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 5 1: arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
2 12 2: arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 0, 2, 4 using the following ranking functions, which are bounded by −15.

3: 0
0: 0
1: 0
2: 0
3: −5
0: −6
1: −7
1_var_snapshot: −7
1*: −7
2: −10
2_var_snapshot: −10
2*: −10
Hints:
6 lexWeak[ [0, 0, 0, 0] ]
13 lexWeak[ [0, 0, 0, 0] ]
1 lexWeak[ [0, 0, 0, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0, 0] ]
0 lexStrict[ [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] ]
4 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* 8 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 6 1_var_snapshot: arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

5 Location Addition

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

2* 15 2: arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

6 Location Addition

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

2 13 2_var_snapshot: arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

7 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

7.1 SCC Subproblem 1/2

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

7.1.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by −21.

2: 1 − arg1
2_var_snapshot: arg1
2*: 2 − arg1
Hints:
13 lexWeak[ [0, 0, 0, 1] ]
15 lexWeak[ [0, 0, 0, 1] ]
3 lexStrict[ [0, 0, 0, 1, 0, 0, 1] , [0, 0, 1, 0, 0, 0, 0] ]

7.1.2 Transition Removal

We remove transitions 13, 15 using the following ranking functions, which are bounded by −2.

2: −1
2_var_snapshot: −2
2*: 0
Hints:
13 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]
15 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]

7.1.3 Splitting Cut-Point Transitions

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

7.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 12.

7.1.3.1.1 Splitting Cut-Point Transitions

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

7.2 SCC Subproblem 2/2

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

7.2.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −298.

1: 1 − 3⋅arg1
1_var_snapshot: −3⋅arg1
1*: 2 − 3⋅arg1
Hints:
6 lexWeak[ [0, 0, 0, 3] ]
8 lexWeak[ [0, 0, 0, 3] ]
1 lexStrict[ [0, 0, 0, 3, 0, 0, 3] , [0, 0, 3, 0, 0, 0, 0] ]

7.2.2 Transition Removal

We remove transitions 6, 8 using the following ranking functions, which are bounded by −1.

1: 0
1_var_snapshot: −1
1*: 1
Hints:
6 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]
8 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]

7.2.3 Splitting Cut-Point Transitions

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

7.2.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 5.

7.2.3.1.1 Splitting Cut-Point Transitions

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

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