# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 6

## Proof

### 1 Invariant Updates

The following invariants are asserted.

 0: TRUE 1: head_19_post ≤ 0 ∧ − head_19_post ≤ 0 ∧ i_17_post ≤ 0 ∧ − i_17_post ≤ 0 ∧ head_19_0 ≤ 0 ∧ − head_19_0 ≤ 0 ∧ i_17_0 ≤ 0 ∧ − i_17_0 ≤ 0 2: −1 + i_17_post ≤ 0 ∧ 1 − i_17_post ≤ 0 ∧ −1 + i_17_0 ≤ 0 ∧ 1 − i_17_0 ≤ 0 ∧ 1 − length_16_0 ≤ 0 3: TRUE 4: 2 − length_16_0 ≤ 0 5: 2 − length_16_0 ≤ 0 6: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) TRUE 1 (1) head_19_post ≤ 0 ∧ − head_19_post ≤ 0 ∧ i_17_post ≤ 0 ∧ − i_17_post ≤ 0 ∧ head_19_0 ≤ 0 ∧ − head_19_0 ≤ 0 ∧ i_17_0 ≤ 0 ∧ − i_17_0 ≤ 0 2 (2) −1 + i_17_post ≤ 0 ∧ 1 − i_17_post ≤ 0 ∧ −1 + i_17_0 ≤ 0 ∧ 1 − i_17_0 ≤ 0 ∧ 1 − length_16_0 ≤ 0 3 (3) TRUE 4 (4) 2 − length_16_0 ≤ 0 5 (5) 2 − length_16_0 ≤ 0 6 (6) TRUE
• initial node: 6
• cover edges:
• transition edges:  0 0 1 1 6 3 1 7 2 2 1 3 2 2 4 4 3 3 4 4 5 5 5 4 6 8 0

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 0, 1, 2, 3, 6, 7, 8 using the following ranking functions, which are bounded by −17.

 6: 0 0: 0 1: 0 2: 0 4: 0 5: 0 3: 0 6: −7 0: −8 1: −9 2: −10 4: −11 5: −11 4_var_snapshot: −11 4*: −11 3: −15

### 4 Location Addition

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

### 5 Location Addition

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

### 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 { 4, 5, 4_var_snapshot, 4* }.

### 6.1.1 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 0.

 4: −2 − 4⋅i_17_0 + 4⋅length_16_0 5: −4⋅i_17_0 + 4⋅length_16_0 4_var_snapshot: −3 − 4⋅i_17_0 + 4⋅length_16_0 4*: −1 − 4⋅i_17_0 + 4⋅length_16_0

### 6.1.2 Transition Removal

We remove transitions 10, 12, 5 using the following ranking functions, which are bounded by −1.

 4: 0 5: 2 4_var_snapshot: −1 4*: 1

### 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 9.

### 6.1.3.1.1 Splitting Cut-Point Transitions

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

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• version: 1.0