LTS Termination Proof

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Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
0 8 0: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp___0_post + tmp___0_post ≤ 0tmp___0_posttmp___0_post ≤ 0tmp___0_0 + tmp___0_0 ≤ 0tmp___0_0tmp___0_0 ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0n_post + n_post ≤ 0n_postn_post ≤ 0n_0 + n_0 ≤ 0n_0n_0 ≤ 0n9_post + n9_post ≤ 0n9_postn9_post ≤ 0n9_0 + n9_0 ≤ 0n9_0n9_0 ≤ 0n12_post + n12_post ≤ 0n12_postn12_post ≤ 0n12_0 + n12_0 ≤ 0n12_0n12_0 ≤ 0i13_post + i13_post ≤ 0i13_posti13_post ≤ 0i13_0 + i13_0 ≤ 0i13_0i13_0 ≤ 0i10_post + i10_post ≤ 0i10_posti10_post ≤ 0i10_0 + i10_0 ≤ 0i10_0i10_0 ≤ 0
4 15 4: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp___0_post + tmp___0_post ≤ 0tmp___0_posttmp___0_post ≤ 0tmp___0_0 + tmp___0_0 ≤ 0tmp___0_0tmp___0_0 ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0n_post + n_post ≤ 0n_postn_post ≤ 0n_0 + n_0 ≤ 0n_0n_0 ≤ 0n9_post + n9_post ≤ 0n9_postn9_post ≤ 0n9_0 + n9_0 ≤ 0n9_0n9_0 ≤ 0n12_post + n12_post ≤ 0n12_postn12_post ≤ 0n12_0 + n12_0 ≤ 0n12_0n12_0 ≤ 0i13_post + i13_post ≤ 0i13_posti13_post ≤ 0i13_0 + i13_0 ≤ 0i13_0i13_0 ≤ 0i10_post + i10_post ≤ 0i10_posti10_post ≤ 0i10_0 + i10_0 ≤ 0i10_0i10_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 1, 4, 6, 7 using the following ranking functions, which are bounded by −17.

6: 0
5: 0
0: 0
1: 0
2: 0
4: 0
3: 0
6: −6
5: −7
0: −8
1: −8
0_var_snapshot: −8
0*: −8
2: −9
4: −9
4_var_snapshot: −9
4*: −9
3: −10
Hints:
9 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
16 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
0 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
5 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
6 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
7 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

3 Location Addition

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

0* 11 0: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp___0_post + tmp___0_post ≤ 0tmp___0_posttmp___0_post ≤ 0tmp___0_0 + tmp___0_0 ≤ 0tmp___0_0tmp___0_0 ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0n_post + n_post ≤ 0n_postn_post ≤ 0n_0 + n_0 ≤ 0n_0n_0 ≤ 0n9_post + n9_post ≤ 0n9_postn9_post ≤ 0n9_0 + n9_0 ≤ 0n9_0n9_0 ≤ 0n12_post + n12_post ≤ 0n12_postn12_post ≤ 0n12_0 + n12_0 ≤ 0n12_0n12_0 ≤ 0i13_post + i13_post ≤ 0i13_posti13_post ≤ 0i13_0 + i13_0 ≤ 0i13_0i13_0 ≤ 0i10_post + i10_post ≤ 0i10_posti10_post ≤ 0i10_0 + i10_0 ≤ 0i10_0i10_0 ≤ 0

4 Location Addition

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

0 9 0_var_snapshot: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp___0_post + tmp___0_post ≤ 0tmp___0_posttmp___0_post ≤ 0tmp___0_0 + tmp___0_0 ≤ 0tmp___0_0tmp___0_0 ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0n_post + n_post ≤ 0n_postn_post ≤ 0n_0 + n_0 ≤ 0n_0n_0 ≤ 0n9_post + n9_post ≤ 0n9_postn9_post ≤ 0n9_0 + n9_0 ≤ 0n9_0n9_0 ≤ 0n12_post + n12_post ≤ 0n12_postn12_post ≤ 0n12_0 + n12_0 ≤ 0n12_0n12_0 ≤ 0i13_post + i13_post ≤ 0i13_posti13_post ≤ 0i13_0 + i13_0 ≤ 0i13_0i13_0 ≤ 0i10_post + i10_post ≤ 0i10_posti10_post ≤ 0i10_0 + i10_0 ≤ 0i10_0i10_0 ≤ 0

5 Location Addition

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

4* 18 4: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp___0_post + tmp___0_post ≤ 0tmp___0_posttmp___0_post ≤ 0tmp___0_0 + tmp___0_0 ≤ 0tmp___0_0tmp___0_0 ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0n_post + n_post ≤ 0n_postn_post ≤ 0n_0 + n_0 ≤ 0n_0n_0 ≤ 0n9_post + n9_post ≤ 0n9_postn9_post ≤ 0n9_0 + n9_0 ≤ 0n9_0n9_0 ≤ 0n12_post + n12_post ≤ 0n12_postn12_post ≤ 0n12_0 + n12_0 ≤ 0n12_0n12_0 ≤ 0i13_post + i13_post ≤ 0i13_posti13_post ≤ 0i13_0 + i13_0 ≤ 0i13_0i13_0 ≤ 0i10_post + i10_post ≤ 0i10_posti10_post ≤ 0i10_0 + i10_0 ≤ 0i10_0i10_0 ≤ 0

6 Location Addition

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

4 16 4_var_snapshot: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp___0_post + tmp___0_post ≤ 0tmp___0_posttmp___0_post ≤ 0tmp___0_0 + tmp___0_0 ≤ 0tmp___0_0tmp___0_0 ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0n_post + n_post ≤ 0n_postn_post ≤ 0n_0 + n_0 ≤ 0n_0n_0 ≤ 0n9_post + n9_post ≤ 0n9_postn9_post ≤ 0n9_0 + n9_0 ≤ 0n9_0n9_0 ≤ 0n12_post + n12_post ≤ 0n12_postn12_post ≤ 0n12_0 + n12_0 ≤ 0n12_0n12_0 ≤ 0i13_post + i13_post ≤ 0i13_posti13_post ≤ 0i13_0 + i13_0 ≤ 0i13_0i13_0 ≤ 0i10_post + i10_post ≤ 0i10_posti10_post ≤ 0i10_0 + i10_0 ≤ 0i10_0i10_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 { 2, 4, 4_var_snapshot, 4* }.

7.1.1 Transition Removal

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

2: −1 − 4⋅i13_0 + 4⋅n12_0
4: 1 − 4⋅i13_0 + 4⋅n12_0
4_var_snapshot: −4⋅i13_0 + 4⋅n12_0
4*: 2 − 4⋅i13_0 + 4⋅n12_0
Hints:
16 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]
18 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]
2 lexStrict[ [0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0] , [0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]

7.1.2 Transition Removal

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

2: −3
4: −1
4_var_snapshot: −2
4*: 0
Hints:
16 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
18 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

7.1.3 Transition Removal

We remove transition 16 using the following ranking functions, which are bounded by −1.

2: 0
4: 0
4_var_snapshot: −1
4*: 0
Hints:
16 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

7.1.4 Splitting Cut-Point Transitions

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

7.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 15.

7.1.4.1.1 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, 1, 0_var_snapshot, 0* }.

7.2.1 Transition Removal

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

0: 2 − 4⋅i10_0 + 4⋅n9_0
1: −4⋅i10_0 + 4⋅n9_0
0_var_snapshot: 1 − 4⋅i10_0 + 4⋅n9_0
0*: 3 − 4⋅i10_0 + 4⋅n9_0
Hints:
9 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4] ]
11 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4] ]
0 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4] ]
5 lexStrict[ [0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

7.2.2 Transition Removal

We remove transitions 11, 0 using the following ranking functions, which are bounded by −3.

0: −1
1: −3
0_var_snapshot: −2
0*: 0
Hints:
9 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
11 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
0 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

7.2.3 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by 0.

0: 1
1: 0
0_var_snapshot: 0
0*: 0
Hints:
9 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

7.2.4 Splitting Cut-Point Transitions

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

7.2.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 8.

7.2.4.1.1 Splitting Cut-Point Transitions

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

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