4.81/2.39	WORST_CASE(Omega(n^1), O(n^1))
4.97/2.40	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
4.97/2.40	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.97/2.40	
4.97/2.40	
4.97/2.40	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
4.97/2.40	
4.97/2.40	(0) CpxIntTrs
4.97/2.40	(1) Koat Proof [FINISHED, 107 ms]
4.97/2.40	(2) BOUNDS(1, n^1)
4.97/2.40	(3) Loat Proof [FINISHED, 701 ms]
4.97/2.40	(4) BOUNDS(n^1, INF)
4.97/2.40	
4.97/2.40	
4.97/2.40	----------------------------------------
4.97/2.40	
4.97/2.40	(0)
4.97/2.40	Obligation:
4.97/2.40	Complexity Int TRS consisting of the following rules:
4.97/2.40	evalNestedSinglestart(A, B, C) -> Com_1(evalNestedSingleentryin(A, B, C)) :|: TRUE
4.97/2.40	evalNestedSingleentryin(A, B, C) -> Com_1(evalNestedSinglebb5in(0, B, C)) :|: TRUE
4.97/2.40	evalNestedSinglebb5in(A, B, C) -> Com_1(evalNestedSinglebb2in(A, B, A)) :|: B >= A + 1
4.97/2.40	evalNestedSinglebb5in(A, B, C) -> Com_1(evalNestedSinglereturnin(A, B, C)) :|: A >= B
4.97/2.40	evalNestedSinglebb2in(A, B, C) -> Com_1(evalNestedSinglebb4in(A, B, C)) :|: C >= B
4.97/2.40	evalNestedSinglebb2in(A, B, C) -> Com_1(evalNestedSinglebb3in(A, B, C)) :|: B >= C + 1
4.97/2.40	evalNestedSinglebb3in(A, B, C) -> Com_1(evalNestedSinglebb1in(A, B, C)) :|: 0 >= D + 1
4.97/2.40	evalNestedSinglebb3in(A, B, C) -> Com_1(evalNestedSinglebb1in(A, B, C)) :|: D >= 1
4.97/2.40	evalNestedSinglebb3in(A, B, C) -> Com_1(evalNestedSinglebb4in(A, B, C)) :|: TRUE
4.97/2.40	evalNestedSinglebb1in(A, B, C) -> Com_1(evalNestedSinglebb2in(A, B, C + 1)) :|: TRUE
4.97/2.40	evalNestedSinglebb4in(A, B, C) -> Com_1(evalNestedSinglebb5in(C + 1, B, C)) :|: TRUE
4.97/2.40	evalNestedSinglereturnin(A, B, C) -> Com_1(evalNestedSinglestop(A, B, C)) :|: TRUE
4.97/2.40	
4.97/2.40	The start-symbols are:[evalNestedSinglestart_3]
4.97/2.40	
4.97/2.40	
4.97/2.40	----------------------------------------
4.97/2.40	
4.97/2.40	(1) Koat Proof (FINISHED)
4.97/2.40	YES(?, 24*ar_1 + 30)
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Initial complexity problem:
4.97/2.40	
4.97/2.40	1:	T:
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglestart(ar_0, ar_1, ar_2) -> Com_1(evalNestedSingleentryin(ar_0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSingleentryin(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb5in(0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb5in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 + 1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb5in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglereturnin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb2in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb2in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 + 1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb3in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb1in(ar_0, ar_1, ar_2)) [ 0 >= d + 1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb3in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb1in(ar_0, ar_1, ar_2)) [ d >= 1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb3in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb4in(ar_0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb1in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb2in(ar_0, ar_1, ar_2 + 1))
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb4in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb5in(ar_2 + 1, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglereturnin(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglestop(ar_0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglestart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.97/2.40	
4.97/2.40		start location:	koat_start
4.97/2.40	
4.97/2.40		leaf cost:	0
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Repeatedly propagating knowledge in problem 1 produces the following problem:
4.97/2.40	
4.97/2.40	2:	T:
4.97/2.40	
4.97/2.40			(Comp: 1, Cost: 1)    evalNestedSinglestart(ar_0, ar_1, ar_2) -> Com_1(evalNestedSingleentryin(ar_0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: 1, Cost: 1)    evalNestedSingleentryin(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb5in(0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb5in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 + 1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb5in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglereturnin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb2in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb2in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 + 1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb3in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb1in(ar_0, ar_1, ar_2)) [ 0 >= d + 1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb3in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb1in(ar_0, ar_1, ar_2)) [ d >= 1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb3in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb4in(ar_0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb1in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb2in(ar_0, ar_1, ar_2 + 1))
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb4in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb5in(ar_2 + 1, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglereturnin(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglestop(ar_0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglestart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.97/2.40	
4.97/2.40		start location:	koat_start
4.97/2.40	
4.97/2.40		leaf cost:	0
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	A polynomial rank function with
4.97/2.40	
4.97/2.40		Pol(evalNestedSinglestart) = 2
4.97/2.40	
4.97/2.40		Pol(evalNestedSingleentryin) = 2
4.97/2.40	
4.97/2.40		Pol(evalNestedSinglebb5in) = 2
4.97/2.40	
4.97/2.40		Pol(evalNestedSinglebb2in) = 2
4.97/2.40	
4.97/2.40		Pol(evalNestedSinglereturnin) = 1
4.97/2.40	
4.97/2.40		Pol(evalNestedSinglebb4in) = 2
4.97/2.40	
4.97/2.40		Pol(evalNestedSinglebb3in) = 2
4.97/2.40	
4.97/2.40		Pol(evalNestedSinglebb1in) = 2
4.97/2.40	
4.97/2.40		Pol(evalNestedSinglestop) = 0
4.97/2.40	
4.97/2.40		Pol(koat_start) = 2
4.97/2.40	
4.97/2.40	orients all transitions weakly and the transitions
4.97/2.40	
4.97/2.40		evalNestedSinglereturnin(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglestop(ar_0, ar_1, ar_2))
4.97/2.40	
4.97/2.40		evalNestedSinglebb5in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglereturnin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
4.97/2.40	
4.97/2.40	strictly and produces the following problem:
4.97/2.40	
4.97/2.40	3:	T:
4.97/2.40	
4.97/2.40			(Comp: 1, Cost: 1)    evalNestedSinglestart(ar_0, ar_1, ar_2) -> Com_1(evalNestedSingleentryin(ar_0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: 1, Cost: 1)    evalNestedSingleentryin(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb5in(0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb5in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 + 1 ]
4.97/2.40	
4.97/2.40			(Comp: 2, Cost: 1)    evalNestedSinglebb5in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglereturnin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb2in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb2in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 + 1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb3in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb1in(ar_0, ar_1, ar_2)) [ 0 >= d + 1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb3in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb1in(ar_0, ar_1, ar_2)) [ d >= 1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb3in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb4in(ar_0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb1in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb2in(ar_0, ar_1, ar_2 + 1))
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)    evalNestedSinglebb4in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb5in(ar_2 + 1, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: 2, Cost: 1)    evalNestedSinglereturnin(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglestop(ar_0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglestart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.97/2.40	
4.97/2.40		start location:	koat_start
4.97/2.40	
4.97/2.40		leaf cost:	0
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	A polynomial rank function with
4.97/2.40	
4.97/2.40		Pol(evalNestedSinglestart) = 2*V_2 + 2
4.97/2.40	
4.97/2.40		Pol(evalNestedSingleentryin) = 2*V_2 + 2
4.97/2.40	
4.97/2.40		Pol(evalNestedSinglebb5in) = -2*V_1 + 2*V_2 + 2
4.97/2.40	
4.97/2.40		Pol(evalNestedSinglebb2in) = 2*V_2 - 2*V_3 + 1
4.97/2.40	
4.97/2.40		Pol(evalNestedSinglereturnin) = -2*V_1 + 2*V_2
4.97/2.40	
4.97/2.40		Pol(evalNestedSinglebb4in) = 2*V_2 - 2*V_3
4.97/2.40	
4.97/2.40		Pol(evalNestedSinglebb3in) = 2*V_2 - 2*V_3
4.97/2.40	
4.97/2.40		Pol(evalNestedSinglebb1in) = 2*V_2 - 2*V_3
4.97/2.40	
4.97/2.40		Pol(evalNestedSinglestop) = -2*V_1 + 2*V_2
4.97/2.40	
4.97/2.40		Pol(koat_start) = 2*V_2 + 2
4.97/2.40	
4.97/2.40	orients all transitions weakly and the transitions
4.97/2.40	
4.97/2.40		evalNestedSinglebb5in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 + 1 ]
4.97/2.40	
4.97/2.40		evalNestedSinglebb2in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 + 1 ]
4.97/2.40	
4.97/2.40	strictly and produces the following problem:
4.97/2.40	
4.97/2.40	4:	T:
4.97/2.40	
4.97/2.40			(Comp: 1, Cost: 1)             evalNestedSinglestart(ar_0, ar_1, ar_2) -> Com_1(evalNestedSingleentryin(ar_0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: 1, Cost: 1)             evalNestedSingleentryin(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb5in(0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: 2*ar_1 + 2, Cost: 1)    evalNestedSinglebb5in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 + 1 ]
4.97/2.40	
4.97/2.40			(Comp: 2, Cost: 1)             evalNestedSinglebb5in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglereturnin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)             evalNestedSinglebb2in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ]
4.97/2.40	
4.97/2.40			(Comp: 2*ar_1 + 2, Cost: 1)    evalNestedSinglebb2in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 + 1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)             evalNestedSinglebb3in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb1in(ar_0, ar_1, ar_2)) [ 0 >= d + 1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)             evalNestedSinglebb3in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb1in(ar_0, ar_1, ar_2)) [ d >= 1 ]
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)             evalNestedSinglebb3in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb4in(ar_0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)             evalNestedSinglebb1in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb2in(ar_0, ar_1, ar_2 + 1))
4.97/2.40	
4.97/2.40			(Comp: ?, Cost: 1)             evalNestedSinglebb4in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb5in(ar_2 + 1, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: 2, Cost: 1)             evalNestedSinglereturnin(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglestop(ar_0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: 1, Cost: 0)             koat_start(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglestart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.97/2.40	
4.97/2.40		start location:	koat_start
4.97/2.40	
4.97/2.40		leaf cost:	0
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Repeatedly propagating knowledge in problem 4 produces the following problem:
4.97/2.40	
4.97/2.40	5:	T:
4.97/2.40	
4.97/2.40			(Comp: 1, Cost: 1)             evalNestedSinglestart(ar_0, ar_1, ar_2) -> Com_1(evalNestedSingleentryin(ar_0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: 1, Cost: 1)             evalNestedSingleentryin(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb5in(0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: 2*ar_1 + 2, Cost: 1)    evalNestedSinglebb5in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 + 1 ]
4.97/2.40	
4.97/2.40			(Comp: 2, Cost: 1)             evalNestedSinglebb5in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglereturnin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
4.97/2.40	
4.97/2.40			(Comp: 4*ar_1 + 4, Cost: 1)    evalNestedSinglebb2in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ]
4.97/2.40	
4.97/2.40			(Comp: 2*ar_1 + 2, Cost: 1)    evalNestedSinglebb2in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 + 1 ]
4.97/2.40	
4.97/2.40			(Comp: 2*ar_1 + 2, Cost: 1)    evalNestedSinglebb3in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb1in(ar_0, ar_1, ar_2)) [ 0 >= d + 1 ]
4.97/2.40	
4.97/2.40			(Comp: 2*ar_1 + 2, Cost: 1)    evalNestedSinglebb3in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb1in(ar_0, ar_1, ar_2)) [ d >= 1 ]
4.97/2.40	
4.97/2.40			(Comp: 2*ar_1 + 2, Cost: 1)    evalNestedSinglebb3in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb4in(ar_0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: 4*ar_1 + 4, Cost: 1)    evalNestedSinglebb1in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb2in(ar_0, ar_1, ar_2 + 1))
4.97/2.40	
4.97/2.40			(Comp: 6*ar_1 + 6, Cost: 1)    evalNestedSinglebb4in(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglebb5in(ar_2 + 1, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: 2, Cost: 1)             evalNestedSinglereturnin(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglestop(ar_0, ar_1, ar_2))
4.97/2.40	
4.97/2.40			(Comp: 1, Cost: 0)             koat_start(ar_0, ar_1, ar_2) -> Com_1(evalNestedSinglestart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.97/2.40	
4.97/2.40		start location:	koat_start
4.97/2.40	
4.97/2.40		leaf cost:	0
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Complexity upper bound 24*ar_1 + 30
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Time: 0.096 sec (SMT: 0.085 sec)
4.97/2.40	
4.97/2.40	
4.97/2.40	----------------------------------------
4.97/2.40	
4.97/2.40	(2)
4.97/2.40	BOUNDS(1, n^1)
4.97/2.40	
4.97/2.40	----------------------------------------
4.97/2.40	
4.97/2.40	(3) Loat Proof (FINISHED)
4.97/2.40	
4.97/2.40	
4.97/2.40	### Pre-processing the ITS problem ###
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Initial linear ITS problem
4.97/2.40	
4.97/2.40	   Start location: evalNestedSinglestart
4.97/2.40	
4.97/2.40	      0: evalNestedSinglestart -> evalNestedSingleentryin : [], cost: 1
4.97/2.40	
4.97/2.40	      1: evalNestedSingleentryin -> evalNestedSinglebb5in : A'=0, [], cost: 1
4.97/2.40	
4.97/2.40	      2: evalNestedSinglebb5in -> evalNestedSinglebb2in : C'=A, [ B>=1+A ], cost: 1
4.97/2.40	
4.97/2.40	      3: evalNestedSinglebb5in -> evalNestedSinglereturnin : [ A>=B ], cost: 1
4.97/2.40	
4.97/2.40	      4: evalNestedSinglebb2in -> evalNestedSinglebb4in : [ C>=B ], cost: 1
4.97/2.40	
4.97/2.40	      5: evalNestedSinglebb2in -> evalNestedSinglebb3in : [ B>=1+C ], cost: 1
4.97/2.40	
4.97/2.40	      6: evalNestedSinglebb3in -> evalNestedSinglebb1in : [ 0>=1+free ], cost: 1
4.97/2.40	
4.97/2.40	      7: evalNestedSinglebb3in -> evalNestedSinglebb1in : [ free_1>=1 ], cost: 1
4.97/2.40	
4.97/2.40	      8: evalNestedSinglebb3in -> evalNestedSinglebb4in : [], cost: 1
4.97/2.40	
4.97/2.40	      9: evalNestedSinglebb1in -> evalNestedSinglebb2in : C'=1+C, [], cost: 1
4.97/2.40	
4.97/2.40	     10: evalNestedSinglebb4in -> evalNestedSinglebb5in : A'=1+C, [], cost: 1
4.97/2.40	
4.97/2.40	     11: evalNestedSinglereturnin -> evalNestedSinglestop : [], cost: 1
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Removed unreachable and leaf rules:
4.97/2.40	
4.97/2.40	   Start location: evalNestedSinglestart
4.97/2.40	
4.97/2.40	      0: evalNestedSinglestart -> evalNestedSingleentryin : [], cost: 1
4.97/2.40	
4.97/2.40	      1: evalNestedSingleentryin -> evalNestedSinglebb5in : A'=0, [], cost: 1
4.97/2.40	
4.97/2.40	      2: evalNestedSinglebb5in -> evalNestedSinglebb2in : C'=A, [ B>=1+A ], cost: 1
4.97/2.40	
4.97/2.40	      4: evalNestedSinglebb2in -> evalNestedSinglebb4in : [ C>=B ], cost: 1
4.97/2.40	
4.97/2.40	      5: evalNestedSinglebb2in -> evalNestedSinglebb3in : [ B>=1+C ], cost: 1
4.97/2.40	
4.97/2.40	      6: evalNestedSinglebb3in -> evalNestedSinglebb1in : [ 0>=1+free ], cost: 1
4.97/2.40	
4.97/2.40	      7: evalNestedSinglebb3in -> evalNestedSinglebb1in : [ free_1>=1 ], cost: 1
4.97/2.40	
4.97/2.40	      8: evalNestedSinglebb3in -> evalNestedSinglebb4in : [], cost: 1
4.97/2.40	
4.97/2.40	      9: evalNestedSinglebb1in -> evalNestedSinglebb2in : C'=1+C, [], cost: 1
4.97/2.40	
4.97/2.40	     10: evalNestedSinglebb4in -> evalNestedSinglebb5in : A'=1+C, [], cost: 1
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Simplified all rules, resulting in:
4.97/2.40	
4.97/2.40	   Start location: evalNestedSinglestart
4.97/2.40	
4.97/2.40	      0: evalNestedSinglestart -> evalNestedSingleentryin : [], cost: 1
4.97/2.40	
4.97/2.40	      1: evalNestedSingleentryin -> evalNestedSinglebb5in : A'=0, [], cost: 1
4.97/2.40	
4.97/2.40	      2: evalNestedSinglebb5in -> evalNestedSinglebb2in : C'=A, [ B>=1+A ], cost: 1
4.97/2.40	
4.97/2.40	      4: evalNestedSinglebb2in -> evalNestedSinglebb4in : [ C>=B ], cost: 1
4.97/2.40	
4.97/2.40	      5: evalNestedSinglebb2in -> evalNestedSinglebb3in : [ B>=1+C ], cost: 1
4.97/2.40	
4.97/2.40	      7: evalNestedSinglebb3in -> evalNestedSinglebb1in : [], cost: 1
4.97/2.40	
4.97/2.40	      8: evalNestedSinglebb3in -> evalNestedSinglebb4in : [], cost: 1
4.97/2.40	
4.97/2.40	      9: evalNestedSinglebb1in -> evalNestedSinglebb2in : C'=1+C, [], cost: 1
4.97/2.40	
4.97/2.40	     10: evalNestedSinglebb4in -> evalNestedSinglebb5in : A'=1+C, [], cost: 1
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	### Simplification by acceleration and chaining ###
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Eliminated locations (on linear paths):
4.97/2.40	
4.97/2.40	   Start location: evalNestedSinglestart
4.97/2.40	
4.97/2.40	     12: evalNestedSinglestart -> evalNestedSinglebb5in : A'=0, [], cost: 2
4.97/2.40	
4.97/2.40	      2: evalNestedSinglebb5in -> evalNestedSinglebb2in : C'=A, [ B>=1+A ], cost: 1
4.97/2.40	
4.97/2.40	      4: evalNestedSinglebb2in -> evalNestedSinglebb4in : [ C>=B ], cost: 1
4.97/2.40	
4.97/2.40	      5: evalNestedSinglebb2in -> evalNestedSinglebb3in : [ B>=1+C ], cost: 1
4.97/2.40	
4.97/2.40	      8: evalNestedSinglebb3in -> evalNestedSinglebb4in : [], cost: 1
4.97/2.40	
4.97/2.40	     13: evalNestedSinglebb3in -> evalNestedSinglebb2in : C'=1+C, [], cost: 2
4.97/2.40	
4.97/2.40	     10: evalNestedSinglebb4in -> evalNestedSinglebb5in : A'=1+C, [], cost: 1
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Eliminated locations (on tree-shaped paths):
4.97/2.40	
4.97/2.40	   Start location: evalNestedSinglestart
4.97/2.40	
4.97/2.40	     12: evalNestedSinglestart -> evalNestedSinglebb5in : A'=0, [], cost: 2
4.97/2.40	
4.97/2.40	      2: evalNestedSinglebb5in -> evalNestedSinglebb2in : C'=A, [ B>=1+A ], cost: 1
4.97/2.40	
4.97/2.40	     15: evalNestedSinglebb2in -> evalNestedSinglebb2in : C'=1+C, [ B>=1+C ], cost: 3
4.97/2.40	
4.97/2.40	     16: evalNestedSinglebb2in -> evalNestedSinglebb5in : A'=1+C, [ C>=B ], cost: 2
4.97/2.40	
4.97/2.40	     17: evalNestedSinglebb2in -> evalNestedSinglebb5in : A'=1+C, [ B>=1+C ], cost: 3
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Accelerating simple loops of location 3.
4.97/2.40	
4.97/2.40	   Accelerating the following rules:
4.97/2.40	
4.97/2.40	     15: evalNestedSinglebb2in -> evalNestedSinglebb2in : C'=1+C, [ B>=1+C ], cost: 3
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	   Accelerated rule 15 with metering function -C+B, yielding the new rule 18.
4.97/2.40	
4.97/2.40	   Removing the simple loops: 15.
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Accelerated all simple loops using metering functions (where possible):
4.97/2.40	
4.97/2.40	   Start location: evalNestedSinglestart
4.97/2.40	
4.97/2.40	     12: evalNestedSinglestart -> evalNestedSinglebb5in : A'=0, [], cost: 2
4.97/2.40	
4.97/2.40	      2: evalNestedSinglebb5in -> evalNestedSinglebb2in : C'=A, [ B>=1+A ], cost: 1
4.97/2.40	
4.97/2.40	     16: evalNestedSinglebb2in -> evalNestedSinglebb5in : A'=1+C, [ C>=B ], cost: 2
4.97/2.40	
4.97/2.40	     17: evalNestedSinglebb2in -> evalNestedSinglebb5in : A'=1+C, [ B>=1+C ], cost: 3
4.97/2.40	
4.97/2.40	     18: evalNestedSinglebb2in -> evalNestedSinglebb2in : C'=B, [ B>=1+C ], cost: -3*C+3*B
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Chained accelerated rules (with incoming rules):
4.97/2.40	
4.97/2.40	   Start location: evalNestedSinglestart
4.97/2.40	
4.97/2.40	     12: evalNestedSinglestart -> evalNestedSinglebb5in : A'=0, [], cost: 2
4.97/2.40	
4.97/2.40	      2: evalNestedSinglebb5in -> evalNestedSinglebb2in : C'=A, [ B>=1+A ], cost: 1
4.97/2.40	
4.97/2.40	     19: evalNestedSinglebb5in -> evalNestedSinglebb2in : C'=B, [ B>=1+A ], cost: 1-3*A+3*B
4.97/2.40	
4.97/2.40	     16: evalNestedSinglebb2in -> evalNestedSinglebb5in : A'=1+C, [ C>=B ], cost: 2
4.97/2.40	
4.97/2.40	     17: evalNestedSinglebb2in -> evalNestedSinglebb5in : A'=1+C, [ B>=1+C ], cost: 3
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Eliminated locations (on tree-shaped paths):
4.97/2.40	
4.97/2.40	   Start location: evalNestedSinglestart
4.97/2.40	
4.97/2.40	     12: evalNestedSinglestart -> evalNestedSinglebb5in : A'=0, [], cost: 2
4.97/2.40	
4.97/2.40	     20: evalNestedSinglebb5in -> evalNestedSinglebb5in : A'=1+A, C'=A, [ B>=1+A ], cost: 4
4.97/2.40	
4.97/2.40	     21: evalNestedSinglebb5in -> evalNestedSinglebb5in : A'=1+B, C'=B, [ B>=1+A ], cost: 3-3*A+3*B
4.97/2.40	
4.97/2.40	     22: evalNestedSinglebb5in -> [10] : [ B>=1+A ], cost: 1-3*A+3*B
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Accelerating simple loops of location 2.
4.97/2.40	
4.97/2.40	   Accelerating the following rules:
4.97/2.40	
4.97/2.40	     20: evalNestedSinglebb5in -> evalNestedSinglebb5in : A'=1+A, C'=A, [ B>=1+A ], cost: 4
4.97/2.40	
4.97/2.40	     21: evalNestedSinglebb5in -> evalNestedSinglebb5in : A'=1+B, C'=B, [ B>=1+A ], cost: 3-3*A+3*B
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	   Accelerated rule 20 with metering function -A+B, yielding the new rule 23.
4.97/2.40	
4.97/2.40	   Found no metering function for rule 21.
4.97/2.40	
4.97/2.40	   Removing the simple loops: 20.
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Accelerated all simple loops using metering functions (where possible):
4.97/2.40	
4.97/2.40	   Start location: evalNestedSinglestart
4.97/2.40	
4.97/2.40	     12: evalNestedSinglestart -> evalNestedSinglebb5in : A'=0, [], cost: 2
4.97/2.40	
4.97/2.40	     21: evalNestedSinglebb5in -> evalNestedSinglebb5in : A'=1+B, C'=B, [ B>=1+A ], cost: 3-3*A+3*B
4.97/2.40	
4.97/2.40	     22: evalNestedSinglebb5in -> [10] : [ B>=1+A ], cost: 1-3*A+3*B
4.97/2.40	
4.97/2.40	     23: evalNestedSinglebb5in -> evalNestedSinglebb5in : A'=B, C'=-1+B, [ B>=1+A ], cost: -4*A+4*B
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Chained accelerated rules (with incoming rules):
4.97/2.40	
4.97/2.40	   Start location: evalNestedSinglestart
4.97/2.40	
4.97/2.40	     12: evalNestedSinglestart -> evalNestedSinglebb5in : A'=0, [], cost: 2
4.97/2.40	
4.97/2.40	     24: evalNestedSinglestart -> evalNestedSinglebb5in : A'=1+B, C'=B, [ B>=1 ], cost: 5+3*B
4.97/2.40	
4.97/2.40	     25: evalNestedSinglestart -> evalNestedSinglebb5in : A'=B, C'=-1+B, [ B>=1 ], cost: 2+4*B
4.97/2.40	
4.97/2.40	     22: evalNestedSinglebb5in -> [10] : [ B>=1+A ], cost: 1-3*A+3*B
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Eliminated locations (on tree-shaped paths):
4.97/2.40	
4.97/2.40	   Start location: evalNestedSinglestart
4.97/2.40	
4.97/2.40	     26: evalNestedSinglestart -> [10] : A'=0, [ B>=1 ], cost: 3+3*B
4.97/2.40	
4.97/2.40	     27: evalNestedSinglestart -> [12] : [ B>=1 ], cost: 5+3*B
4.97/2.40	
4.97/2.40	     28: evalNestedSinglestart -> [12] : [ B>=1 ], cost: 2+4*B
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	### Computing asymptotic complexity ###
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Fully simplified ITS problem
4.97/2.40	
4.97/2.40	   Start location: evalNestedSinglestart
4.97/2.40	
4.97/2.40	     27: evalNestedSinglestart -> [12] : [ B>=1 ], cost: 5+3*B
4.97/2.40	
4.97/2.40	     28: evalNestedSinglestart -> [12] : [ B>=1 ], cost: 2+4*B
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Computing asymptotic complexity for rule 27
4.97/2.40	
4.97/2.40	   Solved the limit problem by the following transformations:
4.97/2.40	
4.97/2.40	      Created initial limit problem:
4.97/2.40	
4.97/2.40	      5+3*B (+), B (+/+!) [not solved]
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	      removing all constraints (solved by SMT)
4.97/2.40	
4.97/2.40	      resulting limit problem: [solved]
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	      applying transformation rule (C) using substitution {B==n}
4.97/2.40	
4.97/2.40	      resulting limit problem:
4.97/2.40	
4.97/2.40	      [solved]
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	   Solution:
4.97/2.40	
4.97/2.40	      B / n
4.97/2.40	
4.97/2.40	   Resulting cost 5+3*n has complexity: Poly(n^1)
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Found new complexity Poly(n^1).
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	Obtained the following overall complexity (w.r.t. the length of the input n):
4.97/2.40	
4.97/2.40	   Complexity:  Poly(n^1)
4.97/2.40	
4.97/2.40	   Cpx degree:  1
4.97/2.40	
4.97/2.40	   Solved cost: 5+3*n
4.97/2.40	
4.97/2.40	   Rule cost:   5+3*B
4.97/2.40	
4.97/2.40	   Rule guard:  [ B>=1 ]
4.97/2.40	
4.97/2.40	
4.97/2.40	
4.97/2.40	WORST_CASE(Omega(n^1),?)
4.97/2.40	
4.97/2.40	
4.97/2.40	----------------------------------------
4.97/2.40	
4.97/2.40	(4)
4.97/2.40	BOUNDS(n^1, INF)
4.97/2.42	EOF