# Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, 2*Arg_0+2*Arg_1+4*Arg_2 {O(n)})

### Initial Complexity Problem (after preprocessing)

Start:f
Program_Vars:Arg_0, Arg_1, Arg_2
Temp_Vars:
Locations:f, g
Transitions:
f(Arg_0,Arg_1,Arg_2) -{0}> g(Arg_0,Arg_1,Arg_2) :|: 0<=Arg_2 && Arg_1<=0 && 0<Arg_1+Arg_2
g(Arg_0,Arg_1,Arg_2) -> 1/2:g(Arg_0-Arg_1,Arg_2,Arg_2) :+: 1/2:g(Arg_0,Arg_1,Arg_2) :|: 0<Arg_0 && 1<=Arg_2 && 1<=Arg_1+Arg_2 && Arg_1<=Arg_2

### Timebounds:

Overall timebound:inf {Infinity}
0,0: f->g: 1 {O(1)}
1,1: g->g: max([-(Arg_1)+Arg_0+Arg_2, 0]) {O(n)}
2,1: g->g: inf {Infinity}

### Expected Timebounds:

Overall expected timebound: 1+2*Arg_0+2*Arg_1+4*Arg_2 {O(n)}
0: f->[1:g]: 1 {O(1)}
1: g->[1/2:g; 1/2:g]: 2*Arg_0+2*Arg_1+4*Arg_2 {O(n)}

### Costbounds:

Overall costbound: inf {Infinity}
0,0: f->g: inf {Infinity}
1,1: g->g: inf {Infinity}
2,1: g->g: inf {Infinity}

### Expected Costbounds:

Overall expected costbound: 2*Arg_0+2*Arg_1+4*Arg_2 {O(n)}
0: f->[1:g]: 0 {O(1)}
1: g->[1/2:g; 1/2:g]: 2*Arg_0+2*Arg_1+4*Arg_2 {O(n)}

### Sizebounds:

0,0: f->g, Arg_0: Arg_0 {O(n)}
0,0: f->g, Arg_1: 0 {O(1)}
0,0: f->g, Arg_2: Arg_2 {O(n)}
1,1: g->g, Arg_0: max([-(Arg_1)+Arg_0+Arg_2, 0])*max([-(Arg_1), 0])+max([max([0, Arg_0]), Arg_2]) {O(n^2)}
1,1: g->g, Arg_1: Arg_2 {O(n)}
1,1: g->g, Arg_2: Arg_2 {O(n)}
2,1: g->g, Arg_0: max([-(Arg_1)+Arg_0+Arg_2, 0])*max([-(Arg_1), 0])+max([max([0, Arg_0]), Arg_2]) {O(n^2)}
2,1: g->g, Arg_1: max([0, Arg_2]) {O(n)}
2,1: g->g, Arg_2: Arg_2 {O(n)}

### ExpSizeBounds:

(0: f->[1:g], g), Arg_0: Arg_0 {O(n)}
(0: f->[1:g], g), Arg_1: Arg_1 {O(n)}
(0: f->[1:g], g), Arg_2: Arg_2 {O(n)}
(1: g->[1/2:g; 1/2:g], g), Arg_0: max([(Arg_0+Arg_1+Arg_2)*Arg_1+max([Arg_0, Arg_2]), 1]) {O(n^2)}
(1: g->[1/2:g; 1/2:g], g), Arg_1: max([max([1, Arg_1]), Arg_2]) {O(n)}
(1: g->[1/2:g; 1/2:g], g), Arg_2: Arg_2 {O(n)}