3.33/1.73	WORST_CASE(?, O(1))
3.33/1.74	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.33/1.74	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.33/1.74	
3.33/1.74	
3.33/1.74	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.33/1.74	
3.33/1.74	(0) CpxIntTrs
3.33/1.74	(1) Koat Proof [FINISHED, 18 ms]
3.33/1.74	(2) BOUNDS(1, 1)
3.33/1.74	
3.33/1.74	
3.33/1.74	----------------------------------------
3.33/1.74	
3.33/1.74	(0)
3.33/1.74	Obligation:
3.33/1.74	Complexity Int TRS consisting of the following rules:
3.33/1.74	f0(A, B) -> Com_1(f6(0, C)) :|: TRUE
3.33/1.74	f6(A, B) -> Com_1(f6(A + 1, B)) :|: 49 >= A
3.33/1.74	f6(A, B) -> Com_1(f14(A, B)) :|: A >= 50
3.33/1.74	
3.33/1.74	The start-symbols are:[f0_2]
3.33/1.74	
3.33/1.74	
3.33/1.74	----------------------------------------
3.33/1.74	
3.33/1.74	(1) Koat Proof (FINISHED)
3.33/1.74	YES(?, 52)
3.33/1.74	
3.33/1.74	
3.33/1.74	
3.33/1.74	Initial complexity problem:
3.33/1.74	
3.33/1.74	1:	T:
3.33/1.74	
3.33/1.74			(Comp: ?, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f6(0, c))
3.33/1.74	
3.33/1.74			(Comp: ?, Cost: 1)    f6(ar_0, ar_1) -> Com_1(f6(ar_0 + 1, ar_1)) [ 49 >= ar_0 ]
3.33/1.74	
3.33/1.74			(Comp: ?, Cost: 1)    f6(ar_0, ar_1) -> Com_1(f14(ar_0, ar_1)) [ ar_0 >= 50 ]
3.33/1.74	
3.33/1.74			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.33/1.74	
3.33/1.74		start location:	koat_start
3.33/1.74	
3.33/1.74		leaf cost:	0
3.33/1.74	
3.33/1.74	
3.33/1.74	
3.33/1.74	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.33/1.74	
3.33/1.74	2:	T:
3.33/1.74	
3.33/1.74			(Comp: 1, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f6(0, c))
3.33/1.74	
3.33/1.74			(Comp: ?, Cost: 1)    f6(ar_0, ar_1) -> Com_1(f6(ar_0 + 1, ar_1)) [ 49 >= ar_0 ]
3.33/1.74	
3.33/1.74			(Comp: ?, Cost: 1)    f6(ar_0, ar_1) -> Com_1(f14(ar_0, ar_1)) [ ar_0 >= 50 ]
3.33/1.74	
3.33/1.74			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.33/1.74	
3.33/1.74		start location:	koat_start
3.33/1.74	
3.33/1.74		leaf cost:	0
3.33/1.74	
3.33/1.74	
3.33/1.74	
3.33/1.74	A polynomial rank function with
3.33/1.74	
3.33/1.74		Pol(f0) = 1
3.33/1.74	
3.33/1.74		Pol(f6) = 1
3.33/1.74	
3.33/1.74		Pol(f14) = 0
3.33/1.74	
3.33/1.74		Pol(koat_start) = 1
3.33/1.74	
3.33/1.74	orients all transitions weakly and the transition
3.33/1.74	
3.33/1.74		f6(ar_0, ar_1) -> Com_1(f14(ar_0, ar_1)) [ ar_0 >= 50 ]
3.33/1.74	
3.33/1.74	strictly and produces the following problem:
3.33/1.74	
3.33/1.74	3:	T:
3.33/1.74	
3.33/1.74			(Comp: 1, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f6(0, c))
3.33/1.74	
3.33/1.74			(Comp: ?, Cost: 1)    f6(ar_0, ar_1) -> Com_1(f6(ar_0 + 1, ar_1)) [ 49 >= ar_0 ]
3.33/1.74	
3.33/1.74			(Comp: 1, Cost: 1)    f6(ar_0, ar_1) -> Com_1(f14(ar_0, ar_1)) [ ar_0 >= 50 ]
3.33/1.74	
3.33/1.74			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.33/1.74	
3.33/1.74		start location:	koat_start
3.33/1.74	
3.33/1.74		leaf cost:	0
3.33/1.74	
3.33/1.74	
3.33/1.74	
3.33/1.74	A polynomial rank function with
3.33/1.74	
3.33/1.74		Pol(f0) = 50
3.33/1.74	
3.33/1.74		Pol(f6) = -V_1 + 50
3.33/1.74	
3.33/1.74		Pol(f14) = -V_1
3.33/1.74	
3.33/1.74		Pol(koat_start) = 50
3.33/1.74	
3.33/1.74	orients all transitions weakly and the transition
3.33/1.74	
3.33/1.74		f6(ar_0, ar_1) -> Com_1(f6(ar_0 + 1, ar_1)) [ 49 >= ar_0 ]
3.33/1.74	
3.33/1.74	strictly and produces the following problem:
3.33/1.74	
3.33/1.74	4:	T:
3.33/1.74	
3.33/1.74			(Comp: 1, Cost: 1)     f0(ar_0, ar_1) -> Com_1(f6(0, c))
3.33/1.74	
3.33/1.74			(Comp: 50, Cost: 1)    f6(ar_0, ar_1) -> Com_1(f6(ar_0 + 1, ar_1)) [ 49 >= ar_0 ]
3.33/1.74	
3.33/1.74			(Comp: 1, Cost: 1)     f6(ar_0, ar_1) -> Com_1(f14(ar_0, ar_1)) [ ar_0 >= 50 ]
3.33/1.74	
3.33/1.74			(Comp: 1, Cost: 0)     koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.33/1.74	
3.33/1.74		start location:	koat_start
3.33/1.74	
3.33/1.74		leaf cost:	0
3.33/1.74	
3.33/1.74	
3.33/1.74	
3.33/1.74	Complexity upper bound 52
3.33/1.74	
3.33/1.74	
3.33/1.74	
3.33/1.74	Time: 0.043 sec (SMT: 0.041 sec)
3.33/1.74	
3.33/1.74	
3.33/1.74	----------------------------------------
3.33/1.74	
3.33/1.74	(2)
3.33/1.74	BOUNDS(1, 1)
3.33/1.76	EOF