LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: x_0 ≤ 0x_post ≤ 0x_1 ≤ 0x_2 ≤ 0x_3 ≤ 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:
2 4 2: x_post + x_post ≤ 0x_postx_post ≤ 0x_3 + x_3 ≤ 0x_3x_3 ≤ 0x_2 + x_2 ≤ 0x_2x_2 ≤ 0x_1 + x_1 ≤ 0x_1x_1 ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

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

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

4 Location Addition

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

2* 7 2: x_post + x_post ≤ 0x_postx_post ≤ 0x_3 + x_3 ≤ 0x_3x_3 ≤ 0x_2 + x_2 ≤ 0x_2x_2 ≤ 0x_1 + x_1 ≤ 0x_1x_1 ≤ 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.

2 5 2_var_snapshot: x_post + x_post ≤ 0x_postx_post ≤ 0x_3 + x_3 ≤ 0x_3x_3 ≤ 0x_2 + x_2 ≤ 0x_2x_2 ≤ 0x_1 + x_1 ≤ 0x_1x_1 ≤ 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, 2, 2_var_snapshot, 2* }.

6.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 2.

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

6.1.2 Transition Removal

We remove transitions 5, 7, 2 using the following ranking functions, which are bounded by −1.

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

6.1.3 Splitting Cut-Point Transitions

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

6.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 4.

6.1.3.1.1 Splitting Cut-Point Transitions

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

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