/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 139 ms]
(2) BOUNDS(1, n^2)
(3) Loat Proof [FINISHED, 625 ms]
(4) BOUNDS(n^2, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
evalSimpleMultipleDepstart(A, B, C, D) -> Com_1(evalSimpleMultipleDepentryin(A, B, C, D)) :|: TRUE
evalSimpleMultipleDepentryin(A, B, C, D) -> Com_1(evalSimpleMultipleDepbb3in(0, 0, C, D)) :|: TRUE
evalSimpleMultipleDepbb3in(A, B, C, D) -> Com_1(evalSimpleMultipleDepbbin(A, B, C, D)) :|: C >= B + 1
evalSimpleMultipleDepbb3in(A, B, C, D) -> Com_1(evalSimpleMultipleDepreturnin(A, B, C, D)) :|: B >= C
evalSimpleMultipleDepbbin(A, B, C, D) -> Com_1(evalSimpleMultipleDepbb1in(A, B, C, D)) :|: D >= A + 1
evalSimpleMultipleDepbbin(A, B, C, D) -> Com_1(evalSimpleMultipleDepbb2in(A, B, C, D)) :|: A >= D
evalSimpleMultipleDepbb1in(A, B, C, D) -> Com_1(evalSimpleMultipleDepbb3in(A + 1, B, C, D)) :|: TRUE
evalSimpleMultipleDepbb2in(A, B, C, D) -> Com_1(evalSimpleMultipleDepbb3in(0, B + 1, C, D)) :|: TRUE
evalSimpleMultipleDepreturnin(A, B, C, D) -> Com_1(evalSimpleMultipleDepstop(A, B, C, D)) :|: TRUE

The start-symbols are:[evalSimpleMultipleDepstart_4]


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(1) Koat Proof (FINISHED)
YES(?, 6*Ar_2 + 6*Ar_3 + 12*Ar_2*Ar_3 + 7)



Initial complexity problem:

1:	T:

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(0, 0, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ]

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ]

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 ]

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(0, Ar_1 + 1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3))

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

		(Comp: 1, Cost: 1)    evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(0, 0, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ]

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ]

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 ]

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(0, Ar_1 + 1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3))

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

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalSimpleMultipleDepstart) = 2

	Pol(evalSimpleMultipleDepentryin) = 2

	Pol(evalSimpleMultipleDepbb3in) = 2

	Pol(evalSimpleMultipleDepbbin) = 2

	Pol(evalSimpleMultipleDepreturnin) = 1

	Pol(evalSimpleMultipleDepbb1in) = 2

	Pol(evalSimpleMultipleDepbb2in) = 2

	Pol(evalSimpleMultipleDepstop) = 0

	Pol(koat_start) = 2

orients all transitions weakly and the transitions

	evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3))

	evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ]

strictly and produces the following problem:

3:	T:

		(Comp: 1, Cost: 1)    evalSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(0, 0, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ]

		(Comp: 2, Cost: 1)    evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ]

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 ]

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(0, Ar_1 + 1, Ar_2, Ar_3))

		(Comp: 2, Cost: 1)    evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3))

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

	start location:	koat_start

	leaf cost:	0



Applied AI with 'oct' on problem 3 to obtain the following invariants:

  For symbol evalSimpleMultipleDepbb1in: 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 evalSimpleMultipleDepbb2in: 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 evalSimpleMultipleDepbb3in: X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0

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





This yielded the following problem:

4:	T:

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

		(Comp: 2, Cost: 1)    evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepstop(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)    evalSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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: ?, Cost: 1)    evalSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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)    evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb2in(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: ?, Cost: 1)    evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb1in(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)    evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepreturnin(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)    evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbbin(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)    evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(0, 0, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepentryin(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(evalSimpleMultipleDepstart) = 2*V_3

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

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

	Pol(evalSimpleMultipleDepbb2in) = -2*V_2 + 2*V_3 - 1

	Pol(evalSimpleMultipleDepbb3in) = -2*V_2 + 2*V_3

	Pol(evalSimpleMultipleDepbb1in) = -2*V_2 + 2*V_3

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

	Pol(evalSimpleMultipleDepentryin) = 2*V_3

orients all transitions weakly and the transitions

	evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb2in(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 ]

	evalSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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:

5:	T:

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

		(Comp: 2, Cost: 1)         evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepstop(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)    evalSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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: ?, Cost: 1)         evalSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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)    evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb2in(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: ?, Cost: 1)         evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb1in(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)         evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepreturnin(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)         evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbbin(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)         evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(0, 0, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)         evalSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3))

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalSimpleMultipleDepbbin) = -2*V_1 + 2*V_4

	Pol(evalSimpleMultipleDepbb1in) = -2*V_1 + 2*V_4 - 1

	Pol(evalSimpleMultipleDepbb3in) = -2*V_1 + 2*V_4

and size complexities

	S("evalSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0

	S("evalSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1

	S("evalSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2

	S("evalSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3

	S("evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(0, 0, Ar_2, Ar_3))", 0-0) = 0

	S("evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(0, 0, Ar_2, Ar_3))", 0-1) = 0

	S("evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(0, 0, Ar_2, Ar_3))", 0-2) = Ar_2

	S("evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(0, 0, Ar_2, Ar_3))", 0-3) = Ar_3

	S("evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_2 >= Ar_1 + 1 ]", 0-0) = Ar_3

	S("evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_2 >= Ar_1 + 1 ]", 0-1) = Ar_2

	S("evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_2 >= Ar_1 + 1 ]", 0-2) = Ar_2

	S("evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_2 >= Ar_1 + 1 ]", 0-3) = Ar_3

	S("evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_1 >= Ar_2 ]", 0-0) = 0

	S("evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_1 >= Ar_2 ]", 0-1) = Ar_2

	S("evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_1 >= Ar_2 ]", 0-2) = Ar_2

	S("evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_1 >= Ar_2 ]", 0-3) = Ar_3

	S("evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb1in(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 ]", 0-0) = Ar_3

	S("evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb1in(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 ]", 0-1) = Ar_2

	S("evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb1in(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 ]", 0-2) = Ar_2

	S("evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb1in(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 ]", 0-3) = Ar_3

	S("evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb2in(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 ]", 0-0) = Ar_3

	S("evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb2in(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 ]", 0-1) = Ar_2

	S("evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb2in(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 ]", 0-2) = Ar_2

	S("evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb2in(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 ]", 0-3) = Ar_3

	S("evalSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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 ]", 0-0) = Ar_3

	S("evalSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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 ]", 0-1) = Ar_2

	S("evalSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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 ]", 0-2) = Ar_2

	S("evalSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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 ]", 0-3) = Ar_3

	S("evalSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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 ]", 0-0) = 0

	S("evalSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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 ]", 0-1) = Ar_2

	S("evalSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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 ]", 0-2) = Ar_2

	S("evalSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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 ]", 0-3) = Ar_3

	S("evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-0) = 0

	S("evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-1) = Ar_2

	S("evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-2) = Ar_2

	S("evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-3) = Ar_3

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

orients the transitions

	evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb1in(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 ]

	evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_1 + 1 ]

	evalSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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 ]

weakly and the transitions

	evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb1in(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 ]

	evalSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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 ]

strictly and produces the following problem:

6:	T:

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

		(Comp: 2, Cost: 1)                       evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepstop(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)                  evalSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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: 2*Ar_3 + 4*Ar_2*Ar_3, Cost: 1)    evalSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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)                  evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb2in(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: 2*Ar_3 + 4*Ar_2*Ar_3, Cost: 1)    evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb1in(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)                       evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepreturnin(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)                       evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbbin(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)                       evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(0, 0, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                       evalSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3))

	start location:	koat_start

	leaf cost:	0



Repeatedly propagating knowledge in problem 6 produces the following problem:

7:	T:

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

		(Comp: 2, Cost: 1)                                    evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepstop(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)                               evalSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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: 2*Ar_3 + 4*Ar_2*Ar_3, Cost: 1)                 evalSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(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)                               evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb2in(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: 2*Ar_3 + 4*Ar_2*Ar_3, Cost: 1)                 evalSimpleMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb1in(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)                                    evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_2 ]

		(Comp: 2*Ar_3 + 4*Ar_2*Ar_3 + 2*Ar_2 + 1, Cost: 1)    evalSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbbin(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)                                    evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepbb3in(0, 0, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                                    evalSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3))

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 6*Ar_2 + 6*Ar_3 + 12*Ar_2*Ar_3 + 7



Time: 0.215 sec (SMT: 0.168 sec)


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

(2)
BOUNDS(1, n^2)

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

(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: evalSimpleMultipleDepstart

      0: evalSimpleMultipleDepstart -> evalSimpleMultipleDepentryin : [], cost: 1

      1: evalSimpleMultipleDepentryin -> evalSimpleMultipleDepbb3in : A'=0, B'=0, [], cost: 1

      2: evalSimpleMultipleDepbb3in -> evalSimpleMultipleDepbbin : [ C>=1+B ], cost: 1

      3: evalSimpleMultipleDepbb3in -> evalSimpleMultipleDepreturnin : [ B>=C ], cost: 1

      4: evalSimpleMultipleDepbbin -> evalSimpleMultipleDepbb1in : [ D>=1+A ], cost: 1

      5: evalSimpleMultipleDepbbin -> evalSimpleMultipleDepbb2in : [ A>=D ], cost: 1

      6: evalSimpleMultipleDepbb1in -> evalSimpleMultipleDepbb3in : A'=1+A, [], cost: 1

      7: evalSimpleMultipleDepbb2in -> evalSimpleMultipleDepbb3in : A'=0, B'=1+B, [], cost: 1

      8: evalSimpleMultipleDepreturnin -> evalSimpleMultipleDepstop : [], cost: 1



Checking for constant complexity:

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

      0: evalSimpleMultipleDepstart -> evalSimpleMultipleDepentryin : [], cost: 1



Removed unreachable and leaf rules:

   Start location: evalSimpleMultipleDepstart

      0: evalSimpleMultipleDepstart -> evalSimpleMultipleDepentryin : [], cost: 1

      1: evalSimpleMultipleDepentryin -> evalSimpleMultipleDepbb3in : A'=0, B'=0, [], cost: 1

      2: evalSimpleMultipleDepbb3in -> evalSimpleMultipleDepbbin : [ C>=1+B ], cost: 1

      4: evalSimpleMultipleDepbbin -> evalSimpleMultipleDepbb1in : [ D>=1+A ], cost: 1

      5: evalSimpleMultipleDepbbin -> evalSimpleMultipleDepbb2in : [ A>=D ], cost: 1

      6: evalSimpleMultipleDepbb1in -> evalSimpleMultipleDepbb3in : A'=1+A, [], cost: 1

      7: evalSimpleMultipleDepbb2in -> evalSimpleMultipleDepbb3in : A'=0, B'=1+B, [], cost: 1



### Simplification by acceleration and chaining ###



Eliminated locations (on linear paths):

   Start location: evalSimpleMultipleDepstart

      9: evalSimpleMultipleDepstart -> evalSimpleMultipleDepbb3in : A'=0, B'=0, [], cost: 2

      2: evalSimpleMultipleDepbb3in -> evalSimpleMultipleDepbbin : [ C>=1+B ], cost: 1

     10: evalSimpleMultipleDepbbin -> evalSimpleMultipleDepbb3in : A'=1+A, [ D>=1+A ], cost: 2

     11: evalSimpleMultipleDepbbin -> evalSimpleMultipleDepbb3in : A'=0, B'=1+B, [ A>=D ], cost: 2



Eliminated locations (on tree-shaped paths):

   Start location: evalSimpleMultipleDepstart

      9: evalSimpleMultipleDepstart -> evalSimpleMultipleDepbb3in : A'=0, B'=0, [], cost: 2

     12: evalSimpleMultipleDepbb3in -> evalSimpleMultipleDepbb3in : A'=1+A, [ C>=1+B && D>=1+A ], cost: 3

     13: evalSimpleMultipleDepbb3in -> evalSimpleMultipleDepbb3in : A'=0, B'=1+B, [ C>=1+B && A>=D ], cost: 3



Accelerating simple loops of location 2.

   Accelerating the following rules:

     12: evalSimpleMultipleDepbb3in -> evalSimpleMultipleDepbb3in : A'=1+A, [ C>=1+B && D>=1+A ], cost: 3

     13: evalSimpleMultipleDepbb3in -> evalSimpleMultipleDepbb3in : A'=0, 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 (after strengthening guard), yielding the new rule 15.

   Nested simple loops 13 (outer loop) and 14 (inner loop) with metering function -1+C-B, resulting in the new rules: 16, 17.

   Removing the simple loops: 12 13.



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

   Start location: evalSimpleMultipleDepstart

      9: evalSimpleMultipleDepstart -> evalSimpleMultipleDepbb3in : A'=0, B'=0, [], cost: 2

     14: evalSimpleMultipleDepbb3in -> evalSimpleMultipleDepbb3in : A'=D, [ C>=1+B && D>=1+A ], cost: 3*D-3*A

     15: evalSimpleMultipleDepbb3in -> evalSimpleMultipleDepbb3in : A'=0, B'=C, [ C>=1+B && A>=D && 0>=D ], cost: 3*C-3*B

     16: evalSimpleMultipleDepbb3in -> evalSimpleMultipleDepbb3in : A'=D, B'=-1+C, [ A>=D && C>=2+B && D>=1 ], cost: -3+3*C+3*D*(-1+C-B)-3*B

     17: evalSimpleMultipleDepbb3in -> evalSimpleMultipleDepbb3in : A'=D, B'=-1+C, [ D>=1+A && C>=2+B && D>=1 ], cost: -3+3*C+3*D*(-1+C-B)+3*D-3*A-3*B



Chained accelerated rules (with incoming rules):

   Start location: evalSimpleMultipleDepstart

      9: evalSimpleMultipleDepstart -> evalSimpleMultipleDepbb3in : A'=0, B'=0, [], cost: 2

     18: evalSimpleMultipleDepstart -> evalSimpleMultipleDepbb3in : A'=D, B'=0, [ C>=1 && D>=1 ], cost: 2+3*D

     19: evalSimpleMultipleDepstart -> evalSimpleMultipleDepbb3in : A'=0, B'=C, [ C>=1 && 0>=D ], cost: 2+3*C

     20: evalSimpleMultipleDepstart -> evalSimpleMultipleDepbb3in : A'=D, B'=-1+C, [ D>=1 && C>=2 ], cost: -1+3*D*(-1+C)+3*C+3*D



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

   Start location: evalSimpleMultipleDepstart

     18: evalSimpleMultipleDepstart -> evalSimpleMultipleDepbb3in : A'=D, B'=0, [ C>=1 && D>=1 ], cost: 2+3*D

     19: evalSimpleMultipleDepstart -> evalSimpleMultipleDepbb3in : A'=0, B'=C, [ C>=1 && 0>=D ], cost: 2+3*C

     20: evalSimpleMultipleDepstart -> evalSimpleMultipleDepbb3in : A'=D, B'=-1+C, [ D>=1 && C>=2 ], cost: -1+3*D*(-1+C)+3*C+3*D



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: evalSimpleMultipleDepstart

     18: evalSimpleMultipleDepstart -> evalSimpleMultipleDepbb3in : A'=D, B'=0, [ C>=1 && D>=1 ], cost: 2+3*D

     19: evalSimpleMultipleDepstart -> evalSimpleMultipleDepbb3in : A'=0, B'=C, [ C>=1 && 0>=D ], cost: 2+3*C

     20: evalSimpleMultipleDepstart -> evalSimpleMultipleDepbb3in : A'=D, B'=-1+C, [ D>=1 && C>=2 ], cost: -1+3*D*(-1+C)+3*C+3*D



Computing asymptotic complexity for rule 18

   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).



Computing asymptotic complexity for rule 20

   Solved the limit problem by the following transformations:

      Created initial limit problem:

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



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



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

      resulting limit problem:

      [solved]



   Solution:

      C / n

      D / n

   Resulting cost -1+3*n+3*n^2 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: -1+3*n+3*n^2

   Rule cost:   -1+3*D*(-1+C)+3*C+3*D

   Rule guard:  [ D>=1 && C>=2 ]



WORST_CASE(Omega(n^2),?)


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(4)
BOUNDS(n^2, INF)