# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f1893_0_createTree_LE, f1938_0_main_InvokeMethod, f1821_0_duplicateRandomPath_NULL, f1_0_main_Load, f1989_0_duplicateRandomPath_NULL, __init, f456_0_createTree_Return
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f1938_0_main_InvokeMethod: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x5 = _arg5 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ x5 = _arg5P ∧ 2 ≤ _arg2P − 1 ∧ 0 ≤ _arg1P − 1 ∧ 0 ≤ _arg1 − 1 ∧ _arg1P ≤ _arg1 f456_0_createTree_Return 2 f1938_0_main_InvokeMethod: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x1 = _x5 ∧ x2 = _x6 ∧ x3 = _x7 ∧ x4 = _x8 ∧ x5 = _x9 ∧ _x3 = _x8 ∧ _x2 = _x7 ∧ _x3 + 2 ≤ _x1 ∧ 2 ≤ _x6 − 1 ∧ 0 ≤ _x5 − 1 ∧ 2 ≤ _x1 − 1 ∧ 0 ≤ _x − 1 ∧ _x6 ≤ _x1 ∧ _x5 + 2 ≤ _x1 ∧ _x5 ≤ _x f1_0_main_Load 3 f1893_0_createTree_LE: x1 = _x10 ∧ x2 = _x11 ∧ x3 = _x12 ∧ x4 = _x13 ∧ x5 = _x14 ∧ x1 = _x15 ∧ x2 = _x16 ∧ x3 = _x17 ∧ x4 = _x18 ∧ x5 = _x19 ∧ 1 = _x18 ∧ 2 ≤ _x16 − 1 ∧ 2 ≤ _x15 − 1 ∧ 0 ≤ _x10 − 1 ∧ _x16 − 2 ≤ _x10 ∧ _x15 − 2 ≤ _x10 ∧ −1 ≤ _x11 − 1 ∧ 0 ≤ _x17 − 1 f1893_0_createTree_LE 4 f1893_0_createTree_LE: x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x22 ∧ x4 = _x23 ∧ x5 = _x24 ∧ x1 = _x25 ∧ x2 = _x26 ∧ x3 = _x27 ∧ x4 = _x28 ∧ x5 = _x29 ∧ _x23 + 1 = _x28 ∧ _x22 − 1 = _x27 ∧ 0 ≤ _x26 − 1 ∧ 2 ≤ _x25 − 1 ∧ 2 ≤ _x21 − 1 ∧ 2 ≤ _x20 − 1 ∧ _x26 + 2 ≤ _x21 ∧ _x25 ≤ _x20 ∧ 0 ≤ _x22 − 1 ∧ −1 ≤ _x23 − 1 f1893_0_createTree_LE 5 f1893_0_createTree_LE: x1 = _x31 ∧ x2 = _x32 ∧ x3 = _x33 ∧ x4 = _x34 ∧ x5 = _x35 ∧ x1 = _x36 ∧ x2 = _x38 ∧ x3 = _x39 ∧ x4 = _x40 ∧ x5 = _x41 ∧ 0 ≤ _x33 − 1 ∧ 0 ≤ _x42 − 1 ∧ −1 ≤ _x34 − 1 ∧ _x36 ≤ _x31 ∧ _x38 + 2 ≤ _x32 ∧ 2 ≤ _x31 − 1 ∧ 2 ≤ _x32 − 1 ∧ 2 ≤ _x36 − 1 ∧ 0 ≤ _x38 − 1 ∧ _x33 − 1 = _x39 ∧ _x34 + 1 = _x40 f1893_0_createTree_LE 6 f1893_0_createTree_LE: x1 = _x43 ∧ x2 = _x44 ∧ x3 = _x45 ∧ x4 = _x46 ∧ x5 = _x47 ∧ x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x4 = _x51 ∧ x5 = _x52 ∧ 0 ≤ _x45 − 1 ∧ 0 ≤ _x53 − 1 ∧ −1 ≤ _x46 − 1 ∧ 2 ≤ _x43 − 1 ∧ 1 ≤ _x44 − 1 ∧ 2 ≤ _x48 − 1 ∧ 2 ≤ _x49 − 1 ∧ _x45 − 1 = _x50 ∧ _x46 + 1 = _x51 f1893_0_createTree_LE 7 f1893_0_createTree_LE: x1 = _x54 ∧ x2 = _x55 ∧ x3 = _x57 ∧ x4 = _x58 ∧ x5 = _x59 ∧ x1 = _x60 ∧ x2 = _x61 ∧ x3 = _x62 ∧ x4 = _x63 ∧ x5 = _x64 ∧ _x58 + 1 = _x63 ∧ _x57 − 1 = _x62 ∧ 2 ≤ _x61 − 1 ∧ 2 ≤ _x60 − 1 ∧ 1 ≤ _x55 − 1 ∧ 2 ≤ _x54 − 1 ∧ 0 ≤ _x57 − 1 ∧ −1 ≤ _x58 − 1 f1893_0_createTree_LE 8 f1893_0_createTree_LE: x1 = _x65 ∧ x2 = _x66 ∧ x3 = _x67 ∧ x4 = _x68 ∧ x5 = _x69 ∧ x1 = _x70 ∧ x2 = _x71 ∧ x3 = _x72 ∧ x4 = _x73 ∧ x5 = _x74 ∧ _x68 + 1 = _x73 ∧ _x67 − 1 = _x72 ∧ 4 ≤ _x71 − 1 ∧ 4 ≤ _x70 − 1 ∧ 2 ≤ _x66 − 1 ∧ 2 ≤ _x65 − 1 ∧ _x71 − 2 ≤ _x66 ∧ _x71 − 2 ≤ _x65 ∧ _x70 − 2 ≤ _x66 ∧ _x70 − 2 ≤ _x65 ∧ 0 ≤ _x67 − 1 ∧ −1 ≤ _x68 − 1 f1893_0_createTree_LE 9 f1893_0_createTree_LE: x1 = _x76 ∧ x2 = _x77 ∧ x3 = _x78 ∧ x4 = _x79 ∧ x5 = _x80 ∧ x1 = _x81 ∧ x2 = _x82 ∧ x3 = _x84 ∧ x4 = _x85 ∧ x5 = _x86 ∧ 0 ≤ _x78 − 1 ∧ 0 ≤ _x87 − 1 ∧ −1 ≤ _x79 − 1 ∧ _x81 − 2 ≤ _x76 ∧ _x81 − 2 ≤ _x77 ∧ _x82 − 2 ≤ _x76 ∧ _x82 − 2 ≤ _x77 ∧ 2 ≤ _x76 − 1 ∧ 2 ≤ _x77 − 1 ∧ 4 ≤ _x81 − 1 ∧ 4 ≤ _x82 − 1 ∧ _x78 − 1 = _x84 ∧ _x79 + 1 = _x85 f1_0_main_Load 10 f1821_0_duplicateRandomPath_NULL: x1 = _x88 ∧ x2 = _x89 ∧ x3 = _x91 ∧ x4 = _x92 ∧ x5 = _x93 ∧ x1 = _x94 ∧ x2 = _x95 ∧ x3 = _x96 ∧ x4 = _x97 ∧ x5 = _x98 ∧ 1 = _x97 ∧ _x89 = _x96 ∧ −1 ≤ _x95 − 1 ∧ −1 ≤ _x94 − 1 ∧ 0 ≤ _x88 − 1 ∧ _x95 + 1 ≤ _x88 ∧ 0 ≤ _x89 − 1 ∧ _x94 + 1 ≤ _x88 f1938_0_main_InvokeMethod 11 f1821_0_duplicateRandomPath_NULL: x1 = _x99 ∧ x2 = _x100 ∧ x3 = _x101 ∧ x4 = _x102 ∧ x5 = _x103 ∧ x1 = _x104 ∧ x2 = _x105 ∧ x3 = _x106 ∧ x4 = _x107 ∧ x5 = _x108 ∧ _x101 = _x107 ∧ _x102 + 2 ≤ _x100 ∧ 2 ≤ _x105 − 1 ∧ 2 ≤ _x104 − 1 ∧ 2 ≤ _x100 − 1 ∧ 0 ≤ _x99 − 1 ∧ _x105 ≤ _x100 ∧ 0 ≤ _x106 − 1 ∧ _x104 ≤ _x100 f1821_0_duplicateRandomPath_NULL 12 f1989_0_duplicateRandomPath_NULL: x1 = _x109 ∧ x2 = _x110 ∧ x3 = _x111 ∧ x4 = _x112 ∧ x5 = _x113 ∧ x1 = _x114 ∧ x2 = _x115 ∧ x3 = _x116 ∧ x4 = _x117 ∧ x5 = _x118 ∧ −1 ≤ _x111 − 1 ∧ 41 ≤ _x119 − 1 ∧ −1 ≤ _x112 − 1 ∧ _x114 ≤ _x109 ∧ _x114 ≤ _x110 ∧ _x115 + 1 ≤ _x109 ∧ _x115 + 1 ≤ _x110 ∧ 0 ≤ _x109 − 1 ∧ 0 ≤ _x110 − 1 ∧ 0 ≤ _x114 − 1 ∧ −1 ≤ _x115 − 1 ∧ _x118 + 2 ≤ _x109 ∧ _x118 + 2 ≤ _x110 ∧ _x111 = _x116 ∧ _x112 + 1 = _x117 f1821_0_duplicateRandomPath_NULL 13 f1989_0_duplicateRandomPath_NULL: x1 = _x120 ∧ x2 = _x121 ∧ x3 = _x122 ∧ x4 = _x123 ∧ x5 = _x124 ∧ x1 = _x125 ∧ x2 = _x126 ∧ x3 = _x127 ∧ x4 = _x128 ∧ x5 = _x129 ∧ −1 ≤ _x122 − 1 ∧ −1 ≤ _x123 − 1 ∧ _x130 ≤ 41 ∧ −1 ≤ _x130 − 1 ∧ _x125 ≤ _x120 ∧ _x125 ≤ _x121 ∧ _x126 + 2 ≤ _x120 ∧ _x126 + 2 ≤ _x121 ∧ 1 ≤ _x120 − 1 ∧ 1 ≤ _x121 − 1 ∧ 1 ≤ _x125 − 1 ∧ −1 ≤ _x126 − 1 ∧ _x129 + 2 ≤ _x120 ∧ _x129 + 2 ≤ _x121 ∧ _x122 = _x127 ∧ _x123 + 1 = _x128 f1821_0_duplicateRandomPath_NULL 14 f1821_0_duplicateRandomPath_NULL: x1 = _x131 ∧ x2 = _x132 ∧ x3 = _x133 ∧ x4 = _x134 ∧ x5 = _x135 ∧ x1 = _x136 ∧ x2 = _x137 ∧ x3 = _x138 ∧ x4 = _x139 ∧ x5 = _x140 ∧ −1 ≤ _x133 − 1 ∧ −1 ≤ _x134 − 1 ∧ _x141 ≤ 41 ∧ −1 ≤ _x141 − 1 ∧ _x136 + 2 ≤ _x131 ∧ _x136 + 2 ≤ _x132 ∧ _x137 + 2 ≤ _x131 ∧ _x137 + 2 ≤ _x132 ∧ 2 ≤ _x131 − 1 ∧ 2 ≤ _x132 − 1 ∧ 0 ≤ _x136 − 1 ∧ 0 ≤ _x137 − 1 ∧ _x133 = _x138 ∧ _x134 + 1 = _x139 f1989_0_duplicateRandomPath_NULL 15 f1821_0_duplicateRandomPath_NULL: x1 = _x142 ∧ x2 = _x143 ∧ x3 = _x144 ∧ x4 = _x145 ∧ x5 = _x146 ∧ x1 = _x147 ∧ x2 = _x148 ∧ x3 = _x149 ∧ x4 = _x150 ∧ x5 = _x151 ∧ _x145 = _x150 ∧ _x144 = _x149 ∧ _x146 + 2 ≤ _x142 ∧ 0 ≤ _x148 − 1 ∧ 0 ≤ _x147 − 1 ∧ 0 ≤ _x143 − 1 ∧ 2 ≤ _x142 − 1 ∧ _x148 ≤ _x143 ∧ _x148 + 2 ≤ _x142 ∧ _x147 ≤ _x143 ∧ _x147 + 2 ≤ _x142 __init 16 f1_0_main_Load: x1 = _x152 ∧ x2 = _x153 ∧ x3 = _x154 ∧ x4 = _x155 ∧ x5 = _x156 ∧ x1 = _x157 ∧ x2 = _x158 ∧ x3 = _x159 ∧ x4 = _x160 ∧ x5 = _x161 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f1893_0_createTree_LE f1893_0_createTree_LE f1893_0_createTree_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 f1938_0_main_InvokeMethod f1938_0_main_InvokeMethod f1938_0_main_InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 f1821_0_duplicateRandomPath_NULL f1821_0_duplicateRandomPath_NULL f1821_0_duplicateRandomPath_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 f1989_0_duplicateRandomPath_NULL f1989_0_duplicateRandomPath_NULL f1989_0_duplicateRandomPath_NULL: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 f456_0_createTree_Return f456_0_createTree_Return f456_0_createTree_Return: 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 { f1893_0_createTree_LE }.

### 2.1.1 Transition Removal

We remove transitions 4, 5, 6, 7, 8, 9 using the following ranking functions, which are bounded by 0.

 f1893_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 { f1821_0_duplicateRandomPath_NULL, f1989_0_duplicateRandomPath_NULL }.

### 2.2.1 Transition Removal

We remove transitions 12, 14, 15, 13 using the following ranking functions, which are bounded by 0.

 f1821_0_duplicateRandomPath_NULL: 1 + x2 f1989_0_duplicateRandomPath_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 % (8 real / 0 unknown / 0 assumptions / 8 total proof steps)