5.68/2.74	WORST_CASE(Omega(n^2), O(n^2))
5.68/2.75	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
5.68/2.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.68/2.75	
5.68/2.75	
5.68/2.75	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).
5.68/2.75	
5.68/2.75	(0) CpxIntTrs
5.68/2.75	(1) Koat Proof [FINISHED, 213 ms]
5.68/2.75	(2) BOUNDS(1, n^2)
5.68/2.75	(3) Loat Proof [FINISHED, 995 ms]
5.68/2.75	(4) BOUNDS(n^2, INF)
5.68/2.75	
5.68/2.75	
5.68/2.75	----------------------------------------
5.68/2.75	
5.68/2.75	(0)
5.68/2.75	Obligation:
5.68/2.75	Complexity Int TRS consisting of the following rules:
5.68/2.75	f0(A, B, C, D, E, F, G) -> Com_1(f10(H, 0, C, D, E, F, G)) :|: TRUE
5.68/2.75	f10(A, B, C, D, E, F, G) -> Com_1(f10(A, B + 1, C, D, E, F, G)) :|: C >= B + 1
5.68/2.75	f18(A, B, C, D, E, F, G) -> Com_1(f21(A, B, C, D, E, 0, G)) :|: D >= 2 + E
5.68/2.75	f21(A, B, C, D, E, F, G) -> Com_1(f21(A, B, C, D, E, F + 1, G)) :|: D >= E + 2 + F
5.68/2.75	f21(A, B, C, D, E, F, G) -> Com_1(f21(A, B, C, D, E, F + 1, H)) :|: D >= E + 2 + F
5.68/2.75	f32(A, B, C, D, E, F, G) -> Com_1(f32(A, B, C, D, E + 1, F, G)) :|: D >= 2 + E
5.68/2.75	f32(A, B, C, D, E, F, G) -> Com_1(f41(A, B, C, D, E, F, G)) :|: E + 1 >= D
5.68/2.75	f21(A, B, C, D, E, F, G) -> Com_1(f18(A, B, C, D, E + 1, F, G)) :|: F + E + 1 >= D
5.68/2.75	f18(A, B, C, D, E, F, G) -> Com_1(f32(A, B, C, D, 0, F, G)) :|: E + 1 >= D
5.68/2.75	f10(A, B, C, D, E, F, G) -> Com_1(f18(A, B, C, C, 0, F, G)) :|: B >= C
5.68/2.75	
5.68/2.75	The start-symbols are:[f0_7]
5.68/2.75	
5.68/2.75	
5.68/2.75	----------------------------------------
5.68/2.75	
5.68/2.75	(1) Koat Proof (FINISHED)
5.68/2.75	YES(?, 8*ar_2 + 24*ar_2^2 + 10)
5.68/2.75	
5.68/2.75	
5.68/2.75	
5.68/2.75	Initial complexity problem:
5.68/2.75	
5.68/2.75	1:	T:
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f10(h, 0, ar_2, ar_3, ar_4, ar_5, ar_6))
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f10(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f10(ar_0, ar_1 + 1, ar_2, ar_3, ar_4, ar_5, ar_6)) [ ar_2 >= ar_1 + 1 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f18(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f21(ar_0, ar_1, ar_2, ar_3, ar_4, 0, ar_6)) [ ar_3 >= ar_4 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f21(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f21(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5 + 1, ar_6)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f21(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f21(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5 + 1, h)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f32(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f32(ar_0, ar_1, ar_2, ar_3, ar_4 + 1, ar_5, ar_6)) [ ar_3 >= ar_4 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f32(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f41(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6)) [ ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f21(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f18(ar_0, ar_1, ar_2, ar_3, ar_4 + 1, ar_5, ar_6)) [ ar_5 + ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f18(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f32(ar_0, ar_1, ar_2, ar_3, 0, ar_5, ar_6)) [ ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f10(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f18(ar_0, ar_1, ar_2, ar_2, 0, ar_5, ar_6)) [ ar_1 >= ar_2 ]
5.68/2.75	
5.68/2.75			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6)) [ 0 <= 0 ]
5.68/2.75	
5.68/2.75		start location:	koat_start
5.68/2.75	
5.68/2.75		leaf cost:	0
5.68/2.75	
5.68/2.75	
5.68/2.75	
5.68/2.75	Slicing away variables that do not contribute to conditions from problem 1 leaves variables [ar_1, ar_2, ar_3, ar_4, ar_5].
5.68/2.75	
5.68/2.75	We thus obtain the following problem:
5.68/2.75	
5.68/2.75	2:	T:
5.68/2.75	
5.68/2.75			(Comp: 1, Cost: 0)    koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))
5.68/2.75	
5.68/2.75		start location:	koat_start
5.68/2.75	
5.68/2.75		leaf cost:	0
5.68/2.75	
5.68/2.75	
5.68/2.75	
5.68/2.75	Repeatedly propagating knowledge in problem 2 produces the following problem:
5.68/2.75	
5.68/2.75	3:	T:
5.68/2.75	
5.68/2.75			(Comp: 1, Cost: 0)    koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]
5.68/2.75	
5.68/2.75			(Comp: 1, Cost: 1)    f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))
5.68/2.75	
5.68/2.75		start location:	koat_start
5.68/2.75	
5.68/2.75		leaf cost:	0
5.68/2.75	
5.68/2.75	
5.68/2.75	
5.68/2.75	A polynomial rank function with
5.68/2.75	
5.68/2.75		Pol(koat_start) = 3
5.68/2.75	
5.68/2.75		Pol(f0) = 3
5.68/2.75	
5.68/2.75		Pol(f10) = 3
5.68/2.75	
5.68/2.75		Pol(f18) = 2
5.68/2.75	
5.68/2.75		Pol(f32) = 1
5.68/2.75	
5.68/2.75		Pol(f21) = 2
5.68/2.75	
5.68/2.75		Pol(f41) = 0
5.68/2.75	
5.68/2.75	orients all transitions weakly and the transitions
5.68/2.75	
5.68/2.75		f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75		f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75		f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]
5.68/2.75	
5.68/2.75	strictly and produces the following problem:
5.68/2.75	
5.68/2.75	4:	T:
5.68/2.75	
5.68/2.75			(Comp: 1, Cost: 0)    koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
5.68/2.75	
5.68/2.75			(Comp: 3, Cost: 1)    f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]
5.68/2.75	
5.68/2.75			(Comp: 3, Cost: 1)    f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: 3, Cost: 1)    f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)    f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]
5.68/2.75	
5.68/2.75			(Comp: 1, Cost: 1)    f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))
5.68/2.75	
5.68/2.75		start location:	koat_start
5.68/2.75	
5.68/2.75		leaf cost:	0
5.68/2.75	
5.68/2.75	
5.68/2.75	
5.68/2.75	A polynomial rank function with
5.68/2.75	
5.68/2.75		Pol(koat_start) = V_2
5.68/2.75	
5.68/2.75		Pol(f0) = V_2
5.68/2.75	
5.68/2.75		Pol(f10) = V_2
5.68/2.75	
5.68/2.75		Pol(f18) = V_3
5.68/2.75	
5.68/2.75		Pol(f32) = V_3 - V_4
5.68/2.75	
5.68/2.75		Pol(f21) = V_3
5.68/2.75	
5.68/2.75		Pol(f41) = V_3 - V_4
5.68/2.75	
5.68/2.75	orients all transitions weakly and the transition
5.68/2.75	
5.68/2.75		f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]
5.68/2.75	
5.68/2.75	strictly and produces the following problem:
5.68/2.75	
5.68/2.75	5:	T:
5.68/2.75	
5.68/2.75			(Comp: 1, Cost: 0)       koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
5.68/2.75	
5.68/2.75			(Comp: 3, Cost: 1)       f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]
5.68/2.75	
5.68/2.75			(Comp: 3, Cost: 1)       f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)       f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: 3, Cost: 1)       f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: ar_2, Cost: 1)    f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)       f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)       f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)       f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)       f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]
5.68/2.75	
5.68/2.75			(Comp: 1, Cost: 1)       f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))
5.68/2.75	
5.68/2.75		start location:	koat_start
5.68/2.75	
5.68/2.75		leaf cost:	0
5.68/2.75	
5.68/2.75	
5.68/2.75	
5.68/2.75	A polynomial rank function with
5.68/2.75	
5.68/2.75		Pol(koat_start) = V_2
5.68/2.75	
5.68/2.75		Pol(f0) = V_2
5.68/2.75	
5.68/2.75		Pol(f10) = -V_1 + V_2
5.68/2.75	
5.68/2.75		Pol(f18) = -V_1 + V_2
5.68/2.75	
5.68/2.75		Pol(f32) = -V_1 + V_2
5.68/2.75	
5.68/2.75		Pol(f21) = -V_1 + V_2
5.68/2.75	
5.68/2.75		Pol(f41) = -V_1 + V_2
5.68/2.75	
5.68/2.75	orients all transitions weakly and the transition
5.68/2.75	
5.68/2.75		f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]
5.68/2.75	
5.68/2.75	strictly and produces the following problem:
5.68/2.75	
5.68/2.75	6:	T:
5.68/2.75	
5.68/2.75			(Comp: 1, Cost: 0)       koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
5.68/2.75	
5.68/2.75			(Comp: 3, Cost: 1)       f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]
5.68/2.75	
5.68/2.75			(Comp: 3, Cost: 1)       f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)       f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: 3, Cost: 1)       f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.75	
5.68/2.75			(Comp: ar_2, Cost: 1)    f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)       f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)       f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ?, Cost: 1)       f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]
5.68/2.75	
5.68/2.75			(Comp: ar_2, Cost: 1)    f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]
5.68/2.75	
5.68/2.75			(Comp: 1, Cost: 1)       f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))
5.68/2.75	
5.68/2.75		start location:	koat_start
5.68/2.75	
5.68/2.75		leaf cost:	0
5.68/2.75	
5.68/2.75	
5.68/2.75	
5.68/2.75	A polynomial rank function with
5.68/2.75	
5.68/2.75		Pol(f21) = V_3 - V_4 - 1
5.68/2.75	
5.68/2.75		Pol(f18) = V_3 - V_4
5.68/2.75	
5.68/2.75	and size complexities
5.68/2.75	
5.68/2.75		S("f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))", 0-0) = 0
5.68/2.75	
5.68/2.75		S("f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))", 0-1) = ar_2
5.68/2.75	
5.68/2.75		S("f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))", 0-2) = ar_3
5.68/2.75	
5.68/2.75		S("f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))", 0-3) = ar_4
5.68/2.75	
5.68/2.75		S("f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))", 0-4) = ar_5
5.68/2.75	
5.68/2.75		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]", 0-0) = ar_2
5.68/2.75	
5.68/2.75		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]", 0-1) = ar_2
5.68/2.75	
5.68/2.75		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]", 0-2) = ar_3
5.68/2.75	
5.68/2.75		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]", 0-3) = ar_4
5.68/2.75	
5.68/2.75		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]", 0-4) = ar_5
5.68/2.75	
5.68/2.75		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]", 0-0) = ar_2
5.68/2.75	
5.68/2.75		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]", 0-1) = ar_2
5.68/2.75	
5.68/2.75		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]", 0-2) = ar_2
5.68/2.75	
5.68/2.75		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]", 0-3) = ?
5.68/2.75	
5.68/2.75		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]", 0-4) = 0
5.68/2.75	
5.68/2.75		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-0) = ar_2
5.68/2.75	
5.68/2.75		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-1) = ar_2
5.68/2.75	
5.68/2.75		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-2) = ar_2
5.68/2.75	
5.68/2.75		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-3) = ?
5.68/2.75	
5.68/2.75		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-4) = ?
5.68/2.75	
5.68/2.75		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-0) = ar_2
5.68/2.75	
5.68/2.75		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-1) = ar_2
5.68/2.75	
5.68/2.75		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-2) = ar_2
5.68/2.75	
5.68/2.75		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-3) = ?
5.68/2.75	
5.68/2.75		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-4) = ?
5.68/2.75	
5.68/2.75		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]", 0-0) = ar_2
5.68/2.75	
5.68/2.75		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]", 0-1) = ar_2
5.68/2.75	
5.68/2.75		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]", 0-2) = ar_2
5.68/2.75	
5.68/2.75		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]", 0-3) = ar_2
5.68/2.75	
5.68/2.75		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]", 0-4) = ?
5.68/2.75	
5.68/2.75		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-0) = ar_2
5.68/2.75	
5.68/2.75		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-1) = ar_2
5.68/2.75	
5.68/2.75		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-2) = ar_2
5.68/2.75	
5.68/2.75		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-3) = ar_2
5.68/2.75	
5.68/2.75		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-4) = ?
5.68/2.75	
5.68/2.75		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]", 0-0) = ar_2
5.68/2.75	
5.68/2.75		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]", 0-1) = ar_2
5.68/2.75	
5.68/2.75		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]", 0-3) = ?
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]", 0-4) = ?
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-3) = 0
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-4) = ?
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]", 0-3) = 0
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]", 0-4) = ar_5
5.68/2.76	
5.68/2.76		S("koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-0) = ar_1
5.68/2.76	
5.68/2.76		S("koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-2) = ar_3
5.68/2.76	
5.68/2.76		S("koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-3) = ar_4
5.68/2.76	
5.68/2.76		S("koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-4) = ar_5
5.68/2.76	
5.68/2.76	orients the transitions
5.68/2.76	
5.68/2.76		f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.76	
5.68/2.76		f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]
5.68/2.76	
5.68/2.76		f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]
5.68/2.76	
5.68/2.76	weakly and the transition
5.68/2.76	
5.68/2.76		f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]
5.68/2.76	
5.68/2.76	strictly and produces the following problem:
5.68/2.76	
5.68/2.76	7:	T:
5.68/2.76	
5.68/2.76			(Comp: 1, Cost: 0)         koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
5.68/2.76	
5.68/2.76			(Comp: 3, Cost: 1)         f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]
5.68/2.76	
5.68/2.76			(Comp: 3, Cost: 1)         f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.76	
5.68/2.76			(Comp: ?, Cost: 1)         f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]
5.68/2.76	
5.68/2.76			(Comp: 3, Cost: 1)         f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.76	
5.68/2.76			(Comp: ar_2, Cost: 1)      f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]
5.68/2.76	
5.68/2.76			(Comp: ?, Cost: 1)         f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.76	
5.68/2.76			(Comp: ?, Cost: 1)         f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.76	
5.68/2.76			(Comp: 3*ar_2, Cost: 1)    f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]
5.68/2.76	
5.68/2.76			(Comp: ar_2, Cost: 1)      f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]
5.68/2.76	
5.68/2.76			(Comp: 1, Cost: 1)         f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))
5.68/2.76	
5.68/2.76		start location:	koat_start
5.68/2.76	
5.68/2.76		leaf cost:	0
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	A polynomial rank function with
5.68/2.76	
5.68/2.76		Pol(f21) = 1
5.68/2.76	
5.68/2.76		Pol(f18) = 0
5.68/2.76	
5.68/2.76	and size complexities
5.68/2.76	
5.68/2.76		S("f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))", 0-0) = 0
5.68/2.76	
5.68/2.76		S("f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))", 0-2) = ar_3
5.68/2.76	
5.68/2.76		S("f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))", 0-3) = ar_4
5.68/2.76	
5.68/2.76		S("f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))", 0-4) = ar_5
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]", 0-2) = ar_3
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]", 0-3) = ar_4
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]", 0-4) = ar_5
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]", 0-3) = ?
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]", 0-4) = 0
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-3) = ?
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-4) = ?
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-3) = ?
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-4) = ?
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]", 0-3) = ar_2
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]", 0-4) = ?
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-3) = ar_2
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-4) = ?
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]", 0-3) = ?
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]", 0-4) = ?
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-3) = 0
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-4) = ?
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]", 0-3) = 0
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]", 0-4) = ar_5
5.68/2.76	
5.68/2.76		S("koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-0) = ar_1
5.68/2.76	
5.68/2.76		S("koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-2) = ar_3
5.68/2.76	
5.68/2.76		S("koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-3) = ar_4
5.68/2.76	
5.68/2.76		S("koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-4) = ar_5
5.68/2.76	
5.68/2.76	orients the transitions
5.68/2.76	
5.68/2.76		f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.76	
5.68/2.76		f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]
5.68/2.76	
5.68/2.76	weakly and the transition
5.68/2.76	
5.68/2.76		f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]
5.68/2.76	
5.68/2.76	strictly and produces the following problem:
5.68/2.76	
5.68/2.76	8:	T:
5.68/2.76	
5.68/2.76			(Comp: 1, Cost: 0)         koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
5.68/2.76	
5.68/2.76			(Comp: 3, Cost: 1)         f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]
5.68/2.76	
5.68/2.76			(Comp: 3, Cost: 1)         f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.76	
5.68/2.76			(Comp: 3*ar_2, Cost: 1)    f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]
5.68/2.76	
5.68/2.76			(Comp: 3, Cost: 1)         f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.76	
5.68/2.76			(Comp: ar_2, Cost: 1)      f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]
5.68/2.76	
5.68/2.76			(Comp: ?, Cost: 1)         f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.76	
5.68/2.76			(Comp: ?, Cost: 1)         f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.76	
5.68/2.76			(Comp: 3*ar_2, Cost: 1)    f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]
5.68/2.76	
5.68/2.76			(Comp: ar_2, Cost: 1)      f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]
5.68/2.76	
5.68/2.76			(Comp: 1, Cost: 1)         f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))
5.68/2.76	
5.68/2.76		start location:	koat_start
5.68/2.76	
5.68/2.76		leaf cost:	0
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	A polynomial rank function with
5.68/2.76	
5.68/2.76		Pol(f21) = V_3 - V_4 - V_5
5.68/2.76	
5.68/2.76	and size complexities
5.68/2.76	
5.68/2.76		S("f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))", 0-0) = 0
5.68/2.76	
5.68/2.76		S("f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))", 0-2) = ar_3
5.68/2.76	
5.68/2.76		S("f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))", 0-3) = ar_4
5.68/2.76	
5.68/2.76		S("f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))", 0-4) = ar_5
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]", 0-2) = ar_3
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]", 0-3) = ar_4
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]", 0-4) = ar_5
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]", 0-3) = 3*ar_2
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]", 0-4) = 0
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-3) = 3*ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-4) = ?
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-3) = 3*ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]", 0-4) = ?
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]", 0-3) = ar_2
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]", 0-4) = ?
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-3) = ar_2
5.68/2.76	
5.68/2.76		S("f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-4) = ?
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]", 0-3) = 3*ar_2
5.68/2.76	
5.68/2.76		S("f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]", 0-4) = ?
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-3) = 0
5.68/2.76	
5.68/2.76		S("f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]", 0-4) = ?
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]", 0-0) = ar_2
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]", 0-2) = ar_2
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]", 0-3) = 0
5.68/2.76	
5.68/2.76		S("f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]", 0-4) = ar_5
5.68/2.76	
5.68/2.76		S("koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-0) = ar_1
5.68/2.76	
5.68/2.76		S("koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-1) = ar_2
5.68/2.76	
5.68/2.76		S("koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-2) = ar_3
5.68/2.76	
5.68/2.76		S("koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-3) = ar_4
5.68/2.76	
5.68/2.76		S("koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]", 0-4) = ar_5
5.68/2.76	
5.68/2.76	orients the transitions
5.68/2.76	
5.68/2.76		f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.76	
5.68/2.76	weakly and the transition
5.68/2.76	
5.68/2.76		f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.76	
5.68/2.76	strictly and produces the following problem:
5.68/2.76	
5.68/2.76	9:	T:
5.68/2.76	
5.68/2.76			(Comp: 1, Cost: 0)            koat_start(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
5.68/2.76	
5.68/2.76			(Comp: 3, Cost: 1)            f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_2, 0, ar_5)) [ ar_1 >= ar_2 ]
5.68/2.76	
5.68/2.76			(Comp: 3, Cost: 1)            f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, 0, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.76	
5.68/2.76			(Comp: 3*ar_2, Cost: 1)       f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_5 + ar_4 + 1 >= ar_3 ]
5.68/2.76	
5.68/2.76			(Comp: 3, Cost: 1)            f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f41(ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 + 1 >= ar_3 ]
5.68/2.76	
5.68/2.76			(Comp: ar_2, Cost: 1)         f32(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f32(ar_1, ar_2, ar_3, ar_4 + 1, ar_5)) [ ar_3 >= ar_4 + 2 ]
5.68/2.76	
5.68/2.76			(Comp: 12*ar_2^2, Cost: 1)    f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.76	
5.68/2.76			(Comp: 12*ar_2^2, Cost: 1)    f21(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, ar_5 + 1)) [ ar_3 >= ar_4 + ar_5 + 2 ]
5.68/2.76	
5.68/2.76			(Comp: 3*ar_2, Cost: 1)       f18(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f21(ar_1, ar_2, ar_3, ar_4, 0)) [ ar_3 >= ar_4 + 2 ]
5.68/2.76	
5.68/2.76			(Comp: ar_2, Cost: 1)         f10(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_1 + 1 ]
5.68/2.76	
5.68/2.76			(Comp: 1, Cost: 1)            f0(ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f10(0, ar_2, ar_3, ar_4, ar_5))
5.68/2.76	
5.68/2.76		start location:	koat_start
5.68/2.76	
5.68/2.76		leaf cost:	0
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Complexity upper bound 8*ar_2 + 24*ar_2^2 + 10
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Time: 0.213 sec (SMT: 0.165 sec)
5.68/2.76	
5.68/2.76	
5.68/2.76	----------------------------------------
5.68/2.76	
5.68/2.76	(2)
5.68/2.76	BOUNDS(1, n^2)
5.68/2.76	
5.68/2.76	----------------------------------------
5.68/2.76	
5.68/2.76	(3) Loat Proof (FINISHED)
5.68/2.76	
5.68/2.76	
5.68/2.76	### Pre-processing the ITS problem ###
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Initial linear ITS problem
5.68/2.76	
5.68/2.76	   Start location: f0
5.68/2.76	
5.68/2.76	      0: f0 -> f10 : A'=free, B'=0, [], cost: 1
5.68/2.76	
5.68/2.76	      1: f10 -> f10 : B'=1+B, [ C>=1+B ], cost: 1
5.68/2.76	
5.68/2.76	      9: f10 -> f18 : D'=C, E'=0, [ B>=C ], cost: 1
5.68/2.76	
5.68/2.76	      2: f18 -> f21 : F'=0, [ D>=2+E ], cost: 1
5.68/2.76	
5.68/2.76	      8: f18 -> f32 : E'=0, [ 1+E>=D ], cost: 1
5.68/2.76	
5.68/2.76	      3: f21 -> f21 : F'=1+F, [ D>=2+F+E ], cost: 1
5.68/2.76	
5.68/2.76	      4: f21 -> f21 : F'=1+F, G'=free_1, [ D>=2+F+E ], cost: 1
5.68/2.76	
5.68/2.76	      7: f21 -> f18 : E'=1+E, [ 1+F+E>=D ], cost: 1
5.68/2.76	
5.68/2.76	      5: f32 -> f32 : E'=1+E, [ D>=2+E ], cost: 1
5.68/2.76	
5.68/2.76	      6: f32 -> f41 : [ 1+E>=D ], cost: 1
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Removed unreachable and leaf rules:
5.68/2.76	
5.68/2.76	   Start location: f0
5.68/2.76	
5.68/2.76	      0: f0 -> f10 : A'=free, B'=0, [], cost: 1
5.68/2.76	
5.68/2.76	      1: f10 -> f10 : B'=1+B, [ C>=1+B ], cost: 1
5.68/2.76	
5.68/2.76	      9: f10 -> f18 : D'=C, E'=0, [ B>=C ], cost: 1
5.68/2.76	
5.68/2.76	      2: f18 -> f21 : F'=0, [ D>=2+E ], cost: 1
5.68/2.76	
5.68/2.76	      8: f18 -> f32 : E'=0, [ 1+E>=D ], cost: 1
5.68/2.76	
5.68/2.76	      3: f21 -> f21 : F'=1+F, [ D>=2+F+E ], cost: 1
5.68/2.76	
5.68/2.76	      4: f21 -> f21 : F'=1+F, G'=free_1, [ D>=2+F+E ], cost: 1
5.68/2.76	
5.68/2.76	      7: f21 -> f18 : E'=1+E, [ 1+F+E>=D ], cost: 1
5.68/2.76	
5.68/2.76	      5: f32 -> f32 : E'=1+E, [ D>=2+E ], cost: 1
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	### Simplification by acceleration and chaining ###
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Accelerating simple loops of location 1.
5.68/2.76	
5.68/2.76	   Accelerating the following rules:
5.68/2.76	
5.68/2.76	      1: f10 -> f10 : B'=1+B, [ C>=1+B ], cost: 1
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	   Accelerated rule 1 with metering function C-B, yielding the new rule 10.
5.68/2.76	
5.68/2.76	   Removing the simple loops: 1.
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Accelerating simple loops of location 3.
5.68/2.76	
5.68/2.76	   Accelerating the following rules:
5.68/2.76	
5.68/2.76	      3: f21 -> f21 : F'=1+F, [ D>=2+F+E ], cost: 1
5.68/2.76	
5.68/2.76	      4: f21 -> f21 : F'=1+F, G'=free_1, [ D>=2+F+E ], cost: 1
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	   Accelerated rule 3 with metering function -1-F+D-E, yielding the new rule 11.
5.68/2.76	
5.68/2.76	   Accelerated rule 4 with metering function -1-F+D-E, yielding the new rule 12.
5.68/2.76	
5.68/2.76	   Removing the simple loops: 3 4.
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Accelerating simple loops of location 4.
5.68/2.76	
5.68/2.76	   Accelerating the following rules:
5.68/2.76	
5.68/2.76	      5: f32 -> f32 : E'=1+E, [ D>=2+E ], cost: 1
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	   Accelerated rule 5 with metering function -1+D-E, yielding the new rule 13.
5.68/2.76	
5.68/2.76	   Removing the simple loops: 5.
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Accelerated all simple loops using metering functions (where possible):
5.68/2.76	
5.68/2.76	   Start location: f0
5.68/2.76	
5.68/2.76	      0: f0 -> f10 : A'=free, B'=0, [], cost: 1
5.68/2.76	
5.68/2.76	      9: f10 -> f18 : D'=C, E'=0, [ B>=C ], cost: 1
5.68/2.76	
5.68/2.76	     10: f10 -> f10 : B'=C, [ C>=1+B ], cost: C-B
5.68/2.76	
5.68/2.76	      2: f18 -> f21 : F'=0, [ D>=2+E ], cost: 1
5.68/2.76	
5.68/2.76	      8: f18 -> f32 : E'=0, [ 1+E>=D ], cost: 1
5.68/2.76	
5.68/2.76	      7: f21 -> f18 : E'=1+E, [ 1+F+E>=D ], cost: 1
5.68/2.76	
5.68/2.76	     11: f21 -> f21 : F'=-1+D-E, [ D>=2+F+E ], cost: -1-F+D-E
5.68/2.76	
5.68/2.76	     12: f21 -> f21 : F'=-1+D-E, G'=free_1, [ D>=2+F+E ], cost: -1-F+D-E
5.68/2.76	
5.68/2.76	     13: f32 -> f32 : E'=-1+D, [ D>=2+E ], cost: -1+D-E
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Chained accelerated rules (with incoming rules):
5.68/2.76	
5.68/2.76	   Start location: f0
5.68/2.76	
5.68/2.76	      0: f0 -> f10 : A'=free, B'=0, [], cost: 1
5.68/2.76	
5.68/2.76	     14: f0 -> f10 : A'=free, B'=C, [ C>=1 ], cost: 1+C
5.68/2.76	
5.68/2.76	      9: f10 -> f18 : D'=C, E'=0, [ B>=C ], cost: 1
5.68/2.76	
5.68/2.76	      2: f18 -> f21 : F'=0, [ D>=2+E ], cost: 1
5.68/2.76	
5.68/2.76	      8: f18 -> f32 : E'=0, [ 1+E>=D ], cost: 1
5.68/2.76	
5.68/2.76	     15: f18 -> f21 : F'=-1+D-E, [ D>=2+E ], cost: D-E
5.68/2.76	
5.68/2.76	     16: f18 -> f21 : F'=-1+D-E, G'=free_1, [ D>=2+E ], cost: D-E
5.68/2.76	
5.68/2.76	     17: f18 -> f32 : E'=-1+D, [ 1+E>=D && D>=2 ], cost: D
5.68/2.76	
5.68/2.76	      7: f21 -> f18 : E'=1+E, [ 1+F+E>=D ], cost: 1
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Removed unreachable locations (and leaf rules with constant cost):
5.68/2.76	
5.68/2.76	   Start location: f0
5.68/2.76	
5.68/2.76	      0: f0 -> f10 : A'=free, B'=0, [], cost: 1
5.68/2.76	
5.68/2.76	     14: f0 -> f10 : A'=free, B'=C, [ C>=1 ], cost: 1+C
5.68/2.76	
5.68/2.76	      9: f10 -> f18 : D'=C, E'=0, [ B>=C ], cost: 1
5.68/2.76	
5.68/2.76	      2: f18 -> f21 : F'=0, [ D>=2+E ], cost: 1
5.68/2.76	
5.68/2.76	     15: f18 -> f21 : F'=-1+D-E, [ D>=2+E ], cost: D-E
5.68/2.76	
5.68/2.76	     16: f18 -> f21 : F'=-1+D-E, G'=free_1, [ D>=2+E ], cost: D-E
5.68/2.76	
5.68/2.76	     17: f18 -> f32 : E'=-1+D, [ 1+E>=D && D>=2 ], cost: D
5.68/2.76	
5.68/2.76	      7: f21 -> f18 : E'=1+E, [ 1+F+E>=D ], cost: 1
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Eliminated locations (on tree-shaped paths):
5.68/2.76	
5.68/2.76	   Start location: f0
5.68/2.76	
5.68/2.76	     18: f0 -> f18 : A'=free, B'=0, D'=C, E'=0, [ 0>=C ], cost: 2
5.68/2.76	
5.68/2.76	     19: f0 -> f18 : A'=free, B'=C, D'=C, E'=0, [ C>=1 ], cost: 2+C
5.68/2.76	
5.68/2.76	     17: f18 -> f32 : E'=-1+D, [ 1+E>=D && D>=2 ], cost: D
5.68/2.76	
5.68/2.76	     20: f18 -> f18 : E'=1+E, F'=-1+D-E, [ D>=2+E ], cost: 1+D-E
5.68/2.76	
5.68/2.76	     21: f18 -> f18 : E'=1+E, F'=-1+D-E, G'=free_1, [ D>=2+E ], cost: 1+D-E
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Accelerating simple loops of location 2.
5.68/2.76	
5.68/2.76	   Accelerating the following rules:
5.68/2.76	
5.68/2.76	     20: f18 -> f18 : E'=1+E, F'=-1+D-E, [ D>=2+E ], cost: 1+D-E
5.68/2.76	
5.68/2.76	     21: f18 -> f18 : E'=1+E, F'=-1+D-E, G'=free_1, [ D>=2+E ], cost: 1+D-E
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	   Accelerated rule 20 with metering function -1+D-E, yielding the new rule 22.
5.68/2.76	
5.68/2.76	   Accelerated rule 21 with metering function -1+D-E, yielding the new rule 23.
5.68/2.76	
5.68/2.76	   Removing the simple loops: 20 21.
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Accelerated all simple loops using metering functions (where possible):
5.68/2.76	
5.68/2.76	   Start location: f0
5.68/2.76	
5.68/2.76	     18: f0 -> f18 : A'=free, B'=0, D'=C, E'=0, [ 0>=C ], cost: 2
5.68/2.76	
5.68/2.76	     19: f0 -> f18 : A'=free, B'=C, D'=C, E'=0, [ C>=1 ], cost: 2+C
5.68/2.76	
5.68/2.76	     17: f18 -> f32 : E'=-1+D, [ 1+E>=D && D>=2 ], cost: D
5.68/2.76	
5.68/2.76	     22: f18 -> f18 : E'=-1+D, F'=1, [ D>=2+E ], cost: -3/2-1/2*(-1+D-E)^2-(-1+D-E)*E+D*(-1+D-E)+3/2*D-3/2*E
5.68/2.76	
5.68/2.76	     23: f18 -> f18 : E'=-1+D, F'=1, G'=free_1, [ D>=2+E ], cost: -3/2-1/2*(-1+D-E)^2-(-1+D-E)*E+D*(-1+D-E)+3/2*D-3/2*E
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Chained accelerated rules (with incoming rules):
5.68/2.76	
5.68/2.76	   Start location: f0
5.68/2.76	
5.68/2.76	     18: f0 -> f18 : A'=free, B'=0, D'=C, E'=0, [ 0>=C ], cost: 2
5.68/2.76	
5.68/2.76	     19: f0 -> f18 : A'=free, B'=C, D'=C, E'=0, [ C>=1 ], cost: 2+C
5.68/2.76	
5.68/2.76	     24: f0 -> f18 : A'=free, B'=C, D'=C, E'=-1+C, F'=1, [ C>=2 ], cost: 1/2+5/2*C+C*(-1+C)-1/2*(-1+C)^2
5.68/2.76	
5.68/2.76	     25: f0 -> f18 : A'=free, B'=C, D'=C, E'=-1+C, F'=1, G'=free_1, [ C>=2 ], cost: 1/2+5/2*C+C*(-1+C)-1/2*(-1+C)^2
5.68/2.76	
5.68/2.76	     17: f18 -> f32 : E'=-1+D, [ 1+E>=D && D>=2 ], cost: D
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Eliminated locations (on tree-shaped paths):
5.68/2.76	
5.68/2.76	   Start location: f0
5.68/2.76	
5.68/2.76	     26: f0 -> f32 : A'=free, B'=C, D'=C, E'=-1+C, F'=1, [ C>=2 ], cost: 1/2+7/2*C+C*(-1+C)-1/2*(-1+C)^2
5.68/2.76	
5.68/2.76	     27: f0 -> f32 : A'=free, B'=C, D'=C, E'=-1+C, F'=1, G'=free_1, [ C>=2 ], cost: 1/2+7/2*C+C*(-1+C)-1/2*(-1+C)^2
5.68/2.76	
5.68/2.76	     28: f0 -> [10] : [ C>=1 ], cost: 2+C
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	### Computing asymptotic complexity ###
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Fully simplified ITS problem
5.68/2.76	
5.68/2.76	   Start location: f0
5.68/2.76	
5.68/2.76	     27: f0 -> f32 : A'=free, B'=C, D'=C, E'=-1+C, F'=1, G'=free_1, [ C>=2 ], cost: 1/2+7/2*C+C*(-1+C)-1/2*(-1+C)^2
5.68/2.76	
5.68/2.76	     28: f0 -> [10] : [ C>=1 ], cost: 2+C
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Computing asymptotic complexity for rule 27
5.68/2.76	
5.68/2.76	   Solved the limit problem by the following transformations:
5.68/2.76	
5.68/2.76	      Created initial limit problem:
5.68/2.76	
5.68/2.76	      1/2*C^2+7/2*C (+), -1+C (+/+!) [not solved]
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	      removing all constraints (solved by SMT)
5.68/2.76	
5.68/2.76	      resulting limit problem: [solved]
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	      applying transformation rule (C) using substitution {C==n}
5.68/2.76	
5.68/2.76	      resulting limit problem:
5.68/2.76	
5.68/2.76	      [solved]
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	   Solution:
5.68/2.76	
5.68/2.76	      C / n
5.68/2.76	
5.68/2.76	   Resulting cost 1/2*n^2+7/2*n has complexity: Poly(n^2)
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Found new complexity Poly(n^2).
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	Obtained the following overall complexity (w.r.t. the length of the input n):
5.68/2.76	
5.68/2.76	   Complexity:  Poly(n^2)
5.68/2.76	
5.68/2.76	   Cpx degree:  2
5.68/2.76	
5.68/2.76	   Solved cost: 1/2*n^2+7/2*n
5.68/2.76	
5.68/2.76	   Rule cost:   1/2+7/2*C+C*(-1+C)-1/2*(-1+C)^2
5.68/2.76	
5.68/2.76	   Rule guard:  [ C>=2 ]
5.68/2.76	
5.68/2.76	
5.68/2.76	
5.68/2.76	WORST_CASE(Omega(n^2),?)
5.68/2.76	
5.68/2.76	
5.68/2.76	----------------------------------------
5.68/2.76	
5.68/2.76	(4)
5.68/2.76	BOUNDS(n^2, INF)
5.84/2.78	EOF