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 = x4x5 = x5
l4 l4 l4: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5
l6 l6 l6: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5
l1 l1 l1: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5
l0 l0 l0: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5
l2 l2 l2: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5
and for every transition t, a duplicate t is considered.

2 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

2.1 SCC Subproblem 1/2

Here we consider the SCC { l1, l0 }.

2.1.1 Transition Removal

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

l0: 9 − x1
l1: 9 − x1

2.1.2 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 0.

l0: 0
l1: −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 2/2

Here we consider the SCC { l4, l2 }.

2.2.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by −18.

l4: −2⋅x2 + 1
l2: −2⋅x2

2.2.2 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 0.

l4: 0
l2: −1

2.2.3 Trivial Cooperation Program

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

Tool configuration

AProVE