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 12 1: x0_post + x0_post ≤ 0x0_postx0_post ≤ 0x0_0 + x0_0 ≤ 0x0_0x0_0 ≤ 0oldX1_post + oldX1_post ≤ 0oldX1_postoldX1_post ≤ 0oldX1_0 + oldX1_0 ≤ 0oldX1_0oldX1_0 ≤ 0oldX0_post + oldX0_post ≤ 0oldX0_postoldX0_post ≤ 0oldX0_0 + oldX0_0 ≤ 0oldX0_0oldX0_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 1, 3, 4, 5, 6, 7, 8, 9, 10, 11 using the following ranking functions, which are bounded by −17.

6: 0
5: 0
4: 0
0: 0
1: 0
2: 0
3: 0
6: −7
5: −8
4: −9
0: −10
1: −10
1_var_snapshot: −10
1*: −10
2: −14
3: −15

3 Location Addition

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

1* 15 1: x0_post + x0_post ≤ 0x0_postx0_post ≤ 0x0_0 + x0_0 ≤ 0x0_0x0_0 ≤ 0oldX1_post + oldX1_post ≤ 0oldX1_postoldX1_post ≤ 0oldX1_0 + oldX1_0 ≤ 0oldX1_0oldX1_0 ≤ 0oldX0_post + oldX0_post ≤ 0oldX0_postoldX0_post ≤ 0oldX0_0 + oldX0_0 ≤ 0oldX0_0oldX0_0 ≤ 0

4 Location Addition

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

1 13 1_var_snapshot: x0_post + x0_post ≤ 0x0_postx0_post ≤ 0x0_0 + x0_0 ≤ 0x0_0x0_0 ≤ 0oldX1_post + oldX1_post ≤ 0oldX1_postoldX1_post ≤ 0oldX1_0 + oldX1_0 ≤ 0oldX1_0oldX1_0 ≤ 0oldX0_post + oldX0_post ≤ 0oldX0_postoldX0_post ≤ 0oldX0_0 + oldX0_0 ≤ 0oldX0_0oldX0_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, 1_var_snapshot, 1* }.

5.1.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by −21.

0: −1 − 2⋅x0_0
1: −2⋅x0_0
1_var_snapshot: −2⋅x0_0
1*: −2⋅x0_0

5.1.2 Transition Removal

We remove transitions 13, 15, 0 using the following ranking functions, which are bounded by −3.

0: 0
1: −2
1_var_snapshot: −3
1*: −1

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 12.

5.1.3.1.1 Splitting Cut-Point Transitions

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

Tool configuration

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