LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 9 1: y_6_0 + y_6_0 ≤ 0y_6_0y_6_0 ≤ 0x_5_post + x_5_post ≤ 0x_5_postx_5_post ≤ 0x_5_0 + x_5_0 ≤ 0x_5_0x_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, 1, 2, 3, 8 using the following ranking functions, which are bounded by −17.

7: 0
0: 0
1: 0
5: 0
6: 0
2: 0
3: 0
4: 0
7: −7
0: −8
1: −9
5: −9
6: −9
1_var_snapshot: −9
1*: −9
2: −13
3: −14
4: −15
Hints:
10 lexWeak[ [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] ]
5 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
6 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
7 lexWeak[ [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] ]
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] ]
2 lexStrict[ [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] ]
8 lexStrict[ [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.

1* 12 1: y_6_0 + y_6_0 ≤ 0y_6_0y_6_0 ≤ 0x_5_post + x_5_post ≤ 0x_5_postx_5_post ≤ 0x_5_0 + x_5_0 ≤ 0x_5_0x_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.

1 10 1_var_snapshot: y_6_0 + y_6_0 ≤ 0y_6_0y_6_0 ≤ 0x_5_post + x_5_post ≤ 0x_5_postx_5_post ≤ 0x_5_0 + x_5_0 ≤ 0x_5_0x_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 { 1, 5, 6, 1_var_snapshot, 1* }.

5.1.1 Transition Removal

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

1: −2 − 4⋅x_5_0 + 4⋅y_6_0
5: −4⋅x_5_0 + 4⋅y_6_0
6: −4⋅x_5_0 + 4⋅y_6_0
1_var_snapshot: −3 − 4⋅x_5_0 + 4⋅y_6_0
1*: −1 − 4⋅x_5_0 + 4⋅y_6_0
Hints:
10 lexWeak[ [4, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]
12 lexWeak[ [4, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]
4 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 4, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
5 lexWeak[ [4, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]
6 lexStrict[ [0, 0, 0, 0, 4, 0, 4, 4, 0, 0, 0, 0, 0] , [0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
7 lexWeak[ [4, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]

5.1.2 Transition Removal

We remove transitions 10, 12, 5, 7 using the following ranking functions, which are bounded by −3.

1: −2
5: 0
6: 0
1_var_snapshot: −3
1*: −1
Hints:
10 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
12 lexStrict[ [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] ]
7 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

5.1.3 Splitting Cut-Point Transitions

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

5.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 9.

5.1.3.1.1 Splitting Cut-Point Transitions

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

Tool configuration

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