# Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, 1/5*Arg_1+9/5 {O(n)})

### Initial Complexity Problem (after preprocessing)

Start:f
Program_Vars:Arg_0, Arg_1
Temp_Vars:
Locations:f, g
Transitions:
f(Arg_0,Arg_1) -{0}> g(0,Arg_1)
g(Arg_0,Arg_1) -> g(Binomial (10, 1/2),Arg_1) :|: Arg_0<Arg_1 && 0<=Arg_0

### Timebounds:

Overall timebound:inf {Infinity}
0,0: f->g: 1 {O(1)}
1,1: g->g: inf {Infinity}

### Expected Timebounds:

Overall expected timebound: 1/5*Arg_1+14/5 {O(n)}
0: f->[1:g]: 1 {O(1)}
1: g->[1:g]: 1/5*Arg_1+9/5 {O(n)}

### Costbounds:

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

### Expected Costbounds:

Overall expected costbound: 1/5*Arg_1+9/5 {O(n)}
0: f->[1:g]: 0 {O(1)}
1: g->[1:g]: 1/5*Arg_1+9/5 {O(n)}

### Sizebounds:

0,0: f->g, Arg_0: 0 {O(1)}
0,0: f->g, Arg_1: Arg_1 {O(n)}
1,1: g->g, Arg_1: Arg_1 {O(n)}

### ExpSizeBounds:

(0: f->[1:g], g), Arg_0: 0 {O(1)}
(0: f->[1:g], g), Arg_1: Arg_1 {O(n)}
(1: g->[1:g], g), Arg_0: 5*(1/5*Arg_1+9/5) {O(n)}
(1: g->[1:g], g), Arg_1: Arg_1 {O(n)}