3.11/1.69	WORST_CASE(?, O(1))
3.11/1.70	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.11/1.70	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.11/1.70	
3.11/1.70	
3.11/1.70	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.11/1.70	
3.11/1.70	(0) CpxIntTrs
3.11/1.70	(1) Koat Proof [FINISHED, 22 ms]
3.11/1.70	(2) BOUNDS(1, 1)
3.11/1.70	
3.11/1.70	
3.11/1.70	----------------------------------------
3.11/1.70	
3.11/1.70	(0)
3.11/1.70	Obligation:
3.11/1.70	Complexity Int TRS consisting of the following rules:
3.11/1.70	f0(A) -> Com_1(f1(300)) :|: TRUE
3.11/1.70	f1(A) -> Com_1(f1(A - 1)) :|: A >= 102
3.11/1.70	f1(A) -> Com_1(f1(A - 1)) :|: 100 >= A
3.11/1.70	
3.11/1.70	The start-symbols are:[f0_1]
3.11/1.70	
3.11/1.70	
3.11/1.70	----------------------------------------
3.11/1.70	
3.11/1.70	(1) Koat Proof (FINISHED)
3.11/1.70	YES(?, 200)
3.11/1.70	
3.11/1.70	
3.11/1.70	
3.11/1.70	Initial complexity problem:
3.11/1.70	
3.11/1.70	1:	T:
3.11/1.70	
3.11/1.70			(Comp: ?, Cost: 1)    f0(ar_0) -> Com_1(f1(300))
3.11/1.70	
3.11/1.70			(Comp: ?, Cost: 1)    f1(ar_0) -> Com_1(f1(ar_0 - 1)) [ ar_0 >= 102 ]
3.11/1.70	
3.11/1.70			(Comp: ?, Cost: 1)    f1(ar_0) -> Com_1(f1(ar_0 - 1)) [ 100 >= ar_0 ]
3.11/1.70	
3.11/1.70			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(f0(ar_0)) [ 0 <= 0 ]
3.11/1.70	
3.11/1.70		start location:	koat_start
3.11/1.70	
3.11/1.70		leaf cost:	0
3.11/1.70	
3.11/1.70	
3.11/1.70	
3.11/1.70	Testing for reachability in the complexity graph removes the following transition from problem 1:
3.40/1.70	
3.40/1.70		f1(ar_0) -> Com_1(f1(ar_0 - 1)) [ 100 >= ar_0 ]
3.40/1.70	
3.40/1.70	We thus obtain the following problem:
3.40/1.70	
3.40/1.70	2:	T:
3.40/1.70	
3.40/1.70			(Comp: ?, Cost: 1)    f1(ar_0) -> Com_1(f1(ar_0 - 1)) [ ar_0 >= 102 ]
3.40/1.70	
3.40/1.70			(Comp: ?, Cost: 1)    f0(ar_0) -> Com_1(f1(300))
3.40/1.70	
3.40/1.70			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(f0(ar_0)) [ 0 <= 0 ]
3.40/1.70	
3.40/1.70		start location:	koat_start
3.40/1.70	
3.40/1.70		leaf cost:	0
3.40/1.70	
3.40/1.70	
3.40/1.70	
3.40/1.70	Repeatedly propagating knowledge in problem 2 produces the following problem:
3.40/1.70	
3.40/1.70	3:	T:
3.40/1.70	
3.40/1.70			(Comp: ?, Cost: 1)    f1(ar_0) -> Com_1(f1(ar_0 - 1)) [ ar_0 >= 102 ]
3.40/1.70	
3.40/1.70			(Comp: 1, Cost: 1)    f0(ar_0) -> Com_1(f1(300))
3.40/1.70	
3.40/1.70			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(f0(ar_0)) [ 0 <= 0 ]
3.40/1.70	
3.40/1.70		start location:	koat_start
3.40/1.70	
3.40/1.70		leaf cost:	0
3.40/1.70	
3.40/1.70	
3.40/1.70	
3.40/1.70	A polynomial rank function with
3.40/1.70	
3.40/1.70		Pol(f1) = V_1 - 101
3.40/1.70	
3.40/1.70		Pol(f0) = 199
3.40/1.70	
3.40/1.70		Pol(koat_start) = 199
3.40/1.70	
3.40/1.70	orients all transitions weakly and the transition
3.40/1.70	
3.40/1.70		f1(ar_0) -> Com_1(f1(ar_0 - 1)) [ ar_0 >= 102 ]
3.40/1.70	
3.40/1.70	strictly and produces the following problem:
3.40/1.70	
3.40/1.70	4:	T:
3.40/1.70	
3.40/1.70			(Comp: 199, Cost: 1)    f1(ar_0) -> Com_1(f1(ar_0 - 1)) [ ar_0 >= 102 ]
3.40/1.70	
3.40/1.70			(Comp: 1, Cost: 1)      f0(ar_0) -> Com_1(f1(300))
3.40/1.70	
3.40/1.70			(Comp: 1, Cost: 0)      koat_start(ar_0) -> Com_1(f0(ar_0)) [ 0 <= 0 ]
3.40/1.70	
3.40/1.70		start location:	koat_start
3.40/1.70	
3.40/1.70		leaf cost:	0
3.40/1.70	
3.40/1.70	
3.40/1.70	
3.40/1.70	Complexity upper bound 200
3.40/1.70	
3.40/1.70	
3.40/1.70	
3.40/1.70	Time: 0.030 sec (SMT: 0.028 sec)
3.40/1.70	
3.40/1.70	
3.40/1.70	----------------------------------------
3.40/1.70	
3.40/1.70	(2)
3.40/1.70	BOUNDS(1, 1)
3.40/1.71	EOF