4.62/2.22	WORST_CASE(Omega(n^2), O(n^2))
4.62/2.23	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
4.62/2.23	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.62/2.23	
4.62/2.23	
4.62/2.23	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).
4.62/2.23	
4.62/2.23	(0) CpxIntTrs
4.62/2.23	(1) Koat Proof [FINISHED, 22 ms]
4.62/2.23	(2) BOUNDS(1, n^2)
4.62/2.23	(3) Loat Proof [FINISHED, 539 ms]
4.62/2.23	(4) BOUNDS(n^2, INF)
4.62/2.23	
4.62/2.23	
4.62/2.23	----------------------------------------
4.62/2.23	
4.62/2.23	(0)
4.62/2.23	Obligation:
4.62/2.23	Complexity Int TRS consisting of the following rules:
4.62/2.23	evalfstart(A, B, C) -> Com_1(evalfentryin(A, B, C)) :|: TRUE
4.62/2.23	evalfentryin(A, B, C) -> Com_1(evalfbb4in(B, A, C)) :|: TRUE
4.62/2.23	evalfbb4in(A, B, C) -> Com_1(evalfbb2in(A, B, A)) :|: B >= 1
4.62/2.23	evalfbb4in(A, B, C) -> Com_1(evalfreturnin(A, B, C)) :|: 0 >= B
4.62/2.23	evalfbb2in(A, B, C) -> Com_1(evalfbb1in(A, B, C)) :|: C >= 1
4.62/2.23	evalfbb2in(A, B, C) -> Com_1(evalfbb3in(A, B, C)) :|: 0 >= C
4.62/2.23	evalfbb1in(A, B, C) -> Com_1(evalfbb2in(A, B, C - 1)) :|: TRUE
4.62/2.23	evalfbb3in(A, B, C) -> Com_1(evalfbb4in(A, B - 1, C)) :|: TRUE
4.62/2.23	evalfreturnin(A, B, C) -> Com_1(evalfstop(A, B, C)) :|: TRUE
4.62/2.23	
4.62/2.23	The start-symbols are:[evalfstart_3]
4.62/2.23	
4.62/2.23	
4.62/2.23	----------------------------------------
4.62/2.23	
4.62/2.23	(1) Koat Proof (FINISHED)
4.62/2.23	YES(?, 7*ar_0 + 2*ar_0*ar_1 + 2*ar_1 + 13)
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	Initial complexity problem:
4.62/2.23	
4.62/2.23	1:	T:
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_1, ar_0, ar_2))
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ ar_1 >= 1 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 - 1))
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1 - 1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.62/2.23	
4.62/2.23		start location:	koat_start
4.62/2.23	
4.62/2.23		leaf cost:	0
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	Repeatedly propagating knowledge in problem 1 produces the following problem:
4.62/2.23	
4.62/2.23	2:	T:
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 1)    evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_1, ar_0, ar_2))
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ ar_1 >= 1 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 - 1))
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1 - 1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.62/2.23	
4.62/2.23		start location:	koat_start
4.62/2.23	
4.62/2.23		leaf cost:	0
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	A polynomial rank function with
4.62/2.23	
4.62/2.23		Pol(evalfstart) = 2
4.62/2.23	
4.62/2.23		Pol(evalfentryin) = 2
4.62/2.23	
4.62/2.23		Pol(evalfbb4in) = 2
4.62/2.23	
4.62/2.23		Pol(evalfbb2in) = 2
4.62/2.23	
4.62/2.23		Pol(evalfreturnin) = 1
4.62/2.23	
4.62/2.23		Pol(evalfbb1in) = 2
4.62/2.23	
4.62/2.23		Pol(evalfbb3in) = 2
4.62/2.23	
4.62/2.23		Pol(evalfstop) = 0
4.62/2.23	
4.62/2.23		Pol(koat_start) = 2
4.62/2.23	
4.62/2.23	orients all transitions weakly and the transitions
4.62/2.23	
4.62/2.23		evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
4.62/2.23	
4.62/2.23		evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ]
4.62/2.23	
4.62/2.23	strictly and produces the following problem:
4.62/2.23	
4.62/2.23	3:	T:
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 1)    evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_1, ar_0, ar_2))
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ ar_1 >= 1 ]
4.62/2.23	
4.62/2.23			(Comp: 2, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 - 1))
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1 - 1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 2, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.62/2.23	
4.62/2.23		start location:	koat_start
4.62/2.23	
4.62/2.23		leaf cost:	0
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	A polynomial rank function with
4.62/2.23	
4.62/2.23		Pol(evalfstart) = V_1 + 1
4.62/2.23	
4.62/2.23		Pol(evalfentryin) = V_1 + 1
4.62/2.23	
4.62/2.23		Pol(evalfbb4in) = V_2 + 1
4.62/2.23	
4.62/2.23		Pol(evalfbb2in) = V_2
4.62/2.23	
4.62/2.23		Pol(evalfreturnin) = V_2
4.62/2.23	
4.62/2.23		Pol(evalfbb1in) = V_2
4.62/2.23	
4.62/2.23		Pol(evalfbb3in) = V_2
4.62/2.23	
4.62/2.23		Pol(evalfstop) = V_2
4.62/2.23	
4.62/2.23		Pol(koat_start) = V_1 + 1
4.62/2.23	
4.62/2.23	orients all transitions weakly and the transition
4.62/2.23	
4.62/2.23		evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ ar_1 >= 1 ]
4.62/2.23	
4.62/2.23	strictly and produces the following problem:
4.62/2.23	
4.62/2.23	4:	T:
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 1)           evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 1)           evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_1, ar_0, ar_2))
4.62/2.23	
4.62/2.23			(Comp: ar_0 + 1, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ ar_1 >= 1 ]
4.62/2.23	
4.62/2.23			(Comp: 2, Cost: 1)           evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)           evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)           evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)           evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 - 1))
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)           evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1 - 1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 2, Cost: 1)           evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.62/2.23	
4.62/2.23		start location:	koat_start
4.62/2.23	
4.62/2.23		leaf cost:	0
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	A polynomial rank function with
4.62/2.23	
4.62/2.23		Pol(evalfbb3in) = 1
4.62/2.23	
4.62/2.23		Pol(evalfbb4in) = 0
4.62/2.23	
4.62/2.23		Pol(evalfbb2in) = 2
4.62/2.23	
4.62/2.23		Pol(evalfbb1in) = 2
4.62/2.23	
4.62/2.23	and size complexities
4.62/2.23	
4.62/2.23		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
4.62/2.23	
4.62/2.23		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
4.62/2.23	
4.62/2.23		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
4.62/2.23	
4.62/2.23		S("evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))", 0-0) = ar_1
4.62/2.23	
4.62/2.23		S("evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))", 0-1) = ?
4.62/2.23	
4.62/2.23		S("evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))", 0-2) = ?
4.62/2.23	
4.62/2.23		S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1 - 1, ar_2))", 0-0) = ar_1
4.62/2.23	
4.62/2.23		S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1 - 1, ar_2))", 0-1) = ?
4.62/2.23	
4.62/2.23		S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1 - 1, ar_2))", 0-2) = ?
4.62/2.23	
4.62/2.23		S("evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 - 1))", 0-0) = ar_1
4.62/2.23	
4.62/2.23		S("evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 - 1))", 0-1) = ?
4.62/2.23	
4.62/2.23		S("evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 - 1))", 0-2) = ?
4.62/2.23	
4.62/2.23		S("evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-0) = ar_1
4.62/2.23	
4.62/2.23		S("evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-1) = ?
4.62/2.23	
4.62/2.23		S("evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-2) = ?
4.62/2.23	
4.62/2.23		S("evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-0) = ar_1
4.62/2.23	
4.62/2.23		S("evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-1) = ?
4.62/2.23	
4.62/2.23		S("evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-2) = ?
4.62/2.23	
4.62/2.23		S("evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ]", 0-0) = ar_1
4.62/2.23	
4.62/2.23		S("evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ]", 0-1) = ?
4.62/2.23	
4.62/2.23		S("evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ]", 0-2) = ?
4.62/2.23	
4.62/2.23		S("evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ ar_1 >= 1 ]", 0-0) = ar_1
4.62/2.23	
4.62/2.23		S("evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ ar_1 >= 1 ]", 0-1) = ?
4.62/2.23	
4.62/2.23		S("evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ ar_1 >= 1 ]", 0-2) = ar_1
4.62/2.23	
4.62/2.23		S("evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_1, ar_0, ar_2))", 0-0) = ar_1
4.62/2.23	
4.62/2.23		S("evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_1, ar_0, ar_2))", 0-1) = ar_0
4.62/2.23	
4.62/2.23		S("evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_1, ar_0, ar_2))", 0-2) = ar_2
4.62/2.23	
4.62/2.23		S("evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))", 0-0) = ar_0
4.62/2.23	
4.62/2.23		S("evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))", 0-1) = ar_1
4.62/2.23	
4.62/2.23		S("evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))", 0-2) = ar_2
4.62/2.23	
4.62/2.23	orients the transitions
4.62/2.23	
4.62/2.23		evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1 - 1, ar_2))
4.62/2.23	
4.62/2.23		evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.62/2.23	
4.62/2.23		evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.62/2.23	
4.62/2.23		evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 - 1))
4.62/2.23	
4.62/2.23	weakly and the transitions
4.62/2.23	
4.62/2.23		evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1 - 1, ar_2))
4.62/2.23	
4.62/2.23		evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.62/2.23	
4.62/2.23	strictly and produces the following problem:
4.62/2.23	
4.62/2.23	5:	T:
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 1)             evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 1)             evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_1, ar_0, ar_2))
4.62/2.23	
4.62/2.23			(Comp: ar_0 + 1, Cost: 1)      evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ ar_1 >= 1 ]
4.62/2.23	
4.62/2.23			(Comp: 2, Cost: 1)             evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)             evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.62/2.23	
4.62/2.23			(Comp: 2*ar_0 + 2, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)             evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 - 1))
4.62/2.23	
4.62/2.23			(Comp: 2*ar_0 + 2, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1 - 1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 2, Cost: 1)             evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 0)             koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.62/2.23	
4.62/2.23		start location:	koat_start
4.62/2.23	
4.62/2.23		leaf cost:	0
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	A polynomial rank function with
4.62/2.23	
4.62/2.23		Pol(evalfbb2in) = V_3 + 1
4.62/2.23	
4.62/2.23		Pol(evalfbb1in) = V_3
4.62/2.23	
4.62/2.23	and size complexities
4.62/2.23	
4.62/2.23		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
4.62/2.23	
4.62/2.23		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
4.62/2.23	
4.62/2.23		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
4.62/2.23	
4.62/2.23		S("evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))", 0-0) = ar_1
4.62/2.23	
4.62/2.23		S("evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))", 0-1) = 3*ar_0 + 162
4.62/2.23	
4.62/2.23		S("evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))", 0-2) = ?
4.62/2.23	
4.62/2.23		S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1 - 1, ar_2))", 0-0) = ar_1
4.62/2.23	
4.62/2.23		S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1 - 1, ar_2))", 0-1) = 3*ar_0 + 18
4.62/2.23	
4.62/2.23		S("evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1 - 1, ar_2))", 0-2) = ?
4.62/2.23	
4.62/2.23		S("evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 - 1))", 0-0) = ar_1
4.62/2.23	
4.62/2.23		S("evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 - 1))", 0-1) = 3*ar_0 + 18
4.62/2.23	
4.62/2.23		S("evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 - 1))", 0-2) = ?
4.62/2.23	
4.62/2.23		S("evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-0) = ar_1
4.62/2.23	
4.62/2.23		S("evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-1) = 3*ar_0 + 18
4.62/2.23	
4.62/2.23		S("evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-2) = ?
4.62/2.23	
4.62/2.23		S("evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-0) = ar_1
4.62/2.23	
4.62/2.23		S("evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-1) = 3*ar_0 + 18
4.62/2.23	
4.62/2.23		S("evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-2) = ?
4.62/2.23	
4.62/2.23		S("evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ]", 0-0) = ar_1
4.62/2.23	
4.62/2.23		S("evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ]", 0-1) = 3*ar_0 + 54
4.62/2.23	
4.62/2.23		S("evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ]", 0-2) = ?
4.62/2.23	
4.62/2.23		S("evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ ar_1 >= 1 ]", 0-0) = ar_1
4.62/2.23	
4.62/2.23		S("evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ ar_1 >= 1 ]", 0-1) = 3*ar_0 + 18
4.62/2.23	
4.62/2.23		S("evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ ar_1 >= 1 ]", 0-2) = ar_1
4.62/2.23	
4.62/2.23		S("evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_1, ar_0, ar_2))", 0-0) = ar_1
4.62/2.23	
4.62/2.23		S("evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_1, ar_0, ar_2))", 0-1) = ar_0
4.62/2.23	
4.62/2.23		S("evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_1, ar_0, ar_2))", 0-2) = ar_2
4.62/2.23	
4.62/2.23		S("evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))", 0-0) = ar_0
4.62/2.23	
4.62/2.23		S("evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))", 0-1) = ar_1
4.62/2.23	
4.62/2.23		S("evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))", 0-2) = ar_2
4.62/2.23	
4.62/2.23	orients the transitions
4.62/2.23	
4.62/2.23		evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.62/2.23	
4.62/2.23		evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 - 1))
4.62/2.23	
4.62/2.23	weakly and the transition
4.62/2.23	
4.62/2.23		evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.62/2.23	
4.62/2.23	strictly and produces the following problem:
4.62/2.23	
4.62/2.23	6:	T:
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 1)                              evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 1)                              evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_1, ar_0, ar_2))
4.62/2.23	
4.62/2.23			(Comp: ar_0 + 1, Cost: 1)                       evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ ar_1 >= 1 ]
4.62/2.23	
4.62/2.23			(Comp: 2, Cost: 1)                              evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ]
4.62/2.23	
4.62/2.23			(Comp: ar_0*ar_1 + ar_1 + ar_0 + 1, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.62/2.23	
4.62/2.23			(Comp: 2*ar_0 + 2, Cost: 1)                     evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.62/2.23	
4.62/2.23			(Comp: ?, Cost: 1)                              evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 - 1))
4.62/2.23	
4.62/2.23			(Comp: 2*ar_0 + 2, Cost: 1)                     evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1 - 1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 2, Cost: 1)                              evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 0)                              koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.62/2.23	
4.62/2.23		start location:	koat_start
4.62/2.23	
4.62/2.23		leaf cost:	0
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	Repeatedly propagating knowledge in problem 6 produces the following problem:
4.62/2.23	
4.62/2.23	7:	T:
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 1)                              evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 1)                              evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_1, ar_0, ar_2))
4.62/2.23	
4.62/2.23			(Comp: ar_0 + 1, Cost: 1)                       evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_0)) [ ar_1 >= 1 ]
4.62/2.23	
4.62/2.23			(Comp: 2, Cost: 1)                              evalfbb4in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ]
4.62/2.23	
4.62/2.23			(Comp: ar_0*ar_1 + ar_1 + ar_0 + 1, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.62/2.23	
4.62/2.23			(Comp: 2*ar_0 + 2, Cost: 1)                     evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.62/2.23	
4.62/2.23			(Comp: ar_0*ar_1 + ar_1 + ar_0 + 1, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 - 1))
4.62/2.23	
4.62/2.23			(Comp: 2*ar_0 + 2, Cost: 1)                     evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbb4in(ar_0, ar_1 - 1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 2, Cost: 1)                              evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
4.62/2.23	
4.62/2.23			(Comp: 1, Cost: 0)                              koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.62/2.23	
4.62/2.23		start location:	koat_start
4.62/2.23	
4.62/2.23		leaf cost:	0
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	Complexity upper bound 7*ar_0 + 2*ar_0*ar_1 + 2*ar_1 + 13
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	Time: 0.091 sec (SMT: 0.081 sec)
4.62/2.23	
4.62/2.23	
4.62/2.23	----------------------------------------
4.62/2.23	
4.62/2.23	(2)
4.62/2.23	BOUNDS(1, n^2)
4.62/2.23	
4.62/2.23	----------------------------------------
4.62/2.23	
4.62/2.23	(3) Loat Proof (FINISHED)
4.62/2.23	
4.62/2.23	
4.62/2.23	### Pre-processing the ITS problem ###
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	Initial linear ITS problem
4.62/2.23	
4.62/2.23	   Start location: evalfstart
4.62/2.23	
4.62/2.23	      0: evalfstart -> evalfentryin : [], cost: 1
4.62/2.23	
4.62/2.23	      1: evalfentryin -> evalfbb4in : A'=B, B'=A, [], cost: 1
4.62/2.23	
4.62/2.23	      2: evalfbb4in -> evalfbb2in : C'=A, [ B>=1 ], cost: 1
4.62/2.23	
4.62/2.23	      3: evalfbb4in -> evalfreturnin : [ 0>=B ], cost: 1
4.62/2.23	
4.62/2.23	      4: evalfbb2in -> evalfbb1in : [ C>=1 ], cost: 1
4.62/2.23	
4.62/2.23	      5: evalfbb2in -> evalfbb3in : [ 0>=C ], cost: 1
4.62/2.23	
4.62/2.23	      6: evalfbb1in -> evalfbb2in : C'=-1+C, [], cost: 1
4.62/2.23	
4.62/2.23	      7: evalfbb3in -> evalfbb4in : B'=-1+B, [], cost: 1
4.62/2.23	
4.62/2.23	      8: evalfreturnin -> evalfstop : [], cost: 1
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	Removed unreachable and leaf rules:
4.62/2.23	
4.62/2.23	   Start location: evalfstart
4.62/2.23	
4.62/2.23	      0: evalfstart -> evalfentryin : [], cost: 1
4.62/2.23	
4.62/2.23	      1: evalfentryin -> evalfbb4in : A'=B, B'=A, [], cost: 1
4.62/2.23	
4.62/2.23	      2: evalfbb4in -> evalfbb2in : C'=A, [ B>=1 ], cost: 1
4.62/2.23	
4.62/2.23	      4: evalfbb2in -> evalfbb1in : [ C>=1 ], cost: 1
4.62/2.23	
4.62/2.23	      5: evalfbb2in -> evalfbb3in : [ 0>=C ], cost: 1
4.62/2.23	
4.62/2.23	      6: evalfbb1in -> evalfbb2in : C'=-1+C, [], cost: 1
4.62/2.23	
4.62/2.23	      7: evalfbb3in -> evalfbb4in : B'=-1+B, [], cost: 1
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	### Simplification by acceleration and chaining ###
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	Eliminated locations (on linear paths):
4.62/2.23	
4.62/2.23	   Start location: evalfstart
4.62/2.23	
4.62/2.23	      9: evalfstart -> evalfbb4in : A'=B, B'=A, [], cost: 2
4.62/2.23	
4.62/2.23	      2: evalfbb4in -> evalfbb2in : C'=A, [ B>=1 ], cost: 1
4.62/2.23	
4.62/2.23	     10: evalfbb2in -> evalfbb2in : C'=-1+C, [ C>=1 ], cost: 2
4.62/2.23	
4.62/2.23	     11: evalfbb2in -> evalfbb4in : B'=-1+B, [ 0>=C ], cost: 2
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	Accelerating simple loops of location 3.
4.62/2.23	
4.62/2.23	   Accelerating the following rules:
4.62/2.23	
4.62/2.23	     10: evalfbb2in -> evalfbb2in : C'=-1+C, [ C>=1 ], cost: 2
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	   Accelerated rule 10 with metering function C, yielding the new rule 12.
4.62/2.23	
4.62/2.23	   Removing the simple loops: 10.
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	Accelerated all simple loops using metering functions (where possible):
4.62/2.23	
4.62/2.23	   Start location: evalfstart
4.62/2.23	
4.62/2.23	      9: evalfstart -> evalfbb4in : A'=B, B'=A, [], cost: 2
4.62/2.23	
4.62/2.23	      2: evalfbb4in -> evalfbb2in : C'=A, [ B>=1 ], cost: 1
4.62/2.23	
4.62/2.23	     11: evalfbb2in -> evalfbb4in : B'=-1+B, [ 0>=C ], cost: 2
4.62/2.23	
4.62/2.23	     12: evalfbb2in -> evalfbb2in : C'=0, [ C>=1 ], cost: 2*C
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	Chained accelerated rules (with incoming rules):
4.62/2.23	
4.62/2.23	   Start location: evalfstart
4.62/2.23	
4.62/2.23	      9: evalfstart -> evalfbb4in : A'=B, B'=A, [], cost: 2
4.62/2.23	
4.62/2.23	      2: evalfbb4in -> evalfbb2in : C'=A, [ B>=1 ], cost: 1
4.62/2.23	
4.62/2.23	     13: evalfbb4in -> evalfbb2in : C'=0, [ B>=1 && A>=1 ], cost: 1+2*A
4.62/2.23	
4.62/2.23	     11: evalfbb2in -> evalfbb4in : B'=-1+B, [ 0>=C ], cost: 2
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	Eliminated locations (on tree-shaped paths):
4.62/2.23	
4.62/2.23	   Start location: evalfstart
4.62/2.23	
4.62/2.23	      9: evalfstart -> evalfbb4in : A'=B, B'=A, [], cost: 2
4.62/2.23	
4.62/2.23	     14: evalfbb4in -> evalfbb4in : B'=-1+B, C'=A, [ B>=1 && 0>=A ], cost: 3
4.62/2.23	
4.62/2.23	     15: evalfbb4in -> evalfbb4in : B'=-1+B, C'=0, [ B>=1 && A>=1 ], cost: 3+2*A
4.62/2.23	
4.62/2.23	
4.62/2.23	
4.62/2.23	Accelerating simple loops of location 2.
4.62/2.23	
4.62/2.23	   Accelerating the following rules:
4.62/2.23	
4.62/2.23	     14: evalfbb4in -> evalfbb4in : B'=-1+B, C'=A, [ B>=1 && 0>=A ], cost: 3
4.62/2.23	
4.62/2.23	     15: evalfbb4in -> evalfbb4in : B'=-1+B, C'=0, [ B>=1 && A>=1 ], cost: 3+2*A
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	   Accelerated rule 14 with metering function B, yielding the new rule 16.
4.62/2.24	
4.62/2.24	   Accelerated rule 15 with metering function B, yielding the new rule 17.
4.62/2.24	
4.62/2.24	   Removing the simple loops: 14 15.
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	Accelerated all simple loops using metering functions (where possible):
4.62/2.24	
4.62/2.24	   Start location: evalfstart
4.62/2.24	
4.62/2.24	      9: evalfstart -> evalfbb4in : A'=B, B'=A, [], cost: 2
4.62/2.24	
4.62/2.24	     16: evalfbb4in -> evalfbb4in : B'=0, C'=A, [ B>=1 && 0>=A ], cost: 3*B
4.62/2.24	
4.62/2.24	     17: evalfbb4in -> evalfbb4in : B'=0, C'=0, [ B>=1 && A>=1 ], cost: 2*A*B+3*B
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	Chained accelerated rules (with incoming rules):
4.62/2.24	
4.62/2.24	   Start location: evalfstart
4.62/2.24	
4.62/2.24	      9: evalfstart -> evalfbb4in : A'=B, B'=A, [], cost: 2
4.62/2.24	
4.62/2.24	     18: evalfstart -> evalfbb4in : A'=B, B'=0, C'=B, [ A>=1 && 0>=B ], cost: 2+3*A
4.62/2.24	
4.62/2.24	     19: evalfstart -> evalfbb4in : A'=B, B'=0, C'=0, [ A>=1 && B>=1 ], cost: 2+3*A+2*A*B
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	Removed unreachable locations (and leaf rules with constant cost):
4.62/2.24	
4.62/2.24	   Start location: evalfstart
4.62/2.24	
4.62/2.24	     18: evalfstart -> evalfbb4in : A'=B, B'=0, C'=B, [ A>=1 && 0>=B ], cost: 2+3*A
4.62/2.24	
4.62/2.24	     19: evalfstart -> evalfbb4in : A'=B, B'=0, C'=0, [ A>=1 && B>=1 ], cost: 2+3*A+2*A*B
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	### Computing asymptotic complexity ###
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	Fully simplified ITS problem
4.62/2.24	
4.62/2.24	   Start location: evalfstart
4.62/2.24	
4.62/2.24	     18: evalfstart -> evalfbb4in : A'=B, B'=0, C'=B, [ A>=1 && 0>=B ], cost: 2+3*A
4.62/2.24	
4.62/2.24	     19: evalfstart -> evalfbb4in : A'=B, B'=0, C'=0, [ A>=1 && B>=1 ], cost: 2+3*A+2*A*B
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	Computing asymptotic complexity for rule 18
4.62/2.24	
4.62/2.24	   Solved the limit problem by the following transformations:
4.62/2.24	
4.62/2.24	      Created initial limit problem:
4.62/2.24	
4.62/2.24	      1-B (+/+!), A (+/+!), 2+3*A (+) [not solved]
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	      removing all constraints (solved by SMT)
4.62/2.24	
4.62/2.24	      resulting limit problem: [solved]
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	      applying transformation rule (C) using substitution {A==n,B==0}
4.62/2.24	
4.62/2.24	      resulting limit problem:
4.62/2.24	
4.62/2.24	      [solved]
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	   Solution:
4.62/2.24	
4.62/2.24	      A / n
4.62/2.24	
4.62/2.24	      B / 0
4.62/2.24	
4.62/2.24	   Resulting cost 2+3*n has complexity: Poly(n^1)
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	Found new complexity Poly(n^1).
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	Computing asymptotic complexity for rule 19
4.62/2.24	
4.62/2.24	   Solved the limit problem by the following transformations:
4.62/2.24	
4.62/2.24	      Created initial limit problem:
4.62/2.24	
4.62/2.24	      A (+/+!), 2+3*A+2*A*B (+), B (+/+!) [not solved]
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	      removing all constraints (solved by SMT)
4.62/2.24	
4.62/2.24	      resulting limit problem: [solved]
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	      applying transformation rule (C) using substitution {A==n,B==n}
4.62/2.24	
4.62/2.24	      resulting limit problem:
4.62/2.24	
4.62/2.24	      [solved]
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	   Solution:
4.62/2.24	
4.62/2.24	      A / n
4.62/2.24	
4.62/2.24	      B / n
4.62/2.24	
4.62/2.24	   Resulting cost 2+3*n+2*n^2 has complexity: Poly(n^2)
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	Found new complexity Poly(n^2).
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	Obtained the following overall complexity (w.r.t. the length of the input n):
4.62/2.24	
4.62/2.24	   Complexity:  Poly(n^2)
4.62/2.24	
4.62/2.24	   Cpx degree:  2
4.62/2.24	
4.62/2.24	   Solved cost: 2+3*n+2*n^2
4.62/2.24	
4.62/2.24	   Rule cost:   2+3*A+2*A*B
4.62/2.24	
4.62/2.24	   Rule guard:  [ A>=1 && B>=1 ]
4.62/2.24	
4.62/2.24	
4.62/2.24	
4.62/2.24	WORST_CASE(Omega(n^2),?)
4.62/2.24	
4.62/2.24	
4.62/2.24	----------------------------------------
4.62/2.24	
4.62/2.24	(4)
4.62/2.24	BOUNDS(n^2, INF)
4.62/2.25	EOF