/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^1), O(n^1))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1, n^1).

(0) CpxIntTrs
(1) Koat Proof [FINISHED, 334 ms]
(2) BOUNDS(1, n^1)
(3) Loat Proof [FINISHED, 1127 ms]
(4) BOUNDS(n^1, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
evalfstart(A, B, C, D) -> Com_1(evalfentryin(A, B, C, D)) :|: TRUE
evalfentryin(A, B, C, D) -> Com_1(evalfbb6in(0, B, C, D)) :|: TRUE
evalfbb6in(A, B, C, D) -> Com_1(evalfbbin(A, B, C, D)) :|: B >= A + 1
evalfbb6in(A, B, C, D) -> Com_1(evalfreturnin(A, B, C, D)) :|: A >= B
evalfbbin(A, B, C, D) -> Com_1(evalfbb2in(A, B, 0, A + 1)) :|: TRUE
evalfbb2in(A, B, C, D) -> Com_1(evalfbb4in(A, B, C, D)) :|: D >= B
evalfbb2in(A, B, C, D) -> Com_1(evalfbb3in(A, B, C, D)) :|: B >= D + 1
evalfbb3in(A, B, C, D) -> Com_1(evalfbb1in(A, B, C, D)) :|: 0 >= E + 1
evalfbb3in(A, B, C, D) -> Com_1(evalfbb1in(A, B, C, D)) :|: E >= 1
evalfbb3in(A, B, C, D) -> Com_1(evalfbb4in(A, B, C, D)) :|: TRUE
evalfbb1in(A, B, C, D) -> Com_1(evalfbb2in(A, B, C + 1, D + 1)) :|: TRUE
evalfbb4in(A, B, C, D) -> Com_1(evalfbb6in(D - 1, B, C, D)) :|: C >= 1
evalfbb4in(A, B, C, D) -> Com_1(evalfbb6in(D, B, C, D)) :|: 0 >= C
evalfreturnin(A, B, C, D) -> Com_1(evalfstop(A, B, C, D)) :|: TRUE

The start-symbols are:[evalfstart_4]


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(1) Koat Proof (FINISHED)
YES(?, 80*Ar_1 + 6)



Initial complexity problem:

1:	T:

		(Comp: ?, Cost: 1)    evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalfbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    evalfbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0, Ar_0 + 1))

		(Comp: ?, Cost: 1)    evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 + 1 ]

		(Comp: ?, Cost: 1)    evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ]

		(Comp: ?, Cost: 1)    evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= 1 ]

		(Comp: ?, Cost: 1)    evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + 1))

		(Comp: ?, Cost: 1)    evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(Ar_3 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]

		(Comp: ?, Cost: 1)    evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(Ar_3, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ]

		(Comp: ?, Cost: 1)    evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3))

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

	start location:	koat_start

	leaf cost:	0



Repeatedly propagating knowledge in problem 1 produces the following problem:

2:	T:

		(Comp: 1, Cost: 1)    evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalfbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    evalfbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0, Ar_0 + 1))

		(Comp: ?, Cost: 1)    evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 + 1 ]

		(Comp: ?, Cost: 1)    evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ]

		(Comp: ?, Cost: 1)    evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= 1 ]

		(Comp: ?, Cost: 1)    evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + 1))

		(Comp: ?, Cost: 1)    evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(Ar_3 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]

		(Comp: ?, Cost: 1)    evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(Ar_3, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ]

		(Comp: ?, Cost: 1)    evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3))

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

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalfstart) = 2

	Pol(evalfentryin) = 2

	Pol(evalfbb6in) = 2

	Pol(evalfbbin) = 2

	Pol(evalfreturnin) = 1

	Pol(evalfbb2in) = 2

	Pol(evalfbb4in) = 2

	Pol(evalfbb3in) = 2

	Pol(evalfbb1in) = 2

	Pol(evalfstop) = 0

	Pol(koat_start) = 2

orients all transitions weakly and the transitions

	evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3))

	evalfbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]

strictly and produces the following problem:

3:	T:

		(Comp: 1, Cost: 1)    evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalfbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ]

		(Comp: 2, Cost: 1)    evalfbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0, Ar_0 + 1))

		(Comp: ?, Cost: 1)    evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 + 1 ]

		(Comp: ?, Cost: 1)    evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ]

		(Comp: ?, Cost: 1)    evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= 1 ]

		(Comp: ?, Cost: 1)    evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + 1))

		(Comp: ?, Cost: 1)    evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(Ar_3 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]

		(Comp: ?, Cost: 1)    evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(Ar_3, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ]

		(Comp: 2, Cost: 1)    evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3))

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

	start location:	koat_start

	leaf cost:	0



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

  For symbol evalfbb1in: X_2 - X_4 - 1 >= 0 /\ X_4 - 1 >= 0 /\ X_3 + X_4 - 1 >= 0 /\ -X_3 + X_4 - 1 >= 0 /\ X_2 + X_4 - 3 >= 0 /\ X_1 + X_4 - 1 >= 0 /\ -X_1 + X_4 - 1 >= 0 /\ X_2 - X_3 - 2 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 - 2 >= 0 /\ X_1 + X_3 >= 0 /\ X_2 - 2 >= 0 /\ X_1 + X_2 - 2 >= 0 /\ -X_1 + X_2 - 2 >= 0 /\ X_1 >= 0

  For symbol evalfbb2in: X_2 - X_4 >= 0 /\ X_4 - 1 >= 0 /\ X_3 + X_4 - 1 >= 0 /\ -X_3 + X_4 - 1 >= 0 /\ X_2 + X_4 - 2 >= 0 /\ X_1 + X_4 - 1 >= 0 /\ -X_1 + X_4 - 1 >= 0 /\ X_2 - X_3 - 1 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 >= 0 /\ X_2 - 1 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 >= 0

  For symbol evalfbb3in: X_2 - X_4 - 1 >= 0 /\ X_4 - 1 >= 0 /\ X_3 + X_4 - 1 >= 0 /\ -X_3 + X_4 - 1 >= 0 /\ X_2 + X_4 - 3 >= 0 /\ X_1 + X_4 - 1 >= 0 /\ -X_1 + X_4 - 1 >= 0 /\ X_2 - X_3 - 2 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 - 2 >= 0 /\ X_1 + X_3 >= 0 /\ X_2 - 2 >= 0 /\ X_1 + X_2 - 2 >= 0 /\ -X_1 + X_2 - 2 >= 0 /\ X_1 >= 0

  For symbol evalfbb4in: X_2 - X_4 >= 0 /\ X_4 - 1 >= 0 /\ X_3 + X_4 - 1 >= 0 /\ -X_3 + X_4 - 1 >= 0 /\ X_2 + X_4 - 2 >= 0 /\ X_1 + X_4 - 1 >= 0 /\ -X_1 + X_4 - 1 >= 0 /\ X_2 - X_3 - 1 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 >= 0 /\ X_2 - 1 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 >= 0

  For symbol evalfbb6in: X_1 >= 0

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

  For symbol evalfreturnin: 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(evalfstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

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

		(Comp: ?, Cost: 1)    evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ 0 >= Ar_2 ]

		(Comp: ?, Cost: 1)    evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(Ar_3 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= 1 ]

		(Comp: ?, Cost: 1)    evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + 1)) [ Ar_1 - Ar_3 - 1 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 2 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 ]

		(Comp: ?, Cost: 1)    evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 - 1 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 2 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 ]

		(Comp: ?, Cost: 1)    evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 - 1 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 2 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 /\ E >= 1 ]

		(Comp: ?, Cost: 1)    evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 - 1 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 2 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 /\ 0 >= E + 1 ]

		(Comp: ?, Cost: 1)    evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_3 + 1 ]

		(Comp: ?, Cost: 1)    evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0, Ar_0 + 1)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 ]

		(Comp: 2, Cost: 1)    evalfbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\ Ar_0 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalfbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 ]

		(Comp: 1, Cost: 1)    evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3))

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(koat_start) = 8*V_2

	Pol(evalfstart) = 8*V_2

	Pol(evalfreturnin) = -8*V_1 + 8*V_2

	Pol(evalfstop) = -8*V_1 + 8*V_2

	Pol(evalfbb4in) = 8*V_2 + 5*V_3 - 8*V_4 + 4

	Pol(evalfbb6in) = -8*V_1 + 8*V_2

	Pol(evalfbb1in) = 8*V_2 + 5*V_3 - 8*V_4 + 4

	Pol(evalfbb2in) = 8*V_2 + 5*V_3 - 8*V_4 + 6

	Pol(evalfbb3in) = 8*V_2 + 5*V_3 - 8*V_4 + 5

	Pol(evalfbbin) = -8*V_1 + 8*V_2 - 1

	Pol(evalfentryin) = 8*V_2

orients all transitions weakly and the transitions

	evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0, Ar_0 + 1)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 ]

	evalfbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 ]

	evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ 0 >= Ar_2 ]

	evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(Ar_3 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= 1 ]

	evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 - 1 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 2 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 ]

	evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 - 1 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 2 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 /\ E >= 1 ]

	evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 - 1 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 2 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 /\ 0 >= E + 1 ]

	evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_1 ]

	evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_3 + 1 ]

	evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + 1)) [ Ar_1 - Ar_3 - 1 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 2 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 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(evalfstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

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

		(Comp: 8*Ar_1, Cost: 1)    evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ 0 >= Ar_2 ]

		(Comp: 8*Ar_1, Cost: 1)    evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(Ar_3 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= 1 ]

		(Comp: 8*Ar_1, Cost: 1)    evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + 1)) [ Ar_1 - Ar_3 - 1 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 2 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 ]

		(Comp: 8*Ar_1, Cost: 1)    evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 - 1 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 2 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 ]

		(Comp: 8*Ar_1, Cost: 1)    evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 - 1 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 2 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 /\ E >= 1 ]

		(Comp: 8*Ar_1, Cost: 1)    evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 - 1 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 3 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 2 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 /\ 0 >= E + 1 ]

		(Comp: 8*Ar_1, Cost: 1)    evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_3 + 1 ]

		(Comp: 8*Ar_1, Cost: 1)    evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 1 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_1 ]

		(Comp: 8*Ar_1, Cost: 1)    evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0, Ar_0 + 1)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 ]

		(Comp: 2, Cost: 1)         evalfbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\ Ar_0 >= Ar_1 ]

		(Comp: 8*Ar_1, Cost: 1)    evalfbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 ]

		(Comp: 1, Cost: 1)         evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb6in(0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)         evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3))

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 80*Ar_1 + 6



Time: 0.377 sec (SMT: 0.301 sec)


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

(2)
BOUNDS(1, n^1)

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

(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: evalfstart

      0: evalfstart -> evalfentryin : [], cost: 1

      1: evalfentryin -> evalfbb6in : A'=0, [], cost: 1

      2: evalfbb6in -> evalfbbin : [ B>=1+A ], cost: 1

      3: evalfbb6in -> evalfreturnin : [ A>=B ], cost: 1

      4: evalfbbin -> evalfbb2in : C'=0, D'=1+A, [], cost: 1

      5: evalfbb2in -> evalfbb4in : [ D>=B ], cost: 1

      6: evalfbb2in -> evalfbb3in : [ B>=1+D ], cost: 1

      7: evalfbb3in -> evalfbb1in : [ 0>=1+free ], cost: 1

      8: evalfbb3in -> evalfbb1in : [ free_1>=1 ], cost: 1

      9: evalfbb3in -> evalfbb4in : [], cost: 1

     10: evalfbb1in -> evalfbb2in : C'=1+C, D'=1+D, [], cost: 1

     11: evalfbb4in -> evalfbb6in : A'=-1+D, [ C>=1 ], cost: 1

     12: evalfbb4in -> evalfbb6in : A'=D, [ 0>=C ], cost: 1

     13: evalfreturnin -> evalfstop : [], cost: 1



Checking for constant complexity:

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

      0: evalfstart -> evalfentryin : [], cost: 1



Removed unreachable and leaf rules:

   Start location: evalfstart

      0: evalfstart -> evalfentryin : [], cost: 1

      1: evalfentryin -> evalfbb6in : A'=0, [], cost: 1

      2: evalfbb6in -> evalfbbin : [ B>=1+A ], cost: 1

      4: evalfbbin -> evalfbb2in : C'=0, D'=1+A, [], cost: 1

      5: evalfbb2in -> evalfbb4in : [ D>=B ], cost: 1

      6: evalfbb2in -> evalfbb3in : [ B>=1+D ], cost: 1

      7: evalfbb3in -> evalfbb1in : [ 0>=1+free ], cost: 1

      8: evalfbb3in -> evalfbb1in : [ free_1>=1 ], cost: 1

      9: evalfbb3in -> evalfbb4in : [], cost: 1

     10: evalfbb1in -> evalfbb2in : C'=1+C, D'=1+D, [], cost: 1

     11: evalfbb4in -> evalfbb6in : A'=-1+D, [ C>=1 ], cost: 1

     12: evalfbb4in -> evalfbb6in : A'=D, [ 0>=C ], cost: 1



Simplified all rules, resulting in:

   Start location: evalfstart

      0: evalfstart -> evalfentryin : [], cost: 1

      1: evalfentryin -> evalfbb6in : A'=0, [], cost: 1

      2: evalfbb6in -> evalfbbin : [ B>=1+A ], cost: 1

      4: evalfbbin -> evalfbb2in : C'=0, D'=1+A, [], cost: 1

      5: evalfbb2in -> evalfbb4in : [ D>=B ], cost: 1

      6: evalfbb2in -> evalfbb3in : [ B>=1+D ], cost: 1

      8: evalfbb3in -> evalfbb1in : [], cost: 1

      9: evalfbb3in -> evalfbb4in : [], cost: 1

     10: evalfbb1in -> evalfbb2in : C'=1+C, D'=1+D, [], cost: 1

     11: evalfbb4in -> evalfbb6in : A'=-1+D, [ C>=1 ], cost: 1

     12: evalfbb4in -> evalfbb6in : A'=D, [ 0>=C ], cost: 1



### Simplification by acceleration and chaining ###



Eliminated locations (on linear paths):

   Start location: evalfstart

     14: evalfstart -> evalfbb6in : A'=0, [], cost: 2

     15: evalfbb6in -> evalfbb2in : C'=0, D'=1+A, [ B>=1+A ], cost: 2

      5: evalfbb2in -> evalfbb4in : [ D>=B ], cost: 1

      6: evalfbb2in -> evalfbb3in : [ B>=1+D ], cost: 1

      9: evalfbb3in -> evalfbb4in : [], cost: 1

     16: evalfbb3in -> evalfbb2in : C'=1+C, D'=1+D, [], cost: 2

     11: evalfbb4in -> evalfbb6in : A'=-1+D, [ C>=1 ], cost: 1

     12: evalfbb4in -> evalfbb6in : A'=D, [ 0>=C ], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: evalfstart

     14: evalfstart -> evalfbb6in : A'=0, [], cost: 2

     15: evalfbb6in -> evalfbb2in : C'=0, D'=1+A, [ B>=1+A ], cost: 2

     18: evalfbb2in -> evalfbb2in : C'=1+C, D'=1+D, [ B>=1+D ], cost: 3

     19: evalfbb2in -> evalfbb6in : A'=-1+D, [ D>=B && C>=1 ], cost: 2

     20: evalfbb2in -> evalfbb6in : A'=D, [ D>=B && 0>=C ], cost: 2

     21: evalfbb2in -> evalfbb6in : A'=-1+D, [ B>=1+D && C>=1 ], cost: 3

     22: evalfbb2in -> evalfbb6in : A'=D, [ B>=1+D && 0>=C ], cost: 3



Accelerating simple loops of location 4.

   Accelerating the following rules:

     18: evalfbb2in -> evalfbb2in : C'=1+C, D'=1+D, [ B>=1+D ], cost: 3



   Accelerated rule 18 with metering function -D+B, yielding the new rule 23.

   Removing the simple loops: 18.



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

   Start location: evalfstart

     14: evalfstart -> evalfbb6in : A'=0, [], cost: 2

     15: evalfbb6in -> evalfbb2in : C'=0, D'=1+A, [ B>=1+A ], cost: 2

     19: evalfbb2in -> evalfbb6in : A'=-1+D, [ D>=B && C>=1 ], cost: 2

     20: evalfbb2in -> evalfbb6in : A'=D, [ D>=B && 0>=C ], cost: 2

     21: evalfbb2in -> evalfbb6in : A'=-1+D, [ B>=1+D && C>=1 ], cost: 3

     22: evalfbb2in -> evalfbb6in : A'=D, [ B>=1+D && 0>=C ], cost: 3

     23: evalfbb2in -> evalfbb2in : C'=C-D+B, D'=B, [ B>=1+D ], cost: -3*D+3*B



Chained accelerated rules (with incoming rules):

   Start location: evalfstart

     14: evalfstart -> evalfbb6in : A'=0, [], cost: 2

     15: evalfbb6in -> evalfbb2in : C'=0, D'=1+A, [ B>=1+A ], cost: 2

     24: evalfbb6in -> evalfbb2in : C'=-1-A+B, D'=B, [ B>=2+A ], cost: -1-3*A+3*B

     19: evalfbb2in -> evalfbb6in : A'=-1+D, [ D>=B && C>=1 ], cost: 2

     20: evalfbb2in -> evalfbb6in : A'=D, [ D>=B && 0>=C ], cost: 2

     21: evalfbb2in -> evalfbb6in : A'=-1+D, [ B>=1+D && C>=1 ], cost: 3

     22: evalfbb2in -> evalfbb6in : A'=D, [ B>=1+D && 0>=C ], cost: 3



Eliminated locations (on tree-shaped paths):

   Start location: evalfstart

     14: evalfstart -> evalfbb6in : A'=0, [], cost: 2

     25: evalfbb6in -> evalfbb6in : A'=1+A, C'=0, D'=1+A, [ B>=1+A && 1+A>=B ], cost: 4

     26: evalfbb6in -> evalfbb6in : A'=1+A, C'=0, D'=1+A, [ B>=2+A ], cost: 5

     27: evalfbb6in -> evalfbb6in : A'=-1+B, C'=-1-A+B, D'=B, [ B>=2+A ], cost: 1-3*A+3*B

     28: evalfbb6in -> [11] : [ B>=2+A ], cost: -1-3*A+3*B



Accelerating simple loops of location 2.

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

   Accelerating the following rules:

     25: evalfbb6in -> evalfbb6in : A'=1+A, C'=0, D'=1+A, [ 1+A-B==0 ], cost: 4

     26: evalfbb6in -> evalfbb6in : A'=1+A, C'=0, D'=1+A, [ B>=2+A ], cost: 5

     27: evalfbb6in -> evalfbb6in : A'=-1+B, C'=-1-A+B, D'=B, [ B>=2+A ], cost: 1-3*A+3*B



   Accelerated rule 25 with metering function -A+B, yielding the new rule 29.

   Accelerated rule 26 with metering function -1-A+B, yielding the new rule 30.

   Found no metering function for rule 27.

   Removing the simple loops: 25 26.



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

   Start location: evalfstart

     14: evalfstart -> evalfbb6in : A'=0, [], cost: 2

     27: evalfbb6in -> evalfbb6in : A'=-1+B, C'=-1-A+B, D'=B, [ B>=2+A ], cost: 1-3*A+3*B

     28: evalfbb6in -> [11] : [ B>=2+A ], cost: -1-3*A+3*B

     29: evalfbb6in -> evalfbb6in : A'=B, C'=0, D'=B, [ 1+A-B==0 ], cost: -4*A+4*B

     30: evalfbb6in -> evalfbb6in : A'=-1+B, C'=0, D'=-1+B, [ B>=2+A ], cost: -5-5*A+5*B



Chained accelerated rules (with incoming rules):

   Start location: evalfstart

     14: evalfstart -> evalfbb6in : A'=0, [], cost: 2

     31: evalfstart -> evalfbb6in : A'=-1+B, C'=-1+B, D'=B, [ B>=2 ], cost: 3+3*B

     32: evalfstart -> evalfbb6in : A'=B, C'=0, D'=B, [ 1-B==0 ], cost: 2+4*B

     33: evalfstart -> evalfbb6in : A'=-1+B, C'=0, D'=-1+B, [ B>=2 ], cost: -3+5*B

     28: evalfbb6in -> [11] : [ B>=2+A ], cost: -1-3*A+3*B



Eliminated locations (on tree-shaped paths):

   Start location: evalfstart

     34: evalfstart -> [11] : A'=0, [ B>=2 ], cost: 1+3*B

     35: evalfstart -> [13] : [ B>=2 ], cost: 3+3*B

     36: evalfstart -> [13] : [ 1-B==0 ], cost: 2+4*B

     37: evalfstart -> [13] : [ B>=2 ], cost: -3+5*B



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: evalfstart

     35: evalfstart -> [13] : [ B>=2 ], cost: 3+3*B

     36: evalfstart -> [13] : [ 1-B==0 ], cost: 2+4*B

     37: evalfstart -> [13] : [ B>=2 ], cost: -3+5*B



Computing asymptotic complexity for rule 35

   Solved the limit problem by the following transformations:

      Created initial limit problem:

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



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



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

      resulting limit problem:

      [solved]



   Solution:

      B / n

   Resulting cost 3+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: 3+3*n

   Rule cost:   3+3*B

   Rule guard:  [ B>=2 ]



WORST_CASE(Omega(n^1),?)


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

(4)
BOUNDS(n^1, INF)