
Initial complexity problem:
1:	T:
		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2))
		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_2, Ar_2)) [ Fresh_2 >= 1 /\ Ar_0 >= 0 ]
		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_1, Ar_2)) [ 0 >= Fresh_1 + 1 /\ Ar_0 >= 0 ]
		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, 0, Ar_2)) [ Ar_0 >= 0 ]
		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_0)) [ 0 >= Ar_0 + 1 ]
		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f3(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)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2))
		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_2, Ar_2)) [ Fresh_2 >= 1 /\ Ar_0 >= 0 ]
		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_1, Ar_2)) [ 0 >= Fresh_1 + 1 /\ Ar_0 >= 0 ]
		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, 0, Ar_2)) [ Ar_0 >= 0 ]
		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_0)) [ 0 >= Ar_0 + 1 ]
		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]
	start location:	koat_start
	leaf cost:	0

A polynomial rank function with
	Pol(f3) = 1
	Pol(f2) = 1
	Pol(f300) = 0
	Pol(koat_start) = 1
orients all transitions weakly and the transition
	f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_0)) [ 0 >= Ar_0 + 1 ]
strictly and produces the following problem:
3:	T:
		(Comp: 1, Cost: 1)    f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2))
		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_2, Ar_2)) [ Fresh_2 >= 1 /\ Ar_0 >= 0 ]
		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_1, Ar_2)) [ 0 >= Fresh_1 + 1 /\ Ar_0 >= 0 ]
		(Comp: ?, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, 0, Ar_2)) [ Ar_0 >= 0 ]
		(Comp: 1, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_0)) [ 0 >= Ar_0 + 1 ]
		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]
	start location:	koat_start
	leaf cost:	0

A polynomial rank function with
	Pol(f3) = V_1 + 1
	Pol(f2) = V_1 + 1
	Pol(f300) = V_1
	Pol(koat_start) = V_1 + 1
orients all transitions weakly and the transitions
	f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_2, Ar_2)) [ Fresh_2 >= 1 /\ Ar_0 >= 0 ]
	f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_1, Ar_2)) [ 0 >= Fresh_1 + 1 /\ Ar_0 >= 0 ]
strictly and produces the following problem:
4:	T:
		(Comp: 1, Cost: 1)           f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2))
		(Comp: Ar_0 + 1, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_2, Ar_2)) [ Fresh_2 >= 1 /\ Ar_0 >= 0 ]
		(Comp: Ar_0 + 1, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_1, Ar_2)) [ 0 >= Fresh_1 + 1 /\ Ar_0 >= 0 ]
		(Comp: ?, Cost: 1)           f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, 0, Ar_2)) [ Ar_0 >= 0 ]
		(Comp: 1, Cost: 1)           f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_0)) [ 0 >= Ar_0 + 1 ]
		(Comp: 1, Cost: 0)           koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]
	start location:	koat_start
	leaf cost:	0

By chaining the transition koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] with all transitions in problem 4, the following new transition is obtained:
	koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]
We thus obtain the following problem:
5:	T:
		(Comp: 1, Cost: 1)           koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]
		(Comp: 1, Cost: 1)           f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2))
		(Comp: Ar_0 + 1, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_2, Ar_2)) [ Fresh_2 >= 1 /\ Ar_0 >= 0 ]
		(Comp: Ar_0 + 1, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_1, Ar_2)) [ 0 >= Fresh_1 + 1 /\ Ar_0 >= 0 ]
		(Comp: ?, Cost: 1)           f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, 0, Ar_2)) [ Ar_0 >= 0 ]
		(Comp: 1, Cost: 1)           f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_0)) [ 0 >= Ar_0 + 1 ]
	start location:	koat_start
	leaf cost:	0

Testing for reachability in the complexity graph removes the following transition from problem 5:
	f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2))
We thus obtain the following problem:
6:	T:
		(Comp: 1, Cost: 1)           f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_0)) [ 0 >= Ar_0 + 1 ]
		(Comp: ?, Cost: 1)           f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, 0, Ar_2)) [ Ar_0 >= 0 ]
		(Comp: Ar_0 + 1, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_1, Ar_2)) [ 0 >= Fresh_1 + 1 /\ Ar_0 >= 0 ]
		(Comp: Ar_0 + 1, Cost: 1)    f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_2, Ar_2)) [ Fresh_2 >= 1 /\ Ar_0 >= 0 ]
		(Comp: 1, Cost: 1)           koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]
	start location:	koat_start
	leaf cost:	0

Complexity upper bound ?

Time: 0.056 sec (SMT: 0.046 sec)
