/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1, n^1).

(0) CpxIntTrs
(1) Koat Proof [FINISHED, 143 ms]
(2) BOUNDS(1, n^1)
(3) Loat Proof [FINISHED, 518 ms]
(4) BOUNDS(n^1, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
evalSimpleMultiplestart(A, B, C, D) -> Com_1(evalSimpleMultipleentryin(A, B, C, D)) :|: TRUE
evalSimpleMultipleentryin(A, B, C, D) -> Com_1(evalSimpleMultiplebb3in(0, 0, C, D)) :|: TRUE
evalSimpleMultiplebb3in(A, B, C, D) -> Com_1(evalSimpleMultiplebbin(A, B, C, D)) :|: C >= B + 1
evalSimpleMultiplebb3in(A, B, C, D) -> Com_1(evalSimpleMultiplereturnin(A, B, C, D)) :|: B >= C
evalSimpleMultiplebbin(A, B, C, D) -> Com_1(evalSimpleMultiplebb1in(A, B, C, D)) :|: D >= A + 1
evalSimpleMultiplebbin(A, B, C, D) -> Com_1(evalSimpleMultiplebb2in(A, B, C, D)) :|: A >= D
evalSimpleMultiplebb1in(A, B, C, D) -> Com_1(evalSimpleMultiplebb3in(A + 1, B, C, D)) :|: TRUE
evalSimpleMultiplebb2in(A, B, C, D) -> Com_1(evalSimpleMultiplebb3in(A, B + 1, C, D)) :|: TRUE
evalSimpleMultiplereturnin(A, B, C, D) -> Com_1(evalSimpleMultiplestop(A, B, C, D)) :|: TRUE

The start-symbols are:[evalSimpleMultiplestart_4]


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

(1) Koat Proof (FINISHED)
YES(?, 6*Ar_2 + 3*Ar_3 + 7)



Initial complexity problem:

1:	T:

		(Comp: ?, Cost: 1)    evalSimpleMultiplestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultipleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(0, 0, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ]

		(Comp: ?, Cost: 1)    evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ]

		(Comp: ?, Cost: 1)    evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 ]

		(Comp: ?, Cost: 1)    evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0, Ar_1 + 1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplestop(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplestart(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)    evalSimpleMultiplestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalSimpleMultipleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(0, 0, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ]

		(Comp: ?, Cost: 1)    evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ]

		(Comp: ?, Cost: 1)    evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 ]

		(Comp: ?, Cost: 1)    evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0, Ar_1 + 1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplestop(Ar_0, Ar_1, Ar_2, Ar_3))

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

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalSimpleMultiplestart) = 2

	Pol(evalSimpleMultipleentryin) = 2

	Pol(evalSimpleMultiplebb3in) = 2

	Pol(evalSimpleMultiplebbin) = 2

	Pol(evalSimpleMultiplereturnin) = 1

	Pol(evalSimpleMultiplebb1in) = 2

	Pol(evalSimpleMultiplebb2in) = 2

	Pol(evalSimpleMultiplestop) = 0

	Pol(koat_start) = 2

orients all transitions weakly and the transitions

	evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplestop(Ar_0, Ar_1, Ar_2, Ar_3))

	evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ]

strictly and produces the following problem:

3:	T:

		(Comp: 1, Cost: 1)    evalSimpleMultiplestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalSimpleMultipleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(0, 0, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ]

		(Comp: 2, Cost: 1)    evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ]

		(Comp: ?, Cost: 1)    evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 ]

		(Comp: ?, Cost: 1)    evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0, Ar_1 + 1, Ar_2, Ar_3))

		(Comp: 2, Cost: 1)    evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplestop(Ar_0, Ar_1, Ar_2, Ar_3))

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

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalSimpleMultiplestart) = V_4

	Pol(evalSimpleMultipleentryin) = V_4

	Pol(evalSimpleMultiplebb3in) = -V_1 + V_4

	Pol(evalSimpleMultiplebbin) = -V_1 + V_4

	Pol(evalSimpleMultiplereturnin) = -V_1 + V_4

	Pol(evalSimpleMultiplebb1in) = -V_1 + V_4 - 1

	Pol(evalSimpleMultiplebb2in) = -V_1 + V_4

	Pol(evalSimpleMultiplestop) = -V_1 + V_4

	Pol(koat_start) = V_4

orients all transitions weakly and the transition

	evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ]

strictly and produces the following problem:

4:	T:

		(Comp: 1, Cost: 1)       evalSimpleMultiplestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalSimpleMultipleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(0, 0, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)       evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ]

		(Comp: 2, Cost: 1)       evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ]

		(Comp: Ar_3, Cost: 1)    evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)       evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 ]

		(Comp: ?, Cost: 1)       evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)       evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0, Ar_1 + 1, Ar_2, Ar_3))

		(Comp: 2, Cost: 1)       evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplestop(Ar_0, Ar_1, Ar_2, Ar_3))

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

	start location:	koat_start

	leaf cost:	0



Repeatedly propagating knowledge in problem 4 produces the following problem:

5:	T:

		(Comp: 1, Cost: 1)       evalSimpleMultiplestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalSimpleMultipleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(0, 0, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)       evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ]

		(Comp: 2, Cost: 1)       evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ]

		(Comp: Ar_3, Cost: 1)    evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)       evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 ]

		(Comp: Ar_3, Cost: 1)    evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)       evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0, Ar_1 + 1, Ar_2, Ar_3))

		(Comp: 2, Cost: 1)       evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplestop(Ar_0, Ar_1, Ar_2, Ar_3))

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

	start location:	koat_start

	leaf cost:	0



Applied AI with 'oct' on problem 5 to obtain the following invariants:

  For symbol evalSimpleMultiplebb1in: X_4 - 1 >= 0 /\ X_3 + X_4 - 2 >= 0 /\ X_2 + X_4 - 1 >= 0 /\ X_1 + X_4 - 1 >= 0 /\ -X_1 + X_4 - 1 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ -X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0

  For symbol evalSimpleMultiplebb2in: X_1 - X_4 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ -X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0

  For symbol evalSimpleMultiplebb3in: X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0

  For symbol evalSimpleMultiplebbin: X_3 - 1 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ -X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0

  For symbol evalSimpleMultiplereturnin: X_2 - X_3 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0





This yielded the following problem:

6:	T:

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

		(Comp: 2, Cost: 1)       evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplestop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ]

		(Comp: ?, Cost: 1)       evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_0 - Ar_3 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ]

		(Comp: Ar_3, Cost: 1)    evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ Ar_1 + Ar_3 - 1 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ]

		(Comp: ?, Cost: 1)       evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= Ar_3 ]

		(Comp: Ar_3, Cost: 1)    evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_0 + 1 ]

		(Comp: 2, Cost: 1)       evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_2 ]

		(Comp: ?, Cost: 1)       evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_1 + 1 ]

		(Comp: 1, Cost: 1)       evalSimpleMultipleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(0, 0, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalSimpleMultiplestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleentryin(Ar_0, Ar_1, Ar_2, Ar_3))

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(koat_start) = 2*V_3

	Pol(evalSimpleMultiplestart) = 2*V_3

	Pol(evalSimpleMultiplereturnin) = -2*V_2 + 2*V_3

	Pol(evalSimpleMultiplestop) = -2*V_2 + 2*V_3

	Pol(evalSimpleMultiplebb2in) = -2*V_2 + 2*V_3 - 1

	Pol(evalSimpleMultiplebb3in) = -2*V_2 + 2*V_3

	Pol(evalSimpleMultiplebb1in) = -2*V_2 + 2*V_3

	Pol(evalSimpleMultiplebbin) = -2*V_2 + 2*V_3

	Pol(evalSimpleMultipleentryin) = 2*V_3

orients all transitions weakly and the transitions

	evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= Ar_3 ]

	evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_0 - Ar_3 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ]

strictly and produces the following problem:

7:	T:

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

		(Comp: 2, Cost: 1)         evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplestop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ]

		(Comp: 2*Ar_2, Cost: 1)    evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_0 - Ar_3 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ]

		(Comp: Ar_3, Cost: 1)      evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ Ar_1 + Ar_3 - 1 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ]

		(Comp: 2*Ar_2, Cost: 1)    evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= Ar_3 ]

		(Comp: Ar_3, Cost: 1)      evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_0 + 1 ]

		(Comp: 2, Cost: 1)         evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_2 ]

		(Comp: ?, Cost: 1)         evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_1 + 1 ]

		(Comp: 1, Cost: 1)         evalSimpleMultipleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(0, 0, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)         evalSimpleMultiplestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleentryin(Ar_0, Ar_1, Ar_2, Ar_3))

	start location:	koat_start

	leaf cost:	0



Repeatedly propagating knowledge in problem 7 produces the following problem:

8:	T:

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

		(Comp: 2, Cost: 1)                    evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplestop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ]

		(Comp: 2*Ar_2, Cost: 1)               evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_0 - Ar_3 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ]

		(Comp: Ar_3, Cost: 1)                 evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ Ar_1 + Ar_3 - 1 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ]

		(Comp: 2*Ar_2, Cost: 1)               evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= Ar_3 ]

		(Comp: Ar_3, Cost: 1)                 evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_0 + 1 ]

		(Comp: 2, Cost: 1)                    evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplereturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_2 ]

		(Comp: Ar_3 + 2*Ar_2 + 1, Cost: 1)    evalSimpleMultiplebb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_1 + 1 ]

		(Comp: 1, Cost: 1)                    evalSimpleMultipleentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultiplebb3in(0, 0, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                    evalSimpleMultiplestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleentryin(Ar_0, Ar_1, Ar_2, Ar_3))

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 6*Ar_2 + 3*Ar_3 + 7



Time: 0.194 sec (SMT: 0.155 sec)


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

(2)
BOUNDS(1, n^1)

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

(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: evalSimpleMultiplestart

      0: evalSimpleMultiplestart -> evalSimpleMultipleentryin : [], cost: 1

      1: evalSimpleMultipleentryin -> evalSimpleMultiplebb3in : A'=0, B'=0, [], cost: 1

      2: evalSimpleMultiplebb3in -> evalSimpleMultiplebbin : [ C>=1+B ], cost: 1

      3: evalSimpleMultiplebb3in -> evalSimpleMultiplereturnin : [ B>=C ], cost: 1

      4: evalSimpleMultiplebbin -> evalSimpleMultiplebb1in : [ D>=1+A ], cost: 1

      5: evalSimpleMultiplebbin -> evalSimpleMultiplebb2in : [ A>=D ], cost: 1

      6: evalSimpleMultiplebb1in -> evalSimpleMultiplebb3in : A'=1+A, [], cost: 1

      7: evalSimpleMultiplebb2in -> evalSimpleMultiplebb3in : B'=1+B, [], cost: 1

      8: evalSimpleMultiplereturnin -> evalSimpleMultiplestop : [], cost: 1



Checking for constant complexity:

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

      0: evalSimpleMultiplestart -> evalSimpleMultipleentryin : [], cost: 1



Removed unreachable and leaf rules:

   Start location: evalSimpleMultiplestart

      0: evalSimpleMultiplestart -> evalSimpleMultipleentryin : [], cost: 1

      1: evalSimpleMultipleentryin -> evalSimpleMultiplebb3in : A'=0, B'=0, [], cost: 1

      2: evalSimpleMultiplebb3in -> evalSimpleMultiplebbin : [ C>=1+B ], cost: 1

      4: evalSimpleMultiplebbin -> evalSimpleMultiplebb1in : [ D>=1+A ], cost: 1

      5: evalSimpleMultiplebbin -> evalSimpleMultiplebb2in : [ A>=D ], cost: 1

      6: evalSimpleMultiplebb1in -> evalSimpleMultiplebb3in : A'=1+A, [], cost: 1

      7: evalSimpleMultiplebb2in -> evalSimpleMultiplebb3in : B'=1+B, [], cost: 1



### Simplification by acceleration and chaining ###



Eliminated locations (on linear paths):

   Start location: evalSimpleMultiplestart

      9: evalSimpleMultiplestart -> evalSimpleMultiplebb3in : A'=0, B'=0, [], cost: 2

      2: evalSimpleMultiplebb3in -> evalSimpleMultiplebbin : [ C>=1+B ], cost: 1

     10: evalSimpleMultiplebbin -> evalSimpleMultiplebb3in : A'=1+A, [ D>=1+A ], cost: 2

     11: evalSimpleMultiplebbin -> evalSimpleMultiplebb3in : B'=1+B, [ A>=D ], cost: 2



Eliminated locations (on tree-shaped paths):

   Start location: evalSimpleMultiplestart

      9: evalSimpleMultiplestart -> evalSimpleMultiplebb3in : A'=0, B'=0, [], cost: 2

     12: evalSimpleMultiplebb3in -> evalSimpleMultiplebb3in : A'=1+A, [ C>=1+B && D>=1+A ], cost: 3

     13: evalSimpleMultiplebb3in -> evalSimpleMultiplebb3in : B'=1+B, [ C>=1+B && A>=D ], cost: 3



Accelerating simple loops of location 2.

   Accelerating the following rules:

     12: evalSimpleMultiplebb3in -> evalSimpleMultiplebb3in : A'=1+A, [ C>=1+B && D>=1+A ], cost: 3

     13: evalSimpleMultiplebb3in -> evalSimpleMultiplebb3in : B'=1+B, [ C>=1+B && A>=D ], cost: 3



   Accelerated rule 12 with metering function D-A, yielding the new rule 14.

   Accelerated rule 13 with metering function C-B, yielding the new rule 15.

   Removing the simple loops: 12 13.



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

   Start location: evalSimpleMultiplestart

      9: evalSimpleMultiplestart -> evalSimpleMultiplebb3in : A'=0, B'=0, [], cost: 2

     14: evalSimpleMultiplebb3in -> evalSimpleMultiplebb3in : A'=D, [ C>=1+B && D>=1+A ], cost: 3*D-3*A

     15: evalSimpleMultiplebb3in -> evalSimpleMultiplebb3in : B'=C, [ C>=1+B && A>=D ], cost: 3*C-3*B



Chained accelerated rules (with incoming rules):

   Start location: evalSimpleMultiplestart

      9: evalSimpleMultiplestart -> evalSimpleMultiplebb3in : A'=0, B'=0, [], cost: 2

     16: evalSimpleMultiplestart -> evalSimpleMultiplebb3in : A'=D, B'=0, [ C>=1 && D>=1 ], cost: 2+3*D

     17: evalSimpleMultiplestart -> evalSimpleMultiplebb3in : A'=0, B'=C, [ C>=1 && 0>=D ], cost: 2+3*C



Removed unreachable locations (and leaf rules with constant cost):

   Start location: evalSimpleMultiplestart

     16: evalSimpleMultiplestart -> evalSimpleMultiplebb3in : A'=D, B'=0, [ C>=1 && D>=1 ], cost: 2+3*D

     17: evalSimpleMultiplestart -> evalSimpleMultiplebb3in : A'=0, B'=C, [ C>=1 && 0>=D ], cost: 2+3*C



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: evalSimpleMultiplestart

     16: evalSimpleMultiplestart -> evalSimpleMultiplebb3in : A'=D, B'=0, [ C>=1 && D>=1 ], cost: 2+3*D

     17: evalSimpleMultiplestart -> evalSimpleMultiplebb3in : A'=0, B'=C, [ C>=1 && 0>=D ], cost: 2+3*C



Computing asymptotic complexity for rule 16

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      C (+/+!), D (+/+!), 2+3*D (+) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {C==1,D==n}

      resulting limit problem:

      [solved]



   Solution:

      C / 1

      D / n

   Resulting cost 2+3*n has complexity: Poly(n^1)



Found new complexity Poly(n^1).



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

   Complexity:  Poly(n^1)

   Cpx degree:  1

   Solved cost: 2+3*n

   Rule cost:   2+3*D

   Rule guard:  [ C>=1 && D>=1 ]



WORST_CASE(Omega(n^1),?)


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

(4)
BOUNDS(n^1, INF)