3.73/1.76	WORST_CASE(?, O(1))
3.73/1.77	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
3.73/1.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.73/1.77	
3.73/1.77	
3.73/1.77	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.73/1.77	
3.73/1.77	(0) CpxIntTrs
3.73/1.77	(1) Koat Proof [FINISHED, 8 ms]
3.73/1.77	(2) BOUNDS(1, 1)
3.73/1.77	
3.73/1.77	
3.73/1.77	----------------------------------------
3.73/1.77	
3.73/1.77	(0)
3.73/1.77	Obligation:
3.73/1.77	Complexity Int TRS consisting of the following rules:
3.73/1.77	f0(A, B, C) -> Com_1(f8(D, 0, C)) :|: TRUE
3.73/1.77	f8(A, B, C) -> Com_1(f8(A, B + 1, C)) :|: 9 >= B
3.73/1.77	f19(A, B, C) -> Com_1(f19(A, B, C + 1)) :|: 9 >= C
3.73/1.77	f19(A, B, C) -> Com_1(f29(A, B, C)) :|: C >= 10
3.73/1.77	f8(A, B, C) -> Com_1(f19(A, B, 0)) :|: B >= 10
3.73/1.77	
3.73/1.77	The start-symbols are:[f0_3]
3.73/1.77	
3.73/1.77	
3.73/1.77	----------------------------------------
3.73/1.77	
3.73/1.77	(1) Koat Proof (FINISHED)
3.73/1.77	YES(?, 25)
3.73/1.77	
3.73/1.77	
3.73/1.77	
3.73/1.77	Initial complexity problem:
3.73/1.77	
3.73/1.77	1:	T:
3.73/1.77	
3.73/1.77			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f8(d, 0, ar_2))
3.73/1.77	
3.73/1.77			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f8(ar_0, ar_1 + 1, ar_2)) [ 9 >= ar_1 ]
3.73/1.77	
3.73/1.77			(Comp: ?, Cost: 1)    f19(ar_0, ar_1, ar_2) -> Com_1(f19(ar_0, ar_1, ar_2 + 1)) [ 9 >= ar_2 ]
3.73/1.77	
3.73/1.77			(Comp: ?, Cost: 1)    f19(ar_0, ar_1, ar_2) -> Com_1(f29(ar_0, ar_1, ar_2)) [ ar_2 >= 10 ]
3.73/1.77	
3.73/1.77			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f19(ar_0, ar_1, 0)) [ ar_1 >= 10 ]
3.73/1.77	
3.73/1.77			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.73/1.77	
3.73/1.77		start location:	koat_start
3.73/1.77	
3.73/1.77		leaf cost:	0
3.73/1.77	
3.73/1.77	
3.73/1.77	
3.73/1.77	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.73/1.77	
3.73/1.77	2:	T:
3.73/1.77	
3.73/1.77			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f8(d, 0, ar_2))
3.73/1.77	
3.73/1.77			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f8(ar_0, ar_1 + 1, ar_2)) [ 9 >= ar_1 ]
3.73/1.77	
3.73/1.77			(Comp: ?, Cost: 1)    f19(ar_0, ar_1, ar_2) -> Com_1(f19(ar_0, ar_1, ar_2 + 1)) [ 9 >= ar_2 ]
3.73/1.77	
3.73/1.77			(Comp: ?, Cost: 1)    f19(ar_0, ar_1, ar_2) -> Com_1(f29(ar_0, ar_1, ar_2)) [ ar_2 >= 10 ]
3.73/1.77	
3.73/1.77			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f19(ar_0, ar_1, 0)) [ ar_1 >= 10 ]
3.73/1.77	
3.73/1.77			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.73/1.77	
3.73/1.77		start location:	koat_start
3.73/1.77	
3.73/1.77		leaf cost:	0
3.73/1.77	
3.73/1.77	
3.73/1.77	
3.73/1.77	A polynomial rank function with
3.73/1.77	
3.73/1.77		Pol(f0) = 2
3.73/1.77	
3.73/1.77		Pol(f8) = 2
3.73/1.77	
3.73/1.77		Pol(f19) = 1
3.73/1.77	
3.73/1.77		Pol(f29) = 0
3.73/1.77	
3.73/1.77		Pol(koat_start) = 2
3.73/1.77	
3.73/1.77	orients all transitions weakly and the transitions
3.73/1.77	
3.73/1.77		f8(ar_0, ar_1, ar_2) -> Com_1(f19(ar_0, ar_1, 0)) [ ar_1 >= 10 ]
3.73/1.77	
3.73/1.77		f19(ar_0, ar_1, ar_2) -> Com_1(f29(ar_0, ar_1, ar_2)) [ ar_2 >= 10 ]
3.73/1.77	
3.73/1.77	strictly and produces the following problem:
3.73/1.77	
3.73/1.77	3:	T:
3.73/1.77	
3.73/1.77			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f8(d, 0, ar_2))
3.73/1.77	
3.73/1.77			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f8(ar_0, ar_1 + 1, ar_2)) [ 9 >= ar_1 ]
3.73/1.77	
3.73/1.77			(Comp: ?, Cost: 1)    f19(ar_0, ar_1, ar_2) -> Com_1(f19(ar_0, ar_1, ar_2 + 1)) [ 9 >= ar_2 ]
3.73/1.77	
3.73/1.77			(Comp: 2, Cost: 1)    f19(ar_0, ar_1, ar_2) -> Com_1(f29(ar_0, ar_1, ar_2)) [ ar_2 >= 10 ]
3.73/1.77	
3.73/1.77			(Comp: 2, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f19(ar_0, ar_1, 0)) [ ar_1 >= 10 ]
3.73/1.77	
3.73/1.77			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.73/1.77	
3.73/1.77		start location:	koat_start
3.73/1.77	
3.73/1.77		leaf cost:	0
3.73/1.77	
3.73/1.77	
3.73/1.77	
3.73/1.77	A polynomial rank function with
3.73/1.77	
3.73/1.77		Pol(f0) = 10
3.73/1.77	
3.73/1.77		Pol(f8) = -V_2 + 10
3.73/1.77	
3.73/1.77		Pol(f19) = -V_2
3.73/1.77	
3.73/1.77		Pol(f29) = -V_2
3.73/1.77	
3.73/1.77		Pol(koat_start) = 10
3.73/1.77	
3.73/1.77	orients all transitions weakly and the transition
3.73/1.77	
3.73/1.77		f8(ar_0, ar_1, ar_2) -> Com_1(f8(ar_0, ar_1 + 1, ar_2)) [ 9 >= ar_1 ]
3.73/1.77	
3.73/1.77	strictly and produces the following problem:
3.73/1.77	
3.73/1.77	4:	T:
3.73/1.77	
3.73/1.77			(Comp: 1, Cost: 1)     f0(ar_0, ar_1, ar_2) -> Com_1(f8(d, 0, ar_2))
3.73/1.77	
3.73/1.77			(Comp: 10, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f8(ar_0, ar_1 + 1, ar_2)) [ 9 >= ar_1 ]
3.73/1.77	
3.73/1.77			(Comp: ?, Cost: 1)     f19(ar_0, ar_1, ar_2) -> Com_1(f19(ar_0, ar_1, ar_2 + 1)) [ 9 >= ar_2 ]
3.73/1.77	
3.73/1.77			(Comp: 2, Cost: 1)     f19(ar_0, ar_1, ar_2) -> Com_1(f29(ar_0, ar_1, ar_2)) [ ar_2 >= 10 ]
3.73/1.77	
3.73/1.77			(Comp: 2, Cost: 1)     f8(ar_0, ar_1, ar_2) -> Com_1(f19(ar_0, ar_1, 0)) [ ar_1 >= 10 ]
3.73/1.77	
3.73/1.77			(Comp: 1, Cost: 0)     koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.73/1.77	
3.73/1.77		start location:	koat_start
3.73/1.77	
3.73/1.77		leaf cost:	0
3.73/1.77	
3.73/1.77	
3.73/1.77	
3.73/1.77	A polynomial rank function with
3.73/1.77	
3.73/1.77		Pol(f0) = 10
3.73/1.77	
3.73/1.77		Pol(f8) = 10
3.73/1.77	
3.73/1.77		Pol(f19) = -V_3 + 10
3.73/1.77	
3.73/1.77		Pol(f29) = -V_3
3.73/1.77	
3.73/1.77		Pol(koat_start) = 10
3.73/1.77	
3.73/1.77	orients all transitions weakly and the transition
3.73/1.77	
3.73/1.77		f19(ar_0, ar_1, ar_2) -> Com_1(f19(ar_0, ar_1, ar_2 + 1)) [ 9 >= ar_2 ]
3.73/1.77	
3.73/1.77	strictly and produces the following problem:
3.73/1.77	
3.73/1.77	5:	T:
3.73/1.77	
3.73/1.77			(Comp: 1, Cost: 1)     f0(ar_0, ar_1, ar_2) -> Com_1(f8(d, 0, ar_2))
3.73/1.77	
3.73/1.77			(Comp: 10, Cost: 1)    f8(ar_0, ar_1, ar_2) -> Com_1(f8(ar_0, ar_1 + 1, ar_2)) [ 9 >= ar_1 ]
3.73/1.77	
3.73/1.77			(Comp: 10, Cost: 1)    f19(ar_0, ar_1, ar_2) -> Com_1(f19(ar_0, ar_1, ar_2 + 1)) [ 9 >= ar_2 ]
3.73/1.77	
3.73/1.77			(Comp: 2, Cost: 1)     f19(ar_0, ar_1, ar_2) -> Com_1(f29(ar_0, ar_1, ar_2)) [ ar_2 >= 10 ]
3.73/1.77	
3.73/1.77			(Comp: 2, Cost: 1)     f8(ar_0, ar_1, ar_2) -> Com_1(f19(ar_0, ar_1, 0)) [ ar_1 >= 10 ]
3.73/1.77	
3.73/1.77			(Comp: 1, Cost: 0)     koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.73/1.77	
3.73/1.77		start location:	koat_start
3.73/1.77	
3.73/1.77		leaf cost:	0
3.73/1.77	
3.73/1.77	
3.73/1.77	
3.73/1.77	Complexity upper bound 25
3.73/1.77	
3.73/1.77	
3.73/1.77	
3.73/1.77	Time: 0.085 sec (SMT: 0.078 sec)
3.73/1.77	
3.73/1.77	
3.73/1.77	----------------------------------------
3.73/1.77	
3.73/1.77	(2)
3.73/1.77	BOUNDS(1, 1)
3.73/1.79	EOF