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 10 1: i1_post + i1_post ≤ 0i1_posti1_post ≤ 0i1_0 + i1_0 ≤ 0i1_0i1_0 ≤ 0___const_42_0 + ___const_42_0 ≤ 0___const_42_0___const_42_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 5, 8, 9 using the following ranking functions, which are bounded by −13.

7: 0
6: 0
0: 0
1: 0
2: 0
3: 0
4: 0
5: 0
7: −5
6: −6
0: −7
1: −7
2: −7
3: −7
4: −7
1_var_snapshot: −7
1*: −7
5: −8
Hints:
11 lexWeak[ [0, 0, 0, 0, 0, 0] ]
0 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ]
1 lexWeak[ [0, 0, 0, 0, 0, 0] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0] ]
4 lexWeak[ [0, 0, 0, 0, 0, 0] ]
6 lexWeak[ [0, 0, 0, 0, 0, 0, 0] ]
7 lexWeak[ [0, 0, 0, 0, 0, 0] ]
5 lexStrict[ [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] ]
9 lexStrict[ [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.

1* 13 1: i1_post + i1_post ≤ 0i1_posti1_post ≤ 0i1_0 + i1_0 ≤ 0i1_0i1_0 ≤ 0___const_42_0 + ___const_42_0 ≤ 0___const_42_0___const_42_0 ≤ 0

4 Location Addition

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

1 11 1_var_snapshot: i1_post + i1_post ≤ 0i1_posti1_post ≤ 0i1_0 + i1_0 ≤ 0i1_0i1_0 ≤ 0___const_42_0 + ___const_42_0 ≤ 0___const_42_0___const_42_0 ≤ 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 { 0, 1, 2, 3, 4, 1_var_snapshot, 1* }.

5.1.1 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by 5.

0: −4 + 7⋅___const_42_0 − 7⋅i1_0
1: 1 + 7⋅___const_42_0 − 7⋅i1_0
2: −3 + 7⋅___const_42_0 − 7⋅i1_0
3: −2 + 7⋅___const_42_0 − 7⋅i1_0
4: −1 + 7⋅___const_42_0 − 7⋅i1_0
1_var_snapshot: 7⋅___const_42_0 − 7⋅i1_0
1*: 2 + 7⋅___const_42_0 − 7⋅i1_0
Hints:
11 lexWeak[ [0, 0, 0, 7, 7, 0] ]
13 lexWeak[ [0, 0, 0, 7, 7, 0] ]
0 lexWeak[ [0, 0, 0, 7, 0, 7, 7, 0] ]
1 lexWeak[ [0, 0, 0, 7, 7, 0] ]
2 lexWeak[ [0, 0, 0, 7, 7, 0] ]
3 lexWeak[ [0, 0, 0, 7, 7, 0] ]
4 lexWeak[ [0, 0, 0, 7, 7, 0] ]
6 lexStrict[ [0, 0, 0, 0, 7, 7, 0] , [7, 0, 0, 0, 0, 0, 0] ]
7 lexWeak[ [0, 0, 0, 7, 7, 0] ]

5.1.2 Transition Removal

We remove transitions 11, 13, 0, 1, 2, 3, 4, 7 using the following ranking functions, which are bounded by −6.

0: −2
1: −4
2: −1
3: 0
4: −6
1_var_snapshot: −5
1*: −3
Hints:
11 lexStrict[ [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ]
13 lexStrict[ [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ]
0 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ]
1 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] ]
3 lexStrict[ [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] ]
7 lexStrict[ [0, 0, 0, 0, 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 10.

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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