
Initial complexity problem:
1:	T:
		(Comp: ?, Cost: 1)    eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, 0)) [ Ar_0 >= 1 ]
		(Comp: ?, Cost: 1)    eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 1 /\ Ar_1 >= 0 /\ Ar_0 >= Ar_1 + 1 ]
		(Comp: ?, Cost: 1)    eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 0 /\ Ar_1 >= Ar_0 ]
		(Comp: ?, Cost: 1)    start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1))
		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1) -> Com_1(start(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: ?, Cost: 1)    eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, 0)) [ Ar_0 >= 1 ]
		(Comp: ?, Cost: 1)    eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 1 /\ Ar_1 >= 0 /\ Ar_0 >= Ar_1 + 1 ]
		(Comp: ?, Cost: 1)    eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 0 /\ Ar_1 >= Ar_0 ]
		(Comp: 1, Cost: 1)    start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1))
		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ]
	start location:	koat_start
	leaf cost:	0

A polynomial rank function with
	Pol(eval1) = 2*V_1
	Pol(eval2) = 2*V_1 - 1
	Pol(start) = 2*V_1
	Pol(koat_start) = 2*V_1
orients all transitions weakly and the transitions
	eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 0 /\ Ar_1 >= Ar_0 ]
	eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, 0)) [ Ar_0 >= 1 ]
strictly and produces the following problem:
3:	T:
		(Comp: 2*Ar_0, Cost: 1)    eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, 0)) [ Ar_0 >= 1 ]
		(Comp: ?, Cost: 1)         eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 1 /\ Ar_1 >= 0 /\ Ar_0 >= Ar_1 + 1 ]
		(Comp: 2*Ar_0, Cost: 1)    eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 0 /\ Ar_1 >= Ar_0 ]
		(Comp: 1, Cost: 1)         start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1))
		(Comp: 1, Cost: 0)         koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ]
	start location:	koat_start
	leaf cost:	0

A polynomial rank function with
	Pol(eval2) = V_1 - V_2
and size complexities
	S("koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0
	S("koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1
	S("start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1))", 0-0) = Ar_0
	S("start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1))", 0-1) = Ar_1
	S("eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 0 /\\ Ar_1 >= Ar_0 ]", 0-0) = Ar_0
	S("eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 0 /\\ Ar_1 >= Ar_0 ]", 0-1) = Ar_0
	S("eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 1 /\\ Ar_1 >= 0 /\\ Ar_0 >= Ar_1 + 1 ]", 0-0) = Ar_0
	S("eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 1 /\\ Ar_1 >= 0 /\\ Ar_0 >= Ar_1 + 1 ]", 0-1) = Ar_0
	S("eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, 0)) [ Ar_0 >= 1 ]", 0-0) = Ar_0
	S("eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, 0)) [ Ar_0 >= 1 ]", 0-1) = 0
orients the transitions
	eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 1 /\ Ar_1 >= 0 /\ Ar_0 >= Ar_1 + 1 ]
weakly and the transition
	eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 1 /\ Ar_1 >= 0 /\ Ar_0 >= Ar_1 + 1 ]
strictly and produces the following problem:
4:	T:
		(Comp: 2*Ar_0, Cost: 1)      eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, 0)) [ Ar_0 >= 1 ]
		(Comp: 2*Ar_0^2, Cost: 1)    eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 1 /\ Ar_1 >= 0 /\ Ar_0 >= Ar_1 + 1 ]
		(Comp: 2*Ar_0, Cost: 1)      eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 0 /\ Ar_1 >= Ar_0 ]
		(Comp: 1, Cost: 1)           start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1))
		(Comp: 1, Cost: 0)           koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ]
	start location:	koat_start
	leaf cost:	0

Complexity upper bound 4*Ar_0 + 2*Ar_0^2 + 1

Time: 0.038 sec (SMT: 0.031 sec)
