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: tmp_7_post ≤ 0 ∧ − tmp_7_post ≤ 0 ∧ tmp_7_0 ≤ 0 ∧ − tmp_7_0 ≤ 0
3: TRUE
4: TRUE
5: TRUE
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:
0 9 0: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − tmp_7_post + tmp_7_post ≤ 0 ∧ tmp_7_post − tmp_7_post ≤ 0 ∧ − tmp_7_0 + tmp_7_0 ≤ 0 ∧ tmp_7_0 − tmp_7_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 7, 8 using the following ranking functions, which are bounded by −13.

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

4 Location Addition

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

0* 12 0: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − tmp_7_post + tmp_7_post ≤ 0 ∧ tmp_7_post − tmp_7_post ≤ 0 ∧ − tmp_7_0 + tmp_7_0 ≤ 0 ∧ tmp_7_0 − tmp_7_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0

5 Location Addition

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

0 10 0_var_snapshot: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − tmp_7_post + tmp_7_post ≤ 0 ∧ tmp_7_post − tmp_7_post ≤ 0 ∧ − tmp_7_0 + tmp_7_0 ≤ 0 ∧ tmp_7_0 − tmp_7_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_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, 3, 4, 5, 0_var_snapshot, 0* }.

6.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 0.

0: −2 − 3⋅x_5_0 + 3⋅y_6_0
2: −1 − 3⋅x_5_0 + 3⋅y_6_0
3: −2 − 3⋅x_5_0 + 3⋅y_6_0
4: −3 − 3⋅x_5_0 + 3⋅y_6_0
5: −3⋅x_5_0 + 3⋅y_6_0
0_var_snapshot: −2 − 3⋅x_5_0 + 3⋅y_6_0
0*: −1 − 3⋅x_5_0 + 3⋅y_6_0

6.1.2 Transition Removal

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

0: −2 − 6⋅x_5_0 + 6⋅y_6_0
2: −6⋅x_5_0 + 6⋅y_6_0
3: −4 − 6⋅x_5_0 + 6⋅y_6_0
4: −5 − 6⋅x_5_0 + 6⋅y_6_0
5: −6⋅x_5_0 + 6⋅y_6_0
0_var_snapshot: −3 − 6⋅x_5_0 + 6⋅y_6_0
0*: −1 − 6⋅x_5_0 + 6⋅y_6_0

6.1.3 Transition Removal

We remove transitions 10, 12, 2, 5, 6 using the following ranking functions, which are bounded by −2.

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

6.1.4 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by −1.

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