LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: −10 + length4_post ≤ 010 − length4_post ≤ 0−10 + length4_0 ≤ 010 − length4_0 ≤ 0
1: −10 + length4_post ≤ 010 − length4_post ≤ 010 − length4_0 ≤ 0
2: −10 + length4_post ≤ 010 − length4_post ≤ 0−10 + length4_0 ≤ 010 − length4_0 ≤ 0
3: TRUE
4: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
2 5 2: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp___08_post + tmp___08_post ≤ 0tmp___08_posttmp___08_post ≤ 0tmp___08_0 + tmp___08_0 ≤ 0tmp___08_0tmp___08_0 ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0s_post + s_post ≤ 0s_posts_post ≤ 0s_0 + s_0 ≤ 0s_0s_0 ≤ 0length4_post + length4_post ≤ 0length4_postlength4_post ≤ 0length4_0 + length4_0 ≤ 0length4_0length4_0 ≤ 0i5_post + i5_post ≤ 0i5_posti5_post ≤ 0i5_0 + i5_0 ≤ 0i5_0i5_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 3, 4 using the following ranking functions, which are bounded by −13.

4: 0
3: 0
0: 0
2: 0
1: 0
4: −5
3: −6
0: −7
2: −7
2_var_snapshot: −7
2*: −7
1: −8
Hints:
6 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] ]
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] ]
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 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] ]
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] ]
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] ]

4 Location Addition

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

2* 8 2: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp___08_post + tmp___08_post ≤ 0tmp___08_posttmp___08_post ≤ 0tmp___08_0 + tmp___08_0 ≤ 0tmp___08_0tmp___08_0 ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0s_post + s_post ≤ 0s_posts_post ≤ 0s_0 + s_0 ≤ 0s_0s_0 ≤ 0length4_post + length4_post ≤ 0length4_postlength4_post ≤ 0length4_0 + length4_0 ≤ 0length4_0length4_0 ≤ 0i5_post + i5_post ≤ 0i5_posti5_post ≤ 0i5_0 + i5_0 ≤ 0i5_0i5_0 ≤ 0

5 Location Addition

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

2 6 2_var_snapshot: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp___08_post + tmp___08_post ≤ 0tmp___08_posttmp___08_post ≤ 0tmp___08_0 + tmp___08_0 ≤ 0tmp___08_0tmp___08_0 ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0s_post + s_post ≤ 0s_posts_post ≤ 0s_0 + s_0 ≤ 0s_0s_0 ≤ 0length4_post + length4_post ≤ 0length4_postlength4_post ≤ 0length4_0 + length4_0 ≤ 0length4_0length4_0 ≤ 0i5_post + i5_post ≤ 0i5_posti5_post ≤ 0i5_0 + i5_0 ≤ 0i5_0i5_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, 2, 2_var_snapshot, 2* }.

6.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −128.

0: −13⋅i5_0length4_0
2: 1 − 13⋅i5_0
2_var_snapshot: −13⋅i5_0
2*: 2 − 13⋅i5_0
Hints:
6 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13] ]
8 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13] ]
1 lexStrict[ [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 13, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 14, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2 lexWeak[ [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 13] ]

6.1.2 Transition Removal

We remove transitions 8, 2 using the following ranking functions, which are bounded by −11.

0: length4_0length4_post
2: 0
2_var_snapshot: length4_0
2*: length4_post
Hints:
6 lexWeak[ [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0] ]
8 lexStrict[ [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

6.1.3 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by 9.

0: 0
2: length4_0
2_var_snapshot: 0
2*: 0
Hints:
6 lexStrict[ [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

6.1.4 Splitting Cut-Point Transitions

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

6.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 5.

6.1.4.1.1 Splitting Cut-Point Transitions

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

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