# Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, 6*Arg_0+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)
g(Arg_0,Arg_1) -{0}> 1/4:h(Arg_0-1,Arg_1+1) :+: 3/4:h(Arg_0-1,Arg_1) :|: 0<Arg_0
h(Arg_0,Arg_1) -> h(Arg_0,Arg_1-1) :|: 0<Arg_1 && 0<=Arg_0
h(Arg_0,Arg_1) -{2}> g(Arg_0,Arg_1) :|: Arg_1<=0 && 0<=Arg_0

### Timebounds:

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

### Expected Timebounds:

Overall expected timebound: 1+Arg_1+6*Arg_0 {O(n)}
0: f->[1:g]: 1 {O(1)}
1: g->[1/4:h; 3/4:h]: 2*Arg_0 {O(n)}
2: h->[1:h]: 2*Arg_0+Arg_1 {O(n)}
3: 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->h: inf {Infinity}
4,3: h->g: inf {Infinity}

### Expected Costbounds:

Overall expected costbound: 6*Arg_0+Arg_1 {O(n)}
0: f->[1:g]: 0 {O(1)}
1: g->[1/4:h; 3/4:h]: 0 {O(1)}
2: h->[1:h]: 2*Arg_0+Arg_1 {O(n)}
3: h->[1:g]: 4*Arg_0 {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->h, Arg_0: Arg_0 {O(n)}
1,1: g->h, Arg_1: max([1, 1+Arg_1]) {O(n)}
2,1: g->h, Arg_0: Arg_0 {O(n)}
2,1: g->h, Arg_1: max([0, Arg_1]) {O(n)}
3,2: h->h, Arg_0: Arg_0 {O(n)}
3,2: h->h, Arg_1: max([1, 1+Arg_1]) {O(n)}
4,3: h->g, Arg_0: Arg_0 {O(n)}
4,3: h->g, Arg_1: 0 {O(1)}

### 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->[1/4:h; 3/4:h], h), Arg_0: Arg_0 {O(n)}
(1: g->[1/4:h; 3/4:h], h), Arg_1: 1+Arg_1 {O(n)}
(2: h->[1:h], h), Arg_0: Arg_0 {O(n)}
(2: h->[1:h], h), Arg_1: 1+Arg_1 {O(n)}
(3: h->[1:g], g), Arg_0: Arg_0 {O(n)}
(3: h->[1:g], g), Arg_1: Arg_1 {O(n)}