LTS Termination Proof

by AProVE

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
l5 l5 l5: x1 = x1x2 = x2x3 = x3x4 = x4
l4 l4 l4: x1 = x1x2 = x2x3 = x3x4 = x4
l6 l6 l6: x1 = x1x2 = x2x3 = x3x4 = x4
l10 l10 l10: x1 = x1x2 = x2x3 = x3x4 = x4
l8 l8 l8: x1 = x1x2 = x2x3 = x3x4 = x4
l1 l1 l1: x1 = x1x2 = x2x3 = x3x4 = x4
l3 l3 l3: x1 = x1x2 = x2x3 = x3x4 = x4
l0 l0 l0: x1 = x1x2 = x2x3 = x3x4 = x4
l2 l2 l2: x1 = x1x2 = x2x3 = x3x4 = x4
l9 l9 l9: x1 = x1x2 = x2x3 = x3x4 = x4
and for every transition t, a duplicate t is considered.

2 SCC Decomposition

We consider subproblems for each of the 3 SCC(s) of the program graph.

2.1 SCC Subproblem 1/3

Here we consider the SCC { l0, l2 }.

2.1.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by 0.

l0: 2⋅x2
l2: 2⋅x2 + 1

2.1.2 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by 0.

l2: 0
l0: −1

2.1.3 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

2.2 SCC Subproblem 2/3

Here we consider the SCC { l5, l4 }.

2.2.1 Transition Removal

We remove transition 12 using the following ranking functions, which are bounded by 0.

l4: 2⋅x3 + 1
l5: 2⋅x3

2.2.2 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by 0.

l4: 0
l5: −1

2.2.3 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

2.3 SCC Subproblem 3/3

Here we consider the SCC { l6, l8 }.

2.3.1 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by 0.

l8: x4
l6: x4

2.3.2 Transition Removal

We remove transition 10 using the following ranking functions, which are bounded by 0.

l8: 0
l6: −1

2.3.3 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

Tool configuration

AProVE