# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f323_0_createTree_Return, f977_0_random_ArrayAccess, f521_0_createNode_Return, f1_0_main_Load, f931_0_random_ArrayAccess, f2251_0_createTree_LE, f2233_0_randomlyDuplicate_NULL, f588_0_createNode_Return, f1843_0_main_InvokeMethod, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f1843_0_main_InvokeMethod: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ −1 ≤ _arg2P − 1 ∧ 0 ≤ _arg1P − 1 ∧ 0 ≤ _arg1 − 1 ∧ _arg1P ≤ _arg1 f323_0_createTree_Return 2 f1843_0_main_InvokeMethod: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x1 = _x4 ∧ x2 = _x5 ∧ x3 = _x6 ∧ x4 = _x7 ∧ 2 = _x6 ∧ −1 ≤ _x5 − 1 ∧ 0 ≤ _x4 − 1 ∧ −1 ≤ _x1 − 1 ∧ 0 ≤ _x − 1 ∧ _x5 ≤ _x1 ∧ _x4 − 1 ≤ _x1 ∧ _x4 ≤ _x f521_0_createNode_Return 3 f931_0_random_ArrayAccess: x1 = _x8 ∧ x2 = _x9 ∧ x3 = _x10 ∧ x4 = _x11 ∧ x1 = _x12 ∧ x2 = _x13 ∧ x3 = _x14 ∧ x4 = _x15 ∧ 0 ≤ _x12 − 1 f1_0_main_Load 4 f931_0_random_ArrayAccess: x1 = _x16 ∧ x2 = _x17 ∧ x3 = _x18 ∧ x4 = _x19 ∧ x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x22 ∧ x4 = _x23 ∧ 0 ≤ _x20 − 1 ∧ 0 ≤ _x16 − 1 ∧ _x20 ≤ _x16 f588_0_createNode_Return 5 f977_0_random_ArrayAccess: x1 = _x24 ∧ x2 = _x26 ∧ x3 = _x27 ∧ x4 = _x28 ∧ x1 = _x29 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x4 = _x33 ∧ 0 ≤ _x29 − 1 f1_0_main_Load 6 f977_0_random_ArrayAccess: x1 = _x34 ∧ x2 = _x35 ∧ x3 = _x36 ∧ x4 = _x37 ∧ x1 = _x38 ∧ x2 = _x39 ∧ x3 = _x40 ∧ x4 = _x41 ∧ 0 ≤ _x38 − 1 ∧ 0 ≤ _x34 − 1 ∧ _x38 ≤ _x34 f977_0_random_ArrayAccess 7 f2251_0_createTree_LE: x1 = _x42 ∧ x2 = _x43 ∧ x3 = _x44 ∧ x4 = _x46 ∧ x1 = _x47 ∧ x2 = _x48 ∧ x3 = _x49 ∧ x4 = _x50 ∧ 0 ≤ _x51 − 1 ∧ −1 ≤ _x48 − 1 ∧ _x47 − 3 ≤ _x42 ∧ 0 ≤ _x42 − 1 ∧ 3 ≤ _x47 − 1 ∧ _x51 + 1 = _x50 f931_0_random_ArrayAccess 8 f2251_0_createTree_LE: x1 = _x52 ∧ x2 = _x53 ∧ x3 = _x54 ∧ x4 = _x55 ∧ x1 = _x56 ∧ x2 = _x57 ∧ x3 = _x58 ∧ x4 = _x60 ∧ 0 ≤ _x61 − 1 ∧ −1 ≤ _x57 − 1 ∧ _x56 − 1 ≤ _x52 ∧ 0 ≤ _x52 − 1 ∧ 1 ≤ _x56 − 1 ∧ _x61 + 1 = _x60 f2251_0_createTree_LE 9 f2251_0_createTree_LE: x1 = _x62 ∧ x2 = _x63 ∧ x3 = _x64 ∧ x4 = _x65 ∧ x1 = _x66 ∧ x2 = _x67 ∧ x3 = _x68 ∧ x4 = _x69 ∧ _x65 = _x69 ∧ _x64 = _x68 ∧ _x63 − 1 = _x67 ∧ −1 ≤ _x66 − 1 ∧ 1 ≤ _x62 − 1 ∧ 0 ≤ _x63 − 1 ∧ _x66 + 2 ≤ _x62 f2251_0_createTree_LE 10 f2251_0_createTree_LE: x1 = _x70 ∧ x2 = _x71 ∧ x3 = _x72 ∧ x4 = _x73 ∧ x1 = _x74 ∧ x2 = _x75 ∧ x3 = _x76 ∧ x4 = _x77 ∧ 0 ≤ _x71 − 1 ∧ 0 ≤ _x73 − 1 ∧ 0 ≤ _x78 − 1 ∧ _x74 − 2 ≤ _x70 ∧ 2 ≤ _x70 − 1 ∧ 3 ≤ _x74 − 1 ∧ _x71 − 1 = _x75 ∧ _x72 = _x76 f2251_0_createTree_LE 11 f2251_0_createTree_LE: x1 = _x79 ∧ x2 = _x80 ∧ x3 = _x81 ∧ x4 = _x82 ∧ x1 = _x83 ∧ x2 = _x84 ∧ x3 = _x85 ∧ x4 = _x86 ∧ 0 ≤ _x80 − 1 ∧ 0 ≤ _x82 − 1 ∧ 0 ≤ _x87 − 1 ∧ _x83 − 3 ≤ _x79 ∧ 2 ≤ _x79 − 1 ∧ 5 ≤ _x83 − 1 ∧ _x80 − 1 = _x84 ∧ _x81 = _x85 f1_0_main_Load 12 f1257_0_random_ArrayAccess: x1 = _x88 ∧ x2 = _x89 ∧ x3 = _x90 ∧ x4 = _x91 ∧ x1 = _x92 ∧ x2 = _x94 ∧ x3 = _x95 ∧ x4 = _x96 ∧ 0 = _x95 ∧ 0 ≤ _x94 − 1 ∧ 0 ≤ _x92 − 1 ∧ 0 ≤ _x88 − 1 ∧ _x94 ≤ _x88 ∧ −1 ≤ _x89 − 1 ∧ _x92 ≤ _x88 f2251_0_createTree_LE 13 f1257_0_random_ArrayAccess: x1 = _x97 ∧ x2 = _x99 ∧ x3 = _x100 ∧ x4 = _x101 ∧ x1 = _x102 ∧ x2 = _x103 ∧ x3 = _x104 ∧ x4 = _x105 ∧ _x101 = _x104 ∧ 0 ≤ _x103 − 1 ∧ 2 ≤ _x97 − 1 ∧ _x103 + 2 ≤ _x97 ∧ −1 ≤ _x100 − 1 ∧ 0 ≤ _x101 − 1 ∧ 0 ≤ _x99 − 1 f2251_0_createTree_LE 14 f1257_0_random_ArrayAccess: x1 = _x106 ∧ x2 = _x107 ∧ x3 = _x108 ∧ x4 = _x109 ∧ x1 = _x110 ∧ x2 = _x111 ∧ x3 = _x112 ∧ x4 = _x113 ∧ 0 ≤ _x111 − 1 ∧ 2 ≤ _x106 − 1 ∧ _x111 + 2 ≤ _x106 ∧ −1 ≤ _x108 − 1 ∧ 0 ≤ _x112 − 1 ∧ 0 ≤ _x109 − 1 ∧ 0 ≤ _x107 − 1 f1843_0_main_InvokeMethod 15 f2233_0_randomlyDuplicate_NULL: x1 = _x114 ∧ x2 = _x115 ∧ x3 = _x116 ∧ x4 = _x117 ∧ x1 = _x118 ∧ x2 = _x119 ∧ x3 = _x120 ∧ x4 = _x121 ∧ _x116 = _x120 ∧ −1 ≤ _x118 − 1 ∧ −1 ≤ _x115 − 1 ∧ 0 ≤ _x114 − 1 ∧ _x118 ≤ _x115 f2233_0_randomlyDuplicate_NULL 16 f2233_0_randomlyDuplicate_NULL: x1 = _x122 ∧ x2 = _x123 ∧ x3 = _x124 ∧ x4 = _x125 ∧ x1 = _x126 ∧ x2 = _x127 ∧ x3 = _x128 ∧ x4 = _x129 ∧ −1 ≤ _x123 − 1 ∧ −1 ≤ _x124 − 1 ∧ _x130 ≤ 42 ∧ −1 ≤ _x130 − 1 ∧ _x126 + 1 ≤ _x122 ∧ 0 ≤ _x122 − 1 ∧ −1 ≤ _x126 − 1 ∧ _x123 = _x127 ∧ _x124 + 1 = _x128 f2233_0_randomlyDuplicate_NULL 17 f2233_0_randomlyDuplicate_NULL: x1 = _x131 ∧ x2 = _x132 ∧ x3 = _x133 ∧ x4 = _x134 ∧ x1 = _x135 ∧ x2 = _x136 ∧ x3 = _x137 ∧ x4 = _x138 ∧ −1 ≤ _x132 − 1 ∧ 42 ≤ _x139 − 1 ∧ −1 ≤ _x133 − 1 ∧ _x135 + 1 ≤ _x131 ∧ 0 ≤ _x131 − 1 ∧ −1 ≤ _x135 − 1 ∧ _x132 = _x136 ∧ _x133 + 1 = _x137 __init 18 f1_0_main_Load: x1 = _x140 ∧ x2 = _x141 ∧ x3 = _x142 ∧ x4 = _x143 ∧ x1 = _x144 ∧ x2 = _x145 ∧ x3 = _x146 ∧ x4 = _x147 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f323_0_createTree_Return f323_0_createTree_Return f323_0_createTree_Return: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f977_0_random_ArrayAccess f977_0_random_ArrayAccess f977_0_random_ArrayAccess: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f521_0_createNode_Return f521_0_createNode_Return f521_0_createNode_Return: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f931_0_random_ArrayAccess f931_0_random_ArrayAccess f931_0_random_ArrayAccess: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f2251_0_createTree_LE f2251_0_createTree_LE f2251_0_createTree_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f2233_0_randomlyDuplicate_NULL f2233_0_randomlyDuplicate_NULL f2233_0_randomlyDuplicate_NULL: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f588_0_createNode_Return f588_0_createNode_Return f588_0_createNode_Return: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f1843_0_main_InvokeMethod f1843_0_main_InvokeMethod f1843_0_main_InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
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 { f2251_0_createTree_LE }.

### 2.1.1 Transition Removal

We remove transitions 9, 10, 11 using the following ranking functions, which are bounded by 0.

 f2251_0_createTree_LE: x2

### 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 { f2233_0_randomlyDuplicate_NULL }.

### 2.2.1 Transition Removal

We remove transitions 16, 17 using the following ranking functions, which are bounded by 0.

 f2233_0_randomlyDuplicate_NULL: 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)