3.46/1.66	WORST_CASE(?, O(1))
3.46/1.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.46/1.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.46/1.67	
3.46/1.67	
3.46/1.67	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.46/1.67	
3.46/1.67	(0) CpxIntTrs
3.46/1.67	(1) Koat Proof [FINISHED, 8 ms]
3.46/1.67	(2) BOUNDS(1, 1)
3.46/1.67	
3.46/1.67	
3.46/1.67	----------------------------------------
3.46/1.67	
3.46/1.67	(0)
3.46/1.67	Obligation:
3.46/1.67	Complexity Int TRS consisting of the following rules:
3.46/1.67	f0(A, B, C) -> Com_1(f8(0, 10, 0)) :|: TRUE
3.46/1.67	f8(A, B, C) -> Com_1(f8(A + 2, B, C + 1)) :|: B >= C + 1
3.46/1.67	f8(A, B, C) -> Com_1(f6(A, B, C)) :|: 2 * B >= A + 1 && C >= B
3.46/1.67	f8(A, B, C) -> Com_1(f6(A, B, C)) :|: A >= 2 * B && C >= B
3.46/1.67	
3.46/1.67	The start-symbols are:[f0_3]
3.46/1.67	
3.46/1.67	
3.46/1.67	----------------------------------------
3.46/1.67	
3.46/1.67	(1) Koat Proof (FINISHED)
3.46/1.67	YES(?, 13)
3.46/1.67	
3.46/1.67	
3.46/1.67	
3.46/1.67	Initial complexity problem:
3.46/1.67	
3.46/1.67	1:	T:
3.46/1.67	
3.46/1.67			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f8(0, 10, 0))
3.46/1.67	
3.46/1.67			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f8(ar_0 + 2, ar_1, ar_2 + 1)) [ ar_1 >= ar_2 + 1 ]
3.46/1.67	
3.46/1.67			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f6(ar_0, ar_1, ar_2)) [ 2*ar_1 >= ar_0 + 1 /\ ar_2 >= ar_1 ]
3.46/1.67	
3.46/1.67			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f6(ar_0, ar_1, ar_2)) [ ar_0 >= 2*ar_1 /\ ar_2 >= ar_1 ]
3.46/1.67	
3.46/1.67			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.46/1.67	
3.46/1.67		start location:	koat_start
3.46/1.67	
3.46/1.67		leaf cost:	0
3.46/1.67	
3.46/1.67	
3.46/1.67	
3.46/1.67	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.46/1.67	
3.46/1.67	2:	T:
3.46/1.67	
3.46/1.67			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f8(0, 10, 0))
3.46/1.67	
3.46/1.67			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f8(ar_0 + 2, ar_1, ar_2 + 1)) [ ar_1 >= ar_2 + 1 ]
3.46/1.67	
3.46/1.67			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f6(ar_0, ar_1, ar_2)) [ 2*ar_1 >= ar_0 + 1 /\ ar_2 >= ar_1 ]
3.46/1.67	
3.46/1.67			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f6(ar_0, ar_1, ar_2)) [ ar_0 >= 2*ar_1 /\ ar_2 >= ar_1 ]
3.46/1.67	
3.46/1.67			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.46/1.67	
3.46/1.67		start location:	koat_start
3.46/1.67	
3.46/1.67		leaf cost:	0
3.46/1.67	
3.46/1.67	
3.46/1.67	
3.46/1.67	A polynomial rank function with
3.46/1.67	
3.46/1.67		Pol(f0) = 1
3.46/1.67	
3.46/1.67		Pol(f8) = 1
3.46/1.67	
3.46/1.67		Pol(f6) = 0
3.46/1.67	
3.46/1.67		Pol(koat_start) = 1
3.46/1.67	
3.46/1.67	orients all transitions weakly and the transitions
3.46/1.67	
3.46/1.67		f8(ar_0, ar_1, ar_2) -> Com_1(f6(ar_0, ar_1, ar_2)) [ 2*ar_1 >= ar_0 + 1 /\ ar_2 >= ar_1 ]
3.46/1.67	
3.46/1.67		f8(ar_0, ar_1, ar_2) -> Com_1(f6(ar_0, ar_1, ar_2)) [ ar_0 >= 2*ar_1 /\ ar_2 >= ar_1 ]
3.46/1.67	
3.46/1.67	strictly and produces the following problem:
3.46/1.67	
3.46/1.67	3:	T:
3.46/1.67	
3.46/1.67			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f8(0, 10, 0))
3.46/1.67	
3.46/1.67			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f8(ar_0 + 2, ar_1, ar_2 + 1)) [ ar_1 >= ar_2 + 1 ]
3.46/1.67	
3.46/1.67			(Comp: 1, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f6(ar_0, ar_1, ar_2)) [ 2*ar_1 >= ar_0 + 1 /\ ar_2 >= ar_1 ]
3.46/1.67	
3.46/1.67			(Comp: 1, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f6(ar_0, ar_1, ar_2)) [ ar_0 >= 2*ar_1 /\ ar_2 >= ar_1 ]
3.46/1.67	
3.46/1.67			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.46/1.67	
3.46/1.67		start location:	koat_start
3.46/1.67	
3.46/1.67		leaf cost:	0
3.46/1.67	
3.46/1.67	
3.46/1.67	
3.46/1.67	A polynomial rank function with
3.46/1.67	
3.46/1.67		Pol(f0) = 10
3.46/1.67	
3.46/1.67		Pol(f8) = V_2 - V_3
3.46/1.67	
3.46/1.67		Pol(f6) = V_2 - V_3
3.46/1.67	
3.46/1.67		Pol(koat_start) = 10
3.46/1.67	
3.46/1.67	orients all transitions weakly and the transition
3.46/1.67	
3.46/1.67		f8(ar_0, ar_1, ar_2) -> Com_1(f8(ar_0 + 2, ar_1, ar_2 + 1)) [ ar_1 >= ar_2 + 1 ]
3.46/1.67	
3.46/1.67	strictly and produces the following problem:
3.46/1.67	
3.46/1.67	4:	T:
3.46/1.67	
3.46/1.67			(Comp: 1, Cost: 1)     f0(ar_0, ar_1, ar_2) -> Com_1(f8(0, 10, 0))
3.46/1.67	
3.46/1.67			(Comp: 10, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f8(ar_0 + 2, ar_1, ar_2 + 1)) [ ar_1 >= ar_2 + 1 ]
3.46/1.67	
3.46/1.67			(Comp: 1, Cost: 1)     f8(ar_0, ar_1, ar_2) -> Com_1(f6(ar_0, ar_1, ar_2)) [ 2*ar_1 >= ar_0 + 1 /\ ar_2 >= ar_1 ]
3.46/1.67	
3.46/1.67			(Comp: 1, Cost: 1)     f8(ar_0, ar_1, ar_2) -> Com_1(f6(ar_0, ar_1, ar_2)) [ ar_0 >= 2*ar_1 /\ ar_2 >= ar_1 ]
3.46/1.67	
3.46/1.67			(Comp: 1, Cost: 0)     koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.46/1.67	
3.46/1.67		start location:	koat_start
3.46/1.67	
3.46/1.67		leaf cost:	0
3.46/1.67	
3.46/1.67	
3.46/1.67	
3.46/1.67	Complexity upper bound 13
3.46/1.67	
3.46/1.67	
3.46/1.67	
3.46/1.67	Time: 0.045 sec (SMT: 0.041 sec)
3.46/1.67	
3.46/1.67	
3.46/1.67	----------------------------------------
3.46/1.67	
3.46/1.67	(2)
3.46/1.67	BOUNDS(1, 1)
3.58/1.70	EOF