/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 123 ms]
(2) BOUNDS(1, n^1)
(3) Loat Proof [FINISHED, 1037 ms]
(4) BOUNDS(n^1, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
eval_start_start(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_bb0_in(v_1, v_3, v_n, v_x_0, v_x_0_sink)) :|: TRUE
eval_start_bb0_in(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_0(v_1, v_3, v_n, v_x_0, v_x_0_sink)) :|: TRUE
eval_start_0(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_1(v_1, v_3, v_n, v_x_0, v_x_0_sink)) :|: TRUE
eval_start_1(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_2(v_1, v_3, v_n, v_x_0, v_x_0_sink)) :|: TRUE
eval_start_2(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_3(v_1, v_3, v_n, v_x_0, v_x_0_sink)) :|: TRUE
eval_start_3(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_bb1_in(v_1, v_3, v_n, 0, v_x_0_sink)) :|: TRUE
eval_start_bb1_in(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_bb2_in(v_1, v_3, v_n, v_x_0, v_x_0)) :|: v_x_0 < v_n
eval_start_bb1_in(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_bb4_in(v_1, v_3, v_n, v_x_0, v_x_0_sink)) :|: v_x_0 >= v_n
eval_start_bb2_in(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_bb3_in(v_x_0_sink + 1, v_3, v_n, v_x_0, v_x_0_sink)) :|: v_x_0_sink + 1 < v_n
eval_start_bb2_in(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_bb1_in(v_1, v_3, v_n, v_x_0_sink + 1, v_x_0_sink)) :|: v_x_0_sink + 1 >= v_n
eval_start_bb3_in(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_6(v_1, v_3, v_n, v_x_0, v_x_0_sink)) :|: TRUE
eval_start_6(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_7(v_1, nondef_0, v_n, v_x_0, v_x_0_sink)) :|: TRUE
eval_start_7(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_bb1_in(v_1, v_3, v_n, v_1, v_x_0_sink)) :|: v_3 > 0
eval_start_7(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_bb2_in(v_1, v_3, v_n, v_x_0, v_1)) :|: v_3 <= 0
eval_start_bb4_in(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_stop(v_1, v_3, v_n, v_x_0, v_x_0_sink)) :|: TRUE

The start-symbols are:[eval_start_start_5]


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

(1) Koat Proof (FINISHED)
YES(?, 16*Ar_1 + 10)



Initial complexity problem:

1:	T:

		(Comp: ?, Cost: 1)    evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: ?, Cost: 1)    evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: ?, Cost: 1)    evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: ?, Cost: 1)    evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: ?, Cost: 1)    evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: ?, Cost: 1)    evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: ?, Cost: 1)    evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_0, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_2 + 1, Ar_4)) [ Ar_1 >= Ar_2 + 2 ]

		(Comp: ?, Cost: 1)    evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_2 + 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_2 + 1 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: ?, Cost: 1)    evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart7(Ar_0, Ar_1, Ar_2, Ar_3, Fresh_0))

		(Comp: ?, Cost: 1)    evalstart7(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= 1 ]

		(Comp: ?, Cost: 1)    evalstart7(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_3, Ar_3, Ar_4)) [ 0 >= Ar_4 ]

		(Comp: ?, Cost: 1)    evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 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)    evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)    evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)    evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)    evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)    evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)    evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: ?, Cost: 1)    evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_0, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_2 + 1, Ar_4)) [ Ar_1 >= Ar_2 + 2 ]

		(Comp: ?, Cost: 1)    evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_2 + 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_2 + 1 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: ?, Cost: 1)    evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart7(Ar_0, Ar_1, Ar_2, Ar_3, Fresh_0))

		(Comp: ?, Cost: 1)    evalstart7(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= 1 ]

		(Comp: ?, Cost: 1)    evalstart7(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_3, Ar_3, Ar_4)) [ 0 >= Ar_4 ]

		(Comp: ?, Cost: 1)    evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalstartstart) = 2

	Pol(evalstartbb0in) = 2

	Pol(evalstart0) = 2

	Pol(evalstart1) = 2

	Pol(evalstart2) = 2

	Pol(evalstart3) = 2

	Pol(evalstartbb1in) = 2

	Pol(evalstartbb2in) = 2

	Pol(evalstartbb4in) = 1

	Pol(evalstartbb3in) = 2

	Pol(evalstart6) = 2

	Pol(evalstart7) = 2

	Pol(evalstartstop) = 0

	Pol(koat_start) = 2

orients all transitions weakly and the transitions

	evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

	evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ]

strictly and produces the following problem:

3:	T:

		(Comp: 1, Cost: 1)    evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)    evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)    evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)    evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)    evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)    evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: ?, Cost: 1)    evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_0, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ]

		(Comp: 2, Cost: 1)    evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_2 + 1, Ar_4)) [ Ar_1 >= Ar_2 + 2 ]

		(Comp: ?, Cost: 1)    evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_2 + 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_2 + 1 >= Ar_1 ]

		(Comp: ?, Cost: 1)    evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: ?, Cost: 1)    evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart7(Ar_0, Ar_1, Ar_2, Ar_3, Fresh_0))

		(Comp: ?, Cost: 1)    evalstart7(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= 1 ]

		(Comp: ?, Cost: 1)    evalstart7(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_3, Ar_3, Ar_4)) [ 0 >= Ar_4 ]

		(Comp: 2, Cost: 1)    evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalstartstart) = 2*V_2

	Pol(evalstartbb0in) = 2*V_2

	Pol(evalstart0) = 2*V_2

	Pol(evalstart1) = 2*V_2

	Pol(evalstart2) = 2*V_2

	Pol(evalstart3) = 2*V_2

	Pol(evalstartbb1in) = -2*V_1 + 2*V_2

	Pol(evalstartbb2in) = 2*V_2 - 2*V_3 - 1

	Pol(evalstartbb4in) = -2*V_1 + 2*V_2

	Pol(evalstartbb3in) = 2*V_2 - 2*V_4

	Pol(evalstart6) = 2*V_2 - 2*V_4

	Pol(evalstart7) = 2*V_2 - 2*V_4

	Pol(evalstartstop) = -2*V_1 + 2*V_2

	Pol(koat_start) = 2*V_2

orients all transitions weakly and the transitions

	evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_2 + 1, Ar_4)) [ Ar_1 >= Ar_2 + 2 ]

	evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_0, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ]

strictly and produces the following problem:

4:	T:

		(Comp: 1, Cost: 1)         evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)         evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)         evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)         evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)         evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)         evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 2*Ar_1, Cost: 1)    evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_0, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ]

		(Comp: 2, Cost: 1)         evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ]

		(Comp: 2*Ar_1, Cost: 1)    evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_2 + 1, Ar_4)) [ Ar_1 >= Ar_2 + 2 ]

		(Comp: ?, Cost: 1)         evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_2 + 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_2 + 1 >= Ar_1 ]

		(Comp: ?, Cost: 1)         evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: ?, Cost: 1)         evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart7(Ar_0, Ar_1, Ar_2, Ar_3, Fresh_0))

		(Comp: ?, Cost: 1)         evalstart7(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= 1 ]

		(Comp: ?, Cost: 1)         evalstart7(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_3, Ar_3, Ar_4)) [ 0 >= Ar_4 ]

		(Comp: 2, Cost: 1)         evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 0)         koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 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)         evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)         evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)         evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)         evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)         evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 1)         evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 2*Ar_1, Cost: 1)    evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_0, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ]

		(Comp: 2, Cost: 1)         evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ]

		(Comp: 2*Ar_1, Cost: 1)    evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_2 + 1, Ar_4)) [ Ar_1 >= Ar_2 + 2 ]

		(Comp: 4*Ar_1, Cost: 1)    evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_2 + 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_2 + 1 >= Ar_1 ]

		(Comp: 2*Ar_1, Cost: 1)    evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 2*Ar_1, Cost: 1)    evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart7(Ar_0, Ar_1, Ar_2, Ar_3, Fresh_0))

		(Comp: 2*Ar_1, Cost: 1)    evalstart7(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= 1 ]

		(Comp: 2*Ar_1, Cost: 1)    evalstart7(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_3, Ar_3, Ar_4)) [ 0 >= Ar_4 ]

		(Comp: 2, Cost: 1)         evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))

		(Comp: 1, Cost: 0)         koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 16*Ar_1 + 10



Time: 0.134 sec (SMT: 0.096 sec)


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

(2)
BOUNDS(1, n^1)

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

(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: evalstartstart

      0: evalstartstart -> evalstartbb0in : [], cost: 1

      1: evalstartbb0in -> evalstart0 : [], cost: 1

      2: evalstart0 -> evalstart1 : [], cost: 1

      3: evalstart1 -> evalstart2 : [], cost: 1

      4: evalstart2 -> evalstart3 : [], cost: 1

      5: evalstart3 -> evalstartbb1in : A'=0, [], cost: 1

      6: evalstartbb1in -> evalstartbb2in : C'=A, [ B>=1+A ], cost: 1

      7: evalstartbb1in -> evalstartbb4in : [ A>=B ], cost: 1

      8: evalstartbb2in -> evalstartbb3in : D'=1+C, [ B>=2+C ], cost: 1

      9: evalstartbb2in -> evalstartbb1in : A'=1+C, [ 1+C>=B ], cost: 1

     10: evalstartbb3in -> evalstart6 : [], cost: 1

     11: evalstart6 -> evalstart7 : E'=free, [], cost: 1

     12: evalstart7 -> evalstartbb1in : A'=D, [ E>=1 ], cost: 1

     13: evalstart7 -> evalstartbb2in : C'=D, [ 0>=E ], cost: 1

     14: evalstartbb4in -> evalstartstop : [], cost: 1



Checking for constant complexity:

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

      0: evalstartstart -> evalstartbb0in : [], cost: 1



Removed unreachable and leaf rules:

   Start location: evalstartstart

      0: evalstartstart -> evalstartbb0in : [], cost: 1

      1: evalstartbb0in -> evalstart0 : [], cost: 1

      2: evalstart0 -> evalstart1 : [], cost: 1

      3: evalstart1 -> evalstart2 : [], cost: 1

      4: evalstart2 -> evalstart3 : [], cost: 1

      5: evalstart3 -> evalstartbb1in : A'=0, [], cost: 1

      6: evalstartbb1in -> evalstartbb2in : C'=A, [ B>=1+A ], cost: 1

      8: evalstartbb2in -> evalstartbb3in : D'=1+C, [ B>=2+C ], cost: 1

      9: evalstartbb2in -> evalstartbb1in : A'=1+C, [ 1+C>=B ], cost: 1

     10: evalstartbb3in -> evalstart6 : [], cost: 1

     11: evalstart6 -> evalstart7 : E'=free, [], cost: 1

     12: evalstart7 -> evalstartbb1in : A'=D, [ E>=1 ], cost: 1

     13: evalstart7 -> evalstartbb2in : C'=D, [ 0>=E ], cost: 1



### Simplification by acceleration and chaining ###



Eliminated locations (on linear paths):

   Start location: evalstartstart

     19: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6

      6: evalstartbb1in -> evalstartbb2in : C'=A, [ B>=1+A ], cost: 1

      9: evalstartbb2in -> evalstartbb1in : A'=1+C, [ 1+C>=B ], cost: 1

     21: evalstartbb2in -> evalstart7 : D'=1+C, E'=free, [ B>=2+C ], cost: 3

     12: evalstart7 -> evalstartbb1in : A'=D, [ E>=1 ], cost: 1

     13: evalstart7 -> evalstartbb2in : C'=D, [ 0>=E ], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: evalstartstart

     19: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6

      6: evalstartbb1in -> evalstartbb2in : C'=A, [ B>=1+A ], cost: 1

      9: evalstartbb2in -> evalstartbb1in : A'=1+C, [ 1+C>=B ], cost: 1

     22: evalstartbb2in -> evalstartbb1in : A'=1+C, D'=1+C, E'=free, [ B>=2+C && free>=1 ], cost: 4

     23: evalstartbb2in -> evalstartbb2in : C'=1+C, D'=1+C, E'=free, [ B>=2+C && 0>=free ], cost: 4



Accelerating simple loops of location 7.

   Accelerating the following rules:

     23: evalstartbb2in -> evalstartbb2in : C'=1+C, D'=1+C, E'=free, [ B>=2+C && 0>=free ], cost: 4



   Accelerated rule 23 with metering function -1-C+B, yielding the new rule 24.

   Removing the simple loops: 23.



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

   Start location: evalstartstart

     19: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6

      6: evalstartbb1in -> evalstartbb2in : C'=A, [ B>=1+A ], cost: 1

      9: evalstartbb2in -> evalstartbb1in : A'=1+C, [ 1+C>=B ], cost: 1

     22: evalstartbb2in -> evalstartbb1in : A'=1+C, D'=1+C, E'=free, [ B>=2+C && free>=1 ], cost: 4

     24: evalstartbb2in -> evalstartbb2in : C'=-1+B, D'=-1+B, E'=free, [ B>=2+C && 0>=free ], cost: -4-4*C+4*B



Chained accelerated rules (with incoming rules):

   Start location: evalstartstart

     19: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6

      6: evalstartbb1in -> evalstartbb2in : C'=A, [ B>=1+A ], cost: 1

     25: evalstartbb1in -> evalstartbb2in : C'=-1+B, D'=-1+B, E'=free, [ B>=2+A && 0>=free ], cost: -3-4*A+4*B

      9: evalstartbb2in -> evalstartbb1in : A'=1+C, [ 1+C>=B ], cost: 1

     22: evalstartbb2in -> evalstartbb1in : A'=1+C, D'=1+C, E'=free, [ B>=2+C && free>=1 ], cost: 4



Eliminated locations (on tree-shaped paths):

   Start location: evalstartstart

     19: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6

     26: evalstartbb1in -> evalstartbb1in : A'=1+A, C'=A, [ B>=1+A && 1+A>=B ], cost: 2

     27: evalstartbb1in -> evalstartbb1in : A'=1+A, C'=A, D'=1+A, E'=free, [ B>=2+A && free>=1 ], cost: 5

     28: evalstartbb1in -> evalstartbb1in : A'=B, C'=-1+B, D'=-1+B, E'=free, [ B>=2+A && 0>=free ], cost: -2-4*A+4*B

     29: evalstartbb1in -> [14] : [ B>=2+A && 0>=free ], cost: -3-4*A+4*B



Accelerating simple loops of location 6.

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

   Accelerating the following rules:

     26: evalstartbb1in -> evalstartbb1in : A'=1+A, C'=A, [ 1+A-B==0 ], cost: 2

     27: evalstartbb1in -> evalstartbb1in : A'=1+A, C'=A, D'=1+A, E'=free, [ B>=2+A && free>=1 ], cost: 5

     28: evalstartbb1in -> evalstartbb1in : A'=B, C'=-1+B, D'=-1+B, E'=free, [ B>=2+A && 0>=free ], cost: -2-4*A+4*B



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

   Accelerated rule 27 with metering function -1-A+B, yielding the new rule 31.

   Found no metering function for rule 28.

   Removing the simple loops: 26 27.



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

   Start location: evalstartstart

     19: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6

     28: evalstartbb1in -> evalstartbb1in : A'=B, C'=-1+B, D'=-1+B, E'=free, [ B>=2+A && 0>=free ], cost: -2-4*A+4*B

     29: evalstartbb1in -> [14] : [ B>=2+A && 0>=free ], cost: -3-4*A+4*B

     30: evalstartbb1in -> evalstartbb1in : A'=B, C'=-1+B, [ 1+A-B==0 ], cost: -2*A+2*B

     31: evalstartbb1in -> evalstartbb1in : A'=-1+B, C'=-2+B, D'=-1+B, E'=free, [ B>=2+A && free>=1 ], cost: -5-5*A+5*B



Chained accelerated rules (with incoming rules):

   Start location: evalstartstart

     19: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6

     32: evalstartstart -> evalstartbb1in : A'=B, C'=-1+B, D'=-1+B, E'=free, [ B>=2 && 0>=free ], cost: 4+4*B

     33: evalstartstart -> evalstartbb1in : A'=B, C'=-1+B, [ 1-B==0 ], cost: 6+2*B

     34: evalstartstart -> evalstartbb1in : A'=-1+B, C'=-2+B, D'=-1+B, E'=free, [ B>=2 && free>=1 ], cost: 1+5*B

     29: evalstartbb1in -> [14] : [ B>=2+A && 0>=free ], cost: -3-4*A+4*B



Eliminated locations (on tree-shaped paths):

   Start location: evalstartstart

     35: evalstartstart -> [14] : A'=0, [ B>=2 && 0>=free ], cost: 3+4*B

     36: evalstartstart -> [16] : [ B>=2 && 0>=free ], cost: 4+4*B

     37: evalstartstart -> [16] : [ 1-B==0 ], cost: 6+2*B

     38: evalstartstart -> [16] : [ B>=2 && free>=1 ], cost: 1+5*B



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: evalstartstart

     36: evalstartstart -> [16] : [ B>=2 && 0>=free ], cost: 4+4*B

     37: evalstartstart -> [16] : [ 1-B==0 ], cost: 6+2*B

     38: evalstartstart -> [16] : [ B>=2 && free>=1 ], cost: 1+5*B



Computing asymptotic complexity for rule 36

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -1+B (+/+!), 1-free (+/+!), 4+4*B (+) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {free==0,B==n}

      resulting limit problem:

      [solved]



   Solution:

      free / 0

      B / n

   Resulting cost 4+4*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: 4+4*n

   Rule cost:   4+4*B

   Rule guard:  [ B>=2 && 0>=free ]



WORST_CASE(Omega(n^1),?)


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

(4)
BOUNDS(n^1, INF)