# Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, 151+Arg_1+2*Arg_0 {O(n)})

### Initial Complexity Problem (after preprocessing)

Start:f
Program_Vars:Arg_0, Arg_1
Temp_Vars:
Locations:f, g, h
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) :|: 100<Arg_0
g(Arg_0,Arg_1) -{0}> h(Arg_0+Arg_1+50,Arg_1) :|: Arg_0<=100
h(Arg_0,Arg_1) -> h(Arg_0-1,Arg_1) :|: 0<=Arg_0 && Arg_0<=150+Arg_1

### Timebounds:

Overall timebound:inf {Infinity}
0,0: f->g: 1 {O(1)}
1,1: g->g: max([0, Arg_0]) {O(n)}
2,1: g->g: inf {Infinity}
3,2: g->h: 1 {O(1)}
4,3: h->h: max([0, 151+Arg_1]) {O(n)}

### Expected Timebounds:

Overall expected timebound: 153+Arg_1+2*Arg_0 {O(n)}
0: f->[1:g]: 1 {O(1)}
1: g->[1/2:g; 1/2:g]: 2*Arg_0 {O(n)}
2: g->[1:h]: 1 {O(1)}
3: h->[1:h]: 151+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}
3,2: g->h: inf {Infinity}
4,3: h->h: inf {Infinity}

### Expected Costbounds:

Overall expected costbound: 151+Arg_1+2*Arg_0 {O(n)}
0: f->[1:g]: 0 {O(1)}
1: g->[1/2:g; 1/2:g]: 2*Arg_0 {O(n)}
2: g->[1:h]: 0 {O(1)}
3: h->[1:h]: 151+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)}
3,2: g->h, Arg_0: 150+Arg_1 {O(n)}
3,2: g->h, Arg_1: Arg_1 {O(n)}
4,3: h->h, Arg_0: 150+Arg_1 {O(n)}
4,3: h->h, 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)}
(2: g->[1:h], h), Arg_0: 150+Arg_1 {O(n)}
(2: g->[1:h], h), Arg_1: Arg_1 {O(n)}
(3: h->[1:h], h), Arg_0: 150+Arg_1 {O(n)}
(3: h->[1:h], h), Arg_1: Arg_1 {O(n)}