# Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, Arg_0+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) -{0}> g(Arg_0,Arg_1)
g(Arg_0,Arg_1) -> 1/2:g(Arg_0-1,Arg_1) :+: 1/2:g(Arg_0,Arg_1-1) :|: 0<Arg_0 && 0<Arg_1

### Timebounds:

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

### Expected Timebounds:

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