# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 2
• Transitions: (pre-variables and post-variables)  0 0 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0 ∧ − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ arg1P + arg2P − arg3P ≤ 0 ∧ − arg1P − arg2P + arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 1 1 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg3 ≤ 0 ∧ 1 − arg1 + arg2 ≤ 0 ∧ −1 − arg1P + arg1 ≤ 0 ∧ 1 + arg1P − arg1 ≤ 0 ∧ −1 + arg1 + arg2 − arg3P ≤ 0 ∧ 1 − arg1 − arg2 + arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 1 2 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg3 ≤ 0 ∧ 1 + arg1 − arg2 ≤ 0 ∧ −1 − arg2P + arg2 ≤ 0 ∧ 1 + arg2P − arg2 ≤ 0 ∧ −1 + arg1 + arg2 − arg3P ≤ 0 ∧ 1 − arg1 − arg2 + arg3P ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0 1 3 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg3 ≤ 0 ∧ arg1 − arg2 ≤ 0 ∧ − arg1 + arg2 ≤ 0 ∧ −1 − arg1P + arg1 ≤ 0 ∧ 1 + arg1P − arg1 ≤ 0 ∧ arg1 − arg2P ≤ 0 ∧ − arg1 + arg2P ≤ 0 ∧ −1 + 2⋅arg1 − arg3P ≤ 0 ∧ 1 − 2⋅arg1 + arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 2 4 0: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 5 1: − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 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
and for every transition t, a duplicate t is considered.

### 2 Transition Removal

We remove transitions 0, 4 using the following ranking functions, which are bounded by −11.

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

### 3 Location Addition

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

1* 8 1: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

### 4 Location Addition

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

1 6 1_var_snapshot: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

### 5 SCC Decomposition

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

### 5.1 SCC Subproblem 1/1

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

### 5.1.1 Splitting Cut-Point Transitions

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

### 5.1.1.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 5.

### 5.1.1.1.1 Fresh Variable Addition

The new variable __snapshot_1_arg3P is introduced. The transition formulas are extended as follows:

 6: __snapshot_1_arg3P ≤ arg3P ∧ arg3P ≤ __snapshot_1_arg3P 8: __snapshot_1_arg3P ≤ __snapshot_1_arg3P ∧ __snapshot_1_arg3P ≤ __snapshot_1_arg3P 1: __snapshot_1_arg3P ≤ __snapshot_1_arg3P ∧ __snapshot_1_arg3P ≤ __snapshot_1_arg3P 2: __snapshot_1_arg3P ≤ __snapshot_1_arg3P ∧ __snapshot_1_arg3P ≤ __snapshot_1_arg3P 3: __snapshot_1_arg3P ≤ __snapshot_1_arg3P ∧ __snapshot_1_arg3P ≤ __snapshot_1_arg3P

### 5.1.1.1.2 Fresh Variable Addition

The new variable __snapshot_1_arg3 is introduced. The transition formulas are extended as follows:

 6: __snapshot_1_arg3 ≤ arg3 ∧ arg3 ≤ __snapshot_1_arg3 8: __snapshot_1_arg3 ≤ __snapshot_1_arg3 ∧ __snapshot_1_arg3 ≤ __snapshot_1_arg3 1: __snapshot_1_arg3 ≤ __snapshot_1_arg3 ∧ __snapshot_1_arg3 ≤ __snapshot_1_arg3 2: __snapshot_1_arg3 ≤ __snapshot_1_arg3 ∧ __snapshot_1_arg3 ≤ __snapshot_1_arg3 3: __snapshot_1_arg3 ≤ __snapshot_1_arg3 ∧ __snapshot_1_arg3 ≤ __snapshot_1_arg3

### 5.1.1.1.3 Fresh Variable Addition

The new variable __snapshot_1_arg2P is introduced. The transition formulas are extended as follows:

 6: __snapshot_1_arg2P ≤ arg2P ∧ arg2P ≤ __snapshot_1_arg2P 8: __snapshot_1_arg2P ≤ __snapshot_1_arg2P ∧ __snapshot_1_arg2P ≤ __snapshot_1_arg2P 1: __snapshot_1_arg2P ≤ __snapshot_1_arg2P ∧ __snapshot_1_arg2P ≤ __snapshot_1_arg2P 2: __snapshot_1_arg2P ≤ __snapshot_1_arg2P ∧ __snapshot_1_arg2P ≤ __snapshot_1_arg2P 3: __snapshot_1_arg2P ≤ __snapshot_1_arg2P ∧ __snapshot_1_arg2P ≤ __snapshot_1_arg2P

### 5.1.1.1.4 Fresh Variable Addition

The new variable __snapshot_1_arg2 is introduced. The transition formulas are extended as follows:

 6: __snapshot_1_arg2 ≤ arg2 ∧ arg2 ≤ __snapshot_1_arg2 8: __snapshot_1_arg2 ≤ __snapshot_1_arg2 ∧ __snapshot_1_arg2 ≤ __snapshot_1_arg2 1: __snapshot_1_arg2 ≤ __snapshot_1_arg2 ∧ __snapshot_1_arg2 ≤ __snapshot_1_arg2 2: __snapshot_1_arg2 ≤ __snapshot_1_arg2 ∧ __snapshot_1_arg2 ≤ __snapshot_1_arg2 3: __snapshot_1_arg2 ≤ __snapshot_1_arg2 ∧ __snapshot_1_arg2 ≤ __snapshot_1_arg2

### 5.1.1.1.5 Fresh Variable Addition

The new variable __snapshot_1_arg1P is introduced. The transition formulas are extended as follows:

 6: __snapshot_1_arg1P ≤ arg1P ∧ arg1P ≤ __snapshot_1_arg1P 8: __snapshot_1_arg1P ≤ __snapshot_1_arg1P ∧ __snapshot_1_arg1P ≤ __snapshot_1_arg1P 1: __snapshot_1_arg1P ≤ __snapshot_1_arg1P ∧ __snapshot_1_arg1P ≤ __snapshot_1_arg1P 2: __snapshot_1_arg1P ≤ __snapshot_1_arg1P ∧ __snapshot_1_arg1P ≤ __snapshot_1_arg1P 3: __snapshot_1_arg1P ≤ __snapshot_1_arg1P ∧ __snapshot_1_arg1P ≤ __snapshot_1_arg1P

### 5.1.1.1.6 Fresh Variable Addition

The new variable __snapshot_1_arg1 is introduced. The transition formulas are extended as follows:

 6: __snapshot_1_arg1 ≤ arg1 ∧ arg1 ≤ __snapshot_1_arg1 8: __snapshot_1_arg1 ≤ __snapshot_1_arg1 ∧ __snapshot_1_arg1 ≤ __snapshot_1_arg1 1: __snapshot_1_arg1 ≤ __snapshot_1_arg1 ∧ __snapshot_1_arg1 ≤ __snapshot_1_arg1 2: __snapshot_1_arg1 ≤ __snapshot_1_arg1 ∧ __snapshot_1_arg1 ≤ __snapshot_1_arg1 3: __snapshot_1_arg1 ≤ __snapshot_1_arg1 ∧ __snapshot_1_arg1 ≤ __snapshot_1_arg1

### 5.1.1.1.7 Invariant Updates

The following invariants are asserted.

 0: TRUE 1: −1 − arg1 − arg2 + arg3 ≤ 0 ∧ −1 − 2⋅arg1 ≤ 0 ∨ −2 − 2⋅arg1 + arg3 ≤ 0 ∧ −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 + arg1 − arg2 ≤ 0 ∨ 1 ≤ 0 2: TRUE 1: −1 − arg1 − arg2 + arg3 ≤ 0 ∧ −1 − 2⋅arg1 ≤ 0 ∨ −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ 1 − __snapshot_1_arg1 + __snapshot_1_arg2 + arg1 − arg2 ≤ 0 ∧ 1 + arg1 − arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 ∧ −2⋅__snapshot_1_arg1 + 2⋅__snapshot_1_arg2 ≤ 0 ∨ −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 ∧ arg1 − arg2 ≤ 0 ∨ −1 − arg1 − arg2 + arg3 ≤ 0 ∧ arg1 − arg2 ≤ 0 ∨ −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 1_var_snapshot: − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg3 ≤ 0 ∧ −1 − __snapshot_1_arg1 − __snapshot_1_arg2 − arg1 + arg2 ≤ 0 ∧ −2⋅__snapshot_1_arg1 + 2⋅__snapshot_1_arg2 + 2⋅arg1 − 2⋅arg2 ≤ 0 ∨ − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg3 ≤ 0 ∧ −2⋅__snapshot_1_arg1 + 2⋅__snapshot_1_arg2 + 2⋅arg1 − 2⋅arg2 ≤ 0 ∧ 1 + arg1 − arg2 ≤ 0 ∨ − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg3 ≤ 0 ∧ −2⋅__snapshot_1_arg1 + 2⋅__snapshot_1_arg2 + 2⋅arg1 − 2⋅arg2 ≤ 0 ∧ arg1 − arg2 ≤ 0 ∨ − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg3 ≤ 0 ∧ −2⋅__snapshot_1_arg1 + 2⋅__snapshot_1_arg2 + 2⋅arg1 − 2⋅arg2 ≤ 0 1*: −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 ∨ −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 ∧ arg1 − arg2 ≤ 0 ∨ −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ 1 − __snapshot_1_arg1 + __snapshot_1_arg2 + arg1 − arg2 ≤ 0 ∧ 1 + arg1 − arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 ∧ −2⋅__snapshot_1_arg1 + 2⋅__snapshot_1_arg2 ≤ 0 ∨ 1 ≤ 0

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (2) TRUE 1 (0) TRUE 2 (1) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ −1 − 2⋅arg1 ≤ 0 3 (1) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ −1 − 2⋅arg1 ≤ 0 4 (1) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ −1 − 2⋅arg1 ≤ 0 5 (1) −2 − 2⋅arg1 + arg3 ≤ 0 ∧ −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 + arg1 − arg2 ≤ 0 6 (1) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ −1 − 2⋅arg1 ≤ 0 7 (1_var_snapshot) − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg3 ≤ 0 ∧ −1 − __snapshot_1_arg1 − __snapshot_1_arg2 − arg1 + arg2 ≤ 0 ∧ −2⋅__snapshot_1_arg1 + 2⋅__snapshot_1_arg2 + 2⋅arg1 − 2⋅arg2 ≤ 0 12 (1*) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 13 (1*) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 ∧ arg1 − arg2 ≤ 0 14 (1*) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ 1 − __snapshot_1_arg1 + __snapshot_1_arg2 + arg1 − arg2 ≤ 0 ∧ 1 + arg1 − arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 ∧ −2⋅__snapshot_1_arg1 + 2⋅__snapshot_1_arg2 ≤ 0 15 (1) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ 1 − __snapshot_1_arg1 + __snapshot_1_arg2 + arg1 − arg2 ≤ 0 ∧ 1 + arg1 − arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 ∧ −2⋅__snapshot_1_arg1 + 2⋅__snapshot_1_arg2 ≤ 0 16 (1_var_snapshot) − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg3 ≤ 0 ∧ −2⋅__snapshot_1_arg1 + 2⋅__snapshot_1_arg2 + 2⋅arg1 − 2⋅arg2 ≤ 0 ∧ 1 + arg1 − arg2 ≤ 0 17 (1_var_snapshot) − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg3 ≤ 0 ∧ −2⋅__snapshot_1_arg1 + 2⋅__snapshot_1_arg2 + 2⋅arg1 − 2⋅arg2 ≤ 0 ∧ 1 + arg1 − arg2 ≤ 0 25 (1) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 ∧ arg1 − arg2 ≤ 0 26 (1_var_snapshot) − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg3 ≤ 0 ∧ −2⋅__snapshot_1_arg1 + 2⋅__snapshot_1_arg2 + 2⋅arg1 − 2⋅arg2 ≤ 0 ∧ arg1 − arg2 ≤ 0 27 (1_var_snapshot) − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg3 ≤ 0 ∧ −2⋅__snapshot_1_arg1 + 2⋅__snapshot_1_arg2 + 2⋅arg1 − 2⋅arg2 ≤ 0 ∧ arg1 − arg2 ≤ 0 51 (1) 1 ≤ 0 52 (1) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ −1 − 2⋅arg1 ≤ 0 53 (1) 1 ≤ 0 54 (1) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ arg1 − arg2 ≤ 0 55 (1_var_snapshot) − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg3 ≤ 0 ∧ −2⋅__snapshot_1_arg1 + 2⋅__snapshot_1_arg2 + 2⋅arg1 − 2⋅arg2 ≤ 0 ∧ arg1 − arg2 ≤ 0 58 (1*) 1 ≤ 0 59 (1*) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 60 (1*) 1 ≤ 0 61 (1*) 1 ≤ 0 62 (1*) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 63 (1*) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 64 (1) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 65 (1_var_snapshot) − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg3 ≤ 0 ∧ −2⋅__snapshot_1_arg1 + 2⋅__snapshot_1_arg2 + 2⋅arg1 − 2⋅arg2 ≤ 0 73 (1*) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 74 (1*) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0 75 (1*) −1 − arg1 − arg2 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg1 − __snapshot_1_arg2 + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1 − __snapshot_1_arg2 ≤ 0
• initial node: 0
• cover edges:  3 → 2 4 → 2 16 → 17 26 → 27 52 → 2 55 → 27 59 → 12 62 → 12 63 → 12 73 → 12 74 → 12 75 → 12
• transition edges:  0 4 1 1 0 2 2 1 3 2 2 4 2 3 5 2 5 6 5 1 51 5 2 52 5 3 53 5 5 54 6 6 7 7 1 12 7 2 13 7 3 14 12 8 64 13 8 25 14 8 15 15 6 16 15 6 17 17 1 58 17 2 59 17 3 60 25 6 26 25 6 27 27 1 61 27 2 62 27 3 63 54 6 55 64 6 65 65 1 73 65 2 74 65 3 75

### 5.1.1.1.8 Transition Removal

We remove transition 8 using the following lexicographic ranking functions, which are bounded by [−2, −2].

 1: [arg1 + arg2, 2⋅arg1 − 2⋅arg2] 1_var_snapshot: [__snapshot_1_arg1 + __snapshot_1_arg2, 2⋅__snapshot_1_arg1 − 2⋅__snapshot_1_arg2] 1*: [__snapshot_1_arg1 + __snapshot_1_arg2, 2⋅__snapshot_1_arg1 − 2⋅__snapshot_1_arg2]

### 5.1.1.1.9 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by −5.

 1: −1 1_var_snapshot: −2 1*: −3

### 5.1.1.1.10 Splitting Cut-Point Transitions

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

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