/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(?, O(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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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(1, 1).

(0) CpxIntTrs
(1) Koat Proof [FINISHED, 20 ms]
(2) BOUNDS(1, 1)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
start(A) -> Com_1(a(A)) :|: A >= 1
start(A) -> Com_1(a(A)) :|: A >= 2
start(A) -> Com_1(a(A)) :|: A >= 4
a(A) -> Com_1(a(A * B)) :|: 1 >= 2 * B && 3 * B >= 2 && A >= 2

The start-symbols are:[start_1]


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(1) Koat Proof (FINISHED)
YES(?, 3)



Initial complexity problem:

1:	T:

		(Comp: ?, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 1 ]

		(Comp: ?, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 2 ]

		(Comp: ?, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 4 ]

		(Comp: ?, Cost: 1)    a(ar_0) -> Com_1(a(ar_0*b)) [ 1 >= 2*b /\ 3*b >= 2 /\ ar_0 >= 2 ]

		(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(start(ar_0)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



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

	a(ar_0) -> Com_1(a(ar_0*b)) [ 1 >= 2*b /\ 3*b >= 2 /\ ar_0 >= 2 ]

We thus obtain the following problem:

2:	T:

		(Comp: ?, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 4 ]

		(Comp: ?, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 2 ]

		(Comp: ?, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 1 ]

		(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(start(ar_0)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Repeatedly propagating knowledge in problem 2 produces the following problem:

3:	T:

		(Comp: 1, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 4 ]

		(Comp: 1, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 2 ]

		(Comp: 1, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 1 ]

		(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(start(ar_0)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 3



Time: 0.010 sec (SMT: 0.010 sec)


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(2)
BOUNDS(1, 1)