/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 13 ms]
(2) BOUNDS(1, n^1)
(3) Loat Proof [FINISHED, 634 ms]
(4) BOUNDS(n^1, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
evalwcet2start(A, B) -> Com_1(evalwcet2entryin(A, B)) :|: TRUE
evalwcet2entryin(A, B) -> Com_1(evalwcet2bb5in(A, B)) :|: TRUE
evalwcet2bb5in(A, B) -> Com_1(evalwcet2bb2in(A, 0)) :|: 4 >= A
evalwcet2bb5in(A, B) -> Com_1(evalwcet2returnin(A, B)) :|: A >= 5
evalwcet2bb2in(A, B) -> Com_1(evalwcet2bb1in(A, B)) :|: A >= 3 && 9 >= B
evalwcet2bb2in(A, B) -> Com_1(evalwcet2bb4in(A, B)) :|: 2 >= A
evalwcet2bb2in(A, B) -> Com_1(evalwcet2bb4in(A, B)) :|: B >= 10
evalwcet2bb1in(A, B) -> Com_1(evalwcet2bb2in(A, B + 1)) :|: TRUE
evalwcet2bb4in(A, B) -> Com_1(evalwcet2bb5in(A + 1, B)) :|: TRUE
evalwcet2returnin(A, B) -> Com_1(evalwcet2stop(A, B)) :|: TRUE

The start-symbols are:[evalwcet2start_2]


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

(1) Koat Proof (FINISHED)
YES(?, 56*Ar_0 + 258)



Initial complexity problem:

1:	T:

		(Comp: ?, Cost: 1)    evalwcet2start(Ar_0, Ar_1) -> Com_1(evalwcet2entryin(Ar_0, Ar_1))

		(Comp: ?, Cost: 1)    evalwcet2entryin(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0, Ar_1))

		(Comp: ?, Cost: 1)    evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, 0)) [ 4 >= Ar_0 ]

		(Comp: ?, Cost: 1)    evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2returnin(Ar_0, Ar_1)) [ Ar_0 >= 5 ]

		(Comp: ?, Cost: 1)    evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb1in(Ar_0, Ar_1)) [ Ar_0 >= 3 /\ 9 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ 2 >= Ar_0 ]

		(Comp: ?, Cost: 1)    evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ Ar_1 >= 10 ]

		(Comp: ?, Cost: 1)    evalwcet2bb1in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, Ar_1 + 1))

		(Comp: ?, Cost: 1)    evalwcet2bb4in(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0 + 1, Ar_1))

		(Comp: ?, Cost: 1)    evalwcet2returnin(Ar_0, Ar_1) -> Com_1(evalwcet2stop(Ar_0, Ar_1))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1) -> Com_1(evalwcet2start(Ar_0, Ar_1)) [ 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)    evalwcet2start(Ar_0, Ar_1) -> Com_1(evalwcet2entryin(Ar_0, Ar_1))

		(Comp: 1, Cost: 1)    evalwcet2entryin(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0, Ar_1))

		(Comp: ?, Cost: 1)    evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, 0)) [ 4 >= Ar_0 ]

		(Comp: ?, Cost: 1)    evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2returnin(Ar_0, Ar_1)) [ Ar_0 >= 5 ]

		(Comp: ?, Cost: 1)    evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb1in(Ar_0, Ar_1)) [ Ar_0 >= 3 /\ 9 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ 2 >= Ar_0 ]

		(Comp: ?, Cost: 1)    evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ Ar_1 >= 10 ]

		(Comp: ?, Cost: 1)    evalwcet2bb1in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, Ar_1 + 1))

		(Comp: ?, Cost: 1)    evalwcet2bb4in(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0 + 1, Ar_1))

		(Comp: ?, Cost: 1)    evalwcet2returnin(Ar_0, Ar_1) -> Com_1(evalwcet2stop(Ar_0, Ar_1))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1) -> Com_1(evalwcet2start(Ar_0, Ar_1)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalwcet2start) = 2

	Pol(evalwcet2entryin) = 2

	Pol(evalwcet2bb5in) = 2

	Pol(evalwcet2bb2in) = 2

	Pol(evalwcet2returnin) = 1

	Pol(evalwcet2bb1in) = 2

	Pol(evalwcet2bb4in) = 2

	Pol(evalwcet2stop) = 0

	Pol(koat_start) = 2

orients all transitions weakly and the transitions

	evalwcet2returnin(Ar_0, Ar_1) -> Com_1(evalwcet2stop(Ar_0, Ar_1))

	evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2returnin(Ar_0, Ar_1)) [ Ar_0 >= 5 ]

strictly and produces the following problem:

3:	T:

		(Comp: 1, Cost: 1)    evalwcet2start(Ar_0, Ar_1) -> Com_1(evalwcet2entryin(Ar_0, Ar_1))

		(Comp: 1, Cost: 1)    evalwcet2entryin(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0, Ar_1))

		(Comp: ?, Cost: 1)    evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, 0)) [ 4 >= Ar_0 ]

		(Comp: 2, Cost: 1)    evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2returnin(Ar_0, Ar_1)) [ Ar_0 >= 5 ]

		(Comp: ?, Cost: 1)    evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb1in(Ar_0, Ar_1)) [ Ar_0 >= 3 /\ 9 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ 2 >= Ar_0 ]

		(Comp: ?, Cost: 1)    evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ Ar_1 >= 10 ]

		(Comp: ?, Cost: 1)    evalwcet2bb1in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, Ar_1 + 1))

		(Comp: ?, Cost: 1)    evalwcet2bb4in(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0 + 1, Ar_1))

		(Comp: 2, Cost: 1)    evalwcet2returnin(Ar_0, Ar_1) -> Com_1(evalwcet2stop(Ar_0, Ar_1))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1) -> Com_1(evalwcet2start(Ar_0, Ar_1)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalwcet2start) = -2*V_1 + 9

	Pol(evalwcet2entryin) = -2*V_1 + 9

	Pol(evalwcet2bb5in) = -2*V_1 + 9

	Pol(evalwcet2bb2in) = -2*V_1 + 8

	Pol(evalwcet2returnin) = -2*V_1

	Pol(evalwcet2bb1in) = -2*V_1 + 8

	Pol(evalwcet2bb4in) = -2*V_1 + 7

	Pol(evalwcet2stop) = -2*V_1

	Pol(koat_start) = -2*V_1 + 9

orients all transitions weakly and the transitions

	evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, 0)) [ 4 >= Ar_0 ]

	evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ 2 >= Ar_0 ]

strictly and produces the following problem:

4:	T:

		(Comp: 1, Cost: 1)             evalwcet2start(Ar_0, Ar_1) -> Com_1(evalwcet2entryin(Ar_0, Ar_1))

		(Comp: 1, Cost: 1)             evalwcet2entryin(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0, Ar_1))

		(Comp: 2*Ar_0 + 9, Cost: 1)    evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, 0)) [ 4 >= Ar_0 ]

		(Comp: 2, Cost: 1)             evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2returnin(Ar_0, Ar_1)) [ Ar_0 >= 5 ]

		(Comp: ?, Cost: 1)             evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb1in(Ar_0, Ar_1)) [ Ar_0 >= 3 /\ 9 >= Ar_1 ]

		(Comp: 2*Ar_0 + 9, Cost: 1)    evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ 2 >= Ar_0 ]

		(Comp: ?, Cost: 1)             evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ Ar_1 >= 10 ]

		(Comp: ?, Cost: 1)             evalwcet2bb1in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, Ar_1 + 1))

		(Comp: ?, Cost: 1)             evalwcet2bb4in(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0 + 1, Ar_1))

		(Comp: 2, Cost: 1)             evalwcet2returnin(Ar_0, Ar_1) -> Com_1(evalwcet2stop(Ar_0, Ar_1))

		(Comp: 1, Cost: 0)             koat_start(Ar_0, Ar_1) -> Com_1(evalwcet2start(Ar_0, Ar_1)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalwcet2bb4in) = 1

	Pol(evalwcet2bb5in) = 0

	Pol(evalwcet2bb2in) = 2

	Pol(evalwcet2bb1in) = 2

and size complexities

	S("koat_start(Ar_0, Ar_1) -> Com_1(evalwcet2start(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0

	S("koat_start(Ar_0, Ar_1) -> Com_1(evalwcet2start(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1

	S("evalwcet2returnin(Ar_0, Ar_1) -> Com_1(evalwcet2stop(Ar_0, Ar_1))", 0-0) = ?

	S("evalwcet2returnin(Ar_0, Ar_1) -> Com_1(evalwcet2stop(Ar_0, Ar_1))", 0-1) = ?

	S("evalwcet2bb4in(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0 + 1, Ar_1))", 0-0) = ?

	S("evalwcet2bb4in(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0 + 1, Ar_1))", 0-1) = ?

	S("evalwcet2bb1in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, Ar_1 + 1))", 0-0) = ?

	S("evalwcet2bb1in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, Ar_1 + 1))", 0-1) = ?

	S("evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ Ar_1 >= 10 ]", 0-0) = ?

	S("evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ Ar_1 >= 10 ]", 0-1) = ?

	S("evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ 2 >= Ar_0 ]", 0-0) = ?

	S("evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ 2 >= Ar_0 ]", 0-1) = ?

	S("evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb1in(Ar_0, Ar_1)) [ Ar_0 >= 3 /\\ 9 >= Ar_1 ]", 0-0) = ?

	S("evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb1in(Ar_0, Ar_1)) [ Ar_0 >= 3 /\\ 9 >= Ar_1 ]", 0-1) = ?

	S("evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2returnin(Ar_0, Ar_1)) [ Ar_0 >= 5 ]", 0-0) = ?

	S("evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2returnin(Ar_0, Ar_1)) [ Ar_0 >= 5 ]", 0-1) = ?

	S("evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, 0)) [ 4 >= Ar_0 ]", 0-0) = ?

	S("evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, 0)) [ 4 >= Ar_0 ]", 0-1) = 0

	S("evalwcet2entryin(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0, Ar_1))", 0-0) = Ar_0

	S("evalwcet2entryin(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0, Ar_1))", 0-1) = Ar_1

	S("evalwcet2start(Ar_0, Ar_1) -> Com_1(evalwcet2entryin(Ar_0, Ar_1))", 0-0) = Ar_0

	S("evalwcet2start(Ar_0, Ar_1) -> Com_1(evalwcet2entryin(Ar_0, Ar_1))", 0-1) = Ar_1

orients the transitions

	evalwcet2bb4in(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0 + 1, Ar_1))

	evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ Ar_1 >= 10 ]

	evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb1in(Ar_0, Ar_1)) [ Ar_0 >= 3 /\ 9 >= Ar_1 ]

	evalwcet2bb1in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, Ar_1 + 1))

weakly and the transitions

	evalwcet2bb4in(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0 + 1, Ar_1))

	evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ Ar_1 >= 10 ]

strictly and produces the following problem:

5:	T:

		(Comp: 1, Cost: 1)              evalwcet2start(Ar_0, Ar_1) -> Com_1(evalwcet2entryin(Ar_0, Ar_1))

		(Comp: 1, Cost: 1)              evalwcet2entryin(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0, Ar_1))

		(Comp: 2*Ar_0 + 9, Cost: 1)     evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, 0)) [ 4 >= Ar_0 ]

		(Comp: 2, Cost: 1)              evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2returnin(Ar_0, Ar_1)) [ Ar_0 >= 5 ]

		(Comp: ?, Cost: 1)              evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb1in(Ar_0, Ar_1)) [ Ar_0 >= 3 /\ 9 >= Ar_1 ]

		(Comp: 2*Ar_0 + 9, Cost: 1)     evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ 2 >= Ar_0 ]

		(Comp: 6*Ar_0 + 27, Cost: 1)    evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ Ar_1 >= 10 ]

		(Comp: ?, Cost: 1)              evalwcet2bb1in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, Ar_1 + 1))

		(Comp: 6*Ar_0 + 27, Cost: 1)    evalwcet2bb4in(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0 + 1, Ar_1))

		(Comp: 2, Cost: 1)              evalwcet2returnin(Ar_0, Ar_1) -> Com_1(evalwcet2stop(Ar_0, Ar_1))

		(Comp: 1, Cost: 0)              koat_start(Ar_0, Ar_1) -> Com_1(evalwcet2start(Ar_0, Ar_1)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalwcet2bb2in) = -V_2 + 10

	Pol(evalwcet2bb1in) = -V_2 + 9

and size complexities

	S("koat_start(Ar_0, Ar_1) -> Com_1(evalwcet2start(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0

	S("koat_start(Ar_0, Ar_1) -> Com_1(evalwcet2start(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1

	S("evalwcet2returnin(Ar_0, Ar_1) -> Com_1(evalwcet2stop(Ar_0, Ar_1))", 0-0) = 7*Ar_0 + 64827

	S("evalwcet2returnin(Ar_0, Ar_1) -> Com_1(evalwcet2stop(Ar_0, Ar_1))", 0-1) = ?

	S("evalwcet2bb4in(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0 + 1, Ar_1))", 0-0) = 7*Ar_0 + 1323

	S("evalwcet2bb4in(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0 + 1, Ar_1))", 0-1) = ?

	S("evalwcet2bb1in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, Ar_1 + 1))", 0-0) = 7*Ar_0 + 1323

	S("evalwcet2bb1in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, Ar_1 + 1))", 0-1) = ?

	S("evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ Ar_1 >= 10 ]", 0-0) = 7*Ar_0 + 1323

	S("evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ Ar_1 >= 10 ]", 0-1) = ?

	S("evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ 2 >= Ar_0 ]", 0-0) = 7*Ar_0 + 1323

	S("evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ 2 >= Ar_0 ]", 0-1) = ?

	S("evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb1in(Ar_0, Ar_1)) [ Ar_0 >= 3 /\\ 9 >= Ar_1 ]", 0-0) = 7*Ar_0 + 1323

	S("evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb1in(Ar_0, Ar_1)) [ Ar_0 >= 3 /\\ 9 >= Ar_1 ]", 0-1) = ?

	S("evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2returnin(Ar_0, Ar_1)) [ Ar_0 >= 5 ]", 0-0) = 7*Ar_0 + 9261

	S("evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2returnin(Ar_0, Ar_1)) [ Ar_0 >= 5 ]", 0-1) = ?

	S("evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, 0)) [ 4 >= Ar_0 ]", 0-0) = 7*Ar_0 + 1323

	S("evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, 0)) [ 4 >= Ar_0 ]", 0-1) = 0

	S("evalwcet2entryin(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0, Ar_1))", 0-0) = Ar_0

	S("evalwcet2entryin(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0, Ar_1))", 0-1) = Ar_1

	S("evalwcet2start(Ar_0, Ar_1) -> Com_1(evalwcet2entryin(Ar_0, Ar_1))", 0-0) = Ar_0

	S("evalwcet2start(Ar_0, Ar_1) -> Com_1(evalwcet2entryin(Ar_0, Ar_1))", 0-1) = Ar_1

orients the transitions

	evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb1in(Ar_0, Ar_1)) [ Ar_0 >= 3 /\ 9 >= Ar_1 ]

	evalwcet2bb1in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, Ar_1 + 1))

weakly and the transition

	evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb1in(Ar_0, Ar_1)) [ Ar_0 >= 3 /\ 9 >= Ar_1 ]

strictly and produces the following problem:

6:	T:

		(Comp: 1, Cost: 1)               evalwcet2start(Ar_0, Ar_1) -> Com_1(evalwcet2entryin(Ar_0, Ar_1))

		(Comp: 1, Cost: 1)               evalwcet2entryin(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0, Ar_1))

		(Comp: 2*Ar_0 + 9, Cost: 1)      evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, 0)) [ 4 >= Ar_0 ]

		(Comp: 2, Cost: 1)               evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2returnin(Ar_0, Ar_1)) [ Ar_0 >= 5 ]

		(Comp: 20*Ar_0 + 90, Cost: 1)    evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb1in(Ar_0, Ar_1)) [ Ar_0 >= 3 /\ 9 >= Ar_1 ]

		(Comp: 2*Ar_0 + 9, Cost: 1)      evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ 2 >= Ar_0 ]

		(Comp: 6*Ar_0 + 27, Cost: 1)     evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ Ar_1 >= 10 ]

		(Comp: ?, Cost: 1)               evalwcet2bb1in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, Ar_1 + 1))

		(Comp: 6*Ar_0 + 27, Cost: 1)     evalwcet2bb4in(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0 + 1, Ar_1))

		(Comp: 2, Cost: 1)               evalwcet2returnin(Ar_0, Ar_1) -> Com_1(evalwcet2stop(Ar_0, Ar_1))

		(Comp: 1, Cost: 0)               koat_start(Ar_0, Ar_1) -> Com_1(evalwcet2start(Ar_0, Ar_1)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Repeatedly propagating knowledge in problem 6 produces the following problem:

7:	T:

		(Comp: 1, Cost: 1)               evalwcet2start(Ar_0, Ar_1) -> Com_1(evalwcet2entryin(Ar_0, Ar_1))

		(Comp: 1, Cost: 1)               evalwcet2entryin(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0, Ar_1))

		(Comp: 2*Ar_0 + 9, Cost: 1)      evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, 0)) [ 4 >= Ar_0 ]

		(Comp: 2, Cost: 1)               evalwcet2bb5in(Ar_0, Ar_1) -> Com_1(evalwcet2returnin(Ar_0, Ar_1)) [ Ar_0 >= 5 ]

		(Comp: 20*Ar_0 + 90, Cost: 1)    evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb1in(Ar_0, Ar_1)) [ Ar_0 >= 3 /\ 9 >= Ar_1 ]

		(Comp: 2*Ar_0 + 9, Cost: 1)      evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ 2 >= Ar_0 ]

		(Comp: 6*Ar_0 + 27, Cost: 1)     evalwcet2bb2in(Ar_0, Ar_1) -> Com_1(evalwcet2bb4in(Ar_0, Ar_1)) [ Ar_1 >= 10 ]

		(Comp: 20*Ar_0 + 90, Cost: 1)    evalwcet2bb1in(Ar_0, Ar_1) -> Com_1(evalwcet2bb2in(Ar_0, Ar_1 + 1))

		(Comp: 6*Ar_0 + 27, Cost: 1)     evalwcet2bb4in(Ar_0, Ar_1) -> Com_1(evalwcet2bb5in(Ar_0 + 1, Ar_1))

		(Comp: 2, Cost: 1)               evalwcet2returnin(Ar_0, Ar_1) -> Com_1(evalwcet2stop(Ar_0, Ar_1))

		(Comp: 1, Cost: 0)               koat_start(Ar_0, Ar_1) -> Com_1(evalwcet2start(Ar_0, Ar_1)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 56*Ar_0 + 258



Time: 0.078 sec (SMT: 0.064 sec)


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

(2)
BOUNDS(1, n^1)

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

(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: evalwcet2start

      0: evalwcet2start -> evalwcet2entryin : [], cost: 1

      1: evalwcet2entryin -> evalwcet2bb5in : [], cost: 1

      2: evalwcet2bb5in -> evalwcet2bb2in : B'=0, [ 4>=A ], cost: 1

      3: evalwcet2bb5in -> evalwcet2returnin : [ A>=5 ], cost: 1

      4: evalwcet2bb2in -> evalwcet2bb1in : [ A>=3 && 9>=B ], cost: 1

      5: evalwcet2bb2in -> evalwcet2bb4in : [ 2>=A ], cost: 1

      6: evalwcet2bb2in -> evalwcet2bb4in : [ B>=10 ], cost: 1

      7: evalwcet2bb1in -> evalwcet2bb2in : B'=1+B, [], cost: 1

      8: evalwcet2bb4in -> evalwcet2bb5in : A'=1+A, [], cost: 1

      9: evalwcet2returnin -> evalwcet2stop : [], cost: 1



Checking for constant complexity:

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

      0: evalwcet2start -> evalwcet2entryin : [], cost: 1



Removed unreachable and leaf rules:

   Start location: evalwcet2start

      0: evalwcet2start -> evalwcet2entryin : [], cost: 1

      1: evalwcet2entryin -> evalwcet2bb5in : [], cost: 1

      2: evalwcet2bb5in -> evalwcet2bb2in : B'=0, [ 4>=A ], cost: 1

      4: evalwcet2bb2in -> evalwcet2bb1in : [ A>=3 && 9>=B ], cost: 1

      5: evalwcet2bb2in -> evalwcet2bb4in : [ 2>=A ], cost: 1

      6: evalwcet2bb2in -> evalwcet2bb4in : [ B>=10 ], cost: 1

      7: evalwcet2bb1in -> evalwcet2bb2in : B'=1+B, [], cost: 1

      8: evalwcet2bb4in -> evalwcet2bb5in : A'=1+A, [], cost: 1



### Simplification by acceleration and chaining ###



Eliminated locations (on linear paths):

   Start location: evalwcet2start

     10: evalwcet2start -> evalwcet2bb5in : [], cost: 2

      2: evalwcet2bb5in -> evalwcet2bb2in : B'=0, [ 4>=A ], cost: 1

      5: evalwcet2bb2in -> evalwcet2bb4in : [ 2>=A ], cost: 1

      6: evalwcet2bb2in -> evalwcet2bb4in : [ B>=10 ], cost: 1

     11: evalwcet2bb2in -> evalwcet2bb2in : B'=1+B, [ A>=3 && 9>=B ], cost: 2

      8: evalwcet2bb4in -> evalwcet2bb5in : A'=1+A, [], cost: 1



Accelerating simple loops of location 3.

   Accelerating the following rules:

     11: evalwcet2bb2in -> evalwcet2bb2in : B'=1+B, [ A>=3 && 9>=B ], cost: 2



   Accelerated rule 11 with metering function 10-B, yielding the new rule 12.

   Removing the simple loops: 11.



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

   Start location: evalwcet2start

     10: evalwcet2start -> evalwcet2bb5in : [], cost: 2

      2: evalwcet2bb5in -> evalwcet2bb2in : B'=0, [ 4>=A ], cost: 1

      5: evalwcet2bb2in -> evalwcet2bb4in : [ 2>=A ], cost: 1

      6: evalwcet2bb2in -> evalwcet2bb4in : [ B>=10 ], cost: 1

     12: evalwcet2bb2in -> evalwcet2bb2in : B'=10, [ A>=3 && 9>=B ], cost: 20-2*B

      8: evalwcet2bb4in -> evalwcet2bb5in : A'=1+A, [], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: evalwcet2start

     10: evalwcet2start -> evalwcet2bb5in : [], cost: 2

      2: evalwcet2bb5in -> evalwcet2bb2in : B'=0, [ 4>=A ], cost: 1

     13: evalwcet2bb5in -> evalwcet2bb2in : B'=10, [ 4>=A && A>=3 ], cost: 21

      5: evalwcet2bb2in -> evalwcet2bb4in : [ 2>=A ], cost: 1

      6: evalwcet2bb2in -> evalwcet2bb4in : [ B>=10 ], cost: 1

      8: evalwcet2bb4in -> evalwcet2bb5in : A'=1+A, [], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: evalwcet2start

     10: evalwcet2start -> evalwcet2bb5in : [], cost: 2

     14: evalwcet2bb5in -> evalwcet2bb4in : B'=0, [ 2>=A ], cost: 2

     15: evalwcet2bb5in -> evalwcet2bb4in : B'=10, [ 4>=A && A>=3 ], cost: 22

      8: evalwcet2bb4in -> evalwcet2bb5in : A'=1+A, [], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: evalwcet2start

     10: evalwcet2start -> evalwcet2bb5in : [], cost: 2

     16: evalwcet2bb5in -> evalwcet2bb5in : A'=1+A, B'=0, [ 2>=A ], cost: 3

     17: evalwcet2bb5in -> evalwcet2bb5in : A'=1+A, B'=10, [ 4>=A && A>=3 ], cost: 23



Accelerating simple loops of location 2.

   Accelerating the following rules:

     16: evalwcet2bb5in -> evalwcet2bb5in : A'=1+A, B'=0, [ 2>=A ], cost: 3

     17: evalwcet2bb5in -> evalwcet2bb5in : A'=1+A, B'=10, [ 4>=A && A>=3 ], cost: 23



   Accelerated rule 16 with metering function 3-A, yielding the new rule 18.

   Accelerated rule 17 with metering function 5-A, yielding the new rule 19.

   Removing the simple loops: 16 17.



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

   Start location: evalwcet2start

     10: evalwcet2start -> evalwcet2bb5in : [], cost: 2

     18: evalwcet2bb5in -> evalwcet2bb5in : A'=3, B'=0, [ 2>=A ], cost: 9-3*A

     19: evalwcet2bb5in -> evalwcet2bb5in : A'=5, B'=10, [ 4>=A && A>=3 ], cost: 115-23*A



Chained accelerated rules (with incoming rules):

   Start location: evalwcet2start

     10: evalwcet2start -> evalwcet2bb5in : [], cost: 2

     20: evalwcet2start -> evalwcet2bb5in : A'=3, B'=0, [ 2>=A ], cost: 11-3*A

     21: evalwcet2start -> evalwcet2bb5in : A'=5, B'=10, [ 4>=A && A>=3 ], cost: 117-23*A



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

   Start location: evalwcet2start

     20: evalwcet2start -> evalwcet2bb5in : A'=3, B'=0, [ 2>=A ], cost: 11-3*A

     21: evalwcet2start -> evalwcet2bb5in : A'=5, B'=10, [ 4>=A && A>=3 ], cost: 117-23*A



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: evalwcet2start

     20: evalwcet2start -> evalwcet2bb5in : A'=3, B'=0, [ 2>=A ], cost: 11-3*A

     21: evalwcet2start -> evalwcet2bb5in : A'=5, B'=10, [ 4>=A && A>=3 ], cost: 117-23*A



Computing asymptotic complexity for rule 20

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      3-A (+/+!), 11-3*A (+) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



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

      resulting limit problem:

      [solved]



   Solution:

      A / -n

   Resulting cost 11+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: 11+3*n

   Rule cost:   11-3*A

   Rule guard:  [ 2>=A ]



WORST_CASE(Omega(n^1),?)


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

(4)
BOUNDS(n^1, INF)