5.03/2.44	WORST_CASE(Omega(n^1), O(n^1))
5.03/2.45	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
5.03/2.45	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.03/2.45	
5.03/2.45	
5.03/2.45	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
5.03/2.45	
5.03/2.45	(0) CpxIntTrs
5.03/2.45	(1) Koat Proof [FINISHED, 113 ms]
5.03/2.45	(2) BOUNDS(1, n^1)
5.03/2.45	(3) Loat Proof [FINISHED, 703 ms]
5.03/2.45	(4) BOUNDS(n^1, INF)
5.03/2.45	
5.03/2.45	
5.03/2.45	----------------------------------------
5.03/2.45	
5.03/2.45	(0)
5.03/2.45	Obligation:
5.03/2.45	Complexity Int TRS consisting of the following rules:
5.03/2.45	evalfstart(A, B, C) -> Com_1(evalfentryin(A, B, C)) :|: TRUE
5.03/2.45	evalfentryin(A, B, C) -> Com_1(evalfbb5in(0, B, C)) :|: TRUE
5.03/2.45	evalfbb5in(A, B, C) -> Com_1(evalfreturnin(A, B, C)) :|: A >= B
5.03/2.45	evalfbb5in(A, B, C) -> Com_1(evalfbb6in(A, B, C)) :|: B >= A + 1
5.03/2.45	evalfbb6in(A, B, C) -> Com_1(evalfbb2in(A, B, A)) :|: 0 >= D + 1
5.03/2.45	evalfbb6in(A, B, C) -> Com_1(evalfbb2in(A, B, A)) :|: D >= 1
5.03/2.45	evalfbb6in(A, B, C) -> Com_1(evalfreturnin(A, B, C)) :|: TRUE
5.03/2.45	evalfbb2in(A, B, C) -> Com_1(evalfbb4in(A, B, C)) :|: C >= B
5.03/2.45	evalfbb2in(A, B, C) -> Com_1(evalfbb3in(A, B, C)) :|: B >= C + 1
5.03/2.45	evalfbb3in(A, B, C) -> Com_1(evalfbb1in(A, B, C)) :|: 0 >= D + 1
5.03/2.45	evalfbb3in(A, B, C) -> Com_1(evalfbb1in(A, B, C)) :|: D >= 1
5.03/2.45	evalfbb3in(A, B, C) -> Com_1(evalfbb4in(A, B, C)) :|: TRUE
5.03/2.45	evalfbb1in(A, B, C) -> Com_1(evalfbb2in(A, B, C + 1)) :|: TRUE
5.03/2.45	evalfbb4in(A, B, C) -> Com_1(evalfbb5in(C + 1, B, C)) :|: TRUE
5.03/2.45	evalfreturnin(A, B, C) -> Com_1(evalfstop(A, B, C)) :|: TRUE
5.03/2.45	
5.03/2.45	The start-symbols are:[evalfstart_3]
5.03/2.45	
5.03/2.45	
5.03/2.45	----------------------------------------
5.03/2.45	
5.03/2.45	(1) Koat Proof (FINISHED)
5.03/2.45	YES(?, 36*ar_1 + 44)
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Initial complexity problem:
5.03/2.45	
5.03/2.45	1:	T:
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb5in(0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb5in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb5in(ar_0, ar_1, ar_2) -> Com_1(evalfbb6in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 + 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb6in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ 0 >= d + 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb6in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ d >= 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb6in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 + 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ 0 >= d + 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ d >= 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb5in(ar_2 + 1, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
5.03/2.45	
5.03/2.45		start location:	koat_start
5.03/2.45	
5.03/2.45		leaf cost:	0
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Repeatedly propagating knowledge in problem 1 produces the following problem:
5.03/2.45	
5.03/2.45	2:	T:
5.03/2.45	
5.03/2.45			(Comp: 1, Cost: 1)    evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 1, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb5in(0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb5in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb5in(ar_0, ar_1, ar_2) -> Com_1(evalfbb6in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 + 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb6in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ 0 >= d + 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb6in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ d >= 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb6in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 + 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ 0 >= d + 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ d >= 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb5in(ar_2 + 1, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
5.03/2.45	
5.03/2.45		start location:	koat_start
5.03/2.45	
5.03/2.45		leaf cost:	0
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	A polynomial rank function with
5.03/2.45	
5.03/2.45		Pol(evalfstart) = 2
5.03/2.45	
5.03/2.45		Pol(evalfentryin) = 2
5.03/2.45	
5.03/2.45		Pol(evalfbb5in) = 2
5.03/2.45	
5.03/2.45		Pol(evalfreturnin) = 1
5.03/2.45	
5.03/2.45		Pol(evalfbb6in) = 2
5.03/2.45	
5.03/2.45		Pol(evalfbb2in) = 2
5.03/2.45	
5.03/2.45		Pol(evalfbb4in) = 2
5.03/2.45	
5.03/2.45		Pol(evalfbb3in) = 2
5.03/2.45	
5.03/2.45		Pol(evalfbb1in) = 2
5.03/2.45	
5.03/2.45		Pol(evalfstop) = 0
5.03/2.45	
5.03/2.45		Pol(koat_start) = 2
5.03/2.45	
5.03/2.45	orients all transitions weakly and the transitions
5.03/2.45	
5.03/2.45		evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45		evalfbb6in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45		evalfbb5in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
5.03/2.45	
5.03/2.45	strictly and produces the following problem:
5.03/2.45	
5.03/2.45	3:	T:
5.03/2.45	
5.03/2.45			(Comp: 1, Cost: 1)    evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 1, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb5in(0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 2, Cost: 1)    evalfbb5in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb5in(ar_0, ar_1, ar_2) -> Com_1(evalfbb6in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 + 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb6in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ 0 >= d + 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb6in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ d >= 1 ]
5.03/2.45	
5.03/2.45			(Comp: 2, Cost: 1)    evalfbb6in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 + 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ 0 >= d + 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ d >= 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb5in(ar_2 + 1, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 2, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
5.03/2.45	
5.03/2.45		start location:	koat_start
5.03/2.45	
5.03/2.45		leaf cost:	0
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	A polynomial rank function with
5.03/2.45	
5.03/2.45		Pol(evalfstart) = 2*V_2 + 2
5.03/2.45	
5.03/2.45		Pol(evalfentryin) = 2*V_2 + 2
5.03/2.45	
5.03/2.45		Pol(evalfbb5in) = -2*V_1 + 2*V_2 + 2
5.03/2.45	
5.03/2.45		Pol(evalfreturnin) = -2*V_1 + 2*V_2
5.03/2.45	
5.03/2.45		Pol(evalfbb6in) = -2*V_1 + 2*V_2 + 1
5.03/2.45	
5.03/2.45		Pol(evalfbb2in) = 2*V_2 - 2*V_3 + 1
5.03/2.45	
5.03/2.45		Pol(evalfbb4in) = 2*V_2 - 2*V_3
5.03/2.45	
5.03/2.45		Pol(evalfbb3in) = 2*V_2 - 2*V_3
5.03/2.45	
5.03/2.45		Pol(evalfbb1in) = 2*V_2 - 2*V_3
5.03/2.45	
5.03/2.45		Pol(evalfstop) = -2*V_1 + 2*V_2
5.03/2.45	
5.03/2.45		Pol(koat_start) = 2*V_2 + 2
5.03/2.45	
5.03/2.45	orients all transitions weakly and the transitions
5.03/2.45	
5.03/2.45		evalfbb5in(ar_0, ar_1, ar_2) -> Com_1(evalfbb6in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 + 1 ]
5.03/2.45	
5.03/2.45		evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 + 1 ]
5.03/2.45	
5.03/2.45	strictly and produces the following problem:
5.03/2.45	
5.03/2.45	4:	T:
5.03/2.45	
5.03/2.45			(Comp: 1, Cost: 1)             evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 1, Cost: 1)             evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb5in(0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 2, Cost: 1)             evalfbb5in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
5.03/2.45	
5.03/2.45			(Comp: 2*ar_1 + 2, Cost: 1)    evalfbb5in(ar_0, ar_1, ar_2) -> Com_1(evalfbb6in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 + 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)             evalfbb6in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ 0 >= d + 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)             evalfbb6in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ d >= 1 ]
5.03/2.45	
5.03/2.45			(Comp: 2, Cost: 1)             evalfbb6in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)             evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ]
5.03/2.45	
5.03/2.45			(Comp: 2*ar_1 + 2, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 + 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)             evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ 0 >= d + 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)             evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ d >= 1 ]
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)             evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)             evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1))
5.03/2.45	
5.03/2.45			(Comp: ?, Cost: 1)             evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb5in(ar_2 + 1, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 2, Cost: 1)             evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 1, Cost: 0)             koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
5.03/2.45	
5.03/2.45		start location:	koat_start
5.03/2.45	
5.03/2.45		leaf cost:	0
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Repeatedly propagating knowledge in problem 4 produces the following problem:
5.03/2.45	
5.03/2.45	5:	T:
5.03/2.45	
5.03/2.45			(Comp: 1, Cost: 1)               evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 1, Cost: 1)               evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb5in(0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 2, Cost: 1)               evalfbb5in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
5.03/2.45	
5.03/2.45			(Comp: 2*ar_1 + 2, Cost: 1)      evalfbb5in(ar_0, ar_1, ar_2) -> Com_1(evalfbb6in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 + 1 ]
5.03/2.45	
5.03/2.45			(Comp: 2*ar_1 + 2, Cost: 1)      evalfbb6in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ 0 >= d + 1 ]
5.03/2.45	
5.03/2.45			(Comp: 2*ar_1 + 2, Cost: 1)      evalfbb6in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ d >= 1 ]
5.03/2.45	
5.03/2.45			(Comp: 2, Cost: 1)               evalfbb6in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 8*ar_1 + 8, Cost: 1)      evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ]
5.03/2.45	
5.03/2.45			(Comp: 2*ar_1 + 2, Cost: 1)      evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 + 1 ]
5.03/2.45	
5.03/2.45			(Comp: 2*ar_1 + 2, Cost: 1)      evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ 0 >= d + 1 ]
5.03/2.45	
5.03/2.45			(Comp: 2*ar_1 + 2, Cost: 1)      evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ d >= 1 ]
5.03/2.45	
5.03/2.45			(Comp: 2*ar_1 + 2, Cost: 1)      evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 4*ar_1 + 4, Cost: 1)      evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1))
5.03/2.45	
5.03/2.45			(Comp: 10*ar_1 + 10, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb5in(ar_2 + 1, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 2, Cost: 1)               evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
5.03/2.45	
5.03/2.45			(Comp: 1, Cost: 0)               koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
5.03/2.45	
5.03/2.45		start location:	koat_start
5.03/2.45	
5.03/2.45		leaf cost:	0
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Complexity upper bound 36*ar_1 + 44
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Time: 0.118 sec (SMT: 0.102 sec)
5.03/2.45	
5.03/2.45	
5.03/2.45	----------------------------------------
5.03/2.45	
5.03/2.45	(2)
5.03/2.45	BOUNDS(1, n^1)
5.03/2.45	
5.03/2.45	----------------------------------------
5.03/2.45	
5.03/2.45	(3) Loat Proof (FINISHED)
5.03/2.45	
5.03/2.45	
5.03/2.45	### Pre-processing the ITS problem ###
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Initial linear ITS problem
5.03/2.45	
5.03/2.45	   Start location: evalfstart
5.03/2.45	
5.03/2.45	      0: evalfstart -> evalfentryin : [], cost: 1
5.03/2.45	
5.03/2.45	      1: evalfentryin -> evalfbb5in : A'=0, [], cost: 1
5.03/2.45	
5.03/2.45	      2: evalfbb5in -> evalfreturnin : [ A>=B ], cost: 1
5.03/2.45	
5.03/2.45	      3: evalfbb5in -> evalfbb6in : [ B>=1+A ], cost: 1
5.03/2.45	
5.03/2.45	      4: evalfbb6in -> evalfbb2in : C'=A, [ 0>=1+free ], cost: 1
5.03/2.45	
5.03/2.45	      5: evalfbb6in -> evalfbb2in : C'=A, [ free_1>=1 ], cost: 1
5.03/2.45	
5.03/2.45	      6: evalfbb6in -> evalfreturnin : [], cost: 1
5.03/2.45	
5.03/2.45	      7: evalfbb2in -> evalfbb4in : [ C>=B ], cost: 1
5.03/2.45	
5.03/2.45	      8: evalfbb2in -> evalfbb3in : [ B>=1+C ], cost: 1
5.03/2.45	
5.03/2.45	      9: evalfbb3in -> evalfbb1in : [ 0>=1+free_2 ], cost: 1
5.03/2.45	
5.03/2.45	     10: evalfbb3in -> evalfbb1in : [ free_3>=1 ], cost: 1
5.03/2.45	
5.03/2.45	     11: evalfbb3in -> evalfbb4in : [], cost: 1
5.03/2.45	
5.03/2.45	     12: evalfbb1in -> evalfbb2in : C'=1+C, [], cost: 1
5.03/2.45	
5.03/2.45	     13: evalfbb4in -> evalfbb5in : A'=1+C, [], cost: 1
5.03/2.45	
5.03/2.45	     14: evalfreturnin -> evalfstop : [], cost: 1
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Removed unreachable and leaf rules:
5.03/2.45	
5.03/2.45	   Start location: evalfstart
5.03/2.45	
5.03/2.45	      0: evalfstart -> evalfentryin : [], cost: 1
5.03/2.45	
5.03/2.45	      1: evalfentryin -> evalfbb5in : A'=0, [], cost: 1
5.03/2.45	
5.03/2.45	      3: evalfbb5in -> evalfbb6in : [ B>=1+A ], cost: 1
5.03/2.45	
5.03/2.45	      4: evalfbb6in -> evalfbb2in : C'=A, [ 0>=1+free ], cost: 1
5.03/2.45	
5.03/2.45	      5: evalfbb6in -> evalfbb2in : C'=A, [ free_1>=1 ], cost: 1
5.03/2.45	
5.03/2.45	      7: evalfbb2in -> evalfbb4in : [ C>=B ], cost: 1
5.03/2.45	
5.03/2.45	      8: evalfbb2in -> evalfbb3in : [ B>=1+C ], cost: 1
5.03/2.45	
5.03/2.45	      9: evalfbb3in -> evalfbb1in : [ 0>=1+free_2 ], cost: 1
5.03/2.45	
5.03/2.45	     10: evalfbb3in -> evalfbb1in : [ free_3>=1 ], cost: 1
5.03/2.45	
5.03/2.45	     11: evalfbb3in -> evalfbb4in : [], cost: 1
5.03/2.45	
5.03/2.45	     12: evalfbb1in -> evalfbb2in : C'=1+C, [], cost: 1
5.03/2.45	
5.03/2.45	     13: evalfbb4in -> evalfbb5in : A'=1+C, [], cost: 1
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Simplified all rules, resulting in:
5.03/2.45	
5.03/2.45	   Start location: evalfstart
5.03/2.45	
5.03/2.45	      0: evalfstart -> evalfentryin : [], cost: 1
5.03/2.45	
5.03/2.45	      1: evalfentryin -> evalfbb5in : A'=0, [], cost: 1
5.03/2.45	
5.03/2.45	      3: evalfbb5in -> evalfbb6in : [ B>=1+A ], cost: 1
5.03/2.45	
5.03/2.45	      5: evalfbb6in -> evalfbb2in : C'=A, [], cost: 1
5.03/2.45	
5.03/2.45	      7: evalfbb2in -> evalfbb4in : [ C>=B ], cost: 1
5.03/2.45	
5.03/2.45	      8: evalfbb2in -> evalfbb3in : [ B>=1+C ], cost: 1
5.03/2.45	
5.03/2.45	     10: evalfbb3in -> evalfbb1in : [], cost: 1
5.03/2.45	
5.03/2.45	     11: evalfbb3in -> evalfbb4in : [], cost: 1
5.03/2.45	
5.03/2.45	     12: evalfbb1in -> evalfbb2in : C'=1+C, [], cost: 1
5.03/2.45	
5.03/2.45	     13: evalfbb4in -> evalfbb5in : A'=1+C, [], cost: 1
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	### Simplification by acceleration and chaining ###
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Eliminated locations (on linear paths):
5.03/2.45	
5.03/2.45	   Start location: evalfstart
5.03/2.45	
5.03/2.45	     15: evalfstart -> evalfbb5in : A'=0, [], cost: 2
5.03/2.45	
5.03/2.45	     16: evalfbb5in -> evalfbb2in : C'=A, [ B>=1+A ], cost: 2
5.03/2.45	
5.03/2.45	      7: evalfbb2in -> evalfbb4in : [ C>=B ], cost: 1
5.03/2.45	
5.03/2.45	      8: evalfbb2in -> evalfbb3in : [ B>=1+C ], cost: 1
5.03/2.45	
5.03/2.45	     11: evalfbb3in -> evalfbb4in : [], cost: 1
5.03/2.45	
5.03/2.45	     17: evalfbb3in -> evalfbb2in : C'=1+C, [], cost: 2
5.03/2.45	
5.03/2.45	     13: evalfbb4in -> evalfbb5in : A'=1+C, [], cost: 1
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Eliminated locations (on tree-shaped paths):
5.03/2.45	
5.03/2.45	   Start location: evalfstart
5.03/2.45	
5.03/2.45	     15: evalfstart -> evalfbb5in : A'=0, [], cost: 2
5.03/2.45	
5.03/2.45	     16: evalfbb5in -> evalfbb2in : C'=A, [ B>=1+A ], cost: 2
5.03/2.45	
5.03/2.45	     19: evalfbb2in -> evalfbb2in : C'=1+C, [ B>=1+C ], cost: 3
5.03/2.45	
5.03/2.45	     20: evalfbb2in -> evalfbb5in : A'=1+C, [ C>=B ], cost: 2
5.03/2.45	
5.03/2.45	     21: evalfbb2in -> evalfbb5in : A'=1+C, [ B>=1+C ], cost: 3
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Accelerating simple loops of location 4.
5.03/2.45	
5.03/2.45	   Accelerating the following rules:
5.03/2.45	
5.03/2.45	     19: evalfbb2in -> evalfbb2in : C'=1+C, [ B>=1+C ], cost: 3
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	   Accelerated rule 19 with metering function -C+B, yielding the new rule 22.
5.03/2.45	
5.03/2.45	   Removing the simple loops: 19.
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Accelerated all simple loops using metering functions (where possible):
5.03/2.45	
5.03/2.45	   Start location: evalfstart
5.03/2.45	
5.03/2.45	     15: evalfstart -> evalfbb5in : A'=0, [], cost: 2
5.03/2.45	
5.03/2.45	     16: evalfbb5in -> evalfbb2in : C'=A, [ B>=1+A ], cost: 2
5.03/2.45	
5.03/2.45	     20: evalfbb2in -> evalfbb5in : A'=1+C, [ C>=B ], cost: 2
5.03/2.45	
5.03/2.45	     21: evalfbb2in -> evalfbb5in : A'=1+C, [ B>=1+C ], cost: 3
5.03/2.45	
5.03/2.45	     22: evalfbb2in -> evalfbb2in : C'=B, [ B>=1+C ], cost: -3*C+3*B
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Chained accelerated rules (with incoming rules):
5.03/2.45	
5.03/2.45	   Start location: evalfstart
5.03/2.45	
5.03/2.45	     15: evalfstart -> evalfbb5in : A'=0, [], cost: 2
5.03/2.45	
5.03/2.45	     16: evalfbb5in -> evalfbb2in : C'=A, [ B>=1+A ], cost: 2
5.03/2.45	
5.03/2.45	     23: evalfbb5in -> evalfbb2in : C'=B, [ B>=1+A ], cost: 2-3*A+3*B
5.03/2.45	
5.03/2.45	     20: evalfbb2in -> evalfbb5in : A'=1+C, [ C>=B ], cost: 2
5.03/2.45	
5.03/2.45	     21: evalfbb2in -> evalfbb5in : A'=1+C, [ B>=1+C ], cost: 3
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Eliminated locations (on tree-shaped paths):
5.03/2.45	
5.03/2.45	   Start location: evalfstart
5.03/2.45	
5.03/2.45	     15: evalfstart -> evalfbb5in : A'=0, [], cost: 2
5.03/2.45	
5.03/2.45	     24: evalfbb5in -> evalfbb5in : A'=1+A, C'=A, [ B>=1+A ], cost: 5
5.03/2.45	
5.03/2.45	     25: evalfbb5in -> evalfbb5in : A'=1+B, C'=B, [ B>=1+A ], cost: 4-3*A+3*B
5.03/2.45	
5.03/2.45	     26: evalfbb5in -> [11] : [ B>=1+A ], cost: 2-3*A+3*B
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Accelerating simple loops of location 2.
5.03/2.45	
5.03/2.45	   Accelerating the following rules:
5.03/2.45	
5.03/2.45	     24: evalfbb5in -> evalfbb5in : A'=1+A, C'=A, [ B>=1+A ], cost: 5
5.03/2.45	
5.03/2.45	     25: evalfbb5in -> evalfbb5in : A'=1+B, C'=B, [ B>=1+A ], cost: 4-3*A+3*B
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	   Accelerated rule 24 with metering function -A+B, yielding the new rule 27.
5.03/2.45	
5.03/2.45	   Found no metering function for rule 25.
5.03/2.45	
5.03/2.45	   Removing the simple loops: 24.
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Accelerated all simple loops using metering functions (where possible):
5.03/2.45	
5.03/2.45	   Start location: evalfstart
5.03/2.45	
5.03/2.45	     15: evalfstart -> evalfbb5in : A'=0, [], cost: 2
5.03/2.45	
5.03/2.45	     25: evalfbb5in -> evalfbb5in : A'=1+B, C'=B, [ B>=1+A ], cost: 4-3*A+3*B
5.03/2.45	
5.03/2.45	     26: evalfbb5in -> [11] : [ B>=1+A ], cost: 2-3*A+3*B
5.03/2.45	
5.03/2.45	     27: evalfbb5in -> evalfbb5in : A'=B, C'=-1+B, [ B>=1+A ], cost: -5*A+5*B
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Chained accelerated rules (with incoming rules):
5.03/2.45	
5.03/2.45	   Start location: evalfstart
5.03/2.45	
5.03/2.45	     15: evalfstart -> evalfbb5in : A'=0, [], cost: 2
5.03/2.45	
5.03/2.45	     28: evalfstart -> evalfbb5in : A'=1+B, C'=B, [ B>=1 ], cost: 6+3*B
5.03/2.45	
5.03/2.45	     29: evalfstart -> evalfbb5in : A'=B, C'=-1+B, [ B>=1 ], cost: 2+5*B
5.03/2.45	
5.03/2.45	     26: evalfbb5in -> [11] : [ B>=1+A ], cost: 2-3*A+3*B
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Eliminated locations (on tree-shaped paths):
5.03/2.45	
5.03/2.45	   Start location: evalfstart
5.03/2.45	
5.03/2.45	     30: evalfstart -> [11] : A'=0, [ B>=1 ], cost: 4+3*B
5.03/2.45	
5.03/2.45	     31: evalfstart -> [13] : [ B>=1 ], cost: 6+3*B
5.03/2.45	
5.03/2.45	     32: evalfstart -> [13] : [ B>=1 ], cost: 2+5*B
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	### Computing asymptotic complexity ###
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Fully simplified ITS problem
5.03/2.45	
5.03/2.45	   Start location: evalfstart
5.03/2.45	
5.03/2.45	     31: evalfstart -> [13] : [ B>=1 ], cost: 6+3*B
5.03/2.45	
5.03/2.45	     32: evalfstart -> [13] : [ B>=1 ], cost: 2+5*B
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Computing asymptotic complexity for rule 31
5.03/2.45	
5.03/2.45	   Solved the limit problem by the following transformations:
5.03/2.45	
5.03/2.45	      Created initial limit problem:
5.03/2.45	
5.03/2.45	      6+3*B (+), B (+/+!) [not solved]
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	      removing all constraints (solved by SMT)
5.03/2.45	
5.03/2.45	      resulting limit problem: [solved]
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	      applying transformation rule (C) using substitution {B==n}
5.03/2.45	
5.03/2.45	      resulting limit problem:
5.03/2.45	
5.03/2.45	      [solved]
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	   Solution:
5.03/2.45	
5.03/2.45	      B / n
5.03/2.45	
5.03/2.45	   Resulting cost 6+3*n has complexity: Poly(n^1)
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Found new complexity Poly(n^1).
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	Obtained the following overall complexity (w.r.t. the length of the input n):
5.03/2.45	
5.03/2.45	   Complexity:  Poly(n^1)
5.03/2.45	
5.03/2.45	   Cpx degree:  1
5.03/2.45	
5.03/2.45	   Solved cost: 6+3*n
5.03/2.45	
5.03/2.45	   Rule cost:   6+3*B
5.03/2.45	
5.03/2.45	   Rule guard:  [ B>=1 ]
5.03/2.45	
5.03/2.45	
5.03/2.45	
5.03/2.45	WORST_CASE(Omega(n^1),?)
5.03/2.45	
5.03/2.45	
5.03/2.45	----------------------------------------
5.03/2.45	
5.03/2.45	(4)
5.03/2.45	BOUNDS(n^1, INF)
5.03/2.47	EOF