LTS Termination Proof

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Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 6 1: z_0 + z_0 ≤ 0z_0z_0 ≤ 0y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0
2 13 2: z_0 + z_0 ≤ 0z_0z_0 ≤ 0y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 0, 5 using the following ranking functions, which are bounded by −13.

4: 0
0: 0
1: 0
2: 0
3: 0
4: −4
0: −5
1: −6
2: −6
3: −6
1_var_snapshot: −6
1*: −6
2_var_snapshot: −6
2*: −6

3 Location Addition

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

1* 9 1: z_0 + z_0 ≤ 0z_0z_0 ≤ 0y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0

4 Location Addition

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

1 7 1_var_snapshot: z_0 + z_0 ≤ 0z_0z_0 ≤ 0y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0

5 Location Addition

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

2* 16 2: z_0 + z_0 ≤ 0z_0z_0 ≤ 0y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0

6 Location Addition

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

2 14 2_var_snapshot: z_0 + z_0 ≤ 0z_0z_0 ≤ 0y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0

7 SCC Decomposition

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

7.1 SCC Subproblem 1/1

Here we consider the SCC { 1, 2, 3, 1_var_snapshot, 1*, 2_var_snapshot, 2* }.

7.1.1 Transition Removal

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

1: y_0 + z_0
2: y_0 + z_0
3: y_0 + z_0
1_var_snapshot: y_0 + z_0
1*: y_0 + z_0
2_var_snapshot: y_0 + z_0
2*: y_0 + z_0

7.1.2 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 7.

1: 1 − 8⋅x_0 + 8⋅y_0
2: −4 − 8⋅x_0 + 8⋅y_0
3: −2 − 8⋅x_0 + 8⋅y_0
1_var_snapshot: −8⋅x_0 + 8⋅y_0
1*: 2 − 8⋅x_0 + 8⋅y_0
2_var_snapshot: −5 − 8⋅x_0 + 8⋅y_0
2*: −3 − 8⋅x_0 + 8⋅y_0

7.1.3 Transition Removal

We remove transitions 9, 14, 16, 1, 3 using the following ranking functions, which are bounded by −6.

1: −5
2: −2
3: 0
1_var_snapshot: −6
1*: −4
2_var_snapshot: −3
2*: −1

7.1.4 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by −1.

1: 0
2: 0
3: 0
1_var_snapshot: −1
1*: 0
2_var_snapshot: 0
2*: 0

7.1.5 Splitting Cut-Point Transitions

We consider 2 subproblems corresponding to sets of cut-point transitions as follows.

7.1.5.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 6.

7.1.5.1.1 Splitting Cut-Point Transitions

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

7.1.5.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 13.

7.1.5.2.1 Splitting Cut-Point Transitions

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

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