# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f389_0_dupList_NONNULL, f117_0_createList_GE, f1_0_main_Load, __init, f232_0_main_InvokeMethod
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f117_0_createList_GE: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ 0 ≤ _arg1 − 1 ∧ 0 ≤ _arg2 − 1 ∧ −1 ≤ _arg1P − 1 f1_0_main_Load 2 f232_0_main_InvokeMethod: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x1 = _x3 ∧ x2 = _x4 ∧ x3 = _x5 ∧ −1 ≤ _x6 − 1 ∧ 0 ≤ _x1 − 1 ∧ _x3 ≤ _x ∧ 0 ≤ _x − 1 ∧ 0 ≤ _x3 − 1 ∧ 3 ≤ _x4 − 1 ∧ 0 = _x5 f1_0_main_Load 3 f232_0_main_InvokeMethod: x1 = _x8 ∧ x2 = _x9 ∧ x3 = _x10 ∧ x1 = _x11 ∧ x2 = _x13 ∧ x3 = _x14 ∧ −1 ≤ _x16 − 1 ∧ 0 ≤ _x9 − 1 ∧ _x11 ≤ _x8 ∧ _x13 − 1 ≤ _x8 ∧ 0 ≤ _x8 − 1 ∧ 0 ≤ _x11 − 1 ∧ 1 ≤ _x13 − 1 ∧ 0 = _x14 f117_0_createList_GE 4 f117_0_createList_GE: x1 = _x17 ∧ x2 = _x18 ∧ x3 = _x19 ∧ x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x22 ∧ _x17 − 1 = _x20 ∧ _x17 − 1 ≤ _x17 − 1 ∧ −1 ≤ _x17 − 1 f232_0_main_InvokeMethod 5 f389_0_dupList_NONNULL: x1 = _x23 ∧ x2 = _x24 ∧ x3 = _x25 ∧ x1 = _x26 ∧ x2 = _x27 ∧ x3 = _x28 ∧ _x26 ≤ _x24 ∧ 0 ≤ _x29 − 1 ∧ _x27 + 2 ≤ _x24 ∧ 0 ≤ _x23 − 1 ∧ 1 ≤ _x24 − 1 ∧ 1 ≤ _x26 − 1 ∧ −1 ≤ _x27 − 1 ∧ 0 = _x25 ∧ 0 = _x28 f389_0_dupList_NONNULL 6 f389_0_dupList_NONNULL: x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x1 = _x33 ∧ x2 = _x34 ∧ x3 = _x35 ∧ _x35 + 2 ≤ _x31 ∧ _x32 + 2 ≤ _x30 ∧ _x35 + 4 ≤ _x30 ∧ −1 ≤ _x34 − 1 ∧ 0 ≤ _x33 − 1 ∧ 0 ≤ _x31 − 1 ∧ 2 ≤ _x30 − 1 ∧ _x34 + 1 ≤ _x31 ∧ _x34 + 3 ≤ _x30 ∧ _x33 ≤ _x31 ∧ _x32 ≤ −1 ∧ _x33 + 2 ≤ _x30 f389_0_dupList_NONNULL 7 f389_0_dupList_NONNULL: x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x1 = _x39 ∧ x2 = _x40 ∧ x3 = _x41 ∧ _x41 + 2 ≤ _x37 ∧ _x38 + 2 ≤ _x36 ∧ _x41 + 4 ≤ _x36 ∧ −1 ≤ _x40 − 1 ∧ 0 ≤ _x39 − 1 ∧ 0 ≤ _x37 − 1 ∧ 2 ≤ _x36 − 1 ∧ _x40 + 1 ≤ _x37 ∧ _x40 + 3 ≤ _x36 ∧ _x39 ≤ _x37 ∧ 0 ≤ _x38 − 1 ∧ _x39 + 2 ≤ _x36 f389_0_dupList_NONNULL 8 f389_0_dupList_NONNULL: x1 = _x42 ∧ x2 = _x43 ∧ x3 = _x44 ∧ x1 = _x45 ∧ x2 = _x46 ∧ x3 = _x47 ∧ 1 = _x47 ∧ 0 = _x44 ∧ 0 ≤ _x46 − 1 ∧ 2 ≤ _x45 − 1 ∧ 0 ≤ _x43 − 1 ∧ 2 ≤ _x42 − 1 ∧ _x46 ≤ _x43 ∧ _x46 + 2 ≤ _x42 ∧ _x45 − 2 ≤ _x43 ∧ _x45 ≤ _x42 __init 9 f1_0_main_Load: x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x1 = _x51 ∧ x2 = _x52 ∧ x3 = _x53 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f389_0_dupList_NONNULL f389_0_dupList_NONNULL f389_0_dupList_NONNULL: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f117_0_createList_GE f117_0_createList_GE f117_0_createList_GE: 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 f232_0_main_InvokeMethod f232_0_main_InvokeMethod f232_0_main_InvokeMethod: 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 { f389_0_dupList_NONNULL }.

### 2.1.1 Transition Removal

We remove transitions 6, 7 using the following ranking functions, which are bounded by 0.

 f389_0_dupList_NONNULL: −1 + x1

### 2.1.2 Transition Removal

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

 f389_0_dupList_NONNULL: − x3

### 2.1.3 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 { f117_0_createList_GE }.

### 2.2.1 Transition Removal

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

 f117_0_createList_GE: 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)