# Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, 1+41*(1+7*Arg_2)+49*Arg_2 {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(1,0,Arg_2)
g(Arg_0,Arg_1,Arg_2) -> h(Arg_0,Arg_1+1,Arg_2) :|: Arg_1<Arg_2 && 0<=Arg_1 && 1<=Arg_0+Arg_1 && Arg_0<=5+Arg_1 && Arg_0<=6 && 1<=Arg_0
h(Arg_0,Arg_1,Arg_2) -{41}> g(1,Arg_1,Arg_2) :|: 4<=Arg_0 && 1<=Arg_2 && 2<=Arg_1+Arg_2 && Arg_1<=Arg_2 && 2<=Arg_0+Arg_2 && Arg_0<=4+Arg_2 && 1<=Arg_1 && 2<=Arg_0+Arg_1 && Arg_0<=4+Arg_1 && Arg_0<=6 && 1<=Arg_0
h(Arg_0,Arg_1,Arg_2) -> g(Uniform (1, 3),Arg_1,Arg_2) :|: Arg_0<4 && 1<=Arg_2 && 2<=Arg_1+Arg_2 && Arg_1<=Arg_2 && 2<=Arg_0+Arg_2 && Arg_0<=4+Arg_2 && 1<=Arg_1 && 2<=Arg_0+Arg_1 && Arg_0<=4+Arg_1 && Arg_0<=6 && 1<=Arg_0

### Timebounds:

Overall timebound:2*max([(-1)+7*Arg_2, 0])+max([1, 1+42*Arg_2]) {O(n)}
0,0: f->g: 1 {O(1)}
1,1: g->h: max([0, 42*Arg_2]) {O(n)}
2,2: h->g: max([(-1)+7*Arg_2, 0]) {O(n)}
3,3: h->g: max([(-1)+7*Arg_2, 0]) {O(n)}

### Expected Timebounds:

Overall expected timebound: 3+56*Arg_2 {O(n)}
0: f->[1:g]: 1 {O(1)}
1: g->[1:h]: 42*Arg_2 {O(n)}
2: h->[1:g]: 1+7*Arg_2 {O(n)}
3: h->[1:g]: 1+7*Arg_2 {O(n)}

### Costbounds:

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

### Expected Costbounds:

Overall expected costbound: 1+41*(1+7*Arg_2)+49*Arg_2 {O(n)}
0: f->[1:g]: 0 {O(1)}
1: g->[1:h]: 42*Arg_2 {O(n)}
2: h->[1:g]: 41*(1+7*Arg_2) {O(n)}
3: h->[1:g]: 1+7*Arg_2 {O(n)}

### Sizebounds:

0,0: f->g, Arg_0: 1 {O(1)}
0,0: f->g, Arg_1: 0 {O(1)}
0,0: f->g, Arg_2: Arg_2 {O(n)}
1,1: g->h, Arg_0: 6 {O(1)}
1,1: g->h, Arg_1: max([0, 42*Arg_2]) {O(n)}
1,1: g->h, Arg_2: Arg_2 {O(n)}
2,2: h->g, Arg_0: 1 {O(1)}
2,2: h->g, Arg_1: max([0, 42*Arg_2]) {O(n)}
2,2: h->g, Arg_2: Arg_2 {O(n)}
3,3: h->g, Arg_0: 6 {O(1)}
3,3: h->g, Arg_1: max([0, 42*Arg_2]) {O(n)}
3,3: 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: 0 {O(1)}
(0: f->[1:g], g), Arg_2: Arg_2 {O(n)}
(1: g->[1:h], h), Arg_0: 6 {O(1)}
(1: g->[1:h], h), Arg_1: 42*Arg_2 {O(n)}
(1: g->[1:h], h), Arg_2: Arg_2 {O(n)}
(2: h->[1:g], g), Arg_0: 1 {O(1)}
(2: h->[1:g], g), Arg_1: 42*Arg_2 {O(n)}
(2: h->[1:g], g), Arg_2: Arg_2 {O(n)}
(3: h->[1:g], g), Arg_0: 6 {O(1)}
(3: h->[1:g], g), Arg_1: 42*Arg_2 {O(n)}
(3: h->[1:g], g), Arg_2: Arg_2 {O(n)}