LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: TRUE
2: x_13_0 ≤ 0
3: rv_15_post ≤ 0rv_15_post ≤ 02 − y_17_post ≤ 01 − x_13_0 ≤ 02 − y_17_0 ≤ 0rv_15_0 ≤ 0rv_15_0 ≤ 0
4: 1 − x_13_0 ≤ 0
5: 1 − x_13_0 ≤ 0
6: TRUE
7: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 9 1: y_17_post + y_17_post ≤ 0y_17_posty_17_post ≤ 0y_17_0 + y_17_0 ≤ 0y_17_0y_17_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0st_16_post + st_16_post ≤ 0st_16_postst_16_post ≤ 0st_16_0 + st_16_0 ≤ 0st_16_0st_16_0 ≤ 0st_14_0 + st_14_0 ≤ 0st_14_0st_14_0 ≤ 0rv_15_post + rv_15_post ≤ 0rv_15_postrv_15_post ≤ 0rv_15_0 + rv_15_0 ≤ 0rv_15_0rv_15_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0nd_12_post + nd_12_post ≤ 0nd_12_postnd_12_post ≤ 0nd_12_1 + nd_12_1 ≤ 0nd_12_1nd_12_1 ≤ 0nd_12_0 + nd_12_0 ≤ 0nd_12_0nd_12_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 1, 8 using the following ranking functions, which are bounded by −13.

7: 0
0: 0
1: 0
3: 0
4: 0
5: 0
6: 0
2: 0
7: −5
0: −6
1: −7
3: −7
4: −7
5: −7
6: −7
1_var_snapshot: −7
1*: −7
2: −11

4 Location Addition

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

1* 12 1: y_17_post + y_17_post ≤ 0y_17_posty_17_post ≤ 0y_17_0 + y_17_0 ≤ 0y_17_0y_17_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0st_16_post + st_16_post ≤ 0st_16_postst_16_post ≤ 0st_16_0 + st_16_0 ≤ 0st_16_0st_16_0 ≤ 0st_14_0 + st_14_0 ≤ 0st_14_0st_14_0 ≤ 0rv_15_post + rv_15_post ≤ 0rv_15_postrv_15_post ≤ 0rv_15_0 + rv_15_0 ≤ 0rv_15_0rv_15_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0nd_12_post + nd_12_post ≤ 0nd_12_postnd_12_post ≤ 0nd_12_1 + nd_12_1 ≤ 0nd_12_1nd_12_1 ≤ 0nd_12_0 + nd_12_0 ≤ 0nd_12_0nd_12_0 ≤ 0

5 Location Addition

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

1 10 1_var_snapshot: y_17_post + y_17_post ≤ 0y_17_posty_17_post ≤ 0y_17_0 + y_17_0 ≤ 0y_17_0y_17_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0st_16_post + st_16_post ≤ 0st_16_postst_16_post ≤ 0st_16_0 + st_16_0 ≤ 0st_16_0st_16_0 ≤ 0st_14_0 + st_14_0 ≤ 0st_14_0st_14_0 ≤ 0rv_15_post + rv_15_post ≤ 0rv_15_postrv_15_post ≤ 0rv_15_0 + rv_15_0 ≤ 0rv_15_0rv_15_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0nd_12_post + nd_12_post ≤ 0nd_12_postnd_12_post ≤ 0nd_12_1 + nd_12_1 ≤ 0nd_12_1nd_12_1 ≤ 0nd_12_0 + nd_12_0 ≤ 0nd_12_0nd_12_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 { 1, 3, 4, 5, 6, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

We remove transitions 5, 6 using the following ranking functions, which are bounded by 0.

1: −1 + 3⋅x_13_0
3: −1 + 3⋅x_13_0
4: −1 + 3⋅x_13_0
5: −2 + 3⋅x_13_0
6: 3⋅x_13_0
1_var_snapshot: −1 + 3⋅x_13_0
1*: −1 + 3⋅x_13_0

6.1.2 Transition Removal

We remove transitions 4, 7 using the following ranking functions, which are bounded by −2.

1: −1
3: −1
4: −2
5: 0
6: 0
1_var_snapshot: −1
1*: −1

6.1.3 Transition Removal

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

1: −1 + 4⋅y_17_0
3: 1 + 4⋅y_17_0
4: 0
5: 0
6: 0
1_var_snapshot: −2 + 4⋅y_17_0
1*: 4⋅y_17_0

6.1.4 Transition Removal

We remove transitions 10, 12 using the following ranking functions, which are bounded by −2.

1: −1
3: 0
4: 0
5: 0
6: 0
1_var_snapshot: −2
1*: 0

6.1.5 Splitting Cut-Point Transitions

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

6.1.5.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 9.

6.1.5.1.1 Splitting Cut-Point Transitions

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

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