3.19/1.70	WORST_CASE(?, O(1))
3.54/1.71	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
3.54/1.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.54/1.71	
3.54/1.71	
3.54/1.71	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.54/1.71	
3.54/1.71	(0) CpxIntTrs
3.54/1.71	(1) Koat Proof [FINISHED, 5 ms]
3.54/1.71	(2) BOUNDS(1, 1)
3.54/1.71	
3.54/1.71	
3.54/1.71	----------------------------------------
3.54/1.71	
3.54/1.71	(0)
3.54/1.71	Obligation:
3.54/1.71	Complexity Int TRS consisting of the following rules:
3.54/1.71	f0(A, B) -> Com_1(f6(C, 0)) :|: TRUE
3.54/1.71	f6(A, B) -> Com_1(f6(A, B + 1)) :|: 9 >= B
3.54/1.71	f6(A, B) -> Com_1(f15(A, B)) :|: B >= 10
3.54/1.71	
3.54/1.71	The start-symbols are:[f0_2]
3.54/1.71	
3.54/1.71	
3.54/1.71	----------------------------------------
3.54/1.71	
3.54/1.71	(1) Koat Proof (FINISHED)
3.54/1.71	YES(?, 12)
3.54/1.71	
3.54/1.71	
3.54/1.71	
3.54/1.71	Initial complexity problem:
3.54/1.71	
3.54/1.71	1:	T:
3.54/1.71	
3.54/1.71			(Comp: ?, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f6(c, 0))
3.54/1.71	
3.54/1.71			(Comp: ?, Cost: 1)    f6(ar_0, ar_1) -> Com_1(f6(ar_0, ar_1 + 1)) [ 9 >= ar_1 ]
3.54/1.71	
3.54/1.71			(Comp: ?, Cost: 1)    f6(ar_0, ar_1) -> Com_1(f15(ar_0, ar_1)) [ ar_1 >= 10 ]
3.54/1.71	
3.54/1.71			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.54/1.71	
3.54/1.71		start location:	koat_start
3.54/1.71	
3.54/1.71		leaf cost:	0
3.54/1.71	
3.54/1.71	
3.54/1.71	
3.54/1.71	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.54/1.71	
3.54/1.71	2:	T:
3.54/1.71	
3.54/1.71			(Comp: 1, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f6(c, 0))
3.54/1.71	
3.54/1.71			(Comp: ?, Cost: 1)    f6(ar_0, ar_1) -> Com_1(f6(ar_0, ar_1 + 1)) [ 9 >= ar_1 ]
3.54/1.71	
3.54/1.71			(Comp: ?, Cost: 1)    f6(ar_0, ar_1) -> Com_1(f15(ar_0, ar_1)) [ ar_1 >= 10 ]
3.54/1.71	
3.54/1.71			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.54/1.71	
3.54/1.71		start location:	koat_start
3.54/1.71	
3.54/1.71		leaf cost:	0
3.54/1.71	
3.54/1.71	
3.54/1.71	
3.54/1.71	A polynomial rank function with
3.54/1.71	
3.54/1.71		Pol(f0) = 1
3.54/1.71	
3.54/1.71		Pol(f6) = 1
3.54/1.71	
3.54/1.71		Pol(f15) = 0
3.54/1.71	
3.54/1.71		Pol(koat_start) = 1
3.54/1.71	
3.54/1.71	orients all transitions weakly and the transition
3.54/1.71	
3.54/1.71		f6(ar_0, ar_1) -> Com_1(f15(ar_0, ar_1)) [ ar_1 >= 10 ]
3.54/1.71	
3.54/1.71	strictly and produces the following problem:
3.54/1.71	
3.54/1.71	3:	T:
3.54/1.71	
3.54/1.71			(Comp: 1, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f6(c, 0))
3.54/1.71	
3.54/1.71			(Comp: ?, Cost: 1)    f6(ar_0, ar_1) -> Com_1(f6(ar_0, ar_1 + 1)) [ 9 >= ar_1 ]
3.54/1.71	
3.54/1.71			(Comp: 1, Cost: 1)    f6(ar_0, ar_1) -> Com_1(f15(ar_0, ar_1)) [ ar_1 >= 10 ]
3.54/1.71	
3.54/1.71			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.54/1.71	
3.54/1.71		start location:	koat_start
3.54/1.71	
3.54/1.71		leaf cost:	0
3.54/1.71	
3.54/1.71	
3.54/1.71	
3.54/1.71	A polynomial rank function with
3.54/1.71	
3.54/1.71		Pol(f0) = 10
3.54/1.71	
3.54/1.71		Pol(f6) = -V_2 + 10
3.54/1.71	
3.54/1.71		Pol(f15) = -V_2
3.54/1.71	
3.54/1.71		Pol(koat_start) = 10
3.54/1.71	
3.54/1.71	orients all transitions weakly and the transition
3.54/1.71	
3.54/1.71		f6(ar_0, ar_1) -> Com_1(f6(ar_0, ar_1 + 1)) [ 9 >= ar_1 ]
3.54/1.71	
3.54/1.71	strictly and produces the following problem:
3.54/1.71	
3.54/1.71	4:	T:
3.54/1.71	
3.54/1.71			(Comp: 1, Cost: 1)     f0(ar_0, ar_1) -> Com_1(f6(c, 0))
3.54/1.71	
3.54/1.71			(Comp: 10, Cost: 1)    f6(ar_0, ar_1) -> Com_1(f6(ar_0, ar_1 + 1)) [ 9 >= ar_1 ]
3.54/1.71	
3.54/1.71			(Comp: 1, Cost: 1)     f6(ar_0, ar_1) -> Com_1(f15(ar_0, ar_1)) [ ar_1 >= 10 ]
3.54/1.71	
3.54/1.71			(Comp: 1, Cost: 0)     koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.54/1.71	
3.54/1.71		start location:	koat_start
3.54/1.71	
3.54/1.71		leaf cost:	0
3.54/1.71	
3.54/1.71	
3.54/1.71	
3.54/1.71	Complexity upper bound 12
3.54/1.71	
3.54/1.71	
3.54/1.71	
3.54/1.71	Time: 0.046 sec (SMT: 0.044 sec)
3.54/1.71	
3.54/1.71	
3.54/1.71	----------------------------------------
3.54/1.71	
3.54/1.71	(2)
3.54/1.71	BOUNDS(1, 1)
3.54/1.73	EOF