# Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, 2/3*Arg_0+7+2/3*Arg_1 {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) -> g(Arg_0,Arg_1)
g(Arg_0,Arg_1) -> 1/2:g(Arg_0,Arg_1+1) :+: 1/2:g(Uniform (0, 10),Arg_1+1) :|: Arg_0<=Arg_1

### Timebounds:

Overall timebound:inf {Infinity}
0,0: f->g: 1 {O(1)}
1,1: g->g: inf {Infinity}
2,1: g->g: inf {Infinity}

### Expected Timebounds:

Overall expected timebound: 2/3*Arg_0+7+2/3*Arg_1 {O(n)}
0: f->[1:g]: 1 {O(1)}
1: g->[1/2:g; 1/2:g]: 2/3*Arg_0+6+2/3*Arg_1 {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: 2/3*Arg_0+7+2/3*Arg_1 {O(n)}
0: f->[1:g]: 1 {O(1)}
1: g->[1/2:g; 1/2:g]: 2/3*Arg_0+6+2/3*Arg_1 {O(n)}

### Sizebounds:

0,0: f->g, Arg_0: Arg_0 {O(n)}
0,0: f->g, Arg_1: Arg_1 {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)}
(1: g->[1/2:g; 1/2:g], g), Arg_0: 5/2*(2/3*Arg_0+6+2/3*Arg_1)+Arg_0 {O(n)}
(1: g->[1/2:g; 1/2:g], g), Arg_1: 2/3*Arg_0+6+5/3*Arg_1 {O(n)}