3.27/1.58	WORST_CASE(?, O(1))
3.27/1.59	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.27/1.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.27/1.59	
3.27/1.59	
3.27/1.59	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.27/1.59	
3.27/1.59	(0) CpxIntTrs
3.27/1.59	(1) Koat Proof [FINISHED, 8 ms]
3.27/1.59	(2) BOUNDS(1, 1)
3.27/1.59	
3.27/1.59	
3.27/1.59	----------------------------------------
3.27/1.59	
3.27/1.59	(0)
3.27/1.59	Obligation:
3.27/1.59	Complexity Int TRS consisting of the following rules:
3.27/1.59	start(A) -> Com_1(a(A)) :|: A >= 1
3.27/1.59	start(A) -> Com_1(a(A)) :|: A >= 2
3.27/1.59	start(A) -> Com_1(a(A)) :|: A >= 4
3.27/1.59	a(A) -> Com_1(a(A * B)) :|: 1 >= 2 * B && 3 * B >= 2 && A >= 2
3.27/1.59	
3.27/1.59	The start-symbols are:[start_1]
3.27/1.59	
3.27/1.59	
3.27/1.59	----------------------------------------
3.27/1.59	
3.27/1.59	(1) Koat Proof (FINISHED)
3.27/1.59	YES(?, 3)
3.27/1.59	
3.27/1.59	
3.27/1.59	
3.27/1.59	Initial complexity problem:
3.27/1.59	
3.27/1.59	1:	T:
3.27/1.59	
3.27/1.59			(Comp: ?, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 1 ]
3.27/1.59	
3.27/1.59			(Comp: ?, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 2 ]
3.27/1.59	
3.27/1.59			(Comp: ?, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 4 ]
3.27/1.59	
3.27/1.59			(Comp: ?, Cost: 1)    a(ar_0) -> Com_1(a(ar_0*b)) [ 1 >= 2*b /\ 3*b >= 2 /\ ar_0 >= 2 ]
3.27/1.59	
3.27/1.59			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(start(ar_0)) [ 0 <= 0 ]
3.27/1.59	
3.27/1.59		start location:	koat_start
3.27/1.59	
3.27/1.59		leaf cost:	0
3.27/1.59	
3.27/1.59	
3.27/1.59	
3.27/1.59	Testing for reachability in the complexity graph removes the following transition from problem 1:
3.27/1.59	
3.27/1.59		a(ar_0) -> Com_1(a(ar_0*b)) [ 1 >= 2*b /\ 3*b >= 2 /\ ar_0 >= 2 ]
3.27/1.59	
3.27/1.59	We thus obtain the following problem:
3.27/1.59	
3.27/1.59	2:	T:
3.27/1.59	
3.27/1.59			(Comp: ?, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 4 ]
3.27/1.59	
3.27/1.59			(Comp: ?, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 2 ]
3.27/1.59	
3.27/1.59			(Comp: ?, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 1 ]
3.27/1.59	
3.27/1.59			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(start(ar_0)) [ 0 <= 0 ]
3.27/1.59	
3.27/1.59		start location:	koat_start
3.27/1.59	
3.27/1.59		leaf cost:	0
3.27/1.59	
3.27/1.59	
3.27/1.59	
3.27/1.59	Repeatedly propagating knowledge in problem 2 produces the following problem:
3.27/1.59	
3.27/1.59	3:	T:
3.27/1.59	
3.27/1.59			(Comp: 1, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 4 ]
3.27/1.59	
3.27/1.59			(Comp: 1, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 2 ]
3.27/1.59	
3.27/1.59			(Comp: 1, Cost: 1)    start(ar_0) -> Com_1(a(ar_0)) [ ar_0 >= 1 ]
3.27/1.59	
3.27/1.59			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(start(ar_0)) [ 0 <= 0 ]
3.27/1.59	
3.27/1.59		start location:	koat_start
3.27/1.59	
3.27/1.59		leaf cost:	0
3.27/1.59	
3.27/1.59	
3.27/1.59	
3.27/1.59	Complexity upper bound 3
3.27/1.59	
3.27/1.59	
3.27/1.59	
3.27/1.59	Time: 0.012 sec (SMT: 0.011 sec)
3.27/1.59	
3.27/1.59	
3.27/1.59	----------------------------------------
3.27/1.59	
3.27/1.59	(2)
3.27/1.59	BOUNDS(1, 1)
3.36/1.64	EOF