/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).

(0) CpxIntTrs
(1) Koat Proof [FINISHED, 488 ms]
(2) BOUNDS(1, n^1)
(3) Loat Proof [FINISHED, 1190 ms]
(4) BOUNDS(n^1, INF)


----------------------------------------

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
start(A, B, C, D, E, F, G, H) -> Com_1(stop(A, B, C, D, 0, F, D, H)) :|: 0 >= A && B >= C && B <= C && D >= A && D <= A && E >= F && E <= F && G >= H && G <= H
start(A, B, C, D, E, F, G, H) -> Com_1(cut(A, B, C, D, 0, F, D - 1, H)) :|: A >= 2 && B >= C && B <= C && D >= A && D <= A && E >= F && E <= F && G >= H && G <= H
start(A, B, C, D, E, F, G, H) -> Com_1(stop(A, B, C, D, 0, F, D - 1, H)) :|: D >= 1 && D <= 1 && B >= C && B <= C && A >= 1 && A <= 1 && E >= F && E <= F && G >= H && G <= H
start(A, B, C, D, E, F, G, H) -> Com_1(cut(A, 0, C, D, 1, F, D - 1, H)) :|: A >= 2 && B >= C && B <= C && D >= A && D <= A && E >= F && E <= F && G >= H && G <= H
start(A, B, C, D, E, F, G, H) -> Com_1(stop(A, 0, C, D, 0, F, D - 1, H)) :|: D >= 1 && D <= 1 && B >= C && B <= C && A >= 1 && A <= 1 && E >= F && E <= F && G >= H && G <= H
cut(A, B, C, D, E, F, G, H) -> Com_1(cut(A, B, C, D, E - 1, F, G - 1, H)) :|: G >= 2 && E >= 2 && G >= 1 && E >= 0 && A >= G + 1 && A >= G + E && D >= A && D <= A
cut(A, B, C, D, E, F, G, H) -> Com_1(stop(A, B, C, D, E - 1, F, G - 1, H)) :|: E >= 2 && E >= 0 && A >= 2 && A >= 1 + E && G >= 1 && G <= 1 && D >= A && D <= A
cut(A, B, C, D, E, F, G, H) -> Com_1(cut(A, B, C, D, 0, F, G - 1, H)) :|: G >= 2 && 1 >= E && G >= 1 && E >= 0 && A >= G + 1 && A >= G + E && D >= A && D <= A
cut(A, B, C, D, E, F, G, H) -> Com_1(stop(A, B, C, D, 0, F, G - 1, H)) :|: 1 >= E && E >= 0 && A >= 2 && A >= 1 + E && G >= 1 && G <= 1 && D >= A && D <= A
cut(A, B, C, D, E, F, G, H) -> Com_1(cut(A, E, C, D, 1 + E, F, G - 1, H)) :|: G >= 2 && A >= E + 2 && G >= 1 && E >= 0 && A >= G + 1 && A >= G + E && D >= A && D <= A
cut(A, B, C, D, E, F, G, H) -> Com_1(stop(A, E, C, D, 1 + E, F, G - 1, H)) :|: A >= E + 2 && E >= 0 && A >= 2 && A >= 1 + E && G >= 1 && G <= 1 && D >= A && D <= A
cut(A, B, C, D, E, F, G, H) -> Com_1(cut(A, E, C, D, 0, F, G - 1, H)) :|: G >= 2 && E + 1 >= A && G >= 1 && E >= 0 && A >= G + 1 && A >= G + E && D >= A && D <= A
cut(A, B, C, D, E, F, G, H) -> Com_1(stop(A, E, C, D, 0, F, G - 1, H)) :|: A >= 1 && A >= 2 && G >= 1 && G <= 1 && E + 1 >= A && E + 1 <= A && D >= A && D <= A
start0(A, B, C, D, E, F, G, H) -> Com_1(start(A, C, C, A, F, F, H, H)) :|: TRUE

The start-symbols are:[start0_8]


----------------------------------------

(1) Koat Proof (FINISHED)
YES(?, 3*Ar_0 + 10)



Initial complexity problem:

1:	T:

		(Comp: ?, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_3, Ar_7)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: ?, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_0 >= 2 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: ?, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_3 = 1 /\ Ar_1 = Ar_2 /\ Ar_0 = 1 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: ?, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, 0, Ar_2, Ar_3, 1, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_0 >= 2 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: ?, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, 0, Ar_2, Ar_3, 0, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_3 = 1 /\ Ar_1 = Ar_2 /\ Ar_0 = 1 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ Ar_4 >= 2 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_4 >= 2 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ 1 >= Ar_4 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ 1 >= Ar_4 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_4, Ar_2, Ar_3, Ar_4 + 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ Ar_0 >= Ar_4 + 2 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_4, Ar_2, Ar_3, Ar_4 + 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_0 >= Ar_4 + 2 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_4, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ Ar_4 + 1 >= Ar_0 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_4, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_0 >= 1 /\ Ar_0 >= 2 /\ Ar_6 = 1 /\ Ar_4 + 1 = Ar_0 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_0, Ar_5, Ar_5, Ar_7, Ar_7))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Testing for reachability in the complexity graph removes the following transition from problem 1:

	cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_4, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ Ar_4 + 1 >= Ar_0 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

We thus obtain the following problem:

2:	T:

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_4 >= 2 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ Ar_4 >= 2 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_4, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_0 >= 1 /\ Ar_0 >= 2 /\ Ar_6 = 1 /\ Ar_4 + 1 = Ar_0 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_4, Ar_2, Ar_3, Ar_4 + 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_0 >= Ar_4 + 2 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_4, Ar_2, Ar_3, Ar_4 + 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ Ar_0 >= Ar_4 + 2 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ 1 >= Ar_4 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ 1 >= Ar_4 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, 0, Ar_2, Ar_3, 0, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_3 = 1 /\ Ar_1 = Ar_2 /\ Ar_0 = 1 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: ?, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, 0, Ar_2, Ar_3, 1, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_0 >= 2 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: ?, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_3 = 1 /\ Ar_1 = Ar_2 /\ Ar_0 = 1 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: ?, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_0 >= 2 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: ?, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_3, Ar_7)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: ?, Cost: 1)    start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_0, Ar_5, Ar_5, Ar_7, Ar_7))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Repeatedly propagating knowledge in problem 2 produces the following problem:

3:	T:

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_4 >= 2 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ Ar_4 >= 2 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_4, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_0 >= 1 /\ Ar_0 >= 2 /\ Ar_6 = 1 /\ Ar_4 + 1 = Ar_0 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_4, Ar_2, Ar_3, Ar_4 + 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_0 >= Ar_4 + 2 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_4, Ar_2, Ar_3, Ar_4 + 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ Ar_0 >= Ar_4 + 2 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ 1 >= Ar_4 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ 1 >= Ar_4 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

		(Comp: 1, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, 0, Ar_2, Ar_3, 0, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_3 = 1 /\ Ar_1 = Ar_2 /\ Ar_0 = 1 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: 1, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, 0, Ar_2, Ar_3, 1, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_0 >= 2 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: 1, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_3 = 1 /\ Ar_1 = Ar_2 /\ Ar_0 = 1 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: 1, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_0 >= 2 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: 1, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_3, Ar_7)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: 1, Cost: 1)    start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_0, Ar_5, Ar_5, Ar_7, Ar_7))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(cut) = 1

	Pol(stop) = 0

	Pol(start) = 1

	Pol(start0) = 1

	Pol(koat_start) = 1

orients all transitions weakly and the transitions

	cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_4, Ar_2, Ar_3, Ar_4 + 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_0 >= Ar_4 + 2 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

	cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_4, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_0 >= 1 /\ Ar_0 >= 2 /\ Ar_6 = 1 /\ Ar_4 + 1 = Ar_0 /\ Ar_3 = Ar_0 ]

	cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ 1 >= Ar_4 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

	cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_4 >= 2 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

strictly and produces the following problem:

4:	T:

		(Comp: 1, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_4 >= 2 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ Ar_4 >= 2 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

		(Comp: 1, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_4, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_0 >= 1 /\ Ar_0 >= 2 /\ Ar_6 = 1 /\ Ar_4 + 1 = Ar_0 /\ Ar_3 = Ar_0 ]

		(Comp: 1, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_4, Ar_2, Ar_3, Ar_4 + 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_0 >= Ar_4 + 2 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_4, Ar_2, Ar_3, Ar_4 + 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ Ar_0 >= Ar_4 + 2 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

		(Comp: 1, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ 1 >= Ar_4 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ 1 >= Ar_4 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

		(Comp: 1, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, 0, Ar_2, Ar_3, 0, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_3 = 1 /\ Ar_1 = Ar_2 /\ Ar_0 = 1 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: 1, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, 0, Ar_2, Ar_3, 1, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_0 >= 2 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: 1, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_3 = 1 /\ Ar_1 = Ar_2 /\ Ar_0 = 1 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: 1, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_0 >= 2 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: 1, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_3, Ar_7)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: 1, Cost: 1)    start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_0, Ar_5, Ar_5, Ar_7, Ar_7))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(cut) = V_7

	Pol(stop) = V_7

	Pol(start) = V_1

	Pol(start0) = V_1

	Pol(koat_start) = V_1

orients all transitions weakly and the transitions

	cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_4, Ar_2, Ar_3, Ar_4 + 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ Ar_0 >= Ar_4 + 2 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

	cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ 1 >= Ar_4 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

	cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ Ar_4 >= 2 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

strictly and produces the following problem:

5:	T:

		(Comp: 1, Cost: 1)       cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_4 >= 2 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

		(Comp: Ar_0, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ Ar_4 >= 2 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

		(Comp: 1, Cost: 1)       cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_4, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_0 >= 1 /\ Ar_0 >= 2 /\ Ar_6 = 1 /\ Ar_4 + 1 = Ar_0 /\ Ar_3 = Ar_0 ]

		(Comp: 1, Cost: 1)       cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_4, Ar_2, Ar_3, Ar_4 + 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_0 >= Ar_4 + 2 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

		(Comp: Ar_0, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_4, Ar_2, Ar_3, Ar_4 + 1, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ Ar_0 >= Ar_4 + 2 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

		(Comp: 1, Cost: 1)       cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ 1 >= Ar_4 /\ Ar_4 >= 0 /\ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 1 /\ Ar_6 = 1 /\ Ar_3 = Ar_0 ]

		(Comp: Ar_0, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_6 - 1, Ar_7)) [ Ar_6 >= 2 /\ 1 >= Ar_4 /\ Ar_6 >= 1 /\ Ar_4 >= 0 /\ Ar_0 >= Ar_6 + 1 /\ Ar_0 >= Ar_6 + Ar_4 /\ Ar_3 = Ar_0 ]

		(Comp: 1, Cost: 1)       start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, 0, Ar_2, Ar_3, 0, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_3 = 1 /\ Ar_1 = Ar_2 /\ Ar_0 = 1 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: 1, Cost: 1)       start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, 0, Ar_2, Ar_3, 1, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_0 >= 2 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: 1, Cost: 1)       start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_3 = 1 /\ Ar_1 = Ar_2 /\ Ar_0 = 1 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: 1, Cost: 1)       start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_3 - 1, Ar_7)) [ Ar_0 >= 2 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: 1, Cost: 1)       start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_3, Ar_7)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 /\ Ar_6 = Ar_7 ]

		(Comp: 1, Cost: 1)       start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_0, Ar_5, Ar_5, Ar_7, Ar_7))

		(Comp: 1, Cost: 0)       koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6, Ar_7)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 3*Ar_0 + 10



Time: 0.468 sec (SMT: 0.339 sec)


----------------------------------------

(2)
BOUNDS(1, n^1)

----------------------------------------

(3) Loat Proof (FINISHED)


### Pre-processing the ITS problem ###



Initial linear ITS problem

   Start location: start0

      0: start -> stop : E'=0, G'=D, [ 0>=A && B==C && D==A && E==F && G==H ], cost: 1

      1: start -> cut : E'=0, G'=-1+D, [ A>=2 && B==C && D==A && E==F && G==H ], cost: 1

      2: start -> stop : E'=0, G'=-1+D, [ D==1 && B==C && A==1 && E==F && G==H ], cost: 1

      3: start -> cut : B'=0, E'=1, G'=-1+D, [ A>=2 && B==C && D==A && E==F && G==H ], cost: 1

      4: start -> stop : B'=0, E'=0, G'=-1+D, [ D==1 && B==C && A==1 && E==F && G==H ], cost: 1

      5: cut -> cut : E'=-1+E, G'=-1+G, [ G>=2 && E>=2 && G>=1 && E>=0 && A>=1+G && A>=G+E && D==A ], cost: 1

      6: cut -> stop : E'=-1+E, G'=-1+G, [ E>=2 && E>=0 && A>=2 && A>=1+E && G==1 && D==A ], cost: 1

      7: cut -> cut : E'=0, G'=-1+G, [ G>=2 && 1>=E && G>=1 && E>=0 && A>=1+G && A>=G+E && D==A ], cost: 1

      8: cut -> stop : E'=0, G'=-1+G, [ 1>=E && E>=0 && A>=2 && A>=1+E && G==1 && D==A ], cost: 1

      9: cut -> cut : B'=E, E'=1+E, G'=-1+G, [ G>=2 && A>=2+E && G>=1 && E>=0 && A>=1+G && A>=G+E && D==A ], cost: 1

     10: cut -> stop : B'=E, E'=1+E, G'=-1+G, [ A>=2+E && E>=0 && A>=2 && A>=1+E && G==1 && D==A ], cost: 1

     11: cut -> cut : B'=E, E'=0, G'=-1+G, [ G>=2 && 1+E>=A && G>=1 && E>=0 && A>=1+G && A>=G+E && D==A ], cost: 1

     12: cut -> stop : B'=E, E'=0, G'=-1+G, [ A>=1 && A>=2 && G==1 && 1+E==A && D==A ], cost: 1

     13: start0 -> start : B'=C, D'=A, E'=F, G'=H, [], cost: 1



Checking for constant complexity:

   The following rule is satisfiable with cost >= 1, yielding constant complexity:

     13: start0 -> start : B'=C, D'=A, E'=F, G'=H, [], cost: 1



Removed unreachable and leaf rules:

   Start location: start0

      1: start -> cut : E'=0, G'=-1+D, [ A>=2 && B==C && D==A && E==F && G==H ], cost: 1

      3: start -> cut : B'=0, E'=1, G'=-1+D, [ A>=2 && B==C && D==A && E==F && G==H ], cost: 1

      5: cut -> cut : E'=-1+E, G'=-1+G, [ G>=2 && E>=2 && G>=1 && E>=0 && A>=1+G && A>=G+E && D==A ], cost: 1

      7: cut -> cut : E'=0, G'=-1+G, [ G>=2 && 1>=E && G>=1 && E>=0 && A>=1+G && A>=G+E && D==A ], cost: 1

      9: cut -> cut : B'=E, E'=1+E, G'=-1+G, [ G>=2 && A>=2+E && G>=1 && E>=0 && A>=1+G && A>=G+E && D==A ], cost: 1

     11: cut -> cut : B'=E, E'=0, G'=-1+G, [ G>=2 && 1+E>=A && G>=1 && E>=0 && A>=1+G && A>=G+E && D==A ], cost: 1

     13: start0 -> start : B'=C, D'=A, E'=F, G'=H, [], cost: 1



Removed rules with unsatisfiable guard:

   Start location: start0

      1: start -> cut : E'=0, G'=-1+D, [ A>=2 && B==C && D==A && E==F && G==H ], cost: 1

      3: start -> cut : B'=0, E'=1, G'=-1+D, [ A>=2 && B==C && D==A && E==F && G==H ], cost: 1

      5: cut -> cut : E'=-1+E, G'=-1+G, [ G>=2 && E>=2 && G>=1 && E>=0 && A>=1+G && A>=G+E && D==A ], cost: 1

      7: cut -> cut : E'=0, G'=-1+G, [ G>=2 && 1>=E && G>=1 && E>=0 && A>=1+G && A>=G+E && D==A ], cost: 1

      9: cut -> cut : B'=E, E'=1+E, G'=-1+G, [ G>=2 && A>=2+E && G>=1 && E>=0 && A>=1+G && A>=G+E && D==A ], cost: 1

     13: start0 -> start : B'=C, D'=A, E'=F, G'=H, [], cost: 1



Simplified all rules, resulting in:

   Start location: start0

      1: start -> cut : E'=0, G'=-1+D, [ A>=2 && B==C && D==A && E==F && G==H ], cost: 1

      3: start -> cut : B'=0, E'=1, G'=-1+D, [ A>=2 && B==C && D==A && E==F && G==H ], cost: 1

      5: cut -> cut : E'=-1+E, G'=-1+G, [ G>=2 && E>=2 && A>=1+G && A>=G+E && D==A ], cost: 1

      7: cut -> cut : E'=0, G'=-1+G, [ G>=2 && 1>=E && E>=0 && A>=1+G && D==A ], cost: 1

      9: cut -> cut : B'=E, E'=1+E, G'=-1+G, [ G>=2 && A>=2+E && E>=0 && A>=1+G && A>=G+E && D==A ], cost: 1

     13: start0 -> start : B'=C, D'=A, E'=F, G'=H, [], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      5: cut -> cut : E'=-1+E, G'=-1+G, [ G>=2 && E>=2 && A>=1+G && A>=G+E && D==A ], cost: 1

      7: cut -> cut : E'=0, G'=-1+G, [ G>=2 && 1>=E && E>=0 && A>=1+G && D==A ], cost: 1

      9: cut -> cut : B'=E, E'=1+E, G'=-1+G, [ G>=2 && A>=2+E && E>=0 && A>=1+G && A>=G+E && D==A ], cost: 1



   Accelerated rule 5 with metering function -1+G (after adding E>=G), yielding the new rule 14.

   Accelerated rule 5 with metering function -1+E (after adding E<=G), yielding the new rule 15.

   Accelerated rule 7 with metering function -1+G, yielding the new rule 16.

   Accelerated rule 9 with metering function -1+G, yielding the new rule 17.

   Removing the simple loops: 5 7 9.



Accelerated all simple loops using metering functions (where possible):

   Start location: start0

      1: start -> cut : E'=0, G'=-1+D, [ A>=2 && B==C && D==A && E==F && G==H ], cost: 1

      3: start -> cut : B'=0, E'=1, G'=-1+D, [ A>=2 && B==C && D==A && E==F && G==H ], cost: 1

     14: cut -> cut : E'=1-G+E, G'=1, [ G>=2 && E>=2 && A>=1+G && A>=G+E && D==A && E>=G ], cost: -1+G

     15: cut -> cut : E'=1, G'=1+G-E, [ G>=2 && E>=2 && A>=1+G && A>=G+E && D==A && E<=G ], cost: -1+E

     16: cut -> cut : E'=0, G'=1, [ G>=2 && 1>=E && E>=0 && A>=1+G && D==A ], cost: -1+G

     17: cut -> cut : B'=-2+G+E, E'=-1+G+E, G'=1, [ G>=2 && A>=2+E && E>=0 && A>=1+G && A>=G+E && D==A ], cost: -1+G

     13: start0 -> start : B'=C, D'=A, E'=F, G'=H, [], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: start0

      1: start -> cut : E'=0, G'=-1+D, [ A>=2 && B==C && D==A && E==F && G==H ], cost: 1

      3: start -> cut : B'=0, E'=1, G'=-1+D, [ A>=2 && B==C && D==A && E==F && G==H ], cost: 1

     18: start -> cut : E'=0, G'=1, [ A>=2 && B==C && D==A && E==F && G==H && -1+D>=2 ], cost: -1+D

     19: start -> cut : B'=0, E'=0, G'=1, [ A>=2 && B==C && D==A && E==F && G==H && -1+D>=2 ], cost: -1+D

     20: start -> cut : B'=-3+D, E'=-2+D, G'=1, [ A>=2 && B==C && D==A && E==F && G==H && -1+D>=2 ], cost: -1+D

     21: start -> cut : B'=-2+D, E'=-1+D, G'=1, [ B==C && D==A && E==F && G==H && -1+D>=2 && A>=3 ], cost: -1+D

     13: start0 -> start : B'=C, D'=A, E'=F, G'=H, [], cost: 1



Removed unreachable locations (and leaf rules with constant cost):

   Start location: start0

     18: start -> cut : E'=0, G'=1, [ A>=2 && B==C && D==A && E==F && G==H && -1+D>=2 ], cost: -1+D

     19: start -> cut : B'=0, E'=0, G'=1, [ A>=2 && B==C && D==A && E==F && G==H && -1+D>=2 ], cost: -1+D

     20: start -> cut : B'=-3+D, E'=-2+D, G'=1, [ A>=2 && B==C && D==A && E==F && G==H && -1+D>=2 ], cost: -1+D

     21: start -> cut : B'=-2+D, E'=-1+D, G'=1, [ B==C && D==A && E==F && G==H && -1+D>=2 && A>=3 ], cost: -1+D

     13: start0 -> start : B'=C, D'=A, E'=F, G'=H, [], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: start0

     22: start0 -> cut : B'=C, D'=A, E'=0, G'=1, [ -1+A>=2 ], cost: A

     23: start0 -> cut : B'=0, D'=A, E'=0, G'=1, [ -1+A>=2 ], cost: A

     24: start0 -> cut : B'=-3+A, D'=A, E'=-2+A, G'=1, [ -1+A>=2 ], cost: A

     25: start0 -> cut : B'=-2+A, D'=A, E'=-1+A, G'=1, [ -1+A>=2 ], cost: A



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: start0

     25: start0 -> cut : B'=-2+A, D'=A, E'=-1+A, G'=1, [ -1+A>=2 ], cost: A



Computing asymptotic complexity for rule 25

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      A (+), -2+A (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {A==n}

      resulting limit problem:

      [solved]



   Solution:

      A / n

   Resulting cost n has complexity: Poly(n^1)



Found new complexity Poly(n^1).



Obtained the following overall complexity (w.r.t. the length of the input n):

   Complexity:  Poly(n^1)

   Cpx degree:  1

   Solved cost: n

   Rule cost:   A

   Rule guard:  [ -1+A>=2 ]



WORST_CASE(Omega(n^1),?)


----------------------------------------

(4)
BOUNDS(n^1, INF)