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 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l1 l1 l1: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l13 l13 l13: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l18 l18 l18: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l17 l17 l17: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l21 l21 l21: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l14 l14 l14: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l9 l9 l9: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l25 l25 l25: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l6 l6 l6: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l8 l8 l8: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l0 l0 l0: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l12 l12 l12: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l19 l19 l19: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l7 l7 l7: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l24 l24 l24: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l11 l11 l11: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l3 l3 l3: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l20 l20 l20: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l2 l2 l2: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l23 l23 l23: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l4 l4 l4: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l10 l10 l10: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l15 l15 l15: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
l16 l16 l16: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 ∧ x21 = x21 ∧ x22 = x22 ∧ x23 = x23 ∧ x24 = x24
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, l4, l7, l3, l2 }.

2.1.1 Transition Removal

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

l2: 6⋅x17 − 6⋅x18
l3: 6⋅x17 − 6⋅x18 + 3
l5: 6⋅x17 − 6⋅x18 + 1
l4: 6⋅x17 − 6⋅x18 + 1
l7: 6⋅x17 − 6⋅x18 + 2

2.1.2 Transition Removal

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

l2: 3⋅x17 + 3⋅x18 − 6⋅x19 + 7
l3: −6⋅x9 + 3⋅x17 + 3⋅x18 + 3
l5: 3⋅x18 − 3⋅x19 + 2
l4: 3⋅x18 − 3⋅x19 + 1
l7: 0

2.1.3 Transition Removal

We remove transitions 2, 4 using the following ranking functions, which are bounded by 0.

l2: 0
l3: x8 − x18
l5: 1
l4: 1

2.1.4 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by −3.

l4: 3⋅x17 − 3⋅x19 − 1
l5: 3⋅x17 − 3⋅x19

2.1.5 Transition Removal

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

l4: 0
l5: −1

2.1.6 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 { l1, l0 }.

2.2.1 Transition Removal

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

l0: 3⋅x17 − 3⋅x19 − 1
l1: 3⋅x17 − 3⋅x19

2.2.2 Transition Removal

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

l0: 0
l1: −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 3/4

Here we consider the SCC { l23, l6 }.

2.3.1 Transition Removal

We remove transition 32 using the following ranking functions, which are bounded by −3.

l6: 3⋅x17 − 3⋅x20
l23: 3⋅x17 − 3⋅x20 − 1

2.3.2 Transition Removal

We remove transition 30 using the following ranking functions, which are bounded by 0.

l23: 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.

2.4 SCC Subproblem 4/4

Here we consider the SCC { l10, l11, l8, l13, l20, l16, l18, l17, l19, l12, l9, l14 }.

2.4.1 Transition Removal

We remove transition 29 using the following ranking functions, which are bounded by 0.

l19: 6⋅x17 − 6⋅x21 + 3
l9: 6⋅x17 − 6⋅x21 + 2
l18: 6⋅x17 − 6⋅x21 + 1
l8: 6⋅x17 − 6⋅x21 + 2
l11: 6⋅x17 − 6⋅x21 + 2
l10: 6⋅x17 − 6⋅x21 + 2
l16: 6⋅x17 − 6⋅x21 + 2
l13: 6⋅x17 − 6⋅x21 + 2
l12: 6⋅x17 − 6⋅x21 + 2
l17: 6⋅x17 − 6⋅x21 + 2
l14: 6⋅x17 − 6⋅x21 + 2
l20: 6⋅x17 − 6⋅x21 + 2

2.4.2 Transition Removal

We remove transitions 23, 26 using the following ranking functions, which are bounded by 0.

l18: 0
l19: x11 − x21
l9: 1
l8: 1
l11: 1
l10: 1
l16: 1
l13: 1
l12: 1
l17: 1
l14: 1
l20: 1

2.4.3 Transition Removal

We remove transition 27 using the following ranking functions, which are bounded by 0.

l8: 6⋅x17 − 6⋅x21 − 6⋅x22 − 2
l9: 6⋅x17 − 6⋅x21 − 6⋅x22 + 3
l11: 6⋅x17 − 6⋅x21 − 6⋅x22 − 1
l10: 6⋅x17 − 6⋅x21 − 6⋅x22 − 1
l16: 6⋅x17 − 6⋅x21 − 6⋅x22
l13: 6⋅x17 − 6⋅x21 − 6⋅x22 + 1
l12: 6⋅x17 − 6⋅x21 − 6⋅x22 + 1
l17: 6⋅x17 − 6⋅x21 − 6⋅x22 + 1
l14: 6⋅x17 − 6⋅x21 − 6⋅x22 + 1
l20: 6⋅x17 − 6⋅x21 − 6⋅x22 + 2

2.4.4 Transition Removal

We remove transition 24 using the following ranking functions, which are bounded by 0.

l8: −1 + 2⋅x22 − x23 − x24
l9: 3⋅x12 − x13 − x14 − x22
l11: −1 + 2⋅x22 − x23 − x24
l10: −1 + 2⋅x22 − x23 − x24
l16: −1 + x22 − x23
l13: −1 + x22 − x23
l12: −1 + x22 − x23
l17: −1 + x22 − x23
l14: −1 + x22 − x23
l20: 0

2.4.5 Transition Removal

We remove transitions 11, 13, 18, 21 using the following ranking functions, which are bounded by 0.

l8: 0
l9: x12 − x22
l11: 1
l10: 1
l16: 2
l13: 3
l12: 3
l17: 3
l14: 3

2.4.6 Transition Removal

We remove transitions 14, 22 using the following ranking functions, which are bounded by 0.

l10: 3⋅x21 + 3⋅x22 − 3⋅x24 − 1
l11: 3⋅x21 + 3⋅x22 − 3⋅x24
l12: 4⋅x21 + 4⋅x22 − 4⋅x23 − 1
l13: 4⋅x21 + 4⋅x22 − 4⋅x23 + 2
l17: 4⋅x21 + 4⋅x22 − 4⋅x23 + 1
l14: 4⋅x21 + 4⋅x22 − 4⋅x23

2.4.7 Transition Removal

We remove transitions 12, 15, 19, 16, 20 using the following ranking functions, which are bounded by 0.

l10: 0
l11: −1
l12: 0
l13: x13 − x23
l17: 2
l14: 1

2.4.8 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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