/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), O(n^1))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1, n^1).

(0) CpxIntTrs
(1) Koat Proof [FINISHED, 148 ms]
(2) BOUNDS(1, n^1)
(3) Loat Proof [FINISHED, 619 ms]
(4) BOUNDS(n^1, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f0(A, B, C, D, E) -> Com_1(f1(A, B, C, D, E)) :|: TRUE
f1(A, B, C, D, E) -> Com_1(f1(A, B + 1, C, D, E)) :|: A >= B + 1
f1(A, B, C, D, E) -> Com_1(f2(A, B, B, D, E)) :|: B >= A
f2(A, B, C, D, E) -> Com_1(f2(A, B, C - 1, D, E)) :|: C >= 1
f2(A, B, C, D, E) -> Com_1(f3(A, B, C, C, E)) :|: 0 >= C
f3(A, B, C, D, E) -> Com_1(f3(A, B, C, D + 1, E)) :|: A >= D + 1
f3(A, B, C, D, E) -> Com_1(f4(A, B, C, D, D)) :|: D >= A
f4(A, B, C, D, E) -> Com_1(f4(A, B, C, D, E - 1)) :|: E >= 1

The start-symbols are:[f0_5]


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

(1) Koat Proof (FINISHED)
YES(?, 88*Ar_0 + 85*Ar_1 + 10)



Initial complexity problem:

1:	T:

		(Comp: ?, Cost: 1)    f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: ?, Cost: 1)    f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]

		(Comp: ?, Cost: 1)    f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]

		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]

		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]

		(Comp: ?, Cost: 1)    f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 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, Ar_4) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: ?, Cost: 1)    f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]

		(Comp: ?, Cost: 1)    f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]

		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]

		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]

		(Comp: ?, Cost: 1)    f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(f0) = 3

	Pol(f1) = 3

	Pol(f2) = 2

	Pol(f3) = 1

	Pol(f4) = 0

	Pol(koat_start) = 3

orients all transitions weakly and the transitions

	f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]

	f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]

	f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]

strictly and produces the following problem:

3:	T:

		(Comp: 1, Cost: 1)    f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: ?, Cost: 1)    f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]

		(Comp: 3, Cost: 1)    f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]

		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]

		(Comp: 3, Cost: 1)    f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]

		(Comp: 3, Cost: 1)    f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]

		(Comp: ?, Cost: 1)    f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(f0) = V_1 - V_2

	Pol(f1) = V_1 - V_2

	Pol(f2) = V_1 - V_2

	Pol(f3) = V_1 - V_2 + V_3 - V_4

	Pol(f4) = V_1 - V_2 + V_3 - V_4

	Pol(koat_start) = V_1 - V_2

orients all transitions weakly and the transition

	f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]

strictly and produces the following problem:

4:	T:

		(Comp: 1, Cost: 1)              f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: Ar_0 + Ar_1, Cost: 1)    f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]

		(Comp: 3, Cost: 1)              f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]

		(Comp: ?, Cost: 1)              f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]

		(Comp: 3, Cost: 1)              f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]

		(Comp: ?, Cost: 1)              f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]

		(Comp: 3, Cost: 1)              f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]

		(Comp: ?, Cost: 1)              f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]

		(Comp: 1, Cost: 0)              koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(f3) = V_1 - V_4

and size complexities

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-0) = Ar_0

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-1) = Ar_1

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-2) = Ar_2

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-3) = Ar_3

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-4) = Ar_4

	S("f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]", 0-0) = Ar_0

	S("f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]", 0-1) = 2*Ar_0 + 2*Ar_1

	S("f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]", 0-2) = 2*Ar_0 + 2*Ar_1

	S("f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]", 0-3) = ?

	S("f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]", 0-4) = ?

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]", 0-0) = Ar_0

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]", 0-1) = 2*Ar_0 + 2*Ar_1

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]", 0-2) = 2*Ar_0 + 2*Ar_1

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]", 0-3) = ?

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]", 0-4) = ?

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]", 0-0) = Ar_0

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]", 0-1) = 2*Ar_0 + 2*Ar_1

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]", 0-2) = 2*Ar_0 + 2*Ar_1

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]", 0-3) = ?

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]", 0-4) = Ar_4

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]", 0-0) = Ar_0

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]", 0-1) = 2*Ar_0 + 2*Ar_1

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]", 0-2) = 2*Ar_0 + 2*Ar_1

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]", 0-3) = 2*Ar_0 + 2*Ar_1

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]", 0-4) = Ar_4

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]", 0-0) = Ar_0

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]", 0-1) = 2*Ar_0 + 2*Ar_1

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]", 0-2) = 2*Ar_0 + 2*Ar_1

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]", 0-3) = Ar_3

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]", 0-4) = Ar_4

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]", 0-0) = Ar_0

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]", 0-1) = 2*Ar_0 + 2*Ar_1

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]", 0-2) = 2*Ar_0 + 2*Ar_1

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]", 0-3) = Ar_3

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]", 0-4) = Ar_4

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]", 0-0) = Ar_0

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = 2*Ar_0 + 2*Ar_1

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]", 0-2) = Ar_2

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]", 0-3) = Ar_3

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]", 0-4) = Ar_4

	S("f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-0) = Ar_0

	S("f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-1) = Ar_1

	S("f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-2) = Ar_2

	S("f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-3) = Ar_3

	S("f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-4) = Ar_4

orients the transitions

	f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]

weakly and the transition

	f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]

strictly and produces the following problem:

5:	T:

		(Comp: 1, Cost: 1)                  f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: Ar_0 + Ar_1, Cost: 1)        f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]

		(Comp: 3, Cost: 1)                  f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]

		(Comp: ?, Cost: 1)                  f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]

		(Comp: 3, Cost: 1)                  f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]

		(Comp: 9*Ar_0 + 6*Ar_1, Cost: 1)    f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]

		(Comp: 3, Cost: 1)                  f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]

		(Comp: ?, Cost: 1)                  f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]

		(Comp: 1, Cost: 0)                  koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(f4) = V_5

	Pol(f2) = V_3

and size complexities

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-0) = Ar_0

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-1) = Ar_1

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-2) = Ar_2

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-3) = Ar_3

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-4) = Ar_4

	S("f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]", 0-0) = Ar_0

	S("f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]", 0-1) = 2*Ar_0 + 2*Ar_1

	S("f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]", 0-2) = 2*Ar_0 + 2*Ar_1

	S("f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]", 0-3) = 11*Ar_0 + 11*Ar_1

	S("f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]", 0-4) = 11*Ar_0 + 11*Ar_1

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]", 0-0) = Ar_0

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]", 0-1) = 2*Ar_0 + 2*Ar_1

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]", 0-2) = 2*Ar_0 + 2*Ar_1

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]", 0-3) = 11*Ar_0 + 11*Ar_1

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]", 0-4) = 11*Ar_0 + 11*Ar_1

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]", 0-0) = Ar_0

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]", 0-1) = 2*Ar_0 + 2*Ar_1

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]", 0-2) = 2*Ar_0 + 2*Ar_1

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]", 0-3) = 11*Ar_0 + 11*Ar_1

	S("f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]", 0-4) = Ar_4

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]", 0-0) = Ar_0

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]", 0-1) = 2*Ar_0 + 2*Ar_1

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]", 0-2) = 2*Ar_0 + 2*Ar_1

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]", 0-3) = 2*Ar_0 + 2*Ar_1

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]", 0-4) = Ar_4

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]", 0-0) = Ar_0

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]", 0-1) = 2*Ar_0 + 2*Ar_1

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]", 0-2) = 2*Ar_0 + 2*Ar_1

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]", 0-3) = Ar_3

	S("f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]", 0-4) = Ar_4

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]", 0-0) = Ar_0

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]", 0-1) = 2*Ar_0 + 2*Ar_1

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]", 0-2) = 2*Ar_0 + 2*Ar_1

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]", 0-3) = Ar_3

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]", 0-4) = Ar_4

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]", 0-0) = Ar_0

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = 2*Ar_0 + 2*Ar_1

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]", 0-2) = Ar_2

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]", 0-3) = Ar_3

	S("f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]", 0-4) = Ar_4

	S("f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-0) = Ar_0

	S("f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-1) = Ar_1

	S("f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-2) = Ar_2

	S("f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-3) = Ar_3

	S("f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-4) = Ar_4

orients the transitions

	f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]

	f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]

weakly and the transitions

	f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]

	f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]

strictly and produces the following problem:

6:	T:

		(Comp: 1, Cost: 1)                    f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: Ar_0 + Ar_1, Cost: 1)          f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 + 1 ]

		(Comp: 3, Cost: 1)                    f1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_1, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 ]

		(Comp: 39*Ar_0 + 39*Ar_1, Cost: 1)    f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2 - 1, Ar_3, Ar_4)) [ Ar_2 >= 1 ]

		(Comp: 3, Cost: 1)                    f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_2, Ar_4)) [ 0 >= Ar_2 ]

		(Comp: 9*Ar_0 + 6*Ar_1, Cost: 1)      f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) [ Ar_0 >= Ar_3 + 1 ]

		(Comp: 3, Cost: 1)                    f3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_3)) [ Ar_3 >= Ar_0 ]

		(Comp: 39*Ar_0 + 39*Ar_1, Cost: 1)    f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) [ Ar_4 >= 1 ]

		(Comp: 1, Cost: 0)                    koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 88*Ar_0 + 85*Ar_1 + 10



Time: 0.145 sec (SMT: 0.112 sec)


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

(2)
BOUNDS(1, n^1)

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

(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f0

      0: f0 -> f1 : [], cost: 1

      1: f1 -> f1 : B'=1+B, [ A>=1+B ], cost: 1

      2: f1 -> f2 : C'=B, [ B>=A ], cost: 1

      3: f2 -> f2 : C'=-1+C, [ C>=1 ], cost: 1

      4: f2 -> f3 : D'=C, [ 0>=C ], cost: 1

      5: f3 -> f3 : D'=1+D, [ A>=1+D ], cost: 1

      6: f3 -> f4 : E'=D, [ D>=A ], cost: 1

      7: f4 -> f4 : E'=-1+E, [ E>=1 ], cost: 1



Checking for constant complexity:

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

      0: f0 -> f1 : [], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: f1 -> f1 : B'=1+B, [ A>=1+B ], cost: 1



   Accelerated rule 1 with metering function A-B, yielding the new rule 8.

   Removing the simple loops: 1.



Accelerating simple loops of location 2.

   Accelerating the following rules:

      3: f2 -> f2 : C'=-1+C, [ C>=1 ], cost: 1



   Accelerated rule 3 with metering function C, yielding the new rule 9.

   Removing the simple loops: 3.



Accelerating simple loops of location 3.

   Accelerating the following rules:

      5: f3 -> f3 : D'=1+D, [ A>=1+D ], cost: 1



   Accelerated rule 5 with metering function -D+A, yielding the new rule 10.

   Removing the simple loops: 5.



Accelerating simple loops of location 4.

   Accelerating the following rules:

      7: f4 -> f4 : E'=-1+E, [ E>=1 ], cost: 1



   Accelerated rule 7 with metering function E, yielding the new rule 11.

   Removing the simple loops: 7.



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

   Start location: f0

      0: f0 -> f1 : [], cost: 1

      2: f1 -> f2 : C'=B, [ B>=A ], cost: 1

      8: f1 -> f1 : B'=A, [ A>=1+B ], cost: A-B

      4: f2 -> f3 : D'=C, [ 0>=C ], cost: 1

      9: f2 -> f2 : C'=0, [ C>=1 ], cost: C

      6: f3 -> f4 : E'=D, [ D>=A ], cost: 1

     10: f3 -> f3 : D'=A, [ A>=1+D ], cost: -D+A

     11: f4 -> f4 : E'=0, [ E>=1 ], cost: E



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f0 -> f1 : [], cost: 1

     12: f0 -> f1 : B'=A, [ A>=1+B ], cost: 1+A-B

      2: f1 -> f2 : C'=B, [ B>=A ], cost: 1

     13: f1 -> f2 : C'=0, [ B>=A && B>=1 ], cost: 1+B

      4: f2 -> f3 : D'=C, [ 0>=C ], cost: 1

     14: f2 -> f3 : D'=A, [ 0>=C && A>=1+C ], cost: 1-C+A

      6: f3 -> f4 : E'=D, [ D>=A ], cost: 1

     15: f3 -> f4 : E'=0, [ D>=A && D>=1 ], cost: 1+D



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

   Start location: f0

      0: f0 -> f1 : [], cost: 1

     12: f0 -> f1 : B'=A, [ A>=1+B ], cost: 1+A-B

      2: f1 -> f2 : C'=B, [ B>=A ], cost: 1

     13: f1 -> f2 : C'=0, [ B>=A && B>=1 ], cost: 1+B

      4: f2 -> f3 : D'=C, [ 0>=C ], cost: 1

     14: f2 -> f3 : D'=A, [ 0>=C && A>=1+C ], cost: 1-C+A

     15: f3 -> f4 : E'=0, [ D>=A && D>=1 ], cost: 1+D



Eliminated locations (on tree-shaped paths):

   Start location: f0

     16: f0 -> f2 : C'=B, [ B>=A ], cost: 2

     17: f0 -> f2 : C'=0, [ B>=A && B>=1 ], cost: 2+B

     18: f0 -> f2 : B'=A, C'=A, [ A>=1+B ], cost: 2+A-B

     19: f0 -> f2 : B'=A, C'=0, [ A>=1+B && A>=1 ], cost: 2+2*A-B

     20: f2 -> f4 : D'=A, E'=0, [ 0>=C && A>=1+C && A>=1 ], cost: 2-C+2*A



Eliminated locations (on tree-shaped paths):

   Start location: f0

     21: f0 -> f4 : C'=0, D'=A, E'=0, [ B>=A && B>=1 && A>=1 ], cost: 4+2*A+B

     22: f0 -> f4 : B'=A, C'=0, D'=A, E'=0, [ A>=1+B && A>=1 ], cost: 4+4*A-B

     23: f0 -> [9] : [ A>=1+B ], cost: 2+A-B



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f0

     21: f0 -> f4 : C'=0, D'=A, E'=0, [ B>=A && B>=1 && A>=1 ], cost: 4+2*A+B

     22: f0 -> f4 : B'=A, C'=0, D'=A, E'=0, [ A>=1+B && A>=1 ], cost: 4+4*A-B

     23: f0 -> [9] : [ A>=1+B ], cost: 2+A-B



Computing asymptotic complexity for rule 21

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      4+2*A+B (+), A (+/+!), B (+/+!), 1-A+B (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



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

      resulting limit problem:

      [solved]



   Solution:

      A / n

      B / n

   Resulting cost 4+3*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: 4+3*n

   Rule cost:   4+2*A+B

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



WORST_CASE(Omega(n^1),?)


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

(4)
BOUNDS(n^1, INF)