/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 496 ms]
(2) BOUNDS(1, n^1)
(3) Loat Proof [FINISHED, 516 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, I, J) -> Com_1(stop(A, B, C, D, 1, F, G, H, I, J)) :|: A >= 1 && B >= C + 1 && D >= B && D <= B && E >= F && E <= F && G >= C && G <= C && H >= I && H <= I && J >= A && J <= A
start(A, B, C, D, E, F, G, H, I, J) -> Com_1(lbl71(A, B, C, 1 + D, 1, F, G, H, I, J)) :|: A >= 1 && C >= B && D >= B && D <= B && E >= F && E <= F && G >= C && G <= C && H >= I && H <= I && J >= A && J <= A
start(A, B, C, D, E, F, G, H, I, J) -> Com_1(stop(A, B, C, D, -(1), F, G, H, I, J)) :|: B >= C + 1 && 0 >= A && D >= B && D <= B && E >= F && E <= F && G >= C && G <= C && H >= I && H <= I && J >= A && J <= A
start(A, B, C, D, E, F, G, H, I, J) -> Com_1(lbl81(A, B, C, 1 + D, -(1), F, G, H, I, J)) :|: C >= B && 0 >= A && D >= B && D <= B && E >= F && E <= F && G >= C && G <= C && H >= I && H <= I && J >= A && J <= A
lbl71(A, B, C, D, E, F, G, H, I, J) -> Com_1(stop(A, B, C, D, E, F, G, H, I, J)) :|: C >= B && A >= 1 && D >= C + 1 && D <= C + 1 && E >= 1 && E <= 1 && J >= A && J <= A && H >= I && H <= I && G >= C && G <= C
lbl71(A, B, C, D, E, F, G, H, I, J) -> Com_1(lbl71(A, B, C, E + D, E, F, G, H, I, J)) :|: A >= 1 && C >= D && D >= B + 1 && C + 1 >= D && E >= 1 && E <= 1 && J >= A && J <= A && H >= I && H <= I && G >= C && G <= C
lbl71(A, B, C, D, E, F, G, H, I, J) -> Com_1(lbl81(A, B, C, D - E, E, F, G, H, I, J)) :|: C >= D && 0 >= A && D >= B + 1 && C + 1 >= D && A >= 1 && E >= 1 && E <= 1 && J >= A && J <= A && H >= I && H <= I && G >= C && G <= C
lbl81(A, B, C, D, E, F, G, H, I, J) -> Com_1(stop(A, B, C, D, E, F, G, H, I, J)) :|: 0 >= A && C >= B && D >= C + 1 && D <= C + 1 && E + 1 >= 0 && E + 1 <= 0 && J >= A && J <= A && H >= I && H <= I && G >= C && G <= C
lbl81(A, B, C, D, E, F, G, H, I, J) -> Com_1(lbl71(A, B, C, E + D, E, F, G, H, I, J)) :|: A >= 1 && C >= D && 0 >= A && D >= B + 1 && C + 1 >= D && E + 1 >= 0 && E + 1 <= 0 && J >= A && J <= A && H >= I && H <= I && G >= C && G <= C
lbl81(A, B, C, D, E, F, G, H, I, J) -> Com_1(lbl81(A, B, C, D - E, E, F, G, H, I, J)) :|: C >= D && 0 >= A && D >= B + 1 && C + 1 >= D && E + 1 >= 0 && E + 1 <= 0 && J >= A && J <= A && H >= I && H <= I && G >= C && G <= C
start0(A, B, C, D, E, F, G, H, I, J) -> Com_1(start(A, B, C, B, F, F, C, I, I, A)) :|: TRUE

The start-symbols are:[start0_10]


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

(1) Koat Proof (FINISHED)
YES(?, 2*ar_1 + 2*ar_2 + 7)



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, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, 1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl71(ar_0, ar_1, ar_2, ar_3 + 1, 1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_2 >= ar_1 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, -1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_1 >= ar_2 + 1 /\ 0 >= ar_0 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1, -1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_1 /\ 0 >= ar_0 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: ?, Cost: 1)    lbl71(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_1 /\ ar_0 >= 1 /\ ar_3 = ar_2 + 1 /\ ar_4 = 1 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: ?, Cost: 1)    lbl71(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl71(ar_0, ar_1, ar_2, ar_4 + ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_2 >= ar_3 /\ ar_3 >= ar_1 + 1 /\ ar_2 + 1 >= ar_3 /\ ar_4 = 1 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: ?, Cost: 1)    lbl71(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 - ar_4, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_3 /\ 0 >= ar_0 /\ ar_3 >= ar_1 + 1 /\ ar_2 + 1 >= ar_3 /\ ar_0 >= 1 /\ ar_4 = 1 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: ?, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ 0 >= ar_0 /\ ar_2 >= ar_1 /\ ar_3 = ar_2 + 1 /\ ar_4 + 1 = 0 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: ?, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl71(ar_0, ar_1, ar_2, ar_4 + ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_2 >= ar_3 /\ 0 >= ar_0 /\ ar_3 >= ar_1 + 1 /\ ar_2 + 1 >= ar_3 /\ ar_4 + 1 = 0 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: ?, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 - ar_4, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_3 /\ 0 >= ar_0 /\ ar_3 >= ar_1 + 1 /\ ar_2 + 1 >= ar_3 /\ ar_4 + 1 = 0 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: ?, Cost: 1)    start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(start(ar_0, ar_1, ar_2, ar_1, ar_5, ar_5, ar_2, ar_8, ar_8, ar_0))

		(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



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

	lbl71(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 - ar_4, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_3 /\ 0 >= ar_0 /\ ar_3 >= ar_1 + 1 /\ ar_2 + 1 >= ar_3 /\ ar_0 >= 1 /\ ar_4 = 1 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

	lbl81(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl71(ar_0, ar_1, ar_2, ar_4 + ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_2 >= ar_3 /\ 0 >= ar_0 /\ ar_3 >= ar_1 + 1 /\ ar_2 + 1 >= ar_3 /\ ar_4 + 1 = 0 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

We thus obtain the following problem:

2:	T:

		(Comp: ?, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 - ar_4, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_3 /\ 0 >= ar_0 /\ ar_3 >= ar_1 + 1 /\ ar_2 + 1 >= ar_3 /\ ar_4 + 1 = 0 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: ?, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ 0 >= ar_0 /\ ar_2 >= ar_1 /\ ar_3 = ar_2 + 1 /\ ar_4 + 1 = 0 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: ?, Cost: 1)    lbl71(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl71(ar_0, ar_1, ar_2, ar_4 + ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_2 >= ar_3 /\ ar_3 >= ar_1 + 1 /\ ar_2 + 1 >= ar_3 /\ ar_4 = 1 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: ?, Cost: 1)    lbl71(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_1 /\ ar_0 >= 1 /\ ar_3 = ar_2 + 1 /\ ar_4 = 1 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1, -1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_1 /\ 0 >= ar_0 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, -1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_1 >= ar_2 + 1 /\ 0 >= ar_0 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl71(ar_0, ar_1, ar_2, ar_3 + 1, 1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_2 >= ar_1 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, 1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: ?, Cost: 1)    start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(start(ar_0, ar_1, ar_2, ar_1, ar_5, ar_5, ar_2, ar_8, ar_8, ar_0))

		(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Repeatedly propagating knowledge in problem 2 produces the following problem:

3:	T:

		(Comp: ?, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 - ar_4, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_3 /\ 0 >= ar_0 /\ ar_3 >= ar_1 + 1 /\ ar_2 + 1 >= ar_3 /\ ar_4 + 1 = 0 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: ?, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ 0 >= ar_0 /\ ar_2 >= ar_1 /\ ar_3 = ar_2 + 1 /\ ar_4 + 1 = 0 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: ?, Cost: 1)    lbl71(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl71(ar_0, ar_1, ar_2, ar_4 + ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_2 >= ar_3 /\ ar_3 >= ar_1 + 1 /\ ar_2 + 1 >= ar_3 /\ ar_4 = 1 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: ?, Cost: 1)    lbl71(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_1 /\ ar_0 >= 1 /\ ar_3 = ar_2 + 1 /\ ar_4 = 1 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1, -1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_1 /\ 0 >= ar_0 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, -1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_1 >= ar_2 + 1 /\ 0 >= ar_0 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl71(ar_0, ar_1, ar_2, ar_3 + 1, 1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_2 >= ar_1 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, 1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: 1, Cost: 1)    start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(start(ar_0, ar_1, ar_2, ar_1, ar_5, ar_5, ar_2, ar_8, ar_8, ar_0))

		(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(lbl81) = 1

	Pol(stop) = 0

	Pol(lbl71) = 1

	Pol(start) = 1

	Pol(start0) = 1

	Pol(koat_start) = 1

orients all transitions weakly and the transitions

	lbl81(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ 0 >= ar_0 /\ ar_2 >= ar_1 /\ ar_3 = ar_2 + 1 /\ ar_4 + 1 = 0 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

	lbl71(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_1 /\ ar_0 >= 1 /\ ar_3 = ar_2 + 1 /\ ar_4 = 1 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

strictly and produces the following problem:

4:	T:

		(Comp: ?, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 - ar_4, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_3 /\ 0 >= ar_0 /\ ar_3 >= ar_1 + 1 /\ ar_2 + 1 >= ar_3 /\ ar_4 + 1 = 0 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: 1, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ 0 >= ar_0 /\ ar_2 >= ar_1 /\ ar_3 = ar_2 + 1 /\ ar_4 + 1 = 0 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: ?, Cost: 1)    lbl71(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl71(ar_0, ar_1, ar_2, ar_4 + ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_2 >= ar_3 /\ ar_3 >= ar_1 + 1 /\ ar_2 + 1 >= ar_3 /\ ar_4 = 1 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: 1, Cost: 1)    lbl71(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_1 /\ ar_0 >= 1 /\ ar_3 = ar_2 + 1 /\ ar_4 = 1 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1, -1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_1 /\ 0 >= ar_0 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, -1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_1 >= ar_2 + 1 /\ 0 >= ar_0 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl71(ar_0, ar_1, ar_2, ar_3 + 1, 1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_2 >= ar_1 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, 1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: 1, Cost: 1)    start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(start(ar_0, ar_1, ar_2, ar_1, ar_5, ar_5, ar_2, ar_8, ar_8, ar_0))

		(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(lbl81) = -V_4 - V_5 + V_7

	Pol(stop) = V_3 - V_4

	Pol(lbl71) = -V_4 + V_7 + 1

	Pol(start) = -V_2 + V_7

	Pol(start0) = -V_2 + V_3

	Pol(koat_start) = -V_2 + V_3

orients all transitions weakly and the transitions

	lbl81(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 - ar_4, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_3 /\ 0 >= ar_0 /\ ar_3 >= ar_1 + 1 /\ ar_2 + 1 >= ar_3 /\ ar_4 + 1 = 0 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

	lbl71(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl71(ar_0, ar_1, ar_2, ar_4 + ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_2 >= ar_3 /\ ar_3 >= ar_1 + 1 /\ ar_2 + 1 >= ar_3 /\ ar_4 = 1 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

strictly and produces the following problem:

5:	T:

		(Comp: ar_1 + ar_2, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 - ar_4, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_3 /\ 0 >= ar_0 /\ ar_3 >= ar_1 + 1 /\ ar_2 + 1 >= ar_3 /\ ar_4 + 1 = 0 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: 1, Cost: 1)              lbl81(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ 0 >= ar_0 /\ ar_2 >= ar_1 /\ ar_3 = ar_2 + 1 /\ ar_4 + 1 = 0 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: ar_1 + ar_2, Cost: 1)    lbl71(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl71(ar_0, ar_1, ar_2, ar_4 + ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_2 >= ar_3 /\ ar_3 >= ar_1 + 1 /\ ar_2 + 1 >= ar_3 /\ ar_4 = 1 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: 1, Cost: 1)              lbl71(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_1 /\ ar_0 >= 1 /\ ar_3 = ar_2 + 1 /\ ar_4 = 1 /\ ar_9 = ar_0 /\ ar_7 = ar_8 /\ ar_6 = ar_2 ]

		(Comp: 1, Cost: 1)              start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1, -1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_2 >= ar_1 /\ 0 >= ar_0 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: 1, Cost: 1)              start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, -1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_1 >= ar_2 + 1 /\ 0 >= ar_0 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: 1, Cost: 1)              start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(lbl71(ar_0, ar_1, ar_2, ar_3 + 1, 1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_2 >= ar_1 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: 1, Cost: 1)              start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3, 1, ar_5, ar_6, ar_7, ar_8, ar_9)) [ ar_0 >= 1 /\ ar_1 >= ar_2 + 1 /\ ar_3 = ar_1 /\ ar_4 = ar_5 /\ ar_6 = ar_2 /\ ar_7 = ar_8 /\ ar_9 = ar_0 ]

		(Comp: 1, Cost: 1)              start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(start(ar_0, ar_1, ar_2, ar_1, ar_5, ar_5, ar_2, ar_8, ar_8, ar_0))

		(Comp: 1, Cost: 0)              koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 2*ar_1 + 2*ar_2 + 7



Time: 0.438 sec (SMT: 0.332 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'=1, [ A>=1 && B>=1+C && D==B && E==F && G==C && H==Q && J==A ], cost: 1

      1: start -> lbl71 : D'=1+D, E'=1, [ A>=1 && C>=B && D==B && E==F && G==C && H==Q && J==A ], cost: 1

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

      3: start -> lbl81 : D'=1+D, E'=-1, [ C>=B && 0>=A && D==B && E==F && G==C && H==Q && J==A ], cost: 1

      4: lbl71 -> stop : [ C>=B && A>=1 && D==1+C && E==1 && J==A && H==Q && G==C ], cost: 1

      5: lbl71 -> lbl71 : D'=D+E, [ A>=1 && C>=D && D>=1+B && 1+C>=D && E==1 && J==A && H==Q && G==C ], cost: 1

      6: lbl71 -> lbl81 : D'=D-E, [ C>=D && 0>=A && D>=1+B && 1+C>=D && A>=1 && E==1 && J==A && H==Q && G==C ], cost: 1

      7: lbl81 -> stop : [ 0>=A && C>=B && D==1+C && 1+E==0 && J==A && H==Q && G==C ], cost: 1

      8: lbl81 -> lbl71 : D'=D+E, [ A>=1 && C>=D && 0>=A && D>=1+B && 1+C>=D && 1+E==0 && J==A && H==Q && G==C ], cost: 1

      9: lbl81 -> lbl81 : D'=D-E, [ C>=D && 0>=A && D>=1+B && 1+C>=D && 1+E==0 && J==A && H==Q && G==C ], cost: 1

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



Removed unreachable and leaf rules:

   Start location: start0

      1: start -> lbl71 : D'=1+D, E'=1, [ A>=1 && C>=B && D==B && E==F && G==C && H==Q && J==A ], cost: 1

      3: start -> lbl81 : D'=1+D, E'=-1, [ C>=B && 0>=A && D==B && E==F && G==C && H==Q && J==A ], cost: 1

      5: lbl71 -> lbl71 : D'=D+E, [ A>=1 && C>=D && D>=1+B && 1+C>=D && E==1 && J==A && H==Q && G==C ], cost: 1

      6: lbl71 -> lbl81 : D'=D-E, [ C>=D && 0>=A && D>=1+B && 1+C>=D && A>=1 && E==1 && J==A && H==Q && G==C ], cost: 1

      8: lbl81 -> lbl71 : D'=D+E, [ A>=1 && C>=D && 0>=A && D>=1+B && 1+C>=D && 1+E==0 && J==A && H==Q && G==C ], cost: 1

      9: lbl81 -> lbl81 : D'=D-E, [ C>=D && 0>=A && D>=1+B && 1+C>=D && 1+E==0 && J==A && H==Q && G==C ], cost: 1

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



Removed rules with unsatisfiable guard:

   Start location: start0

      1: start -> lbl71 : D'=1+D, E'=1, [ A>=1 && C>=B && D==B && E==F && G==C && H==Q && J==A ], cost: 1

      3: start -> lbl81 : D'=1+D, E'=-1, [ C>=B && 0>=A && D==B && E==F && G==C && H==Q && J==A ], cost: 1

      5: lbl71 -> lbl71 : D'=D+E, [ A>=1 && C>=D && D>=1+B && 1+C>=D && E==1 && J==A && H==Q && G==C ], cost: 1

      9: lbl81 -> lbl81 : D'=D-E, [ C>=D && 0>=A && D>=1+B && 1+C>=D && 1+E==0 && J==A && H==Q && G==C ], cost: 1

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



Simplified all rules, resulting in:

   Start location: start0

      1: start -> lbl71 : D'=1+D, E'=1, [ A>=1 && C>=B && D==B && E==F && G==C && H==Q && J==A ], cost: 1

      3: start -> lbl81 : D'=1+D, E'=-1, [ C>=B && 0>=A && D==B && E==F && G==C && H==Q && J==A ], cost: 1

      5: lbl71 -> lbl71 : D'=D+E, [ A>=1 && C>=D && D>=1+B && E==1 && J==A && H==Q && G==C ], cost: 1

      9: lbl81 -> lbl81 : D'=D-E, [ C>=D && 0>=A && D>=1+B && 1+E==0 && J==A && H==Q && G==C ], cost: 1

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



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      5: lbl71 -> lbl71 : D'=D+E, [ A>=1 && C>=D && D>=1+B && E==1 && J==A && H==Q && G==C ], cost: 1



   Accelerated rule 5 with metering function 2+C-D-E, yielding the new rule 11.

   Removing the simple loops: 5.



Accelerating simple loops of location 2.

   Accelerating the following rules:

      9: lbl81 -> lbl81 : D'=D-E, [ C>=D && 0>=A && D>=1+B && 1+E==0 && J==A && H==Q && G==C ], cost: 1



   Accelerated rule 9 with metering function C-D-E, yielding the new rule 12.

   Removing the simple loops: 9.



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

   Start location: start0

      1: start -> lbl71 : D'=1+D, E'=1, [ A>=1 && C>=B && D==B && E==F && G==C && H==Q && J==A ], cost: 1

      3: start -> lbl81 : D'=1+D, E'=-1, [ C>=B && 0>=A && D==B && E==F && G==C && H==Q && J==A ], cost: 1

     11: lbl71 -> lbl71 : D'=(2+C-D-E)*E+D, [ A>=1 && C>=D && D>=1+B && E==1 && J==A && H==Q && G==C && 2+C-D-E>=1 ], cost: 2+C-D-E

     12: lbl81 -> lbl81 : D'=-(C-D-E)*E+D, [ C>=D && 0>=A && D>=1+B && 1+E==0 && J==A && H==Q && G==C && C-D-E>=1 ], cost: C-D-E

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



Chained accelerated rules (with incoming rules):

   Start location: start0

      1: start -> lbl71 : D'=1+D, E'=1, [ A>=1 && C>=B && D==B && E==F && G==C && H==Q && J==A ], cost: 1

      3: start -> lbl81 : D'=1+D, E'=-1, [ C>=B && 0>=A && D==B && E==F && G==C && H==Q && J==A ], cost: 1

     13: start -> lbl71 : D'=1+C, E'=1, [ A>=1 && C>=B && D==B && E==F && G==C && H==Q && J==A && C>=1+D ], cost: 1+C-D

     14: start -> lbl81 : D'=1+C, E'=-1, [ C>=B && 0>=A && D==B && E==F && G==C && H==Q && J==A && C>=1+D ], cost: 1+C-D

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



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

   Start location: start0

     13: start -> lbl71 : D'=1+C, E'=1, [ A>=1 && C>=B && D==B && E==F && G==C && H==Q && J==A && C>=1+D ], cost: 1+C-D

     14: start -> lbl81 : D'=1+C, E'=-1, [ C>=B && 0>=A && D==B && E==F && G==C && H==Q && J==A && C>=1+D ], cost: 1+C-D

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



Eliminated locations (on tree-shaped paths):

   Start location: start0

     15: start0 -> lbl71 : D'=1+C, E'=1, G'=C, H'=Q, J'=A, [ A>=1 && C>=1+B ], cost: 2+C-B

     16: start0 -> lbl81 : D'=1+C, E'=-1, G'=C, H'=Q, J'=A, [ 0>=A && C>=1+B ], cost: 2+C-B



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: start0

     15: start0 -> lbl71 : D'=1+C, E'=1, G'=C, H'=Q, J'=A, [ A>=1 && C>=1+B ], cost: 2+C-B

     16: start0 -> lbl81 : D'=1+C, E'=-1, G'=C, H'=Q, J'=A, [ 0>=A && C>=1+B ], cost: 2+C-B



Computing asymptotic complexity for rule 15

   Solved the limit problem by the following transformations:

      Created initial limit problem:

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



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {C==0,A==n,B==-n}

      resulting limit problem:

      [solved]



   Solution:

      C / 0

      A / n

      B / -n

   Resulting cost 2+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: 2+n

   Rule cost:   2+C-B

   Rule guard:  [ A>=1 && C>=1+B ]



WORST_CASE(Omega(n^1),?)


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

(4)
BOUNDS(n^1, INF)