4.39/2.09	WORST_CASE(Omega(n^2), O(n^2))
4.39/2.10	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
4.39/2.10	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.39/2.10	
4.39/2.10	
4.39/2.10	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).
4.39/2.10	
4.39/2.10	(0) CpxIntTrs
4.39/2.10	(1) Koat Proof [FINISHED, 18 ms]
4.39/2.10	(2) BOUNDS(1, n^2)
4.39/2.10	(3) Loat Proof [FINISHED, 424 ms]
4.39/2.10	(4) BOUNDS(n^2, INF)
4.39/2.10	
4.39/2.10	
4.39/2.10	----------------------------------------
4.39/2.10	
4.39/2.10	(0)
4.39/2.10	Obligation:
4.39/2.10	Complexity Int TRS consisting of the following rules:
4.39/2.10	evalwhile2start(A, B, C) -> Com_1(evalwhile2entryin(A, B, C)) :|: TRUE
4.39/2.10	evalwhile2entryin(A, B, C) -> Com_1(evalwhile2bb4in(B, B, C)) :|: TRUE
4.39/2.10	evalwhile2bb4in(A, B, C) -> Com_1(evalwhile2bb2in(A, B, B)) :|: A >= 1
4.39/2.10	evalwhile2bb4in(A, B, C) -> Com_1(evalwhile2returnin(A, B, C)) :|: 0 >= A
4.39/2.10	evalwhile2bb2in(A, B, C) -> Com_1(evalwhile2bb1in(A, B, C)) :|: C >= 1
4.39/2.10	evalwhile2bb2in(A, B, C) -> Com_1(evalwhile2bb3in(A, B, C)) :|: 0 >= C
4.39/2.10	evalwhile2bb1in(A, B, C) -> Com_1(evalwhile2bb2in(A, B, C - 1)) :|: TRUE
4.39/2.10	evalwhile2bb3in(A, B, C) -> Com_1(evalwhile2bb4in(A - 1, B, C)) :|: TRUE
4.39/2.10	evalwhile2returnin(A, B, C) -> Com_1(evalwhile2stop(A, B, C)) :|: TRUE
4.39/2.10	
4.39/2.10	The start-symbols are:[evalwhile2start_3]
4.39/2.10	
4.39/2.10	
4.39/2.10	----------------------------------------
4.39/2.10	
4.39/2.10	(1) Koat Proof (FINISHED)
4.39/2.10	YES(?, 9*ar_1 + 2*ar_1^2 + 13)
4.39/2.10	
4.39/2.10	
4.39/2.10	
4.39/2.10	Initial complexity problem:
4.39/2.10	
4.39/2.10	1:	T:
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2entryin(ar_0, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2entryin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_1, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2returnin(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0 - 1, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2returnin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.39/2.10	
4.39/2.10		start location:	koat_start
4.39/2.10	
4.39/2.10		leaf cost:	0
4.39/2.10	
4.39/2.10	
4.39/2.10	
4.39/2.10	Repeatedly propagating knowledge in problem 1 produces the following problem:
4.39/2.10	
4.39/2.10	2:	T:
4.39/2.10	
4.39/2.10			(Comp: 1, Cost: 1)    evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2entryin(ar_0, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: 1, Cost: 1)    evalwhile2entryin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_1, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2returnin(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0 - 1, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2returnin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.39/2.10	
4.39/2.10		start location:	koat_start
4.39/2.10	
4.39/2.10		leaf cost:	0
4.39/2.10	
4.39/2.10	
4.39/2.10	
4.39/2.10	A polynomial rank function with
4.39/2.10	
4.39/2.10		Pol(evalwhile2start) = 2
4.39/2.10	
4.39/2.10		Pol(evalwhile2entryin) = 2
4.39/2.10	
4.39/2.10		Pol(evalwhile2bb4in) = 2
4.39/2.10	
4.39/2.10		Pol(evalwhile2bb2in) = 2
4.39/2.10	
4.39/2.10		Pol(evalwhile2returnin) = 1
4.39/2.10	
4.39/2.10		Pol(evalwhile2bb1in) = 2
4.39/2.10	
4.39/2.10		Pol(evalwhile2bb3in) = 2
4.39/2.10	
4.39/2.10		Pol(evalwhile2stop) = 0
4.39/2.10	
4.39/2.10		Pol(koat_start) = 2
4.39/2.10	
4.39/2.10	orients all transitions weakly and the transitions
4.39/2.10	
4.39/2.10		evalwhile2returnin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))
4.39/2.10	
4.39/2.10		evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2returnin(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
4.39/2.10	
4.39/2.10	strictly and produces the following problem:
4.39/2.10	
4.39/2.10	3:	T:
4.39/2.10	
4.39/2.10			(Comp: 1, Cost: 1)    evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2entryin(ar_0, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: 1, Cost: 1)    evalwhile2entryin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_1, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]
4.39/2.10	
4.39/2.10			(Comp: 2, Cost: 1)    evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2returnin(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)    evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0 - 1, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: 2, Cost: 1)    evalwhile2returnin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.39/2.10	
4.39/2.10		start location:	koat_start
4.39/2.10	
4.39/2.10		leaf cost:	0
4.39/2.10	
4.39/2.10	
4.39/2.10	
4.39/2.10	A polynomial rank function with
4.39/2.10	
4.39/2.10		Pol(evalwhile2start) = V_2 + 1
4.39/2.10	
4.39/2.10		Pol(evalwhile2entryin) = V_2 + 1
4.39/2.10	
4.39/2.10		Pol(evalwhile2bb4in) = V_1 + 1
4.39/2.10	
4.39/2.10		Pol(evalwhile2bb2in) = V_1
4.39/2.10	
4.39/2.10		Pol(evalwhile2returnin) = V_1
4.39/2.10	
4.39/2.10		Pol(evalwhile2bb1in) = V_1
4.39/2.10	
4.39/2.10		Pol(evalwhile2bb3in) = V_1
4.39/2.10	
4.39/2.10		Pol(evalwhile2stop) = V_1
4.39/2.10	
4.39/2.10		Pol(koat_start) = V_2 + 1
4.39/2.10	
4.39/2.10	orients all transitions weakly and the transition
4.39/2.10	
4.39/2.10		evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]
4.39/2.10	
4.39/2.10	strictly and produces the following problem:
4.39/2.10	
4.39/2.10	4:	T:
4.39/2.10	
4.39/2.10			(Comp: 1, Cost: 1)           evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2entryin(ar_0, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: 1, Cost: 1)           evalwhile2entryin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_1, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: ar_1 + 1, Cost: 1)    evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]
4.39/2.10	
4.39/2.10			(Comp: 2, Cost: 1)           evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2returnin(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)           evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)           evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)           evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)           evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0 - 1, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: 2, Cost: 1)           evalwhile2returnin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.39/2.10	
4.39/2.10		start location:	koat_start
4.39/2.10	
4.39/2.10		leaf cost:	0
4.39/2.10	
4.39/2.10	
4.39/2.10	
4.39/2.10	A polynomial rank function with
4.39/2.10	
4.39/2.10		Pol(evalwhile2bb3in) = 1
4.39/2.10	
4.39/2.10		Pol(evalwhile2bb4in) = 0
4.39/2.10	
4.39/2.10		Pol(evalwhile2bb2in) = 2
4.39/2.10	
4.39/2.10		Pol(evalwhile2bb1in) = 2
4.39/2.10	
4.39/2.10	and size complexities
4.39/2.10	
4.39/2.10		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
4.39/2.10	
4.39/2.10		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
4.39/2.10	
4.39/2.10		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
4.39/2.10	
4.39/2.10		S("evalwhile2returnin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-0) = ?
4.39/2.10	
4.39/2.10		S("evalwhile2returnin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-1) = ar_1
4.39/2.10	
4.39/2.10		S("evalwhile2returnin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-2) = ?
4.39/2.10	
4.39/2.10		S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0 - 1, ar_1, ar_2))", 0-0) = ?
4.39/2.10	
4.39/2.10		S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0 - 1, ar_1, ar_2))", 0-1) = ar_1
4.39/2.10	
4.39/2.10		S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0 - 1, ar_1, ar_2))", 0-2) = ?
4.39/2.10	
4.39/2.10		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-0) = ?
4.39/2.10	
4.39/2.10		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-1) = ar_1
4.39/2.10	
4.39/2.10		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-2) = ?
4.39/2.10	
4.39/2.10		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-0) = ?
4.39/2.10	
4.39/2.10		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-1) = ar_1
4.39/2.10	
4.39/2.10		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-2) = ?
4.39/2.10	
4.39/2.10		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-0) = ?
4.39/2.10	
4.39/2.10		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-1) = ar_1
4.39/2.10	
4.39/2.10		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-2) = ?
4.39/2.10	
4.39/2.10		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2returnin(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-0) = ?
4.39/2.10	
4.39/2.10		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2returnin(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-1) = ar_1
4.39/2.10	
4.39/2.10		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2returnin(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-2) = ?
4.39/2.10	
4.39/2.10		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-0) = ?
4.39/2.10	
4.39/2.10		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-1) = ar_1
4.39/2.10	
4.39/2.10		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-2) = ar_1
4.39/2.10	
4.39/2.10		S("evalwhile2entryin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_1, ar_1, ar_2))", 0-0) = ar_1
4.39/2.10	
4.39/2.10		S("evalwhile2entryin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_1, ar_1, ar_2))", 0-1) = ar_1
4.39/2.10	
4.39/2.10		S("evalwhile2entryin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_1, ar_1, ar_2))", 0-2) = ar_2
4.39/2.10	
4.39/2.10		S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2entryin(ar_0, ar_1, ar_2))", 0-0) = ar_0
4.39/2.10	
4.39/2.10		S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2entryin(ar_0, ar_1, ar_2))", 0-1) = ar_1
4.39/2.10	
4.39/2.10		S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2entryin(ar_0, ar_1, ar_2))", 0-2) = ar_2
4.39/2.10	
4.39/2.10	orients the transitions
4.39/2.10	
4.39/2.10		evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0 - 1, ar_1, ar_2))
4.39/2.10	
4.39/2.10		evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.39/2.10	
4.39/2.10		evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.39/2.10	
4.39/2.10		evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
4.39/2.10	
4.39/2.10	weakly and the transitions
4.39/2.10	
4.39/2.10		evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0 - 1, ar_1, ar_2))
4.39/2.10	
4.39/2.10		evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.39/2.10	
4.39/2.10	strictly and produces the following problem:
4.39/2.10	
4.39/2.10	5:	T:
4.39/2.10	
4.39/2.10			(Comp: 1, Cost: 1)             evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2entryin(ar_0, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: 1, Cost: 1)             evalwhile2entryin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_1, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: ar_1 + 1, Cost: 1)      evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]
4.39/2.10	
4.39/2.10			(Comp: 2, Cost: 1)             evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2returnin(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)             evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.39/2.10	
4.39/2.10			(Comp: 2*ar_1 + 2, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.39/2.10	
4.39/2.10			(Comp: ?, Cost: 1)             evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
4.39/2.10	
4.39/2.10			(Comp: 2*ar_1 + 2, Cost: 1)    evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0 - 1, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: 2, Cost: 1)             evalwhile2returnin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))
4.39/2.10	
4.39/2.10			(Comp: 1, Cost: 0)             koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.39/2.10	
4.39/2.10		start location:	koat_start
4.39/2.11	
4.39/2.11		leaf cost:	0
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	A polynomial rank function with
4.39/2.11	
4.39/2.11		Pol(evalwhile2bb2in) = V_3 + 1
4.39/2.11	
4.39/2.11		Pol(evalwhile2bb1in) = V_3
4.39/2.11	
4.39/2.11	and size complexities
4.39/2.11	
4.39/2.11		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
4.39/2.11	
4.39/2.11		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
4.39/2.11	
4.39/2.11		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
4.39/2.11	
4.39/2.11		S("evalwhile2returnin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-0) = 3*ar_1 + 162
4.39/2.11	
4.39/2.11		S("evalwhile2returnin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-1) = ar_1
4.39/2.11	
4.39/2.11		S("evalwhile2returnin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-2) = ?
4.39/2.11	
4.39/2.11		S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0 - 1, ar_1, ar_2))", 0-0) = 3*ar_1 + 18
4.39/2.11	
4.39/2.11		S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0 - 1, ar_1, ar_2))", 0-1) = ar_1
4.39/2.11	
4.39/2.11		S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0 - 1, ar_1, ar_2))", 0-2) = ?
4.39/2.11	
4.39/2.11		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-0) = 3*ar_1 + 18
4.39/2.11	
4.39/2.11		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-1) = ar_1
4.39/2.11	
4.39/2.11		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-2) = ?
4.39/2.11	
4.39/2.11		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-0) = 3*ar_1 + 18
4.39/2.11	
4.39/2.11		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-1) = ar_1
4.39/2.11	
4.39/2.11		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-2) = ?
4.39/2.11	
4.39/2.11		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-0) = 3*ar_1 + 18
4.39/2.11	
4.39/2.11		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-1) = ar_1
4.39/2.11	
4.39/2.11		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-2) = ?
4.39/2.11	
4.39/2.11		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2returnin(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-0) = 3*ar_1 + 54
4.39/2.11	
4.39/2.11		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2returnin(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-1) = ar_1
4.39/2.11	
4.39/2.11		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2returnin(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-2) = ?
4.39/2.11	
4.39/2.11		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-0) = 3*ar_1 + 18
4.39/2.11	
4.39/2.11		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-1) = ar_1
4.39/2.11	
4.39/2.11		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-2) = ar_1
4.39/2.11	
4.39/2.11		S("evalwhile2entryin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_1, ar_1, ar_2))", 0-0) = ar_1
4.39/2.11	
4.39/2.11		S("evalwhile2entryin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_1, ar_1, ar_2))", 0-1) = ar_1
4.39/2.11	
4.39/2.11		S("evalwhile2entryin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_1, ar_1, ar_2))", 0-2) = ar_2
4.39/2.11	
4.39/2.11		S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2entryin(ar_0, ar_1, ar_2))", 0-0) = ar_0
4.39/2.11	
4.39/2.11		S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2entryin(ar_0, ar_1, ar_2))", 0-1) = ar_1
4.39/2.11	
4.39/2.11		S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2entryin(ar_0, ar_1, ar_2))", 0-2) = ar_2
4.39/2.11	
4.39/2.11	orients the transitions
4.39/2.11	
4.39/2.11		evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.39/2.11	
4.39/2.11		evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
4.39/2.11	
4.39/2.11	weakly and the transition
4.39/2.11	
4.39/2.11		evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.39/2.11	
4.39/2.11	strictly and produces the following problem:
4.39/2.11	
4.39/2.11	6:	T:
4.39/2.11	
4.39/2.11			(Comp: 1, Cost: 1)                      evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2entryin(ar_0, ar_1, ar_2))
4.39/2.11	
4.39/2.11			(Comp: 1, Cost: 1)                      evalwhile2entryin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_1, ar_1, ar_2))
4.39/2.11	
4.39/2.11			(Comp: ar_1 + 1, Cost: 1)               evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]
4.39/2.11	
4.39/2.11			(Comp: 2, Cost: 1)                      evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2returnin(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
4.39/2.11	
4.39/2.11			(Comp: ar_1^2 + 2*ar_1 + 1, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.39/2.11	
4.39/2.11			(Comp: 2*ar_1 + 2, Cost: 1)             evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.39/2.11	
4.39/2.11			(Comp: ?, Cost: 1)                      evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
4.39/2.11	
4.39/2.11			(Comp: 2*ar_1 + 2, Cost: 1)             evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0 - 1, ar_1, ar_2))
4.39/2.11	
4.39/2.11			(Comp: 2, Cost: 1)                      evalwhile2returnin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))
4.39/2.11	
4.39/2.11			(Comp: 1, Cost: 0)                      koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.39/2.11	
4.39/2.11		start location:	koat_start
4.39/2.11	
4.39/2.11		leaf cost:	0
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Repeatedly propagating knowledge in problem 6 produces the following problem:
4.39/2.11	
4.39/2.11	7:	T:
4.39/2.11	
4.39/2.11			(Comp: 1, Cost: 1)                      evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2entryin(ar_0, ar_1, ar_2))
4.39/2.11	
4.39/2.11			(Comp: 1, Cost: 1)                      evalwhile2entryin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_1, ar_1, ar_2))
4.39/2.11	
4.39/2.11			(Comp: ar_1 + 1, Cost: 1)               evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]
4.39/2.11	
4.39/2.11			(Comp: 2, Cost: 1)                      evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2returnin(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
4.39/2.11	
4.39/2.11			(Comp: ar_1^2 + 2*ar_1 + 1, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.39/2.11	
4.39/2.11			(Comp: 2*ar_1 + 2, Cost: 1)             evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.39/2.11	
4.39/2.11			(Comp: ar_1^2 + 2*ar_1 + 1, Cost: 1)    evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
4.39/2.11	
4.39/2.11			(Comp: 2*ar_1 + 2, Cost: 1)             evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0 - 1, ar_1, ar_2))
4.39/2.11	
4.39/2.11			(Comp: 2, Cost: 1)                      evalwhile2returnin(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))
4.39/2.11	
4.39/2.11			(Comp: 1, Cost: 0)                      koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.39/2.11	
4.39/2.11		start location:	koat_start
4.39/2.11	
4.39/2.11		leaf cost:	0
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Complexity upper bound 9*ar_1 + 2*ar_1^2 + 13
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Time: 0.089 sec (SMT: 0.078 sec)
4.39/2.11	
4.39/2.11	
4.39/2.11	----------------------------------------
4.39/2.11	
4.39/2.11	(2)
4.39/2.11	BOUNDS(1, n^2)
4.39/2.11	
4.39/2.11	----------------------------------------
4.39/2.11	
4.39/2.11	(3) Loat Proof (FINISHED)
4.39/2.11	
4.39/2.11	
4.39/2.11	### Pre-processing the ITS problem ###
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Initial linear ITS problem
4.39/2.11	
4.39/2.11	   Start location: evalwhile2start
4.39/2.11	
4.39/2.11	      0: evalwhile2start -> evalwhile2entryin : [], cost: 1
4.39/2.11	
4.39/2.11	      1: evalwhile2entryin -> evalwhile2bb4in : A'=B, [], cost: 1
4.39/2.11	
4.39/2.11	      2: evalwhile2bb4in -> evalwhile2bb2in : C'=B, [ A>=1 ], cost: 1
4.39/2.11	
4.39/2.11	      3: evalwhile2bb4in -> evalwhile2returnin : [ 0>=A ], cost: 1
4.39/2.11	
4.39/2.11	      4: evalwhile2bb2in -> evalwhile2bb1in : [ C>=1 ], cost: 1
4.39/2.11	
4.39/2.11	      5: evalwhile2bb2in -> evalwhile2bb3in : [ 0>=C ], cost: 1
4.39/2.11	
4.39/2.11	      6: evalwhile2bb1in -> evalwhile2bb2in : C'=-1+C, [], cost: 1
4.39/2.11	
4.39/2.11	      7: evalwhile2bb3in -> evalwhile2bb4in : A'=-1+A, [], cost: 1
4.39/2.11	
4.39/2.11	      8: evalwhile2returnin -> evalwhile2stop : [], cost: 1
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Removed unreachable and leaf rules:
4.39/2.11	
4.39/2.11	   Start location: evalwhile2start
4.39/2.11	
4.39/2.11	      0: evalwhile2start -> evalwhile2entryin : [], cost: 1
4.39/2.11	
4.39/2.11	      1: evalwhile2entryin -> evalwhile2bb4in : A'=B, [], cost: 1
4.39/2.11	
4.39/2.11	      2: evalwhile2bb4in -> evalwhile2bb2in : C'=B, [ A>=1 ], cost: 1
4.39/2.11	
4.39/2.11	      4: evalwhile2bb2in -> evalwhile2bb1in : [ C>=1 ], cost: 1
4.39/2.11	
4.39/2.11	      5: evalwhile2bb2in -> evalwhile2bb3in : [ 0>=C ], cost: 1
4.39/2.11	
4.39/2.11	      6: evalwhile2bb1in -> evalwhile2bb2in : C'=-1+C, [], cost: 1
4.39/2.11	
4.39/2.11	      7: evalwhile2bb3in -> evalwhile2bb4in : A'=-1+A, [], cost: 1
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	### Simplification by acceleration and chaining ###
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Eliminated locations (on linear paths):
4.39/2.11	
4.39/2.11	   Start location: evalwhile2start
4.39/2.11	
4.39/2.11	      9: evalwhile2start -> evalwhile2bb4in : A'=B, [], cost: 2
4.39/2.11	
4.39/2.11	      2: evalwhile2bb4in -> evalwhile2bb2in : C'=B, [ A>=1 ], cost: 1
4.39/2.11	
4.39/2.11	     10: evalwhile2bb2in -> evalwhile2bb2in : C'=-1+C, [ C>=1 ], cost: 2
4.39/2.11	
4.39/2.11	     11: evalwhile2bb2in -> evalwhile2bb4in : A'=-1+A, [ 0>=C ], cost: 2
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Accelerating simple loops of location 3.
4.39/2.11	
4.39/2.11	   Accelerating the following rules:
4.39/2.11	
4.39/2.11	     10: evalwhile2bb2in -> evalwhile2bb2in : C'=-1+C, [ C>=1 ], cost: 2
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	   Accelerated rule 10 with metering function C, yielding the new rule 12.
4.39/2.11	
4.39/2.11	   Removing the simple loops: 10.
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Accelerated all simple loops using metering functions (where possible):
4.39/2.11	
4.39/2.11	   Start location: evalwhile2start
4.39/2.11	
4.39/2.11	      9: evalwhile2start -> evalwhile2bb4in : A'=B, [], cost: 2
4.39/2.11	
4.39/2.11	      2: evalwhile2bb4in -> evalwhile2bb2in : C'=B, [ A>=1 ], cost: 1
4.39/2.11	
4.39/2.11	     11: evalwhile2bb2in -> evalwhile2bb4in : A'=-1+A, [ 0>=C ], cost: 2
4.39/2.11	
4.39/2.11	     12: evalwhile2bb2in -> evalwhile2bb2in : C'=0, [ C>=1 ], cost: 2*C
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Chained accelerated rules (with incoming rules):
4.39/2.11	
4.39/2.11	   Start location: evalwhile2start
4.39/2.11	
4.39/2.11	      9: evalwhile2start -> evalwhile2bb4in : A'=B, [], cost: 2
4.39/2.11	
4.39/2.11	      2: evalwhile2bb4in -> evalwhile2bb2in : C'=B, [ A>=1 ], cost: 1
4.39/2.11	
4.39/2.11	     13: evalwhile2bb4in -> evalwhile2bb2in : C'=0, [ A>=1 && B>=1 ], cost: 1+2*B
4.39/2.11	
4.39/2.11	     11: evalwhile2bb2in -> evalwhile2bb4in : A'=-1+A, [ 0>=C ], cost: 2
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Eliminated locations (on tree-shaped paths):
4.39/2.11	
4.39/2.11	   Start location: evalwhile2start
4.39/2.11	
4.39/2.11	      9: evalwhile2start -> evalwhile2bb4in : A'=B, [], cost: 2
4.39/2.11	
4.39/2.11	     14: evalwhile2bb4in -> evalwhile2bb4in : A'=-1+A, C'=B, [ A>=1 && 0>=B ], cost: 3
4.39/2.11	
4.39/2.11	     15: evalwhile2bb4in -> evalwhile2bb4in : A'=-1+A, C'=0, [ A>=1 && B>=1 ], cost: 3+2*B
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Accelerating simple loops of location 2.
4.39/2.11	
4.39/2.11	   Accelerating the following rules:
4.39/2.11	
4.39/2.11	     14: evalwhile2bb4in -> evalwhile2bb4in : A'=-1+A, C'=B, [ A>=1 && 0>=B ], cost: 3
4.39/2.11	
4.39/2.11	     15: evalwhile2bb4in -> evalwhile2bb4in : A'=-1+A, C'=0, [ A>=1 && B>=1 ], cost: 3+2*B
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	   Accelerated rule 14 with metering function A, yielding the new rule 16.
4.39/2.11	
4.39/2.11	   Accelerated rule 15 with metering function A, yielding the new rule 17.
4.39/2.11	
4.39/2.11	   Removing the simple loops: 14 15.
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Accelerated all simple loops using metering functions (where possible):
4.39/2.11	
4.39/2.11	   Start location: evalwhile2start
4.39/2.11	
4.39/2.11	      9: evalwhile2start -> evalwhile2bb4in : A'=B, [], cost: 2
4.39/2.11	
4.39/2.11	     16: evalwhile2bb4in -> evalwhile2bb4in : A'=0, C'=B, [ A>=1 && 0>=B ], cost: 3*A
4.39/2.11	
4.39/2.11	     17: evalwhile2bb4in -> evalwhile2bb4in : A'=0, C'=0, [ A>=1 && B>=1 ], cost: 3*A+2*A*B
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Chained accelerated rules (with incoming rules):
4.39/2.11	
4.39/2.11	   Start location: evalwhile2start
4.39/2.11	
4.39/2.11	      9: evalwhile2start -> evalwhile2bb4in : A'=B, [], cost: 2
4.39/2.11	
4.39/2.11	     18: evalwhile2start -> evalwhile2bb4in : A'=0, C'=0, [ B>=1 ], cost: 2+2*B^2+3*B
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Removed unreachable locations (and leaf rules with constant cost):
4.39/2.11	
4.39/2.11	   Start location: evalwhile2start
4.39/2.11	
4.39/2.11	     18: evalwhile2start -> evalwhile2bb4in : A'=0, C'=0, [ B>=1 ], cost: 2+2*B^2+3*B
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	### Computing asymptotic complexity ###
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Fully simplified ITS problem
4.39/2.11	
4.39/2.11	   Start location: evalwhile2start
4.39/2.11	
4.39/2.11	     18: evalwhile2start -> evalwhile2bb4in : A'=0, C'=0, [ B>=1 ], cost: 2+2*B^2+3*B
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Computing asymptotic complexity for rule 18
4.39/2.11	
4.39/2.11	   Solved the limit problem by the following transformations:
4.39/2.11	
4.39/2.11	      Created initial limit problem:
4.39/2.11	
4.39/2.11	      2+2*B^2+3*B (+), B (+/+!) [not solved]
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	      removing all constraints (solved by SMT)
4.39/2.11	
4.39/2.11	      resulting limit problem: [solved]
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	      applying transformation rule (C) using substitution {B==n}
4.39/2.11	
4.39/2.11	      resulting limit problem:
4.39/2.11	
4.39/2.11	      [solved]
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	   Solution:
4.39/2.11	
4.39/2.11	      B / n
4.39/2.11	
4.39/2.11	   Resulting cost 2+2*n^2+3*n has complexity: Poly(n^2)
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Found new complexity Poly(n^2).
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	Obtained the following overall complexity (w.r.t. the length of the input n):
4.39/2.11	
4.39/2.11	   Complexity:  Poly(n^2)
4.39/2.11	
4.39/2.11	   Cpx degree:  2
4.39/2.11	
4.39/2.11	   Solved cost: 2+2*n^2+3*n
4.39/2.11	
4.39/2.11	   Rule cost:   2+2*B^2+3*B
4.39/2.11	
4.39/2.11	   Rule guard:  [ B>=1 ]
4.39/2.11	
4.39/2.11	
4.39/2.11	
4.39/2.11	WORST_CASE(Omega(n^2),?)
4.39/2.11	
4.39/2.11	
4.39/2.11	----------------------------------------
4.39/2.11	
4.39/2.11	(4)
4.39/2.11	BOUNDS(n^2, INF)
4.46/2.12	EOF