (sos_01) }
5.32/5.55	  (X ==> (Y ==> 1)) >= (Y ==> (X ==> ((Y + X) + (X ==> (Y ==> 1)))))
5.32/5.55	= { by axiom 18 (sos_02) }
5.32/5.55	  (X ==> (Y ==> 1)) >= (Y ==> (X ==> ((X ==> (Y ==> 1)) + (Y + X))))
5.32/5.55	= { by lemma 27 }
5.32/5.55	  (X ==> (Y ==> 1)) >= (Y ==> (X ==> (Y + ((X ==> (Y ==> 1)) + X))))
5.32/5.55	= { by axiom 10 (sos_01) }
5.32/5.55	  (X ==> (Y ==> 1)) >= (Y ==> (X ==> ((Y + (X ==> (Y ==> 1))) + X)))
5.32/5.55	= { by axiom 21 (sos_07_1) }
5.32/5.55	  $$fresh4((Y + (X ==> (Y ==> 1))) >= (X ==> ((Y + (X ==> (Y ==> 1))) + X)), $$true, Y, X ==> (Y ==> 1), X ==> ((Y + (X ==> (Y ==> 1))) + X))
5.32/5.55	= { by lemma 28 }
5.32/5.55	  $$fresh4($$true, $$true, Y, X ==> (Y ==> 1), X ==> ((Y + (X ==> (Y ==> 1))) + X))
5.32/5.55	= { by axiom 6 (sos_07_1) }
5.32/5.55	  $$true
5.32/5.55	
5.32/5.55	Lemma 34: X ==> (Y ==> 1) = Y ==> (X ==> 1).
5.32/5.55	Proof:
5.32/5.55	  X ==> (Y ==> 1)
5.32/5.55	= { by axiom 3 (sos_06) }
5.32/5.55	  $$fresh($$true, $$true, X ==> (Y ==> 1), Y ==> (X ==> 1))
5.32/5.55	= { by lemma 33 }
5.32/5.55	  $$fresh((X ==> (Y ==> 1)) >= (Y ==> (X ==> 1)), $$true, X ==> (Y ==> 1), Y ==> (X ==> 1))
5.32/5.55	= { by axiom 20 (sos_06) }
5.32/5.55	  $$fresh2((Y ==> (X ==> 1)) >= (X ==> (Y ==> 1)), $$true, X ==> (Y ==> 1), Y ==> (X ==> 1))
5.32/5.55	= { by lemma 33 }
5.32/5.55	  $$fresh2($$true, $$true, X ==> (Y ==> 1), Y ==> (X ==> 1))
5.32/5.55	= { by axiom 4 (sos_06) }
5.32/5.55	  Y ==> (X ==> 1)
5.32/5.55	
5.32/5.55	Lemma 35: (X ==> 1) ==> (Y ==> 1) = Y ==> X.
5.32/5.55	Proof:
5.32/5.55	  (X ==> 1) ==> (Y ==> 1)
5.32/5.55	= { by lemma 34 }
5.32/5.55	  Y ==> ((X ==> 1) ==> 1)
5.32/5.55	= { by axiom 12 (sos_13) }
5.32/5.55	  Y ==> X
5.32/5.55	
5.32/5.55	Lemma 36: (X ==> (Y ==> Z)) >= (Y ==> (X ==> Z)) = $$true.
5.32/5.55	Proof:
5.32/5.55	  (X ==> (Y ==> Z)) >= (Y ==> (X ==> Z))
5.32/5.55	= { by axiom 21 (sos_07_1) }
5.32/5.55	  $$fresh4((Y + (X ==> (Y ==> Z))) >= (X ==> Z), $$true, Y, X ==> (Y ==> Z), X ==> Z)
5.32/5.55	= { by axiom 21 (sos_07_1) }
5.32/5.55	  $$fresh4($$fresh4((X + (Y + (X ==> (Y ==> Z)))) >= Z, $$true, X, Y + (X ==> (Y ==> Z)), Z), $$true, Y, X ==> (Y ==> Z), X ==> Z)
5.32/5.55	= { by lemma 31 }
5.32/5.55	  $$fresh4($$fresh4((Y + ((Y ==> Z) + ((Y ==> Z) ==> X))) >= Z, $$true, X, Y + (X ==> (Y ==> Z)), Z), $$true, Y, X ==> (Y ==> Z), X ==> Z)
5.32/5.55	= { by lemma 29 }
5.32/5.55	  $$fresh4($$fresh4((Z + (((Y ==> Z) ==> X) + (Z ==> Y))) >= Z, $$true, X, Y + (X ==> (Y ==> Z)), Z), $$true, Y, X ==> (Y ==> Z), X ==> Z)
5.32/5.55	= { by lemma 26 }
5.32/5.55	  $$fresh4($$fresh4($$true, $$true, X, Y + (X ==> (Y ==> Z)), Z), $$true, Y, X ==> (Y ==> Z), X ==> Z)
5.32/5.55	= { by axiom 6 (sos_07_1) }
5.32/5.55	  $$fresh4($$true, $$true, Y, X ==> (Y ==> Z), X ==> Z)
5.32/5.55	= { by axiom 6 (sos_07_1) }
5.32/5.55	  $$true
5.32/5.55	
5.32/5.55	Lemma 37: X ==> (Y ==> Z) = Y ==> (X ==> Z).
5.32/5.55	Proof:
5.32/5.55	  X ==> (Y ==> Z)
5.32/5.55	= { by axiom 3 (sos_06) }
5.32/5.55	  $$fresh($$true, $$true, X ==> (Y ==> Z), Y ==> (X ==> Z))
5.32/5.55	= { by lemma 36 }
5.32/5.55	  $$fresh((X ==> (Y ==> Z)) >= (Y ==> (X ==> Z)), $$true, X ==> (Y ==> Z), Y ==> (X ==> Z))
5.32/5.55	= { by axiom 20 (sos_06) }
5.32/5.55	  $$fresh2((Y ==> (X ==> Z)) >= (X ==> (Y ==> Z)), $$true, X ==> (Y ==> Z), Y ==> (X ==> Z))
5.32/5.55	= { by lemma 36 }
5.32/5.55	  $$fresh2($$true, $$true, X ==> (Y ==> Z), Y ==> (X ==> Z))
5.32/5.55	= { by axiom 4 (sos_06) }
5.32/5.55	  Y ==> (X ==> Z)
5.32/5.55	
5.32/5.55	Lemma 38: X ==> (Y ==> (X + Y)) = 0.
5.32/5.55	Proof:
5.32/5.55	  X ==> (Y ==> (X + Y))
5.32/5.55	= { by lemma 37 }
5.32/5.55	  Y ==> (X ==> (X + Y))
5.32/5.55	= { by axiom 18 (sos_02) }
5.32/5.55	  Y ==> (X ==> (Y + X))
5.32/5.55	= { by lemma 32 }
5.32/5.55	  0
5.32/5.55	
5.32/5.55	Lemma 39: ((X ==> Z) ==> 1) ==> Y = X ==> ((Y ==> 1) ==> Z).
5.32/5.55	Proof:
5.32/5.55	  ((X ==> Z) ==> 1) ==> Y
5.32/5.55	= { by axiom 12 (sos_13) }
5.32/5.55	  ((X ==> Z) ==> 1) ==> ((Y ==> 1) ==> 1)
5.32/5.55	= { by lemma 35 }
5.32/5.55	  (Y ==> 1) ==> (X ==> Z)
5.32/5.55	= { by lemma 37 }
5.40/5.63	  X ==> ((Y ==> 1) ==> Z)
5.40/5.63	
5.40/5.63	Lemma 40: (X + Y) ==> 1 = X ==> (Y ==> 1).
5.40/5.63	Proof:
5.40/5.63	  (X + Y) ==> 1
5.40/5.63	= { by lemma 25 }
5.40/5.63	  (0 + (X + Y)) ==> 1
5.40/5.63	= { by lemma 27 }
5.40/5.63	  (X + (0 + Y)) ==> 1
5.40/5.63	= { by axiom 18 (sos_02) }
5.40/5.63	  (X + (Y + 0)) ==> 1
5.40/5.63	= { by lemma 27 }
5.40/5.63	  (Y + (X + 0)) ==> 1
5.40/5.63	= { by axiom 4 (sos_06) }
5.40/5.63	  (Y + (X + $$fresh2($$true, $$true, 0 ==> 0, 0))) ==> 1
5.40/5.63	= { by lemma 28 }
5.40/5.63	  (Y + (X + $$fresh2(0 >= (0 ==> (0 + 0)), $$true, 0 ==> 0, 0))) ==> 1
5.40/5.63	= { by axiom 19 (sos_03) }
5.40/5.63	  (Y + (X + $$fresh2(0 >= (0 ==> 0), $$true, 0 ==> 0, 0))) ==> 1
5.40/5.63	= { by axiom 20 (sos_06) }
5.40/5.63	  (Y + (X + $$fresh((0 ==> 0) >= 0, $$true, 0 ==> 0, 0))) ==> 1
5.40/5.63	= { by lemma 25 }
5.40/5.63	  (Y + (X + $$fresh((0 + (0 ==> 0)) >= 0, $$true, 0 ==> 0, 0))) ==> 1
5.40/5.63	= { by lemma 26 }
5.40/5.63	  (Y + (X + $$fresh($$true, $$true, 0 ==> 0, 0))) ==> 1
5.40/5.63	= { by axiom 3 (sos_06) }
5.40/5.63	  (Y + (X + (0 ==> 0))) ==> 1
5.40/5.63	= { by lemma 38 }
5.40/5.63	  (Y + (X + ((Y ==> (X ==> (Y + X))) ==> 0))) ==> 1
5.40/5.63	= { by lemma 35 }
5.40/5.63	  (Y + (X + ((Y ==> (((Y + X) ==> 1) ==> (X ==> 1))) ==> 0))) ==> 1
5.40/5.63	= { by lemma 39 }
5.40/5.63	  (Y + (X + ((((Y ==> (X ==> 1)) ==> 1) ==> (Y + X)) ==> 0))) ==> 1
5.40/5.63	= { by lemma 25 }
5.40/5.63	  (Y + (X + (0 + ((((Y ==> (X ==> 1)) ==> 1) ==> (Y + X)) ==> 0)))) ==> 1
5.40/5.63	= { by axiom 10 (sos_01) }
5.40/5.63	  ((Y + X) + (0 + ((((Y ==> (X ==> 1)) ==> 1) ==> (Y + X)) ==> 0))) ==> 1
5.40/5.63	= { by lemma 32 }
5.40/5.63	  ((Y + X) + (((Y + X) ==> ((Y ==> (X ==> 1)) ==> ((Y + X) + (Y ==> (X ==> 1))))) + ((((Y ==> (X ==> 1)) ==> 1) ==> (Y + X)) ==> 0))) ==> 1
5.40/5.63	= { by axiom 10 (sos_01) }
5.40/5.63	  ((Y + X) + (((Y + X) ==> ((Y ==> (X ==> 1)) ==> (Y + (X + (Y ==> (X ==> 1)))))) + ((((Y ==> (X ==> 1)) ==> 1) ==> (Y + X)) ==> 0))) ==> 1
5.40/5.63	= { by lemma 31 }
5.40/5.63	  ((Y + X) + (((Y + X) ==> ((Y ==> (X ==> 1)) ==> (X + ((X ==> 1) + ((X ==> 1) ==> Y))))) + ((((Y ==> (X ==> 1)) ==> 1) ==> (Y + X)) ==> 0))) ==> 1
5.40/5.63	= { by lemma 30 }
5.40/5.63	  ((Y + X) + (((Y + X) ==> ((Y ==> (X ==> 1)) ==> 1)) + ((((Y ==> (X ==> 1)) ==> 1) ==> (Y + X)) ==> 0))) ==> 1
5.40/5.63	= { by axiom 18 (sos_02) }
5.40/5.63	  ((Y + X) + (((((Y ==> (X ==> 1)) ==> 1) ==> (Y + X)) ==> 0) + ((Y + X) ==> ((Y ==> (X ==> 1)) ==> 1)))) ==> 1
5.40/5.63	= { by lemma 29 }
5.40/5.63	  (((Y ==> (X ==> 1)) ==> 1) + ((((Y ==> (X ==> 1)) ==> 1) ==> (Y + X)) + ((((Y ==> (X ==> 1)) ==> 1) ==> (Y + X)) ==> 0))) ==> 1
5.40/5.63	= { by axiom 24 (sos_12) }
5.40/5.63	  (((Y ==> (X ==> 1)) ==> 1) + (0 + (0 ==> (((Y ==> (X ==> 1)) ==> 1) ==> (Y + X))))) ==> 1
5.40/5.63	= { by lemma 39 }
5.40/5.63	  (((Y ==> (X ==> 1)) ==> 1) + (0 + (0 ==> (Y ==> (((Y + X) ==> 1) ==> (X ==> 1)))))) ==> 1
5.40/5.63	= { by lemma 35 }
5.40/5.63	  (((Y ==> (X ==> 1)) ==> 1) + (0 + (0 ==> (Y ==> (X ==> (Y + X)))))) ==> 1
5.40/5.63	= { by lemma 38 }
5.40/5.63	  (((Y ==> (X ==> 1)) ==> 1) + (0 + (0 ==> 0))) ==> 1
5.40/5.63	= { by axiom 4 (sos_06) }
5.40/5.63	  (((Y ==> (X ==> 1)) ==> 1) + (0 + $$fresh2($$true, $$true, 0, 0 ==> 0))) ==> 1
5.40/5.63	= { by axiom 13 (sos_08) }
5.40/5.63	  (((Y ==> (X ==> 1)) ==> 1) + (0 + $$fresh2((0 ==> 0) >= 0, $$true, 0, 0 ==> 0))) ==> 1
5.40/5.63	= { by axiom 20 (sos_06) }
5.40/5.63	  (((Y ==> (X ==> 1)) ==> 1) + (0 + $$fresh(0 >= (0 ==> 0), $$true, 0, 0 ==> 0))) ==> 1
5.40/5.63	= { by axiom 21 (sos_07_1) }
5.40/5.63	  (((Y ==> (X ==> 1)) ==> 1) + (0 + $$fresh($$fresh4((0 + 0) >= 0, $$true, 0, 0, 0), $$true, 0, 0 ==> 0))) ==> 1
5.40/5.63	= { by axiom 13 (sos_08) }
5.40/5.63	  (((Y ==> (X ==> 1)) ==> 1) + (0 + $$fresh($$fresh4($$true, $$true, 0, 0, 0), $$true, 0, 0 ==> 0))) ==> 1
5.40/5.63	= { by axiom 6 (sos_07_1) }
5.40/5.63	  (((Y ==> (X ==> 1)) ==> 1) + (0 + $$fresh($$true, $$true, 0, 0 ==> 0))) ==> 1
5.40/5.63	= { by axiom 3 (sos_06) }
5.40/5.63	  (((Y ==> (X ==> 1)) ==> 1) + (0 + 0)) ==> 1
5.40/5.63	= { by axiom 19 (sos_03) }
5.40/5.63	  (((Y ==> (X ==> 1)) ==> 1) + 0) ==> 1
5.40/5.63	= { by axiom 18 (sos_02) }
5.40/5.63	  (0 + ((Y ==> (X ==> 1)) ==> 1)) ==> 1
5.40/5.63	= { by lemma 34 }
5.40/5.63	  (0 + ((X ==> (Y ==> 1)) ==> 1)) ==> 1
5.40/5.63	= { by lemma 25 }
5.40/5.63	  ((X ==> (Y ==> 1)) ==> 1) ==> 1
5.40/5.63	= { by axiom 12 (sos_13) }
5.40/5.63	  X ==> (Y ==> 1)
5.40/5.63	
5.40/5.63	Goal 1 (goals_15): (sK2_goals_15_X17 ==> sK1_goals_15_X18) ==> sK1_goals_15_X18 = (sK1_goals_15_X18 ==> sK2_goals_15_X17) ==> sK2_goals_15_X17.
5.40/5.63	Proof:
5.40/5.63	  (sK2_goals_15_X17 ==> sK1_goals_15_X18) ==> sK1_goals_15_X18
5.40/5.63	= { by axiom 12 (sos_13) }
5.40/5.63	  (sK2_goals_15_X17 ==> sK1_goals_15_X18) ==> ((sK1_goals_15_X18 ==> 1) ==> 1)
5.40/5.63	= { by lemma 40 }
5.40/5.63	  ((sK2_goals_15_X17 ==> sK1_goals_15_X18) + (sK1_goals_15_X18 ==> 1)) ==> 1
5.40/5.63	= { by axiom 18 (sos_02) }
5.40/5.63	  ((sK1_goals_15_X18 ==> 1) + (sK2_goals_15_X17 ==> sK1_goals_15_X18)) ==> 1
5.40/5.63	= { by lemma 35 }
5.40/5.63	  ((sK1_goals_15_X18 ==> 1) + ((sK1_goals_15_X18 ==> 1) ==> (sK2_goals_15_X17 ==> 1))) ==> 1
5.40/5.63	= { by axiom 24 (sos_12) }
5.40/5.63	  ((sK2_goals_15_X17 ==> 1) + ((sK2_goals_15_X17 ==> 1) ==> (sK1_goals_15_X18 ==> 1))) ==> 1
5.40/5.63	= { by lemma 35 }
5.40/5.63	  ((sK2_goals_15_X17 ==> 1) + (sK1_goals_15_X18 ==> sK2_goals_15_X17)) ==> 1
5.40/5.63	= { by lemma 40 }
5.40/5.63	  (sK2_goals_15_X17 ==> 1) ==> ((sK1_goals_15_X18 ==> sK2_goals_15_X17) ==> 1)
5.40/5.63	= { by lemma 35 }
5.40/5.63	  (sK1_goals_15_X18 ==> sK2_goals_15_X17) ==> sK2_goals_15_X17
5.40/5.63	% SZS output end Proof
5.40/5.63	
5.40/5.63	RESULT: Theorem (the conjecture is true).
5.40/5.64	EOF