4.43/2.20	WORST_CASE(Omega(n^2), O(n^2))
4.43/2.21	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.43/2.21	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.43/2.21	
4.43/2.21	
4.43/2.21	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).
4.43/2.21	
4.43/2.21	(0) CpxIntTrs
4.43/2.21	(1) Koat Proof [FINISHED, 157 ms]
4.43/2.21	(2) BOUNDS(1, n^2)
4.43/2.21	(3) Loat Proof [FINISHED, 439 ms]
4.43/2.21	(4) BOUNDS(n^2, INF)
4.43/2.21	
4.43/2.21	
4.43/2.21	----------------------------------------
4.43/2.21	
4.43/2.21	(0)
4.43/2.21	Obligation:
4.43/2.21	Complexity Int TRS consisting of the following rules:
4.43/2.21	l0(A, B, C, D) -> Com_1(l1(0, B, C, D)) :|: TRUE
4.43/2.21	l1(A, B, C, D) -> Com_1(l1(A + 1, B - 1, C, D)) :|: B >= 1
4.43/2.21	l1(A, B, C, D) -> Com_1(l2(A, B, A, D)) :|: 0 >= B
4.43/2.21	l2(A, B, C, D) -> Com_1(l3(A, B, C, C)) :|: C >= 1
4.43/2.21	l3(A, B, C, D) -> Com_1(l3(A, B, C, D - 1)) :|: D >= 1 && C >= 1
4.43/2.21	l3(A, B, C, D) -> Com_1(l2(A, B, C - 1, D)) :|: 0 >= D && C >= 1
4.43/2.21	
4.43/2.21	The start-symbols are:[l0_4]
4.43/2.21	
4.43/2.21	
4.43/2.21	----------------------------------------
4.43/2.21	
4.43/2.21	(1) Koat Proof (FINISHED)
4.43/2.21	YES(?, 5*ar_1 + 2*ar_1^2 + 2)
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Initial complexity problem:
4.43/2.21	
4.43/2.21	1:	T:
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)    l0(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(0, ar_1, ar_2, ar_3))
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)    l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(ar_0 + 1, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)    l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_0, ar_3)) [ 0 >= ar_1 ]
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)    l2(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_2)) [ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)    l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)    l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_2 - 1, ar_3)) [ 0 >= ar_3 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(l0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
4.43/2.21	
4.43/2.21		start location:	koat_start
4.43/2.21	
4.43/2.21		leaf cost:	0
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Repeatedly propagating knowledge in problem 1 produces the following problem:
4.43/2.21	
4.43/2.21	2:	T:
4.43/2.21	
4.43/2.21			(Comp: 1, Cost: 1)    l0(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(0, ar_1, ar_2, ar_3))
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)    l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(ar_0 + 1, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)    l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_0, ar_3)) [ 0 >= ar_1 ]
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)    l2(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_2)) [ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)    l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)    l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_2 - 1, ar_3)) [ 0 >= ar_3 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(l0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
4.43/2.21	
4.43/2.21		start location:	koat_start
4.43/2.21	
4.43/2.21		leaf cost:	0
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	A polynomial rank function with
4.43/2.21	
4.43/2.21		Pol(l0) = 1
4.43/2.21	
4.43/2.21		Pol(l1) = 1
4.43/2.21	
4.43/2.21		Pol(l2) = 0
4.43/2.21	
4.43/2.21		Pol(l3) = 0
4.43/2.21	
4.43/2.21		Pol(koat_start) = 1
4.43/2.21	
4.43/2.21	orients all transitions weakly and the transition
4.43/2.21	
4.43/2.21		l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_0, ar_3)) [ 0 >= ar_1 ]
4.43/2.21	
4.43/2.21	strictly and produces the following problem:
4.43/2.21	
4.43/2.21	3:	T:
4.43/2.21	
4.43/2.21			(Comp: 1, Cost: 1)    l0(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(0, ar_1, ar_2, ar_3))
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)    l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(ar_0 + 1, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: 1, Cost: 1)    l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_0, ar_3)) [ 0 >= ar_1 ]
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)    l2(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_2)) [ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)    l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)    l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_2 - 1, ar_3)) [ 0 >= ar_3 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(l0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
4.43/2.21	
4.43/2.21		start location:	koat_start
4.43/2.21	
4.43/2.21		leaf cost:	0
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	A polynomial rank function with
4.43/2.21	
4.43/2.21		Pol(l0) = V_2
4.43/2.21	
4.43/2.21		Pol(l1) = V_2
4.43/2.21	
4.43/2.21		Pol(l2) = V_2
4.43/2.21	
4.43/2.21		Pol(l3) = V_2
4.43/2.21	
4.43/2.21		Pol(koat_start) = V_2
4.43/2.21	
4.43/2.21	orients all transitions weakly and the transition
4.43/2.21	
4.43/2.21		l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(ar_0 + 1, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 1 ]
4.43/2.21	
4.43/2.21	strictly and produces the following problem:
4.43/2.21	
4.43/2.21	4:	T:
4.43/2.21	
4.43/2.21			(Comp: 1, Cost: 1)       l0(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(0, ar_1, ar_2, ar_3))
4.43/2.21	
4.43/2.21			(Comp: ar_1, Cost: 1)    l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(ar_0 + 1, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: 1, Cost: 1)       l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_0, ar_3)) [ 0 >= ar_1 ]
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)       l2(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_2)) [ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)       l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)       l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_2 - 1, ar_3)) [ 0 >= ar_3 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(l0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
4.43/2.21	
4.43/2.21		start location:	koat_start
4.43/2.21	
4.43/2.21		leaf cost:	0
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	A polynomial rank function with
4.43/2.21	
4.43/2.21		Pol(l3) = 2*V_3 - 1
4.43/2.21	
4.43/2.21		Pol(l2) = 2*V_3
4.43/2.21	
4.43/2.21	and size complexities
4.43/2.21	
4.43/2.21		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(l0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-0) = ar_0
4.43/2.21	
4.43/2.21		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(l0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-1) = ar_1
4.43/2.21	
4.43/2.21		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(l0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-2) = ar_2
4.43/2.21	
4.43/2.21		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(l0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-3) = ar_3
4.43/2.21	
4.43/2.21		S("l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_2 - 1, ar_3)) [ 0 >= ar_3 /\\ ar_2 >= 1 ]", 0-0) = ar_1
4.43/2.21	
4.43/2.21		S("l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_2 - 1, ar_3)) [ 0 >= ar_3 /\\ ar_2 >= 1 ]", 0-1) = ar_1
4.43/2.21	
4.43/2.21		S("l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_2 - 1, ar_3)) [ 0 >= ar_3 /\\ ar_2 >= 1 ]", 0-2) = ar_1
4.43/2.21	
4.43/2.21		S("l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_2 - 1, ar_3)) [ 0 >= ar_3 /\\ ar_2 >= 1 ]", 0-3) = ar_1
4.43/2.21	
4.43/2.21		S("l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\\ ar_2 >= 1 ]", 0-0) = ar_1
4.43/2.21	
4.43/2.21		S("l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\\ ar_2 >= 1 ]", 0-1) = ar_1
4.43/2.21	
4.43/2.21		S("l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\\ ar_2 >= 1 ]", 0-2) = ar_1
4.43/2.21	
4.43/2.21		S("l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\\ ar_2 >= 1 ]", 0-3) = ar_1
4.43/2.21	
4.43/2.21		S("l2(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_2)) [ ar_2 >= 1 ]", 0-0) = ar_1
4.43/2.21	
4.43/2.21		S("l2(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_2)) [ ar_2 >= 1 ]", 0-1) = ar_1
4.43/2.21	
4.43/2.21		S("l2(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_2)) [ ar_2 >= 1 ]", 0-2) = ar_1
4.43/2.21	
4.43/2.21		S("l2(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_2)) [ ar_2 >= 1 ]", 0-3) = ar_1
4.43/2.21	
4.43/2.21		S("l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_0, ar_3)) [ 0 >= ar_1 ]", 0-0) = ar_1
4.43/2.21	
4.43/2.21		S("l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_0, ar_3)) [ 0 >= ar_1 ]", 0-1) = ar_1
4.43/2.21	
4.43/2.21		S("l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_0, ar_3)) [ 0 >= ar_1 ]", 0-2) = ar_1
4.43/2.21	
4.43/2.21		S("l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_0, ar_3)) [ 0 >= ar_1 ]", 0-3) = ar_3
4.43/2.21	
4.43/2.21		S("l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(ar_0 + 1, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 1 ]", 0-0) = ar_1
4.43/2.21	
4.43/2.21		S("l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(ar_0 + 1, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 1 ]", 0-1) = ar_1
4.43/2.21	
4.43/2.21		S("l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(ar_0 + 1, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 1 ]", 0-2) = ar_2
4.43/2.21	
4.43/2.21		S("l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(ar_0 + 1, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 1 ]", 0-3) = ar_3
4.43/2.21	
4.43/2.21		S("l0(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(0, ar_1, ar_2, ar_3))", 0-0) = 0
4.43/2.21	
4.43/2.21		S("l0(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(0, ar_1, ar_2, ar_3))", 0-1) = ar_1
4.43/2.21	
4.43/2.21		S("l0(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(0, ar_1, ar_2, ar_3))", 0-2) = ar_2
4.43/2.21	
4.43/2.21		S("l0(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(0, ar_1, ar_2, ar_3))", 0-3) = ar_3
4.43/2.21	
4.43/2.21	orients the transitions
4.43/2.21	
4.43/2.21		l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21		l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_2 - 1, ar_3)) [ 0 >= ar_3 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21		l2(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_2)) [ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21	weakly and the transitions
4.43/2.21	
4.43/2.21		l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_2 - 1, ar_3)) [ 0 >= ar_3 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21		l2(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_2)) [ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21	strictly and produces the following problem:
4.43/2.21	
4.43/2.21	5:	T:
4.43/2.21	
4.43/2.21			(Comp: 1, Cost: 1)         l0(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(0, ar_1, ar_2, ar_3))
4.43/2.21	
4.43/2.21			(Comp: ar_1, Cost: 1)      l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(ar_0 + 1, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: 1, Cost: 1)         l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_0, ar_3)) [ 0 >= ar_1 ]
4.43/2.21	
4.43/2.21			(Comp: 2*ar_1, Cost: 1)    l2(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_2)) [ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: ?, Cost: 1)         l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: 2*ar_1, Cost: 1)    l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_2 - 1, ar_3)) [ 0 >= ar_3 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: 1, Cost: 0)         koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(l0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
4.43/2.21	
4.43/2.21		start location:	koat_start
4.43/2.21	
4.43/2.21		leaf cost:	0
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	A polynomial rank function with
4.43/2.21	
4.43/2.21		Pol(l3) = V_4
4.43/2.21	
4.43/2.21	and size complexities
4.43/2.21	
4.43/2.21		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(l0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-0) = ar_0
4.43/2.21	
4.43/2.21		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(l0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-1) = ar_1
4.43/2.21	
4.43/2.21		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(l0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-2) = ar_2
4.43/2.21	
4.43/2.21		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(l0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-3) = ar_3
4.43/2.21	
4.43/2.21		S("l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_2 - 1, ar_3)) [ 0 >= ar_3 /\\ ar_2 >= 1 ]", 0-0) = ar_1
4.43/2.21	
4.43/2.21		S("l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_2 - 1, ar_3)) [ 0 >= ar_3 /\\ ar_2 >= 1 ]", 0-1) = ar_1
4.43/2.21	
4.43/2.21		S("l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_2 - 1, ar_3)) [ 0 >= ar_3 /\\ ar_2 >= 1 ]", 0-2) = ar_1
4.43/2.21	
4.43/2.21		S("l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_2 - 1, ar_3)) [ 0 >= ar_3 /\\ ar_2 >= 1 ]", 0-3) = ar_1
4.43/2.21	
4.43/2.21		S("l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\\ ar_2 >= 1 ]", 0-0) = ar_1
4.43/2.21	
4.43/2.21		S("l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\\ ar_2 >= 1 ]", 0-1) = ar_1
4.43/2.21	
4.43/2.21		S("l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\\ ar_2 >= 1 ]", 0-2) = ar_1
4.43/2.21	
4.43/2.21		S("l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\\ ar_2 >= 1 ]", 0-3) = ar_1
4.43/2.21	
4.43/2.21		S("l2(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_2)) [ ar_2 >= 1 ]", 0-0) = ar_1
4.43/2.21	
4.43/2.21		S("l2(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_2)) [ ar_2 >= 1 ]", 0-1) = ar_1
4.43/2.21	
4.43/2.21		S("l2(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_2)) [ ar_2 >= 1 ]", 0-2) = ar_1
4.43/2.21	
4.43/2.21		S("l2(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_2)) [ ar_2 >= 1 ]", 0-3) = ar_1
4.43/2.21	
4.43/2.21		S("l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_0, ar_3)) [ 0 >= ar_1 ]", 0-0) = ar_1
4.43/2.21	
4.43/2.21		S("l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_0, ar_3)) [ 0 >= ar_1 ]", 0-1) = ar_1
4.43/2.21	
4.43/2.21		S("l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_0, ar_3)) [ 0 >= ar_1 ]", 0-2) = ar_1
4.43/2.21	
4.43/2.21		S("l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_0, ar_3)) [ 0 >= ar_1 ]", 0-3) = ar_3
4.43/2.21	
4.43/2.21		S("l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(ar_0 + 1, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 1 ]", 0-0) = ar_1
4.43/2.21	
4.43/2.21		S("l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(ar_0 + 1, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 1 ]", 0-1) = ar_1
4.43/2.21	
4.43/2.21		S("l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(ar_0 + 1, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 1 ]", 0-2) = ar_2
4.43/2.21	
4.43/2.21		S("l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(ar_0 + 1, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 1 ]", 0-3) = ar_3
4.43/2.21	
4.43/2.21		S("l0(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(0, ar_1, ar_2, ar_3))", 0-0) = 0
4.43/2.21	
4.43/2.21		S("l0(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(0, ar_1, ar_2, ar_3))", 0-1) = ar_1
4.43/2.21	
4.43/2.21		S("l0(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(0, ar_1, ar_2, ar_3))", 0-2) = ar_2
4.43/2.21	
4.43/2.21		S("l0(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(0, ar_1, ar_2, ar_3))", 0-3) = ar_3
4.43/2.21	
4.43/2.21	orients the transitions
4.43/2.21	
4.43/2.21		l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21	weakly and the transition
4.43/2.21	
4.43/2.21		l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21	strictly and produces the following problem:
4.43/2.21	
4.43/2.21	6:	T:
4.43/2.21	
4.43/2.21			(Comp: 1, Cost: 1)           l0(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(0, ar_1, ar_2, ar_3))
4.43/2.21	
4.43/2.21			(Comp: ar_1, Cost: 1)        l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l1(ar_0 + 1, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: 1, Cost: 1)           l1(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_0, ar_3)) [ 0 >= ar_1 ]
4.43/2.21	
4.43/2.21			(Comp: 2*ar_1, Cost: 1)      l2(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_2)) [ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: 2*ar_1^2, Cost: 1)    l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l3(ar_0, ar_1, ar_2, ar_3 - 1)) [ ar_3 >= 1 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: 2*ar_1, Cost: 1)      l3(ar_0, ar_1, ar_2, ar_3) -> Com_1(l2(ar_0, ar_1, ar_2 - 1, ar_3)) [ 0 >= ar_3 /\ ar_2 >= 1 ]
4.43/2.21	
4.43/2.21			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(l0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
4.43/2.21	
4.43/2.21		start location:	koat_start
4.43/2.21	
4.43/2.21		leaf cost:	0
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Complexity upper bound 5*ar_1 + 2*ar_1^2 + 2
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Time: 0.141 sec (SMT: 0.123 sec)
4.43/2.21	
4.43/2.21	
4.43/2.21	----------------------------------------
4.43/2.21	
4.43/2.21	(2)
4.43/2.21	BOUNDS(1, n^2)
4.43/2.21	
4.43/2.21	----------------------------------------
4.43/2.21	
4.43/2.21	(3) Loat Proof (FINISHED)
4.43/2.21	
4.43/2.21	
4.43/2.21	### Pre-processing the ITS problem ###
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Initial linear ITS problem
4.43/2.21	
4.43/2.21	   Start location: l0
4.43/2.21	
4.43/2.21	      0: l0 -> l1 : A'=0, [], cost: 1
4.43/2.21	
4.43/2.21	      1: l1 -> l1 : A'=1+A, B'=-1+B, [ B>=1 ], cost: 1
4.43/2.21	
4.43/2.21	      2: l1 -> l2 : C'=A, [ 0>=B ], cost: 1
4.43/2.21	
4.43/2.21	      3: l2 -> l3 : D'=C, [ C>=1 ], cost: 1
4.43/2.21	
4.43/2.21	      4: l3 -> l3 : D'=-1+D, [ D>=1 && C>=1 ], cost: 1
4.43/2.21	
4.43/2.21	      5: l3 -> l2 : C'=-1+C, [ 0>=D && C>=1 ], cost: 1
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	### Simplification by acceleration and chaining ###
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Accelerating simple loops of location 1.
4.43/2.21	
4.43/2.21	   Accelerating the following rules:
4.43/2.21	
4.43/2.21	      1: l1 -> l1 : A'=1+A, B'=-1+B, [ B>=1 ], cost: 1
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	   Accelerated rule 1 with metering function B, yielding the new rule 6.
4.43/2.21	
4.43/2.21	   Removing the simple loops: 1.
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Accelerating simple loops of location 3.
4.43/2.21	
4.43/2.21	   Accelerating the following rules:
4.43/2.21	
4.43/2.21	      4: l3 -> l3 : D'=-1+D, [ D>=1 && C>=1 ], cost: 1
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	   Accelerated rule 4 with metering function D, yielding the new rule 7.
4.43/2.21	
4.43/2.21	   Removing the simple loops: 4.
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Accelerated all simple loops using metering functions (where possible):
4.43/2.21	
4.43/2.21	   Start location: l0
4.43/2.21	
4.43/2.21	      0: l0 -> l1 : A'=0, [], cost: 1
4.43/2.21	
4.43/2.21	      2: l1 -> l2 : C'=A, [ 0>=B ], cost: 1
4.43/2.21	
4.43/2.21	      6: l1 -> l1 : A'=A+B, B'=0, [ B>=1 ], cost: B
4.43/2.21	
4.43/2.21	      3: l2 -> l3 : D'=C, [ C>=1 ], cost: 1
4.43/2.21	
4.43/2.21	      5: l3 -> l2 : C'=-1+C, [ 0>=D && C>=1 ], cost: 1
4.43/2.21	
4.43/2.21	      7: l3 -> l3 : D'=0, [ D>=1 && C>=1 ], cost: D
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Chained accelerated rules (with incoming rules):
4.43/2.21	
4.43/2.21	   Start location: l0
4.43/2.21	
4.43/2.21	      0: l0 -> l1 : A'=0, [], cost: 1
4.43/2.21	
4.43/2.21	      8: l0 -> l1 : A'=B, B'=0, [ B>=1 ], cost: 1+B
4.43/2.21	
4.43/2.21	      2: l1 -> l2 : C'=A, [ 0>=B ], cost: 1
4.43/2.21	
4.43/2.21	      3: l2 -> l3 : D'=C, [ C>=1 ], cost: 1
4.43/2.21	
4.43/2.21	      9: l2 -> l3 : D'=0, [ C>=1 ], cost: 1+C
4.43/2.21	
4.43/2.21	      5: l3 -> l2 : C'=-1+C, [ 0>=D && C>=1 ], cost: 1
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Eliminated locations (on tree-shaped paths):
4.43/2.21	
4.43/2.21	   Start location: l0
4.43/2.21	
4.43/2.21	     10: l0 -> l2 : A'=0, C'=0, [ 0>=B ], cost: 2
4.43/2.21	
4.43/2.21	     11: l0 -> l2 : A'=B, B'=0, C'=B, [ B>=1 ], cost: 2+B
4.43/2.21	
4.43/2.21	     12: l2 -> l2 : C'=-1+C, D'=0, [ C>=1 ], cost: 2+C
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Accelerating simple loops of location 2.
4.43/2.21	
4.43/2.21	   Accelerating the following rules:
4.43/2.21	
4.43/2.21	     12: l2 -> l2 : C'=-1+C, D'=0, [ C>=1 ], cost: 2+C
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	   Accelerated rule 12 with metering function C, yielding the new rule 13.
4.43/2.21	
4.43/2.21	   Removing the simple loops: 12.
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Accelerated all simple loops using metering functions (where possible):
4.43/2.21	
4.43/2.21	   Start location: l0
4.43/2.21	
4.43/2.21	     10: l0 -> l2 : A'=0, C'=0, [ 0>=B ], cost: 2
4.43/2.21	
4.43/2.21	     11: l0 -> l2 : A'=B, B'=0, C'=B, [ B>=1 ], cost: 2+B
4.43/2.21	
4.43/2.21	     13: l2 -> l2 : C'=0, D'=0, [ C>=1 ], cost: 1/2*C^2+5/2*C
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Chained accelerated rules (with incoming rules):
4.43/2.21	
4.43/2.21	   Start location: l0
4.43/2.21	
4.43/2.21	     10: l0 -> l2 : A'=0, C'=0, [ 0>=B ], cost: 2
4.43/2.21	
4.43/2.21	     11: l0 -> l2 : A'=B, B'=0, C'=B, [ B>=1 ], cost: 2+B
4.43/2.21	
4.43/2.21	     14: l0 -> l2 : A'=B, B'=0, C'=0, D'=0, [ B>=1 ], cost: 2+1/2*B^2+7/2*B
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Removed unreachable locations (and leaf rules with constant cost):
4.43/2.21	
4.43/2.21	   Start location: l0
4.43/2.21	
4.43/2.21	     11: l0 -> l2 : A'=B, B'=0, C'=B, [ B>=1 ], cost: 2+B
4.43/2.21	
4.43/2.21	     14: l0 -> l2 : A'=B, B'=0, C'=0, D'=0, [ B>=1 ], cost: 2+1/2*B^2+7/2*B
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	### Computing asymptotic complexity ###
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Fully simplified ITS problem
4.43/2.21	
4.43/2.21	   Start location: l0
4.43/2.21	
4.43/2.21	     11: l0 -> l2 : A'=B, B'=0, C'=B, [ B>=1 ], cost: 2+B
4.43/2.21	
4.43/2.21	     14: l0 -> l2 : A'=B, B'=0, C'=0, D'=0, [ B>=1 ], cost: 2+1/2*B^2+7/2*B
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Computing asymptotic complexity for rule 11
4.43/2.21	
4.43/2.21	   Solved the limit problem by the following transformations:
4.43/2.21	
4.43/2.21	      Created initial limit problem:
4.43/2.21	
4.43/2.21	      2+B (+), B (+/+!) [not solved]
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	      removing all constraints (solved by SMT)
4.43/2.21	
4.43/2.21	      resulting limit problem: [solved]
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	      applying transformation rule (C) using substitution {B==n}
4.43/2.21	
4.43/2.21	      resulting limit problem:
4.43/2.21	
4.43/2.21	      [solved]
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	   Solution:
4.43/2.21	
4.43/2.21	      B / n
4.43/2.21	
4.43/2.21	   Resulting cost 2+n has complexity: Poly(n^1)
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Found new complexity Poly(n^1).
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Computing asymptotic complexity for rule 14
4.43/2.21	
4.43/2.21	   Solved the limit problem by the following transformations:
4.43/2.21	
4.43/2.21	      Created initial limit problem:
4.43/2.21	
4.43/2.21	      2+1/2*B^2+7/2*B (+), B (+/+!) [not solved]
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	      removing all constraints (solved by SMT)
4.43/2.21	
4.43/2.21	      resulting limit problem: [solved]
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	      applying transformation rule (C) using substitution {B==n}
4.43/2.21	
4.43/2.21	      resulting limit problem:
4.43/2.21	
4.43/2.21	      [solved]
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	   Solution:
4.43/2.21	
4.43/2.21	      B / n
4.43/2.21	
4.43/2.21	   Resulting cost 2+7/2*n+1/2*n^2 has complexity: Poly(n^2)
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Found new complexity Poly(n^2).
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	Obtained the following overall complexity (w.r.t. the length of the input n):
4.43/2.21	
4.43/2.21	   Complexity:  Poly(n^2)
4.43/2.21	
4.43/2.21	   Cpx degree:  2
4.43/2.21	
4.43/2.21	   Solved cost: 2+7/2*n+1/2*n^2
4.43/2.21	
4.43/2.21	   Rule cost:   2+1/2*B^2+7/2*B
4.43/2.21	
4.43/2.21	   Rule guard:  [ B>=1 ]
4.43/2.21	
4.43/2.21	
4.43/2.21	
4.43/2.21	WORST_CASE(Omega(n^2),?)
4.43/2.21	
4.43/2.21	
4.43/2.21	----------------------------------------
4.43/2.21	
4.43/2.21	(4)
4.43/2.21	BOUNDS(n^2, INF)
4.63/2.50	EOF