5.21/2.37	WORST_CASE(Omega(n^2), O(n^2))
5.21/2.38	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
5.21/2.38	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.21/2.38	
5.21/2.38	
5.21/2.38	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).
5.21/2.38	
5.21/2.38	(0) CpxIntTrs
5.21/2.38	(1) Koat Proof [FINISHED, 298 ms]
5.21/2.38	(2) BOUNDS(1, n^2)
5.21/2.38	(3) Loat Proof [FINISHED, 680 ms]
5.21/2.38	(4) BOUNDS(n^2, INF)
5.21/2.38	
5.21/2.38	
5.21/2.38	----------------------------------------
5.21/2.38	
5.21/2.38	(0)
5.21/2.38	Obligation:
5.21/2.38	Complexity Int TRS consisting of the following rules:
5.21/2.38	start(A, B, C, D, E, F) -> Com_1(stop(A, B, C, F, E, F)) :|: 0 >= A && B >= C && B <= C && D >= E && D <= E && F >= A && F <= A
5.21/2.38	start(A, B, C, D, E, F) -> Com_1(lbl62(A, F - 1, C, F, E, F)) :|: A >= 1 && B >= C && B <= C && D >= E && D <= E && F >= A && F <= A
5.21/2.38	lbl72(A, B, C, D, E, F) -> Com_1(stop(A, B, C, D, E, F)) :|: A >= 1 && D >= 0 && D <= 0 && B >= 0 && B <= 0 && F >= A && F <= A
5.21/2.38	lbl72(A, B, C, D, E, F) -> Com_1(lbl72(A, F, C, D - 1, E, F)) :|: D >= 1 && 0 >= A && D >= 0 && A >= D + 1 && B >= 0 && B <= 0 && F >= A && F <= A
5.21/2.38	lbl72(A, B, C, D, E, F) -> Com_1(lbl62(A, F - 1, C, D, E, F)) :|: A >= 1 && D >= 1 && D >= 0 && A >= D + 1 && B >= 0 && B <= 0 && F >= A && F <= A
5.21/2.38	lbl62(A, B, C, D, E, F) -> Com_1(lbl72(A, B, C, D - 1, E, F)) :|: A >= D && A >= 1 && D >= 1 && B >= 0 && B <= 0 && F >= A && F <= A
5.21/2.38	lbl62(A, B, C, D, E, F) -> Com_1(lbl62(A, B - 1, C, D, E, F)) :|: B >= 1 && A >= D && A >= B + 1 && B >= 0 && D >= 1 && F >= A && F <= A
5.21/2.38	start0(A, B, C, D, E, F) -> Com_1(start(A, C, C, E, E, A)) :|: TRUE
5.21/2.38	
5.21/2.38	The start-symbols are:[start0_6]
5.21/2.38	
5.21/2.38	
5.21/2.38	----------------------------------------
5.21/2.38	
5.21/2.38	(1) Koat Proof (FINISHED)
5.21/2.38	YES(?, 7*ar_0 + 2*ar_0^2 + 5)
5.21/2.38	
5.21/2.38	
5.21/2.38	
5.21/2.38	Initial complexity problem:
5.21/2.38	
5.21/2.38	1:	T:
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_5, ar_4, ar_5)) [ 0 >= ar_0 /\ ar_1 = ar_2 /\ ar_3 = ar_4 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_5, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_1 = ar_2 /\ ar_3 = ar_4 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_3 = 0 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl72(ar_0, ar_5, ar_2, ar_3 - 1, ar_4, ar_5)) [ ar_3 >= 1 /\ 0 >= ar_0 /\ ar_3 >= 0 /\ ar_0 >= ar_3 + 1 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_3 >= 1 /\ ar_3 >= 0 /\ ar_0 >= ar_3 + 1 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl72(ar_0, ar_1, ar_2, ar_3 - 1, ar_4, ar_5)) [ ar_0 >= ar_3 /\ ar_0 >= 1 /\ ar_3 >= 1 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_1 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= 1 /\ ar_0 >= ar_3 /\ ar_0 >= ar_1 + 1 /\ ar_1 >= 0 /\ ar_3 >= 1 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start(ar_0, ar_2, ar_2, ar_4, ar_4, ar_0))
5.21/2.38	
5.21/2.38			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
5.21/2.38	
5.21/2.38		start location:	koat_start
5.21/2.38	
5.21/2.38		leaf cost:	0
5.21/2.38	
5.21/2.38	
5.21/2.38	
5.21/2.38	Testing for reachability in the complexity graph removes the following transition from problem 1:
5.21/2.38	
5.21/2.38		lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl72(ar_0, ar_5, ar_2, ar_3 - 1, ar_4, ar_5)) [ ar_3 >= 1 /\ 0 >= ar_0 /\ ar_3 >= 0 /\ ar_0 >= ar_3 + 1 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38	We thus obtain the following problem:
5.21/2.38	
5.21/2.38	2:	T:
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_3 >= 1 /\ ar_3 >= 0 /\ ar_0 >= ar_3 + 1 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_3 = 0 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_1 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= 1 /\ ar_0 >= ar_3 /\ ar_0 >= ar_1 + 1 /\ ar_1 >= 0 /\ ar_3 >= 1 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl72(ar_0, ar_1, ar_2, ar_3 - 1, ar_4, ar_5)) [ ar_0 >= ar_3 /\ ar_0 >= 1 /\ ar_3 >= 1 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_5, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_1 = ar_2 /\ ar_3 = ar_4 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_5, ar_4, ar_5)) [ 0 >= ar_0 /\ ar_1 = ar_2 /\ ar_3 = ar_4 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start(ar_0, ar_2, ar_2, ar_4, ar_4, ar_0))
5.21/2.38	
5.21/2.38			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
5.21/2.38	
5.21/2.38		start location:	koat_start
5.21/2.38	
5.21/2.38		leaf cost:	0
5.21/2.38	
5.21/2.38	
5.21/2.38	
5.21/2.38	Repeatedly propagating knowledge in problem 2 produces the following problem:
5.21/2.38	
5.21/2.38	3:	T:
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_3 >= 1 /\ ar_3 >= 0 /\ ar_0 >= ar_3 + 1 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_3 = 0 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_1 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= 1 /\ ar_0 >= ar_3 /\ ar_0 >= ar_1 + 1 /\ ar_1 >= 0 /\ ar_3 >= 1 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl72(ar_0, ar_1, ar_2, ar_3 - 1, ar_4, ar_5)) [ ar_0 >= ar_3 /\ ar_0 >= 1 /\ ar_3 >= 1 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_5, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_1 = ar_2 /\ ar_3 = ar_4 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_5, ar_4, ar_5)) [ 0 >= ar_0 /\ ar_1 = ar_2 /\ ar_3 = ar_4 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: 1, Cost: 1)    start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start(ar_0, ar_2, ar_2, ar_4, ar_4, ar_0))
5.21/2.38	
5.21/2.38			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
5.21/2.38	
5.21/2.38		start location:	koat_start
5.21/2.38	
5.21/2.38		leaf cost:	0
5.21/2.38	
5.21/2.38	
5.21/2.38	
5.21/2.38	A polynomial rank function with
5.21/2.38	
5.21/2.38		Pol(lbl72) = 1
5.21/2.38	
5.21/2.38		Pol(lbl62) = 1
5.21/2.38	
5.21/2.38		Pol(stop) = 0
5.21/2.38	
5.21/2.38		Pol(start) = 1
5.21/2.38	
5.21/2.38		Pol(start0) = 1
5.21/2.38	
5.21/2.38		Pol(koat_start) = 1
5.21/2.38	
5.21/2.38	orients all transitions weakly and the transition
5.21/2.38	
5.21/2.38		lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_3 = 0 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38	strictly and produces the following problem:
5.21/2.38	
5.21/2.38	4:	T:
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_3 >= 1 /\ ar_3 >= 0 /\ ar_0 >= ar_3 + 1 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: 1, Cost: 1)    lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_3 = 0 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_1 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= 1 /\ ar_0 >= ar_3 /\ ar_0 >= ar_1 + 1 /\ ar_1 >= 0 /\ ar_3 >= 1 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)    lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl72(ar_0, ar_1, ar_2, ar_3 - 1, ar_4, ar_5)) [ ar_0 >= ar_3 /\ ar_0 >= 1 /\ ar_3 >= 1 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_5, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_1 = ar_2 /\ ar_3 = ar_4 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_5, ar_4, ar_5)) [ 0 >= ar_0 /\ ar_1 = ar_2 /\ ar_3 = ar_4 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: 1, Cost: 1)    start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start(ar_0, ar_2, ar_2, ar_4, ar_4, ar_0))
5.21/2.38	
5.21/2.38			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
5.21/2.38	
5.21/2.38		start location:	koat_start
5.21/2.38	
5.21/2.38		leaf cost:	0
5.21/2.38	
5.21/2.38	
5.21/2.38	
5.21/2.38	A polynomial rank function with
5.21/2.38	
5.21/2.38		Pol(lbl72) = 2*V_4 + 1
5.21/2.38	
5.21/2.38		Pol(lbl62) = 2*V_4
5.21/2.38	
5.21/2.38		Pol(stop) = 2*V_1 + 2*V_4 - 2*V_6
5.21/2.38	
5.21/2.38		Pol(start) = 2*V_1
5.21/2.38	
5.21/2.38		Pol(start0) = 2*V_1
5.21/2.38	
5.21/2.38		Pol(koat_start) = 2*V_1
5.21/2.38	
5.21/2.38	orients all transitions weakly and the transitions
5.21/2.38	
5.21/2.38		lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_3 >= 1 /\ ar_3 >= 0 /\ ar_0 >= ar_3 + 1 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38		lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl72(ar_0, ar_1, ar_2, ar_3 - 1, ar_4, ar_5)) [ ar_0 >= ar_3 /\ ar_0 >= 1 /\ ar_3 >= 1 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38	strictly and produces the following problem:
5.21/2.38	
5.21/2.38	5:	T:
5.21/2.38	
5.21/2.38			(Comp: 2*ar_0, Cost: 1)    lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_3 >= 1 /\ ar_3 >= 0 /\ ar_0 >= ar_3 + 1 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: 1, Cost: 1)         lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_3 = 0 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: ?, Cost: 1)         lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_1 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= 1 /\ ar_0 >= ar_3 /\ ar_0 >= ar_1 + 1 /\ ar_1 >= 0 /\ ar_3 >= 1 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: 2*ar_0, Cost: 1)    lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl72(ar_0, ar_1, ar_2, ar_3 - 1, ar_4, ar_5)) [ ar_0 >= ar_3 /\ ar_0 >= 1 /\ ar_3 >= 1 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: 1, Cost: 1)         start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_5, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_1 = ar_2 /\ ar_3 = ar_4 /\ ar_5 = ar_0 ]
5.21/2.38	
5.21/2.38			(Comp: 1, Cost: 1)         start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_5, ar_4, ar_5)) [ 0 >= ar_0 /\ ar_1 = ar_2 /\ ar_3 = ar_4 /\ ar_5 = ar_0 ]
5.21/2.39	
5.21/2.39			(Comp: 1, Cost: 1)         start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start(ar_0, ar_2, ar_2, ar_4, ar_4, ar_0))
5.21/2.39	
5.21/2.39			(Comp: 1, Cost: 0)         koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
5.21/2.39	
5.21/2.39		start location:	koat_start
5.21/2.39	
5.21/2.39		leaf cost:	0
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	A polynomial rank function with
5.21/2.39	
5.21/2.39		Pol(lbl62) = V_2 + 1
5.21/2.39	
5.21/2.39	and size complexities
5.21/2.39	
5.21/2.39		S("koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-0) = ar_0
5.21/2.39	
5.21/2.39		S("koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-1) = ar_1
5.21/2.39	
5.21/2.39		S("koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-2) = ar_2
5.21/2.39	
5.21/2.39		S("koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-3) = ar_3
5.21/2.39	
5.21/2.39		S("koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-4) = ar_4
5.21/2.39	
5.21/2.39		S("koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-5) = ar_5
5.21/2.39	
5.21/2.39		S("start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start(ar_0, ar_2, ar_2, ar_4, ar_4, ar_0))", 0-0) = ar_0
5.21/2.39	
5.21/2.39		S("start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start(ar_0, ar_2, ar_2, ar_4, ar_4, ar_0))", 0-1) = ar_2
5.21/2.39	
5.21/2.39		S("start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start(ar_0, ar_2, ar_2, ar_4, ar_4, ar_0))", 0-2) = ar_2
5.21/2.39	
5.21/2.39		S("start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start(ar_0, ar_2, ar_2, ar_4, ar_4, ar_0))", 0-3) = ar_4
5.21/2.39	
5.21/2.39		S("start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start(ar_0, ar_2, ar_2, ar_4, ar_4, ar_0))", 0-4) = ar_4
5.21/2.39	
5.21/2.39		S("start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start(ar_0, ar_2, ar_2, ar_4, ar_4, ar_0))", 0-5) = ar_0
5.21/2.39	
5.21/2.39		S("start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_5, ar_4, ar_5)) [ 0 >= ar_0 /\\ ar_1 = ar_2 /\\ ar_3 = ar_4 /\\ ar_5 = ar_0 ]", 0-0) = ar_0
5.21/2.39	
5.21/2.39		S("start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_5, ar_4, ar_5)) [ 0 >= ar_0 /\\ ar_1 = ar_2 /\\ ar_3 = ar_4 /\\ ar_5 = ar_0 ]", 0-1) = ar_2
5.21/2.39	
5.21/2.39		S("start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_5, ar_4, ar_5)) [ 0 >= ar_0 /\\ ar_1 = ar_2 /\\ ar_3 = ar_4 /\\ ar_5 = ar_0 ]", 0-2) = ar_2
5.21/2.39	
5.21/2.39		S("start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_5, ar_4, ar_5)) [ 0 >= ar_0 /\\ ar_1 = ar_2 /\\ ar_3 = ar_4 /\\ ar_5 = ar_0 ]", 0-3) = ar_0
5.21/2.39	
5.21/2.39		S("start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_5, ar_4, ar_5)) [ 0 >= ar_0 /\\ ar_1 = ar_2 /\\ ar_3 = ar_4 /\\ ar_5 = ar_0 ]", 0-4) = ar_4
5.21/2.39	
5.21/2.39		S("start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_5, ar_4, ar_5)) [ 0 >= ar_0 /\\ ar_1 = ar_2 /\\ ar_3 = ar_4 /\\ ar_5 = ar_0 ]", 0-5) = ar_0
5.21/2.39	
5.21/2.39		S("start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_5, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_1 = ar_2 /\\ ar_3 = ar_4 /\\ ar_5 = ar_0 ]", 0-0) = ar_0
5.21/2.39	
5.21/2.39		S("start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_5, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_1 = ar_2 /\\ ar_3 = ar_4 /\\ ar_5 = ar_0 ]", 0-1) = ar_0
5.21/2.39	
5.21/2.39		S("start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_5, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_1 = ar_2 /\\ ar_3 = ar_4 /\\ ar_5 = ar_0 ]", 0-2) = ar_2
5.21/2.39	
5.21/2.39		S("start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_5, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_1 = ar_2 /\\ ar_3 = ar_4 /\\ ar_5 = ar_0 ]", 0-3) = ar_0
5.21/2.39	
5.21/2.39		S("start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_5, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_1 = ar_2 /\\ ar_3 = ar_4 /\\ ar_5 = ar_0 ]", 0-4) = ar_4
5.21/2.39	
5.21/2.39		S("start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_5, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_1 = ar_2 /\\ ar_3 = ar_4 /\\ ar_5 = ar_0 ]", 0-5) = ar_0
5.21/2.39	
5.21/2.39		S("lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl72(ar_0, ar_1, ar_2, ar_3 - 1, ar_4, ar_5)) [ ar_0 >= ar_3 /\\ ar_0 >= 1 /\\ ar_3 >= 1 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-0) = ar_0
5.21/2.39	
5.21/2.39		S("lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl72(ar_0, ar_1, ar_2, ar_3 - 1, ar_4, ar_5)) [ ar_0 >= ar_3 /\\ ar_0 >= 1 /\\ ar_3 >= 1 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-1) = 0
5.21/2.39	
5.21/2.39		S("lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl72(ar_0, ar_1, ar_2, ar_3 - 1, ar_4, ar_5)) [ ar_0 >= ar_3 /\\ ar_0 >= 1 /\\ ar_3 >= 1 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-2) = ar_2
5.21/2.39	
5.21/2.39		S("lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl72(ar_0, ar_1, ar_2, ar_3 - 1, ar_4, ar_5)) [ ar_0 >= ar_3 /\\ ar_0 >= 1 /\\ ar_3 >= 1 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-3) = ar_0
5.21/2.39	
5.21/2.39		S("lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl72(ar_0, ar_1, ar_2, ar_3 - 1, ar_4, ar_5)) [ ar_0 >= ar_3 /\\ ar_0 >= 1 /\\ ar_3 >= 1 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-4) = ar_4
5.21/2.39	
5.21/2.39		S("lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl72(ar_0, ar_1, ar_2, ar_3 - 1, ar_4, ar_5)) [ ar_0 >= ar_3 /\\ ar_0 >= 1 /\\ ar_3 >= 1 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-5) = ar_0
5.21/2.39	
5.21/2.39		S("lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_1 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= 1 /\\ ar_0 >= ar_3 /\\ ar_0 >= ar_1 + 1 /\\ ar_1 >= 0 /\\ ar_3 >= 1 /\\ ar_5 = ar_0 ]", 0-0) = ar_0
5.21/2.39	
5.21/2.39		S("lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_1 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= 1 /\\ ar_0 >= ar_3 /\\ ar_0 >= ar_1 + 1 /\\ ar_1 >= 0 /\\ ar_3 >= 1 /\\ ar_5 = ar_0 ]", 0-1) = ar_0
5.21/2.39	
5.21/2.39		S("lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_1 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= 1 /\\ ar_0 >= ar_3 /\\ ar_0 >= ar_1 + 1 /\\ ar_1 >= 0 /\\ ar_3 >= 1 /\\ ar_5 = ar_0 ]", 0-2) = ar_2
5.21/2.39	
5.21/2.39		S("lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_1 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= 1 /\\ ar_0 >= ar_3 /\\ ar_0 >= ar_1 + 1 /\\ ar_1 >= 0 /\\ ar_3 >= 1 /\\ ar_5 = ar_0 ]", 0-3) = ar_0
5.21/2.39	
5.21/2.39		S("lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_1 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= 1 /\\ ar_0 >= ar_3 /\\ ar_0 >= ar_1 + 1 /\\ ar_1 >= 0 /\\ ar_3 >= 1 /\\ ar_5 = ar_0 ]", 0-4) = ar_4
5.21/2.39	
5.21/2.39		S("lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_1 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= 1 /\\ ar_0 >= ar_3 /\\ ar_0 >= ar_1 + 1 /\\ ar_1 >= 0 /\\ ar_3 >= 1 /\\ ar_5 = ar_0 ]", 0-5) = ar_0
5.21/2.39	
5.21/2.39		S("lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_3 = 0 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-0) = ar_0
5.21/2.39	
5.21/2.39		S("lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_3 = 0 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-1) = 0
5.21/2.39	
5.21/2.39		S("lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_3 = 0 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-2) = ar_2
5.21/2.39	
5.21/2.39		S("lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_3 = 0 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-3) = 0
5.21/2.39	
5.21/2.39		S("lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_3 = 0 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-4) = ar_4
5.21/2.39	
5.21/2.39		S("lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_3 = 0 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-5) = ar_0
5.21/2.39	
5.21/2.39		S("lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_3 >= 1 /\\ ar_3 >= 0 /\\ ar_0 >= ar_3 + 1 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-0) = ar_0
5.21/2.39	
5.21/2.39		S("lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_3 >= 1 /\\ ar_3 >= 0 /\\ ar_0 >= ar_3 + 1 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-1) = ar_0
5.21/2.39	
5.21/2.39		S("lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_3 >= 1 /\\ ar_3 >= 0 /\\ ar_0 >= ar_3 + 1 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-2) = ar_2
5.21/2.39	
5.21/2.39		S("lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_3 >= 1 /\\ ar_3 >= 0 /\\ ar_0 >= ar_3 + 1 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-3) = ar_0
5.21/2.39	
5.21/2.39		S("lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_3 >= 1 /\\ ar_3 >= 0 /\\ ar_0 >= ar_3 + 1 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-4) = ar_4
5.21/2.39	
5.21/2.39		S("lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\\ ar_3 >= 1 /\\ ar_3 >= 0 /\\ ar_0 >= ar_3 + 1 /\\ ar_1 = 0 /\\ ar_5 = ar_0 ]", 0-5) = ar_0
5.21/2.39	
5.21/2.39	orients the transitions
5.21/2.39	
5.21/2.39		lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_1 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= 1 /\ ar_0 >= ar_3 /\ ar_0 >= ar_1 + 1 /\ ar_1 >= 0 /\ ar_3 >= 1 /\ ar_5 = ar_0 ]
5.21/2.39	
5.21/2.39	weakly and the transition
5.21/2.39	
5.21/2.39		lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_1 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= 1 /\ ar_0 >= ar_3 /\ ar_0 >= ar_1 + 1 /\ ar_1 >= 0 /\ ar_3 >= 1 /\ ar_5 = ar_0 ]
5.21/2.39	
5.21/2.39	strictly and produces the following problem:
5.21/2.39	
5.21/2.39	6:	T:
5.21/2.39	
5.21/2.39			(Comp: 2*ar_0, Cost: 1)                   lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_3 >= 1 /\ ar_3 >= 0 /\ ar_0 >= ar_3 + 1 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.39	
5.21/2.39			(Comp: 1, Cost: 1)                        lbl72(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_3 = 0 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.39	
5.21/2.39			(Comp: 2*ar_0^2 + 3*ar_0 + 1, Cost: 1)    lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_1 - 1, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= 1 /\ ar_0 >= ar_3 /\ ar_0 >= ar_1 + 1 /\ ar_1 >= 0 /\ ar_3 >= 1 /\ ar_5 = ar_0 ]
5.21/2.39	
5.21/2.39			(Comp: 2*ar_0, Cost: 1)                   lbl62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl72(ar_0, ar_1, ar_2, ar_3 - 1, ar_4, ar_5)) [ ar_0 >= ar_3 /\ ar_0 >= 1 /\ ar_3 >= 1 /\ ar_1 = 0 /\ ar_5 = ar_0 ]
5.21/2.39	
5.21/2.39			(Comp: 1, Cost: 1)                        start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(lbl62(ar_0, ar_5 - 1, ar_2, ar_5, ar_4, ar_5)) [ ar_0 >= 1 /\ ar_1 = ar_2 /\ ar_3 = ar_4 /\ ar_5 = ar_0 ]
5.21/2.39	
5.21/2.39			(Comp: 1, Cost: 1)                        start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(stop(ar_0, ar_1, ar_2, ar_5, ar_4, ar_5)) [ 0 >= ar_0 /\ ar_1 = ar_2 /\ ar_3 = ar_4 /\ ar_5 = ar_0 ]
5.21/2.39	
5.21/2.39			(Comp: 1, Cost: 1)                        start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start(ar_0, ar_2, ar_2, ar_4, ar_4, ar_0))
5.21/2.39	
5.21/2.39			(Comp: 1, Cost: 0)                        koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
5.21/2.39	
5.21/2.39		start location:	koat_start
5.21/2.39	
5.21/2.39		leaf cost:	0
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Complexity upper bound 7*ar_0 + 2*ar_0^2 + 5
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Time: 0.264 sec (SMT: 0.227 sec)
5.21/2.39	
5.21/2.39	
5.21/2.39	----------------------------------------
5.21/2.39	
5.21/2.39	(2)
5.21/2.39	BOUNDS(1, n^2)
5.21/2.39	
5.21/2.39	----------------------------------------
5.21/2.39	
5.21/2.39	(3) Loat Proof (FINISHED)
5.21/2.39	
5.21/2.39	
5.21/2.39	### Pre-processing the ITS problem ###
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Initial linear ITS problem
5.21/2.39	
5.21/2.39	   Start location: start0
5.21/2.39	
5.21/2.39	      0: start -> stop : D'=F, [ 0>=A && B==C && D==E && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      1: start -> lbl62 : B'=-1+F, D'=F, [ A>=1 && B==C && D==E && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      2: lbl72 -> stop : [ A>=1 && D==0 && B==0 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      3: lbl72 -> lbl72 : B'=F, D'=-1+D, [ D>=1 && 0>=A && D>=0 && A>=1+D && B==0 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      4: lbl72 -> lbl62 : B'=-1+F, [ A>=1 && D>=1 && D>=0 && A>=1+D && B==0 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      5: lbl62 -> lbl72 : D'=-1+D, [ A>=D && A>=1 && D>=1 && B==0 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      6: lbl62 -> lbl62 : B'=-1+B, [ B>=1 && A>=D && A>=1+B && B>=0 && D>=1 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      7: start0 -> start : B'=C, D'=E, F'=A, [], cost: 1
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Removed unreachable and leaf rules:
5.21/2.39	
5.21/2.39	   Start location: start0
5.21/2.39	
5.21/2.39	      1: start -> lbl62 : B'=-1+F, D'=F, [ A>=1 && B==C && D==E && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      3: lbl72 -> lbl72 : B'=F, D'=-1+D, [ D>=1 && 0>=A && D>=0 && A>=1+D && B==0 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      4: lbl72 -> lbl62 : B'=-1+F, [ A>=1 && D>=1 && D>=0 && A>=1+D && B==0 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      5: lbl62 -> lbl72 : D'=-1+D, [ A>=D && A>=1 && D>=1 && B==0 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      6: lbl62 -> lbl62 : B'=-1+B, [ B>=1 && A>=D && A>=1+B && B>=0 && D>=1 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      7: start0 -> start : B'=C, D'=E, F'=A, [], cost: 1
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Removed rules with unsatisfiable guard:
5.21/2.39	
5.21/2.39	   Start location: start0
5.21/2.39	
5.21/2.39	      1: start -> lbl62 : B'=-1+F, D'=F, [ A>=1 && B==C && D==E && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      4: lbl72 -> lbl62 : B'=-1+F, [ A>=1 && D>=1 && D>=0 && A>=1+D && B==0 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      5: lbl62 -> lbl72 : D'=-1+D, [ A>=D && A>=1 && D>=1 && B==0 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      6: lbl62 -> lbl62 : B'=-1+B, [ B>=1 && A>=D && A>=1+B && B>=0 && D>=1 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      7: start0 -> start : B'=C, D'=E, F'=A, [], cost: 1
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Simplified all rules, resulting in:
5.21/2.39	
5.21/2.39	   Start location: start0
5.21/2.39	
5.21/2.39	      1: start -> lbl62 : B'=-1+F, D'=F, [ A>=1 && B==C && D==E && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      4: lbl72 -> lbl62 : B'=-1+F, [ D>=1 && A>=1+D && B==0 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      5: lbl62 -> lbl72 : D'=-1+D, [ A>=D && A>=1 && D>=1 && B==0 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      6: lbl62 -> lbl62 : B'=-1+B, [ B>=1 && A>=D && A>=1+B && D>=1 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      7: start0 -> start : B'=C, D'=E, F'=A, [], cost: 1
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	### Simplification by acceleration and chaining ###
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Accelerating simple loops of location 2.
5.21/2.39	
5.21/2.39	   Accelerating the following rules:
5.21/2.39	
5.21/2.39	      6: lbl62 -> lbl62 : B'=-1+B, [ B>=1 && A>=D && A>=1+B && D>=1 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	   Accelerated rule 6 with metering function B, yielding the new rule 8.
5.21/2.39	
5.21/2.39	   Removing the simple loops: 6.
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Accelerated all simple loops using metering functions (where possible):
5.21/2.39	
5.21/2.39	   Start location: start0
5.21/2.39	
5.21/2.39	      1: start -> lbl62 : B'=-1+F, D'=F, [ A>=1 && B==C && D==E && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      4: lbl72 -> lbl62 : B'=-1+F, [ D>=1 && A>=1+D && B==0 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      5: lbl62 -> lbl72 : D'=-1+D, [ A>=D && A>=1 && D>=1 && B==0 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      8: lbl62 -> lbl62 : B'=0, [ B>=1 && A>=D && A>=1+B && D>=1 && F==A ], cost: B
5.21/2.39	
5.21/2.39	      7: start0 -> start : B'=C, D'=E, F'=A, [], cost: 1
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Chained accelerated rules (with incoming rules):
5.21/2.39	
5.21/2.39	   Start location: start0
5.21/2.39	
5.21/2.39	      1: start -> lbl62 : B'=-1+F, D'=F, [ A>=1 && B==C && D==E && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      9: start -> lbl62 : B'=0, D'=F, [ A>=1 && B==C && D==E && F==A && -1+F>=1 ], cost: F
5.21/2.39	
5.21/2.39	      4: lbl72 -> lbl62 : B'=-1+F, [ D>=1 && A>=1+D && B==0 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	     10: lbl72 -> lbl62 : B'=0, [ D>=1 && A>=1+D && B==0 && F==A && -1+F>=1 ], cost: F
5.21/2.39	
5.21/2.39	      5: lbl62 -> lbl72 : D'=-1+D, [ A>=D && A>=1 && D>=1 && B==0 && F==A ], cost: 1
5.21/2.39	
5.21/2.39	      7: start0 -> start : B'=C, D'=E, F'=A, [], cost: 1
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Eliminated locations (on tree-shaped paths):
5.21/2.39	
5.21/2.39	   Start location: start0
5.21/2.39	
5.21/2.39	     13: lbl62 -> lbl62 : B'=-1+F, D'=-1+D, [ A>=D && A>=1 && B==0 && F==A && -1+D>=1 ], cost: 2
5.21/2.39	
5.21/2.39	     14: lbl62 -> lbl62 : B'=0, D'=-1+D, [ A>=D && A>=1 && B==0 && F==A && -1+D>=1 && -1+F>=1 ], cost: 1+F
5.21/2.39	
5.21/2.39	     11: start0 -> lbl62 : B'=-1+A, D'=A, F'=A, [ A>=1 ], cost: 2
5.21/2.39	
5.21/2.39	     12: start0 -> lbl62 : B'=0, D'=A, F'=A, [ -1+A>=1 ], cost: 1+A
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Accelerating simple loops of location 2.
5.21/2.39	
5.21/2.39	   Accelerating the following rules:
5.21/2.39	
5.21/2.39	     13: lbl62 -> lbl62 : B'=-1+F, D'=-1+D, [ A>=D && A>=1 && B==0 && F==A && -1+D>=1 ], cost: 2
5.21/2.39	
5.21/2.39	     14: lbl62 -> lbl62 : B'=0, D'=-1+D, [ A>=D && A>=1 && B==0 && F==A && -1+D>=1 && -1+F>=1 ], cost: 1+F
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	   Accelerated rule 13 with NONTERM (after strengthening guard), yielding the new rule 15.
5.21/2.39	
5.21/2.39	   Accelerated rule 14 with metering function -1+D, yielding the new rule 16.
5.21/2.39	
5.21/2.39	   Removing the simple loops: 14.
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Accelerated all simple loops using metering functions (where possible):
5.21/2.39	
5.21/2.39	   Start location: start0
5.21/2.39	
5.21/2.39	     13: lbl62 -> lbl62 : B'=-1+F, D'=-1+D, [ A>=D && A>=1 && B==0 && F==A && -1+D>=1 ], cost: 2
5.21/2.39	
5.21/2.39	     15: lbl62 -> [6] : [ A>=D && A>=1 && B==0 && F==A && -1+D>=1 && -1+F==0 ], cost: INF
5.21/2.39	
5.21/2.39	     16: lbl62 -> lbl62 : B'=0, D'=1, [ A>=D && A>=1 && B==0 && F==A && -1+D>=1 && -1+F>=1 ], cost: -1+D+F*(-1+D)
5.21/2.39	
5.21/2.39	     11: start0 -> lbl62 : B'=-1+A, D'=A, F'=A, [ A>=1 ], cost: 2
5.21/2.39	
5.21/2.39	     12: start0 -> lbl62 : B'=0, D'=A, F'=A, [ -1+A>=1 ], cost: 1+A
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Chained accelerated rules (with incoming rules):
5.21/2.39	
5.21/2.39	   Start location: start0
5.21/2.39	
5.21/2.39	     11: start0 -> lbl62 : B'=-1+A, D'=A, F'=A, [ A>=1 ], cost: 2
5.21/2.39	
5.21/2.39	     12: start0 -> lbl62 : B'=0, D'=A, F'=A, [ -1+A>=1 ], cost: 1+A
5.21/2.39	
5.21/2.39	     17: start0 -> lbl62 : B'=-1+A, D'=-1+A, F'=A, [ -1+A>=1 ], cost: 3+A
5.21/2.39	
5.21/2.39	     18: start0 -> lbl62 : B'=0, D'=1, F'=A, [ -1+A>=1 ], cost: (-1+A)*A+2*A
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Removed unreachable locations (and leaf rules with constant cost):
5.21/2.39	
5.21/2.39	   Start location: start0
5.21/2.39	
5.21/2.39	     12: start0 -> lbl62 : B'=0, D'=A, F'=A, [ -1+A>=1 ], cost: 1+A
5.21/2.39	
5.21/2.39	     17: start0 -> lbl62 : B'=-1+A, D'=-1+A, F'=A, [ -1+A>=1 ], cost: 3+A
5.21/2.39	
5.21/2.39	     18: start0 -> lbl62 : B'=0, D'=1, F'=A, [ -1+A>=1 ], cost: (-1+A)*A+2*A
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	### Computing asymptotic complexity ###
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Fully simplified ITS problem
5.21/2.39	
5.21/2.39	   Start location: start0
5.21/2.39	
5.21/2.39	     17: start0 -> lbl62 : B'=-1+A, D'=-1+A, F'=A, [ -1+A>=1 ], cost: 3+A
5.21/2.39	
5.21/2.39	     18: start0 -> lbl62 : B'=0, D'=1, F'=A, [ -1+A>=1 ], cost: (-1+A)*A+2*A
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Computing asymptotic complexity for rule 17
5.21/2.39	
5.21/2.39	   Solved the limit problem by the following transformations:
5.21/2.39	
5.21/2.39	      Created initial limit problem:
5.21/2.39	
5.21/2.39	      -1+A (+/+!), 3+A (+) [not solved]
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	      removing all constraints (solved by SMT)
5.21/2.39	
5.21/2.39	      resulting limit problem: [solved]
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	      applying transformation rule (C) using substitution {A==n}
5.21/2.39	
5.21/2.39	      resulting limit problem:
5.21/2.39	
5.21/2.39	      [solved]
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	   Solution:
5.21/2.39	
5.21/2.39	      A / n
5.21/2.39	
5.21/2.39	   Resulting cost 3+n has complexity: Poly(n^1)
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Found new complexity Poly(n^1).
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Computing asymptotic complexity for rule 18
5.21/2.39	
5.21/2.39	   Solved the limit problem by the following transformations:
5.21/2.39	
5.21/2.39	      Created initial limit problem:
5.21/2.39	
5.21/2.39	      -1+A (+/+!), A+A^2 (+) [not solved]
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	      removing all constraints (solved by SMT)
5.21/2.39	
5.21/2.39	      resulting limit problem: [solved]
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	      applying transformation rule (C) using substitution {A==n}
5.21/2.39	
5.21/2.39	      resulting limit problem:
5.21/2.39	
5.21/2.39	      [solved]
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	   Solution:
5.21/2.39	
5.21/2.39	      A / n
5.21/2.39	
5.21/2.39	   Resulting cost n^2+n has complexity: Poly(n^2)
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Found new complexity Poly(n^2).
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	Obtained the following overall complexity (w.r.t. the length of the input n):
5.21/2.39	
5.21/2.39	   Complexity:  Poly(n^2)
5.21/2.39	
5.21/2.39	   Cpx degree:  2
5.21/2.39	
5.21/2.39	   Solved cost: n^2+n
5.21/2.39	
5.21/2.39	   Rule cost:   (-1+A)*A+2*A
5.21/2.39	
5.21/2.39	   Rule guard:  [ -1+A>=1 ]
5.21/2.39	
5.21/2.39	
5.21/2.39	
5.21/2.39	WORST_CASE(Omega(n^2),?)
5.21/2.39	
5.21/2.39	
5.21/2.39	----------------------------------------
5.43/2.39	
5.43/2.39	(4)
5.43/2.39	BOUNDS(n^2, INF)
5.43/2.45	EOF