# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f297_0_createIntList_Return, f508_0_random_ArrayAccess, f698_0_nth_LE, f746_0_main_LE, f1_0_main_Load, f658_0_createIntList_LE, __init
• Transitions: (pre-variables and post-variables)  f297_0_createIntList_Return 1 f508_0_random_ArrayAccess: x1 = _arg1 ∧ x2 = _arg2 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ −1 ≤ _arg1P − 1 ∧ −1 ≤ _arg1 − 1 ∧ _arg1P ≤ _arg1 f1_0_main_Load 2 f508_0_random_ArrayAccess: x1 = _x ∧ x2 = _x1 ∧ x1 = _x2 ∧ x2 = _x3 ∧ −1 ≤ _x2 − 1 ∧ 0 ≤ _x − 1 f508_0_random_ArrayAccess 3 f698_0_nth_LE: x1 = _x4 ∧ x2 = _x5 ∧ x1 = _x6 ∧ x2 = _x8 ∧ −1 ≤ _x8 − 1 ∧ 0 ≤ _x9 − 1 ∧ _x6 ≤ _x4 ∧ 0 ≤ _x4 − 1 ∧ 0 ≤ _x6 − 1 f698_0_nth_LE 4 f746_0_main_LE: x1 = _x10 ∧ x2 = _x11 ∧ x1 = _x12 ∧ x2 = _x13 ∧ _x12 + 2 ≤ _x10 ∧ _x11 ≤ 1 ∧ 0 ≤ _x10 − 1 f698_0_nth_LE 5 f698_0_nth_LE: x1 = _x14 ∧ x2 = _x15 ∧ x1 = _x16 ∧ x2 = _x17 ∧ _x15 − 1 = _x17 ∧ −1 ≤ _x16 − 1 ∧ 0 ≤ _x14 − 1 ∧ 1 ≤ _x15 − 1 ∧ _x16 + 1 ≤ _x14 f746_0_main_LE 6 f746_0_main_LE: x1 = _x18 ∧ x2 = _x19 ∧ x1 = _x20 ∧ x2 = _x21 ∧ _x18 − 1 = _x20 ∧ 0 ≤ _x18 − 1 f1_0_main_Load 7 f658_0_createIntList_LE: x1 = _x22 ∧ x2 = _x23 ∧ x1 = _x24 ∧ x2 = _x25 ∧ 1 = _x25 ∧ 0 ≤ _x22 − 1 ∧ −1 ≤ _x24 − 1 ∧ −1 ≤ _x23 − 1 f658_0_createIntList_LE 8 f658_0_createIntList_LE: x1 = _x26 ∧ x2 = _x27 ∧ x1 = _x28 ∧ x2 = _x29 ∧ _x27 + 1 = _x29 ∧ _x26 − 1 = _x28 ∧ 0 ≤ _x27 − 1 ∧ 0 ≤ _x26 − 1 __init 9 f1_0_main_Load: x1 = _x30 ∧ x2 = _x31 ∧ x1 = _x32 ∧ x2 = _x33 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f297_0_createIntList_Return f297_0_createIntList_Return f297_0_createIntList_Return: x1 = x1 ∧ x2 = x2 f508_0_random_ArrayAccess f508_0_random_ArrayAccess f508_0_random_ArrayAccess: x1 = x1 ∧ x2 = x2 f698_0_nth_LE f698_0_nth_LE f698_0_nth_LE: x1 = x1 ∧ x2 = x2 f746_0_main_LE f746_0_main_LE f746_0_main_LE: x1 = x1 ∧ x2 = x2 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 f658_0_createIntList_LE f658_0_createIntList_LE f658_0_createIntList_LE: 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 3 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/3

Here we consider the SCC { f658_0_createIntList_LE }.

### 2.1.1 Transition Removal

We remove transition 8 using the following ranking functions, which are bounded by 0.

 f658_0_createIntList_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 2/3

Here we consider the SCC { f698_0_nth_LE }.

### 2.2.1 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by 0.

 f698_0_nth_LE: x2

### 2.2.2 Trivial Cooperation Program

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

### 2.3 SCC Subproblem 3/3

Here we consider the SCC { f746_0_main_LE }.

### 2.3.1 Transition Removal

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

 f746_0_main_LE: x1

### 2.3.2 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 % (9 real / 0 unknown / 0 assumptions / 9 total proof steps)