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


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WORST_CASE(?, O(1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 8 ms]
(2) BOUNDS(1, 1)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f0(A, B, C) -> Com_1(f5(A, 0, C)) :|: A >= 128
f0(A, B, C) -> Com_1(f7(A, 0, D)) :|: 127 >= A
f7(A, B, C) -> Com_1(f7(A, B + 1, C + 1)) :|: 35 >= B
f7(A, B, C) -> Com_1(f5(A, B, C)) :|: B >= 36

The start-symbols are:[f0_3]


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(1) Koat Proof (FINISHED)
YES(?, 39)



Initial complexity problem:

1:	T:

		(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, 0, ar_2)) [ ar_0 >= 128 ]

		(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, 0, d)) [ 127 >= ar_0 ]

		(Comp: ?, Cost: 1)    f7(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, ar_1 + 1, ar_2 + 1)) [ 35 >= ar_1 ]

		(Comp: ?, Cost: 1)    f7(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, ar_1, ar_2)) [ ar_1 >= 36 ]

		(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Repeatedly propagating knowledge in problem 1 produces the following problem:

2:	T:

		(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, 0, ar_2)) [ ar_0 >= 128 ]

		(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, 0, d)) [ 127 >= ar_0 ]

		(Comp: ?, Cost: 1)    f7(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, ar_1 + 1, ar_2 + 1)) [ 35 >= ar_1 ]

		(Comp: ?, Cost: 1)    f7(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, ar_1, ar_2)) [ ar_1 >= 36 ]

		(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(f0) = 1

	Pol(f5) = 0

	Pol(f7) = 1

	Pol(koat_start) = 1

orients all transitions weakly and the transition

	f7(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, ar_1, ar_2)) [ ar_1 >= 36 ]

strictly and produces the following problem:

3:	T:

		(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, 0, ar_2)) [ ar_0 >= 128 ]

		(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, 0, d)) [ 127 >= ar_0 ]

		(Comp: ?, Cost: 1)    f7(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, ar_1 + 1, ar_2 + 1)) [ 35 >= ar_1 ]

		(Comp: 1, Cost: 1)    f7(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, ar_1, ar_2)) [ ar_1 >= 36 ]

		(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(f0) = 36

	Pol(f5) = -V_2

	Pol(f7) = -V_2 + 36

	Pol(koat_start) = 36

orients all transitions weakly and the transition

	f7(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, ar_1 + 1, ar_2 + 1)) [ 35 >= ar_1 ]

strictly and produces the following problem:

4:	T:

		(Comp: 1, Cost: 1)     f0(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, 0, ar_2)) [ ar_0 >= 128 ]

		(Comp: 1, Cost: 1)     f0(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, 0, d)) [ 127 >= ar_0 ]

		(Comp: 36, Cost: 1)    f7(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, ar_1 + 1, ar_2 + 1)) [ 35 >= ar_1 ]

		(Comp: 1, Cost: 1)     f7(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, ar_1, ar_2)) [ ar_1 >= 36 ]

		(Comp: 1, Cost: 0)     koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 39



Time: 0.067 sec (SMT: 0.064 sec)


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(2)
BOUNDS(1, 1)