# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 4
• Transitions: (pre-variables and post-variables)  0 0 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P ≤ 0 ∧ − arg2 ≤ 0 ∧ − arg2P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 1 1 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1 + arg2 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg1P + arg2 ≤ 0 ∧ arg1P − arg2 ≤ 0 ∧ arg1 − arg2P ≤ 0 ∧ − arg1 + arg2P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 1 2 3: 1 + arg1 − arg2 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0 2 3 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2 ≤ 0 ∧ arg2 ≤ 0 ∧ − arg1P ≤ 0 ∧ arg1P ≤ 0 ∧ arg1 − arg2P ≤ 0 ∧ − arg1 + arg2P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 2 4 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ −1 − arg2P + arg2 ≤ 0 ∧ 1 + arg2P − arg2 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0 3 5 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2 ≤ 0 ∧ arg2 ≤ 0 ∧ − arg2P ≤ 0 ∧ arg2P ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0 3 6 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ −1 − arg2P + arg2 ≤ 0 ∧ 1 + arg2P − arg2 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0 4 7 0: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0

## Proof

The following invariants are asserted.

 0: TRUE 1: − arg1P ≤ 0 ∧ − arg1 ≤ 0 2: TRUE 3: − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0 4: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) TRUE 1 (1) − arg1P ≤ 0 ∧ − arg1 ≤ 0 2 (2) TRUE 3 (3) − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0 4 (4) TRUE
• initial node: 4
• cover edges:
• transition edges:  0 0 1 1 1 2 1 2 3 2 3 1 2 4 2 3 5 1 3 6 3 4 7 0

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 8 1: − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0 2 15 2: − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0 3 22 3: − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 0, 7 using the following ranking functions, which are bounded by −15.

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

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

1* 11 1: arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

1 9 1_var_snapshot: arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

2* 18 2: arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

2 16 2_var_snapshot: arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

3* 25 3: arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

3 23 3_var_snapshot: arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

### 10 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

### 10.1 SCC Subproblem 1/1

Here we consider the SCC { 1, 2, 3, 1_var_snapshot, 1*, 2_var_snapshot, 2*, 3_var_snapshot, 3* }.

### 10.1.1 Transition Removal

We remove transitions 1, 2, 5, 6 using the following ranking functions, which are bounded by 5.

 1: −2 + 7⋅arg1 + 9⋅arg2 2: 9⋅arg1 + 2⋅arg2 3: 14⋅arg1 + 2⋅arg2 1_var_snapshot: −3 + 7⋅arg1 + 9⋅arg2 1*: −1 + 7⋅arg1 + 9⋅arg2 2_var_snapshot: 9⋅arg1 + 2⋅arg2 2*: 1 + 9⋅arg1 + 2⋅arg2 3_var_snapshot: 14⋅arg1 + 2⋅arg2 3*: 1 + 14⋅arg1 + 2⋅arg2

### 10.1.2 Transition Removal

We remove transitions 9, 11, 23, 25, 3, 4 using the following ranking functions, which are bounded by −3.

 1: −2 2: 1 + 3⋅arg2 3: 0 1_var_snapshot: −3 1*: −1 2_var_snapshot: 3⋅arg2 2*: 2 + 3⋅arg2 3_var_snapshot: −1 3*: 1

### 10.1.3 Transition Removal

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

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

### 10.1.4 Splitting Cut-Point Transitions

We consider 3 subproblems corresponding to sets of cut-point transitions as follows.

### 10.1.4.1 Cut-Point Subproblem 1/3

Here we consider cut-point transition 8.

### 10.1.4.1.1 Splitting Cut-Point Transitions

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

### 10.1.4.2 Cut-Point Subproblem 2/3

Here we consider cut-point transition 15.

### 10.1.4.2.1 Splitting Cut-Point Transitions

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

### 10.1.4.3 Cut-Point Subproblem 3/3

Here we consider cut-point transition 22.

### 10.1.4.3.1 Splitting Cut-Point Transitions

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

T2Cert

• version: 1.0