# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f908_0_main_InvokeMethod, f358_0_createTree_Return, f940_0_gopher_NONNULL, f1019_0_insert_GT, f1_0_main_Load, f880_0_createTree_GE, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f908_0_main_InvokeMethod: 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 ≤ _x7 − 1 ∧ 0 ≤ _arg2 − 1 ∧ _arg1P ≤ _arg1 ∧ 0 ≤ _arg1 − 1 ∧ 0 ≤ _arg1P − 1 ∧ 0 ≤ _arg2P − 1 ∧ −1 ≤ _arg3P − 1 f358_0_createTree_Return 2 f908_0_main_InvokeMethod: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x1 = _x6 ∧ x2 = _x8 ∧ x3 = _x9 ∧ x4 = _x10 ∧ x5 = _x11 ∧ x6 = _x12 ∧ _x4 = _x11 ∧ _x3 = _x10 ∧ _x4 + 2 ≤ _x1 ∧ _x3 + 2 ≤ _x1 ∧ −1 ≤ _x9 − 1 ∧ 0 ≤ _x8 − 1 ∧ 0 ≤ _x6 − 1 ∧ −1 ≤ _x2 − 1 ∧ 0 ≤ _x1 − 1 ∧ 0 ≤ _x − 1 ∧ _x9 ≤ _x2 ∧ _x9 + 1 ≤ _x1 ∧ _x8 ≤ _x1 ∧ _x6 − 1 ≤ _x2 ∧ _x6 ≤ _x1 ∧ _x6 ≤ _x f1_0_main_Load 3 f880_0_createTree_GE: x1 = _x13 ∧ x2 = _x14 ∧ x3 = _x15 ∧ x4 = _x16 ∧ x5 = _x17 ∧ x6 = _x18 ∧ x1 = _x19 ∧ x2 = _x20 ∧ x3 = _x21 ∧ x4 = _x22 ∧ x5 = _x23 ∧ x6 = _x24 ∧ 2 = _x23 ∧ _x14 = _x22 ∧ 0 = _x20 ∧ 1 ≤ _x19 − 1 ∧ 0 ≤ _x13 − 1 ∧ −1 ≤ _x21 − 1 ∧ 1 ≤ _x14 − 1 ∧ −1 ≤ _x24 − 1 f880_0_createTree_GE 4 f880_0_createTree_GE: x1 = _x25 ∧ x2 = _x26 ∧ x3 = _x27 ∧ x4 = _x28 ∧ x5 = _x29 ∧ x6 = _x31 ∧ x1 = _x32 ∧ x2 = _x33 ∧ x3 = _x34 ∧ x4 = _x35 ∧ x5 = _x36 ∧ x6 = _x37 ∧ −1 ≤ _x29 − 1 ∧ _x26 ≤ _x27 − 1 ∧ 0 ≤ _x27 − 1 ∧ 1 ≤ _x28 − 1 ∧ −1 ≤ _x38 − 1 ∧ _x29 ≤ _x28 − 1 ∧ 0 ≤ _x25 − 1 ∧ 3 ≤ _x32 − 1 ∧ _x31 + 2 ≤ _x25 ∧ _x26 + 1 = _x33 ∧ _x27 = _x34 ∧ _x28 = _x35 ∧ _x29 + 1 = _x36 f880_0_createTree_GE 5 f880_0_createTree_GE: x1 = _x39 ∧ x2 = _x40 ∧ x3 = _x41 ∧ x4 = _x42 ∧ x5 = _x43 ∧ x6 = _x44 ∧ x1 = _x45 ∧ x2 = _x46 ∧ x3 = _x47 ∧ x4 = _x49 ∧ x5 = _x50 ∧ x6 = _x51 ∧ −1 ≤ _x43 − 1 ∧ _x40 ≤ _x41 − 1 ∧ 0 ≤ _x41 − 1 ∧ 1 ≤ _x42 − 1 ∧ −1 ≤ _x52 − 1 ∧ _x43 ≤ _x42 − 1 ∧ 0 ≤ _x39 − 1 ∧ 2 ≤ _x45 − 1 ∧ _x44 + 2 ≤ _x39 ∧ _x40 + 1 = _x46 ∧ _x41 = _x47 ∧ _x42 = _x49 ∧ _x43 + 1 = _x50 f908_0_main_InvokeMethod 6 f940_0_gopher_NONNULL: x1 = _x53 ∧ x2 = _x54 ∧ x3 = _x55 ∧ x4 = _x56 ∧ x5 = _x57 ∧ x6 = _x59 ∧ x1 = _x60 ∧ x2 = _x61 ∧ x3 = _x62 ∧ x4 = _x63 ∧ x5 = _x64 ∧ x6 = _x65 ∧ _x60 ≤ _x54 ∧ 1 ≤ _x66 − 1 ∧ _x61 + 1 ≤ _x54 ∧ _x61 ≤ _x55 ∧ 0 ≤ _x53 − 1 ∧ 0 ≤ _x54 − 1 ∧ −1 ≤ _x55 − 1 ∧ 0 ≤ _x60 − 1 ∧ −1 ≤ _x61 − 1 ∧ _x56 + 2 ≤ _x54 ∧ _x57 + 2 ≤ _x54 ∧ _x57 = _x62 f940_0_gopher_NONNULL 7 f940_0_gopher_NONNULL: x1 = _x67 ∧ x2 = _x68 ∧ x3 = _x69 ∧ x4 = _x70 ∧ x5 = _x71 ∧ x6 = _x72 ∧ x1 = _x73 ∧ x2 = _x74 ∧ x3 = _x75 ∧ x4 = _x76 ∧ x5 = _x77 ∧ x6 = _x78 ∧ 0 = _x75 ∧ _x69 + 2 ≤ _x67 ∧ −1 ≤ _x74 − 1 ∧ 3 ≤ _x73 − 1 ∧ 0 ≤ _x68 − 1 ∧ 2 ≤ _x67 − 1 ∧ _x74 + 1 ≤ _x68 ∧ _x74 + 3 ≤ _x67 ∧ _x73 − 2 ≤ _x67 f880_0_createTree_GE 8 f1019_0_insert_GT: x1 = _x79 ∧ x2 = _x80 ∧ x3 = _x81 ∧ x4 = _x82 ∧ x5 = _x83 ∧ x6 = _x84 ∧ x1 = _x85 ∧ x2 = _x86 ∧ x3 = _x87 ∧ x4 = _x88 ∧ x5 = _x89 ∧ x6 = _x90 ∧ _x84 = _x88 ∧ _x82 = _x87 ∧ _x84 + 2 ≤ _x79 ∧ 0 ≤ _x85 − 1 ∧ 0 ≤ _x79 − 1 ∧ _x85 ≤ _x79 ∧ _x83 ≤ _x82 − 1 ∧ −1 ≤ _x86 − 1 ∧ 1 ≤ _x82 − 1 ∧ 0 ≤ _x81 − 1 ∧ _x80 ≤ _x81 − 1 ∧ −1 ≤ _x83 − 1 f1019_0_insert_GT 9 f1019_0_insert_GT: x1 = _x91 ∧ x2 = _x92 ∧ x3 = _x93 ∧ x4 = _x94 ∧ x5 = _x95 ∧ x6 = _x96 ∧ x1 = _x97 ∧ x2 = _x98 ∧ x3 = _x99 ∧ x4 = _x100 ∧ x5 = _x101 ∧ x6 = _x102 ∧ _x93 = _x99 ∧ _x92 = _x98 ∧ _x100 + 4 ≤ _x91 ∧ _x94 + 2 ≤ _x91 ∧ 0 ≤ _x97 − 1 ∧ 2 ≤ _x91 − 1 ∧ _x97 + 2 ≤ _x91 ∧ _x94 ≤ _x92 − 1 ∧ 1 ≤ _x93 − 1 f1019_0_insert_GT 10 f1019_0_insert_GT: x1 = _x103 ∧ x2 = _x104 ∧ x3 = _x105 ∧ x4 = _x106 ∧ x5 = _x107 ∧ x6 = _x108 ∧ x1 = _x109 ∧ x2 = _x110 ∧ x3 = _x111 ∧ x4 = _x112 ∧ x5 = _x113 ∧ x6 = _x114 ∧ _x105 = _x111 ∧ _x104 = _x110 ∧ _x112 + 4 ≤ _x103 ∧ _x106 + 2 ≤ _x103 ∧ 0 ≤ _x109 − 1 ∧ 2 ≤ _x103 − 1 ∧ _x109 + 2 ≤ _x103 ∧ _x104 ≤ _x106 ∧ 1 ≤ _x105 − 1 __init 11 f1_0_main_Load: x1 = _x115 ∧ x2 = _x116 ∧ x3 = _x117 ∧ x4 = _x118 ∧ x5 = _x119 ∧ x6 = _x120 ∧ x1 = _x121 ∧ x2 = _x122 ∧ x3 = _x123 ∧ x4 = _x124 ∧ x5 = _x125 ∧ x6 = _x126 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f908_0_main_InvokeMethod f908_0_main_InvokeMethod f908_0_main_InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f358_0_createTree_Return f358_0_createTree_Return f358_0_createTree_Return: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f940_0_gopher_NONNULL f940_0_gopher_NONNULL f940_0_gopher_NONNULL: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f1019_0_insert_GT f1019_0_insert_GT f1019_0_insert_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 f880_0_createTree_GE f880_0_createTree_GE f880_0_createTree_GE: 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 3 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/3

Here we consider the SCC { f880_0_createTree_GE }.

### 2.1.1 Transition Removal

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

 f880_0_createTree_GE: − x2 + 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 2/3

Here we consider the SCC { f1019_0_insert_GT }.

### 2.2.1 Transition Removal

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

 f1019_0_insert_GT: x1

### 2.2.2 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

### 2.3 SCC Subproblem 3/3

Here we consider the SCC { f940_0_gopher_NONNULL }.

### 2.3.1 Transition Removal

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

 f940_0_gopher_NONNULL: x2

### 2.3.2 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 % (10 real / 0 unknown / 0 assumptions / 10 total proof steps)