/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 235 ms]
(2) BOUNDS(1, n^1)
(3) Loat Proof [FINISHED, 748 ms]
(4) BOUNDS(n^1, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
eval_start_start(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb0_in(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
eval_start_bb0_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_0(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
eval_start_0(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_1(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
eval_start_1(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_2(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
eval_start_2(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_3(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
eval_start_3(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_4(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
eval_start_4(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_5(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
eval_start_5(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_6(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
eval_start_6(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb1_in(v_m, v_n, 0, 0)) :|: TRUE
eval_start_bb1_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb2_in(v_m, v_n, v_x_0, v_y_0)) :|: v_x_0 < v_n
eval_start_bb1_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb3_in(v_m, v_n, v_x_0, v_y_0)) :|: v_x_0 >= v_n
eval_start_bb2_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb1_in(v_m, v_n, v_x_0, v_y_0 + 1)) :|: v_y_0 < v_m
eval_start_bb2_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb1_in(v_m, v_n, v_x_0 + 1, v_y_0 + 1)) :|: v_y_0 < v_m && v_y_0 >= v_m
eval_start_bb2_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb1_in(v_m, v_n, v_x_0, v_y_0)) :|: v_y_0 >= v_m && v_y_0 < v_m
eval_start_bb2_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb1_in(v_m, v_n, v_x_0 + 1, v_y_0)) :|: v_y_0 >= v_m
eval_start_bb3_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_stop(v_m, v_n, v_x_0, v_y_0)) :|: TRUE

The start-symbols are:[eval_start_start_4]


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

(1) Koat Proof (FINISHED)
YES(?, 2*Ar_2 + 2*Ar_3 + 14)



Initial complexity problem:

1:	T:

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

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

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

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

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

		(Comp: ?, Cost: 1)    evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalstart6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, 0, Ar_2, Ar_3))

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

		(Comp: ?, Cost: 1)    evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ]

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

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

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

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

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

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

	start location:	koat_start

	leaf cost:	0



Testing for reachability in the complexity graph removes the following transitions from problem 1:

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

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

We thus obtain the following problem:

2:	T:

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

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

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

		(Comp: ?, Cost: 1)    evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ]

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

		(Comp: ?, Cost: 1)    evalstart6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, 0, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3))

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

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

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

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

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

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

	start location:	koat_start

	leaf cost:	0



Repeatedly propagating knowledge in problem 2 produces the following problem:

3:	T:

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

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

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

		(Comp: ?, Cost: 1)    evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ]

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

		(Comp: 1, Cost: 1)    evalstart6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, 0, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3))

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

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

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

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

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

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

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalstartbb3in) = 1

	Pol(evalstartstop) = 0

	Pol(evalstartbb2in) = 2

	Pol(evalstartbb1in) = 2

	Pol(evalstart6) = 2

	Pol(evalstart5) = 2

	Pol(evalstart4) = 2

	Pol(evalstart3) = 2

	Pol(evalstart2) = 2

	Pol(evalstart1) = 2

	Pol(evalstart0) = 2

	Pol(evalstartbb0in) = 2

	Pol(evalstartstart) = 2

	Pol(koat_start) = 2

orients all transitions weakly and the transitions

	evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3))

	evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ]

strictly and produces the following problem:

4:	T:

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

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

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

		(Comp: 2, Cost: 1)    evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ]

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

		(Comp: 1, Cost: 1)    evalstart6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, 0, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3))

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

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

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

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

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

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

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalstartbb3in) = -V_2 + V_4

	Pol(evalstartstop) = -V_2 + V_4

	Pol(evalstartbb2in) = -V_2 + V_4

	Pol(evalstartbb1in) = -V_2 + V_4

	Pol(evalstart6) = V_4

	Pol(evalstart5) = V_4

	Pol(evalstart4) = V_4

	Pol(evalstart3) = V_4

	Pol(evalstart2) = V_4

	Pol(evalstart1) = V_4

	Pol(evalstart0) = V_4

	Pol(evalstartbb0in) = V_4

	Pol(evalstartstart) = V_4

	Pol(koat_start) = V_4

orients all transitions weakly and the transition

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

strictly and produces the following problem:

5:	T:

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

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

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

		(Comp: 2, Cost: 1)       evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ]

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

		(Comp: 1, Cost: 1)       evalstart6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, 0, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3))

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

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

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

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

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

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

	start location:	koat_start

	leaf cost:	0



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

  For symbol evalstartbb1in: X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0

  For symbol evalstartbb2in: X_3 - 1 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ -X_1 + X_3 - 1 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0

  For symbol evalstartbb3in: X_1 - X_3 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0





This yielded the following problem:

6:	T:

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

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

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

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

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

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

		(Comp: 1, Cost: 1)       evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalstart6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, 0, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)       evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_0 + 1 ]

		(Comp: 2, Cost: 1)       evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= Ar_2 ]

		(Comp: Ar_3, Cost: 1)    evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_1 + 1 ]

		(Comp: ?, Cost: 1)       evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_3 ]

		(Comp: 2, Cost: 1)       evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(koat_start) = V_3

	Pol(evalstartstart) = V_3

	Pol(evalstartbb0in) = V_3

	Pol(evalstart0) = V_3

	Pol(evalstart1) = V_3

	Pol(evalstart2) = V_3

	Pol(evalstart3) = V_3

	Pol(evalstart4) = V_3

	Pol(evalstart5) = V_3

	Pol(evalstart6) = V_3

	Pol(evalstartbb1in) = -V_1 + V_3

	Pol(evalstartbb2in) = -V_1 + V_3

	Pol(evalstartbb3in) = -V_1 + V_3

	Pol(evalstartstop) = -V_1 + V_3

orients all transitions weakly and the transition

	evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_3 ]

strictly and produces the following problem:

7:	T:

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

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

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

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

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

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

		(Comp: 1, Cost: 1)       evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalstart6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, 0, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)       evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_0 + 1 ]

		(Comp: 2, Cost: 1)       evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= Ar_2 ]

		(Comp: Ar_3, Cost: 1)    evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_1 + 1 ]

		(Comp: Ar_2, Cost: 1)    evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_3 ]

		(Comp: 2, Cost: 1)       evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ]

	start location:	koat_start

	leaf cost:	0



Repeatedly propagating knowledge in problem 7 produces the following problem:

8:	T:

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

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

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

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

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

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

		(Comp: 1, Cost: 1)                  evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                  evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                  evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                  evalstart6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, 0, Ar_2, Ar_3))

		(Comp: Ar_2 + Ar_3 + 1, Cost: 1)    evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_0 + 1 ]

		(Comp: 2, Cost: 1)                  evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= Ar_2 ]

		(Comp: Ar_3, Cost: 1)               evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_1 + 1 ]

		(Comp: Ar_2, Cost: 1)               evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_3 ]

		(Comp: 2, Cost: 1)                  evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ]

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 2*Ar_2 + 2*Ar_3 + 14



Time: 0.234 sec (SMT: 0.182 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 -> evalstart4 : [], cost: 1

      6: evalstart4 -> evalstart5 : [], cost: 1

      7: evalstart5 -> evalstart6 : [], cost: 1

      8: evalstart6 -> evalstartbb1in : A'=0, B'=0, [], cost: 1

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

     10: evalstartbb1in -> evalstartbb3in : [ A>=C ], cost: 1

     11: evalstartbb2in -> evalstartbb1in : B'=1+B, [ D>=1+B ], cost: 1

     12: evalstartbb2in -> evalstartbb1in : A'=1+A, B'=1+B, [ D>=1+B && B>=D ], cost: 1

     13: evalstartbb2in -> evalstartbb1in : [ B>=D && D>=1+B ], cost: 1

     14: evalstartbb2in -> evalstartbb1in : A'=1+A, [ B>=D ], cost: 1

     15: evalstartbb3in -> 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 -> evalstart4 : [], cost: 1

      6: evalstart4 -> evalstart5 : [], cost: 1

      7: evalstart5 -> evalstart6 : [], cost: 1

      8: evalstart6 -> evalstartbb1in : A'=0, B'=0, [], cost: 1

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

     11: evalstartbb2in -> evalstartbb1in : B'=1+B, [ D>=1+B ], cost: 1

     12: evalstartbb2in -> evalstartbb1in : A'=1+A, B'=1+B, [ D>=1+B && B>=D ], cost: 1

     13: evalstartbb2in -> evalstartbb1in : [ B>=D && D>=1+B ], cost: 1

     14: evalstartbb2in -> evalstartbb1in : A'=1+A, [ B>=D ], cost: 1



Removed rules with unsatisfiable guard:

   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 -> evalstart4 : [], cost: 1

      6: evalstart4 -> evalstart5 : [], cost: 1

      7: evalstart5 -> evalstart6 : [], cost: 1

      8: evalstart6 -> evalstartbb1in : A'=0, B'=0, [], cost: 1

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

     11: evalstartbb2in -> evalstartbb1in : B'=1+B, [ D>=1+B ], cost: 1

     14: evalstartbb2in -> evalstartbb1in : A'=1+A, [ B>=D ], cost: 1



### Simplification by acceleration and chaining ###



Eliminated locations (on linear paths):

   Start location: evalstartstart

     23: evalstartstart -> evalstartbb1in : A'=0, B'=0, [], cost: 9

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

     11: evalstartbb2in -> evalstartbb1in : B'=1+B, [ D>=1+B ], cost: 1

     14: evalstartbb2in -> evalstartbb1in : A'=1+A, [ B>=D ], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: evalstartstart

     23: evalstartstart -> evalstartbb1in : A'=0, B'=0, [], cost: 9

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

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



Accelerating simple loops of location 9.

   Accelerating the following rules:

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

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



   Accelerated rule 24 with metering function D-B, yielding the new rule 26.

   Accelerated rule 25 with metering function C-A, yielding the new rule 27.

   Removing the simple loops: 24 25.



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

   Start location: evalstartstart

     23: evalstartstart -> evalstartbb1in : A'=0, B'=0, [], cost: 9

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

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



Chained accelerated rules (with incoming rules):

   Start location: evalstartstart

     23: evalstartstart -> evalstartbb1in : A'=0, B'=0, [], cost: 9

     28: evalstartstart -> evalstartbb1in : A'=0, B'=D, [ C>=1 && D>=1 ], cost: 9+2*D

     29: evalstartstart -> evalstartbb1in : A'=C, B'=0, [ C>=1 && 0>=D ], cost: 9+2*C



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

   Start location: evalstartstart

     28: evalstartstart -> evalstartbb1in : A'=0, B'=D, [ C>=1 && D>=1 ], cost: 9+2*D

     29: evalstartstart -> evalstartbb1in : A'=C, B'=0, [ C>=1 && 0>=D ], cost: 9+2*C



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: evalstartstart

     28: evalstartstart -> evalstartbb1in : A'=0, B'=D, [ C>=1 && D>=1 ], cost: 9+2*D

     29: evalstartstart -> evalstartbb1in : A'=C, B'=0, [ C>=1 && 0>=D ], cost: 9+2*C



Computing asymptotic complexity for rule 28

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      C (+/+!), D (+/+!), 9+2*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 9+2*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: 9+2*n

   Rule cost:   9+2*D

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



WORST_CASE(Omega(n^1),?)


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

(4)
BOUNDS(n^1, INF)