# Inferring Expected Runtimes Using Sizes

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

### Initial Complexity Problem (after preprocessing)

Start:f
Program_Vars:Arg_0, Arg_1, Arg_2
Temp_Vars:
Locations:f, g, h
Transitions:
f(Arg_0,Arg_1,Arg_2) -{0}> g(Arg_0,1,0)
g(Arg_0,Arg_1,Arg_2) -> 1/2:h(Arg_0,0,Arg_2) :+: 1/2:h(Arg_0,1,Arg_2) :|: Arg_2<Arg_0 && 0<Arg_1 && 0<=Arg_2 && 1<=Arg_1+Arg_2 && Arg_1<=1+Arg_2 && Arg_1<=1 && 0<=Arg_1
h(Arg_0,Arg_1,Arg_2) -> g(Arg_0,Arg_1,Arg_2+1) :|: 1+Arg_2<=Arg_0 && 0<=Arg_2 && 0<=Arg_1+Arg_2 && Arg_1<=1+Arg_2 && 1<=Arg_0+Arg_2 && Arg_1<=1 && Arg_1<=Arg_0 && 0<=Arg_1 && 1<=Arg_0+Arg_1 && 1<=Arg_0

### Timebounds:

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

### Expected Timebounds:

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

### Costbounds:

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

### Expected Costbounds:

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

### Sizebounds:

0,0: f->g, Arg_0: Arg_0 {O(n)}
0,0: f->g, Arg_1: 1 {O(1)}
0,0: f->g, Arg_2: 0 {O(1)}
1,1: g->h, Arg_0: Arg_0 {O(n)}
1,1: g->h, Arg_1: 0 {O(1)}
1,1: g->h, Arg_2: max([0, 2*Arg_0]) {O(n)}
2,1: g->h, Arg_0: Arg_0 {O(n)}
2,1: g->h, Arg_1: 1 {O(1)}
2,1: g->h, Arg_2: max([0, 2*Arg_0]) {O(n)}
3,2: h->g, Arg_0: Arg_0 {O(n)}
3,2: h->g, Arg_1: 1 {O(1)}
3,2: h->g, Arg_2: max([0, 2*Arg_0]) {O(n)}

### ExpSizeBounds:

(0: f->[1:g], g), Arg_0: Arg_0 {O(n)}
(0: f->[1:g], g), Arg_1: 1 {O(1)}
(0: f->[1:g], g), Arg_2: 0 {O(1)}
(1: g->[1/2:h; 1/2:h], h), Arg_0: Arg_0 {O(n)}
(1: g->[1/2:h; 1/2:h], h), Arg_1: 1 {O(1)}
(1: g->[1/2:h; 1/2:h], h), Arg_2: 2*Arg_0 {O(n)}
(2: h->[1:g], g), Arg_0: Arg_0 {O(n)}
(2: h->[1:g], g), Arg_1: 1 {O(1)}
(2: h->[1:g], g), Arg_2: 2*Arg_0 {O(n)}