# Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, 2*(2+2*Arg_2)*(2+2*Arg_2)+5+4*Arg_2 {O(n^2)})

### 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) -> g(1,Arg_1,Arg_2)
g(Arg_0,Arg_1,Arg_2) -> h(Arg_0+1,Arg_0-1,Arg_2) :|: Arg_0<Arg_2 && 1<=Arg_0
h(Arg_0,Arg_1,Arg_2) -> 1/2:h(Arg_0,Arg_1-1,Arg_2) :+: 1/2:h(Arg_0,Arg_1,Arg_2) :|: 0<Arg_1 && 2<=Arg_2 && 2<=Arg_1+Arg_2 && 2+Arg_1<=Arg_2 && 4<=Arg_0+Arg_2 && Arg_0<=Arg_2 && 2+Arg_1<=Arg_0 && 0<=Arg_1 && 2<=Arg_0+Arg_1 && 2<=Arg_0
h(Arg_0,Arg_1,Arg_2) -> g(Arg_0,Arg_1,Arg_2) :|: Arg_1<1 && 2<=Arg_2 && 2<=Arg_1+Arg_2 && 2+Arg_1<=Arg_2 && 4<=Arg_0+Arg_2 && Arg_0<=Arg_2 && 2+Arg_1<=Arg_0 && 0<=Arg_1 && 2<=Arg_0+Arg_1 && 2<=Arg_0

### Timebounds:

Overall timebound:inf {Infinity}
0,0: f->g: 1 {O(1)}
1,1: g->h: max([(-2)+2*Arg_2, 0]) {O(n)}
2,2: h->h: max([(-2)+2*Arg_2, 0])*max([(-2)+2*Arg_2, 0]) {O(n^2)}
3,2: h->h: inf {Infinity}
4,3: h->g: max([(-2)+2*Arg_2, 0]) {O(n)}

### Expected Timebounds:

Overall expected timebound: 2*(2+2*Arg_2)*(2+2*Arg_2)+5+4*Arg_2 {O(n^2)}
0: f->[1:g]: 1 {O(1)}
1: g->[1:h]: 2+2*Arg_2 {O(n)}
2: h->[1/2:h; 1/2:h]: 2*(2+2*Arg_2)*(2+2*Arg_2) {O(n^2)}
3: h->[1:g]: 2+2*Arg_2 {O(n)}

### Costbounds:

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

### Expected Costbounds:

Overall expected costbound: 2*(2+2*Arg_2)*(2+2*Arg_2)+5+4*Arg_2 {O(n^2)}
0: f->[1:g]: 1 {O(1)}
1: g->[1:h]: 2+2*Arg_2 {O(n)}
2: h->[1/2:h; 1/2:h]: 2*(2+2*Arg_2)*(2+2*Arg_2) {O(n^2)}
3: h->[1:g]: 2+2*Arg_2 {O(n)}

### Sizebounds:

0,0: f->g, Arg_0: 1 {O(1)}
0,0: f->g, Arg_1: Arg_1 {O(n)}
0,0: f->g, Arg_2: Arg_2 {O(n)}
1,1: g->h, Arg_0: 1+max([(-2)+2*Arg_2, 0]) {O(n)}
1,1: g->h, Arg_1: max([(-2)+2*Arg_2, 0]) {O(n)}
1,1: g->h, Arg_2: Arg_2 {O(n)}
2,2: h->h, Arg_0: 1+max([(-2)+2*Arg_2, 0]) {O(n)}
2,2: h->h, Arg_1: max([(-2)+2*Arg_2, 0]) {O(n)}
2,2: h->h, Arg_2: Arg_2 {O(n)}
3,2: h->h, Arg_0: 1+max([(-2)+2*Arg_2, 0]) {O(n)}
3,2: h->h, Arg_1: max([(-2)+2*Arg_2, 0]) {O(n)}
3,2: h->h, Arg_2: Arg_2 {O(n)}
4,3: h->g, Arg_0: 1+max([(-2)+2*Arg_2, 0]) {O(n)}
4,3: h->g, Arg_1: 0 {O(1)}
4,3: h->g, Arg_2: Arg_2 {O(n)}

### ExpSizeBounds:

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