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 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l7 l7 l7: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l11 l11 l11: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l3 l3 l3: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l13 l13 l13: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l17 l17 l17: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l2 l2 l2: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l14 l14 l14: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l9 l9 l9: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l4 l4 l4: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l10 l10 l10: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l6 l6 l6: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l8 l8 l8: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l15 l15 l15: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l16 l16 l16: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l0 l0 l0: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l12 l12 l12: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
and for every transition t, a duplicate t is considered.

2 SCC Decomposition

We consider subproblems for each of the 4 SCC(s) of the program graph.

2.1 SCC Subproblem 1/4

Here we consider the SCC { l5, l6, l13, l15, l14 }.

2.1.1 Transition Removal

We remove transition 21 using the following ranking functions, which are bounded by 0.

l5: −1 − x2 + x4
l6: −1 − x2 + x4
l14: −2 − x2 + x4
l15: −2 − x2 + x4
l13: −2 − x2 + x4

2.1.2 Transition Removal

We remove transitions 5, 15, 19, 14, 18, 17 using the following ranking functions, which are bounded by 0.

l5: 0
l6: −1
l14: 1
l15: 3
l13: 2

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 2/4

Here we consider the SCC { l10, l11, l8, l12, l9 }.

2.2.1 Transition Removal

We remove transition 13 using the following ranking functions, which are bounded by 0.

l10: −1 − x2 + x4
l11: −1 − x2 + x4
l12: −2 − x2 + x4
l9: −2 − x2 + x4
l8: −2 − x2 + x4

2.2.2 Transition Removal

We remove transitions 9, 10 using the following ranking functions, which are bounded by 0.

l10: 0
l11: −1
l12: 1
l9: 1
l8: 1

2.2.3 Transition Removal

We remove transition 11 using the following ranking functions, which are bounded by 0.

l9: −1 + x1 − x3
l12: −1 + x1 − x3
l8: −2 + x1 − x3

2.2.4 Transition Removal

We remove transitions 16, 8 using the following ranking functions, which are bounded by 0.

l9: 0
l12: −1
l8: 1

2.2.5 Trivial Cooperation Program

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

2.3 SCC Subproblem 3/4

Here we consider the SCC { l4, l7, l3 }.

2.3.1 Transition Removal

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

l3: 3⋅x1 − 3⋅x2 − 1
l4: 3⋅x1 − 3⋅x2 + 1
l7: −3⋅x2 + 3⋅x1

2.3.2 Transition Removal

We remove transitions 3, 22 using the following ranking functions, which are bounded by 0.

l3: 2
l4: 1
l7: 0

2.3.3 Trivial Cooperation Program

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

2.4 SCC Subproblem 4/4

Here we consider the SCC { l0, l2 }.

2.4.1 Transition Removal

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

l0: −2⋅x2 + 2⋅x4
l2: −2⋅x2 + 2⋅x4 + 1

2.4.2 Transition Removal

We remove transition 23 using the following ranking functions, which are bounded by 0.

l2: 0
l0: −1

2.4.3 Trivial Cooperation Program

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

Tool configuration

AProVE