LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
2: 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
3: − arg2P ≤ 0 ∧ − arg2 ≤ 0
4: 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
5: 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
6: − arg2P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 2 − arg2 ≤ 0 ∧ 1 − x69 ≤ 0
7: TRUE
8: TRUE
9: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 19 1: − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x69 + x69 ≤ 0 ∧ x69 − x69 ≤ 0 ∧ − x45 + x45 ≤ 0 ∧ x45 − x45 ≤ 0 ∧ − x36 + x36 ≤ 0 ∧ x36 − x36 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0
3 26 3: − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x69 + x69 ≤ 0 ∧ x69 − x69 ≤ 0 ∧ − x45 + x45 ≤ 0 ∧ x45 − x45 ≤ 0 ∧ − x36 + x36 ≤ 0 ∧ x36 − x36 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0
7 33 7: − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x69 + x69 ≤ 0 ∧ x69 − x69 ≤ 0 ∧ − x45 + x45 ≤ 0 ∧ x45 − x45 ≤ 0 ∧ − x36 + x36 ≤ 0 ∧ x36 − x36 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 2, 6, 14, 15, 18 using the following ranking functions, which are bounded by −19.

9: 0
0: 0
1: 0
2: 0
4: 0
5: 0
3: 0
6: 0
7: 0
8: 0
9: −6
0: −7
1: −8
2: −8
4: −8
5: −8
1_var_snapshot: −8
1*: −8
7: −9
8: −9
7_var_snapshot: −9
7*: −9
3: −14
6: −14
3_var_snapshot: −14
3*: −14

4 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

1* 22 1: − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x69 + x69 ≤ 0 ∧ x69 − x69 ≤ 0 ∧ − x45 + x45 ≤ 0 ∧ x45 − x45 ≤ 0 ∧ − x36 + x36 ≤ 0 ∧ x36 − x36 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

5 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

1 20 1_var_snapshot: − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x69 + x69 ≤ 0 ∧ x69 − x69 ≤ 0 ∧ − x45 + x45 ≤ 0 ∧ x45 − x45 ≤ 0 ∧ − x36 + x36 ≤ 0 ∧ x36 − x36 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

6 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

3* 29 3: − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x69 + x69 ≤ 0 ∧ x69 − x69 ≤ 0 ∧ − x45 + x45 ≤ 0 ∧ x45 − x45 ≤ 0 ∧ − x36 + x36 ≤ 0 ∧ x36 − x36 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

7 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

3 27 3_var_snapshot: − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x69 + x69 ≤ 0 ∧ x69 − x69 ≤ 0 ∧ − x45 + x45 ≤ 0 ∧ x45 − x45 ≤ 0 ∧ − x36 + x36 ≤ 0 ∧ x36 − x36 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

8 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

7* 36 7: − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x69 + x69 ≤ 0 ∧ x69 − x69 ≤ 0 ∧ − x45 + x45 ≤ 0 ∧ x45 − x45 ≤ 0 ∧ − x36 + x36 ≤ 0 ∧ x36 − x36 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

9 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

7 34 7_var_snapshot: − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x69 + x69 ≤ 0 ∧ x69 − x69 ≤ 0 ∧ − x45 + x45 ≤ 0 ∧ x45 − x45 ≤ 0 ∧ − x36 + x36 ≤ 0 ∧ x36 − x36 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

10 SCC Decomposition

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

10.1 SCC Subproblem 1/3

Here we consider the SCC { 3, 6, 3_var_snapshot, 3* }.

10.1.1 Transition Removal

We remove transitions 29, 12, 13 using the following ranking functions, which are bounded by 0.

3: 2 + 4⋅arg2
6: 4⋅arg2
3_var_snapshot: 1 + 4⋅arg2
3*: 3 + 4⋅arg2

10.1.2 Transition Removal

We remove transition 27 using the following ranking functions, which are bounded by −1.

3: 0
6: 0
3_var_snapshot: −1
3*: 0

10.1.3 Splitting Cut-Point Transitions

We consider 1 subproblems corresponding to sets of cut-point transitions as follows.

10.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 26.

10.1.3.1.1 Splitting Cut-Point Transitions

There remain no cut-point transition to consider. Hence the cooperation termination is trivial.

10.2 SCC Subproblem 2/3

Here we consider the SCC { 7, 8, 7_var_snapshot, 7* }.

10.2.1 Transition Removal

We remove transition 17 using the following ranking functions, which are bounded by 4.

7: −1 + 3⋅arg1
8: −1 + 3⋅arg1
7_var_snapshot: −2 + 3⋅arg1
7*: 3⋅arg1

10.2.2 Transition Removal

We remove transitions 34, 36, 16 using the following ranking functions, which are bounded by −2.

7: 0
8: −2
7_var_snapshot: −1
7*: 1

10.2.3 Splitting Cut-Point Transitions

We consider 1 subproblems corresponding to sets of cut-point transitions as follows.

10.2.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 33.

10.2.3.1.1 Splitting Cut-Point Transitions

There remain no cut-point transition to consider. Hence the cooperation termination is trivial.

10.3 SCC Subproblem 3/3

Here we consider the SCC { 1, 2, 4, 5, 1_var_snapshot, 1* }.

10.3.1 Transition Removal

We remove transitions 1, 3, 4 using the following ranking functions, which are bounded by 6.

1: 3 − 5⋅arg2 + 5⋅arg3
2: 1 − 5⋅arg2 + 5⋅arg4
4: −5⋅arg2 + 5⋅arg3
5: −5⋅arg2 + 5⋅arg4
1_var_snapshot: 2 − 5⋅arg2 + 5⋅arg3
1*: 4 − 5⋅arg2 + 5⋅arg3

10.3.2 Transition Removal

We remove transitions 20, 5, 7, 8, 9, 10, 11 using the following ranking functions, which are bounded by −3.

1: −2
2: arg1 + arg3
4: arg1
5: 0
1_var_snapshot: −3⋅arg1
1*: −1

10.3.3 Transition Removal

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

1: 0
2: 0
4: 0
5: 0
1_var_snapshot: 0
1*: arg1

10.3.4 Splitting Cut-Point Transitions

We consider 1 subproblems corresponding to sets of cut-point transitions as follows.

10.3.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 19.

10.3.4.1.1 Splitting Cut-Point Transitions

There remain no cut-point transition to consider. Hence the cooperation termination is trivial.

Tool configuration

T2Cert