/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^2), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).

(0) CpxIntTrs
(1) Koat Proof [FINISHED, 214 ms]
(2) BOUNDS(1, n^2)
(3) Loat Proof [FINISHED, 820 ms]
(4) BOUNDS(n^2, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
eval_cousot9_start(v__0, v_N, v_i_0, v_j) -> Com_1(eval_cousot9_bb0_in(v__0, v_N, v_i_0, v_j)) :|: TRUE
eval_cousot9_bb0_in(v__0, v_N, v_i_0, v_j) -> Com_1(eval_cousot9_0(v__0, v_N, v_i_0, v_j)) :|: TRUE
eval_cousot9_0(v__0, v_N, v_i_0, v_j) -> Com_1(eval_cousot9_1(v__0, v_N, v_i_0, v_j)) :|: TRUE
eval_cousot9_1(v__0, v_N, v_i_0, v_j) -> Com_1(eval_cousot9_2(v__0, v_N, v_i_0, v_j)) :|: TRUE
eval_cousot9_2(v__0, v_N, v_i_0, v_j) -> Com_1(eval_cousot9_3(v__0, v_N, v_i_0, v_j)) :|: TRUE
eval_cousot9_3(v__0, v_N, v_i_0, v_j) -> Com_1(eval_cousot9_4(v__0, v_N, v_i_0, v_j)) :|: TRUE
eval_cousot9_4(v__0, v_N, v_i_0, v_j) -> Com_1(eval_cousot9_5(v__0, v_N, v_i_0, v_j)) :|: TRUE
eval_cousot9_5(v__0, v_N, v_i_0, v_j) -> Com_1(eval_cousot9_6(v__0, v_N, v_i_0, v_j)) :|: TRUE
eval_cousot9_6(v__0, v_N, v_i_0, v_j) -> Com_1(eval_cousot9_bb1_in(v_j, v_N, v_N, v_j)) :|: TRUE
eval_cousot9_bb1_in(v__0, v_N, v_i_0, v_j) -> Com_1(eval_cousot9_bb2_in(v__0, v_N, v_i_0, v_j)) :|: v_i_0 > 0
eval_cousot9_bb1_in(v__0, v_N, v_i_0, v_j) -> Com_1(eval_cousot9_bb3_in(v__0, v_N, v_i_0, v_j)) :|: v_i_0 <= 0
eval_cousot9_bb2_in(v__0, v_N, v_i_0, v_j) -> Com_1(eval_cousot9_bb1_in(v__0 - 1, v_N, v_i_0, v_j)) :|: v__0 > 0
eval_cousot9_bb2_in(v__0, v_N, v_i_0, v_j) -> Com_1(eval_cousot9_bb1_in(v_N, v_N, v_i_0, v_j)) :|: v__0 > 0 && v__0 <= 0
eval_cousot9_bb2_in(v__0, v_N, v_i_0, v_j) -> Com_1(eval_cousot9_bb1_in(v__0 - 1, v_N, v_i_0 - 1, v_j)) :|: v__0 <= 0 && v__0 > 0
eval_cousot9_bb2_in(v__0, v_N, v_i_0, v_j) -> Com_1(eval_cousot9_bb1_in(v_N, v_N, v_i_0 - 1, v_j)) :|: v__0 <= 0
eval_cousot9_bb3_in(v__0, v_N, v_i_0, v_j) -> Com_1(eval_cousot9_stop(v__0, v_N, v_i_0, v_j)) :|: TRUE

The start-symbols are:[eval_cousot9_start_4]


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

(1) Koat Proof (FINISHED)
YES(?, 2*Ar_3 + 2*Ar_3^2 + 2*Ar_1 + 14)



Initial complexity problem:

1:	T:

		(Comp: ?, Cost: 1)    evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_1, Ar_1, Ar_3, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]

		(Comp: ?, Cost: 1)    evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ]

		(Comp: ?, Cost: 1)    evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 ]

		(Comp: ?, Cost: 1)    evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ 0 >= Ar_0 ]

		(Comp: ?, Cost: 1)    evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_0 - 1, Ar_1, Ar_2 - 1, Ar_3)) [ 0 >= Ar_0 /\ Ar_0 >= 1 ]

		(Comp: ?, Cost: 1)    evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_3, Ar_1, Ar_2 - 1, Ar_3)) [ 0 >= Ar_0 ]

		(Comp: ?, Cost: 1)    evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9stop(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9start(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:

	evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ 0 >= Ar_0 ]

	evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_0 - 1, Ar_1, Ar_2 - 1, Ar_3)) [ 0 >= Ar_0 /\ Ar_0 >= 1 ]

We thus obtain the following problem:

2:	T:

		(Comp: ?, Cost: 1)    evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9stop(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_3, Ar_1, Ar_2 - 1, Ar_3)) [ 0 >= Ar_0 ]

		(Comp: ?, Cost: 1)    evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 ]

		(Comp: ?, Cost: 1)    evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ]

		(Comp: ?, Cost: 1)    evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]

		(Comp: ?, Cost: 1)    evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_1, Ar_1, Ar_3, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9start(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)    evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9stop(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_3, Ar_1, Ar_2 - 1, Ar_3)) [ 0 >= Ar_0 ]

		(Comp: ?, Cost: 1)    evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 ]

		(Comp: ?, Cost: 1)    evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ]

		(Comp: ?, Cost: 1)    evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]

		(Comp: 1, Cost: 1)    evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_1, Ar_1, Ar_3, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalcousot9bb3in) = 1

	Pol(evalcousot9stop) = 0

	Pol(evalcousot9bb2in) = 2

	Pol(evalcousot9bb1in) = 2

	Pol(evalcousot96) = 2

	Pol(evalcousot95) = 2

	Pol(evalcousot94) = 2

	Pol(evalcousot93) = 2

	Pol(evalcousot92) = 2

	Pol(evalcousot91) = 2

	Pol(evalcousot90) = 2

	Pol(evalcousot9bb0in) = 2

	Pol(evalcousot9start) = 2

	Pol(koat_start) = 2

orients all transitions weakly and the transitions

	evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9stop(Ar_0, Ar_1, Ar_2, Ar_3))

	evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ]

strictly and produces the following problem:

4:	T:

		(Comp: 2, Cost: 1)    evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9stop(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_3, Ar_1, Ar_2 - 1, Ar_3)) [ 0 >= Ar_0 ]

		(Comp: ?, Cost: 1)    evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 ]

		(Comp: 2, Cost: 1)    evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ]

		(Comp: ?, Cost: 1)    evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]

		(Comp: 1, Cost: 1)    evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_1, Ar_1, Ar_3, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Applied AI with 'oct' on problem 4 to obtain the following invariants:

  For symbol evalcousot9bb1in: -X_3 + X_4 >= 0

  For symbol evalcousot9bb2in: X_4 - 1 >= 0 /\ X_3 + X_4 - 2 >= 0 /\ -X_3 + X_4 >= 0 /\ X_3 - 1 >= 0

  For symbol evalcousot9bb3in: -X_3 + X_4 >= 0 /\ -X_3 >= 0





This yielded the following problem:

5:	T:

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

		(Comp: 1, Cost: 1)    evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)    evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_1, Ar_1, Ar_3, Ar_3))

		(Comp: ?, Cost: 1)    evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\ Ar_2 >= 1 ]

		(Comp: 2, Cost: 1)    evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\ 0 >= Ar_2 ]

		(Comp: ?, Cost: 1)    evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_0 >= 1 ]

		(Comp: ?, Cost: 1)    evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_3, Ar_1, Ar_2 - 1, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_2 - 1 >= 0 /\ 0 >= Ar_0 ]

		(Comp: 2, Cost: 1)    evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\ -Ar_2 >= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(koat_start) = V_4

	Pol(evalcousot9start) = V_4

	Pol(evalcousot9bb0in) = V_4

	Pol(evalcousot90) = V_4

	Pol(evalcousot91) = V_4

	Pol(evalcousot92) = V_4

	Pol(evalcousot93) = V_4

	Pol(evalcousot94) = V_4

	Pol(evalcousot95) = V_4

	Pol(evalcousot96) = V_4

	Pol(evalcousot9bb1in) = V_3

	Pol(evalcousot9bb2in) = V_3

	Pol(evalcousot9bb3in) = V_3

	Pol(evalcousot9stop) = V_3

orients all transitions weakly and the transition

	evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_3, Ar_1, Ar_2 - 1, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_2 - 1 >= 0 /\ 0 >= Ar_0 ]

strictly and produces the following problem:

6:	T:

		(Comp: 1, Cost: 0)       koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

		(Comp: 1, Cost: 1)       evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)       evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_1, Ar_1, Ar_3, Ar_3))

		(Comp: ?, Cost: 1)       evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\ Ar_2 >= 1 ]

		(Comp: 2, Cost: 1)       evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\ 0 >= Ar_2 ]

		(Comp: ?, Cost: 1)       evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_0 >= 1 ]

		(Comp: Ar_3, Cost: 1)    evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_3, Ar_1, Ar_2 - 1, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_2 - 1 >= 0 /\ 0 >= Ar_0 ]

		(Comp: 2, Cost: 1)       evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\ -Ar_2 >= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(evalcousot9bb2in) = V_1

	Pol(evalcousot9bb1in) = V_1

and size complexities

	S("evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\\ -Ar_2 >= 0 ]", 0-0) = Ar_1 + Ar_3

	S("evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\\ -Ar_2 >= 0 ]", 0-1) = Ar_1

	S("evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\\ -Ar_2 >= 0 ]", 0-2) = Ar_3

	S("evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\\ -Ar_2 >= 0 ]", 0-3) = Ar_3

	S("evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_3, Ar_1, Ar_2 - 1, Ar_3)) [ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_2 - 1 >= 0 /\\ 0 >= Ar_0 ]", 0-0) = Ar_3

	S("evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_3, Ar_1, Ar_2 - 1, Ar_3)) [ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_2 - 1 >= 0 /\\ 0 >= Ar_0 ]", 0-1) = Ar_1

	S("evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_3, Ar_1, Ar_2 - 1, Ar_3)) [ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_2 - 1 >= 0 /\\ 0 >= Ar_0 ]", 0-2) = Ar_3

	S("evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_3, Ar_1, Ar_2 - 1, Ar_3)) [ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_2 - 1 >= 0 /\\ 0 >= Ar_0 ]", 0-3) = Ar_3

	S("evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_0 >= 1 ]", 0-0) = Ar_1 + Ar_3

	S("evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_0 >= 1 ]", 0-1) = Ar_1

	S("evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_0 >= 1 ]", 0-2) = Ar_3

	S("evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_0 >= 1 ]", 0-3) = Ar_3

	S("evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\\ 0 >= Ar_2 ]", 0-0) = Ar_1 + Ar_3

	S("evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\\ 0 >= Ar_2 ]", 0-1) = Ar_1

	S("evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\\ 0 >= Ar_2 ]", 0-2) = Ar_3

	S("evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\\ 0 >= Ar_2 ]", 0-3) = Ar_3

	S("evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\\ Ar_2 >= 1 ]", 0-0) = Ar_1 + Ar_3

	S("evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\\ Ar_2 >= 1 ]", 0-1) = Ar_1

	S("evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\\ Ar_2 >= 1 ]", 0-2) = Ar_3

	S("evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\\ Ar_2 >= 1 ]", 0-3) = Ar_3

	S("evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_1, Ar_1, Ar_3, Ar_3))", 0-0) = Ar_1

	S("evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_1, Ar_1, Ar_3, Ar_3))", 0-1) = Ar_1

	S("evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_1, Ar_1, Ar_3, Ar_3))", 0-2) = Ar_3

	S("evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_1, Ar_1, Ar_3, Ar_3))", 0-3) = Ar_3

	S("evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0

	S("evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1

	S("evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2

	S("evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3

	S("evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0

	S("evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1

	S("evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2

	S("evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3

	S("evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0

	S("evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1

	S("evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2

	S("evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3

	S("evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0

	S("evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1

	S("evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2

	S("evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3

	S("evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0

	S("evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1

	S("evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2

	S("evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3

	S("evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0

	S("evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1

	S("evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2

	S("evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3

	S("evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0

	S("evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1

	S("evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2

	S("evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3

	S("evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0

	S("evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1

	S("evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2

	S("evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-0) = Ar_0

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-1) = Ar_1

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-2) = Ar_2

	S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3

orients the transitions

	evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_0 >= 1 ]

	evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\ Ar_2 >= 1 ]

weakly and the transition

	evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_0 >= 1 ]

strictly and produces the following problem:

7:	T:

		(Comp: 1, Cost: 0)                koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

		(Comp: 1, Cost: 1)                evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_1, Ar_1, Ar_3, Ar_3))

		(Comp: ?, Cost: 1)                evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\ Ar_2 >= 1 ]

		(Comp: 2, Cost: 1)                evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\ 0 >= Ar_2 ]

		(Comp: Ar_3^2 + Ar_1, Cost: 1)    evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_0 >= 1 ]

		(Comp: Ar_3, Cost: 1)             evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_3, Ar_1, Ar_2 - 1, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_2 - 1 >= 0 /\ 0 >= Ar_0 ]

		(Comp: 2, Cost: 1)                evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\ -Ar_2 >= 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(evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]

		(Comp: 1, Cost: 1)                           evalcousot9start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                           evalcousot9bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                           evalcousot90(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                           evalcousot91(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                           evalcousot92(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                           evalcousot93(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                           evalcousot94(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                           evalcousot95(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3))

		(Comp: 1, Cost: 1)                           evalcousot96(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_1, Ar_1, Ar_3, Ar_3))

		(Comp: Ar_3 + Ar_3^2 + Ar_1 + 1, Cost: 1)    evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\ Ar_2 >= 1 ]

		(Comp: 2, Cost: 1)                           evalcousot9bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\ 0 >= Ar_2 ]

		(Comp: Ar_3^2 + Ar_1, Cost: 1)               evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_0 >= 1 ]

		(Comp: Ar_3, Cost: 1)                        evalcousot9bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9bb1in(Ar_3, Ar_1, Ar_2 - 1, Ar_3)) [ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_2 - 1 >= 0 /\ 0 >= Ar_0 ]

		(Comp: 2, Cost: 1)                           evalcousot9bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalcousot9stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ -Ar_2 + Ar_3 >= 0 /\ -Ar_2 >= 0 ]

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 2*Ar_3 + 2*Ar_3^2 + 2*Ar_1 + 14



Time: 0.230 sec (SMT: 0.185 sec)


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

(2)
BOUNDS(1, n^2)

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

(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: evalcousot9start

      0: evalcousot9start -> evalcousot9bb0in : [], cost: 1

      1: evalcousot9bb0in -> evalcousot90 : [], cost: 1

      2: evalcousot90 -> evalcousot91 : [], cost: 1

      3: evalcousot91 -> evalcousot92 : [], cost: 1

      4: evalcousot92 -> evalcousot93 : [], cost: 1

      5: evalcousot93 -> evalcousot94 : [], cost: 1

      6: evalcousot94 -> evalcousot95 : [], cost: 1

      7: evalcousot95 -> evalcousot96 : [], cost: 1

      8: evalcousot96 -> evalcousot9bb1in : A'=B, C'=D, [], cost: 1

      9: evalcousot9bb1in -> evalcousot9bb2in : [ C>=1 ], cost: 1

     10: evalcousot9bb1in -> evalcousot9bb3in : [ 0>=C ], cost: 1

     11: evalcousot9bb2in -> evalcousot9bb1in : A'=-1+A, [ A>=1 ], cost: 1

     12: evalcousot9bb2in -> evalcousot9bb1in : A'=D, [ A>=1 && 0>=A ], cost: 1

     13: evalcousot9bb2in -> evalcousot9bb1in : A'=-1+A, C'=-1+C, [ 0>=A && A>=1 ], cost: 1

     14: evalcousot9bb2in -> evalcousot9bb1in : A'=D, C'=-1+C, [ 0>=A ], cost: 1

     15: evalcousot9bb3in -> evalcousot9stop : [], cost: 1



Checking for constant complexity:

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

      0: evalcousot9start -> evalcousot9bb0in : [], cost: 1



Removed unreachable and leaf rules:

   Start location: evalcousot9start

      0: evalcousot9start -> evalcousot9bb0in : [], cost: 1

      1: evalcousot9bb0in -> evalcousot90 : [], cost: 1

      2: evalcousot90 -> evalcousot91 : [], cost: 1

      3: evalcousot91 -> evalcousot92 : [], cost: 1

      4: evalcousot92 -> evalcousot93 : [], cost: 1

      5: evalcousot93 -> evalcousot94 : [], cost: 1

      6: evalcousot94 -> evalcousot95 : [], cost: 1

      7: evalcousot95 -> evalcousot96 : [], cost: 1

      8: evalcousot96 -> evalcousot9bb1in : A'=B, C'=D, [], cost: 1

      9: evalcousot9bb1in -> evalcousot9bb2in : [ C>=1 ], cost: 1

     11: evalcousot9bb2in -> evalcousot9bb1in : A'=-1+A, [ A>=1 ], cost: 1

     12: evalcousot9bb2in -> evalcousot9bb1in : A'=D, [ A>=1 && 0>=A ], cost: 1

     13: evalcousot9bb2in -> evalcousot9bb1in : A'=-1+A, C'=-1+C, [ 0>=A && A>=1 ], cost: 1

     14: evalcousot9bb2in -> evalcousot9bb1in : A'=D, C'=-1+C, [ 0>=A ], cost: 1



Removed rules with unsatisfiable guard:

   Start location: evalcousot9start

      0: evalcousot9start -> evalcousot9bb0in : [], cost: 1

      1: evalcousot9bb0in -> evalcousot90 : [], cost: 1

      2: evalcousot90 -> evalcousot91 : [], cost: 1

      3: evalcousot91 -> evalcousot92 : [], cost: 1

      4: evalcousot92 -> evalcousot93 : [], cost: 1

      5: evalcousot93 -> evalcousot94 : [], cost: 1

      6: evalcousot94 -> evalcousot95 : [], cost: 1

      7: evalcousot95 -> evalcousot96 : [], cost: 1

      8: evalcousot96 -> evalcousot9bb1in : A'=B, C'=D, [], cost: 1

      9: evalcousot9bb1in -> evalcousot9bb2in : [ C>=1 ], cost: 1

     11: evalcousot9bb2in -> evalcousot9bb1in : A'=-1+A, [ A>=1 ], cost: 1

     14: evalcousot9bb2in -> evalcousot9bb1in : A'=D, C'=-1+C, [ 0>=A ], cost: 1



### Simplification by acceleration and chaining ###



Eliminated locations (on linear paths):

   Start location: evalcousot9start

     23: evalcousot9start -> evalcousot9bb1in : A'=B, C'=D, [], cost: 9

      9: evalcousot9bb1in -> evalcousot9bb2in : [ C>=1 ], cost: 1

     11: evalcousot9bb2in -> evalcousot9bb1in : A'=-1+A, [ A>=1 ], cost: 1

     14: evalcousot9bb2in -> evalcousot9bb1in : A'=D, C'=-1+C, [ 0>=A ], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: evalcousot9start

     23: evalcousot9start -> evalcousot9bb1in : A'=B, C'=D, [], cost: 9

     24: evalcousot9bb1in -> evalcousot9bb1in : A'=-1+A, [ C>=1 && A>=1 ], cost: 2

     25: evalcousot9bb1in -> evalcousot9bb1in : A'=D, C'=-1+C, [ C>=1 && 0>=A ], cost: 2



Accelerating simple loops of location 9.

   Accelerating the following rules:

     24: evalcousot9bb1in -> evalcousot9bb1in : A'=-1+A, [ C>=1 && A>=1 ], cost: 2

     25: evalcousot9bb1in -> evalcousot9bb1in : A'=D, C'=-1+C, [ C>=1 && 0>=A ], cost: 2



   Accelerated rule 24 with metering function A, yielding the new rule 26.

   Accelerated rule 25 with metering function C (after strengthening guard), yielding the new rule 27.

   Nested simple loops 25 (outer loop) and 26 (inner loop) with metering function -1+C, resulting in the new rules: 28, 29.

   Removing the simple loops: 24 25.



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

   Start location: evalcousot9start

     23: evalcousot9start -> evalcousot9bb1in : A'=B, C'=D, [], cost: 9

     26: evalcousot9bb1in -> evalcousot9bb1in : A'=0, [ C>=1 && A>=1 ], cost: 2*A

     27: evalcousot9bb1in -> evalcousot9bb1in : A'=D, C'=0, [ C>=1 && 0>=A && 0>=D ], cost: 2*C

     28: evalcousot9bb1in -> evalcousot9bb1in : A'=0, C'=1, [ 0>=A && -1+C>=1 && D>=1 ], cost: -2+2*D*(-1+C)+2*C

     29: evalcousot9bb1in -> evalcousot9bb1in : A'=0, C'=1, [ A>=1 && -1+C>=1 && D>=1 ], cost: -2+2*D*(-1+C)+2*C+2*A



Chained accelerated rules (with incoming rules):

   Start location: evalcousot9start

     23: evalcousot9start -> evalcousot9bb1in : A'=B, C'=D, [], cost: 9

     30: evalcousot9start -> evalcousot9bb1in : A'=0, C'=D, [ D>=1 && B>=1 ], cost: 9+2*B

     31: evalcousot9start -> evalcousot9bb1in : A'=0, C'=1, [ 0>=B && -1+D>=1 ], cost: 7+2*D+2*(-1+D)*D

     32: evalcousot9start -> evalcousot9bb1in : A'=0, C'=1, [ B>=1 && -1+D>=1 ], cost: 7+2*D+2*(-1+D)*D+2*B



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

   Start location: evalcousot9start

     30: evalcousot9start -> evalcousot9bb1in : A'=0, C'=D, [ D>=1 && B>=1 ], cost: 9+2*B

     31: evalcousot9start -> evalcousot9bb1in : A'=0, C'=1, [ 0>=B && -1+D>=1 ], cost: 7+2*D+2*(-1+D)*D

     32: evalcousot9start -> evalcousot9bb1in : A'=0, C'=1, [ B>=1 && -1+D>=1 ], cost: 7+2*D+2*(-1+D)*D+2*B



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: evalcousot9start

     30: evalcousot9start -> evalcousot9bb1in : A'=0, C'=D, [ D>=1 && B>=1 ], cost: 9+2*B

     31: evalcousot9start -> evalcousot9bb1in : A'=0, C'=1, [ 0>=B && -1+D>=1 ], cost: 7+2*D+2*(-1+D)*D

     32: evalcousot9start -> evalcousot9bb1in : A'=0, C'=1, [ B>=1 && -1+D>=1 ], cost: 7+2*D+2*(-1+D)*D+2*B



Computing asymptotic complexity for rule 30

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      D (+/+!), 9+2*B (+), B (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {D==1,B==n}

      resulting limit problem:

      [solved]



   Solution:

      D / 1

      B / n

   Resulting cost 9+2*n has complexity: Poly(n^1)



Found new complexity Poly(n^1).



Computing asymptotic complexity for rule 31

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -1+D (+/+!), 1-B (+/+!), 7+2*D^2 (+) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



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

      resulting limit problem:

      [solved]



   Solution:

      D / n

      B / -n

   Resulting cost 7+2*n^2 has complexity: Poly(n^2)



Found new complexity Poly(n^2).



Obtained the following overall complexity (w.r.t. the length of the input n):

   Complexity:  Poly(n^2)

   Cpx degree:  2

   Solved cost: 7+2*n^2

   Rule cost:   7+2*D+2*(-1+D)*D

   Rule guard:  [ 0>=B && -1+D>=1 ]



WORST_CASE(Omega(n^2),?)


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

(4)
BOUNDS(n^2, INF)