/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^2), O(n^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, n^2).

(0) CpxIntTrs
(1) Koat Proof [FINISHED, 204 ms]
(2) BOUNDS(1, n^2)
(3) Loat Proof [FINISHED, 668 ms]
(4) BOUNDS(n^2, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
start(A, B, C, D, E, F) -> Com_1(lbl71(A, B, 1, D, 0, F)) :|: A >= 2 && B >= A && B <= A && C >= D && C <= D && E >= F && E <= F
start(A, B, C, D, E, F) -> Com_1(stop(A, B, 0, D, 1, F)) :|: 1 >= A && B >= A && B <= A && C >= D && C <= D && E >= F && E <= F
lbl71(A, B, C, D, E, F) -> Com_1(lbl71(A, B, 1 + C, D, E, F)) :|: A >= C + 2 && A >= C + 1 && A >= E + 2 && C >= 1 && E >= 0 && B >= A && B <= A
lbl71(A, B, C, D, E, F) -> Com_1(cut(A, B, C, D, 1 + E, F)) :|: A >= E + 3 && A >= E + 2 && A >= 2 && E >= 0 && C + 1 >= A && C + 1 <= A && B >= A && B <= A
lbl71(A, B, C, D, E, F) -> Com_1(stop(A, B, C, D, 1 + E, F)) :|: A >= 2 && E + 2 >= A && E + 2 <= A && C + 1 >= A && C + 1 <= A && B >= A && B <= A
cut(A, B, C, D, E, F) -> Com_1(lbl71(A, B, 1, D, E, F)) :|: A >= 2 && A >= 2 + E && E >= 1 && C + 1 >= A && C + 1 <= A && B >= A && B <= A
cut(A, B, C, D, E, F) -> Com_1(cut(A, B, 0, D, 1 + E, F)) :|: 1 >= A && A >= E + 3 && A >= 2 + E && E >= 1 && C + 1 >= A && C + 1 <= A && B >= A && B <= A
cut(A, B, C, D, E, F) -> Com_1(stop(A, B, 0, D, 1 + E, F)) :|: 1 >= A && A >= 3 && C + 1 >= A && C + 1 <= A && E + 2 >= A && E + 2 <= A && B >= A && B <= A
start0(A, B, C, D, E, F) -> Com_1(start(A, A, D, D, F, F)) :|: TRUE

The start-symbols are:[start0_6]


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

(1) Koat Proof (FINISHED)
YES(?, 7*Ar_0 + 2*Ar_0^2 + 5)



Initial complexity problem:

1:	T:

		(Comp: ?, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, 0, Ar_5)) [ Ar_0 >= 2 /\ Ar_1 = Ar_0 /\ Ar_2 = Ar_3 /\ Ar_4 = Ar_5 ]

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

		(Comp: ?, Cost: 1)    lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_2 + 2 /\ Ar_0 >= Ar_2 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_2 >= 1 /\ Ar_4 >= 0 /\ Ar_1 = Ar_0 ]

		(Comp: ?, Cost: 1)    lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_4 + 3 /\ Ar_0 >= Ar_4 + 2 /\ Ar_0 >= 2 /\ Ar_4 >= 0 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: ?, Cost: 1)    lbl71(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 + 1, Ar_5)) [ Ar_0 >= 2 /\ Ar_4 + 2 = Ar_0 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 2 /\ Ar_4 >= 1 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, 0, Ar_3, Ar_4 + 1, Ar_5)) [ 1 >= Ar_0 /\ Ar_0 >= Ar_4 + 3 /\ Ar_0 >= Ar_4 + 2 /\ Ar_4 >= 1 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, 0, Ar_3, Ar_4 + 1, Ar_5)) [ 1 >= Ar_0 /\ Ar_0 >= 3 /\ Ar_2 + 1 = Ar_0 /\ Ar_4 + 2 = Ar_0 /\ Ar_1 = Ar_0 ]

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

		(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 ]

	start location:	koat_start

	leaf cost:	0



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

	cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, 0, Ar_3, Ar_4 + 1, Ar_5)) [ 1 >= Ar_0 /\ Ar_0 >= Ar_4 + 3 /\ Ar_0 >= Ar_4 + 2 /\ Ar_4 >= 1 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

	cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, 0, Ar_3, Ar_4 + 1, Ar_5)) [ 1 >= Ar_0 /\ Ar_0 >= 3 /\ Ar_2 + 1 = Ar_0 /\ Ar_4 + 2 = Ar_0 /\ Ar_1 = 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) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 2 /\ Ar_4 >= 1 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: ?, Cost: 1)    lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_4 + 3 /\ Ar_0 >= Ar_4 + 2 /\ Ar_0 >= 2 /\ Ar_4 >= 0 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: ?, Cost: 1)    lbl71(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 + 1, Ar_5)) [ Ar_0 >= 2 /\ Ar_4 + 2 = Ar_0 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: ?, Cost: 1)    lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_2 + 2 /\ Ar_0 >= Ar_2 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_2 >= 1 /\ Ar_4 >= 0 /\ Ar_1 = Ar_0 ]

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

		(Comp: ?, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, 0, Ar_5)) [ Ar_0 >= 2 /\ Ar_1 = Ar_0 /\ Ar_2 = Ar_3 /\ Ar_4 = Ar_5 ]

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

		(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 ]

	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) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 2 /\ Ar_4 >= 1 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: ?, Cost: 1)    lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_4 + 3 /\ Ar_0 >= Ar_4 + 2 /\ Ar_0 >= 2 /\ Ar_4 >= 0 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: ?, Cost: 1)    lbl71(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 + 1, Ar_5)) [ Ar_0 >= 2 /\ Ar_4 + 2 = Ar_0 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: ?, Cost: 1)    lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_2 + 2 /\ Ar_0 >= Ar_2 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_2 >= 1 /\ Ar_4 >= 0 /\ Ar_1 = Ar_0 ]

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

		(Comp: 1, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, 0, Ar_5)) [ Ar_0 >= 2 /\ Ar_1 = Ar_0 /\ Ar_2 = Ar_3 /\ Ar_4 = Ar_5 ]

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

		(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 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(cut) = 1

	Pol(lbl71) = 1

	Pol(stop) = 0

	Pol(start) = 1

	Pol(start0) = 1

	Pol(koat_start) = 1

orients all transitions weakly and the transition

	lbl71(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 + 1, Ar_5)) [ Ar_0 >= 2 /\ Ar_4 + 2 = Ar_0 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

strictly and produces the following problem:

4:	T:

		(Comp: ?, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 2 /\ Ar_4 >= 1 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: ?, Cost: 1)    lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_4 + 3 /\ Ar_0 >= Ar_4 + 2 /\ Ar_0 >= 2 /\ Ar_4 >= 0 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: 1, Cost: 1)    lbl71(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 + 1, Ar_5)) [ Ar_0 >= 2 /\ Ar_4 + 2 = Ar_0 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: ?, Cost: 1)    lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_2 + 2 /\ Ar_0 >= Ar_2 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_2 >= 1 /\ Ar_4 >= 0 /\ Ar_1 = Ar_0 ]

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

		(Comp: 1, Cost: 1)    start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, 0, Ar_5)) [ Ar_0 >= 2 /\ Ar_1 = Ar_0 /\ Ar_2 = Ar_3 /\ Ar_4 = Ar_5 ]

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

		(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 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(cut) = 2*V_2 - 2*V_5 + 1

	Pol(lbl71) = 2*V_2 - 2*V_5

	Pol(stop) = 2*V_1 - 2*V_5

	Pol(start) = 2*V_1

	Pol(start0) = 2*V_1

	Pol(koat_start) = 2*V_1

orients all transitions weakly and the transitions

	lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_4 + 3 /\ Ar_0 >= Ar_4 + 2 /\ Ar_0 >= 2 /\ Ar_4 >= 0 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

	cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 2 /\ Ar_4 >= 1 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

strictly and produces the following problem:

5:	T:

		(Comp: 2*Ar_0, Cost: 1)    cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 2 /\ Ar_4 >= 1 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: 2*Ar_0, Cost: 1)    lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_4 + 3 /\ Ar_0 >= Ar_4 + 2 /\ Ar_0 >= 2 /\ Ar_4 >= 0 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: 1, Cost: 1)         lbl71(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 + 1, Ar_5)) [ Ar_0 >= 2 /\ Ar_4 + 2 = Ar_0 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: ?, Cost: 1)         lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_2 + 2 /\ Ar_0 >= Ar_2 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_2 >= 1 /\ Ar_4 >= 0 /\ Ar_1 = Ar_0 ]

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

		(Comp: 1, Cost: 1)         start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, 0, Ar_5)) [ Ar_0 >= 2 /\ Ar_1 = Ar_0 /\ Ar_2 = Ar_3 /\ Ar_4 = Ar_5 ]

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

		(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 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(lbl71) = V_1 - V_3

and size complexities

	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

	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

	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

	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

	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

	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

	S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_0, Ar_3, Ar_3, Ar_5, Ar_5))", 0-0) = Ar_0

	S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_0, Ar_3, Ar_3, Ar_5, Ar_5))", 0-1) = Ar_0

	S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_0, Ar_3, Ar_3, Ar_5, Ar_5))", 0-2) = Ar_3

	S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_0, Ar_3, Ar_3, Ar_5, Ar_5))", 0-3) = Ar_3

	S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_0, Ar_3, Ar_3, Ar_5, Ar_5))", 0-4) = Ar_5

	S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_0, Ar_3, Ar_3, Ar_5, Ar_5))", 0-5) = Ar_5

	S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, 0, Ar_5)) [ Ar_0 >= 2 /\\ Ar_1 = Ar_0 /\\ Ar_2 = Ar_3 /\\ Ar_4 = Ar_5 ]", 0-0) = Ar_0

	S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, 0, Ar_5)) [ Ar_0 >= 2 /\\ Ar_1 = Ar_0 /\\ Ar_2 = Ar_3 /\\ Ar_4 = Ar_5 ]", 0-1) = Ar_0

	S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, 0, Ar_5)) [ Ar_0 >= 2 /\\ Ar_1 = Ar_0 /\\ Ar_2 = Ar_3 /\\ Ar_4 = Ar_5 ]", 0-2) = 1

	S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, 0, Ar_5)) [ Ar_0 >= 2 /\\ Ar_1 = Ar_0 /\\ Ar_2 = Ar_3 /\\ Ar_4 = Ar_5 ]", 0-3) = Ar_3

	S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, 0, Ar_5)) [ Ar_0 >= 2 /\\ Ar_1 = Ar_0 /\\ Ar_2 = Ar_3 /\\ Ar_4 = Ar_5 ]", 0-4) = 0

	S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, 0, Ar_5)) [ Ar_0 >= 2 /\\ Ar_1 = Ar_0 /\\ Ar_2 = Ar_3 /\\ Ar_4 = Ar_5 ]", 0-5) = Ar_5

	S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, 0, Ar_3, 1, Ar_5)) [ 1 >= Ar_0 /\\ Ar_1 = Ar_0 /\\ Ar_2 = Ar_3 /\\ Ar_4 = Ar_5 ]", 0-0) = Ar_0

	S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, 0, Ar_3, 1, Ar_5)) [ 1 >= Ar_0 /\\ Ar_1 = Ar_0 /\\ Ar_2 = Ar_3 /\\ Ar_4 = Ar_5 ]", 0-1) = Ar_0

	S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, 0, Ar_3, 1, Ar_5)) [ 1 >= Ar_0 /\\ Ar_1 = Ar_0 /\\ Ar_2 = Ar_3 /\\ Ar_4 = Ar_5 ]", 0-2) = 0

	S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, 0, Ar_3, 1, Ar_5)) [ 1 >= Ar_0 /\\ Ar_1 = Ar_0 /\\ Ar_2 = Ar_3 /\\ Ar_4 = Ar_5 ]", 0-3) = Ar_3

	S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, 0, Ar_3, 1, Ar_5)) [ 1 >= Ar_0 /\\ Ar_1 = Ar_0 /\\ Ar_2 = Ar_3 /\\ Ar_4 = Ar_5 ]", 0-4) = 1

	S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, 0, Ar_3, 1, Ar_5)) [ 1 >= Ar_0 /\\ Ar_1 = Ar_0 /\\ Ar_2 = Ar_3 /\\ Ar_4 = Ar_5 ]", 0-5) = Ar_5

	S("lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_2 + 2 /\\ Ar_0 >= Ar_2 + 1 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_2 >= 1 /\\ Ar_4 >= 0 /\\ Ar_1 = Ar_0 ]", 0-0) = Ar_0

	S("lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_2 + 2 /\\ Ar_0 >= Ar_2 + 1 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_2 >= 1 /\\ Ar_4 >= 0 /\\ Ar_1 = Ar_0 ]", 0-1) = Ar_0

	S("lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_2 + 2 /\\ Ar_0 >= Ar_2 + 1 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_2 >= 1 /\\ Ar_4 >= 0 /\\ Ar_1 = Ar_0 ]", 0-2) = Ar_0

	S("lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_2 + 2 /\\ Ar_0 >= Ar_2 + 1 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_2 >= 1 /\\ Ar_4 >= 0 /\\ Ar_1 = Ar_0 ]", 0-3) = Ar_3

	S("lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_2 + 2 /\\ Ar_0 >= Ar_2 + 1 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_2 >= 1 /\\ Ar_4 >= 0 /\\ Ar_1 = Ar_0 ]", 0-4) = Ar_0

	S("lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_2 + 2 /\\ Ar_0 >= Ar_2 + 1 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_2 >= 1 /\\ Ar_4 >= 0 /\\ Ar_1 = Ar_0 ]", 0-5) = Ar_5

	S("lbl71(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 + 1, Ar_5)) [ Ar_0 >= 2 /\\ Ar_4 + 2 = Ar_0 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-0) = Ar_0

	S("lbl71(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 + 1, Ar_5)) [ Ar_0 >= 2 /\\ Ar_4 + 2 = Ar_0 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-1) = Ar_0

	S("lbl71(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 + 1, Ar_5)) [ Ar_0 >= 2 /\\ Ar_4 + 2 = Ar_0 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-2) = Ar_0 + 1

	S("lbl71(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 + 1, Ar_5)) [ Ar_0 >= 2 /\\ Ar_4 + 2 = Ar_0 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-3) = Ar_3

	S("lbl71(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 + 1, Ar_5)) [ Ar_0 >= 2 /\\ Ar_4 + 2 = Ar_0 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-4) = Ar_0

	S("lbl71(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 + 1, Ar_5)) [ Ar_0 >= 2 /\\ Ar_4 + 2 = Ar_0 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-5) = Ar_5

	S("lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_4 + 3 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_0 >= 2 /\\ Ar_4 >= 0 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-0) = Ar_0

	S("lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_4 + 3 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_0 >= 2 /\\ Ar_4 >= 0 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-1) = Ar_0

	S("lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_4 + 3 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_0 >= 2 /\\ Ar_4 >= 0 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-2) = Ar_0

	S("lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_4 + 3 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_0 >= 2 /\\ Ar_4 >= 0 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-3) = Ar_3

	S("lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_4 + 3 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_0 >= 2 /\\ Ar_4 >= 0 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-4) = Ar_0

	S("lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_4 + 3 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_0 >= 2 /\\ Ar_4 >= 0 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-5) = Ar_5

	S("cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 2 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_4 >= 1 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-0) = Ar_0

	S("cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 2 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_4 >= 1 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-1) = Ar_0

	S("cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 2 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_4 >= 1 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-2) = 1

	S("cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 2 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_4 >= 1 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-3) = Ar_3

	S("cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 2 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_4 >= 1 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-4) = Ar_0

	S("cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 2 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_4 >= 1 /\\ Ar_2 + 1 = Ar_0 /\\ Ar_1 = Ar_0 ]", 0-5) = Ar_5

orients the transitions

	lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_2 + 2 /\ Ar_0 >= Ar_2 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_2 >= 1 /\ Ar_4 >= 0 /\ Ar_1 = Ar_0 ]

weakly and the transition

	lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_2 + 2 /\ Ar_0 >= Ar_2 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_2 >= 1 /\ Ar_4 >= 0 /\ Ar_1 = Ar_0 ]

strictly and produces the following problem:

6:	T:

		(Comp: 2*Ar_0, Cost: 1)                   cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 2 /\ Ar_0 >= Ar_4 + 2 /\ Ar_4 >= 1 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: 2*Ar_0, Cost: 1)                   lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_4 + 3 /\ Ar_0 >= Ar_4 + 2 /\ Ar_0 >= 2 /\ Ar_4 >= 0 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: 1, Cost: 1)                        lbl71(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 + 1, Ar_5)) [ Ar_0 >= 2 /\ Ar_4 + 2 = Ar_0 /\ Ar_2 + 1 = Ar_0 /\ Ar_1 = Ar_0 ]

		(Comp: 2*Ar_0^2 + 3*Ar_0 + 1, Cost: 1)    lbl71(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_2 + 2 /\ Ar_0 >= Ar_2 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_2 >= 1 /\ Ar_4 >= 0 /\ Ar_1 = Ar_0 ]

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

		(Comp: 1, Cost: 1)                        start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl71(Ar_0, Ar_1, 1, Ar_3, 0, Ar_5)) [ Ar_0 >= 2 /\ Ar_1 = Ar_0 /\ Ar_2 = Ar_3 /\ Ar_4 = Ar_5 ]

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

		(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 ]

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 7*Ar_0 + 2*Ar_0^2 + 5



Time: 0.176 sec (SMT: 0.141 sec)


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

(2)
BOUNDS(1, n^2)

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

(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: start0

      0: start -> lbl71 : C'=1, E'=0, [ A>=2 && B==A && C==D && E==F ], cost: 1

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

      2: lbl71 -> lbl71 : C'=1+C, [ A>=2+C && A>=1+C && A>=2+E && C>=1 && E>=0 && B==A ], cost: 1

      3: lbl71 -> cut : E'=1+E, [ A>=3+E && A>=2+E && A>=2 && E>=0 && 1+C==A && B==A ], cost: 1

      4: lbl71 -> stop : E'=1+E, [ A>=2 && 2+E==A && 1+C==A && B==A ], cost: 1

      5: cut -> lbl71 : C'=1, [ A>=2 && A>=2+E && E>=1 && 1+C==A && B==A ], cost: 1

      6: cut -> cut : C'=0, E'=1+E, [ 1>=A && A>=3+E && A>=2+E && E>=1 && 1+C==A && B==A ], cost: 1

      7: cut -> stop : C'=0, E'=1+E, [ 1>=A && A>=3 && 1+C==A && 2+E==A && B==A ], cost: 1

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



Checking for constant complexity:

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

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



Removed unreachable and leaf rules:

   Start location: start0

      0: start -> lbl71 : C'=1, E'=0, [ A>=2 && B==A && C==D && E==F ], cost: 1

      2: lbl71 -> lbl71 : C'=1+C, [ A>=2+C && A>=1+C && A>=2+E && C>=1 && E>=0 && B==A ], cost: 1

      3: lbl71 -> cut : E'=1+E, [ A>=3+E && A>=2+E && A>=2 && E>=0 && 1+C==A && B==A ], cost: 1

      5: cut -> lbl71 : C'=1, [ A>=2 && A>=2+E && E>=1 && 1+C==A && B==A ], cost: 1

      6: cut -> cut : C'=0, E'=1+E, [ 1>=A && A>=3+E && A>=2+E && E>=1 && 1+C==A && B==A ], cost: 1

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



Removed rules with unsatisfiable guard:

   Start location: start0

      0: start -> lbl71 : C'=1, E'=0, [ A>=2 && B==A && C==D && E==F ], cost: 1

      2: lbl71 -> lbl71 : C'=1+C, [ A>=2+C && A>=1+C && A>=2+E && C>=1 && E>=0 && B==A ], cost: 1

      3: lbl71 -> cut : E'=1+E, [ A>=3+E && A>=2+E && A>=2 && E>=0 && 1+C==A && B==A ], cost: 1

      5: cut -> lbl71 : C'=1, [ A>=2 && A>=2+E && E>=1 && 1+C==A && B==A ], cost: 1

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



Simplified all rules, resulting in:

   Start location: start0

      0: start -> lbl71 : C'=1, E'=0, [ A>=2 && B==A && C==D && E==F ], cost: 1

      2: lbl71 -> lbl71 : C'=1+C, [ A>=2+C && A>=2+E && C>=1 && E>=0 && B==A ], cost: 1

      3: lbl71 -> cut : E'=1+E, [ A>=3+E && A>=2 && E>=0 && 1+C==A && B==A ], cost: 1

      5: cut -> lbl71 : C'=1, [ A>=2+E && E>=1 && 1+C==A && B==A ], cost: 1

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



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      2: lbl71 -> lbl71 : C'=1+C, [ A>=2+C && A>=2+E && C>=1 && E>=0 && B==A ], cost: 1



   Accelerated rule 2 with metering function -1-C+A, yielding the new rule 9.

   Removing the simple loops: 2.



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

   Start location: start0

      0: start -> lbl71 : C'=1, E'=0, [ A>=2 && B==A && C==D && E==F ], cost: 1

      3: lbl71 -> cut : E'=1+E, [ A>=3+E && A>=2 && E>=0 && 1+C==A && B==A ], cost: 1

      9: lbl71 -> lbl71 : C'=-1+A, [ A>=2+C && A>=2+E && C>=1 && E>=0 && B==A ], cost: -1-C+A

      5: cut -> lbl71 : C'=1, [ A>=2+E && E>=1 && 1+C==A && B==A ], cost: 1

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



Chained accelerated rules (with incoming rules):

   Start location: start0

      0: start -> lbl71 : C'=1, E'=0, [ A>=2 && B==A && C==D && E==F ], cost: 1

     10: start -> lbl71 : C'=-1+A, E'=0, [ B==A && C==D && E==F && A>=3 ], cost: -1+A

      3: lbl71 -> cut : E'=1+E, [ A>=3+E && A>=2 && E>=0 && 1+C==A && B==A ], cost: 1

      5: cut -> lbl71 : C'=1, [ A>=2+E && E>=1 && 1+C==A && B==A ], cost: 1

     11: cut -> lbl71 : C'=-1+A, [ A>=2+E && E>=1 && 1+C==A && B==A && A>=3 ], cost: -1+A

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



Eliminated locations (on tree-shaped paths):

   Start location: start0

     14: lbl71 -> lbl71 : C'=1, E'=1+E, [ A>=3+E && A>=2 && E>=0 && 1+C==A && B==A ], cost: 2

     15: lbl71 -> lbl71 : C'=-1+A, E'=1+E, [ A>=3+E && E>=0 && 1+C==A && B==A && A>=3 ], cost: A

     12: start0 -> lbl71 : B'=A, C'=1, E'=0, [ A>=2 ], cost: 2

     13: start0 -> lbl71 : B'=A, C'=-1+A, E'=0, [ A>=3 ], cost: A



Accelerating simple loops of location 1.

   Accelerating the following rules:

     14: lbl71 -> lbl71 : C'=1, E'=1+E, [ A>=3+E && A>=2 && E>=0 && 1+C==A && B==A ], cost: 2

     15: lbl71 -> lbl71 : C'=-1+A, E'=1+E, [ A>=3+E && E>=0 && 1+C==A && B==A && A>=3 ], cost: A



   Accelerated rule 14 with NONTERM (after strengthening guard), yielding the new rule 16.

   Accelerated rule 15 with metering function -2+A-E, yielding the new rule 17.

   Removing the simple loops: 15.



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

   Start location: start0

     14: lbl71 -> lbl71 : C'=1, E'=1+E, [ A>=3+E && A>=2 && E>=0 && 1+C==A && B==A ], cost: 2

     16: lbl71 -> [6] : [ A>=3+E && E>=0 && 1+C==A && B==A && 2==A ], cost: NONTERM

     17: lbl71 -> lbl71 : C'=-1+A, E'=-2+A, [ A>=3+E && E>=0 && 1+C==A && B==A && A>=3 ], cost: (-2+A-E)*A

     12: start0 -> lbl71 : B'=A, C'=1, E'=0, [ A>=2 ], cost: 2

     13: start0 -> lbl71 : B'=A, C'=-1+A, E'=0, [ A>=3 ], cost: A



Chained accelerated rules (with incoming rules):

   Start location: start0

     12: start0 -> lbl71 : B'=A, C'=1, E'=0, [ A>=2 ], cost: 2

     13: start0 -> lbl71 : B'=A, C'=-1+A, E'=0, [ A>=3 ], cost: A

     18: start0 -> lbl71 : B'=A, C'=1, E'=1, [ A>=3 ], cost: 2+A

     19: start0 -> lbl71 : B'=A, C'=-1+A, E'=-2+A, [ A>=3 ], cost: A*(-2+A)+A



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

   Start location: start0

     13: start0 -> lbl71 : B'=A, C'=-1+A, E'=0, [ A>=3 ], cost: A

     18: start0 -> lbl71 : B'=A, C'=1, E'=1, [ A>=3 ], cost: 2+A

     19: start0 -> lbl71 : B'=A, C'=-1+A, E'=-2+A, [ A>=3 ], cost: A*(-2+A)+A



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: start0

     18: start0 -> lbl71 : B'=A, C'=1, E'=1, [ A>=3 ], cost: 2+A

     19: start0 -> lbl71 : B'=A, C'=-1+A, E'=-2+A, [ A>=3 ], cost: A*(-2+A)+A



Computing asymptotic complexity for rule 18

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      2+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 2+n has complexity: Poly(n^1)



Found new complexity Poly(n^1).



Computing asymptotic complexity for rule 19

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -A+A^2 (+), -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^2-n has complexity: Poly(n^2)



Found new complexity Poly(n^2).



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

   Complexity:  Poly(n^2)

   Cpx degree:  2

   Solved cost: n^2-n

   Rule cost:   A*(-2+A)+A

   Rule guard:  [ A>=3 ]



WORST_CASE(Omega(n^2),?)


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

(4)
BOUNDS(n^2, INF)