Initial Problem

Start: start
Program_Vars: Arg_0, Arg_1
Temp_Vars:
Locations: eval1, eval2, eval3, start
Transitions:
0:eval1(Arg_0,Arg_1) -> eval2(Arg_0,Arg_1):|:1<=Arg_0 && 1<=Arg_1 && Arg_1+1<=Arg_0
1:eval1(Arg_0,Arg_1) -> eval3(Arg_0,Arg_1):|:1<=Arg_0 && 1<=Arg_1 && Arg_0<=Arg_1
3:eval2(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1):|:Arg_0<=0
2:eval2(Arg_0,Arg_1) -> eval2(Arg_0-1,Arg_1):|:1<=Arg_0
5:eval3(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1):|:Arg_1<=0
4:eval3(Arg_0,Arg_1) -> eval3(Arg_0,Arg_1-1):|:1<=Arg_1
6:start(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1)

Preprocessing

Found invariant 1<=Arg_1 && 1<=Arg_0+Arg_1 && 0<=Arg_0 for location eval2

Found invariant 0<=Arg_1 && 1<=Arg_0+Arg_1 && 1<=Arg_0 for location eval3

Problem after Preprocessing

Start: start
Program_Vars: Arg_0, Arg_1
Temp_Vars:
Locations: eval1, eval2, eval3, start
Transitions:
0:eval1(Arg_0,Arg_1) -> eval2(Arg_0,Arg_1):|:1<=Arg_0 && 1<=Arg_1 && Arg_1+1<=Arg_0
1:eval1(Arg_0,Arg_1) -> eval3(Arg_0,Arg_1):|:1<=Arg_0 && 1<=Arg_1 && Arg_0<=Arg_1
3:eval2(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1):|:1<=Arg_1 && 1<=Arg_0+Arg_1 && 0<=Arg_0 && Arg_0<=0
2:eval2(Arg_0,Arg_1) -> eval2(Arg_0-1,Arg_1):|:1<=Arg_1 && 1<=Arg_0+Arg_1 && 0<=Arg_0 && 1<=Arg_0
5:eval3(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1):|:0<=Arg_1 && 1<=Arg_0+Arg_1 && 1<=Arg_0 && Arg_1<=0
4:eval3(Arg_0,Arg_1) -> eval3(Arg_0,Arg_1-1):|:0<=Arg_1 && 1<=Arg_0+Arg_1 && 1<=Arg_0 && 1<=Arg_1
6:start(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1)

knowledge_propagation leads to new time bound 1 {O(1)} for transition 0:eval1(Arg_0,Arg_1) -> eval2(Arg_0,Arg_1):|:1<=Arg_0 && 1<=Arg_1 && Arg_1+1<=Arg_0

knowledge_propagation leads to new time bound 1 {O(1)} for transition 1:eval1(Arg_0,Arg_1) -> eval3(Arg_0,Arg_1):|:1<=Arg_0 && 1<=Arg_1 && Arg_0<=Arg_1

MPRF for transition 2:eval2(Arg_0,Arg_1) -> eval2(Arg_0-1,Arg_1):|:1<=Arg_1 && 1<=Arg_0+Arg_1 && 0<=Arg_0 && 1<=Arg_0 of depth 1:

new bound:

Arg_0+1 {O(n)}

MPRF:

eval2 [Arg_0+1 ]
eval3 [Arg_0+1 ]
eval1 [Arg_0+1 ]

MPRF for transition 3:eval2(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1):|:1<=Arg_1 && 1<=Arg_0+Arg_1 && 0<=Arg_0 && Arg_0<=0 of depth 1:

new bound:

Arg_0 {O(n)}

MPRF:

eval2 [1 ]
eval3 [Arg_0 ]
eval1 [Arg_0 ]

MPRF for transition 4:eval3(Arg_0,Arg_1) -> eval3(Arg_0,Arg_1-1):|:0<=Arg_1 && 1<=Arg_0+Arg_1 && 1<=Arg_0 && 1<=Arg_1 of depth 1:

new bound:

2*Arg_1 {O(n)}

MPRF:

eval2 [2*Arg_1 ]
eval3 [Arg_1+1 ]
eval1 [2*Arg_1 ]

MPRF for transition 5:eval3(Arg_0,Arg_1) -> eval1(Arg_0,Arg_1):|:0<=Arg_1 && 1<=Arg_0+Arg_1 && 1<=Arg_0 && Arg_1<=0 of depth 1:

new bound:

Arg_1 {O(n)}

MPRF:

eval2 [Arg_1 ]
eval3 [1 ]
eval1 [Arg_1 ]

All Bounds

Timebounds

Overall timebound:2*Arg_0+3*Arg_1+4 {O(n)}
0: eval1->eval2: 1 {O(1)}
1: eval1->eval3: 1 {O(1)}
2: eval2->eval2: Arg_0+1 {O(n)}
3: eval2->eval1: Arg_0 {O(n)}
4: eval3->eval3: 2*Arg_1 {O(n)}
5: eval3->eval1: Arg_1 {O(n)}
6: start->eval1: 1 {O(1)}

Costbounds

Overall costbound: 2*Arg_0+3*Arg_1+4 {O(n)}
0: eval1->eval2: 1 {O(1)}
1: eval1->eval3: 1 {O(1)}
2: eval2->eval2: Arg_0+1 {O(n)}
3: eval2->eval1: Arg_0 {O(n)}
4: eval3->eval3: 2*Arg_1 {O(n)}
5: eval3->eval1: Arg_1 {O(n)}
6: start->eval1: 1 {O(1)}

Sizebounds

0: eval1->eval2, Arg_0: Arg_0 {O(n)}
0: eval1->eval2, Arg_1: Arg_1 {O(n)}
1: eval1->eval3, Arg_0: Arg_0 {O(n)}
1: eval1->eval3, Arg_1: Arg_1 {O(n)}
2: eval2->eval2, Arg_0: Arg_0 {O(n)}
2: eval2->eval2, Arg_1: Arg_1 {O(n)}
3: eval2->eval1, Arg_0: 0 {O(1)}
3: eval2->eval1, Arg_1: Arg_1 {O(n)}
4: eval3->eval3, Arg_0: Arg_0 {O(n)}
4: eval3->eval3, Arg_1: Arg_1 {O(n)}
5: eval3->eval1, Arg_0: Arg_0 {O(n)}
5: eval3->eval1, Arg_1: 0 {O(1)}
6: start->eval1, Arg_0: Arg_0 {O(n)}
6: start->eval1, Arg_1: Arg_1 {O(n)}