# Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, 2+2*max([1, 2^(Arg_3)]) {O(EXP)})

### 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(0,1,0,Arg_3)
g(Arg_0,Arg_1,Arg_2,Arg_3) -{0}> g(Arg_1,Arg_0+Arg_1,Arg_2+1,Arg_3) :|: Arg_2<Arg_3 && 0<=Arg_2 && 1<=Arg_1+Arg_2 && 0<=Arg_0+Arg_2 && 1<=Arg_1 && 1<=Arg_0+Arg_1 && Arg_0<=Arg_1 && 0<=Arg_0
g(Arg_0,Arg_1,Arg_2,Arg_3) -{0}> h(Arg_0,Arg_1,Arg_2,Arg_3) :|: Arg_3<Arg_2+1 && 0<=Arg_2 && 1<=Arg_1+Arg_2 && 0<=Arg_0+Arg_2 && 1<=Arg_1 && 1<=Arg_0+Arg_1 && Arg_0<=Arg_1 && 0<=Arg_0
h(Arg_0,Arg_1,Arg_2,Arg_3) -> 1/2:h(Arg_0,Arg_1-2,Arg_2,Arg_3) :+: 1/2:h(Arg_0,Arg_1+1,Arg_2,Arg_3) :|: 0<Arg_1 && Arg_3<=Arg_2 && 0<=Arg_2 && 0<=1+Arg_1+Arg_2 && 0<=Arg_0+Arg_2 && 0<=1+Arg_1 && 0<=1+Arg_0+Arg_1 && 0<=Arg_0

### Timebounds:

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

### Expected Timebounds:

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

### Costbounds:

Overall costbound: inf {Infinity}
0,0: f->g: inf {Infinity}
1,1: g->g: inf {Infinity}
2,2: g->h: inf {Infinity}
3,3: h->h: inf {Infinity}
4,3: h->h: inf {Infinity}

### Expected Costbounds:

Overall expected costbound: 2+2*max([1, 2^(Arg_3)]) {O(EXP)}
0: f->[1:g]: 0 {O(1)}
1: g->[1:g]: 0 {O(1)}
2: g->[1:h]: 0 {O(1)}
3: h->[1/2:h; 1/2:h]: 2+2*max([1, 2^(Arg_3)]) {O(EXP)}

### Sizebounds:

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

### ExpSizeBounds:

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