3.45/1.74	WORST_CASE(?, O(1))
3.73/1.75	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.73/1.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.73/1.75	
3.73/1.75	
3.73/1.75	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.73/1.75	
3.73/1.75	(0) CpxIntTrs
3.73/1.75	(1) Koat Proof [FINISHED, 98 ms]
3.73/1.75	(2) BOUNDS(1, 1)
3.73/1.75	
3.73/1.75	
3.73/1.75	----------------------------------------
3.73/1.75	
3.73/1.75	(0)
3.73/1.75	Obligation:
3.73/1.75	Complexity Int TRS consisting of the following rules:
3.73/1.75	f0(A, B, C, D) -> Com_1(f8(0, B, C, D)) :|: TRUE
3.73/1.75	f8(A, B, C, D) -> Com_1(f8(A + 1, B, C, D)) :|: 3 >= A
3.73/1.75	f8(A, B, C, D) -> Com_1(f8(A + 1, A, A + 1, E)) :|: 3 >= A
3.73/1.75	f8(A, B, C, D) -> Com_1(f23(A, B, C, D)) :|: A >= 4 && 0 >= E + 1
3.73/1.75	f8(A, B, C, D) -> Com_1(f23(A, B, C, D)) :|: A >= 4
3.73/1.75	
3.73/1.75	The start-symbols are:[f0_4]
3.73/1.75	
3.73/1.75	
3.73/1.75	----------------------------------------
3.73/1.75	
3.73/1.75	(1) Koat Proof (FINISHED)
3.73/1.75	YES(?, 11)
3.73/1.75	
3.73/1.75	
3.73/1.75	
3.73/1.75	Initial complexity problem:
3.73/1.75	
3.73/1.75	1:	T:
3.73/1.75	
3.73/1.75			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(0, ar_1, ar_2, ar_3))
3.73/1.75	
3.73/1.75			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(ar_0 + 1, ar_1, ar_2, ar_3)) [ 3 >= ar_0 ]
3.73/1.75	
3.73/1.75			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(ar_0 + 1, ar_0, ar_0 + 1, e)) [ 3 >= ar_0 ]
3.73/1.75	
3.73/1.75			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f23(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 4 /\ 0 >= e + 1 ]
3.73/1.75	
3.73/1.75			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f23(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 4 ]
3.73/1.75	
3.73/1.75			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.73/1.75	
3.73/1.75		start location:	koat_start
3.73/1.75	
3.73/1.75		leaf cost:	0
3.73/1.75	
3.73/1.75	
3.73/1.75	
3.73/1.75	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.73/1.75	
3.73/1.75	2:	T:
3.73/1.75	
3.73/1.75			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(0, ar_1, ar_2, ar_3))
3.73/1.75	
3.73/1.75			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(ar_0 + 1, ar_1, ar_2, ar_3)) [ 3 >= ar_0 ]
3.73/1.75	
3.73/1.75			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(ar_0 + 1, ar_0, ar_0 + 1, e)) [ 3 >= ar_0 ]
3.73/1.75	
3.73/1.75			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f23(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 4 /\ 0 >= e + 1 ]
3.73/1.75	
3.73/1.75			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f23(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 4 ]
3.73/1.75	
3.73/1.75			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.73/1.75	
3.73/1.75		start location:	koat_start
3.73/1.75	
3.73/1.75		leaf cost:	0
3.73/1.75	
3.73/1.75	
3.73/1.75	
3.73/1.75	A polynomial rank function with
3.73/1.75	
3.73/1.75		Pol(f0) = 1
3.73/1.75	
3.73/1.75		Pol(f8) = 1
3.73/1.75	
3.73/1.75		Pol(f23) = 0
3.73/1.75	
3.73/1.75		Pol(koat_start) = 1
3.73/1.75	
3.73/1.75	orients all transitions weakly and the transitions
3.73/1.75	
3.73/1.75		f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f23(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 4 /\ 0 >= e + 1 ]
3.73/1.75	
3.73/1.75		f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f23(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 4 ]
3.73/1.75	
3.73/1.75	strictly and produces the following problem:
3.73/1.75	
3.73/1.75	3:	T:
3.73/1.75	
3.73/1.75			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(0, ar_1, ar_2, ar_3))
3.73/1.75	
3.73/1.75			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(ar_0 + 1, ar_1, ar_2, ar_3)) [ 3 >= ar_0 ]
3.73/1.75	
3.73/1.75			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(ar_0 + 1, ar_0, ar_0 + 1, e)) [ 3 >= ar_0 ]
3.73/1.75	
3.73/1.75			(Comp: 1, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f23(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 4 /\ 0 >= e + 1 ]
3.73/1.75	
3.73/1.75			(Comp: 1, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f23(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 4 ]
3.73/1.75	
3.73/1.75			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.73/1.75	
3.73/1.75		start location:	koat_start
3.73/1.75	
3.73/1.75		leaf cost:	0
3.73/1.75	
3.73/1.75	
3.73/1.75	
3.73/1.75	A polynomial rank function with
3.73/1.75	
3.73/1.75		Pol(f0) = 4
3.73/1.75	
3.73/1.75		Pol(f8) = -V_1 + 4
3.73/1.75	
3.73/1.75		Pol(f23) = -V_1
3.73/1.75	
3.73/1.75		Pol(koat_start) = 4
3.73/1.75	
3.73/1.75	orients all transitions weakly and the transitions
3.73/1.75	
3.73/1.75		f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(ar_0 + 1, ar_1, ar_2, ar_3)) [ 3 >= ar_0 ]
3.73/1.75	
3.73/1.75		f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(ar_0 + 1, ar_0, ar_0 + 1, e)) [ 3 >= ar_0 ]
3.73/1.75	
3.73/1.75	strictly and produces the following problem:
3.73/1.75	
3.73/1.75	4:	T:
3.73/1.75	
3.73/1.75			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(0, ar_1, ar_2, ar_3))
3.73/1.75	
3.73/1.75			(Comp: 4, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(ar_0 + 1, ar_1, ar_2, ar_3)) [ 3 >= ar_0 ]
3.73/1.75	
3.73/1.75			(Comp: 4, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(ar_0 + 1, ar_0, ar_0 + 1, e)) [ 3 >= ar_0 ]
3.73/1.75	
3.73/1.75			(Comp: 1, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f23(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 4 /\ 0 >= e + 1 ]
3.73/1.75	
3.73/1.75			(Comp: 1, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f23(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 4 ]
3.73/1.75	
3.73/1.75			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.73/1.75	
3.73/1.75		start location:	koat_start
3.73/1.75	
3.73/1.75		leaf cost:	0
3.73/1.75	
3.73/1.75	
3.73/1.75	
3.73/1.75	Complexity upper bound 11
3.73/1.75	
3.73/1.75	
3.73/1.75	
3.73/1.75	Time: 0.108 sec (SMT: 0.100 sec)
3.73/1.75	
3.73/1.75	
3.73/1.75	----------------------------------------
3.73/1.75	
3.73/1.75	(2)
3.73/1.75	BOUNDS(1, 1)
3.73/1.77	EOF