3.35/1.63	WORST_CASE(?, O(1))
3.35/1.64	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.35/1.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.35/1.64	
3.35/1.64	
3.35/1.64	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.35/1.64	
3.35/1.64	(0) CpxIntTrs
3.35/1.64	(1) Koat Proof [FINISHED, 20 ms]
3.35/1.64	(2) BOUNDS(1, 1)
3.35/1.64	
3.35/1.64	
3.35/1.64	----------------------------------------
3.35/1.64	
3.35/1.64	(0)
3.35/1.64	Obligation:
3.35/1.64	Complexity Int TRS consisting of the following rules:
3.35/1.64	f0(A, B, C) -> Com_1(f5(A, 0, C)) :|: A >= 128
3.35/1.64	f0(A, B, C) -> Com_1(f7(A, 0, D)) :|: 127 >= A
3.35/1.64	f7(A, B, C) -> Com_1(f7(A, B + 1, C + 1)) :|: 35 >= B
3.35/1.64	f7(A, B, C) -> Com_1(f5(A, B, C)) :|: B >= 36
3.35/1.64	
3.35/1.64	The start-symbols are:[f0_3]
3.35/1.64	
3.35/1.64	
3.35/1.64	----------------------------------------
3.35/1.64	
3.35/1.64	(1) Koat Proof (FINISHED)
3.35/1.64	YES(?, 39)
3.35/1.64	
3.35/1.64	
3.35/1.64	
3.35/1.64	Initial complexity problem:
3.35/1.64	
3.35/1.64	1:	T:
3.35/1.64	
3.35/1.64			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, 0, ar_2)) [ ar_0 >= 128 ]
3.35/1.64	
3.35/1.64			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, 0, d)) [ 127 >= ar_0 ]
3.35/1.64	
3.35/1.64			(Comp: ?, Cost: 1)    f7(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, ar_1 + 1, ar_2 + 1)) [ 35 >= ar_1 ]
3.35/1.64	
3.35/1.64			(Comp: ?, Cost: 1)    f7(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, ar_1, ar_2)) [ ar_1 >= 36 ]
3.35/1.64	
3.35/1.64			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.35/1.64	
3.35/1.64		start location:	koat_start
3.35/1.64	
3.35/1.64		leaf cost:	0
3.35/1.64	
3.35/1.64	
3.35/1.64	
3.35/1.64	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.35/1.64	
3.35/1.64	2:	T:
3.35/1.64	
3.35/1.64			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, 0, ar_2)) [ ar_0 >= 128 ]
3.35/1.64	
3.35/1.64			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, 0, d)) [ 127 >= ar_0 ]
3.35/1.64	
3.35/1.64			(Comp: ?, Cost: 1)    f7(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, ar_1 + 1, ar_2 + 1)) [ 35 >= ar_1 ]
3.35/1.64	
3.35/1.64			(Comp: ?, Cost: 1)    f7(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, ar_1, ar_2)) [ ar_1 >= 36 ]
3.35/1.64	
3.35/1.64			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.35/1.64	
3.35/1.64		start location:	koat_start
3.35/1.64	
3.35/1.64		leaf cost:	0
3.35/1.64	
3.35/1.64	
3.35/1.64	
3.35/1.64	A polynomial rank function with
3.35/1.64	
3.35/1.64		Pol(f0) = 1
3.35/1.64	
3.35/1.64		Pol(f5) = 0
3.35/1.64	
3.35/1.64		Pol(f7) = 1
3.35/1.64	
3.35/1.64		Pol(koat_start) = 1
3.35/1.64	
3.35/1.64	orients all transitions weakly and the transition
3.35/1.64	
3.35/1.64		f7(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, ar_1, ar_2)) [ ar_1 >= 36 ]
3.35/1.64	
3.35/1.64	strictly and produces the following problem:
3.35/1.64	
3.35/1.64	3:	T:
3.35/1.64	
3.35/1.64			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, 0, ar_2)) [ ar_0 >= 128 ]
3.35/1.64	
3.35/1.64			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, 0, d)) [ 127 >= ar_0 ]
3.35/1.64	
3.35/1.64			(Comp: ?, Cost: 1)    f7(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, ar_1 + 1, ar_2 + 1)) [ 35 >= ar_1 ]
3.35/1.64	
3.35/1.64			(Comp: 1, Cost: 1)    f7(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, ar_1, ar_2)) [ ar_1 >= 36 ]
3.35/1.64	
3.35/1.64			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.35/1.64	
3.35/1.64		start location:	koat_start
3.35/1.64	
3.35/1.64		leaf cost:	0
3.35/1.64	
3.35/1.64	
3.35/1.64	
3.35/1.64	A polynomial rank function with
3.35/1.64	
3.35/1.64		Pol(f0) = 36
3.35/1.64	
3.35/1.64		Pol(f5) = -V_2
3.35/1.64	
3.35/1.64		Pol(f7) = -V_2 + 36
3.35/1.64	
3.35/1.64		Pol(koat_start) = 36
3.35/1.64	
3.35/1.64	orients all transitions weakly and the transition
3.35/1.64	
3.35/1.64		f7(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, ar_1 + 1, ar_2 + 1)) [ 35 >= ar_1 ]
3.35/1.64	
3.35/1.64	strictly and produces the following problem:
3.35/1.64	
3.35/1.64	4:	T:
3.35/1.64	
3.35/1.64			(Comp: 1, Cost: 1)     f0(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, 0, ar_2)) [ ar_0 >= 128 ]
3.35/1.64	
3.35/1.64			(Comp: 1, Cost: 1)     f0(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, 0, d)) [ 127 >= ar_0 ]
3.35/1.64	
3.35/1.64			(Comp: 36, Cost: 1)    f7(ar_0, ar_1, ar_2) -> Com_1(f7(ar_0, ar_1 + 1, ar_2 + 1)) [ 35 >= ar_1 ]
3.35/1.64	
3.35/1.64			(Comp: 1, Cost: 1)     f7(ar_0, ar_1, ar_2) -> Com_1(f5(ar_0, ar_1, ar_2)) [ ar_1 >= 36 ]
3.35/1.64	
3.35/1.64			(Comp: 1, Cost: 0)     koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.35/1.64	
3.35/1.64		start location:	koat_start
3.35/1.64	
3.35/1.64		leaf cost:	0
3.35/1.64	
3.35/1.64	
3.35/1.64	
3.35/1.64	Complexity upper bound 39
3.35/1.64	
3.35/1.64	
3.35/1.64	
3.35/1.64	Time: 0.055 sec (SMT: 0.051 sec)
3.35/1.64	
3.35/1.64	
3.35/1.64	----------------------------------------
3.35/1.64	
3.35/1.64	(2)
3.35/1.64	BOUNDS(1, 1)
3.35/1.66	EOF