4.60/2.12	WORST_CASE(Omega(n^1), O(n^1))
4.60/2.14	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.60/2.14	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.60/2.14	
4.60/2.14	
4.60/2.14	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
4.60/2.14	
4.60/2.14	(0) CpxIntTrs
4.60/2.14	(1) Koat Proof [FINISHED, 211 ms]
4.60/2.14	(2) BOUNDS(1, n^1)
4.60/2.14	(3) Loat Proof [FINISHED, 411 ms]
4.60/2.14	(4) BOUNDS(n^1, INF)
4.60/2.14	
4.60/2.14	
4.60/2.14	----------------------------------------
4.60/2.14	
4.60/2.14	(0)
4.60/2.14	Obligation:
4.60/2.14	Complexity Int TRS consisting of the following rules:
4.60/2.14	start(A, B, C, D) -> Com_1(stop(A, B, C, D)) :|: 0 >= A + 1 && B >= C && B <= C && D >= A && D <= A
4.60/2.14	start(A, B, C, D) -> Com_1(stop(A, B, C, D)) :|: 0 >= C + 1 && B >= C && B <= C && D >= A && D <= A
4.60/2.14	start(A, B, C, D) -> Com_1(stop(A, B, C, D)) :|: A >= 0 && A + 2 >= C && C >= 0 && C + 2 >= A && B >= C && B <= C && D >= A && D <= A
4.60/2.14	start(A, B, C, D) -> Com_1(lbl81(A, B, C, 1 + D)) :|: A >= 0 && C >= A + 3 && B >= C && B <= C && D >= A && D <= A
4.60/2.14	start(A, B, C, D) -> Com_1(lbl91(A, 1 + B, C, D)) :|: A >= C + 3 && C >= 0 && B >= C && B <= C && D >= A && D <= A
4.60/2.15	lbl81(A, B, C, D) -> Com_1(stop(A, B, C, D)) :|: C >= 3 + A && A >= 0 && D + 2 >= C && D + 2 <= C && B >= C && B <= C
4.60/2.15	lbl81(A, B, C, D) -> Com_1(lbl81(A, B, C, 1 + D)) :|: C >= D + 3 && D >= A + 1 && A >= 0 && C >= D + 2 && B >= C && B <= C
4.60/2.15	lbl81(A, B, C, D) -> Com_1(lbl91(A, 1 + B, C, D)) :|: D >= C + 3 && D >= A + 1 && A >= 0 && C >= D + 2 && B >= C && B <= C
4.60/2.15	lbl91(A, B, C, D) -> Com_1(stop(A, B, C, D)) :|: A >= 3 + C && C >= 0 && B + 2 >= A && B + 2 <= A && D >= A && D <= A
4.60/2.15	lbl91(A, B, C, D) -> Com_1(lbl81(A, B, C, 1 + D)) :|: B >= A + 3 && A >= B + 2 && B >= C + 1 && C >= 0 && D >= A && D <= A
4.60/2.15	lbl91(A, B, C, D) -> Com_1(lbl91(A, 1 + B, C, D)) :|: A >= B + 3 && A >= B + 2 && B >= C + 1 && C >= 0 && D >= A && D <= A
4.60/2.15	start0(A, B, C, D) -> Com_1(start(A, C, C, A)) :|: TRUE
4.60/2.15	
4.60/2.15	The start-symbols are:[start0_4]
4.60/2.15	
4.60/2.15	
4.60/2.15	----------------------------------------
4.60/2.15	
4.60/2.15	(1) Koat Proof (FINISHED)
4.60/2.15	YES(?, 2*ar_0 + 2*ar_2 + 8)
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Initial complexity problem:
4.60/2.15	
4.60/2.15	1:	T:
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 0 /\ ar_0 + 2 >= ar_2 /\ ar_2 >= 0 /\ ar_2 + 2 >= ar_0 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1)) [ ar_0 >= 0 /\ ar_2 >= ar_0 + 3 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl91(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_0 >= ar_2 + 3 /\ ar_2 >= 0 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 3 /\ ar_0 >= 0 /\ ar_3 + 2 = ar_2 /\ ar_1 = ar_2 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1)) [ ar_2 >= ar_3 + 3 /\ ar_3 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_2 >= ar_3 + 2 /\ ar_1 = ar_2 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl91(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_2 + 3 /\ ar_3 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_2 >= ar_3 + 2 /\ ar_1 = ar_2 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    lbl91(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 + 3 /\ ar_2 >= 0 /\ ar_1 + 2 = ar_0 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    lbl91(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1)) [ ar_1 >= ar_0 + 3 /\ ar_0 >= ar_1 + 2 /\ ar_1 >= ar_2 + 1 /\ ar_2 >= 0 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    lbl91(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl91(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_0 >= ar_1 + 3 /\ ar_0 >= ar_1 + 2 /\ ar_1 >= ar_2 + 1 /\ ar_2 >= 0 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    start0(ar_0, ar_1, ar_2, ar_3) -> Com_1(start(ar_0, ar_2, ar_2, ar_0))
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
4.60/2.15	
4.60/2.15		start location:	koat_start
4.60/2.15	
4.60/2.15		leaf cost:	0
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Testing for reachability in the complexity graph removes the following transitions from problem 1:
4.60/2.15	
4.60/2.15		lbl81(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl91(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_2 + 3 /\ ar_3 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_2 >= ar_3 + 2 /\ ar_1 = ar_2 ]
4.60/2.15	
4.60/2.15		lbl91(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1)) [ ar_1 >= ar_0 + 3 /\ ar_0 >= ar_1 + 2 /\ ar_1 >= ar_2 + 1 /\ ar_2 >= 0 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15	We thus obtain the following problem:
4.60/2.15	
4.60/2.15	2:	T:
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    lbl91(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl91(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_0 >= ar_1 + 3 /\ ar_0 >= ar_1 + 2 /\ ar_1 >= ar_2 + 1 /\ ar_2 >= 0 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    lbl91(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 + 3 /\ ar_2 >= 0 /\ ar_1 + 2 = ar_0 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1)) [ ar_2 >= ar_3 + 3 /\ ar_3 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_2 >= ar_3 + 2 /\ ar_1 = ar_2 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 3 /\ ar_0 >= 0 /\ ar_3 + 2 = ar_2 /\ ar_1 = ar_2 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl91(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_0 >= ar_2 + 3 /\ ar_2 >= 0 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1)) [ ar_0 >= 0 /\ ar_2 >= ar_0 + 3 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 0 /\ ar_0 + 2 >= ar_2 /\ ar_2 >= 0 /\ ar_2 + 2 >= ar_0 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    start0(ar_0, ar_1, ar_2, ar_3) -> Com_1(start(ar_0, ar_2, ar_2, ar_0))
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
4.60/2.15	
4.60/2.15		start location:	koat_start
4.60/2.15	
4.60/2.15		leaf cost:	0
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Repeatedly propagating knowledge in problem 2 produces the following problem:
4.60/2.15	
4.60/2.15	3:	T:
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    lbl91(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl91(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_0 >= ar_1 + 3 /\ ar_0 >= ar_1 + 2 /\ ar_1 >= ar_2 + 1 /\ ar_2 >= 0 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    lbl91(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 + 3 /\ ar_2 >= 0 /\ ar_1 + 2 = ar_0 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1)) [ ar_2 >= ar_3 + 3 /\ ar_3 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_2 >= ar_3 + 2 /\ ar_1 = ar_2 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 3 /\ ar_0 >= 0 /\ ar_3 + 2 = ar_2 /\ ar_1 = ar_2 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl91(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_0 >= ar_2 + 3 /\ ar_2 >= 0 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1)) [ ar_0 >= 0 /\ ar_2 >= ar_0 + 3 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 0 /\ ar_0 + 2 >= ar_2 /\ ar_2 >= 0 /\ ar_2 + 2 >= ar_0 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)    start0(ar_0, ar_1, ar_2, ar_3) -> Com_1(start(ar_0, ar_2, ar_2, ar_0))
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
4.60/2.15	
4.60/2.15		start location:	koat_start
4.60/2.15	
4.60/2.15		leaf cost:	0
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	A polynomial rank function with
4.60/2.15	
4.60/2.15		Pol(lbl91) = 1
4.60/2.15	
4.60/2.15		Pol(stop) = 0
4.60/2.15	
4.60/2.15		Pol(lbl81) = 1
4.60/2.15	
4.60/2.15		Pol(start) = 1
4.60/2.15	
4.60/2.15		Pol(start0) = 1
4.60/2.15	
4.60/2.15		Pol(koat_start) = 1
4.60/2.15	
4.60/2.15	orients all transitions weakly and the transitions
4.60/2.15	
4.60/2.15		lbl91(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 + 3 /\ ar_2 >= 0 /\ ar_1 + 2 = ar_0 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15		lbl81(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 3 /\ ar_0 >= 0 /\ ar_3 + 2 = ar_2 /\ ar_1 = ar_2 ]
4.60/2.15	
4.60/2.15	strictly and produces the following problem:
4.60/2.15	
4.60/2.15	4:	T:
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    lbl91(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl91(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_0 >= ar_1 + 3 /\ ar_0 >= ar_1 + 2 /\ ar_1 >= ar_2 + 1 /\ ar_2 >= 0 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)    lbl91(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 + 3 /\ ar_2 >= 0 /\ ar_1 + 2 = ar_0 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ?, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1)) [ ar_2 >= ar_3 + 3 /\ ar_3 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_2 >= ar_3 + 2 /\ ar_1 = ar_2 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 3 /\ ar_0 >= 0 /\ ar_3 + 2 = ar_2 /\ ar_1 = ar_2 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl91(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_0 >= ar_2 + 3 /\ ar_2 >= 0 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1)) [ ar_0 >= 0 /\ ar_2 >= ar_0 + 3 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 0 /\ ar_0 + 2 >= ar_2 /\ ar_2 >= 0 /\ ar_2 + 2 >= ar_0 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)    start0(ar_0, ar_1, ar_2, ar_3) -> Com_1(start(ar_0, ar_2, ar_2, ar_0))
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
4.60/2.15	
4.60/2.15		start location:	koat_start
4.60/2.15	
4.60/2.15		leaf cost:	0
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	A polynomial rank function with
4.60/2.15	
4.60/2.15		Pol(lbl91) = 2*V_1 - V_2 + 2*V_3 - V_4
4.60/2.15	
4.60/2.15		Pol(stop) = 2*V_1 - V_2 + 2*V_3 - V_4
4.60/2.15	
4.60/2.15		Pol(lbl81) = 2*V_1 + V_2 - V_4
4.60/2.15	
4.60/2.15		Pol(start) = V_1 + V_3
4.60/2.15	
4.60/2.15		Pol(start0) = V_1 + V_3
4.60/2.15	
4.60/2.15		Pol(koat_start) = V_1 + V_3
4.60/2.15	
4.60/2.15	orients all transitions weakly and the transitions
4.60/2.15	
4.60/2.15		lbl91(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl91(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_0 >= ar_1 + 3 /\ ar_0 >= ar_1 + 2 /\ ar_1 >= ar_2 + 1 /\ ar_2 >= 0 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15		lbl81(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1)) [ ar_2 >= ar_3 + 3 /\ ar_3 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_2 >= ar_3 + 2 /\ ar_1 = ar_2 ]
4.60/2.15	
4.60/2.15	strictly and produces the following problem:
4.60/2.15	
4.60/2.15	5:	T:
4.60/2.15	
4.60/2.15			(Comp: ar_0 + ar_2, Cost: 1)    lbl91(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl91(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_0 >= ar_1 + 3 /\ ar_0 >= ar_1 + 2 /\ ar_1 >= ar_2 + 1 /\ ar_2 >= 0 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)              lbl91(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 + 3 /\ ar_2 >= 0 /\ ar_1 + 2 = ar_0 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: ar_0 + ar_2, Cost: 1)    lbl81(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1)) [ ar_2 >= ar_3 + 3 /\ ar_3 >= ar_0 + 1 /\ ar_0 >= 0 /\ ar_2 >= ar_3 + 2 /\ ar_1 = ar_2 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)              lbl81(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 3 /\ ar_0 >= 0 /\ ar_3 + 2 = ar_2 /\ ar_1 = ar_2 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)              start(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl91(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_0 >= ar_2 + 3 /\ ar_2 >= 0 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)              start(ar_0, ar_1, ar_2, ar_3) -> Com_1(lbl81(ar_0, ar_1, ar_2, ar_3 + 1)) [ ar_0 >= 0 /\ ar_2 >= ar_0 + 3 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)              start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 0 /\ ar_0 + 2 >= ar_2 /\ ar_2 >= 0 /\ ar_2 + 2 >= ar_0 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)              start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)              start(ar_0, ar_1, ar_2, ar_3) -> Com_1(stop(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 /\ ar_1 = ar_2 /\ ar_3 = ar_0 ]
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 1)              start0(ar_0, ar_1, ar_2, ar_3) -> Com_1(start(ar_0, ar_2, ar_2, ar_0))
4.60/2.15	
4.60/2.15			(Comp: 1, Cost: 0)              koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(start0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
4.60/2.15	
4.60/2.15		start location:	koat_start
4.60/2.15	
4.60/2.15		leaf cost:	0
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Complexity upper bound 2*ar_0 + 2*ar_2 + 8
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Time: 0.199 sec (SMT: 0.179 sec)
4.60/2.15	
4.60/2.15	
4.60/2.15	----------------------------------------
4.60/2.15	
4.60/2.15	(2)
4.60/2.15	BOUNDS(1, n^1)
4.60/2.15	
4.60/2.15	----------------------------------------
4.60/2.15	
4.60/2.15	(3) Loat Proof (FINISHED)
4.60/2.15	
4.60/2.15	
4.60/2.15	### Pre-processing the ITS problem ###
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Initial linear ITS problem
4.60/2.15	
4.60/2.15	   Start location: start0
4.60/2.15	
4.60/2.15	      0: start -> stop : [ 0>=1+A && B==C && D==A ], cost: 1
4.60/2.15	
4.60/2.15	      1: start -> stop : [ 0>=1+C && B==C && D==A ], cost: 1
4.60/2.15	
4.60/2.15	      2: start -> stop : [ A>=0 && 2+A>=C && C>=0 && 2+C>=A && B==C && D==A ], cost: 1
4.60/2.15	
4.60/2.15	      3: start -> lbl81 : D'=1+D, [ A>=0 && C>=3+A && B==C && D==A ], cost: 1
4.60/2.15	
4.60/2.15	      4: start -> lbl91 : B'=1+B, [ A>=3+C && C>=0 && B==C && D==A ], cost: 1
4.60/2.15	
4.60/2.15	      5: lbl81 -> stop : [ C>=3+A && A>=0 && 2+D==C && B==C ], cost: 1
4.60/2.15	
4.60/2.15	      6: lbl81 -> lbl81 : D'=1+D, [ C>=3+D && D>=1+A && A>=0 && C>=2+D && B==C ], cost: 1
4.60/2.15	
4.60/2.15	      7: lbl81 -> lbl91 : B'=1+B, [ D>=3+C && D>=1+A && A>=0 && C>=2+D && B==C ], cost: 1
4.60/2.15	
4.60/2.15	      8: lbl91 -> stop : [ A>=3+C && C>=0 && 2+B==A && D==A ], cost: 1
4.60/2.15	
4.60/2.15	      9: lbl91 -> lbl81 : D'=1+D, [ B>=3+A && A>=2+B && B>=1+C && C>=0 && D==A ], cost: 1
4.60/2.15	
4.60/2.15	     10: lbl91 -> lbl91 : B'=1+B, [ A>=3+B && A>=2+B && B>=1+C && C>=0 && D==A ], cost: 1
4.60/2.15	
4.60/2.15	     11: start0 -> start : B'=C, D'=A, [], cost: 1
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Removed unreachable and leaf rules:
4.60/2.15	
4.60/2.15	   Start location: start0
4.60/2.15	
4.60/2.15	      3: start -> lbl81 : D'=1+D, [ A>=0 && C>=3+A && B==C && D==A ], cost: 1
4.60/2.15	
4.60/2.15	      4: start -> lbl91 : B'=1+B, [ A>=3+C && C>=0 && B==C && D==A ], cost: 1
4.60/2.15	
4.60/2.15	      6: lbl81 -> lbl81 : D'=1+D, [ C>=3+D && D>=1+A && A>=0 && C>=2+D && B==C ], cost: 1
4.60/2.15	
4.60/2.15	      7: lbl81 -> lbl91 : B'=1+B, [ D>=3+C && D>=1+A && A>=0 && C>=2+D && B==C ], cost: 1
4.60/2.15	
4.60/2.15	      9: lbl91 -> lbl81 : D'=1+D, [ B>=3+A && A>=2+B && B>=1+C && C>=0 && D==A ], cost: 1
4.60/2.15	
4.60/2.15	     10: lbl91 -> lbl91 : B'=1+B, [ A>=3+B && A>=2+B && B>=1+C && C>=0 && D==A ], cost: 1
4.60/2.15	
4.60/2.15	     11: start0 -> start : B'=C, D'=A, [], cost: 1
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Removed rules with unsatisfiable guard:
4.60/2.15	
4.60/2.15	   Start location: start0
4.60/2.15	
4.60/2.15	      3: start -> lbl81 : D'=1+D, [ A>=0 && C>=3+A && B==C && D==A ], cost: 1
4.60/2.15	
4.60/2.15	      4: start -> lbl91 : B'=1+B, [ A>=3+C && C>=0 && B==C && D==A ], cost: 1
4.60/2.15	
4.60/2.15	      6: lbl81 -> lbl81 : D'=1+D, [ C>=3+D && D>=1+A && A>=0 && C>=2+D && B==C ], cost: 1
4.60/2.15	
4.60/2.15	     10: lbl91 -> lbl91 : B'=1+B, [ A>=3+B && A>=2+B && B>=1+C && C>=0 && D==A ], cost: 1
4.60/2.15	
4.60/2.15	     11: start0 -> start : B'=C, D'=A, [], cost: 1
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Simplified all rules, resulting in:
4.60/2.15	
4.60/2.15	   Start location: start0
4.60/2.15	
4.60/2.15	      3: start -> lbl81 : D'=1+D, [ A>=0 && C>=3+A && B==C && D==A ], cost: 1
4.60/2.15	
4.60/2.15	      4: start -> lbl91 : B'=1+B, [ A>=3+C && C>=0 && B==C && D==A ], cost: 1
4.60/2.15	
4.60/2.15	      6: lbl81 -> lbl81 : D'=1+D, [ C>=3+D && D>=1+A && A>=0 && B==C ], cost: 1
4.60/2.15	
4.60/2.15	     10: lbl91 -> lbl91 : B'=1+B, [ A>=3+B && B>=1+C && C>=0 && D==A ], cost: 1
4.60/2.15	
4.60/2.15	     11: start0 -> start : B'=C, D'=A, [], cost: 1
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	### Simplification by acceleration and chaining ###
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Accelerating simple loops of location 1.
4.60/2.15	
4.60/2.15	   Accelerating the following rules:
4.60/2.15	
4.60/2.15	      6: lbl81 -> lbl81 : D'=1+D, [ C>=3+D && D>=1+A && A>=0 && B==C ], cost: 1
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	   Accelerated rule 6 with metering function -2+C-D, yielding the new rule 12.
4.60/2.15	
4.60/2.15	   Removing the simple loops: 6.
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Accelerating simple loops of location 2.
4.60/2.15	
4.60/2.15	   Accelerating the following rules:
4.60/2.15	
4.60/2.15	     10: lbl91 -> lbl91 : B'=1+B, [ A>=3+B && B>=1+C && C>=0 && D==A ], cost: 1
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	   Accelerated rule 10 with metering function -2+A-B, yielding the new rule 13.
4.60/2.15	
4.60/2.15	   Removing the simple loops: 10.
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Accelerated all simple loops using metering functions (where possible):
4.60/2.15	
4.60/2.15	   Start location: start0
4.60/2.15	
4.60/2.15	      3: start -> lbl81 : D'=1+D, [ A>=0 && C>=3+A && B==C && D==A ], cost: 1
4.60/2.15	
4.60/2.15	      4: start -> lbl91 : B'=1+B, [ A>=3+C && C>=0 && B==C && D==A ], cost: 1
4.60/2.15	
4.60/2.15	     12: lbl81 -> lbl81 : D'=-2+C, [ C>=3+D && D>=1+A && A>=0 && B==C ], cost: -2+C-D
4.60/2.15	
4.60/2.15	     13: lbl91 -> lbl91 : B'=-2+A, [ A>=3+B && B>=1+C && C>=0 && D==A ], cost: -2+A-B
4.60/2.15	
4.60/2.15	     11: start0 -> start : B'=C, D'=A, [], cost: 1
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Chained accelerated rules (with incoming rules):
4.60/2.15	
4.60/2.15	   Start location: start0
4.60/2.15	
4.60/2.15	      3: start -> lbl81 : D'=1+D, [ A>=0 && C>=3+A && B==C && D==A ], cost: 1
4.60/2.15	
4.60/2.15	      4: start -> lbl91 : B'=1+B, [ A>=3+C && C>=0 && B==C && D==A ], cost: 1
4.60/2.15	
4.60/2.15	     14: start -> lbl81 : D'=-2+C, [ A>=0 && C>=3+A && B==C && D==A && C>=4+D ], cost: -2+C-D
4.60/2.15	
4.60/2.15	     15: start -> lbl91 : B'=-2+A, [ A>=3+C && C>=0 && B==C && D==A && A>=4+B ], cost: -2+A-B
4.60/2.15	
4.60/2.15	     11: start0 -> start : B'=C, D'=A, [], cost: 1
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Removed unreachable locations (and leaf rules with constant cost):
4.60/2.15	
4.60/2.15	   Start location: start0
4.60/2.15	
4.60/2.15	     14: start -> lbl81 : D'=-2+C, [ A>=0 && C>=3+A && B==C && D==A && C>=4+D ], cost: -2+C-D
4.60/2.15	
4.60/2.15	     15: start -> lbl91 : B'=-2+A, [ A>=3+C && C>=0 && B==C && D==A && A>=4+B ], cost: -2+A-B
4.60/2.15	
4.60/2.15	     11: start0 -> start : B'=C, D'=A, [], cost: 1
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Eliminated locations (on tree-shaped paths):
4.60/2.15	
4.60/2.15	   Start location: start0
4.60/2.15	
4.60/2.15	     16: start0 -> lbl81 : B'=C, D'=-2+C, [ A>=0 && C>=4+A ], cost: -1+C-A
4.60/2.15	
4.60/2.15	     17: start0 -> lbl91 : B'=-2+A, D'=A, [ C>=0 && A>=4+C ], cost: -1-C+A
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	### Computing asymptotic complexity ###
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Fully simplified ITS problem
4.60/2.15	
4.60/2.15	   Start location: start0
4.60/2.15	
4.60/2.15	     16: start0 -> lbl81 : B'=C, D'=-2+C, [ A>=0 && C>=4+A ], cost: -1+C-A
4.60/2.15	
4.60/2.15	     17: start0 -> lbl91 : B'=-2+A, D'=A, [ C>=0 && A>=4+C ], cost: -1-C+A
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Computing asymptotic complexity for rule 16
4.60/2.15	
4.60/2.15	   Solved the limit problem by the following transformations:
4.60/2.15	
4.60/2.15	      Created initial limit problem:
4.60/2.15	
4.60/2.15	      -1+C-A (+), -3+C-A (+/+!), 1+A (+/+!) [not solved]
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	      applying transformation rule (C) using substitution {A==0}
4.60/2.15	
4.60/2.15	      resulting limit problem:
4.60/2.15	
4.60/2.15	      1 (+/+!), -3+C (+/+!), -1+C (+) [not solved]
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	      applying transformation rule (C) using substitution {C==4+A}
4.60/2.15	
4.60/2.15	      resulting limit problem:
4.60/2.15	
4.60/2.15	      1 (+/+!), 3+A (+), 1+A (+/+!) [not solved]
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	      applying transformation rule (B), deleting 1 (+/+!)
4.60/2.15	
4.60/2.15	      resulting limit problem:
4.60/2.15	
4.60/2.15	      3+A (+), 1+A (+/+!) [not solved]
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	      applying transformation rule (D), replacing 3+A (+) by A (+)
4.60/2.15	
4.60/2.15	      resulting limit problem:
4.60/2.15	
4.60/2.15	      A (+), 1+A (+/+!) [not solved]
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	      applying transformation rule (D), replacing 1+A (+/+!) by A (+)
4.60/2.15	
4.60/2.15	      resulting limit problem:
4.60/2.15	
4.60/2.15	      A (+) [solved]
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	   Solution:
4.60/2.15	
4.60/2.15	      C / 4+n
4.60/2.15	
4.60/2.15	      A / 0
4.60/2.15	
4.60/2.15	   Resulting cost 3+n has complexity: Poly(n^1)
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Found new complexity Poly(n^1).
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	Obtained the following overall complexity (w.r.t. the length of the input n):
4.60/2.15	
4.60/2.15	   Complexity:  Poly(n^1)
4.60/2.15	
4.60/2.15	   Cpx degree:  1
4.60/2.15	
4.60/2.15	   Solved cost: 3+n
4.60/2.15	
4.60/2.15	   Rule cost:   -1+C-A
4.60/2.15	
4.60/2.15	   Rule guard:  [ A>=0 && C>=4+A ]
4.60/2.15	
4.60/2.15	
4.60/2.15	
4.60/2.15	WORST_CASE(Omega(n^1),?)
4.60/2.15	
4.60/2.15	
4.60/2.15	----------------------------------------
4.60/2.15	
4.60/2.15	(4)
4.60/2.15	BOUNDS(n^1, INF)
4.76/2.17	EOF