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 ∧ x8 = x8
l22 l22 l22: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l7 l7 l7: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l11 l11 l11: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l13 l13 l13: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l3 l3 l3: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l20 l20 l20: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l18 l18 l18: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l17 l17 l17: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l2 l2 l2: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l21 l21 l21: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l14 l14 l14: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l9 l9 l9: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l23 l23 l23: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l4 l4 l4: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l6 l6 l6: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l10 l10 l10: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l8 l8 l8: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l16 l16 l16: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l15 l15 l15: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l0 l0 l0: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l12 l12 l12: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l19 l19 l19: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
and for every transition t, a duplicate t is considered.

2 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

2.1 SCC Subproblem 1/1

Here we consider the SCC { l5, l22, l7, l11, l3, l13, l20, l18, l17, l2, l21, l9, l14, l4, l6, l10, l8, l15, l16, l0, l19, l12 }.

2.1.1 Transition Removal

We remove transitions 35, 26, 24, 11, 9 using the following ranking functions, which are bounded by 0.

l0: − x2 + x4
l2: − x2 + x4
l7: − x2 + x4
l6: − x2 + x4
l19: − x2 + x4
l18: − x2 + x4
l9: − x2 + x4
l20: − x2 + x4
l11: − x2 + x4
l10: − x2 + x4
l5: − x2 + x4
l4: − x2 + x4
l3: − x2 + x4
l21: −1 − x2 + x4
l22: −1 − x2 + x4
l8: − x2 + x4
l17: − x2 + x4
l12: − x2 + x4
l13: − x2 + x4
l14: − x2 + x4
l15: − x2 + x4
l16: − x2 + x4

2.1.2 Transition Removal

We remove transitions 31, 33, 32 using the following ranking functions, which are bounded by 0.

l0: −1
l2: −1
l7: −1
l6: −1
l19: −1
l18: −1
l9: −1
l20: −1
l11: −1
l10: −1
l5: −1
l4: −1
l3: −1
l21: 0
l22: 1
l8: −1
l17: −1
l12: −1
l13: −1
l14: −1
l15: −1
l16: −1

2.1.3 Transition Removal

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

l0: −12⋅x3 + 12⋅x4
l2: −12⋅x3 + 12⋅x4 − 1
l7: −12⋅x3 + 12⋅x4 + 1
l6: −12⋅x3 + 12⋅x4 − 10
l19: −12⋅x3 + 12⋅x4 − 8
l18: 12⋅x4 − 12⋅x3 − 9
l9: −12⋅x3 + 12⋅x4 − 9
l20: −12⋅x3 + 12⋅x4 − 3
l11: 12⋅x4 − 12⋅x3 − 7
l10: −12⋅x3 + 12⋅x4 − 6
l5: 12⋅x4 − 12⋅x3 − 5
l4: −12⋅x3 + 12⋅x4 − 4
l3: 12⋅x4 − 12⋅x3 − 2
l8: −12⋅x3 + 12⋅x4 − 9
l17: −12⋅x3 + 12⋅x4 − 9
l12: −12⋅x3 + 12⋅x4 − 9
l13: −12⋅x3 + 12⋅x4 − 9
l14: 12⋅x4 − 12⋅x3 − 9
l15: −12⋅x3 + 12⋅x4 − 9
l16: −12⋅x3 + 12⋅x4 − 9

2.1.4 Transition Removal

We remove transitions 19, 21, 20, 27, 23, 7, 25, 4, 29, 28, 34, 3 using the following ranking functions, which are bounded by 0.

l7: −1
l0: −1
l6: −1
l19: 0
l18: −1
l9: −1
l20: 5
l11: 1
l10: 2
l5: 3
l4: 4
l3: 6
l2: 7
l8: −1
l17: −1
l12: −1
l13: −1
l14: −1
l15: −1
l16: −1

2.1.5 Transition Removal

We remove transitions 36, 5, 17 using the following ranking functions, which are bounded by 0.

l7: 0
l0: −1
l6: 1
l18: 2
l9: 2
l8: 2
l17: 2
l12: 2
l13: 2
l14: 2
l15: 2
l16: 2

2.1.6 Transition Removal

We remove transition 18 using the following ranking functions, which are bounded by 0.

l9: − x1 + x4
l18: − x1 + x4
l8: −1 − x1 + x4
l17: −1 − x1 + x4
l12: −1 − x1 + x4
l13: −1 − x1 + x4
l14: −1 − x1 + x4
l15: −1 − x1 + x4
l16: −1 − x1 + x4

2.1.7 Transition Removal

We remove transitions 16, 6, 13, 8, 30, 10, 22, 12, 15, 14 using the following ranking functions, which are bounded by 0.

l9: 0
l18: −1
l8: 1
l17: 7
l12: 2
l13: 3
l14: 4
l15: 5
l16: 6

2.1.8 Trivial Cooperation Program

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

Tool configuration

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