
Initial complexity problem:
1:	T:
		(Comp: ?, Cost: 1)    f1(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_0)) [ Ar_0 >= 1 ]
		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ]
		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0 - 1, Ar_0 - 1)) [ 0 >= Ar_1 /\ Ar_0 >= 2 ]
		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ 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) -> Com_1(f3(Ar_0, Ar_0)) [ Ar_0 >= 1 ]
		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ]
		(Comp: ?, Cost: 1)    f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0 - 1, Ar_0 - 1)) [ 0 >= Ar_1 /\ Ar_0 >= 2 ]
		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ 0 <= 0 ]
	start location:	koat_start
	leaf cost:	0

A polynomial rank function with
	Pol(f1) = V_1
	Pol(f3) = V_1
	Pol(koat_start) = V_1
orients all transitions weakly and the transition
	f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0 - 1, Ar_0 - 1)) [ 0 >= Ar_1 /\ Ar_0 >= 2 ]
strictly and produces the following problem:
3:	T:
		(Comp: 1, Cost: 1)       f1(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_0)) [ Ar_0 >= 1 ]
		(Comp: ?, Cost: 1)       f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ]
		(Comp: Ar_0, Cost: 1)    f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0 - 1, Ar_0 - 1)) [ 0 >= Ar_1 /\ Ar_0 >= 2 ]
		(Comp: 1, Cost: 0)       koat_start(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ 0 <= 0 ]
	start location:	koat_start
	leaf cost:	0

A polynomial rank function with
	Pol(f3) = V_2
and size complexities
	S("koat_start(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0
	S("koat_start(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1
	S("f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0 - 1, Ar_0 - 1)) [ 0 >= Ar_1 /\\ Ar_0 >= 2 ]", 0-0) = Ar_0
	S("f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0 - 1, Ar_0 - 1)) [ 0 >= Ar_1 /\\ Ar_0 >= 2 ]", 0-1) = Ar_0
	S("f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ]", 0-0) = Ar_0
	S("f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ]", 0-1) = Ar_0
	S("f1(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_0)) [ Ar_0 >= 1 ]", 0-0) = Ar_0
	S("f1(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_0)) [ Ar_0 >= 1 ]", 0-1) = Ar_0
orients the transitions
	f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ]
weakly and the transition
	f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ]
strictly and produces the following problem:
4:	T:
		(Comp: 1, Cost: 1)                f1(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_0)) [ Ar_0 >= 1 ]
		(Comp: Ar_0^2 + Ar_0, Cost: 1)    f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ]
		(Comp: Ar_0, Cost: 1)             f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0 - 1, Ar_0 - 1)) [ 0 >= Ar_1 /\ Ar_0 >= 2 ]
		(Comp: 1, Cost: 0)                koat_start(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ 0 <= 0 ]
	start location:	koat_start
	leaf cost:	0

Complexity upper bound Ar_0^2 + 2*Ar_0 + 1

Time: 0.034 sec (SMT: 0.032 sec)
