/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^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2).

(0) CpxIntTrs
(1) Koat Proof [FINISHED, 147 ms]
(2) BOUNDS(1, n^2)
(3) Loat Proof [FINISHED, 746 ms]
(4) BOUNDS(n^1, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f1(A, B, C) -> Com_1(f2(A, B, B + 1)) :|: A >= B && A >= 1 && B >= 1
f2(A, B, C) -> Com_1(f3(A, B, C)) :|: B >= C + 1
f2(A, B, C) -> Com_1(f3(A, B, C)) :|: C >= B + 1
f3(A, B, C) -> Com_1(f2(A, B, 0)) :|: A + 1 >= 0 && C >= 1 && C >= A + 1
f3(A, B, C) -> Com_1(f2(A, B, C + 1)) :|: A >= C && C + 1 >= 0

The start-symbols are:[f1_3]


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(1) Koat Proof (FINISHED)
YES(?, 20*Ar_0^2 + 32*Ar_0*Ar_1 + 280*Ar_0 + 260*Ar_1 + 12*Ar_1^2 + 576)



Initial complexity problem:

1:	T:

		(Comp: ?, Cost: 1)    f1(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ Ar_0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 ]

		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]

		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, 0)) [ Ar_0 + 1 >= 0 /\ Ar_2 >= 1 /\ Ar_2 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) [ 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)    f1(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ Ar_0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 ]

		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]

		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, 0)) [ Ar_0 + 1 >= 0 /\ Ar_2 >= 1 /\ Ar_2 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Applied AI with 'oct' on problem 2 to obtain the following invariants:

  For symbol f2: X_1 - X_3 + 1 >= 0 /\ X_1 - X_2 >= 0 /\ X_2 - 1 >= 0 /\ X_1 + X_2 - 2 >= 0 /\ X_1 - 1 >= 0

  For symbol f3: X_1 - X_3 + 1 >= 0 /\ X_1 - X_2 >= 0 /\ X_2 - 1 >= 0 /\ X_1 + X_2 - 2 >= 0 /\ X_1 - 1 >= 0





This yielded the following problem:

3:	T:

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, 0)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 + 1 >= 0 /\ Ar_2 >= 1 /\ Ar_2 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= Ar_1 + 1 ]

		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 ]

		(Comp: 1, Cost: 1)    f1(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ Ar_0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 ]

	start location:	koat_start

	leaf cost:	0



By chaining the transition koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] with all transitions in problem 3, the following new transition is obtained:

	koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ 0 <= 0 /\ Ar_0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 ]

We thus obtain the following problem:

4:	T:

		(Comp: 1, Cost: 1)    koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ 0 <= 0 /\ Ar_0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, 0)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 + 1 >= 0 /\ Ar_2 >= 1 /\ Ar_2 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= Ar_1 + 1 ]

		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 ]

		(Comp: 1, Cost: 1)    f1(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ Ar_0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 ]

	start location:	koat_start

	leaf cost:	0



Testing for reachability in the complexity graph removes the following transition from problem 4:

	f1(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ Ar_0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 ]

We thus obtain the following problem:

5:	T:

		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, 0)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 + 1 >= 0 /\ Ar_2 >= 1 /\ Ar_2 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= Ar_1 + 1 ]

		(Comp: 1, Cost: 1)    koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ 0 <= 0 /\ Ar_0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 ]

	start location:	koat_start

	leaf cost:	0



By chaining the transition f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 ] with all transitions in problem 5, the following new transition is obtained:

	f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 /\ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

We thus obtain the following problem:

6:	T:

		(Comp: ?, Cost: 2)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 /\ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, 0)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 + 1 >= 0 /\ Ar_2 >= 1 /\ Ar_2 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= Ar_1 + 1 ]

		(Comp: 1, Cost: 1)    koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ 0 <= 0 /\ Ar_0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(f3) = 2*V_1 - 2*V_3 + 2

	Pol(f2) = 2*V_1 - 2*V_3 + 3

and size complexities

	S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ 0 <= 0 /\\ Ar_0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0

	S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ 0 <= 0 /\\ Ar_0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1

	S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ 0 <= 0 /\\ Ar_0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-2) = Ar_1 + 1

	S("f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= Ar_1 + 1 ]", 0-0) = Ar_0

	S("f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= Ar_1 + 1 ]", 0-1) = Ar_1

	S("f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= Ar_1 + 1 ]", 0-2) = ?

	S("f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_2 /\\ Ar_2 + 1 >= 0 ]", 0-0) = Ar_0

	S("f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_2 /\\ Ar_2 + 1 >= 0 ]", 0-1) = Ar_1

	S("f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_2 /\\ Ar_2 + 1 >= 0 ]", 0-2) = ?

	S("f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, 0)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 + 1 >= 0 /\\ Ar_2 >= 1 /\\ Ar_2 >= Ar_0 + 1 ]", 0-0) = Ar_0

	S("f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, 0)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 + 1 >= 0 /\\ Ar_2 >= 1 /\\ Ar_2 >= Ar_0 + 1 ]", 0-1) = Ar_1

	S("f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, 0)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 + 1 >= 0 /\\ Ar_2 >= 1 /\\ Ar_2 >= Ar_0 + 1 ]", 0-2) = 0

	S("f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_2 + 1 /\\ Ar_0 >= Ar_2 /\\ Ar_2 + 1 >= 0 ]", 0-0) = Ar_0

	S("f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_2 + 1 /\\ Ar_0 >= Ar_2 /\\ Ar_2 + 1 >= 0 ]", 0-1) = Ar_1

	S("f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_2 + 1 /\\ Ar_0 >= Ar_2 /\\ Ar_2 + 1 >= 0 ]", 0-2) = Ar_0

orients the transitions

	f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

	f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= Ar_1 + 1 ]

weakly and the transitions

	f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

	f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= Ar_1 + 1 ]

strictly and produces the following problem:

7:	T:

		(Comp: ?, Cost: 2)                      f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 /\ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

		(Comp: ?, Cost: 1)                      f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, 0)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 + 1 >= 0 /\ Ar_2 >= 1 /\ Ar_2 >= Ar_0 + 1 ]

		(Comp: 2*Ar_0 + 2*Ar_1 + 5, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

		(Comp: 2*Ar_0 + 2*Ar_1 + 5, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= Ar_1 + 1 ]

		(Comp: 1, Cost: 1)                      koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ 0 <= 0 /\ Ar_0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 ]

	start location:	koat_start

	leaf cost:	0



Repeatedly propagating knowledge in problem 7 produces the following problem:

8:	T:

		(Comp: ?, Cost: 2)                      f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 /\ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

		(Comp: 2*Ar_0 + 2*Ar_1 + 5, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, 0)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 + 1 >= 0 /\ Ar_2 >= 1 /\ Ar_2 >= Ar_0 + 1 ]

		(Comp: 2*Ar_0 + 2*Ar_1 + 5, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

		(Comp: 2*Ar_0 + 2*Ar_1 + 5, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= Ar_1 + 1 ]

		(Comp: 1, Cost: 1)                      koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ 0 <= 0 /\ Ar_0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(f2) = V_1 - V_3 + 1

and size complexities

	S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ 0 <= 0 /\\ Ar_0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0

	S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ 0 <= 0 /\\ Ar_0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1

	S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ 0 <= 0 /\\ Ar_0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-2) = Ar_1 + 1

	S("f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= Ar_1 + 1 ]", 0-0) = Ar_0

	S("f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= Ar_1 + 1 ]", 0-1) = Ar_1

	S("f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= Ar_1 + 1 ]", 0-2) = 3*Ar_0 + 3*Ar_1 + 54

	S("f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_2 /\\ Ar_2 + 1 >= 0 ]", 0-0) = Ar_0

	S("f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_2 /\\ Ar_2 + 1 >= 0 ]", 0-1) = Ar_1

	S("f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_2 /\\ Ar_2 + 1 >= 0 ]", 0-2) = 3*Ar_0 + 3*Ar_1 + 54

	S("f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, 0)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 + 1 >= 0 /\\ Ar_2 >= 1 /\\ Ar_2 >= Ar_0 + 1 ]", 0-0) = Ar_0

	S("f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, 0)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 + 1 >= 0 /\\ Ar_2 >= 1 /\\ Ar_2 >= Ar_0 + 1 ]", 0-1) = Ar_1

	S("f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, 0)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 + 1 >= 0 /\\ Ar_2 >= 1 /\\ Ar_2 >= Ar_0 + 1 ]", 0-2) = 0

	S("f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_2 + 1 /\\ Ar_0 >= Ar_2 /\\ Ar_2 + 1 >= 0 ]", 0-0) = Ar_0

	S("f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_2 + 1 /\\ Ar_0 >= Ar_2 /\\ Ar_2 + 1 >= 0 ]", 0-1) = Ar_1

	S("f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_2 + 1 /\\ Ar_0 >= Ar_2 /\\ Ar_2 + 1 >= 0 ]", 0-2) = Ar_0

orients the transitions

	f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 /\ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

weakly and the transition

	f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 /\ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

strictly and produces the following problem:

9:	T:

		(Comp: 10*Ar_0^2 + 16*Ar_0*Ar_1 + 137*Ar_0 + 127*Ar_1 + 6*Ar_1^2 + 280, Cost: 2)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 /\ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

		(Comp: 2*Ar_0 + 2*Ar_1 + 5, Cost: 1)                                                f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, 0)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 + 1 >= 0 /\ Ar_2 >= 1 /\ Ar_2 >= Ar_0 + 1 ]

		(Comp: 2*Ar_0 + 2*Ar_1 + 5, Cost: 1)                                                f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_2 /\ Ar_2 + 1 >= 0 ]

		(Comp: 2*Ar_0 + 2*Ar_1 + 5, Cost: 1)                                                f2(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_2 + 1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= Ar_1 + 1 ]

		(Comp: 1, Cost: 1)                                                                  koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_1 + 1)) [ 0 <= 0 /\ Ar_0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 ]

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 20*Ar_0^2 + 32*Ar_0*Ar_1 + 280*Ar_0 + 260*Ar_1 + 12*Ar_1^2 + 576



Time: 0.207 sec (SMT: 0.175 sec)


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

(2)
BOUNDS(1, n^2)

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

(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f1

      0: f1 -> f2 : C'=1+B, [ A>=B && A>=1 && B>=1 ], cost: 1

      1: f2 -> f3 : [ B>=1+C ], cost: 1

      2: f2 -> f3 : [ C>=1+B ], cost: 1

      3: f3 -> f2 : C'=0, [ 1+A>=0 && C>=1 && C>=1+A ], cost: 1

      4: f3 -> f2 : C'=1+C, [ A>=C && 1+C>=0 ], cost: 1



Checking for constant complexity:

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

      0: f1 -> f2 : C'=1+B, [ A>=B && A>=1 && B>=1 ], cost: 1



### Simplification by acceleration and chaining ###



Eliminated locations (on tree-shaped paths):

   Start location: f1

      0: f1 -> f2 : C'=1+B, [ A>=B && A>=1 && B>=1 ], cost: 1

      5: f2 -> f2 : C'=0, [ B>=1+C && 1+A>=0 && C>=1 && C>=1+A ], cost: 2

      6: f2 -> f2 : C'=1+C, [ B>=1+C && A>=C && 1+C>=0 ], cost: 2

      7: f2 -> f2 : C'=0, [ C>=1+B && 1+A>=0 && C>=1 && C>=1+A ], cost: 2

      8: f2 -> f2 : C'=1+C, [ C>=1+B && A>=C && 1+C>=0 ], cost: 2



Accelerating simple loops of location 1.

   Accelerating the following rules:

      5: f2 -> f2 : C'=0, [ B>=1+C && 1+A>=0 && C>=1 && C>=1+A ], cost: 2

      6: f2 -> f2 : C'=1+C, [ B>=1+C && A>=C && 1+C>=0 ], cost: 2

      7: f2 -> f2 : C'=0, [ C>=1+B && 1+A>=0 && C>=1 && C>=1+A ], cost: 2

      8: f2 -> f2 : C'=1+C, [ C>=1+B && A>=C && 1+C>=0 ], cost: 2



   Found no metering function for rule 5.

   Found no metering function for rule 6.

   Found no metering function for rule 7.

   Accelerated rule 8 with metering function 1-C+A, yielding the new rule 9.

   Nested simple loops 7 (outer loop) and 9 (inner loop) with NONTERM, resulting in the new rules: 10, 11.

   Removing the simple loops: 7 8.



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

   Start location: f1

      0: f1 -> f2 : C'=1+B, [ A>=B && A>=1 && B>=1 ], cost: 1

      5: f2 -> f2 : C'=0, [ B>=1+C && 1+A>=0 && C>=1 && C>=1+A ], cost: 2

      6: f2 -> f2 : C'=1+C, [ B>=1+C && A>=C && 1+C>=0 ], cost: 2

      9: f2 -> f2 : C'=1+A, [ C>=1+B && A>=C && 1+C>=0 ], cost: 2-2*C+2*A

     10: f2 -> [3] : [ C>=1+B && C>=1 && C>=1+A && 0>=1+B && A>=0 && 4+2*A>=1 ], cost: NONTERM

     11: f2 -> [3] : C'=1+A, [ C>=1+B && A>=C && 1+C>=0 && 1+A>=1+B && 1+A>=1 && 0>=1+B && 4+2*A>=1 ], cost: NONTERM



Chained accelerated rules (with incoming rules):

   Start location: f1

      0: f1 -> f2 : C'=1+B, [ A>=B && A>=1 && B>=1 ], cost: 1

     12: f1 -> f2 : C'=1+A, [ A>=1 && B>=1 && A>=1+B ], cost: 1+2*A-2*B



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

   Start location: f1

     12: f1 -> f2 : C'=1+A, [ A>=1 && B>=1 && A>=1+B ], cost: 1+2*A-2*B



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f1

     12: f1 -> f2 : C'=1+A, [ A>=1 && B>=1 && A>=1+B ], cost: 1+2*A-2*B



Computing asymptotic complexity for rule 12

   Simplified the guard:

     12: f1 -> f2 : C'=1+A, [ B>=1 && A>=1+B ], cost: 1+2*A-2*B

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      1+2*A-2*B (+), A-B (+/+!), B (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



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

      resulting limit problem:

      [solved]



   Solution:

      A / n

      B / 1

   Resulting cost -1+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: -1+2*n

   Rule cost:   1+2*A-2*B

   Rule guard:  [ B>=1 && A>=1+B ]



WORST_CASE(Omega(n^1),?)


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

(4)
BOUNDS(n^1, INF)