LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
0 8 0: x79_post + x79_post ≤ 0x79_postx79_post ≤ 0x79_0 + x79_0 ≤ 0x79_0x79_0 ≤ 0x35_post + x35_post ≤ 0x35_postx35_post ≤ 0x35_0 + x35_0 ≤ 0x35_0x35_0 ≤ 0tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i911_post + i911_post ≤ 0i911_posti911_post ≤ 0i911_0 + i911_0 ≤ 0i911_0i911_0 ≤ 0i57_post + i57_post ≤ 0i57_posti57_post ≤ 0i57_0 + i57_0 ≤ 0i57_0i57_0 ≤ 0___const_10_0 + ___const_10_0 ≤ 0___const_10_0___const_10_0 ≤ 0
4 15 4: x79_post + x79_post ≤ 0x79_postx79_post ≤ 0x79_0 + x79_0 ≤ 0x79_0x79_0 ≤ 0x35_post + x35_post ≤ 0x35_postx35_post ≤ 0x35_0 + x35_0 ≤ 0x35_0x35_0 ≤ 0tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i911_post + i911_post ≤ 0i911_posti911_post ≤ 0i911_0 + i911_0 ≤ 0i911_0i911_0 ≤ 0i57_post + i57_post ≤ 0i57_posti57_post ≤ 0i57_0 + i57_0 ≤ 0i57_0i57_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 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] ]
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 lexWeak[ [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] ]
3 lexWeak[ [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] ]
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] ]
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] ]
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] ]
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] ]

3 Location Addition

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

0* 11 0: x79_post + x79_post ≤ 0x79_postx79_post ≤ 0x79_0 + x79_0 ≤ 0x79_0x79_0 ≤ 0x35_post + x35_post ≤ 0x35_postx35_post ≤ 0x35_0 + x35_0 ≤ 0x35_0x35_0 ≤ 0tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i911_post + i911_post ≤ 0i911_posti911_post ≤ 0i911_0 + i911_0 ≤ 0i911_0i911_0 ≤ 0i57_post + i57_post ≤ 0i57_posti57_post ≤ 0i57_0 + i57_0 ≤ 0i57_0i57_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.

0 9 0_var_snapshot: x79_post + x79_post ≤ 0x79_postx79_post ≤ 0x79_0 + x79_0 ≤ 0x79_0x79_0 ≤ 0x35_post + x35_post ≤ 0x35_postx35_post ≤ 0x35_0 + x35_0 ≤ 0x35_0x35_0 ≤ 0tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i911_post + i911_post ≤ 0i911_posti911_post ≤ 0i911_0 + i911_0 ≤ 0i911_0i911_0 ≤ 0i57_post + i57_post ≤ 0i57_posti57_post ≤ 0i57_0 + i57_0 ≤ 0i57_0i57_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.

4* 18 4: x79_post + x79_post ≤ 0x79_postx79_post ≤ 0x79_0 + x79_0 ≤ 0x79_0x79_0 ≤ 0x35_post + x35_post ≤ 0x35_postx35_post ≤ 0x35_0 + x35_0 ≤ 0x35_0x35_0 ≤ 0tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i911_post + i911_post ≤ 0i911_posti911_post ≤ 0i911_0 + i911_0 ≤ 0i911_0i911_0 ≤ 0i57_post + i57_post ≤ 0i57_posti57_post ≤ 0i57_0 + i57_0 ≤ 0i57_0i57_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.

4 16 4_var_snapshot: x79_post + x79_post ≤ 0x79_postx79_post ≤ 0x79_0 + x79_0 ≤ 0x79_0x79_0 ≤ 0x35_post + x35_post ≤ 0x35_postx35_post ≤ 0x35_0 + x35_0 ≤ 0x35_0x35_0 ≤ 0tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i911_post + i911_post ≤ 0i911_posti911_post ≤ 0i911_0 + i911_0 ≤ 0i911_0i911_0 ≤ 0i57_post + i57_post ≤ 0i57_posti57_post ≤ 0i57_0 + i57_0 ≤ 0i57_0i57_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 { 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⋅___const_10_0 − 4⋅i911_0
4: 1 + 4⋅___const_10_0 − 4⋅i911_0
4_var_snapshot: 4⋅___const_10_0 − 4⋅i911_0
4*: 2 + 4⋅___const_10_0 − 4⋅i911_0
Hints:
16 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0] ]
18 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 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, 4, 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] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0] ]

7.1.2 Transition Removal

We remove transitions 16, 18, 3 using the following ranking functions, which are bounded by −1.

2: −1
4: 1
4_var_snapshot: 0
4*: 2
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] ]
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] ]
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] ]

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 15.

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, 1, 0_var_snapshot, 0* }.

7.2.1 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by 1.

0: 4⋅___const_10_0 − 4⋅i57_0
1: −2 + 4⋅___const_10_0 − 4⋅i57_0
0_var_snapshot: −1 + 4⋅___const_10_0 − 4⋅i57_0
0*: 1 + 4⋅___const_10_0 − 4⋅i57_0
Hints:
9 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0] ]
11 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0] ]
0 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0] ]
5 lexStrict[ [0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 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] ]

7.2.2 Transition Removal

We remove transitions 9, 11, 0 using the following ranking functions, which are bounded by −1.

0: 1
1: −1
0_var_snapshot: 0
0*: 2
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] ]
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 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] ]

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 8.

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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