3.60/2.26	WORST_CASE(Omega(n^2), O(n^2))
3.60/2.27	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.60/2.27	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.60/2.27	
3.60/2.27	
3.60/2.27	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).
3.60/2.27	
3.60/2.27	(0) CpxIntTrs
3.60/2.27	(1) Koat Proof [FINISHED, 117 ms]
3.60/2.27	(2) BOUNDS(1, n^2)
3.60/2.27	(3) Loat Proof [FINISHED, 515 ms]
3.60/2.27	(4) BOUNDS(n^2, INF)
3.60/2.27	
3.60/2.27	
3.60/2.27	----------------------------------------
3.60/2.27	
3.60/2.27	(0)
3.60/2.27	Obligation:
3.60/2.27	Complexity Int TRS consisting of the following rules:
3.60/2.27	evalfstart(A, B, C, D) -> Com_1(evalfentryin(A, B, C, D)) :|: TRUE
3.60/2.27	evalfentryin(A, B, C, D) -> Com_1(evalfbb4in(1, B, C, D)) :|: TRUE
3.60/2.27	evalfbb4in(A, B, C, D) -> Com_1(evalfbb2in(A, B, 1, D)) :|: B >= A
3.60/2.27	evalfbb4in(A, B, C, D) -> Com_1(evalfreturnin(A, B, C, D)) :|: A >= B + 1
3.60/2.27	evalfbb2in(A, B, C, D) -> Com_1(evalfbb1in(A, B, C, D)) :|: D >= C
3.60/2.27	evalfbb2in(A, B, C, D) -> Com_1(evalfbb3in(A, B, C, D)) :|: C >= D + 1
3.60/2.27	evalfbb1in(A, B, C, D) -> Com_1(evalfbb2in(A, B, C + 1, D)) :|: TRUE
3.60/2.27	evalfbb3in(A, B, C, D) -> Com_1(evalfbb4in(A + 1, B, C, D)) :|: TRUE
3.60/2.27	evalfreturnin(A, B, C, D) -> Com_1(evalfstop(A, B, C, D)) :|: TRUE
3.60/2.27	
3.60/2.27	The start-symbols are:[evalfstart_4]
3.60/2.27	
3.60/2.27	
3.60/2.27	----------------------------------------
3.60/2.27	
3.60/2.27	(1) Koat Proof (FINISHED)
3.60/2.27	YES(?, 9*ar_1 + 2*ar_1*ar_3 + 6)
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Initial complexity problem:
3.60/2.27	
3.60/2.27	1:	T:
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.60/2.27	
3.60/2.27		start location:	koat_start
3.60/2.27	
3.60/2.27		leaf cost:	0
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.60/2.27	
3.60/2.27	2:	T:
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 1)    evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.60/2.27	
3.60/2.27		start location:	koat_start
3.60/2.27	
3.60/2.27		leaf cost:	0
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	A polynomial rank function with
3.60/2.27	
3.60/2.27		Pol(evalfstart) = 2
3.60/2.27	
3.60/2.27		Pol(evalfentryin) = 2
3.60/2.27	
3.60/2.27		Pol(evalfbb4in) = 2
3.60/2.27	
3.60/2.27		Pol(evalfbb2in) = 2
3.60/2.27	
3.60/2.27		Pol(evalfreturnin) = 1
3.60/2.27	
3.60/2.27		Pol(evalfbb1in) = 2
3.60/2.27	
3.60/2.27		Pol(evalfbb3in) = 2
3.60/2.27	
3.60/2.27		Pol(evalfstop) = 0
3.60/2.27	
3.60/2.27		Pol(koat_start) = 2
3.60/2.27	
3.60/2.27	orients all transitions weakly and the transitions
3.60/2.27	
3.60/2.27		evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27		evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]
3.60/2.27	
3.60/2.27	strictly and produces the following problem:
3.60/2.27	
3.60/2.27	3:	T:
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 1)    evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 1)    evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]
3.60/2.27	
3.60/2.27			(Comp: 2, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 2, Cost: 1)    evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.60/2.27	
3.60/2.27		start location:	koat_start
3.60/2.27	
3.60/2.27		leaf cost:	0
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	A polynomial rank function with
3.60/2.27	
3.60/2.27		Pol(evalfstart) = V_2
3.60/2.27	
3.60/2.27		Pol(evalfentryin) = V_2
3.60/2.27	
3.60/2.27		Pol(evalfbb4in) = -V_1 + V_2 + 1
3.60/2.27	
3.60/2.27		Pol(evalfbb2in) = -V_1 + V_2
3.60/2.27	
3.60/2.27		Pol(evalfreturnin) = -V_1 + V_2
3.60/2.27	
3.60/2.27		Pol(evalfbb1in) = -V_1 + V_2
3.60/2.27	
3.60/2.27		Pol(evalfbb3in) = -V_1 + V_2
3.60/2.27	
3.60/2.27		Pol(evalfstop) = -V_1 + V_2
3.60/2.27	
3.60/2.27		Pol(koat_start) = V_2
3.60/2.27	
3.60/2.27	orients all transitions weakly and the transition
3.60/2.27	
3.60/2.27		evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]
3.60/2.27	
3.60/2.27	strictly and produces the following problem:
3.60/2.27	
3.60/2.27	4:	T:
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 1)       evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 1)       evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: ar_1, Cost: 1)    evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]
3.60/2.27	
3.60/2.27			(Comp: 2, Cost: 1)       evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)       evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)       evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)       evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)       evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 2, Cost: 1)       evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.60/2.27	
3.60/2.27		start location:	koat_start
3.60/2.27	
3.60/2.27		leaf cost:	0
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	A polynomial rank function with
3.60/2.27	
3.60/2.27		Pol(evalfbb3in) = 1
3.60/2.27	
3.60/2.27		Pol(evalfbb4in) = 0
3.60/2.27	
3.60/2.27		Pol(evalfbb2in) = 2
3.60/2.27	
3.60/2.27		Pol(evalfbb1in) = 2
3.60/2.27	
3.60/2.27	and size complexities
3.60/2.27	
3.60/2.27		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-0) = ar_0
3.60/2.27	
3.60/2.27		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-2) = ar_2
3.60/2.27	
3.60/2.27		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))", 0-0) = ?
3.60/2.27	
3.60/2.27		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))", 0-2) = ?
3.60/2.27	
3.60/2.27		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-0) = ?
3.60/2.27	
3.60/2.27		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-2) = ?
3.60/2.27	
3.60/2.27		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-0) = ?
3.60/2.27	
3.60/2.27		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-2) = ?
3.60/2.27	
3.60/2.27		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-0) = ?
3.60/2.27	
3.60/2.27		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-2) = ?
3.60/2.27	
3.60/2.27		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-0) = ?
3.60/2.27	
3.60/2.27		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-2) = ?
3.60/2.27	
3.60/2.27		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-0) = ?
3.60/2.27	
3.60/2.27		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-2) = ?
3.60/2.27	
3.60/2.27		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-0) = ?
3.60/2.27	
3.60/2.27		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-2) = 1
3.60/2.27	
3.60/2.27		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))", 0-0) = 1
3.60/2.27	
3.60/2.27		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))", 0-2) = ar_2
3.60/2.27	
3.60/2.27		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0
3.60/2.27	
3.60/2.27		S("evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2
3.60/2.27	
3.60/2.27		S("evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3
3.60/2.27	
3.60/2.27	orients the transitions
3.60/2.27	
3.60/2.27		evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27		evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
3.60/2.27	
3.60/2.27		evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
3.60/2.27	
3.60/2.27		evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
3.60/2.27	
3.60/2.27	weakly and the transitions
3.60/2.27	
3.60/2.27		evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27		evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
3.60/2.27	
3.60/2.27	strictly and produces the following problem:
3.60/2.27	
3.60/2.27	5:	T:
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 1)         evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 1)         evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: ar_1, Cost: 1)      evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]
3.60/2.27	
3.60/2.27			(Comp: 2, Cost: 1)         evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)         evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
3.60/2.27	
3.60/2.27			(Comp: 2*ar_1, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)         evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 2*ar_1, Cost: 1)    evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 2, Cost: 1)         evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 0)         koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.60/2.27	
3.60/2.27		start location:	koat_start
3.60/2.27	
3.60/2.27		leaf cost:	0
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	A polynomial rank function with
3.60/2.27	
3.60/2.27		Pol(evalfbb2in) = -V_3 + V_4 + 1
3.60/2.27	
3.60/2.27		Pol(evalfbb1in) = -V_3 + V_4
3.60/2.27	
3.60/2.27	and size complexities
3.60/2.27	
3.60/2.27		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-0) = ar_0
3.60/2.27	
3.60/2.27		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-2) = ar_2
3.60/2.27	
3.60/2.27		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))", 0-0) = 2*ar_1 + 20
3.60/2.27	
3.60/2.27		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))", 0-2) = ?
3.60/2.27	
3.60/2.27		S("evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-0) = 2*ar_1 + 4
3.60/2.27	
3.60/2.27		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-2) = ?
3.60/2.27	
3.60/2.27		S("evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-0) = 2*ar_1 + 4
3.60/2.27	
3.60/2.27		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-2) = ?
3.60/2.27	
3.60/2.27		S("evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-0) = 2*ar_1 + 4
3.60/2.27	
3.60/2.27		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-2) = ?
3.60/2.27	
3.60/2.27		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-0) = 2*ar_1 + 4
3.60/2.27	
3.60/2.27		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-2) = ?
3.60/2.27	
3.60/2.27		S("evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-0) = 2*ar_1 + 10
3.60/2.27	
3.60/2.27		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-2) = ?
3.60/2.27	
3.60/2.27		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-0) = 2*ar_1 + 4
3.60/2.27	
3.60/2.27		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-2) = 1
3.60/2.27	
3.60/2.27		S("evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))", 0-0) = 1
3.60/2.27	
3.60/2.27		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))", 0-2) = ar_2
3.60/2.27	
3.60/2.27		S("evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))", 0-3) = ar_3
3.60/2.27	
3.60/2.27		S("evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0
3.60/2.27	
3.60/2.27		S("evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1
3.60/2.27	
3.60/2.27		S("evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2
3.60/2.27	
3.60/2.27		S("evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3
3.60/2.27	
3.60/2.27	orients the transitions
3.60/2.27	
3.60/2.27		evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
3.60/2.27	
3.60/2.27		evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
3.60/2.27	
3.60/2.27	weakly and the transition
3.60/2.27	
3.60/2.27		evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
3.60/2.27	
3.60/2.27	strictly and produces the following problem:
3.60/2.27	
3.60/2.27	6:	T:
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 1)                     evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 1)                     evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: ar_1, Cost: 1)                  evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]
3.60/2.27	
3.60/2.27			(Comp: 2, Cost: 1)                     evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]
3.60/2.27	
3.60/2.27			(Comp: ar_1*ar_3 + 2*ar_1, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
3.60/2.27	
3.60/2.27			(Comp: 2*ar_1, Cost: 1)                evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
3.60/2.27	
3.60/2.27			(Comp: ?, Cost: 1)                     evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 2*ar_1, Cost: 1)                evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 2, Cost: 1)                     evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 0)                     koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.60/2.27	
3.60/2.27		start location:	koat_start
3.60/2.27	
3.60/2.27		leaf cost:	0
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Repeatedly propagating knowledge in problem 6 produces the following problem:
3.60/2.27	
3.60/2.27	7:	T:
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 1)                     evalfstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfentryin(ar_0, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 1)                     evalfentryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(1, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: ar_1, Cost: 1)                  evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]
3.60/2.27	
3.60/2.27			(Comp: 2, Cost: 1)                     evalfbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfreturnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]
3.60/2.27	
3.60/2.27			(Comp: ar_1*ar_3 + 2*ar_1, Cost: 1)    evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]
3.60/2.27	
3.60/2.27			(Comp: 2*ar_1, Cost: 1)                evalfbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]
3.60/2.27	
3.60/2.27			(Comp: ar_1*ar_3 + 2*ar_1, Cost: 1)    evalfbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb2in(ar_0, ar_1, ar_2 + 1, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 2*ar_1, Cost: 1)                evalfbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfbb4in(ar_0 + 1, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 2, Cost: 1)                     evalfreturnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstop(ar_0, ar_1, ar_2, ar_3))
3.60/2.27	
3.60/2.27			(Comp: 1, Cost: 0)                     koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.60/2.27	
3.60/2.27		start location:	koat_start
3.60/2.27	
3.60/2.27		leaf cost:	0
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Complexity upper bound 9*ar_1 + 2*ar_1*ar_3 + 6
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Time: 0.120 sec (SMT: 0.103 sec)
3.60/2.27	
3.60/2.27	
3.60/2.27	----------------------------------------
3.60/2.27	
3.60/2.27	(2)
3.60/2.27	BOUNDS(1, n^2)
3.60/2.27	
3.60/2.27	----------------------------------------
3.60/2.27	
3.60/2.27	(3) Loat Proof (FINISHED)
3.60/2.27	
3.60/2.27	
3.60/2.27	### Pre-processing the ITS problem ###
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Initial linear ITS problem
3.60/2.27	
3.60/2.27	   Start location: evalfstart
3.60/2.27	
3.60/2.27	      0: evalfstart -> evalfentryin : [], cost: 1
3.60/2.27	
3.60/2.27	      1: evalfentryin -> evalfbb4in : A'=1, [], cost: 1
3.60/2.27	
3.60/2.27	      2: evalfbb4in -> evalfbb2in : C'=1, [ B>=A ], cost: 1
3.60/2.27	
3.60/2.27	      3: evalfbb4in -> evalfreturnin : [ A>=1+B ], cost: 1
3.60/2.27	
3.60/2.27	      4: evalfbb2in -> evalfbb1in : [ D>=C ], cost: 1
3.60/2.27	
3.60/2.27	      5: evalfbb2in -> evalfbb3in : [ C>=1+D ], cost: 1
3.60/2.27	
3.60/2.27	      6: evalfbb1in -> evalfbb2in : C'=1+C, [], cost: 1
3.60/2.27	
3.60/2.27	      7: evalfbb3in -> evalfbb4in : A'=1+A, [], cost: 1
3.60/2.27	
3.60/2.27	      8: evalfreturnin -> evalfstop : [], cost: 1
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Removed unreachable and leaf rules:
3.60/2.27	
3.60/2.27	   Start location: evalfstart
3.60/2.27	
3.60/2.27	      0: evalfstart -> evalfentryin : [], cost: 1
3.60/2.27	
3.60/2.27	      1: evalfentryin -> evalfbb4in : A'=1, [], cost: 1
3.60/2.27	
3.60/2.27	      2: evalfbb4in -> evalfbb2in : C'=1, [ B>=A ], cost: 1
3.60/2.27	
3.60/2.27	      4: evalfbb2in -> evalfbb1in : [ D>=C ], cost: 1
3.60/2.27	
3.60/2.27	      5: evalfbb2in -> evalfbb3in : [ C>=1+D ], cost: 1
3.60/2.27	
3.60/2.27	      6: evalfbb1in -> evalfbb2in : C'=1+C, [], cost: 1
3.60/2.27	
3.60/2.27	      7: evalfbb3in -> evalfbb4in : A'=1+A, [], cost: 1
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	### Simplification by acceleration and chaining ###
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Eliminated locations (on linear paths):
3.60/2.27	
3.60/2.27	   Start location: evalfstart
3.60/2.27	
3.60/2.27	      9: evalfstart -> evalfbb4in : A'=1, [], cost: 2
3.60/2.27	
3.60/2.27	      2: evalfbb4in -> evalfbb2in : C'=1, [ B>=A ], cost: 1
3.60/2.27	
3.60/2.27	     10: evalfbb2in -> evalfbb2in : C'=1+C, [ D>=C ], cost: 2
3.60/2.27	
3.60/2.27	     11: evalfbb2in -> evalfbb4in : A'=1+A, [ C>=1+D ], cost: 2
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Accelerating simple loops of location 3.
3.60/2.27	
3.60/2.27	   Accelerating the following rules:
3.60/2.27	
3.60/2.27	     10: evalfbb2in -> evalfbb2in : C'=1+C, [ D>=C ], cost: 2
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	   Accelerated rule 10 with metering function 1-C+D, yielding the new rule 12.
3.60/2.27	
3.60/2.27	   Removing the simple loops: 10.
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Accelerated all simple loops using metering functions (where possible):
3.60/2.27	
3.60/2.27	   Start location: evalfstart
3.60/2.27	
3.60/2.27	      9: evalfstart -> evalfbb4in : A'=1, [], cost: 2
3.60/2.27	
3.60/2.27	      2: evalfbb4in -> evalfbb2in : C'=1, [ B>=A ], cost: 1
3.60/2.27	
3.60/2.27	     11: evalfbb2in -> evalfbb4in : A'=1+A, [ C>=1+D ], cost: 2
3.60/2.27	
3.60/2.27	     12: evalfbb2in -> evalfbb2in : C'=1+D, [ D>=C ], cost: 2-2*C+2*D
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Chained accelerated rules (with incoming rules):
3.60/2.27	
3.60/2.27	   Start location: evalfstart
3.60/2.27	
3.60/2.27	      9: evalfstart -> evalfbb4in : A'=1, [], cost: 2
3.60/2.27	
3.60/2.27	      2: evalfbb4in -> evalfbb2in : C'=1, [ B>=A ], cost: 1
3.60/2.27	
3.60/2.27	     13: evalfbb4in -> evalfbb2in : C'=1+D, [ B>=A && D>=1 ], cost: 1+2*D
3.60/2.27	
3.60/2.27	     11: evalfbb2in -> evalfbb4in : A'=1+A, [ C>=1+D ], cost: 2
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Eliminated locations (on tree-shaped paths):
3.60/2.27	
3.60/2.27	   Start location: evalfstart
3.60/2.27	
3.60/2.27	      9: evalfstart -> evalfbb4in : A'=1, [], cost: 2
3.60/2.27	
3.60/2.27	     14: evalfbb4in -> evalfbb4in : A'=1+A, C'=1, [ B>=A && 1>=1+D ], cost: 3
3.60/2.27	
3.60/2.27	     15: evalfbb4in -> evalfbb4in : A'=1+A, C'=1+D, [ B>=A && D>=1 ], cost: 3+2*D
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Accelerating simple loops of location 2.
3.60/2.27	
3.60/2.27	   Accelerating the following rules:
3.60/2.27	
3.60/2.27	     14: evalfbb4in -> evalfbb4in : A'=1+A, C'=1, [ B>=A && 1>=1+D ], cost: 3
3.60/2.27	
3.60/2.27	     15: evalfbb4in -> evalfbb4in : A'=1+A, C'=1+D, [ B>=A && D>=1 ], cost: 3+2*D
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	   Accelerated rule 14 with metering function 1-A+B, yielding the new rule 16.
3.60/2.27	
3.60/2.27	   Accelerated rule 15 with metering function 1-A+B, yielding the new rule 17.
3.60/2.27	
3.60/2.27	   Removing the simple loops: 14 15.
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Accelerated all simple loops using metering functions (where possible):
3.60/2.27	
3.60/2.27	   Start location: evalfstart
3.60/2.27	
3.60/2.27	      9: evalfstart -> evalfbb4in : A'=1, [], cost: 2
3.60/2.27	
3.60/2.27	     16: evalfbb4in -> evalfbb4in : A'=1+B, C'=1, [ B>=A && 1>=1+D ], cost: 3-3*A+3*B
3.60/2.27	
3.60/2.27	     17: evalfbb4in -> evalfbb4in : A'=1+B, C'=1+D, [ B>=A && D>=1 ], cost: 3-2*(-1+A-B)*D-3*A+3*B
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Chained accelerated rules (with incoming rules):
3.60/2.27	
3.60/2.27	   Start location: evalfstart
3.60/2.27	
3.60/2.27	      9: evalfstart -> evalfbb4in : A'=1, [], cost: 2
3.60/2.27	
3.60/2.27	     18: evalfstart -> evalfbb4in : A'=1+B, C'=1, [ B>=1 && 1>=1+D ], cost: 2+3*B
3.60/2.27	
3.60/2.27	     19: evalfstart -> evalfbb4in : A'=1+B, C'=1+D, [ B>=1 && D>=1 ], cost: 2+2*D*B+3*B
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Removed unreachable locations (and leaf rules with constant cost):
3.60/2.27	
3.60/2.27	   Start location: evalfstart
3.60/2.27	
3.60/2.27	     18: evalfstart -> evalfbb4in : A'=1+B, C'=1, [ B>=1 && 1>=1+D ], cost: 2+3*B
3.60/2.27	
3.60/2.27	     19: evalfstart -> evalfbb4in : A'=1+B, C'=1+D, [ B>=1 && D>=1 ], cost: 2+2*D*B+3*B
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	### Computing asymptotic complexity ###
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Fully simplified ITS problem
3.60/2.27	
3.60/2.27	   Start location: evalfstart
3.60/2.27	
3.60/2.27	     18: evalfstart -> evalfbb4in : A'=1+B, C'=1, [ B>=1 && 1>=1+D ], cost: 2+3*B
3.60/2.27	
3.60/2.27	     19: evalfstart -> evalfbb4in : A'=1+B, C'=1+D, [ B>=1 && D>=1 ], cost: 2+2*D*B+3*B
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Computing asymptotic complexity for rule 18
3.60/2.27	
3.60/2.27	   Solved the limit problem by the following transformations:
3.60/2.27	
3.60/2.27	      Created initial limit problem:
3.60/2.27	
3.60/2.27	      1-D (+/+!), 2+3*B (+), B (+/+!) [not solved]
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	      removing all constraints (solved by SMT)
3.60/2.27	
3.60/2.27	      resulting limit problem: [solved]
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	      applying transformation rule (C) using substitution {D==0,B==n}
3.60/2.27	
3.60/2.27	      resulting limit problem:
3.60/2.27	
3.60/2.27	      [solved]
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	   Solution:
3.60/2.27	
3.60/2.27	      D / 0
3.60/2.27	
3.60/2.27	      B / n
3.60/2.27	
3.60/2.27	   Resulting cost 2+3*n has complexity: Poly(n^1)
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Found new complexity Poly(n^1).
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Computing asymptotic complexity for rule 19
3.60/2.27	
3.60/2.27	   Solved the limit problem by the following transformations:
3.60/2.27	
3.60/2.27	      Created initial limit problem:
3.60/2.27	
3.60/2.27	      2+2*D*B+3*B (+), D (+/+!), B (+/+!) [not solved]
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	      removing all constraints (solved by SMT)
3.60/2.27	
3.60/2.27	      resulting limit problem: [solved]
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	      applying transformation rule (C) using substitution {D==n,B==n}
3.60/2.27	
3.60/2.27	      resulting limit problem:
3.60/2.27	
3.60/2.27	      [solved]
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	   Solution:
3.60/2.27	
3.60/2.27	      D / n
3.60/2.27	
3.60/2.27	      B / n
3.60/2.27	
3.60/2.27	   Resulting cost 2+3*n+2*n^2 has complexity: Poly(n^2)
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Found new complexity Poly(n^2).
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	Obtained the following overall complexity (w.r.t. the length of the input n):
3.60/2.27	
3.60/2.27	   Complexity:  Poly(n^2)
3.60/2.27	
3.60/2.27	   Cpx degree:  2
3.60/2.27	
3.60/2.27	   Solved cost: 2+3*n+2*n^2
3.60/2.27	
3.60/2.27	   Rule cost:   2+2*D*B+3*B
3.60/2.27	
3.60/2.27	   Rule guard:  [ B>=1 && D>=1 ]
3.60/2.27	
3.60/2.27	
3.60/2.27	
3.60/2.27	WORST_CASE(Omega(n^2),?)
3.60/2.27	
3.60/2.27	
3.60/2.27	----------------------------------------
3.60/2.27	
3.60/2.27	(4)
3.60/2.27	BOUNDS(n^2, INF)
4.82/2.29	EOF