LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: − tmp7_post ≤ 0 ∧ − tmp7_0 ≤ 0
1: TRUE
2: TRUE
3: TRUE
4: 16 − nI6_0 ≤ 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:
1 8 1: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp___0_post + tmp___0_post ≤ 0 ∧ tmp___0_post − tmp___0_post ≤ 0 ∧ − tmp___0_0 + tmp___0_0 ≤ 0 ∧ tmp___0_0 − tmp___0_0 ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − tmp7_post + tmp7_post ≤ 0 ∧ tmp7_post − tmp7_post ≤ 0 ∧ − tmp7_0 + tmp7_0 ≤ 0 ∧ tmp7_0 − tmp7_0 ≤ 0 ∧ − ret_nBC211_post + ret_nBC211_post ≤ 0 ∧ ret_nBC211_post − ret_nBC211_post ≤ 0 ∧ − ret_nBC211_0 + ret_nBC211_0 ≤ 0 ∧ ret_nBC211_0 − ret_nBC211_0 ≤ 0 ∧ − ret_nBC18_post + ret_nBC18_post ≤ 0 ∧ ret_nBC18_post − ret_nBC18_post ≤ 0 ∧ − ret_nBC18_0 + ret_nBC18_0 ≤ 0 ∧ ret_nBC18_0 − ret_nBC18_0 ≤ 0 ∧ − res5_post + res5_post ≤ 0 ∧ res5_post − res5_post ≤ 0 ∧ − res5_0 + res5_0 ≤ 0 ∧ res5_0 − res5_0 ≤ 0 ∧ − res10_post + res10_post ≤ 0 ∧ res10_post − res10_post ≤ 0 ∧ − res10_4 + res10_4 ≤ 0 ∧ res10_4 − res10_4 ≤ 0 ∧ − res10_3 + res10_3 ≤ 0 ∧ res10_3 − res10_3 ≤ 0 ∧ − res10_2 + res10_2 ≤ 0 ∧ res10_2 − res10_2 ≤ 0 ∧ − res10_1 + res10_1 ≤ 0 ∧ res10_1 − res10_1 ≤ 0 ∧ − res10_0 + res10_0 ≤ 0 ∧ res10_0 − res10_0 ≤ 0 ∧ − nX_0 + nX_0 ≤ 0 ∧ nX_0 − nX_0 ≤ 0 ∧ − nX9_post + nX9_post ≤ 0 ∧ nX9_post − nX9_post ≤ 0 ∧ − nX9_0 + nX9_0 ≤ 0 ∧ nX9_0 − nX9_0 ≤ 0 ∧ − nX4_post + nX4_post ≤ 0 ∧ nX4_post − nX4_post ≤ 0 ∧ − nX4_0 + nX4_0 ≤ 0 ∧ nX4_0 − nX4_0 ≤ 0 ∧ − nI6_post + nI6_post ≤ 0 ∧ nI6_post − nI6_post ≤ 0 ∧ − nI6_0 + nI6_0 ≤ 0 ∧ nI6_0 − nI6_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 3, 6, 7 using the following ranking functions, which are bounded by −13.

6: 0
5: 0
0: 0
1: 0
2: 0
3: 0
4: 0
6: −5
5: −6
0: −7
1: −7
2: −7
3: −7
1_var_snapshot: −7
1*: −7
4: −8

4 Location Addition

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

1* 11 1: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp___0_post + tmp___0_post ≤ 0 ∧ tmp___0_post − tmp___0_post ≤ 0 ∧ − tmp___0_0 + tmp___0_0 ≤ 0 ∧ tmp___0_0 − tmp___0_0 ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − tmp7_post + tmp7_post ≤ 0 ∧ tmp7_post − tmp7_post ≤ 0 ∧ − tmp7_0 + tmp7_0 ≤ 0 ∧ tmp7_0 − tmp7_0 ≤ 0 ∧ − ret_nBC211_post + ret_nBC211_post ≤ 0 ∧ ret_nBC211_post − ret_nBC211_post ≤ 0 ∧ − ret_nBC211_0 + ret_nBC211_0 ≤ 0 ∧ ret_nBC211_0 − ret_nBC211_0 ≤ 0 ∧ − ret_nBC18_post + ret_nBC18_post ≤ 0 ∧ ret_nBC18_post − ret_nBC18_post ≤ 0 ∧ − ret_nBC18_0 + ret_nBC18_0 ≤ 0 ∧ ret_nBC18_0 − ret_nBC18_0 ≤ 0 ∧ − res5_post + res5_post ≤ 0 ∧ res5_post − res5_post ≤ 0 ∧ − res5_0 + res5_0 ≤ 0 ∧ res5_0 − res5_0 ≤ 0 ∧ − res10_post + res10_post ≤ 0 ∧ res10_post − res10_post ≤ 0 ∧ − res10_4 + res10_4 ≤ 0 ∧ res10_4 − res10_4 ≤ 0 ∧ − res10_3 + res10_3 ≤ 0 ∧ res10_3 − res10_3 ≤ 0 ∧ − res10_2 + res10_2 ≤ 0 ∧ res10_2 − res10_2 ≤ 0 ∧ − res10_1 + res10_1 ≤ 0 ∧ res10_1 − res10_1 ≤ 0 ∧ − res10_0 + res10_0 ≤ 0 ∧ res10_0 − res10_0 ≤ 0 ∧ − nX_0 + nX_0 ≤ 0 ∧ nX_0 − nX_0 ≤ 0 ∧ − nX9_post + nX9_post ≤ 0 ∧ nX9_post − nX9_post ≤ 0 ∧ − nX9_0 + nX9_0 ≤ 0 ∧ nX9_0 − nX9_0 ≤ 0 ∧ − nX4_post + nX4_post ≤ 0 ∧ nX4_post − nX4_post ≤ 0 ∧ − nX4_0 + nX4_0 ≤ 0 ∧ nX4_0 − nX4_0 ≤ 0 ∧ − nI6_post + nI6_post ≤ 0 ∧ nI6_post − nI6_post ≤ 0 ∧ − nI6_0 + nI6_0 ≤ 0 ∧ nI6_0 − nI6_0 ≤ 0

5 Location Addition

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

1 9 1_var_snapshot: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp___0_post + tmp___0_post ≤ 0 ∧ tmp___0_post − tmp___0_post ≤ 0 ∧ − tmp___0_0 + tmp___0_0 ≤ 0 ∧ tmp___0_0 − tmp___0_0 ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − tmp7_post + tmp7_post ≤ 0 ∧ tmp7_post − tmp7_post ≤ 0 ∧ − tmp7_0 + tmp7_0 ≤ 0 ∧ tmp7_0 − tmp7_0 ≤ 0 ∧ − ret_nBC211_post + ret_nBC211_post ≤ 0 ∧ ret_nBC211_post − ret_nBC211_post ≤ 0 ∧ − ret_nBC211_0 + ret_nBC211_0 ≤ 0 ∧ ret_nBC211_0 − ret_nBC211_0 ≤ 0 ∧ − ret_nBC18_post + ret_nBC18_post ≤ 0 ∧ ret_nBC18_post − ret_nBC18_post ≤ 0 ∧ − ret_nBC18_0 + ret_nBC18_0 ≤ 0 ∧ ret_nBC18_0 − ret_nBC18_0 ≤ 0 ∧ − res5_post + res5_post ≤ 0 ∧ res5_post − res5_post ≤ 0 ∧ − res5_0 + res5_0 ≤ 0 ∧ res5_0 − res5_0 ≤ 0 ∧ − res10_post + res10_post ≤ 0 ∧ res10_post − res10_post ≤ 0 ∧ − res10_4 + res10_4 ≤ 0 ∧ res10_4 − res10_4 ≤ 0 ∧ − res10_3 + res10_3 ≤ 0 ∧ res10_3 − res10_3 ≤ 0 ∧ − res10_2 + res10_2 ≤ 0 ∧ res10_2 − res10_2 ≤ 0 ∧ − res10_1 + res10_1 ≤ 0 ∧ res10_1 − res10_1 ≤ 0 ∧ − res10_0 + res10_0 ≤ 0 ∧ res10_0 − res10_0 ≤ 0 ∧ − nX_0 + nX_0 ≤ 0 ∧ nX_0 − nX_0 ≤ 0 ∧ − nX9_post + nX9_post ≤ 0 ∧ nX9_post − nX9_post ≤ 0 ∧ − nX9_0 + nX9_0 ≤ 0 ∧ nX9_0 − nX9_0 ≤ 0 ∧ − nX4_post + nX4_post ≤ 0 ∧ nX4_post − nX4_post ≤ 0 ∧ − nX4_0 + nX4_0 ≤ 0 ∧ nX4_0 − nX4_0 ≤ 0 ∧ − nI6_post + nI6_post ≤ 0 ∧ nI6_post − nI6_post ≤ 0 ∧ − nI6_0 + nI6_0 ≤ 0 ∧ nI6_0 − nI6_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, 1, 2, 3, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by −92.

0: −3 − 6⋅nI6_0
1: 1 − 6⋅nI6_0
2: −2 − 6⋅nI6_0
3: −1 − 6⋅nI6_0
1_var_snapshot: −6⋅nI6_0
1*: 2 − 6⋅nI6_0

6.1.2 Transition Removal

We remove transitions 9, 11, 0, 1, 2, 5 using the following ranking functions, which are bounded by −5.

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

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 8.

6.1.3.1.1 Splitting Cut-Point Transitions

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

Tool configuration

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