# Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, (1+Arg_3)*(2*Arg_2+4) {O(n^2)})

### Initial Complexity Problem (after preprocessing)

Start:f
Program_Vars:Arg_0, Arg_1, Arg_2, Arg_3
Temp_Vars:
Locations:f, g, h
Transitions:
f(Arg_0,Arg_1,Arg_2,Arg_3) -{0}> g(Arg_0,Arg_1,Arg_2,Arg_3) :|: 0<=Arg_2
g(Arg_0,Arg_1,Arg_2,Arg_3) -{0}> h(0,Arg_2,Arg_2,Arg_3) :|: 0<Arg_3 && 0<=Arg_2
h(Arg_0,Arg_1,Arg_2,Arg_3) -> 1/4:h(Arg_0,Arg_1+1,Arg_2,Arg_3) :+: 3/4:h(Uniform (1, 3),Arg_1+1,Arg_2,Arg_3) :|: Arg_0<=Arg_1 && 1<=Arg_3 && 1<=Arg_2+Arg_3 && 1<=Arg_1+Arg_3 && 1<=Arg_0+Arg_3 && Arg_2<=Arg_1 && 0<=Arg_2 && 0<=Arg_1+Arg_2 && 0<=Arg_0+Arg_2 && 0<=Arg_1 && 0<=Arg_0+Arg_1 && Arg_0<=2+Arg_1 && 0<=Arg_0
h(Arg_0,Arg_1,Arg_2,Arg_3) -{0}> g(Arg_0,Arg_1,Arg_2,Arg_3-1) :|: Arg_1<Arg_0 && 1<=Arg_3 && 1<=Arg_2+Arg_3 && 1<=Arg_1+Arg_3 && 1<=Arg_0+Arg_3 && Arg_2<=Arg_1 && 0<=Arg_2 && 0<=Arg_1+Arg_2 && 0<=Arg_0+Arg_2 && 0<=Arg_1 && 0<=Arg_0+Arg_1 && Arg_0<=2+Arg_1 && 0<=Arg_0

### Timebounds:

Overall timebound:inf {Infinity}
0,0: f->g: 1 {O(1)}
1,1: g->h: max([0, 1+Arg_3]) {O(n)}
2,2: h->h: inf {Infinity}
3,2: h->h: inf {Infinity}
4,3: h->g: max([0, 3*Arg_3]) {O(n)}

### Expected Timebounds:

Overall expected timebound: (1+Arg_3)*(2*Arg_2+4)+2+4*Arg_3 {O(n^2)}
0: f->[1:g]: 1 {O(1)}
1: g->[1:h]: 1+Arg_3 {O(n)}
2: h->[1/4:h; 3/4:h]: (1+Arg_3)*(2*Arg_2+4) {O(n^2)}
3: h->[1:g]: 3*Arg_3 {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: (1+Arg_3)*(2*Arg_2+4) {O(n^2)}
0: f->[1:g]: 0 {O(1)}
1: g->[1:h]: 0 {O(1)}
2: h->[1/4:h; 3/4:h]: (1+Arg_3)*(2*Arg_2+4) {O(n^2)}
3: h->[1:g]: 0 {O(1)}

### Sizebounds:

0,0: f->g, Arg_0: Arg_0 {O(n)}
0,0: f->g, Arg_1: Arg_1 {O(n)}
0,0: f->g, Arg_2: Arg_2 {O(n)}
0,0: f->g, Arg_3: Arg_3 {O(n)}
1,1: g->h, Arg_0: 0 {O(1)}
1,1: g->h, Arg_1: Arg_2 {O(n)}
1,1: g->h, Arg_2: Arg_2 {O(n)}
1,1: g->h, Arg_3: Arg_3 {O(n)}
2,2: h->h, Arg_2: Arg_2 {O(n)}
2,2: h->h, Arg_3: Arg_3 {O(n)}
3,2: h->h, Arg_2: Arg_2 {O(n)}
3,2: h->h, Arg_3: Arg_3 {O(n)}
4,3: h->g, Arg_2: Arg_2 {O(n)}
4,3: h->g, Arg_3: Arg_3 {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)}
(0: f->[1:g], g), Arg_2: Arg_2 {O(n)}
(0: f->[1:g], g), Arg_3: Arg_3 {O(n)}
(1: g->[1:h], h), Arg_0: 0 {O(1)}
(1: g->[1:h], h), Arg_1: Arg_2 {O(n)}
(1: g->[1:h], h), Arg_2: Arg_2 {O(n)}
(1: g->[1:h], h), Arg_3: Arg_3 {O(n)}
(2: h->[1/4:h; 3/4:h], h), Arg_2: Arg_2 {O(n)}
(2: h->[1/4:h; 3/4:h], h), Arg_3: Arg_3 {O(n)}
(3: h->[1:g], g), Arg_2: Arg_2 {O(n)}
(3: h->[1:g], g), Arg_3: Arg_3 {O(n)}