# Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, 3/2*Arg_0 {O(n)})

### Initial Complexity Problem (after preprocessing)

Start:f
Program_Vars:Arg_0
Temp_Vars:
Locations:f, g, h, i
Transitions:
f(Arg_0) -{0}> g(Arg_0)
g(Arg_0) -{0}> h(Arg_0) :|: 0<Arg_0
h(Arg_0) -{0}> i(Arg_0) :|: 1<=Arg_0
h(Arg_0) -> 2/3:g(Arg_0-1) :+: 1/3:g(Arg_0) :|: 1<=Arg_0

### Timebounds:

Overall timebound:inf {Infinity}
0,0: f->g: 1 {O(1)}
1,1: g->h: inf {Infinity}
2,2: h->i: 1 {O(1)}
3,3: h->g: max([0, Arg_0]) {O(n)}
4,3: h->g: inf {Infinity}

### Expected Timebounds:

Overall expected timebound: 3+3*Arg_0 {O(n)}
0: f->[1:g]: 1 {O(1)}
1: g->[1:h]: 1+3/2*Arg_0 {O(n)}
2: h->[1:i]: 1 {O(1)}
3: h->[2/3:g; 1/3:g]: 3/2*Arg_0 {O(n)}

### Costbounds:

Overall costbound: inf {Infinity}
0,0: f->g: inf {Infinity}
1,1: g->h: inf {Infinity}
2,2: h->i: inf {Infinity}
3,3: h->g: inf {Infinity}
4,3: h->g: inf {Infinity}

### Expected Costbounds:

Overall expected costbound: 3/2*Arg_0 {O(n)}
0: f->[1:g]: 0 {O(1)}
1: g->[1:h]: 0 {O(1)}
2: h->[1:i]: 0 {O(1)}
3: h->[2/3:g; 1/3:g]: 3/2*Arg_0 {O(n)}

### Sizebounds:

0,0: f->g, Arg_0: Arg_0 {O(n)}
1,1: g->h, Arg_0: Arg_0 {O(n)}
2,2: h->i, Arg_0: Arg_0 {O(n)}
3,3: h->g, Arg_0: Arg_0 {O(n)}
4,3: h->g, Arg_0: Arg_0 {O(n)}

### ExpSizeBounds:

(0: f->[1:g], g), Arg_0: Arg_0 {O(n)}
(1: g->[1:h], h), Arg_0: Arg_0 {O(n)}
(2: h->[1:i], i), Arg_0: Arg_0 {O(n)}
(3: h->[2/3:g; 1/3:g], g), Arg_0: Arg_0 {O(n)}