/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxIntTrs
(1) Koat Proof [FINISHED, 9 ms]
(2) BOUNDS(1, 1)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f0(A, B, C, D) -> Com_1(f8(0, B, C, D)) :|: TRUE
f8(A, B, C, D) -> Com_1(f8(A + 1, B, C, D)) :|: 3 >= A
f8(A, B, C, D) -> Com_1(f8(A + 1, A, A + 1, E)) :|: 3 >= A
f8(A, B, C, D) -> Com_1(f23(A, B, C, D)) :|: A >= 4 && 0 >= E + 1
f8(A, B, C, D) -> Com_1(f23(A, B, C, D)) :|: A >= 4

The start-symbols are:[f0_4]


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(1) Koat Proof (FINISHED)
YES(?, 11)



Initial complexity problem:

1:	T:

		(Comp: ?, Cost: 1)    f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f8(0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ 3 >= Ar_0 ]

		(Comp: ?, Cost: 1)    f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f8(Ar_0 + 1, Ar_0, Ar_0 + 1, Fresh_0)) [ 3 >= Ar_0 ]

		(Comp: ?, Cost: 1)    f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f23(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 4 /\ 0 >= E + 1 ]

		(Comp: ?, Cost: 1)    f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f23(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 4 ]

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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, Ar_3) -> Com_1(f8(0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ 3 >= Ar_0 ]

		(Comp: ?, Cost: 1)    f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f8(Ar_0 + 1, Ar_0, Ar_0 + 1, Fresh_0)) [ 3 >= Ar_0 ]

		(Comp: ?, Cost: 1)    f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f23(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 4 /\ 0 >= E + 1 ]

		(Comp: ?, Cost: 1)    f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f23(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 4 ]

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(f0) = 1

	Pol(f8) = 1

	Pol(f23) = 0

	Pol(koat_start) = 1

orients all transitions weakly and the transitions

	f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f23(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 4 /\ 0 >= E + 1 ]

	f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f23(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 4 ]

strictly and produces the following problem:

3:	T:

		(Comp: 1, Cost: 1)    f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f8(0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ 3 >= Ar_0 ]

		(Comp: ?, Cost: 1)    f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f8(Ar_0 + 1, Ar_0, Ar_0 + 1, Fresh_0)) [ 3 >= Ar_0 ]

		(Comp: 1, Cost: 1)    f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f23(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 4 /\ 0 >= E + 1 ]

		(Comp: 1, Cost: 1)    f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f23(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 4 ]

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(f0) = 4

	Pol(f8) = -V_1 + 4

	Pol(f23) = -V_1

	Pol(koat_start) = 4

orients all transitions weakly and the transitions

	f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ 3 >= Ar_0 ]

	f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f8(Ar_0 + 1, Ar_0, Ar_0 + 1, Fresh_0)) [ 3 >= Ar_0 ]

strictly and produces the following problem:

4:	T:

		(Comp: 1, Cost: 1)    f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f8(0, Ar_1, Ar_2, Ar_3))

		(Comp: 4, Cost: 1)    f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ 3 >= Ar_0 ]

		(Comp: 4, Cost: 1)    f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f8(Ar_0 + 1, Ar_0, Ar_0 + 1, Fresh_0)) [ 3 >= Ar_0 ]

		(Comp: 1, Cost: 1)    f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f23(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 4 /\ 0 >= E + 1 ]

		(Comp: 1, Cost: 1)    f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f23(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 4 ]

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 11



Time: 0.086 sec (SMT: 0.075 sec)


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(2)
BOUNDS(1, 1)