Initial Problem

Start: f3
Program_Vars: Arg_0, Arg_1, Arg_2
Temp_Vars:
Locations: f1, f3
Transitions:
0:f1(Arg_0,Arg_1,Arg_2) -> f1(Arg_0,Arg_1+1,Arg_2):|:Arg_1+1<=Arg_0
2:f1(Arg_0,Arg_1,Arg_2) -> f1(Arg_0,Arg_1,Arg_2+1):|:Arg_2+2<=Arg_1 && Arg_0<=Arg_1
1:f3(Arg_0,Arg_1,Arg_2) -> f1(Arg_0,Arg_1,Arg_2):|:Arg_2+1<=Arg_1

Preprocessing

Found invariant 1+Arg_2<=Arg_1 for location f1

Problem after Preprocessing

Start: f3
Program_Vars: Arg_0, Arg_1, Arg_2
Temp_Vars:
Locations: f1, f3
Transitions:
0:f1(Arg_0,Arg_1,Arg_2) -> f1(Arg_0,Arg_1+1,Arg_2):|:1+Arg_2<=Arg_1 && Arg_1+1<=Arg_0
2:f1(Arg_0,Arg_1,Arg_2) -> f1(Arg_0,Arg_1,Arg_2+1):|:1+Arg_2<=Arg_1 && Arg_2+2<=Arg_1 && Arg_0<=Arg_1
1:f3(Arg_0,Arg_1,Arg_2) -> f1(Arg_0,Arg_1,Arg_2):|:Arg_2+1<=Arg_1

MPRF for transition 0:f1(Arg_0,Arg_1,Arg_2) -> f1(Arg_0,Arg_1+1,Arg_2):|:1+Arg_2<=Arg_1 && Arg_1+1<=Arg_0 of depth 1:

new bound:

Arg_0+Arg_1 {O(n)}

MPRF:

f1 [Arg_0-Arg_1 ]

MPRF for transition 2:f1(Arg_0,Arg_1,Arg_2) -> f1(Arg_0,Arg_1,Arg_2+1):|:1+Arg_2<=Arg_1 && Arg_2+2<=Arg_1 && Arg_0<=Arg_1 of depth 1:

new bound:

2*Arg_1*Arg_1+3*Arg_0*Arg_1+Arg_0*Arg_0+Arg_0*Arg_2+Arg_1*Arg_2+Arg_1+Arg_2 {O(n^2)}

MPRF:

f1 [Arg_1-Arg_2 ]

Analysing control-flow refined program

Found invariant 2+Arg_2<=Arg_1 && 2+Arg_2<=Arg_0 && Arg_1<=Arg_0 for location n_f1___2

Found invariant 1+Arg_2<=Arg_1 && Arg_0<=Arg_1 for location n_f1___1

Found invariant 1+Arg_2<=Arg_1 for location f1

MPRF for transition 34:n_f1___2(Arg_0,Arg_1,Arg_2) -> n_f1___2(Arg_0,Arg_1+1,Arg_2):|:2+Arg_2<=Arg_1 && 2+Arg_2<=Arg_0 && Arg_1<=Arg_0 && 1+Arg_2<=Arg_1 && 2+Arg_2<=Arg_1 && 2+Arg_2<=Arg_1 && Arg_1<=Arg_0 && 1+Arg_2<=Arg_1 && 1+Arg_1<=Arg_0 && 1+Arg_2<=Arg_1 of depth 1:

new bound:

Arg_0+Arg_1+2 {O(n)}

MPRF:

n_f1___2 [Arg_0+1-Arg_1 ]

MPRF for transition 32:n_f1___1(Arg_0,Arg_1,Arg_2) -> n_f1___1(Arg_0,Arg_1,Arg_2+1):|:1+Arg_2<=Arg_1 && Arg_0<=Arg_1 && 1+Arg_2<=Arg_1 && Arg_0<=Arg_1 && 1+Arg_2<=Arg_1 && Arg_0<=Arg_1 && 1+Arg_2<=Arg_1 && 2+Arg_2<=Arg_1 && Arg_0<=Arg_1 of depth 1:

new bound:

3*Arg_2+4*Arg_1+Arg_0+7 {O(n)}

MPRF:

n_f1___1 [Arg_1-Arg_2 ]

CFR: Improvement to new bound with the following program:

new bound:

2*Arg_0+3*Arg_2+5*Arg_1+9 {O(n)}

cfr-program:

Start: f3
Program_Vars: Arg_0, Arg_1, Arg_2
Temp_Vars:
Locations: f1, f3, n_f1___1, n_f1___2
Transitions:
35:f1(Arg_0,Arg_1,Arg_2) -> n_f1___1(Arg_0,Arg_1,Arg_2+1):|:1+Arg_2<=Arg_1 && 1+Arg_2<=Arg_1 && 1+Arg_2<=Arg_1 && 2+Arg_2<=Arg_1 && Arg_0<=Arg_1
36:f1(Arg_0,Arg_1,Arg_2) -> n_f1___2(Arg_0,Arg_1+1,Arg_2):|:1+Arg_2<=Arg_1 && 1+Arg_2<=Arg_1 && 1+Arg_2<=Arg_1 && 1+Arg_1<=Arg_0 && 1+Arg_2<=Arg_1
1:f3(Arg_0,Arg_1,Arg_2) -> f1(Arg_0,Arg_1,Arg_2):|:Arg_2+1<=Arg_1
32:n_f1___1(Arg_0,Arg_1,Arg_2) -> n_f1___1(Arg_0,Arg_1,Arg_2+1):|:1+Arg_2<=Arg_1 && Arg_0<=Arg_1 && 1+Arg_2<=Arg_1 && Arg_0<=Arg_1 && 1+Arg_2<=Arg_1 && Arg_0<=Arg_1 && 1+Arg_2<=Arg_1 && 2+Arg_2<=Arg_1 && Arg_0<=Arg_1
33:n_f1___2(Arg_0,Arg_1,Arg_2) -> n_f1___1(Arg_0,Arg_1,Arg_2+1):|:2+Arg_2<=Arg_1 && 2+Arg_2<=Arg_0 && Arg_1<=Arg_0 && 1+Arg_2<=Arg_1 && 2+Arg_2<=Arg_1 && 2+Arg_2<=Arg_1 && Arg_1<=Arg_0 && 1+Arg_2<=Arg_1 && 2+Arg_2<=Arg_1 && Arg_0<=Arg_1
34:n_f1___2(Arg_0,Arg_1,Arg_2) -> n_f1___2(Arg_0,Arg_1+1,Arg_2):|:2+Arg_2<=Arg_1 && 2+Arg_2<=Arg_0 && Arg_1<=Arg_0 && 1+Arg_2<=Arg_1 && 2+Arg_2<=Arg_1 && 2+Arg_2<=Arg_1 && Arg_1<=Arg_0 && 1+Arg_2<=Arg_1 && 1+Arg_1<=Arg_0 && 1+Arg_2<=Arg_1

All Bounds

Timebounds

Overall timebound:2*Arg_0+3*Arg_2+5*Arg_1+13 {O(n)}
35: f1->n_f1___1: 1 {O(1)}
36: f1->n_f1___2: 1 {O(1)}
1: f3->f1: 1 {O(1)}
32: n_f1___1->n_f1___1: 3*Arg_2+4*Arg_1+Arg_0+7 {O(n)}
33: n_f1___2->n_f1___1: 1 {O(1)}
34: n_f1___2->n_f1___2: Arg_0+Arg_1+2 {O(n)}

Costbounds

Overall costbound: 2*Arg_0+3*Arg_2+5*Arg_1+13 {O(n)}
35: f1->n_f1___1: 1 {O(1)}
36: f1->n_f1___2: 1 {O(1)}
1: f3->f1: 1 {O(1)}
32: n_f1___1->n_f1___1: 3*Arg_2+4*Arg_1+Arg_0+7 {O(n)}
33: n_f1___2->n_f1___1: 1 {O(1)}
34: n_f1___2->n_f1___2: Arg_0+Arg_1+2 {O(n)}

Sizebounds

35: f1->n_f1___1, Arg_0: Arg_0 {O(n)}
35: f1->n_f1___1, Arg_1: Arg_1 {O(n)}
35: f1->n_f1___1, Arg_2: Arg_2+1 {O(n)}
36: f1->n_f1___2, Arg_0: Arg_0 {O(n)}
36: f1->n_f1___2, Arg_1: Arg_1+1 {O(n)}
36: f1->n_f1___2, Arg_2: Arg_2 {O(n)}
1: f3->f1, Arg_0: Arg_0 {O(n)}
1: f3->f1, Arg_1: Arg_1 {O(n)}
1: f3->f1, Arg_2: Arg_2 {O(n)}
32: n_f1___1->n_f1___1, Arg_0: 3*Arg_0 {O(n)}
32: n_f1___1->n_f1___1, Arg_1: 4*Arg_1+Arg_0+4 {O(n)}
32: n_f1___1->n_f1___1, Arg_2: 4*Arg_1+6*Arg_2+Arg_0+10 {O(n)}
33: n_f1___2->n_f1___1, Arg_0: 2*Arg_0 {O(n)}
33: n_f1___2->n_f1___1, Arg_1: 3*Arg_1+Arg_0+4 {O(n)}
33: n_f1___2->n_f1___1, Arg_2: 2*Arg_2+2 {O(n)}
34: n_f1___2->n_f1___2, Arg_0: Arg_0 {O(n)}
34: n_f1___2->n_f1___2, Arg_1: 2*Arg_1+Arg_0+3 {O(n)}
34: n_f1___2->n_f1___2, Arg_2: Arg_2 {O(n)}