10.26/5.02	WORST_CASE(Omega(n^1), O(n^2))
10.26/5.03	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
10.26/5.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.26/5.03	
10.26/5.03	
10.26/5.03	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^2).
10.26/5.03	
10.26/5.03	(0) CpxIntTrs
10.26/5.03	(1) Koat Proof [FINISHED, 724 ms]
10.26/5.03	(2) BOUNDS(1, n^2)
10.26/5.03	(3) Loat Proof [FINISHED, 3347 ms]
10.26/5.03	(4) BOUNDS(n^1, INF)
10.26/5.03	
10.26/5.03	
10.26/5.03	----------------------------------------
10.26/5.03	
10.26/5.03	(0)
10.26/5.03	Obligation:
10.26/5.03	Complexity Int TRS consisting of the following rules:
10.26/5.03	f0(A, B) -> Com_1(f2(A, B)) :|: A >= 1 && B >= 1
10.26/5.03	f0(A, B) -> Com_1(f2(A, B)) :|: A >= 1 && 0 >= B + 1
10.26/5.03	f0(A, B) -> Com_1(f2(A, B)) :|: 0 >= A + 1 && B >= 1
10.26/5.03	f0(A, B) -> Com_1(f2(A, B)) :|: 0 >= A + 1 && 0 >= B + 1
10.26/5.03	f2(A, B) -> Com_1(f2(A, B - 1)) :|: A >= B && B >= 2 && A >= 1 && A + B >= 2
10.26/5.03	f2(A, B) -> Com_1(f2(A, B - 1)) :|: A >= B && B >= 2 && 0 >= A + 1 && A + B >= 2
10.26/5.03	f2(A, B) -> Com_1(f2(A, B - 1)) :|: A >= B && 0 >= B && A >= 1 && A + B >= 2
10.26/5.03	f2(A, B) -> Com_1(f2(A, B - 1)) :|: A >= B && 0 >= B && 0 >= A + 1 && A + B >= 2
10.26/5.03	f2(A, B) -> Com_1(f2(A - 1, B)) :|: 1 >= A + B && A >= B + 1 && A >= 2 && B >= 1
10.26/5.03	f2(A, B) -> Com_1(f2(A - 1, B)) :|: 1 >= A + B && A >= B + 1 && A >= 2 && 0 >= B + 1
10.26/5.03	f2(A, B) -> Com_1(f2(A - 1, B)) :|: 1 >= A + B && A >= B + 1 && 0 >= A && B >= 1
10.26/5.03	f2(A, B) -> Com_1(f2(A - 1, B)) :|: 1 >= A + B && A >= B + 1 && 0 >= A && 0 >= B + 1
10.26/5.03	f2(A, B) -> Com_1(f2(A, B + 1)) :|: B >= A && B >= 0 && A >= 1 && 0 >= B + A + 1
10.26/5.03	f2(A, B) -> Com_1(f2(A, B + 1)) :|: B >= A && B >= 0 && 0 >= A + 1 && 0 >= B + A + 1
10.26/5.03	f2(A, B) -> Com_1(f2(A, B + 1)) :|: B >= A && 0 >= B + 2 && A >= 1 && 0 >= B + A + 1
10.26/5.03	f2(A, B) -> Com_1(f2(A, B + 1)) :|: B >= A && 0 >= B + 2 && 0 >= A + 1 && 0 >= B + A + 1
10.26/5.03	f2(A, B) -> Com_1(f2(A + 1, B)) :|: B + A >= 0 && B >= A + 1 && A >= 0 && B >= 1
10.26/5.03	f2(A, B) -> Com_1(f2(A + 1, B)) :|: B + A >= 0 && B >= A + 1 && A >= 0 && 0 >= B + 1
10.26/5.03	f2(A, B) -> Com_1(f2(A + 1, B)) :|: B + A >= 0 && B >= A + 1 && 0 >= A + 2 && B >= 1
10.26/5.03	f2(A, B) -> Com_1(f2(A + 1, B)) :|: B + A >= 0 && B >= A + 1 && 0 >= A + 2 && 0 >= B + 1
10.26/5.03	
10.26/5.03	The start-symbols are:[f0_2]
10.26/5.03	
10.26/5.03	
10.26/5.03	----------------------------------------
10.26/5.03	
10.26/5.03	(1) Koat Proof (FINISHED)
10.26/5.03	YES(?, 8*ar_1 + 3*ar_0^2 + 4*ar_0*ar_1 + 6*ar_0 + ar_1^2 + 4)
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	Initial complexity problem:
10.26/5.03	
10.26/5.03	1:	T:
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ ar_1 >= 2 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ ar_1 >= 2 /\ 0 >= ar_0 + 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ 0 >= ar_1 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ 0 >= ar_1 /\ 0 >= ar_0 + 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= 2 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= 2 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ 0 >= ar_0 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ 0 >= ar_0 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ ar_1 >= 0 /\ ar_0 >= 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ ar_1 >= 0 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ 0 >= ar_1 + 2 /\ ar_0 >= 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ 0 >= ar_1 + 2 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 0 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ 0 >= ar_0 + 2 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ 0 >= ar_0 + 2 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
10.26/5.03	
10.26/5.03		start location:	koat_start
10.26/5.03	
10.26/5.03		leaf cost:	0
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	Testing for reachability in the complexity graph removes the following transitions from problem 1:
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ ar_1 >= 2 /\ 0 >= ar_0 + 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ 0 >= ar_1 /\ 0 >= ar_0 + 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= 2 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ 0 >= ar_0 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ ar_1 >= 0 /\ ar_0 >= 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ 0 >= ar_1 + 2 /\ ar_0 >= 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 0 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ 0 >= ar_0 + 2 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03	We thus obtain the following problem:
10.26/5.03	
10.26/5.03	2:	T:
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ 0 >= ar_1 + 2 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ 0 >= ar_0 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ 0 >= ar_0 + 2 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ ar_1 >= 0 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= 2 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ 0 >= ar_1 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ ar_1 >= 2 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
10.26/5.03	
10.26/5.03		start location:	koat_start
10.26/5.03	
10.26/5.03		leaf cost:	0
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	Repeatedly propagating knowledge in problem 2 produces the following problem:
10.26/5.03	
10.26/5.03	3:	T:
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ 0 >= ar_1 + 2 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ 0 >= ar_0 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ 0 >= ar_0 + 2 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ ar_1 >= 0 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= 2 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ 0 >= ar_1 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ ar_1 >= 2 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
10.26/5.03	
10.26/5.03		start location:	koat_start
10.26/5.03	
10.26/5.03		leaf cost:	0
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	A polynomial rank function with
10.26/5.03	
10.26/5.03		Pol(f2) = -V_1 + V_2
10.26/5.03	
10.26/5.03	and size complexities
10.26/5.03	
10.26/5.03		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ ar_1 >= 2 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-0) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ ar_1 >= 2 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-0) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ 0 >= ar_1 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ 0 >= ar_1 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-1) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= 2 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= 2 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ ar_1 >= 0 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ ar_1 >= 0 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-1) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ 0 >= ar_0 + 2 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ 0 >= ar_0 + 2 /\\ ar_1 >= 1 ]", 0-1) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ 0 >= ar_0 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ 0 >= ar_0 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ 0 >= ar_1 + 2 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-0) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ 0 >= ar_1 + 2 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03	orients the transitions
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03	weakly and the transition
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03	strictly and produces the following problem:
10.26/5.03	
10.26/5.03	4:	T:
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)              f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ 0 >= ar_1 + 2 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)              f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ 0 >= ar_0 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)              f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ 0 >= ar_0 + 2 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)              f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ ar_1 >= 0 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)              f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= 2 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)              f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ 0 >= ar_1 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)              f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ ar_1 >= 2 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 0)              koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
10.26/5.03	
10.26/5.03		start location:	koat_start
10.26/5.03	
10.26/5.03		leaf cost:	0
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	A polynomial rank function with
10.26/5.03	
10.26/5.03		Pol(f2) = V_2
10.26/5.03	
10.26/5.03	and size complexities
10.26/5.03	
10.26/5.03		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ ar_1 >= 2 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-0) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ ar_1 >= 2 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-0) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ 0 >= ar_1 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ 0 >= ar_1 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-1) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= 2 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= 2 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ ar_1 >= 0 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ ar_1 >= 0 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-1) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ 0 >= ar_0 + 2 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ 0 >= ar_0 + 2 /\\ ar_1 >= 1 ]", 0-1) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ 0 >= ar_0 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ 0 >= ar_0 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ 0 >= ar_1 + 2 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-0) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ 0 >= ar_1 + 2 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03	orients the transitions
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ ar_1 >= 2 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03	weakly and the transition
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ ar_1 >= 2 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03	strictly and produces the following problem:
10.26/5.03	
10.26/5.03	5:	T:
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                            f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ 0 >= ar_1 + 2 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                            f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ 0 >= ar_0 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                            f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ 0 >= ar_0 + 2 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                            f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ ar_1 >= 0 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                            f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= 2 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                            f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ 0 >= ar_1 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                  f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0*ar_1 + ar_1^2 + ar_1, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ ar_1 >= 2 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                            f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                            f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                            f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                            f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 0)                            koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
10.26/5.03	
10.26/5.03		start location:	koat_start
10.26/5.03	
10.26/5.03		leaf cost:	0
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	A polynomial rank function with
10.26/5.03	
10.26/5.03		Pol(f2) = V_1 + V_2
10.26/5.03	
10.26/5.03	and size complexities
10.26/5.03	
10.26/5.03		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ ar_1 >= 2 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-0) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ ar_1 >= 2 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-0) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ 0 >= ar_1 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ 0 >= ar_1 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-1) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= 2 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= 2 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ ar_1 >= 0 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ ar_1 >= 0 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-1) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ 0 >= ar_0 + 2 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ 0 >= ar_0 + 2 /\\ ar_1 >= 1 ]", 0-1) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ 0 >= ar_0 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ 0 >= ar_0 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ 0 >= ar_1 + 2 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-0) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ 0 >= ar_1 + 2 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03	orients the transitions
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ 0 >= ar_1 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03	weakly and the transition
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ 0 >= ar_1 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03	strictly and produces the following problem:
10.26/5.03	
10.26/5.03	6:	T:
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                            f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ 0 >= ar_1 + 2 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                            f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ 0 >= ar_0 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                            f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ 0 >= ar_0 + 2 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                            f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ ar_1 >= 0 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                            f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= 2 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                  f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ 0 >= ar_1 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                  f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0*ar_1 + ar_1^2 + ar_1, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ ar_1 >= 2 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                            f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                            f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                            f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                            f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 0)                            koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
10.26/5.03	
10.26/5.03		start location:	koat_start
10.26/5.03	
10.26/5.03		leaf cost:	0
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	A polynomial rank function with
10.26/5.03	
10.26/5.03		Pol(f2) = -V_2
10.26/5.03	
10.26/5.03	and size complexities
10.26/5.03	
10.26/5.03		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ ar_1 >= 2 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-0) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ ar_1 >= 2 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-0) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ 0 >= ar_1 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ 0 >= ar_1 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-1) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= 2 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= 2 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ ar_1 >= 0 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ ar_1 >= 0 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-1) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ 0 >= ar_0 + 2 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ 0 >= ar_0 + 2 /\\ ar_1 >= 1 ]", 0-1) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ 0 >= ar_0 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ 0 >= ar_0 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ 0 >= ar_1 + 2 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-0) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ 0 >= ar_1 + 2 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03	orients the transitions
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ 0 >= ar_0 + 2 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ ar_1 >= 0 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ 0 >= ar_1 + 2 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= 2 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ 0 >= ar_0 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03	weakly and the transition
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ 0 >= ar_1 + 2 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03	strictly and produces the following problem:
10.26/5.03	
10.26/5.03	7:	T:
10.26/5.03	
10.26/5.03			(Comp: 3*ar_1 + ar_0^2 + ar_0*ar_1, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ 0 >= ar_1 + 2 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                              f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ 0 >= ar_0 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                              f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ 0 >= ar_0 + 2 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                              f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ ar_1 >= 0 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                              f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= 2 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ 0 >= ar_1 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                    f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0*ar_1 + ar_1^2 + ar_1, Cost: 1)      f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ ar_1 >= 2 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 0)                              koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
10.26/5.03	
10.26/5.03		start location:	koat_start
10.26/5.03	
10.26/5.03		leaf cost:	0
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	A polynomial rank function with
10.26/5.03	
10.26/5.03		Pol(f2) = V_1
10.26/5.03	
10.26/5.03	and size complexities
10.26/5.03	
10.26/5.03		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ ar_1 >= 2 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-0) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ ar_1 >= 2 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-0) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ 0 >= ar_1 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ 0 >= ar_1 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-1) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= 2 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= 2 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ ar_1 >= 0 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ ar_1 >= 0 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-1) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ 0 >= ar_0 + 2 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ 0 >= ar_0 + 2 /\\ ar_1 >= 1 ]", 0-1) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ 0 >= ar_0 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ 0 >= ar_0 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ 0 >= ar_1 + 2 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-0) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ 0 >= ar_1 + 2 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03	orients the transitions
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= 2 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03	weakly and the transition
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= 2 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03	strictly and produces the following problem:
10.26/5.03	
10.26/5.03	8:	T:
10.26/5.03	
10.26/5.03			(Comp: 3*ar_1 + ar_0^2 + ar_0*ar_1, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ 0 >= ar_1 + 2 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                              f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ 0 >= ar_0 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                              f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ 0 >= ar_0 + 2 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                              f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ ar_1 >= 0 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0^2 + ar_0*ar_1 + ar_0, Cost: 1)      f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= 2 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ 0 >= ar_1 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                    f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0*ar_1 + ar_1^2 + ar_1, Cost: 1)      f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ ar_1 >= 2 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 0)                              koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
10.26/5.03	
10.26/5.03		start location:	koat_start
10.26/5.03	
10.26/5.03		leaf cost:	0
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	A polynomial rank function with
10.26/5.03	
10.26/5.03		Pol(f2) = -V_1 - V_2
10.26/5.03	
10.26/5.03	and size complexities
10.26/5.03	
10.26/5.03		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ ar_1 >= 2 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-0) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ ar_1 >= 2 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-0) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ 0 >= ar_1 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ 0 >= ar_1 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-1) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= 2 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= 2 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ ar_1 >= 0 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ ar_1 >= 0 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-1) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ 0 >= ar_0 + 2 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ 0 >= ar_0 + 2 /\\ ar_1 >= 1 ]", 0-1) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ 0 >= ar_0 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ 0 >= ar_0 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ 0 >= ar_1 + 2 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-0) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ 0 >= ar_1 + 2 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03	orients the transitions
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ ar_1 >= 0 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03	weakly and the transition
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ ar_1 >= 0 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03	strictly and produces the following problem:
10.26/5.03	
10.26/5.03	9:	T:
10.26/5.03	
10.26/5.03			(Comp: 3*ar_1 + ar_0^2 + ar_0*ar_1, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ 0 >= ar_1 + 2 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                              f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ 0 >= ar_0 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                              f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ 0 >= ar_0 + 2 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ ar_1 >= 0 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0^2 + ar_0*ar_1 + ar_0, Cost: 1)      f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= 2 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ 0 >= ar_1 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                    f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0*ar_1 + ar_1^2 + ar_1, Cost: 1)      f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ ar_1 >= 2 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 0)                              koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
10.26/5.03	
10.26/5.03		start location:	koat_start
10.26/5.03	
10.26/5.03		leaf cost:	0
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	A polynomial rank function with
10.26/5.03	
10.26/5.03		Pol(f2) = -V_1
10.26/5.03	
10.26/5.03	and size complexities
10.26/5.03	
10.26/5.03		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ ar_1 >= 2 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-0) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ ar_1 >= 2 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-0) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ 0 >= ar_1 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ 0 >= ar_1 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-1) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= 2 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= 2 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ ar_1 >= 0 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ ar_1 >= 0 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-1) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ 0 >= ar_0 + 2 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ 0 >= ar_0 + 2 /\\ ar_1 >= 1 ]", 0-1) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ 0 >= ar_0 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ 0 >= ar_0 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ 0 >= ar_1 + 2 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-0) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ 0 >= ar_1 + 2 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03	orients the transitions
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ 0 >= ar_0 + 2 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03	weakly and the transition
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ 0 >= ar_0 + 2 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03	strictly and produces the following problem:
10.26/5.03	
10.26/5.03	10:	T:
10.26/5.03	
10.26/5.03			(Comp: 3*ar_1 + ar_0^2 + ar_0*ar_1, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ 0 >= ar_1 + 2 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ?, Cost: 1)                              f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ 0 >= ar_0 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0^2 + ar_0*ar_1 + ar_0, Cost: 1)      f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ 0 >= ar_0 + 2 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ ar_1 >= 0 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0^2 + ar_0*ar_1 + ar_0, Cost: 1)      f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= 2 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ 0 >= ar_1 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                    f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0*ar_1 + ar_1^2 + ar_1, Cost: 1)      f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ ar_1 >= 2 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 0)                              koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
10.26/5.03	
10.26/5.03		start location:	koat_start
10.26/5.03	
10.26/5.03		leaf cost:	0
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	A polynomial rank function with
10.26/5.03	
10.26/5.03		Pol(f2) = V_1 - V_2
10.26/5.03	
10.26/5.03	and size complexities
10.26/5.03	
10.26/5.03		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ ar_1 >= 2 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-0) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ ar_1 >= 2 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-0) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ 0 >= ar_1 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\\ 0 >= ar_1 /\\ ar_0 >= 1 /\\ ar_0 + ar_1 >= 2 ]", 0-1) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= 2 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= 2 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ ar_1 >= 0 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ ar_1 >= 0 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-1) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ 0 >= ar_0 + 2 /\\ ar_1 >= 1 ]", 0-0) = ar_0
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ 0 >= ar_0 + 2 /\\ ar_1 >= 1 ]", 0-1) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ 0 >= ar_0 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\\ ar_0 >= ar_1 + 1 /\\ 0 >= ar_0 /\\ 0 >= ar_1 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ 0 >= ar_1 + 2 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-0) = ar_0 + ar_1
10.26/5.03	
10.26/5.03		S("f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\\ 0 >= ar_1 + 2 /\\ 0 >= ar_0 + 1 /\\ 0 >= ar_1 + ar_0 + 1 ]", 0-1) = ar_1
10.26/5.03	
10.26/5.03	orients the transitions
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ 0 >= ar_0 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03	weakly and the transition
10.26/5.03	
10.26/5.03		f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ 0 >= ar_0 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03	strictly and produces the following problem:
10.26/5.03	
10.26/5.03	11:	T:
10.26/5.03	
10.26/5.03			(Comp: 3*ar_1 + ar_0^2 + ar_0*ar_1, Cost: 1)    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ 0 >= ar_1 + 2 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                    f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ 0 >= ar_0 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0^2 + ar_0*ar_1 + ar_0, Cost: 1)      f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ 0 >= ar_0 + 2 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 + 1)) [ ar_1 >= ar_0 /\ ar_1 >= 0 /\ 0 >= ar_0 + 1 /\ 0 >= ar_1 + ar_0 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0^2 + ar_0*ar_1 + ar_0, Cost: 1)      f2(ar_0, ar_1) -> Com_1(f2(ar_0 - 1, ar_1)) [ 1 >= ar_0 + ar_1 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= 2 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                    f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ 0 >= ar_1 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0 + ar_1, Cost: 1)                    f2(ar_0, ar_1) -> Com_1(f2(ar_0 + 1, ar_1)) [ ar_1 + ar_0 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: ar_0*ar_1 + ar_1^2 + ar_1, Cost: 1)      f2(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1 - 1)) [ ar_0 >= ar_1 /\ ar_1 >= 2 /\ ar_0 >= 1 /\ ar_0 + ar_1 >= 2 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ 0 >= ar_0 + 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ 0 >= ar_1 + 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 1)                              f0(ar_0, ar_1) -> Com_1(f2(ar_0, ar_1)) [ ar_0 >= 1 /\ ar_1 >= 1 ]
10.26/5.03	
10.26/5.03			(Comp: 1, Cost: 0)                              koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
10.26/5.03	
10.26/5.03		start location:	koat_start
10.26/5.03	
10.26/5.03		leaf cost:	0
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	Complexity upper bound 8*ar_1 + 3*ar_0^2 + 4*ar_0*ar_1 + 6*ar_0 + ar_1^2 + 4
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	Time: 0.671 sec (SMT: 0.564 sec)
10.26/5.03	
10.26/5.03	
10.26/5.03	----------------------------------------
10.26/5.03	
10.26/5.03	(2)
10.26/5.03	BOUNDS(1, n^2)
10.26/5.03	
10.26/5.03	----------------------------------------
10.26/5.03	
10.26/5.03	(3) Loat Proof (FINISHED)
10.26/5.03	
10.26/5.03	
10.26/5.03	### Pre-processing the ITS problem ###
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	Initial linear ITS problem
10.26/5.03	
10.26/5.03	   Start location: f0
10.26/5.03	
10.26/5.03	      0: f0 -> f2 : [ A>=1 && B>=1 ], cost: 1
10.26/5.03	
10.26/5.03	      1: f0 -> f2 : [ A>=1 && 0>=1+B ], cost: 1
10.26/5.03	
10.26/5.03	      2: f0 -> f2 : [ 0>=1+A && B>=1 ], cost: 1
10.26/5.03	
10.26/5.03	      3: f0 -> f2 : [ 0>=1+A && 0>=1+B ], cost: 1
10.26/5.03	
10.26/5.03	      4: f2 -> f2 : B'=-1+B, [ A>=B && B>=2 && A>=1 && A+B>=2 ], cost: 1
10.26/5.03	
10.26/5.03	      5: f2 -> f2 : B'=-1+B, [ A>=B && B>=2 && 0>=1+A && A+B>=2 ], cost: 1
10.26/5.03	
10.26/5.03	      6: f2 -> f2 : B'=-1+B, [ A>=B && 0>=B && A>=1 && A+B>=2 ], cost: 1
10.26/5.03	
10.26/5.03	      7: f2 -> f2 : B'=-1+B, [ A>=B && 0>=B && 0>=1+A && A+B>=2 ], cost: 1
10.26/5.03	
10.26/5.03	      8: f2 -> f2 : A'=-1+A, [ 1>=A+B && A>=1+B && A>=2 && B>=1 ], cost: 1
10.26/5.03	
10.26/5.03	      9: f2 -> f2 : A'=-1+A, [ 1>=A+B && A>=1+B && A>=2 && 0>=1+B ], cost: 1
10.26/5.03	
10.26/5.03	     10: f2 -> f2 : A'=-1+A, [ 1>=A+B && A>=1+B && 0>=A && B>=1 ], cost: 1
10.26/5.03	
10.26/5.03	     11: f2 -> f2 : A'=-1+A, [ 1>=A+B && A>=1+B && 0>=A && 0>=1+B ], cost: 1
10.26/5.03	
10.26/5.03	     12: f2 -> f2 : B'=1+B, [ B>=A && B>=0 && A>=1 && 0>=1+A+B ], cost: 1
10.26/5.03	
10.26/5.03	     13: f2 -> f2 : B'=1+B, [ B>=A && B>=0 && 0>=1+A && 0>=1+A+B ], cost: 1
10.26/5.03	
10.26/5.03	     14: f2 -> f2 : B'=1+B, [ B>=A && 0>=2+B && A>=1 && 0>=1+A+B ], cost: 1
10.26/5.03	
10.26/5.03	     15: f2 -> f2 : B'=1+B, [ B>=A && 0>=2+B && 0>=1+A && 0>=1+A+B ], cost: 1
10.26/5.03	
10.26/5.03	     16: f2 -> f2 : A'=1+A, [ A+B>=0 && B>=1+A && A>=0 && B>=1 ], cost: 1
10.26/5.03	
10.26/5.03	     17: f2 -> f2 : A'=1+A, [ A+B>=0 && B>=1+A && A>=0 && 0>=1+B ], cost: 1
10.26/5.03	
10.26/5.03	     18: f2 -> f2 : A'=1+A, [ A+B>=0 && B>=1+A && 0>=2+A && B>=1 ], cost: 1
10.26/5.03	
10.26/5.03	     19: f2 -> f2 : A'=1+A, [ A+B>=0 && B>=1+A && 0>=2+A && 0>=1+B ], cost: 1
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	Removed rules with unsatisfiable guard:
10.26/5.03	
10.26/5.03	   Start location: f0
10.26/5.03	
10.26/5.03	      0: f0 -> f2 : [ A>=1 && B>=1 ], cost: 1
10.26/5.03	
10.26/5.03	      1: f0 -> f2 : [ A>=1 && 0>=1+B ], cost: 1
10.26/5.03	
10.26/5.03	      2: f0 -> f2 : [ 0>=1+A && B>=1 ], cost: 1
10.26/5.03	
10.26/5.03	      3: f0 -> f2 : [ 0>=1+A && 0>=1+B ], cost: 1
10.26/5.03	
10.26/5.03	      4: f2 -> f2 : B'=-1+B, [ A>=B && B>=2 && A>=1 && A+B>=2 ], cost: 1
10.26/5.03	
10.26/5.03	      6: f2 -> f2 : B'=-1+B, [ A>=B && 0>=B && A>=1 && A+B>=2 ], cost: 1
10.26/5.03	
10.26/5.03	      9: f2 -> f2 : A'=-1+A, [ 1>=A+B && A>=1+B && A>=2 && 0>=1+B ], cost: 1
10.26/5.03	
10.26/5.03	     11: f2 -> f2 : A'=-1+A, [ 1>=A+B && A>=1+B && 0>=A && 0>=1+B ], cost: 1
10.26/5.03	
10.26/5.03	     13: f2 -> f2 : B'=1+B, [ B>=A && B>=0 && 0>=1+A && 0>=1+A+B ], cost: 1
10.26/5.03	
10.26/5.03	     15: f2 -> f2 : B'=1+B, [ B>=A && 0>=2+B && 0>=1+A && 0>=1+A+B ], cost: 1
10.26/5.03	
10.26/5.03	     16: f2 -> f2 : A'=1+A, [ A+B>=0 && B>=1+A && A>=0 && B>=1 ], cost: 1
10.26/5.03	
10.26/5.03	     18: f2 -> f2 : A'=1+A, [ A+B>=0 && B>=1+A && 0>=2+A && B>=1 ], cost: 1
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	Simplified all rules, resulting in:
10.26/5.03	
10.26/5.03	   Start location: f0
10.26/5.03	
10.26/5.03	      0: f0 -> f2 : [ A>=1 && B>=1 ], cost: 1
10.26/5.03	
10.26/5.03	      1: f0 -> f2 : [ A>=1 && 0>=1+B ], cost: 1
10.26/5.03	
10.26/5.03	      2: f0 -> f2 : [ 0>=1+A && B>=1 ], cost: 1
10.26/5.03	
10.26/5.03	      3: f0 -> f2 : [ 0>=1+A && 0>=1+B ], cost: 1
10.26/5.03	
10.26/5.03	      4: f2 -> f2 : B'=-1+B, [ A>=B && B>=2 ], cost: 1
10.26/5.03	
10.26/5.03	      6: f2 -> f2 : B'=-1+B, [ 0>=B && A>=1 && A+B>=2 ], cost: 1
10.26/5.03	
10.26/5.03	      9: f2 -> f2 : A'=-1+A, [ 1>=A+B && A>=1+B && A>=2 ], cost: 1
10.26/5.03	
10.26/5.03	     11: f2 -> f2 : A'=-1+A, [ A>=1+B && 0>=A ], cost: 1
10.26/5.03	
10.26/5.03	     13: f2 -> f2 : B'=1+B, [ B>=0 && 0>=1+A && 0>=1+A+B ], cost: 1
10.26/5.03	
10.26/5.03	     15: f2 -> f2 : B'=1+B, [ B>=A && 0>=2+B ], cost: 1
10.26/5.03	
10.26/5.03	     16: f2 -> f2 : A'=1+A, [ B>=1+A && A>=0 ], cost: 1
10.26/5.03	
10.26/5.03	     18: f2 -> f2 : A'=1+A, [ A+B>=0 && B>=1+A && 0>=2+A ], cost: 1
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	### Simplification by acceleration and chaining ###
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	Accelerating simple loops of location 1.
10.26/5.03	
10.26/5.03	   Accelerating the following rules:
10.26/5.03	
10.26/5.03	      4: f2 -> f2 : B'=-1+B, [ A>=B && B>=2 ], cost: 1
10.26/5.03	
10.26/5.03	      6: f2 -> f2 : B'=-1+B, [ 0>=B && A>=1 && A+B>=2 ], cost: 1
10.26/5.03	
10.26/5.03	      9: f2 -> f2 : A'=-1+A, [ 1>=A+B && A>=1+B && A>=2 ], cost: 1
10.26/5.03	
10.26/5.03	     11: f2 -> f2 : A'=-1+A, [ A>=1+B && 0>=A ], cost: 1
10.26/5.03	
10.26/5.03	     13: f2 -> f2 : B'=1+B, [ B>=0 && 0>=1+A && 0>=1+A+B ], cost: 1
10.26/5.03	
10.26/5.03	     15: f2 -> f2 : B'=1+B, [ B>=A && 0>=2+B ], cost: 1
10.26/5.03	
10.26/5.03	     16: f2 -> f2 : A'=1+A, [ B>=1+A && A>=0 ], cost: 1
10.26/5.03	
10.26/5.03	     18: f2 -> f2 : A'=1+A, [ A+B>=0 && B>=1+A && 0>=2+A ], cost: 1
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	   Accelerated rule 4 with metering function -1+B, yielding the new rule 20.
10.26/5.03	
10.26/5.03	   Accelerated rule 6 with metering function -1+A+B, yielding the new rule 21.
10.26/5.03	
10.26/5.03	   Accelerated rule 9 with metering function -1+A, yielding the new rule 22.
10.26/5.03	
10.26/5.03	   Accelerated rule 11 with metering function A-B, yielding the new rule 23.
10.26/5.03	
10.26/5.03	   Accelerated rule 13 with metering function -A-B, yielding the new rule 24.
10.26/5.03	
10.26/5.03	   Accelerated rule 15 with metering function -1-B, yielding the new rule 25.
10.26/5.03	
10.26/5.03	   Accelerated rule 16 with metering function -A+B, yielding the new rule 26.
10.26/5.03	
10.26/5.03	   Accelerated rule 18 with metering function -1-A, yielding the new rule 27.
10.26/5.03	
10.26/5.03	   Removing the simple loops: 4 6 9 11 13 15 16 18.
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	Accelerated all simple loops using metering functions (where possible):
10.26/5.03	
10.26/5.03	   Start location: f0
10.26/5.03	
10.26/5.03	      0: f0 -> f2 : [ A>=1 && B>=1 ], cost: 1
10.26/5.03	
10.26/5.03	      1: f0 -> f2 : [ A>=1 && 0>=1+B ], cost: 1
10.26/5.03	
10.26/5.03	      2: f0 -> f2 : [ 0>=1+A && B>=1 ], cost: 1
10.26/5.03	
10.26/5.03	      3: f0 -> f2 : [ 0>=1+A && 0>=1+B ], cost: 1
10.26/5.03	
10.26/5.03	     20: f2 -> f2 : B'=1, [ A>=B && B>=2 ], cost: -1+B
10.26/5.03	
10.26/5.03	     21: f2 -> f2 : B'=1-A, [ 0>=B && A>=1 && A+B>=2 ], cost: -1+A+B
10.26/5.03	
10.26/5.03	     22: f2 -> f2 : A'=1, [ 1>=A+B && A>=1+B && A>=2 ], cost: -1+A
10.26/5.03	
10.26/5.03	     23: f2 -> f2 : A'=B, [ A>=1+B && 0>=A ], cost: A-B
10.26/5.03	
10.26/5.03	     24: f2 -> f2 : B'=-A, [ B>=0 && 0>=1+A && 0>=1+A+B ], cost: -A-B
10.26/5.03	
10.26/5.03	     25: f2 -> f2 : B'=-1, [ B>=A && 0>=2+B ], cost: -1-B
10.26/5.03	
10.26/5.03	     26: f2 -> f2 : A'=B, [ B>=1+A && A>=0 ], cost: -A+B
10.26/5.03	
10.26/5.03	     27: f2 -> f2 : A'=-1, [ A+B>=0 && B>=1+A && 0>=2+A ], cost: -1-A
10.26/5.03	
10.26/5.03	
10.26/5.03	
10.26/5.03	Chained accelerated rules (with incoming rules):
10.26/5.03	
10.26/5.03	   Start location: f0
10.26/5.03	
10.26/5.03	      0: f0 -> f2 : [ A>=1 && B>=1 ], cost: 1
10.26/5.03	
10.26/5.03	      1: f0 -> f2 : [ A>=1 && 0>=1+B ], cost: 1
10.26/5.03	
10.26/5.03	      2: f0 -> f2 : [ 0>=1+A && B>=1 ], cost: 1
10.26/5.03	
10.26/5.03	      3: f0 -> f2 : [ 0>=1+A && 0>=1+B ], cost: 1
10.26/5.03	
10.26/5.03	     28: f0 -> f2 : B'=1, [ A>=1 && A>=B && B>=2 ], cost: B
10.26/5.03	
10.26/5.03	     29: f0 -> f2 : B'=1-A, [ A>=1 && 0>=1+B && A+B>=2 ], cost: A+B
10.26/5.03	
10.26/5.03	     30: f0 -> f2 : A'=1, [ 0>=1+B && 1>=A+B && A>=1+B && A>=2 ], cost: A
10.26/5.03	
10.26/5.03	     31: f0 -> f2 : A'=B, [ 0>=1+A && 0>=1+B && A>=1+B ], cost: 1+A-B
10.26/5.03	
10.26/5.03	     32: f0 -> f2 : B'=-A, [ 0>=1+A && B>=1 && 0>=1+A+B ], cost: 1-A-B
10.26/5.03	
10.26/5.03	     33: f0 -> f2 : B'=-1, [ 0>=1+A && B>=A && 0>=2+B ], cost: -B
10.26/5.03	
10.26/5.03	     34: f0 -> f2 : A'=B, [ A>=1 && B>=1 && B>=1+A ], cost: 1-A+B
10.26/5.03	
10.26/5.03	     35: f0 -> f2 : A'=-1, [ B>=1 && A+B>=0 && B>=1+A && 0>=2+A ], cost: -A
10.26/5.04	
10.26/5.04	
10.26/5.04	
10.26/5.04	Removed unreachable locations (and leaf rules with constant cost):
10.26/5.04	
10.26/5.04	   Start location: f0
10.26/5.04	
10.26/5.04	     28: f0 -> f2 : B'=1, [ A>=1 && A>=B && B>=2 ], cost: B
10.26/5.04	
10.26/5.04	     29: f0 -> f2 : B'=1-A, [ A>=1 && 0>=1+B && A+B>=2 ], cost: A+B
10.26/5.04	
10.26/5.04	     30: f0 -> f2 : A'=1, [ 0>=1+B && 1>=A+B && A>=1+B && A>=2 ], cost: A
10.26/5.04	
10.26/5.04	     31: f0 -> f2 : A'=B, [ 0>=1+A && 0>=1+B && A>=1+B ], cost: 1+A-B
10.26/5.04	
10.26/5.04	     32: f0 -> f2 : B'=-A, [ 0>=1+A && B>=1 && 0>=1+A+B ], cost: 1-A-B
10.26/5.04	
10.26/5.04	     33: f0 -> f2 : B'=-1, [ 0>=1+A && B>=A && 0>=2+B ], cost: -B
10.26/5.04	
10.26/5.04	     34: f0 -> f2 : A'=B, [ A>=1 && B>=1 && B>=1+A ], cost: 1-A+B
10.26/5.04	
10.26/5.04	     35: f0 -> f2 : A'=-1, [ B>=1 && A+B>=0 && B>=1+A && 0>=2+A ], cost: -A
10.26/5.04	
10.26/5.04	
10.26/5.04	
10.26/5.04	### Computing asymptotic complexity ###
10.26/5.04	
10.26/5.04	
10.26/5.04	
10.26/5.04	Fully simplified ITS problem
10.26/5.04	
10.26/5.04	   Start location: f0
10.26/5.04	
10.26/5.04	     28: f0 -> f2 : B'=1, [ A>=1 && A>=B && B>=2 ], cost: B
10.26/5.04	
10.26/5.04	     29: f0 -> f2 : B'=1-A, [ A>=1 && 0>=1+B && A+B>=2 ], cost: A+B
10.26/5.04	
10.26/5.04	     30: f0 -> f2 : A'=1, [ 0>=1+B && 1>=A+B && A>=1+B && A>=2 ], cost: A
10.26/5.04	
10.26/5.04	     31: f0 -> f2 : A'=B, [ 0>=1+A && 0>=1+B && A>=1+B ], cost: 1+A-B
10.26/5.04	
10.26/5.04	     32: f0 -> f2 : B'=-A, [ 0>=1+A && B>=1 && 0>=1+A+B ], cost: 1-A-B
10.26/5.04	
10.26/5.04	     33: f0 -> f2 : B'=-1, [ 0>=1+A && B>=A && 0>=2+B ], cost: -B
10.26/5.04	
10.26/5.04	     34: f0 -> f2 : A'=B, [ A>=1 && B>=1 && B>=1+A ], cost: 1-A+B
10.26/5.04	
10.26/5.04	     35: f0 -> f2 : A'=-1, [ B>=1 && A+B>=0 && B>=1+A && 0>=2+A ], cost: -A
10.26/5.04	
10.26/5.04	
10.26/5.04	
10.26/5.04	Computing asymptotic complexity for rule 28
10.26/5.04	
10.26/5.04	   Simplified the guard:
10.26/5.04	
10.26/5.04	     28: f0 -> f2 : B'=1, [ A>=B && B>=2 ], cost: B
10.26/5.04	
10.26/5.04	   Solved the limit problem by the following transformations:
10.26/5.04	
10.26/5.04	      Created initial limit problem:
10.26/5.04	
10.26/5.04	      -1+B (+/+!), 1+A-B (+/+!), B (+) [not solved]
10.26/5.04	
10.26/5.04	
10.26/5.04	
10.26/5.04	      removing all constraints (solved by SMT)
10.26/5.04	
10.26/5.04	      resulting limit problem: [solved]
10.26/5.04	
10.26/5.04	
10.26/5.04	
10.26/5.04	      applying transformation rule (C) using substitution {A==n,B==n}
10.26/5.04	
10.26/5.04	      resulting limit problem:
10.26/5.04	
10.26/5.04	      [solved]
10.26/5.04	
10.26/5.04	
10.26/5.04	
10.26/5.04	   Solution:
10.26/5.04	
10.26/5.04	      A / n
10.26/5.04	
10.26/5.04	      B / n
10.26/5.04	
10.26/5.04	   Resulting cost n has complexity: Poly(n^1)
10.26/5.04	
10.26/5.04	
10.26/5.04	
10.26/5.04	Found new complexity Poly(n^1).
10.26/5.04	
10.26/5.04	
10.26/5.04	
10.26/5.04	Obtained the following overall complexity (w.r.t. the length of the input n):
10.26/5.04	
10.26/5.04	   Complexity:  Poly(n^1)
10.26/5.04	
10.26/5.04	   Cpx degree:  1
10.26/5.04	
10.26/5.04	   Solved cost: n
10.26/5.04	
10.26/5.04	   Rule cost:   B
10.26/5.04	
10.26/5.04	   Rule guard:  [ A>=B && B>=2 ]
10.26/5.04	
10.26/5.04	
10.26/5.04	
10.26/5.04	WORST_CASE(Omega(n^1),?)
10.26/5.04	
10.26/5.04	
10.26/5.04	----------------------------------------
10.26/5.04	
10.26/5.04	(4)
10.26/5.04	BOUNDS(n^1, INF)
10.26/5.06	EOF