# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f229_0_random_ArrayAccess, f194_0_createList_LE, f169_0_createList_Return, f298_0_appE_NONNULL, f1_0_main_Load, __init
• Transitions: (pre-variables and post-variables)  f169_0_createList_Return 1 f229_0_random_ArrayAccess: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ −1 ≤ _arg1P − 1 ∧ −1 ≤ _arg2 − 1 ∧ 0 ≤ _arg1 − 1 ∧ _arg1P ≤ _arg2 f1_0_main_Load 2 f229_0_random_ArrayAccess: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x1 = _x3 ∧ x2 = _x4 ∧ x3 = _x5 ∧ _x1 = _x4 ∧ −1 ≤ _x3 − 1 ∧ 0 ≤ _x − 1 f1_0_main_Load 3 f194_0_createList_LE: x1 = _x6 ∧ x2 = _x7 ∧ x3 = _x8 ∧ x1 = _x9 ∧ x2 = _x10 ∧ x3 = _x11 ∧ 0 ≤ _x6 − 1 ∧ −1 ≤ _x9 − 1 ∧ −1 ≤ _x7 − 1 f194_0_createList_LE 4 f194_0_createList_LE: x1 = _x12 ∧ x2 = _x13 ∧ x3 = _x14 ∧ x1 = _x15 ∧ x2 = _x16 ∧ x3 = _x17 ∧ _x12 − 1 = _x15 ∧ 0 ≤ _x12 − 1 f229_0_random_ArrayAccess 5 f298_0_appE_NONNULL: x1 = _x18 ∧ x2 = _x19 ∧ x3 = _x20 ∧ x1 = _x21 ∧ x2 = _x22 ∧ x3 = _x23 ∧ −1 ≤ _x23 − 1 ∧ 0 ≤ _x21 − 1 ∧ 0 ≤ _x18 − 1 ∧ _x23 + 1 ≤ _x18 ∧ _x21 ≤ _x18 ∧ 1 ≤ _x19 − 1 ∧ −1 ≤ _x22 − 1 f298_0_appE_NONNULL 6 f298_0_appE_NONNULL: x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x1 = _x27 ∧ x2 = _x28 ∧ x3 = _x29 ∧ _x25 = _x28 ∧ −1 ≤ _x29 − 1 ∧ 0 ≤ _x27 − 1 ∧ 0 ≤ _x26 − 1 ∧ 2 ≤ _x24 − 1 ∧ _x29 + 1 ≤ _x26 ∧ _x29 + 3 ≤ _x24 ∧ _x27 ≤ _x26 ∧ _x27 + 2 ≤ _x24 f298_0_appE_NONNULL 7 f298_0_appE_NONNULL: x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x1 = _x33 ∧ x2 = _x34 ∧ x3 = _x35 ∧ _x31 − 1 = _x34 ∧ −1 ≤ _x35 − 1 ∧ 1 ≤ _x33 − 1 ∧ −1 ≤ _x32 − 1 ∧ 1 ≤ _x30 − 1 ∧ _x35 ≤ _x32 ∧ _x35 + 2 ≤ _x30 ∧ _x33 − 2 ≤ _x32 ∧ 0 ≤ _x31 − 1 ∧ _x33 ≤ _x30 __init 8 f1_0_main_Load: x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x1 = _x39 ∧ x2 = _x40 ∧ x3 = _x41 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f229_0_random_ArrayAccess f229_0_random_ArrayAccess f229_0_random_ArrayAccess: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f194_0_createList_LE f194_0_createList_LE f194_0_createList_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f169_0_createList_Return f169_0_createList_Return f169_0_createList_Return: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f298_0_appE_NONNULL f298_0_appE_NONNULL f298_0_appE_NONNULL: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3
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 { f194_0_createList_LE }.

### 2.1.1 Transition Removal

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

 f194_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 { f298_0_appE_NONNULL }.

### 2.2.1 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by 0.

 f298_0_appE_NONNULL: −1 + x2

### 2.2.2 Transition Removal

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

 f298_0_appE_NONNULL: −1 + x1

### 2.2.3 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)