# Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, 1+2*Arg_0+2*Arg_1 {O(n)})

### Initial Complexity Problem (after preprocessing)

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

### Timebounds:

Overall timebound:inf {Infinity}
0,0: f->g: 1 {O(1)}
4,4: f->h: 1 {O(1)}
2,2: h->i: inf {Infinity}
3,3: i->g: 1 {O(1)}
5,5: i->h: inf {Infinity}

### Expected Timebounds:

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

### Costbounds:

Overall costbound: inf {Infinity}
0,0: f->g: inf {Infinity}
4,4: f->h: inf {Infinity}
2,2: h->i: inf {Infinity}
3,3: i->g: inf {Infinity}
5,5: i->h: inf {Infinity}

### Expected Costbounds:

Overall expected costbound: 1+2*Arg_0+2*Arg_1 {O(n)}
0: f->[1:g]: 0 {O(1)}
2: h->[1:i]: 0 {O(1)}
3: i->[1:g]: 1 {O(1)}
4: f->[1:h]: 0 {O(1)}
5: i->[1:h]: 2*Arg_0+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)}
0,0: f->g, Arg_2: 0 {O(1)}
4,4: f->h, Arg_0: Arg_0 {O(n)}
4,4: f->h, Arg_1: Arg_1 {O(n)}
4,4: f->h, Arg_2: Arg_0 {O(n)}
2,2: h->i, Arg_1: Arg_1 {O(n)}
3,3: i->g, Arg_1: Arg_1 {O(n)}
3,3: i->g, Arg_2: 0 {O(1)}
5,5: i->h, Arg_1: Arg_1 {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: 0 {O(1)}
(2: h->[1:i], i), Arg_1: Arg_1 {O(n)}
(2: h->[1:i], i), Arg_2: 1/2*(2*Arg_0+2*Arg_1)+Arg_0 {O(n)}
(3: i->[1:g], g), Arg_1: Arg_1 {O(n)}
(3: i->[1:g], g), Arg_2: 0 {O(1)}
(4: f->[1:h], h), Arg_0: Arg_0 {O(n)}
(4: f->[1:h], h), Arg_1: Arg_1 {O(n)}
(4: f->[1:h], h), Arg_2: Arg_0 {O(n)}
(5: i->[1:h], h), Arg_1: Arg_1 {O(n)}
(5: i->[1:h], h), Arg_2: 1/2*(2*Arg_0+2*Arg_1)+Arg_0 {O(n)}