LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: 1 − x_0 ≤ 0
1: TRUE
2: 1 − x_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:
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
1 13 1: 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.

3 Transition Removal

We remove transitions 3, 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
2: −6
1: −6
0_var_snapshot: −6
0*: −6
1_var_snapshot: −6
1*: −6
Hints:
7 lexWeak[ [0, 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, 0] ]
1 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4 lexWeak[ [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] ]
5 lexStrict[ [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.

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

5 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

6 Location Addition

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

1* 16 1: 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 Location Addition

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

1 14 1_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

8 SCC Decomposition

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

8.1 SCC Subproblem 1/1

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

8.1.1 Transition Removal

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

0: 4⋅x_0
2: 4⋅x_0
1: 2 + 4⋅x_0
0_var_snapshot: 4⋅x_0
0*: 4⋅x_0
1_var_snapshot: 1 + 4⋅x_0
1*: 3 + 4⋅x_0
Hints:
7 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0] ]
9 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0] ]
14 lexWeak[ [0, 0, 0, 0, 0, 0, 4, 0] ]
16 lexWeak[ [0, 0, 0, 0, 0, 0, 4, 0] ]
0 lexStrict[ [0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0] , [4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0] ]
4 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0] ]

8.1.2 Transition Removal

We remove transitions 14, 4 using the following ranking functions, which are bounded by 2.

0: 1 − 4⋅x_0 + 4⋅y_0
2: 3 − 4⋅x_0 + 4⋅y_0
1: 4
0_var_snapshot: −4⋅x_0 + 4⋅y_0
0*: 2 − 4⋅x_0 + 4⋅y_0
1_var_snapshot: 3
1*: 4
Hints:
7 lexWeak[ [0, 0, 0, 4, 0, 0, 0, 0, 4] ]
9 lexWeak[ [0, 0, 0, 4, 0, 0, 0, 0, 4] ]
14 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ]
16 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ]
1 lexWeak[ [0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 4] ]
2 lexWeak[ [0, 0, 0, 4, 0, 0, 0, 0, 4] ]
4 lexStrict[ [0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 4] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.1.3 Transition Removal

We remove transitions 16, 1 using the following ranking functions, which are bounded by 0.

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

8.1.4 Transition Removal

We remove transitions 7, 9, 2 using the following ranking functions, which are bounded by −1.

0: 0
2: 2⋅x_0
1: 0
0_var_snapshot: −1
0*: 1
1_var_snapshot: 0
1*: 0
Hints:
7 lexStrict[ [0, 0, 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, 0, 0] ]
2 lexStrict[ [2, 0, 0, 0, 0, 0, 0, 0, 0] , [2, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.1.5 Splitting Cut-Point Transitions

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

8.1.5.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 6.

8.1.5.1.1 Splitting Cut-Point Transitions

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

8.1.5.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 13.

8.1.5.2.1 Splitting Cut-Point Transitions

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

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