LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

1: arg1P ≤ 0arg1 ≤ 0
2: TRUE
3: 1 − x7 ≤ 0
4: 1 − x7 ≤ 0
5: TRUE
6: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
3 9 3: x7 + x7 ≤ 0x7x7 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
4 16 4: x7 + x7 ≤ 0x7x7 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
5 23 5: x7 + x7 ≤ 0x7x7 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 1, 2, 3, 6, 8 using the following ranking functions, which are bounded by −21.

6: 0
2: 0
1: 0
3: 0
4: 0
5: 0
6: −7
2: −8
1: −9
3: −10
3_var_snapshot: −10
3*: −10
4: −13
4_var_snapshot: −13
4*: −13
5: −16
5_var_snapshot: −16
5*: −16
Hints:
10 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
17 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
24 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
5 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
7 lexWeak[ [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] , [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, 0, 0, 0] , [0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
6 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0] ]

4 Location Addition

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

3* 12 3: x7 + x7 ≤ 0x7x7 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + 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.

3 10 3_var_snapshot: x7 + x7 ≤ 0x7x7 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + 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.

4* 19 4: x7 + x7 ≤ 0x7x7 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

7 Location Addition

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

4 17 4_var_snapshot: x7 + x7 ≤ 0x7x7 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

8 Location Addition

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

5* 26 5: x7 + x7 ≤ 0x7x7 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

9 Location Addition

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

5 24 5_var_snapshot: x7 + x7 ≤ 0x7x7 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

10 SCC Decomposition

We consider subproblems for each of the 3 SCC(s) of the program graph.

10.1 SCC Subproblem 1/3

Here we consider the SCC { 4, 4_var_snapshot, 4* }.

10.1.1 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by 2.

4: 1 + 3⋅arg1
4_var_snapshot: 3⋅arg1
4*: 2 + 3⋅arg1
Hints:
17 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0] ]
19 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0] ]
5 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0] ]

10.1.2 Transition Removal

We remove transitions 17, 19 using the following ranking functions, which are bounded by −1.

4: 0
4_var_snapshot: −1
4*: 1
Hints:
17 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
19 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

10.1.3 Splitting Cut-Point Transitions

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

10.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 16.

10.1.3.1.1 Splitting Cut-Point Transitions

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

10.2 SCC Subproblem 2/3

Here we consider the SCC { 3, 3_var_snapshot, 3* }.

10.2.1 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 3.

3: 1 + 2⋅arg1 + arg2
3_var_snapshot: 2⋅arg1 + arg2
3*: 2 + 2⋅arg1 + arg2
Hints:
10 lexWeak[ [0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0] ]
12 lexWeak[ [0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0] ]
4 lexStrict[ [0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 2, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

10.2.2 Transition Removal

We remove transitions 10, 12 using the following ranking functions, which are bounded by −1.

3: 0
3_var_snapshot: −1
3*: x7
Hints:
10 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
12 lexStrict[ [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

10.2.3 Splitting Cut-Point Transitions

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

10.2.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 9.

10.2.3.1.1 Splitting Cut-Point Transitions

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

10.3 SCC Subproblem 3/3

Here we consider the SCC { 5, 5_var_snapshot, 5* }.

10.3.1 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by 2.

5: 1 + 3⋅arg1
5_var_snapshot: 3⋅arg1
5*: 2 + 3⋅arg1
Hints:
24 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0] ]
26 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0] ]
7 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

10.3.2 Transition Removal

We remove transitions 24, 26 using the following ranking functions, which are bounded by −2.

5: −1
5_var_snapshot: −2
5*: 0
Hints:
24 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
26 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

10.3.3 Splitting Cut-Point Transitions

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

10.3.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 23.

10.3.3.1.1 Splitting Cut-Point Transitions

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

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