LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
0 6 0: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0
2 13 2: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 4, 5 using the following ranking functions, which are bounded by −13.

4: 0
3: 0
0: 0
1: 0
2: 0
4: −4
3: −5
0: −6
1: −6
2: −6
0_var_snapshot: −6
0*: −6
2_var_snapshot: −6
2*: −6
Hints:
7 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ]
14 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ]
0 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexWeak[ [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] ]
5 lexStrict[ [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* 9 0: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0

4 Location Addition

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

0 7 0_var_snapshot: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0

5 Location Addition

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

2* 16 2: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0

6 Location Addition

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

2 14 2_var_snapshot: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0

7 SCC Decomposition

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

7.1 SCC Subproblem 1/1

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

7.1.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 2.

0: 2 + 3⋅x_0
1: 2 + 3⋅x_0
2: 1 + 3⋅x_0
0_var_snapshot: 2 + 3⋅x_0
0*: 2 + 3⋅x_0
2_var_snapshot: 3⋅x_0
2*: 2 + 3⋅x_0
Hints:
7 lexWeak[ [0, 0, 0, 0, 0, 0, 3, 0] ]
9 lexWeak[ [0, 0, 0, 0, 0, 0, 3, 0] ]
14 lexWeak[ [0, 0, 0, 0, 0, 0, 3, 0] ]
16 lexWeak[ [0, 0, 0, 0, 0, 0, 3, 0] ]
0 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0] ]
1 lexWeak[ [0, 0, 0, 0, 0, 0, 3, 0] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 3, 0] ]
3 lexStrict[ [0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0] , [0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0] ]

7.1.2 Transition Removal

We remove transition 0 using the following ranking functions, which are bounded by 3.

0: 1 + 4⋅y_0
1: 3 + 4⋅y_0
2: −2 + 4⋅y_0
0_var_snapshot: 4⋅y_0
0*: 2 + 4⋅y_0
2_var_snapshot: −3 + 4⋅y_0
2*: −1 + 4⋅y_0
Hints:
7 lexWeak[ [0, 0, 4, 0, 0, 0, 0, 0] ]
9 lexWeak[ [0, 0, 4, 0, 0, 0, 0, 0] ]
14 lexWeak[ [0, 0, 4, 0, 0, 0, 0, 0] ]
16 lexWeak[ [0, 0, 4, 0, 0, 0, 0, 0] ]
0 lexStrict[ [0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0] , [0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0] ]
1 lexWeak[ [0, 0, 4, 0, 0, 0, 0, 0] ]
2 lexWeak[ [0, 0, 0, 4, 0, 0, 0, 0, 0] ]

7.1.3 Transition Removal

We remove transitions 7, 9, 14, 16, 2 using the following ranking functions, which are bounded by −3.

0: 1
1: 3
2: −2
0_var_snapshot: 0
0*: 2
2_var_snapshot: −3
2*: −1
Hints:
7 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ]
9 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ]
14 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ]
16 lexStrict[ [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] ]
2 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0] ]

7.1.4 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 0.

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

7.1.5 Splitting Cut-Point Transitions

We consider 2 subproblems corresponding to sets of cut-point transitions as follows.

7.1.5.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 6.

7.1.5.1.1 Splitting Cut-Point Transitions

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

7.1.5.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 13.

7.1.5.2.1 Splitting Cut-Point Transitions

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

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