# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f408_0_createTree_Return, f1602_0_main_NULL, f1_0_main_Load, f1295_0_createTree_LE, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f1602_0_main_NULL: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x5 = _arg5 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ x5 = _arg5P ∧ −1 ≤ _arg2P − 1 ∧ 2 ≤ _arg1P − 1 ∧ 0 ≤ _arg1 − 1 f408_0_createTree_Return 2 f1602_0_main_NULL: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x1 = _x5 ∧ x2 = _x6 ∧ x3 = _x7 ∧ x4 = _x8 ∧ x5 = _x9 ∧ _x1 + 2 ≤ _x ∧ −1 ≤ _x6 − 1 ∧ 2 ≤ _x5 − 1 ∧ 2 ≤ _x − 1 ∧ _x6 + 3 ≤ _x ∧ _x5 ≤ _x f1602_0_main_NULL 3 f1602_0_main_NULL: x1 = _x10 ∧ x2 = _x11 ∧ x3 = _x12 ∧ x4 = _x13 ∧ x5 = _x14 ∧ x1 = _x15 ∧ x2 = _x16 ∧ x3 = _x17 ∧ x4 = _x18 ∧ x5 = _x19 ∧ −1 ≤ _x16 − 1 ∧ 2 ≤ _x15 − 1 ∧ 0 ≤ _x11 − 1 ∧ 2 ≤ _x10 − 1 ∧ _x16 + 1 ≤ _x11 ∧ _x16 + 3 ≤ _x10 ∧ _x15 − 2 ≤ _x10 f1_0_main_Load 4 f1295_0_createTree_LE: x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x22 ∧ x4 = _x23 ∧ x5 = _x24 ∧ x1 = _x25 ∧ x2 = _x26 ∧ x3 = _x27 ∧ x4 = _x28 ∧ x5 = _x29 ∧ 1 = _x29 ∧ _x21 = _x28 ∧ 2 ≤ _x26 − 1 ∧ 2 ≤ _x25 − 1 ∧ 0 ≤ _x20 − 1 ∧ _x26 − 2 ≤ _x20 ∧ _x25 − 2 ≤ _x20 ∧ 0 ≤ _x27 − 1 ∧ 0 ≤ _x21 − 1 f1295_0_createTree_LE 5 f1295_0_createTree_LE: x1 = _x30 ∧ x2 = _x32 ∧ x3 = _x33 ∧ x4 = _x34 ∧ x5 = _x35 ∧ x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x4 = _x40 ∧ x5 = _x41 ∧ _x35 + 1 = _x41 ∧ _x34 = _x40 ∧ _x33 − 1 = _x38 ∧ 0 ≤ _x37 − 1 ∧ 2 ≤ _x36 − 1 ∧ 2 ≤ _x32 − 1 ∧ 2 ≤ _x30 − 1 ∧ _x37 + 2 ≤ _x32 ∧ _x36 ≤ _x30 ∧ _x35 ≤ _x34 − 1 ∧ −1 ≤ _x35 − 1 ∧ 0 ≤ _x33 − 1 f1295_0_createTree_LE 6 f1295_0_createTree_LE: x1 = _x42 ∧ x2 = _x43 ∧ x3 = _x44 ∧ x4 = _x45 ∧ x5 = _x46 ∧ x1 = _x47 ∧ x2 = _x48 ∧ x3 = _x49 ∧ x4 = _x50 ∧ x5 = _x51 ∧ −1 ≤ _x46 − 1 ∧ 0 ≤ _x52 − 1 ∧ 0 ≤ _x44 − 1 ∧ _x46 ≤ _x45 − 1 ∧ _x47 ≤ _x42 ∧ _x48 + 2 ≤ _x43 ∧ 2 ≤ _x42 − 1 ∧ 2 ≤ _x43 − 1 ∧ 2 ≤ _x47 − 1 ∧ 0 ≤ _x48 − 1 ∧ _x44 − 1 = _x49 ∧ _x45 = _x50 ∧ _x46 + 1 = _x51 f1295_0_createTree_LE 7 f1295_0_createTree_LE: x1 = _x53 ∧ x2 = _x54 ∧ x3 = _x55 ∧ x4 = _x56 ∧ x5 = _x57 ∧ x1 = _x58 ∧ x2 = _x59 ∧ x3 = _x60 ∧ x4 = _x62 ∧ x5 = _x63 ∧ −1 ≤ _x57 − 1 ∧ 0 ≤ _x64 − 1 ∧ 0 ≤ _x55 − 1 ∧ _x57 ≤ _x56 − 1 ∧ 2 ≤ _x53 − 1 ∧ 1 ≤ _x54 − 1 ∧ 2 ≤ _x58 − 1 ∧ 2 ≤ _x59 − 1 ∧ _x55 − 1 = _x60 ∧ _x56 = _x62 ∧ _x57 + 1 = _x63 f1295_0_createTree_LE 8 f1295_0_createTree_LE: x1 = _x65 ∧ x2 = _x66 ∧ x3 = _x67 ∧ x4 = _x68 ∧ x5 = _x69 ∧ x1 = _x70 ∧ x2 = _x71 ∧ x3 = _x72 ∧ x4 = _x73 ∧ x5 = _x74 ∧ _x69 + 1 = _x74 ∧ _x68 = _x73 ∧ _x67 − 1 = _x72 ∧ 2 ≤ _x71 − 1 ∧ 2 ≤ _x70 − 1 ∧ 1 ≤ _x66 − 1 ∧ 2 ≤ _x65 − 1 ∧ _x69 ≤ _x68 − 1 ∧ −1 ≤ _x69 − 1 ∧ 0 ≤ _x67 − 1 f1295_0_createTree_LE 9 f1295_0_createTree_LE: x1 = _x75 ∧ x2 = _x76 ∧ x3 = _x77 ∧ x4 = _x78 ∧ x5 = _x79 ∧ x1 = _x80 ∧ x2 = _x81 ∧ x3 = _x82 ∧ x4 = _x83 ∧ x5 = _x84 ∧ _x79 + 1 = _x84 ∧ _x78 = _x83 ∧ _x77 − 1 = _x82 ∧ 4 ≤ _x81 − 1 ∧ 4 ≤ _x80 − 1 ∧ 2 ≤ _x76 − 1 ∧ 2 ≤ _x75 − 1 ∧ _x81 − 2 ≤ _x76 ∧ _x81 − 2 ≤ _x75 ∧ _x80 − 2 ≤ _x76 ∧ _x80 − 2 ≤ _x75 ∧ _x79 ≤ _x78 − 1 ∧ −1 ≤ _x79 − 1 ∧ 0 ≤ _x77 − 1 f1295_0_createTree_LE 10 f1295_0_createTree_LE: x1 = _x85 ∧ x2 = _x86 ∧ x3 = _x87 ∧ x4 = _x88 ∧ x5 = _x89 ∧ x1 = _x90 ∧ x2 = _x91 ∧ x3 = _x92 ∧ x4 = _x93 ∧ x5 = _x94 ∧ −1 ≤ _x89 − 1 ∧ 0 ≤ _x95 − 1 ∧ 0 ≤ _x87 − 1 ∧ _x89 ≤ _x88 − 1 ∧ _x90 − 2 ≤ _x85 ∧ _x90 − 2 ≤ _x86 ∧ _x91 − 2 ≤ _x85 ∧ _x91 − 2 ≤ _x86 ∧ 2 ≤ _x85 − 1 ∧ 2 ≤ _x86 − 1 ∧ 4 ≤ _x90 − 1 ∧ 4 ≤ _x91 − 1 ∧ _x87 − 1 = _x92 ∧ _x88 = _x93 ∧ _x89 + 1 = _x94 __init 11 f1_0_main_Load: x1 = _x96 ∧ x2 = _x97 ∧ x3 = _x98 ∧ x4 = _x99 ∧ x5 = _x100 ∧ x1 = _x101 ∧ x2 = _x102 ∧ x3 = _x103 ∧ x4 = _x104 ∧ x5 = _x105 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f408_0_createTree_Return f408_0_createTree_Return f408_0_createTree_Return: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 f1602_0_main_NULL f1602_0_main_NULL f1602_0_main_NULL: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 f1295_0_createTree_LE f1295_0_createTree_LE f1295_0_createTree_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
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 { f1295_0_createTree_LE }.

### 2.1.1 Transition Removal

We remove transitions 5, 6, 7, 8, 9, 10 using the following ranking functions, which are bounded by 0.

 f1295_0_createTree_LE: 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 { f1602_0_main_NULL }.

### 2.2.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 0.

 f1602_0_main_NULL: x2

### 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)