3.39/1.75	WORST_CASE(?, O(1))
3.70/1.75	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.70/1.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.70/1.75	
3.70/1.75	
3.70/1.75	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.70/1.75	
3.70/1.75	(0) CpxIntTrs
3.70/1.75	(1) Koat Proof [FINISHED, 22 ms]
3.70/1.75	(2) BOUNDS(1, 1)
3.70/1.75	
3.70/1.75	
3.70/1.75	----------------------------------------
3.70/1.75	
3.70/1.75	(0)
3.70/1.75	Obligation:
3.70/1.75	Complexity Int TRS consisting of the following rules:
3.70/1.75	f0(A, B) -> Com_1(f4(0, B)) :|: TRUE
3.70/1.75	f4(A, B) -> Com_1(f4(A + 1, B)) :|: 1 >= A
3.70/1.75	f10(A, B) -> Com_1(f10(A, B + 1)) :|: 1 >= B
3.70/1.75	f10(A, B) -> Com_1(f18(A, B)) :|: B >= 2 && 0 >= C + 1
3.70/1.75	f10(A, B) -> Com_1(f18(A, B)) :|: B >= 2
3.70/1.75	f4(A, B) -> Com_1(f10(A, 0)) :|: A >= 2
3.70/1.75	
3.70/1.75	The start-symbols are:[f0_2]
3.70/1.75	
3.70/1.75	
3.70/1.75	----------------------------------------
3.70/1.75	
3.70/1.75	(1) Koat Proof (FINISHED)
3.70/1.75	YES(?, 11)
3.70/1.75	
3.70/1.75	
3.70/1.75	
3.70/1.75	Initial complexity problem:
3.70/1.75	
3.70/1.75	1:	T:
3.70/1.75	
3.70/1.75			(Comp: ?, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f4(0, ar_1))
3.70/1.75	
3.70/1.75			(Comp: ?, Cost: 1)    f4(ar_0, ar_1) -> Com_1(f4(ar_0 + 1, ar_1)) [ 1 >= ar_0 ]
3.70/1.75	
3.70/1.75			(Comp: ?, Cost: 1)    f10(ar_0, ar_1) -> Com_1(f10(ar_0, ar_1 + 1)) [ 1 >= ar_1 ]
3.70/1.75	
3.70/1.75			(Comp: ?, Cost: 1)    f10(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1)) [ ar_1 >= 2 /\ 0 >= c + 1 ]
3.70/1.75	
3.70/1.75			(Comp: ?, Cost: 1)    f10(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1)) [ ar_1 >= 2 ]
3.70/1.75	
3.70/1.75			(Comp: ?, Cost: 1)    f4(ar_0, ar_1) -> Com_1(f10(ar_0, 0)) [ ar_0 >= 2 ]
3.70/1.75	
3.70/1.75			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.70/1.75	
3.70/1.75		start location:	koat_start
3.70/1.75	
3.70/1.75		leaf cost:	0
3.70/1.75	
3.70/1.75	
3.70/1.75	
3.70/1.75	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.70/1.75	
3.70/1.75	2:	T:
3.70/1.75	
3.70/1.75			(Comp: 1, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f4(0, ar_1))
3.70/1.75	
3.70/1.75			(Comp: ?, Cost: 1)    f4(ar_0, ar_1) -> Com_1(f4(ar_0 + 1, ar_1)) [ 1 >= ar_0 ]
3.70/1.75	
3.70/1.75			(Comp: ?, Cost: 1)    f10(ar_0, ar_1) -> Com_1(f10(ar_0, ar_1 + 1)) [ 1 >= ar_1 ]
3.70/1.75	
3.70/1.75			(Comp: ?, Cost: 1)    f10(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1)) [ ar_1 >= 2 /\ 0 >= c + 1 ]
3.70/1.75	
3.70/1.75			(Comp: ?, Cost: 1)    f10(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1)) [ ar_1 >= 2 ]
3.70/1.75	
3.70/1.75			(Comp: ?, Cost: 1)    f4(ar_0, ar_1) -> Com_1(f10(ar_0, 0)) [ ar_0 >= 2 ]
3.70/1.75	
3.70/1.75			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.70/1.75	
3.70/1.75		start location:	koat_start
3.70/1.75	
3.70/1.75		leaf cost:	0
3.70/1.75	
3.70/1.75	
3.70/1.75	
3.70/1.75	A polynomial rank function with
3.70/1.75	
3.70/1.75		Pol(f0) = 2
3.70/1.75	
3.70/1.75		Pol(f4) = 2
3.70/1.75	
3.70/1.75		Pol(f10) = 1
3.70/1.75	
3.70/1.75		Pol(f18) = 0
3.70/1.75	
3.70/1.75		Pol(koat_start) = 2
3.70/1.75	
3.70/1.75	orients all transitions weakly and the transitions
3.70/1.75	
3.70/1.75		f4(ar_0, ar_1) -> Com_1(f10(ar_0, 0)) [ ar_0 >= 2 ]
3.70/1.75	
3.70/1.75		f10(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1)) [ ar_1 >= 2 /\ 0 >= c + 1 ]
3.70/1.75	
3.70/1.75		f10(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1)) [ ar_1 >= 2 ]
3.70/1.75	
3.70/1.75	strictly and produces the following problem:
3.70/1.75	
3.70/1.75	3:	T:
3.70/1.75	
3.70/1.75			(Comp: 1, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f4(0, ar_1))
3.70/1.75	
3.70/1.75			(Comp: ?, Cost: 1)    f4(ar_0, ar_1) -> Com_1(f4(ar_0 + 1, ar_1)) [ 1 >= ar_0 ]
3.70/1.75	
3.70/1.75			(Comp: ?, Cost: 1)    f10(ar_0, ar_1) -> Com_1(f10(ar_0, ar_1 + 1)) [ 1 >= ar_1 ]
3.70/1.75	
3.70/1.75			(Comp: 2, Cost: 1)    f10(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1)) [ ar_1 >= 2 /\ 0 >= c + 1 ]
3.70/1.75	
3.70/1.75			(Comp: 2, Cost: 1)    f10(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1)) [ ar_1 >= 2 ]
3.70/1.75	
3.70/1.75			(Comp: 2, Cost: 1)    f4(ar_0, ar_1) -> Com_1(f10(ar_0, 0)) [ ar_0 >= 2 ]
3.70/1.75	
3.70/1.75			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.70/1.75	
3.70/1.75		start location:	koat_start
3.70/1.75	
3.70/1.75		leaf cost:	0
3.70/1.75	
3.70/1.75	
3.70/1.75	
3.70/1.75	A polynomial rank function with
3.70/1.75	
3.70/1.75		Pol(f0) = 2
3.70/1.75	
3.70/1.75		Pol(f4) = -V_1 + 2
3.70/1.75	
3.70/1.75		Pol(f10) = -V_1 - V_2 + 2
3.70/1.75	
3.70/1.75		Pol(f18) = -V_1 - V_2
3.70/1.75	
3.70/1.75		Pol(koat_start) = 2
3.70/1.75	
3.70/1.75	orients all transitions weakly and the transition
3.70/1.75	
3.70/1.75		f4(ar_0, ar_1) -> Com_1(f4(ar_0 + 1, ar_1)) [ 1 >= ar_0 ]
3.70/1.75	
3.70/1.75	strictly and produces the following problem:
3.70/1.75	
3.70/1.75	4:	T:
3.70/1.75	
3.70/1.75			(Comp: 1, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f4(0, ar_1))
3.70/1.75	
3.70/1.75			(Comp: 2, Cost: 1)    f4(ar_0, ar_1) -> Com_1(f4(ar_0 + 1, ar_1)) [ 1 >= ar_0 ]
3.70/1.75	
3.70/1.75			(Comp: ?, Cost: 1)    f10(ar_0, ar_1) -> Com_1(f10(ar_0, ar_1 + 1)) [ 1 >= ar_1 ]
3.70/1.75	
3.70/1.75			(Comp: 2, Cost: 1)    f10(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1)) [ ar_1 >= 2 /\ 0 >= c + 1 ]
3.70/1.75	
3.70/1.75			(Comp: 2, Cost: 1)    f10(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1)) [ ar_1 >= 2 ]
3.70/1.75	
3.70/1.75			(Comp: 2, Cost: 1)    f4(ar_0, ar_1) -> Com_1(f10(ar_0, 0)) [ ar_0 >= 2 ]
3.70/1.75	
3.70/1.75			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.70/1.75	
3.70/1.75		start location:	koat_start
3.70/1.75	
3.70/1.75		leaf cost:	0
3.70/1.75	
3.70/1.75	
3.70/1.75	
3.70/1.75	A polynomial rank function with
3.70/1.75	
3.70/1.75		Pol(f0) = 2
3.70/1.75	
3.70/1.75		Pol(f4) = 2
3.70/1.75	
3.70/1.75		Pol(f10) = -V_2 + 2
3.70/1.75	
3.70/1.75		Pol(f18) = -V_2
3.70/1.75	
3.70/1.75		Pol(koat_start) = 2
3.70/1.75	
3.70/1.75	orients all transitions weakly and the transition
3.70/1.75	
3.70/1.75		f10(ar_0, ar_1) -> Com_1(f10(ar_0, ar_1 + 1)) [ 1 >= ar_1 ]
3.70/1.75	
3.70/1.75	strictly and produces the following problem:
3.70/1.75	
3.70/1.75	5:	T:
3.70/1.75	
3.70/1.75			(Comp: 1, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f4(0, ar_1))
3.70/1.75	
3.70/1.75			(Comp: 2, Cost: 1)    f4(ar_0, ar_1) -> Com_1(f4(ar_0 + 1, ar_1)) [ 1 >= ar_0 ]
3.70/1.75	
3.70/1.75			(Comp: 2, Cost: 1)    f10(ar_0, ar_1) -> Com_1(f10(ar_0, ar_1 + 1)) [ 1 >= ar_1 ]
3.70/1.75	
3.70/1.75			(Comp: 2, Cost: 1)    f10(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1)) [ ar_1 >= 2 /\ 0 >= c + 1 ]
3.70/1.75	
3.70/1.75			(Comp: 2, Cost: 1)    f10(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1)) [ ar_1 >= 2 ]
3.70/1.75	
3.70/1.75			(Comp: 2, Cost: 1)    f4(ar_0, ar_1) -> Com_1(f10(ar_0, 0)) [ ar_0 >= 2 ]
3.70/1.75	
3.70/1.75			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.70/1.75	
3.70/1.75		start location:	koat_start
3.70/1.75	
3.70/1.75		leaf cost:	0
3.70/1.75	
3.70/1.75	
3.70/1.75	
3.70/1.75	Complexity upper bound 11
3.70/1.75	
3.70/1.75	
3.70/1.75	
3.70/1.75	Time: 0.083 sec (SMT: 0.077 sec)
3.70/1.75	
3.70/1.75	
3.70/1.75	----------------------------------------
3.70/1.75	
3.70/1.75	(2)
3.70/1.75	BOUNDS(1, 1)
3.72/1.77	EOF