# Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, 3*(1/2+1/2*Arg_0+1/2*Arg_1)+3/20*(2*(1/2+1/2*Arg_0+1/2*Arg_1)+2*Arg_1) {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) :|: 10<=Arg_1
g(Arg_0,Arg_1) -{3}> g(Arg_0,Binomial (4, 1/2)) :|: 2+Arg_1<Arg_0 && 10<=Arg_1
g(Arg_0,Arg_1) -{0}> h(Arg_0,Arg_1) :|: Arg_0<Arg_1+3 && 10<=Arg_1
h(Arg_0,Arg_1) -> 2/3:h(Arg_0,Arg_1-10) :+: 1/3:h(Arg_0,Arg_1) :|: 9<Arg_1 && 0<=Arg_1

### Timebounds:

Overall timebound:inf {Infinity}
0,0: f->g: 1 {O(1)}
1,1: g->g: inf {Infinity}
2,2: g->h: 1 {O(1)}
3,3: h->h: inf {Infinity}
4,3: h->h: inf {Infinity}

### Expected Timebounds:

Overall expected timebound: 1/2*Arg_0+1/2*Arg_1+5/2+3/20*(2*(1/2+1/2*Arg_0+1/2*Arg_1)+2*Arg_1) {O(n)}
0: f->[1:g]: 1 {O(1)}
1: g->[1:g]: 1/2+1/2*Arg_0+1/2*Arg_1 {O(n)}
2: g->[1:h]: 1 {O(1)}
3: h->[2/3:h; 1/3:h]: 3/20*(2*(1/2+1/2*Arg_0+1/2*Arg_1)+2*Arg_1) {O(n)}

### Costbounds:

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

### Expected Costbounds:

Overall expected costbound: 3*(1/2+1/2*Arg_0+1/2*Arg_1)+3/20*(2*(1/2+1/2*Arg_0+1/2*Arg_1)+2*Arg_1) {O(n)}
0: f->[1:g]: 0 {O(1)}
1: g->[1:g]: 3*(1/2+1/2*Arg_0+1/2*Arg_1) {O(n)}
2: g->[1:h]: 0 {O(1)}
3: h->[2/3:h; 1/3:h]: 3/20*(2*(1/2+1/2*Arg_0+1/2*Arg_1)+2*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)}
2,2: g->h, Arg_0: Arg_0 {O(n)}
3,3: h->h, Arg_0: Arg_0 {O(n)}
4,3: h->h, Arg_0: Arg_0 {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:g], g), Arg_0: Arg_0 {O(n)}
(1: g->[1:g], g), Arg_1: 2*(1/2+1/2*Arg_0+1/2*Arg_1)+Arg_1 {O(n)}
(2: g->[1:h], h), Arg_0: Arg_0 {O(n)}
(2: g->[1:h], h), Arg_1: 2*(1/2+1/2*Arg_0+1/2*Arg_1)+2*Arg_1 {O(n)}
(3: h->[2/3:h; 1/3:h], h), Arg_0: Arg_0 {O(n)}
(3: h->[2/3:h; 1/3:h], h), Arg_1: 4*(1/2+1/2*Arg_0+1/2*Arg_1)+4*Arg_1 {O(n)}