# Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, 3*Arg_0+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
Transitions:
f(Arg_0,Arg_1,Arg_2) -{0}> g(Arg_0,Arg_1,Arg_2) :|: 0<=Arg_1
g(Arg_0,Arg_1,Arg_2) -{0}> h(Arg_0-Arg_1-1,Arg_1,Arg_1) :|: Arg_1<Arg_0 && 0<=Arg_1
h(Arg_0,Arg_1,Arg_2) -> h(Arg_0,Arg_1,Uniform (-2, -1)) :|: 0<Arg_2 && Arg_2<=Arg_1 && 0<=1+Arg_2 && 0<=Arg_1+Arg_2 && 0<=Arg_1 && 0<=Arg_0+Arg_1
h(Arg_0,Arg_1,Arg_2) -> g(Arg_0,Arg_1,Arg_2) :|: Arg_2<=0 && Arg_2<=Arg_1 && 0<=1+Arg_2 && 0<=Arg_1+Arg_2 && 0<=Arg_1 && 0<=Arg_0+Arg_1

### Timebounds:

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

### Expected Timebounds:

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

### Expected Costbounds:

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