LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: 1 − x_0 ≤ 01 − y_0 ≤ 0
2: TRUE
3: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
0 5 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
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 3, 4 using the following ranking functions, which are bounded by −11.

3: 0
2: 0
0: 0
1: 0
3: −4
2: −5
0: −6
1: −6
0_var_snapshot: −6
0*: −6
Hints:
6 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ]
0 lexWeak[ [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, 0, 0, 0] ]
3 lexStrict[ [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] ]

4 Location Addition

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

0* 8 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 6 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 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, 1, 0_var_snapshot, 0* }.

6.1.1 Transition Removal

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

0: y_0
1: y_0
0_var_snapshot: y_0
0*: y_0
Hints:
6 lexWeak[ [0, 0, 1, 0, 0, 0, 0, 0] ]
8 lexWeak[ [0, 0, 1, 0, 0, 0, 0, 0] ]
0 lexWeak[ [0, 0, 0, 0, 1, 0, 0, 0, 0, 0] ]
1 lexStrict[ [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0] , [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0] ]

6.1.2 Transition Removal

We remove transitions 0, 2 using the following ranking functions, which are bounded by 2.

0: 1 + 4⋅x_0
1: −1 + 4⋅x_0
0_var_snapshot: 4⋅x_0
0*: 2 + 4⋅x_0
Hints:
6 lexWeak[ [0, 0, 0, 0, 0, 0, 4, 0] ]
8 lexWeak[ [0, 0, 0, 0, 0, 0, 4, 0] ]
0 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 4, 0] , [4, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2 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] ]

6.1.3 Transition Removal

We remove transitions 6, 8 using the following ranking functions, which are bounded by −2.

0: −1
1: 0
0_var_snapshot: −2
0*: 0
Hints:
6 lexStrict[ [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] ]

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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