4.74/2.36	WORST_CASE(Omega(n^2), O(n^2))
4.74/2.37	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
4.74/2.37	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.74/2.37	
4.74/2.37	
4.74/2.37	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).
4.74/2.37	
4.74/2.37	(0) CpxIntTrs
4.74/2.37	(1) Koat Proof [FINISHED, 110 ms]
4.74/2.37	(2) BOUNDS(1, n^2)
4.74/2.37	(3) Loat Proof [FINISHED, 620 ms]
4.74/2.37	(4) BOUNDS(n^2, INF)
4.74/2.37	
4.74/2.37	
4.74/2.37	----------------------------------------
4.74/2.37	
4.74/2.37	(0)
4.74/2.37	Obligation:
4.74/2.37	Complexity Int TRS consisting of the following rules:
4.74/2.37	evalsipmabubblestart(A, B) -> Com_1(evalsipmabubbleentryin(A, B)) :|: TRUE
4.74/2.37	evalsipmabubbleentryin(A, B) -> Com_1(evalsipmabubblebb6in(A, B)) :|: TRUE
4.74/2.37	evalsipmabubblebb6in(A, B) -> Com_1(evalsipmabubblebb4in(A, 0)) :|: A >= 0
4.74/2.37	evalsipmabubblebb6in(A, B) -> Com_1(evalsipmabubblereturnin(A, B)) :|: 0 >= A + 1
4.74/2.37	evalsipmabubblebb4in(A, B) -> Com_1(evalsipmabubblebb1in(A, B)) :|: A >= 1 + B
4.74/2.37	evalsipmabubblebb4in(A, B) -> Com_1(evalsipmabubblebb5in(A, B)) :|: B >= A
4.74/2.37	evalsipmabubblebb1in(A, B) -> Com_1(evalsipmabubblebb2in(A, B)) :|: C >= D + 1
4.74/2.37	evalsipmabubblebb1in(A, B) -> Com_1(evalsipmabubblebb3in(A, B)) :|: D >= C
4.74/2.37	evalsipmabubblebb2in(A, B) -> Com_1(evalsipmabubblebb3in(A, B)) :|: TRUE
4.74/2.37	evalsipmabubblebb3in(A, B) -> Com_1(evalsipmabubblebb4in(A, B + 1)) :|: TRUE
4.74/2.37	evalsipmabubblebb5in(A, B) -> Com_1(evalsipmabubblebb6in(A - 1, B)) :|: TRUE
4.74/2.37	evalsipmabubblereturnin(A, B) -> Com_1(evalsipmabubblestop(A, B)) :|: TRUE
4.74/2.37	
4.74/2.37	The start-symbols are:[evalsipmabubblestart_2]
4.74/2.37	
4.74/2.37	
4.74/2.37	----------------------------------------
4.74/2.37	
4.74/2.37	(1) Koat Proof (FINISHED)
4.74/2.37	YES(?, 137*ar_0 + 18*ar_0^2 + 125)
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Initial complexity problem:
4.74/2.37	
4.74/2.37	1:	T:
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblestart(ar_0, ar_1) -> Com_1(evalsipmabubbleentryin(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubbleentryin(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, 0)) [ ar_0 >= 0 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblereturnin(ar_0, ar_1)) [ 0 >= ar_0 + 1 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb1in(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb5in(ar_0, ar_1)) [ ar_1 >= ar_0 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb2in(ar_0, ar_1)) [ c >= d + 1 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1)) [ c >= d ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb2in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb3in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, ar_1 + 1))
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb5in(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0 - 1, ar_1))
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblereturnin(ar_0, ar_1) -> Com_1(evalsipmabubblestop(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalsipmabubblestart(ar_0, ar_1)) [ 0 <= 0 ]
4.74/2.37	
4.74/2.37		start location:	koat_start
4.74/2.37	
4.74/2.37		leaf cost:	0
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Repeatedly propagating knowledge in problem 1 produces the following problem:
4.74/2.37	
4.74/2.37	2:	T:
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 1)    evalsipmabubblestart(ar_0, ar_1) -> Com_1(evalsipmabubbleentryin(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 1)    evalsipmabubbleentryin(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, 0)) [ ar_0 >= 0 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblereturnin(ar_0, ar_1)) [ 0 >= ar_0 + 1 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb1in(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb5in(ar_0, ar_1)) [ ar_1 >= ar_0 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb2in(ar_0, ar_1)) [ c >= d + 1 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1)) [ c >= d ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb2in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb3in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, ar_1 + 1))
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb5in(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0 - 1, ar_1))
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblereturnin(ar_0, ar_1) -> Com_1(evalsipmabubblestop(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalsipmabubblestart(ar_0, ar_1)) [ 0 <= 0 ]
4.74/2.37	
4.74/2.37		start location:	koat_start
4.74/2.37	
4.74/2.37		leaf cost:	0
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	A polynomial rank function with
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblestart) = 2
4.74/2.37	
4.74/2.37		Pol(evalsipmabubbleentryin) = 2
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb6in) = 2
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb4in) = 2
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblereturnin) = 1
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb1in) = 2
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb5in) = 2
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb2in) = 2
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb3in) = 2
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblestop) = 0
4.74/2.37	
4.74/2.37		Pol(koat_start) = 2
4.74/2.37	
4.74/2.37	orients all transitions weakly and the transitions
4.74/2.37	
4.74/2.37		evalsipmabubblereturnin(ar_0, ar_1) -> Com_1(evalsipmabubblestop(ar_0, ar_1))
4.74/2.37	
4.74/2.37		evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblereturnin(ar_0, ar_1)) [ 0 >= ar_0 + 1 ]
4.74/2.37	
4.74/2.37	strictly and produces the following problem:
4.74/2.37	
4.74/2.37	3:	T:
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 1)    evalsipmabubblestart(ar_0, ar_1) -> Com_1(evalsipmabubbleentryin(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 1)    evalsipmabubbleentryin(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, 0)) [ ar_0 >= 0 ]
4.74/2.37	
4.74/2.37			(Comp: 2, Cost: 1)    evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblereturnin(ar_0, ar_1)) [ 0 >= ar_0 + 1 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb1in(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb5in(ar_0, ar_1)) [ ar_1 >= ar_0 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb2in(ar_0, ar_1)) [ c >= d + 1 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1)) [ c >= d ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb2in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb3in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, ar_1 + 1))
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)    evalsipmabubblebb5in(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0 - 1, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 2, Cost: 1)    evalsipmabubblereturnin(ar_0, ar_1) -> Com_1(evalsipmabubblestop(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalsipmabubblestart(ar_0, ar_1)) [ 0 <= 0 ]
4.74/2.37	
4.74/2.37		start location:	koat_start
4.74/2.37	
4.74/2.37		leaf cost:	0
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	A polynomial rank function with
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblestart) = V_1 + 1
4.74/2.37	
4.74/2.37		Pol(evalsipmabubbleentryin) = V_1 + 1
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb6in) = V_1 + 1
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb4in) = V_1
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblereturnin) = V_1
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb1in) = V_1
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb5in) = V_1
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb2in) = V_1
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb3in) = V_1
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblestop) = V_1
4.74/2.37	
4.74/2.37		Pol(koat_start) = V_1 + 1
4.74/2.37	
4.74/2.37	orients all transitions weakly and the transition
4.74/2.37	
4.74/2.37		evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, 0)) [ ar_0 >= 0 ]
4.74/2.37	
4.74/2.37	strictly and produces the following problem:
4.74/2.37	
4.74/2.37	4:	T:
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 1)           evalsipmabubblestart(ar_0, ar_1) -> Com_1(evalsipmabubbleentryin(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 1)           evalsipmabubbleentryin(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: ar_0 + 1, Cost: 1)    evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, 0)) [ ar_0 >= 0 ]
4.74/2.37	
4.74/2.37			(Comp: 2, Cost: 1)           evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblereturnin(ar_0, ar_1)) [ 0 >= ar_0 + 1 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)           evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb1in(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)           evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb5in(ar_0, ar_1)) [ ar_1 >= ar_0 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)           evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb2in(ar_0, ar_1)) [ c >= d + 1 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)           evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1)) [ c >= d ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)           evalsipmabubblebb2in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)           evalsipmabubblebb3in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, ar_1 + 1))
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)           evalsipmabubblebb5in(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0 - 1, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 2, Cost: 1)           evalsipmabubblereturnin(ar_0, ar_1) -> Com_1(evalsipmabubblestop(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1) -> Com_1(evalsipmabubblestart(ar_0, ar_1)) [ 0 <= 0 ]
4.74/2.37	
4.74/2.37		start location:	koat_start
4.74/2.37	
4.74/2.37		leaf cost:	0
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	A polynomial rank function with
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb5in) = 1
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb6in) = 0
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb4in) = 2
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb1in) = 2
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb3in) = 2
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb2in) = 2
4.74/2.37	
4.74/2.37	and size complexities
4.74/2.37	
4.74/2.37		S("koat_start(ar_0, ar_1) -> Com_1(evalsipmabubblestart(ar_0, ar_1)) [ 0 <= 0 ]", 0-0) = ar_0
4.74/2.37	
4.74/2.37		S("koat_start(ar_0, ar_1) -> Com_1(evalsipmabubblestart(ar_0, ar_1)) [ 0 <= 0 ]", 0-1) = ar_1
4.74/2.37	
4.74/2.37		S("evalsipmabubblereturnin(ar_0, ar_1) -> Com_1(evalsipmabubblestop(ar_0, ar_1))", 0-0) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblereturnin(ar_0, ar_1) -> Com_1(evalsipmabubblestop(ar_0, ar_1))", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb5in(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0 - 1, ar_1))", 0-0) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb5in(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0 - 1, ar_1))", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb3in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, ar_1 + 1))", 0-0) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb3in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, ar_1 + 1))", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb2in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1))", 0-0) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb2in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1))", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1)) [ c >= d ]", 0-0) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1)) [ c >= d ]", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb2in(ar_0, ar_1)) [ c >= d + 1 ]", 0-0) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb2in(ar_0, ar_1)) [ c >= d + 1 ]", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb5in(ar_0, ar_1)) [ ar_1 >= ar_0 ]", 0-0) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb5in(ar_0, ar_1)) [ ar_1 >= ar_0 ]", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb1in(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]", 0-0) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb1in(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblereturnin(ar_0, ar_1)) [ 0 >= ar_0 + 1 ]", 0-0) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblereturnin(ar_0, ar_1)) [ 0 >= ar_0 + 1 ]", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, 0)) [ ar_0 >= 0 ]", 0-0) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, 0)) [ ar_0 >= 0 ]", 0-1) = 0
4.74/2.37	
4.74/2.37		S("evalsipmabubbleentryin(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0, ar_1))", 0-0) = ar_0
4.74/2.37	
4.74/2.37		S("evalsipmabubbleentryin(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0, ar_1))", 0-1) = ar_1
4.74/2.37	
4.74/2.37		S("evalsipmabubblestart(ar_0, ar_1) -> Com_1(evalsipmabubbleentryin(ar_0, ar_1))", 0-0) = ar_0
4.74/2.37	
4.74/2.37		S("evalsipmabubblestart(ar_0, ar_1) -> Com_1(evalsipmabubbleentryin(ar_0, ar_1))", 0-1) = ar_1
4.74/2.37	
4.74/2.37	orients the transitions
4.74/2.37	
4.74/2.37		evalsipmabubblebb5in(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0 - 1, ar_1))
4.74/2.37	
4.74/2.37		evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb5in(ar_0, ar_1)) [ ar_1 >= ar_0 ]
4.74/2.37	
4.74/2.37		evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb1in(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
4.74/2.37	
4.74/2.37		evalsipmabubblebb3in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, ar_1 + 1))
4.74/2.37	
4.74/2.37		evalsipmabubblebb2in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1))
4.74/2.37	
4.74/2.37		evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1)) [ c >= d ]
4.74/2.37	
4.74/2.37		evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb2in(ar_0, ar_1)) [ c >= d + 1 ]
4.74/2.37	
4.74/2.37	weakly and the transitions
4.74/2.37	
4.74/2.37		evalsipmabubblebb5in(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0 - 1, ar_1))
4.74/2.37	
4.74/2.37		evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb5in(ar_0, ar_1)) [ ar_1 >= ar_0 ]
4.74/2.37	
4.74/2.37	strictly and produces the following problem:
4.74/2.37	
4.74/2.37	5:	T:
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 1)             evalsipmabubblestart(ar_0, ar_1) -> Com_1(evalsipmabubbleentryin(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 1)             evalsipmabubbleentryin(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: ar_0 + 1, Cost: 1)      evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, 0)) [ ar_0 >= 0 ]
4.74/2.37	
4.74/2.37			(Comp: 2, Cost: 1)             evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblereturnin(ar_0, ar_1)) [ 0 >= ar_0 + 1 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)             evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb1in(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
4.74/2.37	
4.74/2.37			(Comp: 2*ar_0 + 2, Cost: 1)    evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb5in(ar_0, ar_1)) [ ar_1 >= ar_0 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)             evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb2in(ar_0, ar_1)) [ c >= d + 1 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)             evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1)) [ c >= d ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)             evalsipmabubblebb2in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)             evalsipmabubblebb3in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, ar_1 + 1))
4.74/2.37	
4.74/2.37			(Comp: 2*ar_0 + 2, Cost: 1)    evalsipmabubblebb5in(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0 - 1, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 2, Cost: 1)             evalsipmabubblereturnin(ar_0, ar_1) -> Com_1(evalsipmabubblestop(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 0)             koat_start(ar_0, ar_1) -> Com_1(evalsipmabubblestart(ar_0, ar_1)) [ 0 <= 0 ]
4.74/2.37	
4.74/2.37		start location:	koat_start
4.74/2.37	
4.74/2.37		leaf cost:	0
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	A polynomial rank function with
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb4in) = V_1 - V_2 + 1
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb1in) = V_1 - V_2
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb3in) = V_1 - V_2
4.74/2.37	
4.74/2.37		Pol(evalsipmabubblebb2in) = V_1 - V_2
4.74/2.37	
4.74/2.37	and size complexities
4.74/2.37	
4.74/2.37		S("koat_start(ar_0, ar_1) -> Com_1(evalsipmabubblestart(ar_0, ar_1)) [ 0 <= 0 ]", 0-0) = ar_0
4.74/2.37	
4.74/2.37		S("koat_start(ar_0, ar_1) -> Com_1(evalsipmabubblestart(ar_0, ar_1)) [ 0 <= 0 ]", 0-1) = ar_1
4.74/2.37	
4.74/2.37		S("evalsipmabubblereturnin(ar_0, ar_1) -> Com_1(evalsipmabubblestop(ar_0, ar_1))", 0-0) = 3*ar_0 + 162
4.74/2.37	
4.74/2.37		S("evalsipmabubblereturnin(ar_0, ar_1) -> Com_1(evalsipmabubblestop(ar_0, ar_1))", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb5in(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0 - 1, ar_1))", 0-0) = 3*ar_0 + 18
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb5in(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0 - 1, ar_1))", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb3in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, ar_1 + 1))", 0-0) = 3*ar_0 + 18
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb3in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, ar_1 + 1))", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb2in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1))", 0-0) = 3*ar_0 + 18
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb2in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1))", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1)) [ c >= d ]", 0-0) = 3*ar_0 + 18
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1)) [ c >= d ]", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb2in(ar_0, ar_1)) [ c >= d + 1 ]", 0-0) = 3*ar_0 + 18
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb2in(ar_0, ar_1)) [ c >= d + 1 ]", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb5in(ar_0, ar_1)) [ ar_1 >= ar_0 ]", 0-0) = 3*ar_0 + 18
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb5in(ar_0, ar_1)) [ ar_1 >= ar_0 ]", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb1in(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]", 0-0) = 3*ar_0 + 18
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb1in(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblereturnin(ar_0, ar_1)) [ 0 >= ar_0 + 1 ]", 0-0) = 3*ar_0 + 54
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblereturnin(ar_0, ar_1)) [ 0 >= ar_0 + 1 ]", 0-1) = ?
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, 0)) [ ar_0 >= 0 ]", 0-0) = 3*ar_0 + 18
4.74/2.37	
4.74/2.37		S("evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, 0)) [ ar_0 >= 0 ]", 0-1) = 0
4.74/2.37	
4.74/2.37		S("evalsipmabubbleentryin(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0, ar_1))", 0-0) = ar_0
4.74/2.37	
4.74/2.37		S("evalsipmabubbleentryin(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0, ar_1))", 0-1) = ar_1
4.74/2.37	
4.74/2.37		S("evalsipmabubblestart(ar_0, ar_1) -> Com_1(evalsipmabubbleentryin(ar_0, ar_1))", 0-0) = ar_0
4.74/2.37	
4.74/2.37		S("evalsipmabubblestart(ar_0, ar_1) -> Com_1(evalsipmabubbleentryin(ar_0, ar_1))", 0-1) = ar_1
4.74/2.37	
4.74/2.37	orients the transitions
4.74/2.37	
4.74/2.37		evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb1in(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
4.74/2.37	
4.74/2.37		evalsipmabubblebb3in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, ar_1 + 1))
4.74/2.37	
4.74/2.37		evalsipmabubblebb2in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1))
4.74/2.37	
4.74/2.37		evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1)) [ c >= d ]
4.74/2.37	
4.74/2.37		evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb2in(ar_0, ar_1)) [ c >= d + 1 ]
4.74/2.37	
4.74/2.37	weakly and the transition
4.74/2.37	
4.74/2.37		evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb1in(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
4.74/2.37	
4.74/2.37	strictly and produces the following problem:
4.74/2.37	
4.74/2.37	6:	T:
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 1)                          evalsipmabubblestart(ar_0, ar_1) -> Com_1(evalsipmabubbleentryin(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 1)                          evalsipmabubbleentryin(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: ar_0 + 1, Cost: 1)                   evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, 0)) [ ar_0 >= 0 ]
4.74/2.37	
4.74/2.37			(Comp: 2, Cost: 1)                          evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblereturnin(ar_0, ar_1)) [ 0 >= ar_0 + 1 ]
4.74/2.37	
4.74/2.37			(Comp: 3*ar_0^2 + 22*ar_0 + 19, Cost: 1)    evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb1in(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
4.74/2.37	
4.74/2.37			(Comp: 2*ar_0 + 2, Cost: 1)                 evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb5in(ar_0, ar_1)) [ ar_1 >= ar_0 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)                          evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb2in(ar_0, ar_1)) [ c >= d + 1 ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)                          evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1)) [ c >= d ]
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)                          evalsipmabubblebb2in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: ?, Cost: 1)                          evalsipmabubblebb3in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, ar_1 + 1))
4.74/2.37	
4.74/2.37			(Comp: 2*ar_0 + 2, Cost: 1)                 evalsipmabubblebb5in(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0 - 1, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 2, Cost: 1)                          evalsipmabubblereturnin(ar_0, ar_1) -> Com_1(evalsipmabubblestop(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 0)                          koat_start(ar_0, ar_1) -> Com_1(evalsipmabubblestart(ar_0, ar_1)) [ 0 <= 0 ]
4.74/2.37	
4.74/2.37		start location:	koat_start
4.74/2.37	
4.74/2.37		leaf cost:	0
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Repeatedly propagating knowledge in problem 6 produces the following problem:
4.74/2.37	
4.74/2.37	7:	T:
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 1)                          evalsipmabubblestart(ar_0, ar_1) -> Com_1(evalsipmabubbleentryin(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 1)                          evalsipmabubbleentryin(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: ar_0 + 1, Cost: 1)                   evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, 0)) [ ar_0 >= 0 ]
4.74/2.37	
4.74/2.37			(Comp: 2, Cost: 1)                          evalsipmabubblebb6in(ar_0, ar_1) -> Com_1(evalsipmabubblereturnin(ar_0, ar_1)) [ 0 >= ar_0 + 1 ]
4.74/2.37	
4.74/2.37			(Comp: 3*ar_0^2 + 22*ar_0 + 19, Cost: 1)    evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb1in(ar_0, ar_1)) [ ar_0 >= ar_1 + 1 ]
4.74/2.37	
4.74/2.37			(Comp: 2*ar_0 + 2, Cost: 1)                 evalsipmabubblebb4in(ar_0, ar_1) -> Com_1(evalsipmabubblebb5in(ar_0, ar_1)) [ ar_1 >= ar_0 ]
4.74/2.37	
4.74/2.37			(Comp: 3*ar_0^2 + 22*ar_0 + 19, Cost: 1)    evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb2in(ar_0, ar_1)) [ c >= d + 1 ]
4.74/2.37	
4.74/2.37			(Comp: 3*ar_0^2 + 22*ar_0 + 19, Cost: 1)    evalsipmabubblebb1in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1)) [ c >= d ]
4.74/2.37	
4.74/2.37			(Comp: 3*ar_0^2 + 22*ar_0 + 19, Cost: 1)    evalsipmabubblebb2in(ar_0, ar_1) -> Com_1(evalsipmabubblebb3in(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 6*ar_0^2 + 44*ar_0 + 38, Cost: 1)    evalsipmabubblebb3in(ar_0, ar_1) -> Com_1(evalsipmabubblebb4in(ar_0, ar_1 + 1))
4.74/2.37	
4.74/2.37			(Comp: 2*ar_0 + 2, Cost: 1)                 evalsipmabubblebb5in(ar_0, ar_1) -> Com_1(evalsipmabubblebb6in(ar_0 - 1, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 2, Cost: 1)                          evalsipmabubblereturnin(ar_0, ar_1) -> Com_1(evalsipmabubblestop(ar_0, ar_1))
4.74/2.37	
4.74/2.37			(Comp: 1, Cost: 0)                          koat_start(ar_0, ar_1) -> Com_1(evalsipmabubblestart(ar_0, ar_1)) [ 0 <= 0 ]
4.74/2.37	
4.74/2.37		start location:	koat_start
4.74/2.37	
4.74/2.37		leaf cost:	0
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Complexity upper bound 137*ar_0 + 18*ar_0^2 + 125
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Time: 0.099 sec (SMT: 0.087 sec)
4.74/2.37	
4.74/2.37	
4.74/2.37	----------------------------------------
4.74/2.37	
4.74/2.37	(2)
4.74/2.37	BOUNDS(1, n^2)
4.74/2.37	
4.74/2.37	----------------------------------------
4.74/2.37	
4.74/2.37	(3) Loat Proof (FINISHED)
4.74/2.37	
4.74/2.37	
4.74/2.37	### Pre-processing the ITS problem ###
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Initial linear ITS problem
4.74/2.37	
4.74/2.37	   Start location: evalsipmabubblestart
4.74/2.37	
4.74/2.37	      0: evalsipmabubblestart -> evalsipmabubbleentryin : [], cost: 1
4.74/2.37	
4.74/2.37	      1: evalsipmabubbleentryin -> evalsipmabubblebb6in : [], cost: 1
4.74/2.37	
4.74/2.37	      2: evalsipmabubblebb6in -> evalsipmabubblebb4in : B'=0, [ A>=0 ], cost: 1
4.74/2.37	
4.74/2.37	      3: evalsipmabubblebb6in -> evalsipmabubblereturnin : [ 0>=1+A ], cost: 1
4.74/2.37	
4.74/2.37	      4: evalsipmabubblebb4in -> evalsipmabubblebb1in : [ A>=1+B ], cost: 1
4.74/2.37	
4.74/2.37	      5: evalsipmabubblebb4in -> evalsipmabubblebb5in : [ B>=A ], cost: 1
4.74/2.37	
4.74/2.37	      6: evalsipmabubblebb1in -> evalsipmabubblebb2in : [ free>=1+free_1 ], cost: 1
4.74/2.37	
4.74/2.37	      7: evalsipmabubblebb1in -> evalsipmabubblebb3in : [ free_2>=free_3 ], cost: 1
4.74/2.37	
4.74/2.37	      8: evalsipmabubblebb2in -> evalsipmabubblebb3in : [], cost: 1
4.74/2.37	
4.74/2.37	      9: evalsipmabubblebb3in -> evalsipmabubblebb4in : B'=1+B, [], cost: 1
4.74/2.37	
4.74/2.37	     10: evalsipmabubblebb5in -> evalsipmabubblebb6in : A'=-1+A, [], cost: 1
4.74/2.37	
4.74/2.37	     11: evalsipmabubblereturnin -> evalsipmabubblestop : [], cost: 1
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Removed unreachable and leaf rules:
4.74/2.37	
4.74/2.37	   Start location: evalsipmabubblestart
4.74/2.37	
4.74/2.37	      0: evalsipmabubblestart -> evalsipmabubbleentryin : [], cost: 1
4.74/2.37	
4.74/2.37	      1: evalsipmabubbleentryin -> evalsipmabubblebb6in : [], cost: 1
4.74/2.37	
4.74/2.37	      2: evalsipmabubblebb6in -> evalsipmabubblebb4in : B'=0, [ A>=0 ], cost: 1
4.74/2.37	
4.74/2.37	      4: evalsipmabubblebb4in -> evalsipmabubblebb1in : [ A>=1+B ], cost: 1
4.74/2.37	
4.74/2.37	      5: evalsipmabubblebb4in -> evalsipmabubblebb5in : [ B>=A ], cost: 1
4.74/2.37	
4.74/2.37	      6: evalsipmabubblebb1in -> evalsipmabubblebb2in : [ free>=1+free_1 ], cost: 1
4.74/2.37	
4.74/2.37	      7: evalsipmabubblebb1in -> evalsipmabubblebb3in : [ free_2>=free_3 ], cost: 1
4.74/2.37	
4.74/2.37	      8: evalsipmabubblebb2in -> evalsipmabubblebb3in : [], cost: 1
4.74/2.37	
4.74/2.37	      9: evalsipmabubblebb3in -> evalsipmabubblebb4in : B'=1+B, [], cost: 1
4.74/2.37	
4.74/2.37	     10: evalsipmabubblebb5in -> evalsipmabubblebb6in : A'=-1+A, [], cost: 1
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Simplified all rules, resulting in:
4.74/2.37	
4.74/2.37	   Start location: evalsipmabubblestart
4.74/2.37	
4.74/2.37	      0: evalsipmabubblestart -> evalsipmabubbleentryin : [], cost: 1
4.74/2.37	
4.74/2.37	      1: evalsipmabubbleentryin -> evalsipmabubblebb6in : [], cost: 1
4.74/2.37	
4.74/2.37	      2: evalsipmabubblebb6in -> evalsipmabubblebb4in : B'=0, [ A>=0 ], cost: 1
4.74/2.37	
4.74/2.37	      4: evalsipmabubblebb4in -> evalsipmabubblebb1in : [ A>=1+B ], cost: 1
4.74/2.37	
4.74/2.37	      5: evalsipmabubblebb4in -> evalsipmabubblebb5in : [ B>=A ], cost: 1
4.74/2.37	
4.74/2.37	      6: evalsipmabubblebb1in -> evalsipmabubblebb2in : [], cost: 1
4.74/2.37	
4.74/2.37	      7: evalsipmabubblebb1in -> evalsipmabubblebb3in : [], cost: 1
4.74/2.37	
4.74/2.37	      8: evalsipmabubblebb2in -> evalsipmabubblebb3in : [], cost: 1
4.74/2.37	
4.74/2.37	      9: evalsipmabubblebb3in -> evalsipmabubblebb4in : B'=1+B, [], cost: 1
4.74/2.37	
4.74/2.37	     10: evalsipmabubblebb5in -> evalsipmabubblebb6in : A'=-1+A, [], cost: 1
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	### Simplification by acceleration and chaining ###
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Eliminated locations (on linear paths):
4.74/2.37	
4.74/2.37	   Start location: evalsipmabubblestart
4.74/2.37	
4.74/2.37	     12: evalsipmabubblestart -> evalsipmabubblebb6in : [], cost: 2
4.74/2.37	
4.74/2.37	      2: evalsipmabubblebb6in -> evalsipmabubblebb4in : B'=0, [ A>=0 ], cost: 1
4.74/2.37	
4.74/2.37	      4: evalsipmabubblebb4in -> evalsipmabubblebb1in : [ A>=1+B ], cost: 1
4.74/2.37	
4.74/2.37	     13: evalsipmabubblebb4in -> evalsipmabubblebb6in : A'=-1+A, [ B>=A ], cost: 2
4.74/2.37	
4.74/2.37	      7: evalsipmabubblebb1in -> evalsipmabubblebb3in : [], cost: 1
4.74/2.37	
4.74/2.37	     14: evalsipmabubblebb1in -> evalsipmabubblebb3in : [], cost: 2
4.74/2.37	
4.74/2.37	      9: evalsipmabubblebb3in -> evalsipmabubblebb4in : B'=1+B, [], cost: 1
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Eliminated locations (on tree-shaped paths):
4.74/2.37	
4.74/2.37	   Start location: evalsipmabubblestart
4.74/2.37	
4.74/2.37	     12: evalsipmabubblestart -> evalsipmabubblebb6in : [], cost: 2
4.74/2.37	
4.74/2.37	      2: evalsipmabubblebb6in -> evalsipmabubblebb4in : B'=0, [ A>=0 ], cost: 1
4.74/2.37	
4.74/2.37	     13: evalsipmabubblebb4in -> evalsipmabubblebb6in : A'=-1+A, [ B>=A ], cost: 2
4.74/2.37	
4.74/2.37	     15: evalsipmabubblebb4in -> evalsipmabubblebb3in : [ A>=1+B ], cost: 2
4.74/2.37	
4.74/2.37	     16: evalsipmabubblebb4in -> evalsipmabubblebb3in : [ A>=1+B ], cost: 3
4.74/2.37	
4.74/2.37	      9: evalsipmabubblebb3in -> evalsipmabubblebb4in : B'=1+B, [], cost: 1
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Eliminated locations (on tree-shaped paths):
4.74/2.37	
4.74/2.37	   Start location: evalsipmabubblestart
4.74/2.37	
4.74/2.37	     12: evalsipmabubblestart -> evalsipmabubblebb6in : [], cost: 2
4.74/2.37	
4.74/2.37	      2: evalsipmabubblebb6in -> evalsipmabubblebb4in : B'=0, [ A>=0 ], cost: 1
4.74/2.37	
4.74/2.37	     13: evalsipmabubblebb4in -> evalsipmabubblebb6in : A'=-1+A, [ B>=A ], cost: 2
4.74/2.37	
4.74/2.37	     17: evalsipmabubblebb4in -> evalsipmabubblebb4in : B'=1+B, [ A>=1+B ], cost: 3
4.74/2.37	
4.74/2.37	     18: evalsipmabubblebb4in -> evalsipmabubblebb4in : B'=1+B, [ A>=1+B ], cost: 4
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Accelerating simple loops of location 3.
4.74/2.37	
4.74/2.37	   Simplified some of the simple loops (and removed duplicate rules).
4.74/2.37	
4.74/2.37	   Accelerating the following rules:
4.74/2.37	
4.74/2.37	     18: evalsipmabubblebb4in -> evalsipmabubblebb4in : B'=1+B, [ A>=1+B ], cost: 4
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	   Accelerated rule 18 with metering function A-B, yielding the new rule 19.
4.74/2.37	
4.74/2.37	   Removing the simple loops: 18.
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Accelerated all simple loops using metering functions (where possible):
4.74/2.37	
4.74/2.37	   Start location: evalsipmabubblestart
4.74/2.37	
4.74/2.37	     12: evalsipmabubblestart -> evalsipmabubblebb6in : [], cost: 2
4.74/2.37	
4.74/2.37	      2: evalsipmabubblebb6in -> evalsipmabubblebb4in : B'=0, [ A>=0 ], cost: 1
4.74/2.37	
4.74/2.37	     13: evalsipmabubblebb4in -> evalsipmabubblebb6in : A'=-1+A, [ B>=A ], cost: 2
4.74/2.37	
4.74/2.37	     19: evalsipmabubblebb4in -> evalsipmabubblebb4in : B'=A, [ A>=1+B ], cost: 4*A-4*B
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Chained accelerated rules (with incoming rules):
4.74/2.37	
4.74/2.37	   Start location: evalsipmabubblestart
4.74/2.37	
4.74/2.37	     12: evalsipmabubblestart -> evalsipmabubblebb6in : [], cost: 2
4.74/2.37	
4.74/2.37	      2: evalsipmabubblebb6in -> evalsipmabubblebb4in : B'=0, [ A>=0 ], cost: 1
4.74/2.37	
4.74/2.37	     20: evalsipmabubblebb6in -> evalsipmabubblebb4in : B'=A, [ A>=1 ], cost: 1+4*A
4.74/2.37	
4.74/2.37	     13: evalsipmabubblebb4in -> evalsipmabubblebb6in : A'=-1+A, [ B>=A ], cost: 2
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Eliminated locations (on tree-shaped paths):
4.74/2.37	
4.74/2.37	   Start location: evalsipmabubblestart
4.74/2.37	
4.74/2.37	     12: evalsipmabubblestart -> evalsipmabubblebb6in : [], cost: 2
4.74/2.37	
4.74/2.37	     21: evalsipmabubblebb6in -> evalsipmabubblebb6in : A'=-1+A, B'=0, [ A>=0 && 0>=A ], cost: 3
4.74/2.37	
4.74/2.37	     22: evalsipmabubblebb6in -> evalsipmabubblebb6in : A'=-1+A, B'=A, [ A>=1 ], cost: 3+4*A
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Accelerating simple loops of location 2.
4.74/2.37	
4.74/2.37	   Simplified some of the simple loops (and removed duplicate rules).
4.74/2.37	
4.74/2.37	   Accelerating the following rules:
4.74/2.37	
4.74/2.37	     21: evalsipmabubblebb6in -> evalsipmabubblebb6in : A'=-1+A, B'=0, [ -A==0 ], cost: 3
4.74/2.37	
4.74/2.37	     22: evalsipmabubblebb6in -> evalsipmabubblebb6in : A'=-1+A, B'=A, [ A>=1 ], cost: 3+4*A
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	   Accelerated rule 21 with metering function 1+A, yielding the new rule 23.
4.74/2.37	
4.74/2.37	   Accelerated rule 22 with metering function A, yielding the new rule 24.
4.74/2.37	
4.74/2.37	   Removing the simple loops: 21 22.
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Accelerated all simple loops using metering functions (where possible):
4.74/2.37	
4.74/2.37	   Start location: evalsipmabubblestart
4.74/2.37	
4.74/2.37	     12: evalsipmabubblestart -> evalsipmabubblebb6in : [], cost: 2
4.74/2.37	
4.74/2.37	     23: evalsipmabubblebb6in -> evalsipmabubblebb6in : A'=-1, B'=0, [ -A==0 ], cost: 3+3*A
4.74/2.37	
4.74/2.37	     24: evalsipmabubblebb6in -> evalsipmabubblebb6in : A'=0, B'=1, [ A>=1 ], cost: 5*A+2*A^2
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Chained accelerated rules (with incoming rules):
4.74/2.37	
4.74/2.37	   Start location: evalsipmabubblestart
4.74/2.37	
4.74/2.37	     12: evalsipmabubblestart -> evalsipmabubblebb6in : [], cost: 2
4.74/2.37	
4.74/2.37	     25: evalsipmabubblestart -> evalsipmabubblebb6in : A'=-1, B'=0, [ -A==0 ], cost: 5+3*A
4.74/2.37	
4.74/2.37	     26: evalsipmabubblestart -> evalsipmabubblebb6in : A'=0, B'=1, [ A>=1 ], cost: 2+5*A+2*A^2
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Removed unreachable locations (and leaf rules with constant cost):
4.74/2.37	
4.74/2.37	   Start location: evalsipmabubblestart
4.74/2.37	
4.74/2.37	     25: evalsipmabubblestart -> evalsipmabubblebb6in : A'=-1, B'=0, [ -A==0 ], cost: 5+3*A
4.74/2.37	
4.74/2.37	     26: evalsipmabubblestart -> evalsipmabubblebb6in : A'=0, B'=1, [ A>=1 ], cost: 2+5*A+2*A^2
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	### Computing asymptotic complexity ###
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Fully simplified ITS problem
4.74/2.37	
4.74/2.37	   Start location: evalsipmabubblestart
4.74/2.37	
4.74/2.37	     25: evalsipmabubblestart -> evalsipmabubblebb6in : A'=-1, B'=0, [ -A==0 ], cost: 5+3*A
4.74/2.37	
4.74/2.37	     26: evalsipmabubblestart -> evalsipmabubblebb6in : A'=0, B'=1, [ A>=1 ], cost: 2+5*A+2*A^2
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Computing asymptotic complexity for rule 25
4.74/2.37	
4.74/2.37	   Could not solve the limit problem.
4.74/2.37	
4.74/2.37	   Resulting cost 0 has complexity: Unknown
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Computing asymptotic complexity for rule 26
4.74/2.37	
4.74/2.37	   Solved the limit problem by the following transformations:
4.74/2.37	
4.74/2.37	      Created initial limit problem:
4.74/2.37	
4.74/2.37	      2+5*A+2*A^2 (+), A (+/+!) [not solved]
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	      removing all constraints (solved by SMT)
4.74/2.37	
4.74/2.37	      resulting limit problem: [solved]
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	      applying transformation rule (C) using substitution {A==n}
4.74/2.37	
4.74/2.37	      resulting limit problem:
4.74/2.37	
4.74/2.37	      [solved]
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	   Solution:
4.74/2.37	
4.74/2.37	      A / n
4.74/2.37	
4.74/2.37	   Resulting cost 2+5*n+2*n^2 has complexity: Poly(n^2)
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Found new complexity Poly(n^2).
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	Obtained the following overall complexity (w.r.t. the length of the input n):
4.74/2.37	
4.74/2.37	   Complexity:  Poly(n^2)
4.74/2.37	
4.74/2.37	   Cpx degree:  2
4.74/2.37	
4.74/2.37	   Solved cost: 2+5*n+2*n^2
4.74/2.37	
4.74/2.37	   Rule cost:   2+5*A+2*A^2
4.74/2.37	
4.74/2.37	   Rule guard:  [ A>=1 ]
4.74/2.37	
4.74/2.37	
4.74/2.37	
4.74/2.37	WORST_CASE(Omega(n^2),?)
4.74/2.37	
4.74/2.37	
4.74/2.37	----------------------------------------
4.74/2.37	
4.74/2.37	(4)
4.74/2.37	BOUNDS(n^2, INF)
4.74/2.39	EOF