0.00/0.10	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.00/0.11	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.10/0.32	% Computer   : n006.cluster.edu
0.10/0.32	% Model      : x86_64 x86_64
0.10/0.32	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.10/0.32	% Memory     : 8042.1875MB
0.10/0.32	% OS         : Linux 3.10.0-693.el7.x86_64
0.10/0.32	% CPULimit   : 1200
0.10/0.32	% WCLimit    : 120
0.10/0.32	% DateTime   : Tue Jul 13 16:17:25 EDT 2021
0.10/0.32	% CPUTime    : 
69.88/9.14	% SZS status Theorem
69.88/9.14	
70.30/9.23	% SZS output start Proof
70.30/9.23	Take the following subset of the input axioms:
70.30/9.23	  fof(goals_13, conjecture, ![X17, X18, X19]: (X17=X19 <= (X17='==>'(X17, X18) & X19='==>'(X19, X18)))).
70.30/9.23	  fof(sos_01, axiom, ![A, B, C]: '+'(A, '+'(B, C))='+'('+'(A, B), C)).
70.30/9.23	  fof(sos_02, axiom, ![A, B]: '+'(A, B)='+'(B, A)).
70.30/9.23	  fof(sos_03, axiom, ![A]: '+'(A, '0')=A).
70.30/9.23	  fof(sos_04, axiom, ![A]: '>='(A, A)).
70.30/9.23	  fof(sos_06, axiom, ![X3, X4]: (X4=X3 <= ('>='(X3, X4) & '>='(X4, X3)))).
70.30/9.23	  fof(sos_07, axiom, ![X5, X6, X7]: ('>='('+'(X5, X6), X7) <=> '>='(X6, '==>'(X5, X7)))).
70.30/9.23	  fof(sos_08, axiom, ![A]: '>='(A, '0')).
70.30/9.23	  fof(sos_09, axiom, ![X8, X9, X10]: ('>='(X8, X9) => '>='('+'(X8, X10), '+'(X9, X10)))).
70.30/9.23	  fof(sos_11, axiom, ![X14, X15, X16]: ('>='(X14, X15) => '>='('==>'(X16, X14), '==>'(X16, X15)))).
70.30/9.23	  fof(sos_12, axiom, ![A, B]: '+'(B, '==>'(B, A))='+'(A, '==>'(A, B))).
70.30/9.23	
70.30/9.23	Now clausify the problem and encode Horn clauses using encoding 3 of
70.30/9.23	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
70.30/9.23	We repeatedly replace C & s=t => u=v by the two clauses:
70.30/9.23	  fresh(y, y, x1...xn) = u
70.30/9.23	  C => fresh(s, t, x1...xn) = v
70.30/9.23	where fresh is a fresh function symbol and x1..xn are the free
70.30/9.23	variables of u and v.
70.30/9.23	A predicate p(X) is encoded as p(X)=true (this is sound, because the
70.30/9.23	input problem has no model of domain size 1).
70.30/9.23	
70.30/9.23	The encoding turns the above axioms into the following unit equations and goals:
70.30/9.23	
70.30/9.24	Axiom 1 (sos_04): X >= X = true.
70.30/9.24	Axiom 2 (sos_08): X >= 0 = true.
70.30/9.24	Axiom 3 (sos_02): X + Y = Y + X.
70.30/9.24	Axiom 4 (sos_03): X + 0 = X.
70.30/9.24	Axiom 5 (goals_13): x19 = x19 ==> x18.
70.30/9.24	Axiom 6 (goals_13_1): x17 = x17 ==> x18.
70.30/9.24	Axiom 7 (sos_12): X + (X ==> Y) = Y + (Y ==> X).
70.30/9.24	Axiom 8 (sos_01): X + (Y + Z) = (X + Y) + Z.
70.30/9.24	Axiom 9 (sos_06): fresh(X, X, Y, Z) = Y.
70.30/9.24	Axiom 10 (sos_06): fresh2(X, X, Y, Z) = Z.
70.30/9.24	Axiom 11 (sos_07_1): fresh6(X, X, Y, Z, W) = true.
70.30/9.24	Axiom 12 (sos_09): fresh5(X, X, Y, Z, W) = true.
70.30/9.24	Axiom 13 (sos_11): fresh3(X, X, Y, Z, W) = true.
70.30/9.24	Axiom 14 (sos_06): fresh2(X >= Y, true, Y, X) = fresh(Y >= X, true, Y, X).
70.30/9.24	Axiom 15 (sos_09): fresh5(X >= Y, true, X, Y, Z) = (X + Z) >= (Y + Z).
70.30/9.24	Axiom 16 (sos_11): fresh3(X >= Y, true, X, Y, Z) = (Z ==> X) >= (Z ==> Y).
70.30/9.24	Axiom 17 (sos_07_1): fresh6((X + Y) >= Z, true, X, Y, Z) = Y >= (X ==> Z).
70.30/9.24	
70.30/9.24	Lemma 18: 0 + X = X.
70.30/9.24	Proof:
70.30/9.24	  0 + X
70.30/9.24	= { by axiom 3 (sos_02) R->L }
70.30/9.24	  X + 0
70.30/9.24	= { by axiom 4 (sos_03) }
70.30/9.24	  X
70.30/9.24	
70.30/9.24	Lemma 19: (X + Y) >= X = true.
70.30/9.24	Proof:
70.30/9.24	  (X + Y) >= X
70.30/9.24	= { by axiom 3 (sos_02) R->L }
70.30/9.24	  (Y + X) >= X
70.30/9.24	= { by lemma 18 R->L }
70.30/9.24	  (Y + X) >= (0 + X)
70.30/9.24	= { by axiom 15 (sos_09) R->L }
70.30/9.24	  fresh5(Y >= 0, true, Y, 0, X)
70.30/9.24	= { by axiom 2 (sos_08) }
70.30/9.24	  fresh5(true, true, Y, 0, X)
70.30/9.24	= { by axiom 12 (sos_09) }
70.30/9.24	  true
70.30/9.24	
70.30/9.24	Lemma 20: (X + ((X + Y) ==> Z)) >= (Y ==> Z) = true.
70.30/9.24	Proof:
70.30/9.24	  (X + ((X + Y) ==> Z)) >= (Y ==> Z)
70.30/9.24	= { by axiom 3 (sos_02) R->L }
70.30/9.24	  (X + ((Y + X) ==> Z)) >= (Y ==> Z)
70.30/9.24	= { by axiom 17 (sos_07_1) R->L }
70.30/9.24	  fresh6((Y + (X + ((Y + X) ==> Z))) >= Z, true, Y, X + ((Y + X) ==> Z), Z)
70.30/9.24	= { by axiom 8 (sos_01) }
70.30/9.24	  fresh6(((Y + X) + ((Y + X) ==> Z)) >= Z, true, Y, X + ((Y + X) ==> Z), Z)
70.30/9.24	= { by axiom 7 (sos_12) R->L }
70.30/9.24	  fresh6((Z + (Z ==> (Y + X))) >= Z, true, Y, X + ((Y + X) ==> Z), Z)
70.30/9.24	= { by lemma 19 }
70.30/9.24	  fresh6(true, true, Y, X + ((Y + X) ==> Z), Z)
70.30/9.24	= { by axiom 11 (sos_07_1) }
70.30/9.24	  true
70.30/9.24	
70.30/9.24	Lemma 21: Z + ((Z ==> X) + Y) = X + (Y + (X ==> Z)).
70.30/9.24	Proof:
70.30/9.24	  Z + ((Z ==> X) + Y)
70.30/9.24	= { by axiom 8 (sos_01) }
70.30/9.24	  (Z + (Z ==> X)) + Y
70.30/9.24	= { by axiom 7 (sos_12) R->L }
70.30/9.24	  (X + (X ==> Z)) + Y
70.30/9.24	= { by axiom 8 (sos_01) R->L }
70.30/9.24	  X + ((X ==> Z) + Y)
70.30/9.24	= { by axiom 3 (sos_02) }
70.30/9.24	  X + (Y + (X ==> Z))
70.30/9.24	
70.30/9.24	Lemma 22: Z + ((Z ==> Y) + (Y ==> X)) = X + ((Y ==> Z) + (X ==> Y)).
70.30/9.24	Proof:
70.30/9.24	  Z + ((Z ==> Y) + (Y ==> X))
70.30/9.24	= { by lemma 21 }
70.30/9.24	  Y + ((Y ==> X) + (Y ==> Z))
70.30/9.24	= { by lemma 21 }
70.30/9.24	  X + ((Y ==> Z) + (X ==> Y))
70.30/9.24	
70.30/9.24	Lemma 23: fresh(0 >= X, true, 0, X) = X.
70.30/9.24	Proof:
70.30/9.24	  fresh(0 >= X, true, 0, X)
70.30/9.24	= { by axiom 14 (sos_06) R->L }
70.30/9.24	  fresh2(X >= 0, true, 0, X)
70.30/9.24	= { by axiom 2 (sos_08) }
70.30/9.24	  fresh2(true, true, 0, X)
70.30/9.24	= { by axiom 10 (sos_06) }
70.30/9.24	  X
70.30/9.24	
70.30/9.24	Lemma 24: Y + (X + Z) = X + (Y + Z).
70.30/9.24	Proof:
70.30/9.24	  Y + (X + Z)
70.30/9.24	= { by axiom 3 (sos_02) R->L }
70.30/9.24	  (X + Z) + Y
70.30/9.24	= { by axiom 8 (sos_01) R->L }
70.30/9.24	  X + (Z + Y)
70.30/9.24	= { by axiom 3 (sos_02) }
70.30/9.24	  X + (Y + Z)
70.30/9.24	
70.30/9.24	Lemma 25: (X + Y) >= (Z ==> X) = true.
70.30/9.24	Proof:
70.30/9.24	  (X + Y) >= (Z ==> X)
70.30/9.24	= { by axiom 17 (sos_07_1) R->L }
70.30/9.24	  fresh6((Z + (X + Y)) >= X, true, Z, X + Y, X)
70.30/9.24	= { by lemma 24 }
70.30/9.24	  fresh6((X + (Z + Y)) >= X, true, Z, X + Y, X)
70.30/9.24	= { by lemma 19 }
70.30/9.24	  fresh6(true, true, Z, X + Y, X)
70.30/9.24	= { by axiom 11 (sos_07_1) }
70.30/9.24	  true
70.30/9.24	
70.30/9.24	Lemma 26: fresh6(X >= Y, true, X, 0, Y) = 0 >= (X ==> Y).
70.30/9.24	Proof:
70.30/9.24	  fresh6(X >= Y, true, X, 0, Y)
70.30/9.24	= { by axiom 4 (sos_03) R->L }
70.30/9.24	  fresh6((X + 0) >= Y, true, X, 0, Y)
70.30/9.24	= { by axiom 17 (sos_07_1) }
70.30/9.24	  0 >= (X ==> Y)
70.30/9.24	
70.30/9.24	Lemma 27: (X + Y) ==> X = 0.
70.30/9.24	Proof:
70.30/9.24	  (X + Y) ==> X
70.30/9.24	= { by axiom 9 (sos_06) R->L }
70.30/9.24	  fresh(true, true, (X + Y) ==> X, 0)
70.30/9.24	= { by axiom 2 (sos_08) R->L }
70.30/9.24	  fresh(((X + Y) ==> X) >= 0, true, (X + Y) ==> X, 0)
70.30/9.24	= { by axiom 14 (sos_06) R->L }
70.30/9.24	  fresh2(0 >= ((X + Y) ==> X), true, (X + Y) ==> X, 0)
70.30/9.24	= { by lemma 26 R->L }
70.30/9.24	  fresh2(fresh6((X + Y) >= X, true, X + Y, 0, X), true, (X + Y) ==> X, 0)
70.30/9.24	= { by lemma 19 }
70.30/9.24	  fresh2(fresh6(true, true, X + Y, 0, X), true, (X + Y) ==> X, 0)
70.30/9.24	= { by axiom 11 (sos_07_1) }
70.30/9.24	  fresh2(true, true, (X + Y) ==> X, 0)
70.30/9.24	= { by axiom 10 (sos_06) }
70.30/9.24	  0
70.30/9.24	
70.30/9.24	Lemma 28: x18 ==> x17 = 0.
70.30/9.24	Proof:
70.30/9.24	  x18 ==> x17
70.30/9.24	= { by lemma 23 R->L }
70.30/9.24	  fresh(0 >= (x18 ==> x17), true, 0, x18 ==> x17)
70.30/9.24	= { by axiom 17 (sos_07_1) R->L }
70.30/9.24	  fresh(fresh6((x18 + 0) >= x17, true, x18, 0, x17), true, 0, x18 ==> x17)
70.30/9.24	= { by axiom 6 (goals_13_1) }
70.30/9.24	  fresh(fresh6((x18 + 0) >= (x17 ==> x18), true, x18, 0, x17), true, 0, x18 ==> x17)
70.30/9.24	= { by lemma 25 }
70.30/9.24	  fresh(fresh6(true, true, x18, 0, x17), true, 0, x18 ==> x17)
70.30/9.24	= { by axiom 11 (sos_07_1) }
70.30/9.24	  fresh(true, true, 0, x18 ==> x17)
70.30/9.24	= { by axiom 9 (sos_06) }
70.30/9.24	  0
70.30/9.24	
70.30/9.24	Lemma 29: x17 ==> x19 = x19 ==> x17.
70.30/9.24	Proof:
70.30/9.24	  x17 ==> x19
70.30/9.24	= { by axiom 10 (sos_06) R->L }
70.30/9.24	  fresh2(true, true, x19 ==> x17, x17 ==> x19)
70.30/9.24	= { by axiom 11 (sos_07_1) R->L }
70.30/9.24	  fresh2(fresh6(true, true, x19, x17 ==> x19, x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.24	= { by lemma 20 R->L }
70.30/9.24	  fresh2(fresh6(((x19 + (x17 ==> x19)) + (((x19 + (x17 ==> x19)) + x17) ==> x18)) >= (x17 ==> x18), true, x19, x17 ==> x19, x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.24	= { by axiom 6 (goals_13_1) R->L }
70.30/9.24	  fresh2(fresh6(((x19 + (x17 ==> x19)) + (((x19 + (x17 ==> x19)) + x17) ==> x18)) >= x17, true, x19, x17 ==> x19, x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.24	= { by axiom 3 (sos_02) }
70.30/9.24	  fresh2(fresh6(((x19 + (x17 ==> x19)) + ((x17 + (x19 + (x17 ==> x19))) ==> x18)) >= x17, true, x19, x17 ==> x19, x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.24	= { by axiom 5 (goals_13) }
70.30/9.24	  fresh2(fresh6(((x19 + (x17 ==> x19)) + ((x17 + ((x19 ==> x18) + (x17 ==> x19))) ==> x18)) >= x17, true, x19, x17 ==> x19, x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.24	= { by lemma 22 R->L }
70.30/9.24	  fresh2(fresh6(((x19 + (x17 ==> x19)) + ((x18 + ((x18 ==> x19) + (x19 ==> x17))) ==> x18)) >= x17, true, x19, x17 ==> x19, x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.24	= { by lemma 23 R->L }
70.30/9.24	  fresh2(fresh6(((x19 + (x17 ==> x19)) + ((x18 + (fresh(0 >= (x18 ==> x19), true, 0, x18 ==> x19) + (x19 ==> x17))) ==> x18)) >= x17, true, x19, x17 ==> x19, x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.24	= { by axiom 17 (sos_07_1) R->L }
70.30/9.25	  fresh2(fresh6(((x19 + (x17 ==> x19)) + ((x18 + (fresh(fresh6((x18 + 0) >= x19, true, x18, 0, x19), true, 0, x18 ==> x19) + (x19 ==> x17))) ==> x18)) >= x17, true, x19, x17 ==> x19, x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.25	= { by axiom 5 (goals_13) }
70.30/9.25	  fresh2(fresh6(((x19 + (x17 ==> x19)) + ((x18 + (fresh(fresh6((x18 + 0) >= (x19 ==> x18), true, x18, 0, x19), true, 0, x18 ==> x19) + (x19 ==> x17))) ==> x18)) >= x17, true, x19, x17 ==> x19, x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.25	= { by lemma 25 }
70.30/9.25	  fresh2(fresh6(((x19 + (x17 ==> x19)) + ((x18 + (fresh(fresh6(true, true, x18, 0, x19), true, 0, x18 ==> x19) + (x19 ==> x17))) ==> x18)) >= x17, true, x19, x17 ==> x19, x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.25	= { by axiom 11 (sos_07_1) }
70.30/9.25	  fresh2(fresh6(((x19 + (x17 ==> x19)) + ((x18 + (fresh(true, true, 0, x18 ==> x19) + (x19 ==> x17))) ==> x18)) >= x17, true, x19, x17 ==> x19, x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.25	= { by axiom 9 (sos_06) }
70.30/9.25	  fresh2(fresh6(((x19 + (x17 ==> x19)) + ((x18 + (0 + (x19 ==> x17))) ==> x18)) >= x17, true, x19, x17 ==> x19, x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.25	= { by lemma 18 }
70.30/9.25	  fresh2(fresh6(((x19 + (x17 ==> x19)) + ((x18 + (x19 ==> x17)) ==> x18)) >= x17, true, x19, x17 ==> x19, x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.25	= { by axiom 8 (sos_01) R->L }
70.30/9.25	  fresh2(fresh6((x19 + ((x17 ==> x19) + ((x18 + (x19 ==> x17)) ==> x18))) >= x17, true, x19, x17 ==> x19, x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.25	= { by lemma 27 }
70.30/9.25	  fresh2(fresh6((x19 + ((x17 ==> x19) + 0)) >= x17, true, x19, x17 ==> x19, x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.25	= { by axiom 4 (sos_03) }
70.30/9.25	  fresh2(fresh6((x19 + (x17 ==> x19)) >= x17, true, x19, x17 ==> x19, x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.25	= { by axiom 17 (sos_07_1) }
70.30/9.25	  fresh2((x17 ==> x19) >= (x19 ==> x17), true, x19 ==> x17, x17 ==> x19)
70.30/9.25	= { by axiom 14 (sos_06) }
70.30/9.25	  fresh((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19)
70.30/9.25	= { by axiom 17 (sos_07_1) R->L }
70.30/9.25	  fresh(fresh6((x17 + (x19 ==> x17)) >= x19, true, x17, x19 ==> x17, x19), true, x19 ==> x17, x17 ==> x19)
70.30/9.25	= { by axiom 4 (sos_03) R->L }
70.30/9.25	  fresh(fresh6((x17 + ((x19 ==> x17) + 0)) >= x19, true, x17, x19 ==> x17, x19), true, x19 ==> x17, x17 ==> x19)
70.30/9.25	= { by lemma 27 R->L }
70.74/9.25	  fresh(fresh6((x17 + ((x19 ==> x17) + ((x18 + (x17 ==> x19)) ==> x18))) >= x19, true, x17, x19 ==> x17, x19), true, x19 ==> x17, x17 ==> x19)
70.74/9.25	= { by axiom 8 (sos_01) }
70.74/9.25	  fresh(fresh6(((x17 + (x19 ==> x17)) + ((x18 + (x17 ==> x19)) ==> x18)) >= x19, true, x17, x19 ==> x17, x19), true, x19 ==> x17, x17 ==> x19)
70.74/9.25	= { by lemma 18 R->L }
70.74/9.25	  fresh(fresh6(((x17 + (x19 ==> x17)) + ((x18 + (0 + (x17 ==> x19))) ==> x18)) >= x19, true, x17, x19 ==> x17, x19), true, x19 ==> x17, x17 ==> x19)
70.74/9.25	= { by lemma 28 R->L }
70.74/9.25	  fresh(fresh6(((x17 + (x19 ==> x17)) + ((x18 + ((x18 ==> x17) + (x17 ==> x19))) ==> x18)) >= x19, true, x17, x19 ==> x17, x19), true, x19 ==> x17, x17 ==> x19)
70.74/9.25	= { by lemma 22 }
70.74/9.25	  fresh(fresh6(((x17 + (x19 ==> x17)) + ((x19 + ((x17 ==> x18) + (x19 ==> x17))) ==> x18)) >= x19, true, x17, x19 ==> x17, x19), true, x19 ==> x17, x17 ==> x19)
70.74/9.25	= { by axiom 6 (goals_13_1) R->L }
70.74/9.25	  fresh(fresh6(((x17 + (x19 ==> x17)) + ((x19 + (x17 + (x19 ==> x17))) ==> x18)) >= x19, true, x17, x19 ==> x17, x19), true, x19 ==> x17, x17 ==> x19)
70.74/9.25	= { by axiom 3 (sos_02) R->L }
70.74/9.25	  fresh(fresh6(((x17 + (x19 ==> x17)) + (((x17 + (x19 ==> x17)) + x19) ==> x18)) >= x19, true, x17, x19 ==> x17, x19), true, x19 ==> x17, x17 ==> x19)
70.74/9.25	= { by axiom 5 (goals_13) }
70.74/9.25	  fresh(fresh6(((x17 + (x19 ==> x17)) + (((x17 + (x19 ==> x17)) + x19) ==> x18)) >= (x19 ==> x18), true, x17, x19 ==> x17, x19), true, x19 ==> x17, x17 ==> x19)
70.74/9.25	= { by lemma 20 }
70.74/9.25	  fresh(fresh6(true, true, x17, x19 ==> x17, x19), true, x19 ==> x17, x17 ==> x19)
70.74/9.25	= { by axiom 11 (sos_07_1) }
70.74/9.25	  fresh(true, true, x19 ==> x17, x17 ==> x19)
70.74/9.25	= { by axiom 9 (sos_06) }
70.74/9.25	  x19 ==> x17
70.74/9.25	
70.74/9.25	Lemma 30: (X + ((X ==> Y) + Z)) ==> Y = 0.
70.74/9.25	Proof:
70.74/9.25	  (X + ((X ==> Y) + Z)) ==> Y
70.74/9.25	= { by lemma 21 }
70.74/9.25	  (Y + (Z + (Y ==> X))) ==> Y
70.74/9.25	= { by lemma 27 }
70.74/9.25	  0
70.74/9.25	
70.74/9.25	Lemma 31: x19 ==> x17 = 0.
70.74/9.25	Proof:
70.74/9.25	  x19 ==> x17
70.74/9.25	= { by lemma 29 R->L }
70.74/9.25	  x17 ==> x19
70.74/9.25	= { by axiom 5 (goals_13) }
70.74/9.25	  x17 ==> (x19 ==> x18)
70.74/9.25	= { by lemma 18 R->L }
70.74/9.25	  x17 ==> (x19 ==> (0 + x18))
70.74/9.25	= { by axiom 4 (sos_03) R->L }
70.74/9.25	  x17 ==> (x19 ==> (0 + (x18 + 0)))
70.74/9.25	= { by lemma 28 R->L }
70.74/9.25	  x17 ==> (x19 ==> (0 + (x18 + (x18 ==> x17))))
70.74/9.25	= { by axiom 7 (sos_12) }
70.74/9.25	  x17 ==> (x19 ==> (0 + (x17 + (x17 ==> x18))))
70.74/9.25	= { by axiom 6 (goals_13_1) R->L }
70.74/9.25	  x17 ==> (x19 ==> (0 + (x17 + x17)))
70.74/9.25	= { by lemma 24 }
70.74/9.25	  x17 ==> (x19 ==> (x17 + (0 + x17)))
70.74/9.25	= { by axiom 4 (sos_03) R->L }
70.74/9.25	  x17 ==> (x19 ==> (x17 + (0 + (x17 + 0))))
70.74/9.25	= { by axiom 8 (sos_01) }
70.74/9.25	  x17 ==> (x19 ==> (x17 + ((0 + x17) + 0)))
70.74/9.25	= { by lemma 30 R->L }
70.74/9.25	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + ((x17 ==> (x17 ==> x19)) + 0)) ==> (x17 ==> x19)))))
70.74/9.26	= { by axiom 9 (sos_06) R->L }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh(true, true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
70.80/9.26	= { by axiom 2 (sos_08) R->L }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh((x17 ==> (x17 ==> x19)) >= 0, true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
70.80/9.26	= { by axiom 14 (sos_06) R->L }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(0 >= (x17 ==> (x17 ==> x19)), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
70.80/9.26	= { by lemma 26 R->L }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6(x17 >= (x17 ==> x19), true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
70.80/9.26	= { by axiom 6 (goals_13_1) }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6((x17 ==> x18) >= (x17 ==> x19), true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
70.80/9.26	= { by axiom 16 (sos_11) R->L }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6(fresh3(x18 >= x19, true, x18, x19, x17), true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
70.80/9.26	= { by axiom 5 (goals_13) }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6(fresh3(x18 >= (x19 ==> x18), true, x18, x19, x17), true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
70.80/9.26	= { by axiom 17 (sos_07_1) R->L }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6(fresh3(fresh6((x19 + x18) >= x18, true, x19, x18, x18), true, x18, x19, x17), true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
70.80/9.26	= { by axiom 3 (sos_02) R->L }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6(fresh3(fresh6((x18 + x19) >= x18, true, x19, x18, x18), true, x18, x19, x17), true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
70.80/9.26	= { by lemma 19 }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6(fresh3(fresh6(true, true, x19, x18, x18), true, x18, x19, x17), true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
70.80/9.26	= { by axiom 11 (sos_07_1) }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6(fresh3(true, true, x18, x19, x17), true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
70.80/9.26	= { by axiom 13 (sos_11) }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6(true, true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
70.80/9.26	= { by axiom 11 (sos_07_1) }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(true, true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
70.80/9.26	= { by axiom 10 (sos_06) }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (0 + 0)) ==> (x17 ==> x19)))))
70.80/9.26	= { by lemma 18 }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + 0) ==> (x17 ==> x19)))))
70.80/9.26	= { by axiom 3 (sos_02) }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((0 + x17) + ((0 + x17) ==> (x17 ==> x19)))))
70.80/9.26	= { by axiom 7 (sos_12) R->L }
70.80/9.26	  x17 ==> (x19 ==> (x17 + ((x17 ==> x19) + ((x17 ==> x19) ==> (0 + x17)))))
70.80/9.26	= { by lemma 21 }
70.80/9.26	  x17 ==> (x19 ==> (x19 + (((x17 ==> x19) ==> (0 + x17)) + (x19 ==> x17))))
70.80/9.26	= { by axiom 3 (sos_02) }
70.80/9.26	  x17 ==> (x19 ==> (x19 + ((x19 ==> x17) + ((x17 ==> x19) ==> (0 + x17)))))
70.80/9.26	= { by lemma 29 }
70.80/9.26	  x17 ==> (x19 ==> (x19 + ((x19 ==> x17) + ((x19 ==> x17) ==> (0 + x17)))))
70.80/9.26	= { by axiom 7 (sos_12) R->L }
70.80/9.26	  x17 ==> (x19 ==> (x19 + ((0 + x17) + ((0 + x17) ==> (x19 ==> x17)))))
70.80/9.26	= { by axiom 8 (sos_01) R->L }
70.80/9.26	  x17 ==> (x19 ==> (x19 + (0 + (x17 + ((0 + x17) ==> (x19 ==> x17))))))
70.80/9.26	= { by axiom 3 (sos_02) R->L }
70.80/9.26	  x17 ==> (x19 ==> (x19 + (0 + (x17 + ((x17 + 0) ==> (x19 ==> x17))))))
70.80/9.26	= { by lemma 18 R->L }
70.80/9.26	  x17 ==> (x19 ==> (x19 + (0 + (x17 + ((x17 + (0 + 0)) ==> (x19 ==> x17))))))
70.80/9.26	= { by axiom 9 (sos_06) R->L }
70.80/9.26	  x17 ==> (x19 ==> (x19 + (0 + (x17 + ((x17 + (fresh(true, true, 0, x17 ==> (x19 ==> x17)) + 0)) ==> (x19 ==> x17))))))
70.80/9.26	= { by axiom 11 (sos_07_1) R->L }
70.80/9.26	  x17 ==> (x19 ==> (x19 + (0 + (x17 + ((x17 + (fresh(fresh6(true, true, x17, 0, x19 ==> x17), true, 0, x17 ==> (x19 ==> x17)) + 0)) ==> (x19 ==> x17))))))
70.80/9.26	= { by lemma 25 R->L }
70.80/9.26	  x17 ==> (x19 ==> (x19 + (0 + (x17 + ((x17 + (fresh(fresh6((x17 + 0) >= (x19 ==> x17), true, x17, 0, x19 ==> x17), true, 0, x17 ==> (x19 ==> x17)) + 0)) ==> (x19 ==> x17))))))
70.80/9.26	= { by axiom 17 (sos_07_1) }
70.80/9.26	  x17 ==> (x19 ==> (x19 + (0 + (x17 + ((x17 + (fresh(0 >= (x17 ==> (x19 ==> x17)), true, 0, x17 ==> (x19 ==> x17)) + 0)) ==> (x19 ==> x17))))))
70.80/9.26	= { by lemma 23 }
70.80/9.26	  x17 ==> (x19 ==> (x19 + (0 + (x17 + ((x17 + ((x17 ==> (x19 ==> x17)) + 0)) ==> (x19 ==> x17))))))
70.80/9.26	= { by lemma 30 }
70.80/9.26	  x17 ==> (x19 ==> (x19 + (0 + (x17 + 0))))
70.80/9.26	= { by axiom 4 (sos_03) }
70.80/9.26	  x17 ==> (x19 ==> (x19 + (0 + x17)))
70.80/9.26	= { by lemma 24 R->L }
70.80/9.26	  x17 ==> (x19 ==> (0 + (x19 + x17)))
70.80/9.26	= { by lemma 18 }
70.80/9.26	  x17 ==> (x19 ==> (x19 + x17))
70.80/9.26	= { by axiom 3 (sos_02) R->L }
70.80/9.26	  x17 ==> (x19 ==> (x17 + x19))
70.80/9.26	= { by axiom 9 (sos_06) R->L }
70.80/9.26	  fresh(true, true, x17 ==> (x19 ==> (x17 + x19)), 0)
70.80/9.26	= { by axiom 2 (sos_08) R->L }
70.80/9.26	  fresh((x17 ==> (x19 ==> (x17 + x19))) >= 0, true, x17 ==> (x19 ==> (x17 + x19)), 0)
70.80/9.26	= { by axiom 14 (sos_06) R->L }
70.80/9.26	  fresh2(0 >= (x17 ==> (x19 ==> (x17 + x19))), true, x17 ==> (x19 ==> (x17 + x19)), 0)
70.80/9.26	= { by lemma 26 R->L }
70.80/9.26	  fresh2(fresh6(x17 >= (x19 ==> (x17 + x19)), true, x17, 0, x19 ==> (x17 + x19)), true, x17 ==> (x19 ==> (x17 + x19)), 0)
70.80/9.26	= { by axiom 3 (sos_02) R->L }
70.80/9.26	  fresh2(fresh6(x17 >= (x19 ==> (x19 + x17)), true, x17, 0, x19 ==> (x17 + x19)), true, x17 ==> (x19 ==> (x17 + x19)), 0)
70.80/9.26	= { by axiom 17 (sos_07_1) R->L }
70.80/9.26	  fresh2(fresh6(fresh6((x19 + x17) >= (x19 + x17), true, x19, x17, x19 + x17), true, x17, 0, x19 ==> (x17 + x19)), true, x17 ==> (x19 ==> (x17 + x19)), 0)
70.80/9.26	= { by axiom 1 (sos_04) }
70.80/9.26	  fresh2(fresh6(fresh6(true, true, x19, x17, x19 + x17), true, x17, 0, x19 ==> (x17 + x19)), true, x17 ==> (x19 ==> (x17 + x19)), 0)
70.80/9.26	= { by axiom 11 (sos_07_1) }
70.80/9.26	  fresh2(fresh6(true, true, x17, 0, x19 ==> (x17 + x19)), true, x17 ==> (x19 ==> (x17 + x19)), 0)
70.80/9.26	= { by axiom 11 (sos_07_1) }
70.80/9.26	  fresh2(true, true, x17 ==> (x19 ==> (x17 + x19)), 0)
70.80/9.26	= { by axiom 10 (sos_06) }
70.80/9.26	  0
70.80/9.26	
70.80/9.26	Goal 1 (goals_13_2): x17 = x19.
70.80/9.26	Proof:
70.80/9.26	  x17
70.80/9.26	= { by axiom 4 (sos_03) R->L }
70.80/9.26	  x17 + 0
70.80/9.26	= { by lemma 31 R->L }
70.80/9.26	  x17 + (x19 ==> x17)
70.80/9.26	= { by lemma 29 R->L }
70.80/9.26	  x17 + (x17 ==> x19)
70.80/9.26	= { by axiom 7 (sos_12) }
70.80/9.26	  x19 + (x19 ==> x17)
70.80/9.26	= { by lemma 31 }
70.80/9.26	  x19 + 0
70.80/9.26	= { by axiom 4 (sos_03) }
70.80/9.26	  x19
70.80/9.26	% SZS output end Proof
70.80/9.26	
70.80/9.26	RESULT: Theorem (the conjecture is true).
70.80/9.28	EOF