LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
0 7 0: x_6_post + x_6_post ≤ 0x_6_postx_6_post ≤ 0x_6_0 + x_6_0 ≤ 0x_6_0x_6_0 ≤ 0w_5_post + w_5_post ≤ 0w_5_postw_5_post ≤ 0w_5_0 + w_5_0 ≤ 0w_5_0w_5_0 ≤ 0Result_4_post + Result_4_post ≤ 0Result_4_postResult_4_post ≤ 0Result_4_0 + Result_4_0 ≤ 0Result_4_0Result_4_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 0, 5, 6 using the following ranking functions, which are bounded by −13.

5: 0
4: 0
0: 0
2: 0
3: 0
1: 0
5: −5
4: −6
0: −7
2: −7
3: −7
0_var_snapshot: −7
0*: −7
1: −11
Hints:
8 lexWeak[ [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] ]
2 lexWeak[ [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] ]
4 lexWeak[ [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] ]
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] ]
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] ]

3 Location Addition

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

0* 10 0: x_6_post + x_6_post ≤ 0x_6_postx_6_post ≤ 0x_6_0 + x_6_0 ≤ 0x_6_0x_6_0 ≤ 0w_5_post + w_5_post ≤ 0w_5_postw_5_post ≤ 0w_5_0 + w_5_0 ≤ 0w_5_0w_5_0 ≤ 0Result_4_post + Result_4_post ≤ 0Result_4_postResult_4_post ≤ 0Result_4_0 + Result_4_0 ≤ 0Result_4_0Result_4_0 ≤ 0

4 Location Addition

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

0 8 0_var_snapshot: x_6_post + x_6_post ≤ 0x_6_postx_6_post ≤ 0x_6_0 + x_6_0 ≤ 0x_6_0x_6_0 ≤ 0w_5_post + w_5_post ≤ 0w_5_postw_5_post ≤ 0w_5_0 + w_5_0 ≤ 0w_5_0w_5_0 ≤ 0Result_4_post + Result_4_post ≤ 0Result_4_postResult_4_post ≤ 0Result_4_0 + Result_4_0 ≤ 0Result_4_0Result_4_0 ≤ 0

5 SCC Decomposition

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

5.1 SCC Subproblem 1/1

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

5.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −12.

0: −2 − 4⋅w_5_0
2: −4⋅w_5_0
3: −4⋅w_5_0
0_var_snapshot: −3 − 4⋅w_5_0
0*: −1 − 4⋅w_5_0
Hints:
8 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]
10 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]
1 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 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] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0] ]
4 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]

5.1.2 Transition Removal

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

0: −1 − 3⋅x_6_0
2: 1 − 3⋅x_6_0
3: 1 − 3⋅x_6_0
0_var_snapshot: −1 − 3⋅x_6_0
0*: −3⋅x_6_0
Hints:
8 lexWeak[ [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0] ]
10 lexWeak[ [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0] ]
2 lexWeak[ [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexStrict[ [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0] , [0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4 lexWeak[ [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0] ]

5.1.3 Transition Removal

We remove transitions 8, 10, 2 using the following ranking functions, which are bounded by −3.

0: −2
2: 0
3: 0
0_var_snapshot: −3
0*: −1
Hints:
8 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] ]
10 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] ]
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] ]
4 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

5.1.4 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by −1.

0: 0
2: 0
3: 0
0_var_snapshot: 0
0*: −1
Hints:
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] ]

5.1.5 Splitting Cut-Point Transitions

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

5.1.5.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 7.

5.1.5.1.1 Splitting Cut-Point Transitions

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

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