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 8 1: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i7_post + i7_post ≤ 0i7_posti7_post ≤ 0i7_0 + i7_0 ≤ 0i7_0i7_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0___const_10_0 + ___const_10_0 ≤ 0___const_10_0___const_10_0 ≤ 0
2 15 2: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i7_post + i7_post ≤ 0i7_posti7_post ≤ 0i7_0 + i7_0 ≤ 0i7_0i7_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0___const_10_0 + ___const_10_0 ≤ 0___const_10_0___const_10_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 0, 4, 6, 7 using the following ranking functions, which are bounded by −17.

6: 0
5: 0
0: 0
2: 0
1: 0
3: 0
4: 0
6: −6
5: −7
0: −8
2: −8
2_var_snapshot: −8
2*: −8
1: −9
3: −9
1_var_snapshot: −9
1*: −9
4: −10

3 Location Addition

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

1* 11 1: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i7_post + i7_post ≤ 0i7_posti7_post ≤ 0i7_0 + i7_0 ≤ 0i7_0i7_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0___const_10_0 + ___const_10_0 ≤ 0___const_10_0___const_10_0 ≤ 0

4 Location Addition

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

1 9 1_var_snapshot: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i7_post + i7_post ≤ 0i7_posti7_post ≤ 0i7_0 + i7_0 ≤ 0i7_0i7_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0___const_10_0 + ___const_10_0 ≤ 0___const_10_0___const_10_0 ≤ 0

5 Location Addition

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

2* 18 2: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i7_post + i7_post ≤ 0i7_posti7_post ≤ 0i7_0 + i7_0 ≤ 0i7_0i7_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0___const_10_0 + ___const_10_0 ≤ 0___const_10_0___const_10_0 ≤ 0

6 Location Addition

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

2 16 2_var_snapshot: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i7_post + i7_post ≤ 0i7_posti7_post ≤ 0i7_0 + i7_0 ≤ 0i7_0i7_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0___const_10_0 + ___const_10_0 ≤ 0___const_10_0___const_10_0 ≤ 0

7 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

7.1 SCC Subproblem 1/2

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

7.1.1 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by 0.

1: 3⋅___const_10_0 − 3⋅i7_0
3: −2 + 3⋅___const_10_0 − 3⋅i7_0
1_var_snapshot: −1 + 3⋅___const_10_0 − 3⋅i7_0
1*: 3⋅___const_10_0 − 3⋅i7_0

7.1.2 Transition Removal

We remove transitions 9, 11, 3 using the following ranking functions, which are bounded by −3.

1: −1
3: −3
1_var_snapshot: −2
1*: 0

7.1.3 Splitting Cut-Point Transitions

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

7.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 8.

7.1.3.1.1 Splitting Cut-Point Transitions

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

7.2 SCC Subproblem 2/2

Here we consider the SCC { 0, 2, 2_var_snapshot, 2* }.

7.2.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 2.

0: −1 + 4⋅___const_10_0 − 4⋅i4_0
2: 1 + 4⋅___const_10_0 − 4⋅i4_0
2_var_snapshot: 4⋅___const_10_0 − 4⋅i4_0
2*: 2 + 4⋅___const_10_0 − 4⋅i4_0

7.2.2 Transition Removal

We remove transitions 16, 18, 2 using the following ranking functions, which are bounded by −3.

0: −3
2: −1
2_var_snapshot: −2
2*: 0

7.2.3 Splitting Cut-Point Transitions

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

7.2.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 15.

7.2.3.1.1 Splitting Cut-Point Transitions

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

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