# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f1116_0_dupTree_FieldAccess, f394_0_createTree_Return, f1349_0_dupList_InvokeMethod, f1081_0_createTree_GE, f1098_0_main_InvokeMethod, f1152_0_dupList_NONNULL, f577_0_createTree_GT, f1_0_main_Load, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f577_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 f1116_0_dupTree_FieldAccess: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x1 = _x6 ∧ x2 = _x7 ∧ x3 = _x8 ∧ x4 = _x9 ∧ x5 = _x10 ∧ x6 = _x12 ∧ −1 ≤ _x6 − 1 ∧ 0 ≤ _x − 1 ∧ 0 ≤ _x1 − 1 ∧ _x6 + 1 ≤ _x f1_0_main_Load 3 f1098_0_main_InvokeMethod: x1 = _x13 ∧ x2 = _x14 ∧ x3 = _x15 ∧ x4 = _x16 ∧ x5 = _x18 ∧ x6 = _x19 ∧ x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x22 ∧ x4 = _x23 ∧ x5 = _x24 ∧ x6 = _x25 ∧ −1 ≤ _x26 − 1 ∧ 0 ≤ _x14 − 1 ∧ _x20 ≤ _x13 ∧ 0 ≤ _x13 − 1 ∧ 0 ≤ _x20 − 1 ∧ 0 ≤ _x21 − 1 f394_0_createTree_Return 4 f1098_0_main_InvokeMethod: x1 = _x27 ∧ x2 = _x28 ∧ x3 = _x29 ∧ x4 = _x30 ∧ x5 = _x31 ∧ x6 = _x32 ∧ x1 = _x33 ∧ x2 = _x34 ∧ x3 = _x35 ∧ x4 = _x36 ∧ x5 = _x37 ∧ x6 = _x38 ∧ _x28 = _x35 ∧ 1 ≤ _x34 − 1 ∧ 0 ≤ _x33 − 1 ∧ 0 ≤ _x27 − 1 ∧ _x34 − 1 ≤ _x27 ∧ _x33 ≤ _x27 f1098_0_main_InvokeMethod 5 f1116_0_dupTree_FieldAccess: x1 = _x39 ∧ x2 = _x40 ∧ x3 = _x41 ∧ x4 = _x42 ∧ x5 = _x43 ∧ x6 = _x44 ∧ x1 = _x45 ∧ x2 = _x46 ∧ x3 = _x47 ∧ x4 = _x48 ∧ x5 = _x49 ∧ x6 = _x50 ∧ 0 ≤ _x51 − 1 ∧ 1 ≤ _x41 − 1 ∧ _x45 ≤ _x40 ∧ 0 ≤ _x39 − 1 ∧ 0 ≤ _x40 − 1 ∧ 0 ≤ _x45 − 1 f577_0_createTree_GT 6 f1081_0_createTree_GE: x1 = _x52 ∧ x2 = _x53 ∧ x3 = _x54 ∧ x4 = _x55 ∧ x5 = _x56 ∧ x6 = _x57 ∧ x1 = _x58 ∧ x2 = _x59 ∧ x3 = _x60 ∧ x4 = _x61 ∧ x5 = _x62 ∧ x6 = _x63 ∧ _x54 + 1 = _x63 ∧ _x53 = _x62 ∧ 0 = _x60 ∧ _x52 − 1 = _x59 ∧ _x52 = _x58 ∧ −1 ≤ _x61 − 1 ∧ _x54 ≤ _x53 − 1 ∧ 0 ≤ _x54 − 1 ∧ 0 ≤ _x52 − 1 ∧ −1 ≤ _x53 − 1 f1081_0_createTree_GE 7 f577_0_createTree_GT: 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 = _x72 ∧ _x68 = _x71 ∧ _x65 − 1 = _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 f1081_0_createTree_GE 8 f1081_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 ∧ _x81 = _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 f1081_0_createTree_GE 9 f1081_0_createTree_GE: x1 = _x88 ∧ x2 = _x89 ∧ x3 = _x90 ∧ x4 = _x91 ∧ x5 = _x92 ∧ x6 = _x93 ∧ x1 = _x94 ∧ x2 = _x95 ∧ x3 = _x96 ∧ x4 = _x97 ∧ x5 = _x98 ∧ x6 = _x99 ∧ _x92 = _x98 ∧ _x91 = _x97 ∧ _x90 + 1 = _x96 ∧ _x89 = _x95 ∧ _x88 = _x94 ∧ _x89 − 1 ≤ _x89 − 1 ∧ −1 ≤ _x89 − 1 ∧ _x89 − 1 ≤ _x88 − 1 ∧ _x89 ≤ _x88 − 1 ∧ 1 ≤ _x93 − 1 ∧ 0 ≤ _x88 − 1 ∧ _x90 ≤ _x91 − 1 ∧ 0 ≤ _x91 − 1 f1116_0_dupTree_FieldAccess 10 f1152_0_dupList_NONNULL: x1 = _x100 ∧ x2 = _x101 ∧ x3 = _x102 ∧ x4 = _x103 ∧ x5 = _x104 ∧ x6 = _x105 ∧ x1 = _x106 ∧ x2 = _x107 ∧ x3 = _x108 ∧ x4 = _x109 ∧ x5 = _x110 ∧ x6 = _x111 ∧ −1 ≤ _x106 − 1 ∧ 0 ≤ _x100 − 1 ∧ _x106 + 1 ≤ _x100 f1152_0_dupList_NONNULL 11 f1116_0_dupTree_FieldAccess: x1 = _x112 ∧ x2 = _x113 ∧ x3 = _x114 ∧ x4 = _x115 ∧ x5 = _x116 ∧ x6 = _x117 ∧ x1 = _x118 ∧ x2 = _x119 ∧ x3 = _x120 ∧ x4 = _x121 ∧ x5 = _x122 ∧ x6 = _x123 ∧ −1 ≤ _x118 − 1 ∧ 0 ≤ _x112 − 1 ∧ _x118 + 1 ≤ _x112 f1152_0_dupList_NONNULL 12 f1349_0_dupList_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 = _x132 ∧ −1 ≤ _x133 − 1 ∧ −1 ≤ _x131 − 1 ∧ 5 ≤ _x130 − 1 ∧ 3 ≤ _x124 − 1 ∧ _x133 + 2 ≤ _x124 ∧ _x131 + 2 ≤ _x124 ∧ _x130 − 2 ≤ _x124 f1152_0_dupList_NONNULL 13 f1349_0_dupList_InvokeMethod: x1 = _x136 ∧ x2 = _x137 ∧ x3 = _x138 ∧ x4 = _x139 ∧ x5 = _x140 ∧ x6 = _x141 ∧ x1 = _x142 ∧ x2 = _x143 ∧ x3 = _x144 ∧ x4 = _x145 ∧ x5 = _x146 ∧ x6 = _x147 ∧ _x137 = _x144 ∧ −1 ≤ _x145 − 1 ∧ −1 ≤ _x143 − 1 ∧ 10 ≤ _x142 − 1 ∧ 0 ≤ _x136 − 1 ∧ _x145 + 1 ≤ _x136 ∧ _x143 + 1 ≤ _x136 f1349_0_dupList_InvokeMethod 14 f1152_0_dupList_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 ∧ _x150 = _x155 ∧ −1 ≤ _x154 − 1 ∧ −1 ≤ _x151 − 1 ∧ −1 ≤ _x149 − 1 ∧ 4 ≤ _x148 − 1 ∧ _x154 ≤ _x151 ∧ _x154 ≤ _x149 ∧ _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:
 f1116_0_dupTree_FieldAccess f1116_0_dupTree_FieldAccess f1116_0_dupTree_FieldAccess: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f394_0_createTree_Return f394_0_createTree_Return f394_0_createTree_Return: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f1349_0_dupList_InvokeMethod f1349_0_dupList_InvokeMethod f1349_0_dupList_InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f1081_0_createTree_GE f1081_0_createTree_GE f1081_0_createTree_GE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f1098_0_main_InvokeMethod f1098_0_main_InvokeMethod f1098_0_main_InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f1152_0_dupList_NONNULL f1152_0_dupList_NONNULL f1152_0_dupList_NONNULL: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f577_0_createTree_GT f577_0_createTree_GT f577_0_createTree_GT: 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 __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 { f1116_0_dupTree_FieldAccess, f1349_0_dupList_InvokeMethod, f1152_0_dupList_NONNULL }.

### 2.1.1 Transition Removal

We remove transitions 10, 11, 14, 13, 12 using the following ranking functions, which are bounded by 0.

 f1116_0_dupTree_FieldAccess: 2⋅x1 − 1 f1152_0_dupList_NONNULL: 2⋅x1 − 1 f1349_0_dupList_InvokeMethod: 2⋅x4

### 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 { f1081_0_createTree_GE, f577_0_createTree_GT }.

### 2.2.1 Transition Removal

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

 f577_0_createTree_GT: 1 + x1 f1081_0_createTree_GE: x1

### 2.2.2 Transition Removal

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

 f1081_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)