# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 6
• Transitions: (pre-variables and post-variables)  0 0 1: − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0 2 1 3: 10 − i911_0 ≤ 0 ∧ − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0 2 2 4: 0 ≤ 0 ∧ 0 ≤ 0 ∧ −9 + i911_0 ≤ 0 ∧ −1 − i911_0 + i911_post ≤ 0 ∧ 1 + i911_0 − i911_post ≤ 0 ∧ i911_0 − i911_post ≤ 0 ∧ − i911_0 + i911_post ≤ 0 ∧ − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0 4 3 2: − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0 1 4 4: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 10 − i57_0 ≤ 0 ∧ i911_post ≤ 0 ∧ − i911_post ≤ 0 ∧ i911_0 − i911_post ≤ 0 ∧ − i911_0 + i911_post ≤ 0 ∧ x79_0 − x79_post ≤ 0 ∧ − x79_0 + x79_post ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0 1 5 0: 0 ≤ 0 ∧ 0 ≤ 0 ∧ −9 + i57_0 ≤ 0 ∧ −1 − i57_0 + i57_post ≤ 0 ∧ 1 + i57_0 − i57_post ≤ 0 ∧ i57_0 − i57_post ≤ 0 ∧ − i57_0 + i57_post ≤ 0 ∧ − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 5 6 0: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ i57_post ≤ 0 ∧ − i57_post ≤ 0 ∧ i57_0 − i57_post ≤ 0 ∧ − i57_0 + i57_post ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0 ∧ x35_0 − x35_post ≤ 0 ∧ − x35_0 + x35_post ≤ 0 ∧ − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 6 7 5: − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0

## Proof

### 1 Invariant Updates

The following invariants are asserted.

 0: TRUE 1: TRUE 2: 10 − i57_0 ≤ 0 3: 10 − i57_0 ≤ 0 ∧ 10 − i911_0 ≤ 0 4: 10 − i57_0 ≤ 0 5: TRUE 6: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) TRUE 1 (1) TRUE 2 (2) 10 − i57_0 ≤ 0 3 (3) 10 − i57_0 ≤ 0 ∧ 10 − i911_0 ≤ 0 4 (4) 10 − i57_0 ≤ 0 5 (5) TRUE 6 (6) TRUE
• initial node: 6
• cover edges:
• transition edges:  0 0 1 1 4 4 1 5 0 2 1 3 2 2 4 4 3 2 5 6 0 6 7 5

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 0 8 0: − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0 4 15 4: − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 1, 4, 6, 7 using the following ranking functions, which are bounded by −17.

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

### 4 Location Addition

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

0* 11 0: x79_post + x79_post ≤ 0x79_postx79_post ≤ 0x79_0 + x79_0 ≤ 0x79_0x79_0 ≤ 0x35_post + x35_post ≤ 0x35_postx35_post ≤ 0x35_0 + x35_0 ≤ 0x35_0x35_0 ≤ 0tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i911_post + i911_post ≤ 0i911_posti911_post ≤ 0i911_0 + i911_0 ≤ 0i911_0i911_0 ≤ 0i57_post + i57_post ≤ 0i57_posti57_post ≤ 0i57_0 + i57_0 ≤ 0i57_0i57_0 ≤ 0

### 5 Location Addition

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

0 9 0_var_snapshot: x79_post + x79_post ≤ 0x79_postx79_post ≤ 0x79_0 + x79_0 ≤ 0x79_0x79_0 ≤ 0x35_post + x35_post ≤ 0x35_postx35_post ≤ 0x35_0 + x35_0 ≤ 0x35_0x35_0 ≤ 0tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i911_post + i911_post ≤ 0i911_posti911_post ≤ 0i911_0 + i911_0 ≤ 0i911_0i911_0 ≤ 0i57_post + i57_post ≤ 0i57_posti57_post ≤ 0i57_0 + i57_0 ≤ 0i57_0i57_0 ≤ 0

### 6 Location Addition

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

4* 18 4: x79_post + x79_post ≤ 0x79_postx79_post ≤ 0x79_0 + x79_0 ≤ 0x79_0x79_0 ≤ 0x35_post + x35_post ≤ 0x35_postx35_post ≤ 0x35_0 + x35_0 ≤ 0x35_0x35_0 ≤ 0tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i911_post + i911_post ≤ 0i911_posti911_post ≤ 0i911_0 + i911_0 ≤ 0i911_0i911_0 ≤ 0i57_post + i57_post ≤ 0i57_posti57_post ≤ 0i57_0 + i57_0 ≤ 0i57_0i57_0 ≤ 0

### 7 Location Addition

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

4 16 4_var_snapshot: x79_post + x79_post ≤ 0x79_postx79_post ≤ 0x79_0 + x79_0 ≤ 0x79_0x79_0 ≤ 0x35_post + x35_post ≤ 0x35_postx35_post ≤ 0x35_0 + x35_0 ≤ 0x35_0x35_0 ≤ 0tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i911_post + i911_post ≤ 0i911_posti911_post ≤ 0i911_0 + i911_0 ≤ 0i911_0i911_0 ≤ 0i57_post + i57_post ≤ 0i57_posti57_post ≤ 0i57_0 + i57_0 ≤ 0i57_0i57_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 { 2, 4, 4_var_snapshot, 4* }.

### 8.1.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by −38.

 2: −1 − 4⋅i911_0 4: 1 − 4⋅i911_0 4_var_snapshot: −4⋅i911_0 4*: 2 − 4⋅i911_0

### 8.1.2 Transition Removal

We remove transitions 16, 18 using the following ranking functions, which are bounded by −11.

 2: −1 − 2⋅i57_0 4: −10 4_var_snapshot: −2⋅i57_0 4*: 0

### 8.1.3 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 9.

 2: 0 4: 0 4_var_snapshot: i57_0 4*: 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 15.

### 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, 1, 0_var_snapshot, 0* }.

### 8.2.1 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by −28.

 0: 2 − 3⋅i57_0 1: −3⋅i57_0 0_var_snapshot: 1 − 3⋅i57_0 0*: 2 − 3⋅i57_0

### 8.2.2 Transition Removal

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

 0: 0 1: −2 0_var_snapshot: −1 0*: 1

### 8.2.3 Transition Removal

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

 0: 0 1: 0 0_var_snapshot: 1 0*: 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 8.

### 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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