3.30/1.66	WORST_CASE(?, O(1))
3.30/1.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.30/1.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.30/1.67	
3.30/1.67	
3.30/1.67	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.30/1.67	
3.30/1.67	(0) CpxIntTrs
3.30/1.67	(1) Koat Proof [FINISHED, 8 ms]
3.30/1.67	(2) BOUNDS(1, 1)
3.30/1.67	
3.30/1.67	
3.30/1.67	----------------------------------------
3.30/1.67	
3.30/1.67	(0)
3.30/1.67	Obligation:
3.30/1.67	Complexity Int TRS consisting of the following rules:
3.30/1.67	f0(A, B, C, D) -> Com_1(f4(A, E, C, D)) :|: A >= 10
3.30/1.67	f0(A, B, C, D) -> Com_1(f0(1 + A, B, A, D)) :|: 9 >= A
3.30/1.67	f1(A, B, C, D) -> Com_1(f0(1, B, C, D)) :|: 9 >= E && A >= 0 && A <= 0
3.30/1.67	f2(A, B, C, D) -> Com_1(f0(2, B, C, 2)) :|: 9 >= A
3.30/1.67	f3(A, B, C, D) -> Com_1(f0(0, B, C, D)) :|: TRUE
3.30/1.67	
3.30/1.67	The start-symbols are:[f3_4]
3.30/1.67	
3.30/1.67	
3.30/1.67	----------------------------------------
3.30/1.67	
3.30/1.67	(1) Koat Proof (FINISHED)
3.30/1.67	YES(?, 12)
3.30/1.67	
3.30/1.67	
3.30/1.67	
3.30/1.67	Initial complexity problem:
3.30/1.67	
3.30/1.67	1:	T:
3.30/1.67	
3.30/1.67			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f4(ar_0, e, ar_2, ar_3)) [ ar_0 >= 10 ]
3.30/1.67	
3.30/1.67			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0 + 1, ar_1, ar_0, ar_3)) [ 9 >= ar_0 ]
3.30/1.67	
3.30/1.67			(Comp: ?, Cost: 1)    f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(1, ar_1, ar_2, ar_3)) [ 9 >= e /\ ar_0 = 0 ]
3.30/1.67	
3.30/1.67			(Comp: ?, Cost: 1)    f2(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(2, ar_1, ar_2, 2)) [ 9 >= ar_0 ]
3.30/1.67	
3.30/1.67			(Comp: ?, Cost: 1)    f3(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(0, ar_1, ar_2, ar_3))
3.30/1.67	
3.30/1.67			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f3(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.30/1.67	
3.30/1.67		start location:	koat_start
3.30/1.67	
3.30/1.67		leaf cost:	0
3.30/1.67	
3.30/1.67	
3.30/1.67	
3.30/1.67	Testing for reachability in the complexity graph removes the following transitions from problem 1:
3.30/1.67	
3.30/1.67		f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(1, ar_1, ar_2, ar_3)) [ 9 >= e /\ ar_0 = 0 ]
3.30/1.67	
3.30/1.67		f2(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(2, ar_1, ar_2, 2)) [ 9 >= ar_0 ]
3.30/1.67	
3.30/1.67	We thus obtain the following problem:
3.30/1.67	
3.30/1.67	2:	T:
3.30/1.67	
3.30/1.67			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f4(ar_0, e, ar_2, ar_3)) [ ar_0 >= 10 ]
3.30/1.67	
3.30/1.67			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0 + 1, ar_1, ar_0, ar_3)) [ 9 >= ar_0 ]
3.30/1.67	
3.30/1.67			(Comp: ?, Cost: 1)    f3(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(0, ar_1, ar_2, ar_3))
3.30/1.67	
3.30/1.67			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f3(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.30/1.67	
3.30/1.67		start location:	koat_start
3.30/1.67	
3.30/1.67		leaf cost:	0
3.30/1.67	
3.30/1.67	
3.30/1.67	
3.30/1.67	Repeatedly propagating knowledge in problem 2 produces the following problem:
3.30/1.67	
3.30/1.67	3:	T:
3.30/1.67	
3.30/1.67			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f4(ar_0, e, ar_2, ar_3)) [ ar_0 >= 10 ]
3.30/1.67	
3.30/1.67			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0 + 1, ar_1, ar_0, ar_3)) [ 9 >= ar_0 ]
3.30/1.67	
3.30/1.67			(Comp: 1, Cost: 1)    f3(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(0, ar_1, ar_2, ar_3))
3.30/1.67	
3.30/1.67			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f3(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.30/1.67	
3.30/1.67		start location:	koat_start
3.30/1.67	
3.30/1.67		leaf cost:	0
3.30/1.67	
3.30/1.67	
3.30/1.67	
3.30/1.67	A polynomial rank function with
3.30/1.67	
3.30/1.67		Pol(f0) = 1
3.30/1.67	
3.30/1.67		Pol(f4) = 0
3.30/1.67	
3.30/1.67		Pol(f3) = 1
3.30/1.67	
3.30/1.67		Pol(koat_start) = 1
3.30/1.67	
3.30/1.67	orients all transitions weakly and the transition
3.30/1.67	
3.30/1.67		f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f4(ar_0, e, ar_2, ar_3)) [ ar_0 >= 10 ]
3.30/1.67	
3.30/1.67	strictly and produces the following problem:
3.30/1.67	
3.30/1.67	4:	T:
3.30/1.67	
3.30/1.67			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f4(ar_0, e, ar_2, ar_3)) [ ar_0 >= 10 ]
3.30/1.67	
3.30/1.67			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0 + 1, ar_1, ar_0, ar_3)) [ 9 >= ar_0 ]
3.30/1.67	
3.30/1.67			(Comp: 1, Cost: 1)    f3(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(0, ar_1, ar_2, ar_3))
3.30/1.67	
3.30/1.67			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f3(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.30/1.67	
3.30/1.67		start location:	koat_start
3.30/1.67	
3.30/1.67		leaf cost:	0
3.30/1.67	
3.30/1.67	
3.30/1.67	
3.30/1.67	A polynomial rank function with
3.30/1.67	
3.30/1.67		Pol(f0) = -V_1 + 10
3.30/1.67	
3.30/1.67		Pol(f4) = -V_1
3.30/1.67	
3.30/1.67		Pol(f3) = 10
3.30/1.67	
3.30/1.67		Pol(koat_start) = 10
3.30/1.67	
3.30/1.67	orients all transitions weakly and the transition
3.30/1.67	
3.30/1.67		f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0 + 1, ar_1, ar_0, ar_3)) [ 9 >= ar_0 ]
3.30/1.67	
3.30/1.67	strictly and produces the following problem:
3.30/1.67	
3.30/1.67	5:	T:
3.30/1.67	
3.30/1.67			(Comp: 1, Cost: 1)     f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f4(ar_0, e, ar_2, ar_3)) [ ar_0 >= 10 ]
3.30/1.67	
3.30/1.67			(Comp: 10, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0 + 1, ar_1, ar_0, ar_3)) [ 9 >= ar_0 ]
3.30/1.67	
3.30/1.67			(Comp: 1, Cost: 1)     f3(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(0, ar_1, ar_2, ar_3))
3.30/1.67	
3.30/1.67			(Comp: 1, Cost: 0)     koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f3(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.30/1.67	
3.30/1.67		start location:	koat_start
3.30/1.67	
3.30/1.67		leaf cost:	0
3.30/1.67	
3.30/1.67	
3.30/1.67	
3.30/1.67	Complexity upper bound 12
3.30/1.67	
3.30/1.67	
3.30/1.67	
3.30/1.67	Time: 0.057 sec (SMT: 0.052 sec)
3.30/1.67	
3.30/1.67	
3.30/1.67	----------------------------------------
3.30/1.67	
3.30/1.67	(2)
3.30/1.67	BOUNDS(1, 1)
3.57/1.70	EOF