LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: 100 − i_0 ≤ 0
2: TRUE
3: 100 − i_0 ≤ 0
4: 100 − i_0 ≤ 0 ∧ 200 − j_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: − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
2 15 2: − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 3, 6, 7 using the following ranking functions, which are bounded by −17.

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

4 Location Addition

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

1* 11 1: − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_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: − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

6 Location Addition

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

2* 18 2: − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

7 Location Addition

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

2 16 2_var_snapshot: − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

8 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

8.1 SCC Subproblem 1/2

Here we consider the SCC { 1, 3, 1_var_snapshot, 1* }.

8.1.1 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by −798.

1: 1 − 4⋅j_0
3: −1 − 4⋅j_0
1_var_snapshot: −4⋅j_0
1*: 2 − 4⋅j_0

8.1.2 Transition Removal

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

1: 0
3: −2⋅i_0
1_var_snapshot: −100
1*: i_0

8.1.3 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by 99.

1: 0
3: 0
1_var_snapshot: i_0
1*: 0

8.1.4 Splitting Cut-Point Transitions

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

8.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 8.

8.1.4.1.1 Splitting Cut-Point Transitions

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

8.2 SCC Subproblem 2/2

Here we consider the SCC { 0, 2, 2_var_snapshot, 2* }.

8.2.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −398.

0: −1 − 4⋅i_0
2: 1 − 4⋅i_0
2_var_snapshot: −4⋅i_0
2*: 2 − 4⋅i_0

8.2.2 Transition Removal

We remove transitions 16, 18 using the following ranking functions, which are bounded by −1.

0: −2
2: 0
2_var_snapshot: −1
2*: 1

8.2.3 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by −1.

0: −1
2: 0
2_var_snapshot: 0
2*: 0

8.2.4 Splitting Cut-Point Transitions

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

8.2.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 15.

8.2.4.1.1 Splitting Cut-Point Transitions

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

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