4.68/2.15	WORST_CASE(Omega(n^1), O(n^1))
4.68/2.16	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.68/2.16	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.68/2.16	
4.68/2.16	
4.68/2.16	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
4.68/2.16	
4.68/2.16	(0) CpxIntTrs
4.68/2.16	(1) Koat Proof [FINISHED, 245 ms]
4.68/2.16	(2) BOUNDS(1, n^1)
4.68/2.16	(3) Loat Proof [FINISHED, 439 ms]
4.68/2.16	(4) BOUNDS(n^1, INF)
4.68/2.16	
4.68/2.16	
4.68/2.16	----------------------------------------
4.68/2.16	
4.68/2.16	(0)
4.68/2.16	Obligation:
4.68/2.16	Complexity Int TRS consisting of the following rules:
4.68/2.16	evalfstart(A, B, C) -> Com_1(evalfentryin(A, B, C)) :|: TRUE
4.68/2.16	evalfentryin(A, B, C) -> Com_1(evalfbb3in(0, 0, C)) :|: TRUE
4.68/2.16	evalfbb3in(A, B, C) -> Com_1(evalfbbin(A, B, C)) :|: 99 >= B
4.68/2.16	evalfbb3in(A, B, C) -> Com_1(evalfreturnin(A, B, C)) :|: B >= 100
4.68/2.16	evalfbbin(A, B, C) -> Com_1(evalfbb1in(A, B, C)) :|: C >= A + 1
4.68/2.16	evalfbbin(A, B, C) -> Com_1(evalfbb2in(A, B, C)) :|: A >= C
4.68/2.16	evalfbb1in(A, B, C) -> Com_1(evalfbb3in(A + 1, B, C)) :|: TRUE
4.68/2.16	evalfbb2in(A, B, C) -> Com_1(evalfbb3in(A, B + 1, C)) :|: TRUE
4.68/2.16	evalfreturnin(A, B, C) -> Com_1(evalfstop(A, B, C)) :|: TRUE
4.68/2.16	
4.68/2.16	The start-symbols are:[evalfstart_3]
4.68/2.16	
4.68/2.16	
4.68/2.16	----------------------------------------
4.68/2.16	
4.68/2.16	(1) Koat Proof (FINISHED)
4.68/2.16	YES(?, 3*ar_2 + 610)
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Initial complexity problem:
4.68/2.16	
4.68/2.16	1:	T:
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(0, 0, ar_2))
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ 99 >= ar_1 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 >= 100 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1 + 1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.68/2.16	
4.68/2.16		start location:	koat_start
4.68/2.16	
4.68/2.16		leaf cost:	0
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Repeatedly propagating knowledge in problem 1 produces the following problem:
4.68/2.16	
4.68/2.16	2:	T:
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 1)    evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(0, 0, ar_2))
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ 99 >= ar_1 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 >= 100 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1 + 1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.68/2.16	
4.68/2.16		start location:	koat_start
4.68/2.16	
4.68/2.16		leaf cost:	0
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	A polynomial rank function with
4.68/2.16	
4.68/2.16		Pol(evalfstart) = 2
4.68/2.16	
4.68/2.16		Pol(evalfentryin) = 2
4.68/2.16	
4.68/2.16		Pol(evalfbb3in) = 2
4.68/2.16	
4.68/2.16		Pol(evalfbbin) = 2
4.68/2.16	
4.68/2.16		Pol(evalfreturnin) = 1
4.68/2.16	
4.68/2.16		Pol(evalfbb1in) = 2
4.68/2.16	
4.68/2.16		Pol(evalfbb2in) = 2
4.68/2.16	
4.68/2.16		Pol(evalfstop) = 0
4.68/2.16	
4.68/2.16		Pol(koat_start) = 2
4.68/2.16	
4.68/2.16	orients all transitions weakly and the transitions
4.68/2.16	
4.68/2.16		evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
4.68/2.16	
4.68/2.16		evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 >= 100 ]
4.68/2.16	
4.68/2.16	strictly and produces the following problem:
4.68/2.16	
4.68/2.16	3:	T:
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 1)    evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(0, 0, ar_2))
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ 99 >= ar_1 ]
4.68/2.16	
4.68/2.16			(Comp: 2, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 >= 100 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1 + 1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: 2, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.68/2.16	
4.68/2.16		start location:	koat_start
4.68/2.16	
4.68/2.16		leaf cost:	0
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	A polynomial rank function with
4.68/2.16	
4.68/2.16		Pol(evalfstart) = V_3 + 1
4.68/2.16	
4.68/2.16		Pol(evalfentryin) = V_3 + 1
4.68/2.16	
4.68/2.16		Pol(evalfbb3in) = -V_1 + V_3 + 1
4.68/2.16	
4.68/2.16		Pol(evalfbbin) = -V_1 + V_3 + 1
4.68/2.16	
4.68/2.16		Pol(evalfreturnin) = -V_1 + V_3
4.68/2.16	
4.68/2.16		Pol(evalfbb1in) = -V_1 + V_3
4.68/2.16	
4.68/2.16		Pol(evalfbb2in) = -V_1 + V_3 + 1
4.68/2.16	
4.68/2.16		Pol(evalfstop) = -V_1 + V_3
4.68/2.16	
4.68/2.16		Pol(koat_start) = V_3 + 1
4.68/2.16	
4.68/2.16	orients all transitions weakly and the transition
4.68/2.16	
4.68/2.16		evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
4.68/2.16	
4.68/2.16	strictly and produces the following problem:
4.68/2.16	
4.68/2.16	4:	T:
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 1)           evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 1)           evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(0, 0, ar_2))
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)           evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ 99 >= ar_1 ]
4.68/2.16	
4.68/2.16			(Comp: 2, Cost: 1)           evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 >= 100 ]
4.68/2.16	
4.68/2.16			(Comp: ar_2 + 1, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)           evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)           evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)           evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1 + 1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: 2, Cost: 1)           evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.68/2.16	
4.68/2.16		start location:	koat_start
4.68/2.16	
4.68/2.16		leaf cost:	0
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Repeatedly propagating knowledge in problem 4 produces the following problem:
4.68/2.16	
4.68/2.16	5:	T:
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 1)           evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 1)           evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(0, 0, ar_2))
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)           evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ 99 >= ar_1 ]
4.68/2.16	
4.68/2.16			(Comp: 2, Cost: 1)           evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 >= 100 ]
4.68/2.16	
4.68/2.16			(Comp: ar_2 + 1, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)           evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
4.68/2.16	
4.68/2.16			(Comp: ar_2 + 1, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)           evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1 + 1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: 2, Cost: 1)           evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2))
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.68/2.16	
4.68/2.16		start location:	koat_start
4.68/2.16	
4.68/2.16		leaf cost:	0
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Applied AI with 'oct' on problem 5 to obtain the following invariants:
4.68/2.16	
4.68/2.16	  For symbol evalfbb1in: X_3 - 1 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ -X_2 + X_3 + 98 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ -X_1 + X_3 - 1 >= 0 /\ -X_2 + 99 >= 0 /\ X_1 - X_2 + 99 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0
4.68/2.16	
4.68/2.16	  For symbol evalfbb2in: X_1 - X_3 >= 0 /\ -X_2 + 99 >= 0 /\ X_1 - X_2 + 99 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0
4.68/2.16	
4.68/2.16	  For symbol evalfbb3in: X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0
4.68/2.16	
4.68/2.16	  For symbol evalfbbin: -X_2 + 99 >= 0 /\ X_1 - X_2 + 99 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0
4.68/2.16	
4.68/2.16	  For symbol evalfreturnin: X_2 - 100 >= 0 /\ X_1 + X_2 - 100 >= 0 /\ X_1 >= 0
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	This yielded the following problem:
4.68/2.16	
4.68/2.16	6:	T:
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.68/2.16	
4.68/2.16			(Comp: 2, Cost: 1)           evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 100 >= 0 /\ ar_0 + ar_1 - 100 >= 0 /\ ar_0 >= 0 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)           evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - ar_2 >= 0 /\ -ar_1 + 99 >= 0 /\ ar_0 - ar_1 + 99 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
4.68/2.16	
4.68/2.16			(Comp: ar_2 + 1, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 + 98 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_1 + 99 >= 0 /\ ar_0 - ar_1 + 99 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)           evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ -ar_1 + 99 >= 0 /\ ar_0 - ar_1 + 99 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_2 ]
4.68/2.16	
4.68/2.16			(Comp: ar_2 + 1, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ -ar_1 + 99 >= 0 /\ ar_0 - ar_1 + 99 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_0 + 1 ]
4.68/2.16	
4.68/2.16			(Comp: 2, Cost: 1)           evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= 100 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)           evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ 99 >= ar_1 ]
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 1)           evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(0, 0, ar_2))
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 1)           evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
4.68/2.16	
4.68/2.16		start location:	koat_start
4.68/2.16	
4.68/2.16		leaf cost:	0
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	A polynomial rank function with
4.68/2.16	
4.68/2.16		Pol(koat_start) = 200
4.68/2.16	
4.68/2.16		Pol(evalfstart) = 200
4.68/2.16	
4.68/2.16		Pol(evalfreturnin) = -2*V_2
4.68/2.16	
4.68/2.16		Pol(evalfstop) = -2*V_2
4.68/2.16	
4.68/2.16		Pol(evalfbb2in) = -2*V_2 + 199
4.68/2.16	
4.68/2.16		Pol(evalfbb3in) = -2*V_2 + 200
4.68/2.16	
4.68/2.16		Pol(evalfbb1in) = -2*V_2 + 200
4.68/2.16	
4.68/2.16		Pol(evalfbbin) = -2*V_2 + 200
4.68/2.16	
4.68/2.16		Pol(evalfentryin) = 200
4.68/2.16	
4.68/2.16	orients all transitions weakly and the transitions
4.68/2.16	
4.68/2.16		evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ -ar_1 + 99 >= 0 /\ ar_0 - ar_1 + 99 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_2 ]
4.68/2.16	
4.68/2.16		evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - ar_2 >= 0 /\ -ar_1 + 99 >= 0 /\ ar_0 - ar_1 + 99 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
4.68/2.16	
4.68/2.16	strictly and produces the following problem:
4.68/2.16	
4.68/2.16	7:	T:
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.68/2.16	
4.68/2.16			(Comp: 2, Cost: 1)           evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 100 >= 0 /\ ar_0 + ar_1 - 100 >= 0 /\ ar_0 >= 0 ]
4.68/2.16	
4.68/2.16			(Comp: 200, Cost: 1)         evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - ar_2 >= 0 /\ -ar_1 + 99 >= 0 /\ ar_0 - ar_1 + 99 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
4.68/2.16	
4.68/2.16			(Comp: ar_2 + 1, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 + 98 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_1 + 99 >= 0 /\ ar_0 - ar_1 + 99 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
4.68/2.16	
4.68/2.16			(Comp: 200, Cost: 1)         evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ -ar_1 + 99 >= 0 /\ ar_0 - ar_1 + 99 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_2 ]
4.68/2.16	
4.68/2.16			(Comp: ar_2 + 1, Cost: 1)    evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ -ar_1 + 99 >= 0 /\ ar_0 - ar_1 + 99 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_0 + 1 ]
4.68/2.16	
4.68/2.16			(Comp: 2, Cost: 1)           evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= 100 ]
4.68/2.16	
4.68/2.16			(Comp: ?, Cost: 1)           evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ 99 >= ar_1 ]
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 1)           evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(0, 0, ar_2))
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 1)           evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
4.68/2.16	
4.68/2.16		start location:	koat_start
4.68/2.16	
4.68/2.16		leaf cost:	0
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Repeatedly propagating knowledge in problem 7 produces the following problem:
4.68/2.16	
4.68/2.16	8:	T:
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 0)             koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.68/2.16	
4.68/2.16			(Comp: 2, Cost: 1)             evalfreturnin(ar_0, ar_1, ar_2) -> Com_1(evalfstop(ar_0, ar_1, ar_2)) [ ar_1 - 100 >= 0 /\ ar_0 + ar_1 - 100 >= 0 /\ ar_0 >= 0 ]
4.68/2.16	
4.68/2.16			(Comp: 200, Cost: 1)           evalfbb2in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - ar_2 >= 0 /\ -ar_1 + 99 >= 0 /\ ar_0 - ar_1 + 99 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
4.68/2.16	
4.68/2.16			(Comp: ar_2 + 1, Cost: 1)      evalfbb1in(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(ar_0 + 1, ar_1, ar_2)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 + 98 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ -ar_1 + 99 >= 0 /\ ar_0 - ar_1 + 99 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
4.68/2.16	
4.68/2.16			(Comp: 200, Cost: 1)           evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2)) [ -ar_1 + 99 >= 0 /\ ar_0 - ar_1 + 99 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_2 ]
4.68/2.16	
4.68/2.16			(Comp: ar_2 + 1, Cost: 1)      evalfbbin(ar_0, ar_1, ar_2) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2)) [ -ar_1 + 99 >= 0 /\ ar_0 - ar_1 + 99 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_0 + 1 ]
4.68/2.16	
4.68/2.16			(Comp: 2, Cost: 1)             evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= 100 ]
4.68/2.16	
4.68/2.16			(Comp: ar_2 + 202, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2) -> Com_1(evalfbbin(ar_0, ar_1, ar_2)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ 99 >= ar_1 ]
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 1)             evalfentryin(ar_0, ar_1, ar_2) -> Com_1(evalfbb3in(0, 0, ar_2))
4.68/2.16	
4.68/2.16			(Comp: 1, Cost: 1)             evalfstart(ar_0, ar_1, ar_2) -> Com_1(evalfentryin(ar_0, ar_1, ar_2))
4.68/2.16	
4.68/2.16		start location:	koat_start
4.68/2.16	
4.68/2.16		leaf cost:	0
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Complexity upper bound 3*ar_2 + 610
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Time: 0.256 sec (SMT: 0.224 sec)
4.68/2.16	
4.68/2.16	
4.68/2.16	----------------------------------------
4.68/2.16	
4.68/2.16	(2)
4.68/2.16	BOUNDS(1, n^1)
4.68/2.16	
4.68/2.16	----------------------------------------
4.68/2.16	
4.68/2.16	(3) Loat Proof (FINISHED)
4.68/2.16	
4.68/2.16	
4.68/2.16	### Pre-processing the ITS problem ###
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Initial linear ITS problem
4.68/2.16	
4.68/2.16	   Start location: evalfstart
4.68/2.16	
4.68/2.16	      0: evalfstart -> evalfentryin : [], cost: 1
4.68/2.16	
4.68/2.16	      1: evalfentryin -> evalfbb3in : A'=0, B'=0, [], cost: 1
4.68/2.16	
4.68/2.16	      2: evalfbb3in -> evalfbbin : [ 99>=B ], cost: 1
4.68/2.16	
4.68/2.16	      3: evalfbb3in -> evalfreturnin : [ B>=100 ], cost: 1
4.68/2.16	
4.68/2.16	      4: evalfbbin -> evalfbb1in : [ C>=1+A ], cost: 1
4.68/2.16	
4.68/2.16	      5: evalfbbin -> evalfbb2in : [ A>=C ], cost: 1
4.68/2.16	
4.68/2.16	      6: evalfbb1in -> evalfbb3in : A'=1+A, [], cost: 1
4.68/2.16	
4.68/2.16	      7: evalfbb2in -> evalfbb3in : B'=1+B, [], cost: 1
4.68/2.16	
4.68/2.16	      8: evalfreturnin -> evalfstop : [], cost: 1
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Removed unreachable and leaf rules:
4.68/2.16	
4.68/2.16	   Start location: evalfstart
4.68/2.16	
4.68/2.16	      0: evalfstart -> evalfentryin : [], cost: 1
4.68/2.16	
4.68/2.16	      1: evalfentryin -> evalfbb3in : A'=0, B'=0, [], cost: 1
4.68/2.16	
4.68/2.16	      2: evalfbb3in -> evalfbbin : [ 99>=B ], cost: 1
4.68/2.16	
4.68/2.16	      4: evalfbbin -> evalfbb1in : [ C>=1+A ], cost: 1
4.68/2.16	
4.68/2.16	      5: evalfbbin -> evalfbb2in : [ A>=C ], cost: 1
4.68/2.16	
4.68/2.16	      6: evalfbb1in -> evalfbb3in : A'=1+A, [], cost: 1
4.68/2.16	
4.68/2.16	      7: evalfbb2in -> evalfbb3in : B'=1+B, [], cost: 1
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	### Simplification by acceleration and chaining ###
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Eliminated locations (on linear paths):
4.68/2.16	
4.68/2.16	   Start location: evalfstart
4.68/2.16	
4.68/2.16	      9: evalfstart -> evalfbb3in : A'=0, B'=0, [], cost: 2
4.68/2.16	
4.68/2.16	      2: evalfbb3in -> evalfbbin : [ 99>=B ], cost: 1
4.68/2.16	
4.68/2.16	     10: evalfbbin -> evalfbb3in : A'=1+A, [ C>=1+A ], cost: 2
4.68/2.16	
4.68/2.16	     11: evalfbbin -> evalfbb3in : B'=1+B, [ A>=C ], cost: 2
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Eliminated locations (on tree-shaped paths):
4.68/2.16	
4.68/2.16	   Start location: evalfstart
4.68/2.16	
4.68/2.16	      9: evalfstart -> evalfbb3in : A'=0, B'=0, [], cost: 2
4.68/2.16	
4.68/2.16	     12: evalfbb3in -> evalfbb3in : A'=1+A, [ 99>=B && C>=1+A ], cost: 3
4.68/2.16	
4.68/2.16	     13: evalfbb3in -> evalfbb3in : B'=1+B, [ 99>=B && A>=C ], cost: 3
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Accelerating simple loops of location 2.
4.68/2.16	
4.68/2.16	   Accelerating the following rules:
4.68/2.16	
4.68/2.16	     12: evalfbb3in -> evalfbb3in : A'=1+A, [ 99>=B && C>=1+A ], cost: 3
4.68/2.16	
4.68/2.16	     13: evalfbb3in -> evalfbb3in : B'=1+B, [ 99>=B && A>=C ], cost: 3
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	   Accelerated rule 12 with metering function C-A, yielding the new rule 14.
4.68/2.16	
4.68/2.16	   Accelerated rule 13 with metering function 100-B, yielding the new rule 15.
4.68/2.16	
4.68/2.16	   Removing the simple loops: 12 13.
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Accelerated all simple loops using metering functions (where possible):
4.68/2.16	
4.68/2.16	   Start location: evalfstart
4.68/2.16	
4.68/2.16	      9: evalfstart -> evalfbb3in : A'=0, B'=0, [], cost: 2
4.68/2.16	
4.68/2.16	     14: evalfbb3in -> evalfbb3in : A'=C, [ 99>=B && C>=1+A ], cost: 3*C-3*A
4.68/2.16	
4.68/2.16	     15: evalfbb3in -> evalfbb3in : B'=100, [ 99>=B && A>=C ], cost: 300-3*B
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Chained accelerated rules (with incoming rules):
4.68/2.16	
4.68/2.16	   Start location: evalfstart
4.68/2.16	
4.68/2.16	      9: evalfstart -> evalfbb3in : A'=0, B'=0, [], cost: 2
4.68/2.16	
4.68/2.16	     16: evalfstart -> evalfbb3in : A'=C, B'=0, [ C>=1 ], cost: 2+3*C
4.68/2.16	
4.68/2.16	     17: evalfstart -> evalfbb3in : A'=0, B'=100, [ 0>=C ], cost: 302
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Removed unreachable locations (and leaf rules with constant cost):
4.68/2.16	
4.68/2.16	   Start location: evalfstart
4.68/2.16	
4.68/2.16	     16: evalfstart -> evalfbb3in : A'=C, B'=0, [ C>=1 ], cost: 2+3*C
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	### Computing asymptotic complexity ###
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Fully simplified ITS problem
4.68/2.16	
4.68/2.16	   Start location: evalfstart
4.68/2.16	
4.68/2.16	     16: evalfstart -> evalfbb3in : A'=C, B'=0, [ C>=1 ], cost: 2+3*C
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Computing asymptotic complexity for rule 16
4.68/2.16	
4.68/2.16	   Solved the limit problem by the following transformations:
4.68/2.16	
4.68/2.16	      Created initial limit problem:
4.68/2.16	
4.68/2.16	      2+3*C (+), C (+/+!) [not solved]
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	      removing all constraints (solved by SMT)
4.68/2.16	
4.68/2.16	      resulting limit problem: [solved]
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	      applying transformation rule (C) using substitution {C==n}
4.68/2.16	
4.68/2.16	      resulting limit problem:
4.68/2.16	
4.68/2.16	      [solved]
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	   Solution:
4.68/2.16	
4.68/2.16	      C / n
4.68/2.16	
4.68/2.16	   Resulting cost 2+3*n has complexity: Poly(n^1)
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Found new complexity Poly(n^1).
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	Obtained the following overall complexity (w.r.t. the length of the input n):
4.68/2.16	
4.68/2.16	   Complexity:  Poly(n^1)
4.68/2.16	
4.68/2.16	   Cpx degree:  1
4.68/2.16	
4.68/2.16	   Solved cost: 2+3*n
4.68/2.16	
4.68/2.16	   Rule cost:   2+3*C
4.68/2.16	
4.68/2.16	   Rule guard:  [ C>=1 ]
4.68/2.16	
4.68/2.16	
4.68/2.16	
4.68/2.16	WORST_CASE(Omega(n^1),?)
4.68/2.16	
4.68/2.16	
4.68/2.16	----------------------------------------
4.68/2.16	
4.68/2.16	(4)
4.68/2.16	BOUNDS(n^1, INF)
4.85/2.18	EOF