LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: tmp7_post ≤ 0tmp7_0 ≤ 0
1: TRUE
2: TRUE
3: TRUE
4: 32 − nI6_0 ≤ 0
5: TRUE
6: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 8 1: 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 ≤ 0tmp7_post + tmp7_post ≤ 0tmp7_posttmp7_post ≤ 0tmp7_0 + tmp7_0 ≤ 0tmp7_0tmp7_0 ≤ 0ret_nBC211_post + ret_nBC211_post ≤ 0ret_nBC211_postret_nBC211_post ≤ 0ret_nBC211_0 + ret_nBC211_0 ≤ 0ret_nBC211_0ret_nBC211_0 ≤ 0ret_nBC18_post + ret_nBC18_post ≤ 0ret_nBC18_postret_nBC18_post ≤ 0ret_nBC18_0 + ret_nBC18_0 ≤ 0ret_nBC18_0ret_nBC18_0 ≤ 0res5_post + res5_post ≤ 0res5_postres5_post ≤ 0res5_0 + res5_0 ≤ 0res5_0res5_0 ≤ 0res10_post + res10_post ≤ 0res10_postres10_post ≤ 0res10_5 + res10_5 ≤ 0res10_5res10_5 ≤ 0res10_4 + res10_4 ≤ 0res10_4res10_4 ≤ 0res10_3 + res10_3 ≤ 0res10_3res10_3 ≤ 0res10_2 + res10_2 ≤ 0res10_2res10_2 ≤ 0res10_1 + res10_1 ≤ 0res10_1res10_1 ≤ 0res10_0 + res10_0 ≤ 0res10_0res10_0 ≤ 0nX_0 + nX_0 ≤ 0nX_0nX_0 ≤ 0nX9_post + nX9_post ≤ 0nX9_postnX9_post ≤ 0nX9_0 + nX9_0 ≤ 0nX9_0nX9_0 ≤ 0nX4_post + nX4_post ≤ 0nX4_postnX4_post ≤ 0nX4_0 + nX4_0 ≤ 0nX4_0nX4_0 ≤ 0nI6_post + nI6_post ≤ 0nI6_postnI6_post ≤ 0nI6_0 + nI6_0 ≤ 0nI6_0nI6_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 3, 6, 7 using the following ranking functions, which are bounded by −13.

6: 0
5: 0
0: 0
1: 0
2: 0
3: 0
4: 0
6: −5
5: −6
0: −7
1: −7
2: −7
3: −7
1_var_snapshot: −7
1*: −7
4: −8
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, 0, 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, 0, 0, 0, 0, 0, 0, 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 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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4 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, 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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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 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_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 ≤ 0tmp7_post + tmp7_post ≤ 0tmp7_posttmp7_post ≤ 0tmp7_0 + tmp7_0 ≤ 0tmp7_0tmp7_0 ≤ 0ret_nBC211_post + ret_nBC211_post ≤ 0ret_nBC211_postret_nBC211_post ≤ 0ret_nBC211_0 + ret_nBC211_0 ≤ 0ret_nBC211_0ret_nBC211_0 ≤ 0ret_nBC18_post + ret_nBC18_post ≤ 0ret_nBC18_postret_nBC18_post ≤ 0ret_nBC18_0 + ret_nBC18_0 ≤ 0ret_nBC18_0ret_nBC18_0 ≤ 0res5_post + res5_post ≤ 0res5_postres5_post ≤ 0res5_0 + res5_0 ≤ 0res5_0res5_0 ≤ 0res10_post + res10_post ≤ 0res10_postres10_post ≤ 0res10_5 + res10_5 ≤ 0res10_5res10_5 ≤ 0res10_4 + res10_4 ≤ 0res10_4res10_4 ≤ 0res10_3 + res10_3 ≤ 0res10_3res10_3 ≤ 0res10_2 + res10_2 ≤ 0res10_2res10_2 ≤ 0res10_1 + res10_1 ≤ 0res10_1res10_1 ≤ 0res10_0 + res10_0 ≤ 0res10_0res10_0 ≤ 0nX_0 + nX_0 ≤ 0nX_0nX_0 ≤ 0nX9_post + nX9_post ≤ 0nX9_postnX9_post ≤ 0nX9_0 + nX9_0 ≤ 0nX9_0nX9_0 ≤ 0nX4_post + nX4_post ≤ 0nX4_postnX4_post ≤ 0nX4_0 + nX4_0 ≤ 0nX4_0nX4_0 ≤ 0nI6_post + nI6_post ≤ 0nI6_postnI6_post ≤ 0nI6_0 + nI6_0 ≤ 0nI6_0nI6_0 ≤ 0

5 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_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 ≤ 0tmp7_post + tmp7_post ≤ 0tmp7_posttmp7_post ≤ 0tmp7_0 + tmp7_0 ≤ 0tmp7_0tmp7_0 ≤ 0ret_nBC211_post + ret_nBC211_post ≤ 0ret_nBC211_postret_nBC211_post ≤ 0ret_nBC211_0 + ret_nBC211_0 ≤ 0ret_nBC211_0ret_nBC211_0 ≤ 0ret_nBC18_post + ret_nBC18_post ≤ 0ret_nBC18_postret_nBC18_post ≤ 0ret_nBC18_0 + ret_nBC18_0 ≤ 0ret_nBC18_0ret_nBC18_0 ≤ 0res5_post + res5_post ≤ 0res5_postres5_post ≤ 0res5_0 + res5_0 ≤ 0res5_0res5_0 ≤ 0res10_post + res10_post ≤ 0res10_postres10_post ≤ 0res10_5 + res10_5 ≤ 0res10_5res10_5 ≤ 0res10_4 + res10_4 ≤ 0res10_4res10_4 ≤ 0res10_3 + res10_3 ≤ 0res10_3res10_3 ≤ 0res10_2 + res10_2 ≤ 0res10_2res10_2 ≤ 0res10_1 + res10_1 ≤ 0res10_1res10_1 ≤ 0res10_0 + res10_0 ≤ 0res10_0res10_0 ≤ 0nX_0 + nX_0 ≤ 0nX_0nX_0 ≤ 0nX9_post + nX9_post ≤ 0nX9_postnX9_post ≤ 0nX9_0 + nX9_0 ≤ 0nX9_0nX9_0 ≤ 0nX4_post + nX4_post ≤ 0nX4_postnX4_post ≤ 0nX4_0 + nX4_0 ≤ 0nX4_0nX4_0 ≤ 0nI6_post + nI6_post ≤ 0nI6_postnI6_post ≤ 0nI6_0 + nI6_0 ≤ 0nI6_0nI6_0 ≤ 0

6 SCC Decomposition

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

6.1 SCC Subproblem 1/1

Here we consider the SCC { 0, 1, 2, 3, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by −188.

0: −3 − 6⋅nI6_0
1: 1 − 6⋅nI6_0
2: −2 − 6⋅nI6_0
3: −1 − 6⋅nI6_0
1_var_snapshot: −6⋅nI6_0
1*: 2 − 6⋅nI6_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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6] ]
11 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6] ]
0 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6] ]
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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6] ]
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, 6] , [6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6] ]

6.1.2 Transition Removal

We remove transitions 9, 11, 0, 1, 2, 5 using the following ranking functions, which are bounded by −5.

0: −1
1: −3
2: 0
3: −5
1_var_snapshot: −4
1*: −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, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

6.1.3 Splitting Cut-Point Transitions

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

6.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 8.

6.1.3.1.1 Splitting Cut-Point Transitions

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

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