# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f1168_0_growTree_Return, f348_0_createTree_Return, f1232_0_growList_InvokeMethod, f495_0_createTree_GT, f998_0_createTree_GE, f1016_0_main_InvokeMethod, f1_0_main_Load, f1034_0_growList_NONNULL, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f495_0_createTree_GT: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x5 = _arg5 ∧ x6 = _arg6 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ x5 = _arg5P ∧ x6 = _arg6P ∧ 1 = _arg3P ∧ _arg2 = _arg2P ∧ 0 ≤ _arg1 − 1 ∧ 0 ≤ _arg2 − 1 ∧ −1 ≤ _arg1P − 1 f1_0_main_Load 2 f1016_0_main_InvokeMethod: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x1 = _x6 ∧ x2 = _x7 ∧ x3 = _x9 ∧ x4 = _x10 ∧ x5 = _x11 ∧ x6 = _x12 ∧ −1 ≤ _x13 − 1 ∧ 0 ≤ _x1 − 1 ∧ _x6 ≤ _x ∧ 0 ≤ _x − 1 ∧ 0 ≤ _x6 − 1 ∧ −1 ≤ _x7 − 1 f348_0_createTree_Return 3 f1016_0_main_InvokeMethod: x1 = _x15 ∧ x2 = _x16 ∧ x3 = _x17 ∧ x4 = _x18 ∧ x5 = _x19 ∧ x6 = _x20 ∧ x1 = _x21 ∧ x2 = _x22 ∧ x3 = _x23 ∧ x4 = _x24 ∧ x5 = _x25 ∧ x6 = _x26 ∧ _x16 = _x23 ∧ −1 ≤ _x22 − 1 ∧ 0 ≤ _x21 − 1 ∧ 0 ≤ _x15 − 1 ∧ _x22 + 1 ≤ _x15 ∧ _x21 ≤ _x15 f1016_0_main_InvokeMethod 4 f1034_0_growList_NONNULL: x1 = _x27 ∧ x2 = _x28 ∧ x3 = _x29 ∧ x4 = _x30 ∧ x5 = _x31 ∧ x6 = _x32 ∧ x1 = _x33 ∧ x2 = _x34 ∧ x3 = _x35 ∧ x4 = _x36 ∧ x5 = _x37 ∧ x6 = _x38 ∧ 0 ≤ _x39 − 1 ∧ 1 ≤ _x29 − 1 ∧ _x33 ≤ _x28 ∧ 0 ≤ _x27 − 1 ∧ −1 ≤ _x28 − 1 ∧ −1 ≤ _x33 − 1 ∧ _x29 = _x34 f495_0_createTree_GT 5 f998_0_createTree_GE: x1 = _x40 ∧ x2 = _x41 ∧ x3 = _x42 ∧ x4 = _x43 ∧ x5 = _x44 ∧ x6 = _x45 ∧ x1 = _x46 ∧ x2 = _x47 ∧ x3 = _x48 ∧ x4 = _x49 ∧ x5 = _x50 ∧ x6 = _x51 ∧ _x42 + 1 = _x51 ∧ _x41 = _x50 ∧ 0 = _x48 ∧ _x40 − 1 = _x47 ∧ _x40 = _x46 ∧ −1 ≤ _x49 − 1 ∧ _x42 ≤ _x41 − 1 ∧ 0 ≤ _x42 − 1 ∧ 0 ≤ _x40 − 1 ∧ −1 ≤ _x41 − 1 f998_0_createTree_GE 6 f495_0_createTree_GT: x1 = _x52 ∧ x2 = _x53 ∧ x3 = _x54 ∧ x4 = _x55 ∧ x5 = _x56 ∧ x6 = _x57 ∧ x1 = _x58 ∧ x2 = _x59 ∧ x3 = _x60 ∧ x4 = _x61 ∧ x5 = _x62 ∧ x6 = _x63 ∧ _x57 = _x60 ∧ _x56 = _x59 ∧ _x53 − 1 = _x58 ∧ _x53 − 1 ≤ _x53 − 1 ∧ −1 ≤ _x53 − 1 ∧ _x53 − 1 ≤ _x52 − 1 ∧ _x53 ≤ _x52 − 1 ∧ 1 ≤ _x57 − 1 ∧ 0 ≤ _x52 − 1 ∧ _x54 ≤ _x55 − 1 ∧ 0 ≤ _x55 − 1 f998_0_createTree_GE 7 f998_0_createTree_GE: x1 = _x64 ∧ x2 = _x65 ∧ x3 = _x66 ∧ x4 = _x67 ∧ x5 = _x68 ∧ x6 = _x69 ∧ x1 = _x70 ∧ x2 = _x71 ∧ x3 = _x72 ∧ x4 = _x73 ∧ x5 = _x74 ∧ x6 = _x75 ∧ _x69 = _x75 ∧ _x68 = _x74 ∧ _x67 = _x73 ∧ _x66 + 1 = _x72 ∧ _x65 = _x71 ∧ _x64 = _x70 ∧ _x65 − 1 ≤ _x65 − 1 ∧ −1 ≤ _x65 − 1 ∧ _x65 − 1 ≤ _x64 − 1 ∧ _x65 ≤ _x64 − 1 ∧ 1 ≤ _x69 − 1 ∧ 0 ≤ _x64 − 1 ∧ _x66 ≤ _x67 − 1 ∧ 0 ≤ _x67 − 1 f998_0_createTree_GE 8 f998_0_createTree_GE: x1 = _x76 ∧ x2 = _x77 ∧ x3 = _x78 ∧ x4 = _x79 ∧ x5 = _x80 ∧ x6 = _x81 ∧ x1 = _x82 ∧ x2 = _x83 ∧ x3 = _x84 ∧ x4 = _x85 ∧ x5 = _x86 ∧ x6 = _x87 ∧ _x80 = _x86 ∧ _x79 = _x85 ∧ _x78 + 1 = _x84 ∧ _x77 = _x83 ∧ _x76 = _x82 ∧ _x77 − 1 ≤ _x77 − 1 ∧ −1 ≤ _x77 − 1 ∧ _x77 − 1 ≤ _x76 − 1 ∧ _x77 ≤ _x76 − 1 ∧ 1 ≤ _x81 − 1 ∧ 0 ≤ _x76 − 1 ∧ _x78 ≤ _x79 − 1 ∧ 0 ≤ _x79 − 1 f1034_0_growList_NONNULL 9 f1034_0_growList_NONNULL: x1 = _x88 ∧ x2 = _x89 ∧ x3 = _x90 ∧ x4 = _x91 ∧ x5 = _x92 ∧ x6 = _x93 ∧ x1 = _x94 ∧ x2 = _x95 ∧ x3 = _x96 ∧ x4 = _x97 ∧ x5 = _x98 ∧ x6 = _x99 ∧ _x89 = _x95 ∧ −1 ≤ _x94 − 1 ∧ 1 ≤ _x88 − 1 ∧ 1 ≤ _x89 − 1 ∧ _x94 + 2 ≤ _x88 f1034_0_growList_NONNULL 10 f1232_0_growList_InvokeMethod: x1 = _x100 ∧ x2 = _x101 ∧ x3 = _x102 ∧ x4 = _x103 ∧ x5 = _x104 ∧ x6 = _x105 ∧ x1 = _x106 ∧ x2 = _x107 ∧ x3 = _x108 ∧ x4 = _x109 ∧ x5 = _x110 ∧ x6 = _x111 ∧ _x101 = _x109 ∧ −1 ≤ _x110 − 1 ∧ −1 ≤ _x107 − 1 ∧ 4 ≤ _x106 − 1 ∧ 0 ≤ _x100 − 1 ∧ _x110 + 1 ≤ _x100 ∧ 1 ≤ _x101 − 1 ∧ _x107 + 1 ≤ _x100 f1168_0_growTree_Return 11 f1232_0_growList_InvokeMethod: x1 = _x112 ∧ x2 = _x113 ∧ x3 = _x114 ∧ x4 = _x115 ∧ x5 = _x116 ∧ x6 = _x117 ∧ x1 = _x118 ∧ x2 = _x119 ∧ x3 = _x120 ∧ x4 = _x121 ∧ x5 = _x122 ∧ x6 = _x123 ∧ _x114 = _x121 ∧ _x113 = _x120 ∧ _x115 + 6 ≤ _x112 ∧ −1 ≤ _x122 − 1 ∧ −1 ≤ _x119 − 1 ∧ 5 ≤ _x118 − 1 ∧ −1 ≤ _x116 − 1 ∧ 5 ≤ _x112 − 1 ∧ _x122 ≤ _x116 ∧ _x122 + 2 ≤ _x112 ∧ _x119 ≤ _x116 ∧ _x119 + 2 ≤ _x112 ∧ _x118 ≤ _x112 f1034_0_growList_NONNULL 12 f1232_0_growList_InvokeMethod: x1 = _x124 ∧ x2 = _x125 ∧ x3 = _x126 ∧ x4 = _x127 ∧ x5 = _x128 ∧ x6 = _x129 ∧ x1 = _x130 ∧ x2 = _x131 ∧ x3 = _x132 ∧ x4 = _x133 ∧ x5 = _x134 ∧ x6 = _x135 ∧ _x125 = _x133 ∧ −1 ≤ _x134 − 1 ∧ −1 ≤ _x131 − 1 ∧ 7 ≤ _x130 − 1 ∧ 0 ≤ _x124 − 1 ∧ _x134 + 1 ≤ _x124 ∧ _x131 + 1 ≤ _x124 ∧ 1 ≤ _x125 − 1 ∧ _x130 − 7 ≤ _x124 f1232_0_growList_InvokeMethod 13 f1034_0_growList_NONNULL: x1 = _x136 ∧ x2 = _x137 ∧ x3 = _x138 ∧ x4 = _x139 ∧ x5 = _x140 ∧ x6 = _x141 ∧ x1 = _x142 ∧ x2 = _x143 ∧ x3 = _x144 ∧ x4 = _x145 ∧ x5 = _x146 ∧ x6 = _x147 ∧ _x139 = _x143 ∧ −1 ≤ _x142 − 1 ∧ −1 ≤ _x140 − 1 ∧ −1 ≤ _x137 − 1 ∧ 4 ≤ _x136 − 1 ∧ _x142 ≤ _x140 ∧ _x142 ≤ _x137 ∧ 1 ≤ _x139 − 1 ∧ _x142 + 2 ≤ _x136 f1034_0_growList_NONNULL 14 f1034_0_growList_NONNULL: x1 = _x148 ∧ x2 = _x149 ∧ x3 = _x150 ∧ x4 = _x151 ∧ x5 = _x152 ∧ x6 = _x153 ∧ x1 = _x154 ∧ x2 = _x155 ∧ x3 = _x156 ∧ x4 = _x157 ∧ x5 = _x158 ∧ x6 = _x159 ∧ _x149 = _x155 ∧ 0 ≤ _x154 − 1 ∧ 4 ≤ _x148 − 1 ∧ 1 ≤ _x149 − 1 ∧ _x154 + 4 ≤ _x148 __init 15 f1_0_main_Load: x1 = _x160 ∧ x2 = _x161 ∧ x3 = _x162 ∧ x4 = _x163 ∧ x5 = _x164 ∧ x6 = _x165 ∧ x1 = _x166 ∧ x2 = _x167 ∧ x3 = _x168 ∧ x4 = _x169 ∧ x5 = _x170 ∧ x6 = _x171 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f1168_0_growTree_Return f1168_0_growTree_Return f1168_0_growTree_Return: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f348_0_createTree_Return f348_0_createTree_Return f348_0_createTree_Return: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f1232_0_growList_InvokeMethod f1232_0_growList_InvokeMethod f1232_0_growList_InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f495_0_createTree_GT f495_0_createTree_GT f495_0_createTree_GT: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f998_0_createTree_GE f998_0_createTree_GE f998_0_createTree_GE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f1016_0_main_InvokeMethod f1016_0_main_InvokeMethod f1016_0_main_InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f1034_0_growList_NONNULL f1034_0_growList_NONNULL f1034_0_growList_NONNULL: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6
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 { f1232_0_growList_InvokeMethod, f1034_0_growList_NONNULL }.

### 2.1.1 Transition Removal

We remove transitions 9, 13, 12, 10, 14 using the following ranking functions, which are bounded by 0.

 f1034_0_growList_NONNULL: 2⋅x1 f1232_0_growList_InvokeMethod: 2⋅x2 + 1

### 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 { f495_0_createTree_GT, f998_0_createTree_GE }.

### 2.2.1 Transition Removal

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

 f495_0_createTree_GT: 1 + x1 f998_0_createTree_GE: x1

### 2.2.2 Transition Removal

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

 f998_0_createTree_GE: − x3 + x4

### 2.2.3 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 % (9 real / 0 unknown / 0 assumptions / 9 total proof steps)