# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f1_0_main_New, f262_0_main_InvokeMethod, f288_0__init__InvokeMethod, f76_0__init__LE, f194_0_height_NONNULL, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_New 1 f262_0_main_InvokeMethod: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ 2 ≤ _arg1P − 1 f1_0_main_New 2 f262_0_main_InvokeMethod: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x1 = _x3 ∧ x2 = _x4 ∧ x3 = _x5 ∧ 1 ≤ _x3 − 1 f1_0_main_New 3 f76_0__init__LE: x1 = _x6 ∧ x2 = _x7 ∧ x3 = _x8 ∧ x1 = _x9 ∧ x2 = _x10 ∧ x3 = _x11 ∧ 5 = _x11 ∧ 5 = _x10 ∧ 5 = _x9 f76_0__init__LE 4 f76_0__init__LE: x1 = _x12 ∧ x2 = _x13 ∧ x3 = _x14 ∧ x1 = _x15 ∧ x2 = _x16 ∧ x3 = _x17 ∧ _x13 − 1 = _x17 ∧ _x13 − 1 = _x16 ∧ _x13 − 1 = _x15 ∧ _x13 = _x14 ∧ 1 ≤ _x13 − 1 ∧ 0 ≤ _x12 − 1 ∧ _x13 − 1 ≤ _x13 − 1 f76_0__init__LE 5 f288_0__init__InvokeMethod: x1 = _x18 ∧ x2 = _x19 ∧ x3 = _x20 ∧ x1 = _x21 ∧ x2 = _x22 ∧ x3 = _x23 ∧ _x19 − 1 = _x23 ∧ _x18 = _x21 ∧ _x19 = _x20 ∧ 0 ≤ _x18 − 1 ∧ 4 ≤ _x22 − 1 ∧ 1 ≤ _x19 − 1 ∧ _x19 − 1 ≤ _x19 − 1 f76_0__init__LE 6 f288_0__init__InvokeMethod: x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x1 = _x27 ∧ x2 = _x28 ∧ x3 = _x29 ∧ _x25 − 1 = _x29 ∧ _x24 = _x27 ∧ _x25 = _x26 ∧ 0 ≤ _x24 − 1 ∧ 3 ≤ _x28 − 1 ∧ 1 ≤ _x25 − 1 ∧ _x25 − 1 ≤ _x25 − 1 f288_0__init__InvokeMethod 7 f76_0__init__LE: x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x1 = _x33 ∧ x2 = _x34 ∧ x3 = _x35 ∧ _x32 = _x35 ∧ _x32 = _x34 ∧ _x32 = _x33 ∧ 2 ≤ _x31 − 1 ∧ 0 ≤ _x30 − 1 ∧ 0 ≤ _x32 − 1 f262_0_main_InvokeMethod 8 f194_0_height_NONNULL: x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x1 = _x39 ∧ x2 = _x40 ∧ x3 = _x41 ∧ −1 ≤ _x40 − 1 ∧ 0 ≤ _x39 − 1 ∧ 0 ≤ _x36 − 1 ∧ _x40 + 1 ≤ _x36 ∧ _x39 ≤ _x36 f194_0_height_NONNULL 9 f194_0_height_NONNULL: x1 = _x42 ∧ x2 = _x43 ∧ x3 = _x44 ∧ x1 = _x45 ∧ x2 = _x46 ∧ x3 = _x47 ∧ −1 ≤ _x46 − 1 ∧ 0 ≤ _x45 − 1 ∧ 0 ≤ _x43 − 1 ∧ 2 ≤ _x42 − 1 ∧ _x46 + 1 ≤ _x43 ∧ _x46 + 3 ≤ _x42 ∧ _x45 ≤ _x43 ∧ _x45 + 2 ≤ _x42 f194_0_height_NONNULL 10 f194_0_height_NONNULL: x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x1 = _x51 ∧ x2 = _x52 ∧ x3 = _x53 ∧ −1 ≤ _x52 − 1 ∧ 0 ≤ _x51 − 1 ∧ −1 ≤ _x49 − 1 ∧ 2 ≤ _x48 − 1 ∧ _x52 + 3 ≤ _x48 ∧ _x51 + 2 ≤ _x48 __init 11 f1_0_main_New: x1 = _x54 ∧ x2 = _x55 ∧ x3 = _x56 ∧ x1 = _x57 ∧ x2 = _x58 ∧ x3 = _x59 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f1_0_main_New f1_0_main_New f1_0_main_New: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f262_0_main_InvokeMethod f262_0_main_InvokeMethod f262_0_main_InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f288_0__init__InvokeMethod f288_0__init__InvokeMethod f288_0__init__InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f76_0__init__LE f76_0__init__LE f76_0__init__LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f194_0_height_NONNULL f194_0_height_NONNULL f194_0_height_NONNULL: 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 { f288_0__init__InvokeMethod, f76_0__init__LE }.

### 2.1.1 Transition Removal

We remove transitions 4, 7, 6, 5 using the following ranking functions, which are bounded by 0.

 f76_0__init__LE: 2⋅x2 f288_0__init__InvokeMethod: 1 + 2⋅x3

### 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 { f194_0_height_NONNULL }.

### 2.2.1 Transition Removal

We remove transitions 9, 10 using the following ranking functions, which are bounded by 0.

 f194_0_height_NONNULL: 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 % (7 real / 0 unknown / 0 assumptions / 7 total proof steps)