LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: −1 + b_16_post ≤ 01 − b_16_post ≤ 0−1 + b_16_0 ≤ 01 − b_16_0 ≤ 0
4: b_16_post ≤ 0b_16_post ≤ 0b_16_0 ≤ 0b_16_0 ≤ 0
5: −1 + b_16_post ≤ 0x_14_0 ≤ 0−1 + b_16_0 ≤ 0
6: −1 + b_16_post ≤ 01 − b_16_post ≤ 0−1 + b_16_0 ≤ 01 − b_16_0 ≤ 0
7: b_16_post ≤ 0b_16_post ≤ 0b_16_0 ≤ 0b_16_0 ≤ 0
8: b_16_post ≤ 0b_16_post ≤ 0b_16_0 ≤ 0b_16_0 ≤ 0
9: b_16_post ≤ 0b_16_post ≤ 0b_16_0 ≤ 0
11: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 17 1: x_14_post + x_14_post ≤ 0x_14_postx_14_post ≤ 0x_14_1 + x_14_1 ≤ 0x_14_1x_14_1 ≤ 0x_14_0 + x_14_0 ≤ 0x_14_0x_14_0 ≤ 0st_15_0 + st_15_0 ≤ 0st_15_0st_15_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0b_16_post + b_16_post ≤ 0b_16_postb_16_post ≤ 0b_16_0 + b_16_0 ≤ 0b_16_0b_16_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 4, 8, 16 using the following ranking functions, which are bounded by −13.

11: 0
0: 0
1: 0
4: 0
6: 0
7: 0
8: 0
9: 0
5: 0
11: −5
0: −6
1: −7
4: −7
6: −7
7: −7
8: −7
9: −7
1_var_snapshot: −7
1*: −7
5: −11

4 Location Addition

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

1* 20 1: x_14_post + x_14_post ≤ 0x_14_postx_14_post ≤ 0x_14_1 + x_14_1 ≤ 0x_14_1x_14_1 ≤ 0x_14_0 + x_14_0 ≤ 0x_14_0x_14_0 ≤ 0st_15_0 + st_15_0 ≤ 0st_15_0st_15_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0b_16_post + b_16_post ≤ 0b_16_postb_16_post ≤ 0b_16_0 + b_16_0 ≤ 0b_16_0b_16_0 ≤ 0

5 Location Addition

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

1 18 1_var_snapshot: x_14_post + x_14_post ≤ 0x_14_postx_14_post ≤ 0x_14_1 + x_14_1 ≤ 0x_14_1x_14_1 ≤ 0x_14_0 + x_14_0 ≤ 0x_14_0x_14_0 ≤ 0st_15_0 + st_15_0 ≤ 0st_15_0st_15_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0b_16_post + b_16_post ≤ 0b_16_postb_16_post ≤ 0b_16_0 + b_16_0 ≤ 0b_16_0b_16_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, 4, 6, 7, 8, 9, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

We remove transition 10 using the following ranking functions, which are bounded by 7.

1: −5 + 8⋅x_14_0
4: 1 + 8⋅x_14_0
6: −6⋅b_16_0 + 8⋅x_14_0
7: −8⋅x_14_0
8: −11 − 8⋅x_14_0
9: −12 − 8⋅x_14_0
1_var_snapshot: −5⋅b_16_post + 8⋅x_14_0
1*: −4⋅b_16_0 + 8⋅x_14_0

6.1.2 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by 3.

1: 2⋅b_16_post + 3⋅x_14_0
4: 2 + 3⋅x_14_0
6: 3⋅x_14_0
7: 1 − 3⋅x_14_0
8: −3⋅x_14_0
9: −3⋅x_14_0
1_var_snapshot: b_16_0 + 3⋅x_14_0
1*: 2 − 2⋅b_16_0 + 2⋅b_16_post + 3⋅x_14_0

6.1.3 Transition Removal

We remove transition 11 using the following ranking functions, which are bounded by −1.

1: 0
4: 0
6: 0
7: 0
8: 0
9: 0
1_var_snapshot: 0
1*: 0

6.1.4 Splitting Cut-Point Transitions

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

6.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 17.

6.1.4.1.1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: 1 − x_14_0 ≤ 0−1 + b_16_post ≤ 01 − b_16_post ≤ 0−1 + b_16_0 ≤ 01 − b_16_0 ≤ 0x_14_0 ≤ 0−1 + b_16_post ≤ 01 − b_16_post ≤ 0−1 + b_16_0 ≤ 01 − b_16_0 ≤ 0
4: x_14_0 ≤ 0b_16_post ≤ 0b_16_post ≤ 0b_16_0 ≤ 0b_16_0 ≤ 0
5: −1 + b_16_post ≤ 0x_14_0 ≤ 0−1 + b_16_0 ≤ 0
6: 1 − x_14_0 ≤ 0−1 + b_16_post ≤ 01 − b_16_post ≤ 0−1 + b_16_0 ≤ 01 − b_16_0 ≤ 01 ≤ 0−1 + b_16_post ≤ 01 − b_16_post ≤ 0−1 + b_16_0 ≤ 01 − b_16_0 ≤ 0
7: x_14_0 ≤ 0b_16_post ≤ 0b_16_post ≤ 0b_16_0 ≤ 0b_16_0 ≤ 0
8: 1 ≤ 0b_16_post ≤ 0b_16_post ≤ 0b_16_0 ≤ 0b_16_0 ≤ 01 + x_14_0 ≤ 0b_16_post ≤ 0b_16_post ≤ 0b_16_0 ≤ 0b_16_0 ≤ 0
9: 1 + x_14_0 ≤ 0b_16_post ≤ 0b_16_post ≤ 0b_16_0 ≤ 0
11: TRUE
1: 1 − x_14_0 ≤ 0−1 + b_16_post ≤ 01 − b_16_post ≤ 0−1 + b_16_0 ≤ 01 − b_16_0 ≤ 0x_14_0 ≤ 0−1 + b_16_post ≤ 01 − b_16_post ≤ 0−1 + b_16_0 ≤ 01 − b_16_0 ≤ 0
4: x_14_0 ≤ 0b_16_post ≤ 0b_16_post ≤ 0b_16_0 ≤ 0b_16_0 ≤ 0
6: 1 − x_14_0 ≤ 0−1 + b_16_post ≤ 01 − b_16_post ≤ 0−1 + b_16_0 ≤ 01 − b_16_0 ≤ 01 ≤ 0−1 + b_16_post ≤ 01 − b_16_post ≤ 0−1 + b_16_0 ≤ 01 − b_16_0 ≤ 0
7: x_14_0 ≤ 0b_16_post ≤ 0b_16_post ≤ 0b_16_0 ≤ 0b_16_0 ≤ 0
8: 1 ≤ 0b_16_post ≤ 0b_16_post ≤ 0b_16_0 ≤ 0b_16_0 ≤ 0
9: 1 ≤ 0b_16_post ≤ 0b_16_post ≤ 0b_16_0 ≤ 0
1_var_snapshot: 1 − x_14_0 ≤ 0−1 + b_16_post ≤ 01 − b_16_post ≤ 0−1 + b_16_0 ≤ 01 − b_16_0 ≤ 0x_14_0 ≤ 0−1 + b_16_post ≤ 01 − b_16_post ≤ 0−1 + b_16_0 ≤ 01 − b_16_0 ≤ 0
1*: 1 ≤ 0−1 + b_16_post ≤ 01 − b_16_post ≤ 0−1 + b_16_0 ≤ 01 − b_16_0 ≤ 0

The invariants are proved as follows.

IMPACT Invariant Proof

6.1.4.1.2 Transition Removal

We remove transition 18 using the following ranking functions, which are bounded by −10.

1: −1
9: −2
1_var_snapshot: −3
6: −4
4: −5
7: −6
8: −7
1*: −8

6.1.4.1.3 Splitting Cut-Point Transitions

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

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