LTS Termination Proof

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Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 4 1: arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 0, 3 using the following ranking functions, which are bounded by −11.

2: 0
0: 0
1: 0
2: −4
0: −5
1: −6
1_var_snapshot: −6
1*: −6
Hints:
5 lexWeak[ [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 lexStrict[ [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ]
3 lexStrict[ [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* 7 1: arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

4 Location Addition

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

1 5 1_var_snapshot: arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 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, 1_var_snapshot, 1* }.

5.1.1 Transition Removal

We remove transitions 1, 2 using the following ranking functions, which are bounded by −20.

1: 1 − arg1
1_var_snapshot: arg1
1*: 1 − arg1
Hints:
5 lexWeak[ [0, 0, 0, 1] ]
7 lexWeak[ [0, 0, 0, 1] ]
1 lexStrict[ [0, 0, 0, 0, 1, 0, 0, 1] , [0, 0, 0, 1, 0, 0, 0, 0] ]
2 lexStrict[ [0, 0, 0, 0, 1, 0, 0, 1] , [0, 0, 0, 1, 0, 0, 0, 0] ]

5.1.2 Transition Removal

We remove transitions 5, 7 using the following ranking functions, which are bounded by −1.

1: 0
1_var_snapshot: −1
1*: 1
Hints:
5 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]
7 lexStrict[ [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 4.

5.1.3.1.1 Splitting Cut-Point Transitions

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

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