# Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, 3*Arg_3+3/4 {O(n)})

### Initial Complexity Problem (after preprocessing)

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

### 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->i: inf {Infinity}
3,2: h->i: inf {Infinity}
9,7: h->g: max([0, Arg_3]) {O(n)}
4,3: i->j: max([0, 1+Arg_3]) {O(n)}
5,3: i->j: inf {Infinity}
6,4: j->h: inf {Infinity}
7,5: j->h: max([0, Arg_3]) {O(n)}

### Expected Timebounds:

Overall expected timebound: 17/4+8*Arg_3 {O(n)}
0: f->[1:g]: 1 {O(1)}
1: g->[1:h]: 1+Arg_3 {O(n)}
2: h->[1/2:i; 1/2:i]: 2*Arg_3+5/4 {O(n)}
3: i->[1/2:j; 1/2:j]: 1/4+2*Arg_3 {O(n)}
4: j->[1:h]: 3/4+Arg_3 {O(n)}
5: j->[1:h]: Arg_3 {O(n)}
7: h->[1:g]: Arg_3 {O(n)}

### Costbounds:

Overall costbound: inf {Infinity}
0,0: f->g: inf {Infinity}
1,1: g->h: inf {Infinity}
2,2: h->i: inf {Infinity}
3,2: h->i: inf {Infinity}
9,7: h->g: inf {Infinity}
4,3: i->j: inf {Infinity}
5,3: i->j: inf {Infinity}
6,4: j->h: inf {Infinity}
7,5: j->h: inf {Infinity}

### Expected Costbounds:

Overall expected costbound: 3*Arg_3+3/4 {O(n)}
0: f->[1:g]: 0 {O(1)}
1: g->[1:h]: 0 {O(1)}
2: h->[1/2:i; 1/2:i]: 0 {O(1)}
3: i->[1/2:j; 1/2:j]: 0 {O(1)}
4: j->[1:h]: 3/4+Arg_3 {O(n)}
5: j->[1:h]: Arg_3 {O(n)}
7: h->[1:g]: Arg_3 {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)}
0,0: f->g, Arg_3: Arg_3 {O(n)}
1,1: g->h, Arg_0: max([1, Arg_0]) {O(n)}
1,1: g->h, Arg_1: max([0, Arg_1]) {O(n)}
1,1: g->h, Arg_2: 1 {O(1)}
1,1: g->h, Arg_3: Arg_3 {O(n)}
2,2: h->i, Arg_0: 1 {O(1)}
2,2: h->i, Arg_1: max([1, Arg_1]) {O(n)}
2,2: h->i, Arg_2: 1 {O(1)}
2,2: h->i, Arg_3: Arg_3 {O(n)}
3,2: h->i, Arg_0: 1 {O(1)}
3,2: h->i, Arg_1: max([1, Arg_1]) {O(n)}
3,2: h->i, Arg_2: 1 {O(1)}
3,2: h->i, Arg_3: Arg_3 {O(n)}
9,7: h->g, Arg_0: 1 {O(1)}
9,7: h->g, Arg_1: 0 {O(1)}
9,7: h->g, Arg_2: 0 {O(1)}
9,7: h->g, Arg_3: Arg_3 {O(n)}
4,3: i->j, Arg_0: 1 {O(1)}
4,3: i->j, Arg_1: 0 {O(1)}
4,3: i->j, Arg_2: 1 {O(1)}
4,3: i->j, Arg_3: Arg_3 {O(n)}
5,3: i->j, Arg_0: 1 {O(1)}
5,3: i->j, Arg_1: 1 {O(1)}
5,3: i->j, Arg_2: 1 {O(1)}
5,3: i->j, Arg_3: Arg_3 {O(n)}
6,4: j->h, Arg_0: 1 {O(1)}
6,4: j->h, Arg_1: 1 {O(1)}
6,4: j->h, Arg_2: 1 {O(1)}
6,4: j->h, Arg_3: Arg_3 {O(n)}
7,5: j->h, Arg_0: 1 {O(1)}
7,5: j->h, Arg_1: 0 {O(1)}
7,5: j->h, Arg_2: 0 {O(1)}
7,5: j->h, 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: max([1, Arg_0]) {O(n)}
(1: g->[1:h], h), Arg_1: Arg_1 {O(n)}
(1: g->[1:h], h), Arg_2: 1 {O(1)}
(1: g->[1:h], h), Arg_3: Arg_3 {O(n)}
(2: h->[1/2:i; 1/2:i], i), Arg_0: 1 {O(1)}
(2: h->[1/2:i; 1/2:i], i), Arg_1: max([1, Arg_1]) {O(n)}
(2: h->[1/2:i; 1/2:i], i), Arg_2: 1 {O(1)}
(2: h->[1/2:i; 1/2:i], i), Arg_3: Arg_3 {O(n)}
(3: i->[1/2:j; 1/2:j], j), Arg_0: 1 {O(1)}
(3: i->[1/2:j; 1/2:j], j), Arg_1: 1 {O(1)}
(3: i->[1/2:j; 1/2:j], j), Arg_2: 1 {O(1)}
(3: i->[1/2:j; 1/2:j], j), Arg_3: Arg_3 {O(n)}
(4: j->[1:h], h), Arg_0: 1 {O(1)}
(4: j->[1:h], h), Arg_1: 1 {O(1)}
(4: j->[1:h], h), Arg_2: 1 {O(1)}
(4: j->[1:h], h), Arg_3: Arg_3 {O(n)}
(5: j->[1:h], h), Arg_0: 1 {O(1)}
(5: j->[1:h], h), Arg_1: 0 {O(1)}
(5: j->[1:h], h), Arg_2: 0 {O(1)}
(5: j->[1:h], h), Arg_3: Arg_3 {O(n)}
(7: h->[1:g], g), Arg_0: 1 {O(1)}
(7: h->[1:g], g), Arg_1: 0 {O(1)}
(7: h->[1:g], g), Arg_2: 0 {O(1)}
(7: h->[1:g], g), Arg_3: Arg_3 {O(n)}