# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f231_0_random_ArrayAccess, f171_0_createList_Return, f352_0_appE_GT, f197_0_createList_LE, f1_0_main_Load, __init
• Transitions: (pre-variables and post-variables)  f171_0_createList_Return 1 f231_0_random_ArrayAccess: x1 = _arg1 ∧ x2 = _arg2 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ 0 ≤ _arg1 − 1 f1_0_main_Load 2 f231_0_random_ArrayAccess: x1 = _x ∧ x2 = _x1 ∧ x1 = _x2 ∧ x2 = _x3 ∧ _x1 = _x2 ∧ 0 ≤ _x − 1 f1_0_main_Load 3 f197_0_createList_LE: x1 = _x4 ∧ x2 = _x5 ∧ x1 = _x6 ∧ x2 = _x7 ∧ 0 ≤ _x4 − 1 ∧ −1 ≤ _x6 − 1 ∧ −1 ≤ _x5 − 1 f197_0_createList_LE 4 f197_0_createList_LE: x1 = _x8 ∧ x2 = _x9 ∧ x1 = _x10 ∧ x2 = _x11 ∧ _x8 − 1 = _x10 ∧ 0 ≤ _x8 − 1 f231_0_random_ArrayAccess 5 f352_0_appE_GT: x1 = _x12 ∧ x2 = _x13 ∧ x1 = _x14 ∧ x2 = _x15 ∧ 1 ≤ _x12 − 1 ∧ 0 ≤ _x14 − 1 f352_0_appE_GT 6 f352_0_appE_GT: x1 = _x16 ∧ x2 = _x17 ∧ x1 = _x18 ∧ x2 = _x19 ∧ _x16 − 1 = _x18 ∧ 0 ≤ _x16 − 1 __init 7 f1_0_main_Load: x1 = _x20 ∧ x2 = _x21 ∧ x1 = _x22 ∧ x2 = _x23 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f231_0_random_ArrayAccess f231_0_random_ArrayAccess f231_0_random_ArrayAccess: x1 = x1 ∧ x2 = x2 f171_0_createList_Return f171_0_createList_Return f171_0_createList_Return: x1 = x1 ∧ x2 = x2 f352_0_appE_GT f352_0_appE_GT f352_0_appE_GT: x1 = x1 ∧ x2 = x2 f197_0_createList_LE f197_0_createList_LE f197_0_createList_LE: x1 = x1 ∧ x2 = x2 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 __init __init __init: x1 = x1 ∧ x2 = x2
and for every transition t, a duplicate t is considered.

### 2 SCC Decomposition

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

### 2.1 SCC Subproblem 1/2

Here we consider the SCC { f197_0_createList_LE }.

### 2.1.1 Transition Removal

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

 f197_0_createList_LE: x1

### 2.1.2 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

### 2.2 SCC Subproblem 2/2

Here we consider the SCC { f352_0_appE_GT }.

### 2.2.1 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by 0.

 f352_0_appE_GT: x1

### 2.2.2 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

## Tool configuration

AProVE

• version: AProVE Commit ID: unknown
• strategy: Statistics for single proof: 100.00 % (8 real / 0 unknown / 0 assumptions / 8 total proof steps)