/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^2), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).

(0) CpxIntTrs
(1) Koat Proof [FINISHED, 124 ms]
(2) BOUNDS(1, n^2)
(3) Loat Proof [FINISHED, 924 ms]
(4) BOUNDS(n^2, INF)


----------------------------------------

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
evalrealbubblestart(A, B, C, D) -> Com_1(evalrealbubbleentryin(A, B, C, D)) :|: TRUE
evalrealbubbleentryin(A, B, C, D) -> Com_1(evalrealbubblebb7in(A - 1, B, C, D)) :|: TRUE
evalrealbubblebb7in(A, B, C, D) -> Com_1(evalrealbubblebb4in(A, 0, 0, D)) :|: A >= 1
evalrealbubblebb7in(A, B, C, D) -> Com_1(evalrealbubblereturnin(A, B, C, D)) :|: 0 >= A
evalrealbubblebb4in(A, B, C, D) -> Com_1(evalrealbubblebb1in(A, B, C, D)) :|: A >= B + 1
evalrealbubblebb4in(A, B, C, D) -> Com_1(evalrealbubblebb5in(A, B, C, D)) :|: B >= A
evalrealbubblebb1in(A, B, C, D) -> Com_1(evalrealbubblebb2in(A, B, C, D)) :|: E >= F + 1
evalrealbubblebb1in(A, B, C, D) -> Com_1(evalrealbubblebb3in(A, B, C, C)) :|: F >= E
evalrealbubblebb2in(A, B, C, D) -> Com_1(evalrealbubblebb3in(A, B, C, 1)) :|: TRUE
evalrealbubblebb3in(A, B, C, D) -> Com_1(evalrealbubblebb4in(A, B + 1, D, D)) :|: TRUE
evalrealbubblebb5in(A, B, C, D) -> Com_1(evalrealbubblereturnin(A, B, C, D)) :|: C >= 0 && C <= 0
evalrealbubblebb5in(A, B, C, D) -> Com_1(evalrealbubblebb6in(A, B, C, D)) :|: 0 >= C + 1
evalrealbubblebb5in(A, B, C, D) -> Com_1(evalrealbubblebb6in(A, B, C, D)) :|: C >= 1
evalrealbubblebb6in(A, B, C, D) -> Com_1(evalrealbubblebb7in(A - 1, B, C, D)) :|: TRUE
evalrealbubblereturnin(A, B, C, D) -> Com_1(evalrealbubblestop(A, B, C, D)) :|: TRUE

The start-symbols are:[evalrealbubblestart_4]


----------------------------------------

(1) Koat Proof (FINISHED)
YES(?, 115*Ar_0 + 24*Ar_0^2 + 8)



Initial complexity problem:

1:	T:

		(Comp: ?, Cost: 1)    evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, 0, 0, Ar_3)) [ Ar_0 >= 1 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]

		(Comp: ?, Cost: 1)    evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))

		(Comp: ?, Cost: 1)    evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))

		(Comp: ?, Cost: 1)    evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 = 0 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestop(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Repeatedly propagating knowledge in problem 1 produces the following problem:

2:	T:

		(Comp: 1, Cost: 1)    evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, 0, 0, Ar_3)) [ Ar_0 >= 1 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]

		(Comp: ?, Cost: 1)    evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))

		(Comp: ?, Cost: 1)    evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))

		(Comp: ?, Cost: 1)    evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 = 0 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestop(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalrealbubblestart) = 2

	Pol(evalrealbubbleentryin) = 2

	Pol(evalrealbubblebb7in) = 2

	Pol(evalrealbubblebb4in) = 2

	Pol(evalrealbubblereturnin) = 1

	Pol(evalrealbubblebb1in) = 2

	Pol(evalrealbubblebb5in) = 2

	Pol(evalrealbubblebb2in) = 2

	Pol(evalrealbubblebb3in) = 2

	Pol(evalrealbubblebb6in) = 2

	Pol(evalrealbubblestop) = 0

	Pol(koat_start) = 2

orients all transitions weakly and the transitions

	evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestop(Ar_0, Ar_1, Ar_2, Ar_3))

	evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]

	evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 = 0 ]

strictly and produces the following problem:

3:	T:

		(Comp: 1, Cost: 1)    evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, 0, 0, Ar_3)) [ Ar_0 >= 1 ]

		(Comp: 2, Cost: 1)    evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]

		(Comp: ?, Cost: 1)    evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))

		(Comp: ?, Cost: 1)    evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))

		(Comp: 2, Cost: 1)    evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 = 0 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]

		(Comp: ?, Cost: 1)    evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))

		(Comp: 2, Cost: 1)    evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestop(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalrealbubblestart) = V_1

	Pol(evalrealbubbleentryin) = V_1

	Pol(evalrealbubblebb7in) = V_1 + 1

	Pol(evalrealbubblebb4in) = V_1

	Pol(evalrealbubblereturnin) = V_1

	Pol(evalrealbubblebb1in) = V_1

	Pol(evalrealbubblebb5in) = V_1

	Pol(evalrealbubblebb2in) = V_1

	Pol(evalrealbubblebb3in) = V_1

	Pol(evalrealbubblebb6in) = V_1

	Pol(evalrealbubblestop) = V_1

	Pol(koat_start) = V_1

orients all transitions weakly and the transition

	evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, 0, 0, Ar_3)) [ Ar_0 >= 1 ]

strictly and produces the following problem:

4:	T:

		(Comp: 1, Cost: 1)       evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))

		(Comp: Ar_0, Cost: 1)    evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, 0, 0, Ar_3)) [ Ar_0 >= 1 ]

		(Comp: 2, Cost: 1)       evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]

		(Comp: ?, Cost: 1)       evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]

		(Comp: ?, Cost: 1)       evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]

		(Comp: ?, Cost: 1)       evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]

		(Comp: ?, Cost: 1)       evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]

		(Comp: ?, Cost: 1)       evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))

		(Comp: ?, Cost: 1)       evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))

		(Comp: 2, Cost: 1)       evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 = 0 ]

		(Comp: ?, Cost: 1)       evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]

		(Comp: ?, Cost: 1)       evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]

		(Comp: ?, Cost: 1)       evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))

		(Comp: 2, Cost: 1)       evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestop(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 0)       koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalrealbubblebb6in) = 1

	Pol(evalrealbubblebb7in) = 0

	Pol(evalrealbubblebb5in) = 2

	Pol(evalrealbubblebb4in) = 3

	Pol(evalrealbubblebb1in) = 3

	Pol(evalrealbubblebb3in) = 3

	Pol(evalrealbubblebb2in) = 3

and size complexities

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-0) = Ar_0

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-1) = Ar_1

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-2) = Ar_2

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3

	S("evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = ?

	S("evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = ?

	S("evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 + 1

	S("evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 + 2

	S("evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))", 0-0) = ?

	S("evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))", 0-1) = ?

	S("evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))", 0-2) = 1

	S("evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))", 0-3) = 1

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]", 0-0) = ?

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]", 0-1) = ?

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]", 0-2) = 1

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]", 0-3) = 1

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]", 0-0) = ?

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]", 0-1) = ?

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]", 0-2) = 1

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]", 0-3) = 1

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 = 0 ]", 0-0) = ?

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 = 0 ]", 0-1) = ?

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 = 0 ]", 0-2) = 0

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 = 0 ]", 0-3) = 1

	S("evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))", 0-0) = ?

	S("evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))", 0-1) = ?

	S("evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))", 0-2) = 1

	S("evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))", 0-3) = 1

	S("evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))", 0-0) = ?

	S("evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))", 0-1) = ?

	S("evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))", 0-2) = 1

	S("evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))", 0-3) = 1

	S("evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]", 0-0) = ?

	S("evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]", 0-1) = ?

	S("evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]", 0-2) = 1

	S("evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]", 0-3) = 1

	S("evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]", 0-0) = ?

	S("evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]", 0-1) = ?

	S("evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]", 0-2) = 1

	S("evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]", 0-3) = Ar_3 + 2

	S("evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-0) = ?

	S("evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-1) = ?

	S("evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-2) = 1

	S("evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-3) = 1

	S("evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]", 0-0) = ?

	S("evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = ?

	S("evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]", 0-2) = 1

	S("evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]", 0-3) = Ar_3 + 2

	S("evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-0) = ?

	S("evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-1) = ?

	S("evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-2) = Ar_2 + 1

	S("evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-3) = Ar_3 + 1

	S("evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, 0, 0, Ar_3)) [ Ar_0 >= 1 ]", 0-0) = ?

	S("evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, 0, 0, Ar_3)) [ Ar_0 >= 1 ]", 0-1) = 0

	S("evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, 0, 0, Ar_3)) [ Ar_0 >= 1 ]", 0-2) = 0

	S("evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, 0, 0, Ar_3)) [ Ar_0 >= 1 ]", 0-3) = Ar_3 + 1

	S("evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 + 1

	S("evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1

	S("evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2

	S("evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3

	S("evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0

	S("evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1

	S("evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2

	S("evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3

orients the transitions

	evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))

	evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]

	evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]

	evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]

	evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]

	evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))

	evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))

	evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]

	evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]

weakly and the transitions

	evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))

	evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]

	evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]

	evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]

strictly and produces the following problem:

5:	T:

		(Comp: 1, Cost: 1)         evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)         evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))

		(Comp: Ar_0, Cost: 1)      evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, 0, 0, Ar_3)) [ Ar_0 >= 1 ]

		(Comp: 2, Cost: 1)         evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]

		(Comp: ?, Cost: 1)         evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]

		(Comp: 3*Ar_0, Cost: 1)    evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]

		(Comp: ?, Cost: 1)         evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]

		(Comp: ?, Cost: 1)         evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]

		(Comp: ?, Cost: 1)         evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))

		(Comp: ?, Cost: 1)         evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))

		(Comp: 2, Cost: 1)         evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 = 0 ]

		(Comp: 3*Ar_0, Cost: 1)    evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]

		(Comp: 3*Ar_0, Cost: 1)    evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]

		(Comp: 3*Ar_0, Cost: 1)    evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))

		(Comp: 2, Cost: 1)         evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestop(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 0)         koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalrealbubblebb4in) = V_1 - V_2 + 1

	Pol(evalrealbubblebb1in) = V_1 - V_2

	Pol(evalrealbubblebb3in) = V_1 - V_2

	Pol(evalrealbubblebb2in) = V_1 - V_2

and size complexities

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-0) = Ar_0

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-1) = Ar_1

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-2) = Ar_2

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3

	S("evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = 4*Ar_0 + 256

	S("evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = ?

	S("evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 + 1

	S("evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 + 2

	S("evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))", 0-0) = 4*Ar_0 + 16

	S("evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))", 0-1) = ?

	S("evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))", 0-2) = 1

	S("evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))", 0-3) = 1

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]", 0-0) = 4*Ar_0 + 16

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]", 0-1) = ?

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]", 0-2) = 1

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]", 0-3) = 1

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]", 0-0) = 4*Ar_0 + 16

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]", 0-1) = ?

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]", 0-2) = 1

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]", 0-3) = 1

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 = 0 ]", 0-0) = 4*Ar_0 + 64

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 = 0 ]", 0-1) = ?

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 = 0 ]", 0-2) = 0

	S("evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 = 0 ]", 0-3) = 1

	S("evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))", 0-0) = 4*Ar_0 + 16

	S("evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))", 0-1) = ?

	S("evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))", 0-2) = 1

	S("evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))", 0-3) = 1

	S("evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))", 0-0) = 4*Ar_0 + 16

	S("evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))", 0-1) = ?

	S("evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))", 0-2) = 1

	S("evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))", 0-3) = 1

	S("evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]", 0-0) = 4*Ar_0 + 16

	S("evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]", 0-1) = ?

	S("evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]", 0-2) = 1

	S("evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]", 0-3) = 1

	S("evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]", 0-0) = 4*Ar_0 + 16

	S("evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]", 0-1) = ?

	S("evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]", 0-2) = 1

	S("evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]", 0-3) = Ar_3 + 2

	S("evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-0) = 4*Ar_0 + 16

	S("evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-1) = ?

	S("evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-2) = 1

	S("evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-3) = 1

	S("evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]", 0-0) = 4*Ar_0 + 16

	S("evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = ?

	S("evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]", 0-2) = 1

	S("evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]", 0-3) = Ar_3 + 2

	S("evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-0) = 4*Ar_0 + 64

	S("evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-1) = ?

	S("evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-2) = Ar_2 + 1

	S("evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-3) = Ar_3 + 1

	S("evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, 0, 0, Ar_3)) [ Ar_0 >= 1 ]", 0-0) = 4*Ar_0 + 16

	S("evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, 0, 0, Ar_3)) [ Ar_0 >= 1 ]", 0-1) = 0

	S("evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, 0, 0, Ar_3)) [ Ar_0 >= 1 ]", 0-2) = 0

	S("evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, 0, 0, Ar_3)) [ Ar_0 >= 1 ]", 0-3) = Ar_3 + 1

	S("evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 + 1

	S("evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1

	S("evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2

	S("evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3

	S("evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0

	S("evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1

	S("evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2

	S("evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3

orients the transitions

	evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]

	evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))

	evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))

	evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]

	evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]

weakly and the transition

	evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]

strictly and produces the following problem:

6:	T:

		(Comp: 1, Cost: 1)                     evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                     evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))

		(Comp: Ar_0, Cost: 1)                  evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, 0, 0, Ar_3)) [ Ar_0 >= 1 ]

		(Comp: 2, Cost: 1)                     evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]

		(Comp: 4*Ar_0^2 + 17*Ar_0, Cost: 1)    evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]

		(Comp: 3*Ar_0, Cost: 1)                evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]

		(Comp: ?, Cost: 1)                     evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]

		(Comp: ?, Cost: 1)                     evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]

		(Comp: ?, Cost: 1)                     evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))

		(Comp: ?, Cost: 1)                     evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))

		(Comp: 2, Cost: 1)                     evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 = 0 ]

		(Comp: 3*Ar_0, Cost: 1)                evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]

		(Comp: 3*Ar_0, Cost: 1)                evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]

		(Comp: 3*Ar_0, Cost: 1)                evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))

		(Comp: 2, Cost: 1)                     evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestop(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 0)                     koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Repeatedly propagating knowledge in problem 6 produces the following problem:

7:	T:

		(Comp: 1, Cost: 1)                     evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                     evalrealbubbleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))

		(Comp: Ar_0, Cost: 1)                  evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, 0, 0, Ar_3)) [ Ar_0 >= 1 ]

		(Comp: 2, Cost: 1)                     evalrealbubblebb7in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]

		(Comp: 4*Ar_0^2 + 17*Ar_0, Cost: 1)    evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]

		(Comp: 3*Ar_0, Cost: 1)                evalrealbubblebb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 ]

		(Comp: 4*Ar_0^2 + 17*Ar_0, Cost: 1)    evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= F + 1 ]

		(Comp: 4*Ar_0^2 + 17*Ar_0, Cost: 1)    evalrealbubblebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ E >= F ]

		(Comp: 4*Ar_0^2 + 17*Ar_0, Cost: 1)    evalrealbubblebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, 1))

		(Comp: 8*Ar_0^2 + 34*Ar_0, Cost: 1)    evalrealbubblebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb4in(Ar_0, Ar_1 + 1, Ar_3, Ar_3))

		(Comp: 2, Cost: 1)                     evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 = 0 ]

		(Comp: 3*Ar_0, Cost: 1)                evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 + 1 ]

		(Comp: 3*Ar_0, Cost: 1)                evalrealbubblebb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]

		(Comp: 3*Ar_0, Cost: 1)                evalrealbubblebb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblebb7in(Ar_0 - 1, Ar_1, Ar_2, Ar_3))

		(Comp: 2, Cost: 1)                     evalrealbubblereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestop(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 0)                     koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalrealbubblestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 115*Ar_0 + 24*Ar_0^2 + 8



Time: 0.150 sec (SMT: 0.116 sec)


----------------------------------------

(2)
BOUNDS(1, n^2)

----------------------------------------

(3) Loat Proof (FINISHED)


### Pre-processing the ITS problem ###



Initial linear ITS problem

   Start location: evalrealbubblestart

      0: evalrealbubblestart -> evalrealbubbleentryin : [], cost: 1

      1: evalrealbubbleentryin -> evalrealbubblebb7in : A'=-1+A, [], cost: 1

      2: evalrealbubblebb7in -> evalrealbubblebb4in : B'=0, C'=0, [ A>=1 ], cost: 1

      3: evalrealbubblebb7in -> evalrealbubblereturnin : [ 0>=A ], cost: 1

      4: evalrealbubblebb4in -> evalrealbubblebb1in : [ A>=1+B ], cost: 1

      5: evalrealbubblebb4in -> evalrealbubblebb5in : [ B>=A ], cost: 1

      6: evalrealbubblebb1in -> evalrealbubblebb2in : [ free_1>=1+free ], cost: 1

      7: evalrealbubblebb1in -> evalrealbubblebb3in : D'=C, [ free_3>=free_2 ], cost: 1

      8: evalrealbubblebb2in -> evalrealbubblebb3in : D'=1, [], cost: 1

      9: evalrealbubblebb3in -> evalrealbubblebb4in : B'=1+B, C'=D, [], cost: 1

     10: evalrealbubblebb5in -> evalrealbubblereturnin : [ C==0 ], cost: 1

     11: evalrealbubblebb5in -> evalrealbubblebb6in : [ 0>=1+C ], cost: 1

     12: evalrealbubblebb5in -> evalrealbubblebb6in : [ C>=1 ], cost: 1

     13: evalrealbubblebb6in -> evalrealbubblebb7in : A'=-1+A, [], cost: 1

     14: evalrealbubblereturnin -> evalrealbubblestop : [], cost: 1



Checking for constant complexity:

   The following rule is satisfiable with cost >= 1, yielding constant complexity:

      0: evalrealbubblestart -> evalrealbubbleentryin : [], cost: 1



Removed unreachable and leaf rules:

   Start location: evalrealbubblestart

      0: evalrealbubblestart -> evalrealbubbleentryin : [], cost: 1

      1: evalrealbubbleentryin -> evalrealbubblebb7in : A'=-1+A, [], cost: 1

      2: evalrealbubblebb7in -> evalrealbubblebb4in : B'=0, C'=0, [ A>=1 ], cost: 1

      4: evalrealbubblebb4in -> evalrealbubblebb1in : [ A>=1+B ], cost: 1

      5: evalrealbubblebb4in -> evalrealbubblebb5in : [ B>=A ], cost: 1

      6: evalrealbubblebb1in -> evalrealbubblebb2in : [ free_1>=1+free ], cost: 1

      7: evalrealbubblebb1in -> evalrealbubblebb3in : D'=C, [ free_3>=free_2 ], cost: 1

      8: evalrealbubblebb2in -> evalrealbubblebb3in : D'=1, [], cost: 1

      9: evalrealbubblebb3in -> evalrealbubblebb4in : B'=1+B, C'=D, [], cost: 1

     11: evalrealbubblebb5in -> evalrealbubblebb6in : [ 0>=1+C ], cost: 1

     12: evalrealbubblebb5in -> evalrealbubblebb6in : [ C>=1 ], cost: 1

     13: evalrealbubblebb6in -> evalrealbubblebb7in : A'=-1+A, [], cost: 1



Simplified all rules, resulting in:

   Start location: evalrealbubblestart

      0: evalrealbubblestart -> evalrealbubbleentryin : [], cost: 1

      1: evalrealbubbleentryin -> evalrealbubblebb7in : A'=-1+A, [], cost: 1

      2: evalrealbubblebb7in -> evalrealbubblebb4in : B'=0, C'=0, [ A>=1 ], cost: 1

      4: evalrealbubblebb4in -> evalrealbubblebb1in : [ A>=1+B ], cost: 1

      5: evalrealbubblebb4in -> evalrealbubblebb5in : [ B>=A ], cost: 1

      6: evalrealbubblebb1in -> evalrealbubblebb2in : [], cost: 1

      7: evalrealbubblebb1in -> evalrealbubblebb3in : D'=C, [], cost: 1

      8: evalrealbubblebb2in -> evalrealbubblebb3in : D'=1, [], cost: 1

      9: evalrealbubblebb3in -> evalrealbubblebb4in : B'=1+B, C'=D, [], cost: 1

     11: evalrealbubblebb5in -> evalrealbubblebb6in : [ 0>=1+C ], cost: 1

     12: evalrealbubblebb5in -> evalrealbubblebb6in : [ C>=1 ], cost: 1

     13: evalrealbubblebb6in -> evalrealbubblebb7in : A'=-1+A, [], cost: 1



### Simplification by acceleration and chaining ###



Eliminated locations (on linear paths):

   Start location: evalrealbubblestart

     15: evalrealbubblestart -> evalrealbubblebb7in : A'=-1+A, [], cost: 2

      2: evalrealbubblebb7in -> evalrealbubblebb4in : B'=0, C'=0, [ A>=1 ], cost: 1

      4: evalrealbubblebb4in -> evalrealbubblebb1in : [ A>=1+B ], cost: 1

      5: evalrealbubblebb4in -> evalrealbubblebb5in : [ B>=A ], cost: 1

      7: evalrealbubblebb1in -> evalrealbubblebb3in : D'=C, [], cost: 1

     16: evalrealbubblebb1in -> evalrealbubblebb3in : D'=1, [], cost: 2

      9: evalrealbubblebb3in -> evalrealbubblebb4in : B'=1+B, C'=D, [], cost: 1

     11: evalrealbubblebb5in -> evalrealbubblebb6in : [ 0>=1+C ], cost: 1

     12: evalrealbubblebb5in -> evalrealbubblebb6in : [ C>=1 ], cost: 1

     13: evalrealbubblebb6in -> evalrealbubblebb7in : A'=-1+A, [], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: evalrealbubblestart

     15: evalrealbubblestart -> evalrealbubblebb7in : A'=-1+A, [], cost: 2

      2: evalrealbubblebb7in -> evalrealbubblebb4in : B'=0, C'=0, [ A>=1 ], cost: 1

     17: evalrealbubblebb4in -> evalrealbubblebb3in : D'=C, [ A>=1+B ], cost: 2

     18: evalrealbubblebb4in -> evalrealbubblebb3in : D'=1, [ A>=1+B ], cost: 3

     19: evalrealbubblebb4in -> evalrealbubblebb6in : [ B>=A && 0>=1+C ], cost: 2

     20: evalrealbubblebb4in -> evalrealbubblebb6in : [ B>=A && C>=1 ], cost: 2

      9: evalrealbubblebb3in -> evalrealbubblebb4in : B'=1+B, C'=D, [], cost: 1

     13: evalrealbubblebb6in -> evalrealbubblebb7in : A'=-1+A, [], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: evalrealbubblestart

     15: evalrealbubblestart -> evalrealbubblebb7in : A'=-1+A, [], cost: 2

      2: evalrealbubblebb7in -> evalrealbubblebb4in : B'=0, C'=0, [ A>=1 ], cost: 1

     21: evalrealbubblebb4in -> evalrealbubblebb4in : B'=1+B, C'=C, D'=C, [ A>=1+B ], cost: 3

     22: evalrealbubblebb4in -> evalrealbubblebb4in : B'=1+B, C'=1, D'=1, [ A>=1+B ], cost: 4

     23: evalrealbubblebb4in -> evalrealbubblebb7in : A'=-1+A, [ B>=A && 0>=1+C ], cost: 3

     24: evalrealbubblebb4in -> evalrealbubblebb7in : A'=-1+A, [ B>=A && C>=1 ], cost: 3



Accelerating simple loops of location 3.

   Simplified some of the simple loops (and removed duplicate rules).

   Accelerating the following rules:

     21: evalrealbubblebb4in -> evalrealbubblebb4in : B'=1+B, D'=C, [ A>=1+B ], cost: 3

     22: evalrealbubblebb4in -> evalrealbubblebb4in : B'=1+B, C'=1, D'=1, [ A>=1+B ], cost: 4



   Accelerated rule 21 with metering function A-B, yielding the new rule 25.

   Accelerated rule 22 with metering function A-B, yielding the new rule 26.

   Removing the simple loops: 21 22.



Accelerated all simple loops using metering functions (where possible):

   Start location: evalrealbubblestart

     15: evalrealbubblestart -> evalrealbubblebb7in : A'=-1+A, [], cost: 2

      2: evalrealbubblebb7in -> evalrealbubblebb4in : B'=0, C'=0, [ A>=1 ], cost: 1

     23: evalrealbubblebb4in -> evalrealbubblebb7in : A'=-1+A, [ B>=A && 0>=1+C ], cost: 3

     24: evalrealbubblebb4in -> evalrealbubblebb7in : A'=-1+A, [ B>=A && C>=1 ], cost: 3

     25: evalrealbubblebb4in -> evalrealbubblebb4in : B'=A, D'=C, [ A>=1+B ], cost: 3*A-3*B

     26: evalrealbubblebb4in -> evalrealbubblebb4in : B'=A, C'=1, D'=1, [ A>=1+B ], cost: 4*A-4*B



Chained accelerated rules (with incoming rules):

   Start location: evalrealbubblestart

     15: evalrealbubblestart -> evalrealbubblebb7in : A'=-1+A, [], cost: 2

      2: evalrealbubblebb7in -> evalrealbubblebb4in : B'=0, C'=0, [ A>=1 ], cost: 1

     27: evalrealbubblebb7in -> evalrealbubblebb4in : B'=A, C'=0, D'=0, [ A>=1 ], cost: 1+3*A

     28: evalrealbubblebb7in -> evalrealbubblebb4in : B'=A, C'=1, D'=1, [ A>=1 ], cost: 1+4*A

     23: evalrealbubblebb4in -> evalrealbubblebb7in : A'=-1+A, [ B>=A && 0>=1+C ], cost: 3

     24: evalrealbubblebb4in -> evalrealbubblebb7in : A'=-1+A, [ B>=A && C>=1 ], cost: 3



Eliminated locations (on tree-shaped paths):

   Start location: evalrealbubblestart

     15: evalrealbubblestart -> evalrealbubblebb7in : A'=-1+A, [], cost: 2

     29: evalrealbubblebb7in -> evalrealbubblebb7in : A'=-1+A, B'=A, C'=1, D'=1, [ A>=1 ], cost: 4+4*A

     30: evalrealbubblebb7in -> [12] : [ A>=1 ], cost: 1+3*A

     31: evalrealbubblebb7in -> [12] : [ A>=1 ], cost: 1+4*A



Accelerating simple loops of location 2.

   Accelerating the following rules:

     29: evalrealbubblebb7in -> evalrealbubblebb7in : A'=-1+A, B'=A, C'=1, D'=1, [ A>=1 ], cost: 4+4*A



   Accelerated rule 29 with metering function A, yielding the new rule 32.

   Removing the simple loops: 29.



Accelerated all simple loops using metering functions (where possible):

   Start location: evalrealbubblestart

     15: evalrealbubblestart -> evalrealbubblebb7in : A'=-1+A, [], cost: 2

     30: evalrealbubblebb7in -> [12] : [ A>=1 ], cost: 1+3*A

     31: evalrealbubblebb7in -> [12] : [ A>=1 ], cost: 1+4*A

     32: evalrealbubblebb7in -> evalrealbubblebb7in : A'=0, B'=1, C'=1, D'=1, [ A>=1 ], cost: 6*A+2*A^2



Chained accelerated rules (with incoming rules):

   Start location: evalrealbubblestart

     15: evalrealbubblestart -> evalrealbubblebb7in : A'=-1+A, [], cost: 2

     33: evalrealbubblestart -> evalrealbubblebb7in : A'=0, B'=1, C'=1, D'=1, [ -1+A>=1 ], cost: -4+2*(-1+A)^2+6*A

     30: evalrealbubblebb7in -> [12] : [ A>=1 ], cost: 1+3*A

     31: evalrealbubblebb7in -> [12] : [ A>=1 ], cost: 1+4*A



Eliminated locations (on tree-shaped paths):

   Start location: evalrealbubblestart

     34: evalrealbubblestart -> [12] : A'=-1+A, [ -1+A>=1 ], cost: 3*A

     35: evalrealbubblestart -> [12] : A'=-1+A, [ -1+A>=1 ], cost: -1+4*A

     36: evalrealbubblestart -> [14] : [ -1+A>=1 ], cost: -4+2*(-1+A)^2+6*A



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: evalrealbubblestart

     34: evalrealbubblestart -> [12] : A'=-1+A, [ -1+A>=1 ], cost: 3*A

     35: evalrealbubblestart -> [12] : A'=-1+A, [ -1+A>=1 ], cost: -1+4*A

     36: evalrealbubblestart -> [14] : [ -1+A>=1 ], cost: -4+2*(-1+A)^2+6*A



Computing asymptotic complexity for rule 34

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -1+A (+/+!), 3*A (+) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {A==n}

      resulting limit problem:

      [solved]



   Solution:

      A / n

   Resulting cost 3*n has complexity: Poly(n^1)



Found new complexity Poly(n^1).



Computing asymptotic complexity for rule 36

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -1+A (+/+!), -2+2*A+2*A^2 (+) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {A==n}

      resulting limit problem:

      [solved]



   Solution:

      A / n

   Resulting cost -2+2*n^2+2*n has complexity: Poly(n^2)



Found new complexity Poly(n^2).



Obtained the following overall complexity (w.r.t. the length of the input n):

   Complexity:  Poly(n^2)

   Cpx degree:  2

   Solved cost: -2+2*n^2+2*n

   Rule cost:   -4+2*(-1+A)^2+6*A

   Rule guard:  [ -1+A>=1 ]



WORST_CASE(Omega(n^2),?)


----------------------------------------

(4)
BOUNDS(n^2, INF)