# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f458_0_reverse_NULL, f488_0_reverse_FieldAccess, f266_0_createList_LE, f473_0_reverse_FieldAccess, f1_0_main_Load, f231_0_createList_LE, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f231_0_createList_LE: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ −1 ≤ _arg1P − 1 ∧ 1 ≤ _arg2 − 1 ∧ −1 ≤ _x5 − 1 ∧ _arg2P − 1 ≤ _arg1 ∧ _arg3P − 1 ≤ _arg1 ∧ 0 ≤ _arg1 − 1 ∧ 1 ≤ _arg2P − 1 ∧ 1 ≤ _arg3P − 1 ∧ _x5 − 2 = _arg4P f231_0_createList_LE 2 f266_0_createList_LE: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x1 = _x4 ∧ x2 = _x6 ∧ x3 = _x7 ∧ x4 = _x8 ∧ _x − 1 = _x6 ∧ 0 ≤ _x4 − 1 ∧ 0 ≤ _x2 − 1 ∧ _x3 ≤ 0 ∧ 1 ≤ _x1 − 1 f231_0_createList_LE 3 f231_0_createList_LE: x1 = _x9 ∧ x2 = _x10 ∧ x3 = _x11 ∧ x4 = _x12 ∧ x1 = _x13 ∧ x2 = _x14 ∧ x3 = _x15 ∧ x4 = _x16 ∧ _x12 − 1 = _x16 ∧ _x9 = _x13 ∧ 2 ≤ _x15 − 1 ∧ 0 ≤ _x14 − 1 ∧ 0 ≤ _x11 − 1 ∧ 0 ≤ _x10 − 1 ∧ _x15 − 2 ≤ _x11 ∧ 0 ≤ _x12 − 1 ∧ _x14 ≤ _x10 f266_0_createList_LE 4 f266_0_createList_LE: x1 = _x17 ∧ x2 = _x18 ∧ x3 = _x19 ∧ x4 = _x20 ∧ x1 = _x21 ∧ x2 = _x22 ∧ x3 = _x23 ∧ x4 = _x24 ∧ _x18 − 1 = _x22 ∧ 2 ≤ _x21 − 1 ∧ 0 ≤ _x18 − 1 ∧ 0 ≤ _x17 − 1 f231_0_createList_LE 5 f266_0_createList_LE: x1 = _x25 ∧ x2 = _x26 ∧ x3 = _x27 ∧ x4 = _x28 ∧ x1 = _x29 ∧ x2 = _x30 ∧ x3 = _x31 ∧ x4 = _x32 ∧ _x25 − 2 = _x30 ∧ 4 ≤ _x29 − 1 ∧ 1 ≤ _x27 − 1 ∧ 1 ≤ _x26 − 1 ∧ _x29 − 3 ≤ _x27 ∧ _x29 − 3 ≤ _x26 ∧ _x28 ≤ 0 ∧ 1 ≤ _x25 − 1 f231_0_createList_LE 6 f458_0_reverse_NULL: x1 = _x33 ∧ x2 = _x34 ∧ x3 = _x35 ∧ x4 = _x36 ∧ x1 = _x37 ∧ x2 = _x38 ∧ x3 = _x39 ∧ x4 = _x40 ∧ 1 ≤ _x38 − 1 ∧ 1 ≤ _x37 − 1 ∧ 1 ≤ _x35 − 1 ∧ 1 ≤ _x34 − 1 ∧ _x38 ≤ _x35 ∧ _x38 ≤ _x34 ∧ _x37 ≤ _x35 ∧ _x37 ≤ _x34 ∧ _x36 ≤ 0 ∧ _x33 ≤ 1 f266_0_createList_LE 7 f458_0_reverse_NULL: x1 = _x41 ∧ x2 = _x42 ∧ x3 = _x43 ∧ x4 = _x44 ∧ x1 = _x45 ∧ x2 = _x46 ∧ x3 = _x47 ∧ x4 = _x48 ∧ −1 ≤ _x46 − 1 ∧ 1 ≤ _x45 − 1 ∧ 0 ≤ _x41 − 1 ∧ _x42 ≤ 0 ∧ _x45 − 1 ≤ _x41 f266_0_createList_LE 8 f458_0_reverse_NULL: x1 = _x49 ∧ x2 = _x50 ∧ x3 = _x51 ∧ x4 = _x52 ∧ x1 = _x53 ∧ x2 = _x54 ∧ x3 = _x55 ∧ x4 = _x56 ∧ 0 ≤ _x54 − 1 ∧ 1 ≤ _x53 − 1 ∧ 2 ≤ _x49 − 1 ∧ _x50 ≤ 0 ∧ _x53 + 1 ≤ _x49 f266_0_createList_LE 9 f473_0_reverse_FieldAccess: x1 = _x57 ∧ x2 = _x58 ∧ x3 = _x59 ∧ x4 = _x60 ∧ x1 = _x61 ∧ x2 = _x62 ∧ x3 = _x63 ∧ x4 = _x64 ∧ 0 ≤ _x62 − 1 ∧ 1 ≤ _x61 − 1 ∧ 2 ≤ _x57 − 1 ∧ _x58 ≤ 0 ∧ _x61 + 1 ≤ _x57 f458_0_reverse_NULL 10 f488_0_reverse_FieldAccess: x1 = _x65 ∧ x2 = _x66 ∧ x3 = _x67 ∧ x4 = _x68 ∧ x1 = _x69 ∧ x2 = _x70 ∧ x3 = _x71 ∧ x4 = _x72 ∧ −1 ≤ _x72 − 1 ∧ 0 ≤ _x71 − 1 ∧ 0 ≤ _x70 − 1 ∧ −1 ≤ _x69 − 1 ∧ 0 ≤ _x66 − 1 ∧ 0 ≤ _x65 − 1 ∧ _x72 + 1 ≤ _x66 ∧ _x71 ≤ _x65 ∧ _x70 ≤ _x66 ∧ _x69 + 1 ≤ _x66 f488_0_reverse_FieldAccess 11 f458_0_reverse_NULL: x1 = _x73 ∧ x2 = _x74 ∧ x3 = _x75 ∧ x4 = _x76 ∧ x1 = _x77 ∧ x2 = _x80 ∧ x3 = _x81 ∧ x4 = _x82 ∧ −1 ≤ _x80 − 1 ∧ 2 ≤ _x77 − 1 ∧ −1 ≤ _x76 − 1 ∧ 0 ≤ _x75 − 1 ∧ 0 ≤ _x74 − 1 ∧ −1 ≤ _x73 − 1 ∧ _x80 ≤ _x76 ∧ _x80 + 1 ≤ _x74 ∧ _x80 ≤ _x73 f488_0_reverse_FieldAccess 12 f458_0_reverse_NULL: x1 = _x83 ∧ x2 = _x84 ∧ x3 = _x85 ∧ x4 = _x88 ∧ x1 = _x89 ∧ x2 = _x90 ∧ x3 = _x91 ∧ x4 = _x92 ∧ −1 ≤ _x90 − 1 ∧ 2 ≤ _x89 − 1 ∧ −1 ≤ _x88 − 1 ∧ 0 ≤ _x85 − 1 ∧ 0 ≤ _x84 − 1 ∧ −1 ≤ _x83 − 1 ∧ _x90 ≤ _x88 ∧ _x90 + 1 ≤ _x85 ∧ _x90 + 1 ≤ _x84 ∧ _x90 ≤ _x83 ∧ _x89 − 3 ≤ _x88 ∧ _x89 − 2 ≤ _x85 ∧ _x89 − 2 ≤ _x84 ∧ _x89 − 3 ≤ _x83 f473_0_reverse_FieldAccess 13 f473_0_reverse_FieldAccess: x1 = _x93 ∧ x2 = _x94 ∧ x3 = _x95 ∧ x4 = _x96 ∧ x1 = _x97 ∧ x2 = _x98 ∧ x3 = _x99 ∧ x4 = _x100 ∧ _x95 = _x99 ∧ 0 ≤ _x98 − 1 ∧ 2 ≤ _x97 − 1 ∧ 2 ≤ _x94 − 1 ∧ 0 ≤ _x93 − 1 ∧ _x97 − 2 ≤ _x93 ∧ _x100 ≤ _x96 − 1 ∧ 0 ≤ _x96 − 1 ∧ 0 ≤ _x95 − 1 f473_0_reverse_FieldAccess 14 f488_0_reverse_FieldAccess: x1 = _x101 ∧ x2 = _x102 ∧ x3 = _x103 ∧ x4 = _x104 ∧ x1 = _x105 ∧ x2 = _x106 ∧ x3 = _x107 ∧ x4 = _x108 ∧ 0 ≤ _x109 − 1 ∧ −1 ≤ _x110 − 1 ∧ _x109 ≤ _x110 − 1 ∧ _x107 − 2 ≤ _x101 ∧ 0 ≤ _x101 − 1 ∧ 2 ≤ _x102 − 1 ∧ −1 ≤ _x105 − 1 ∧ 0 ≤ _x106 − 1 ∧ 2 ≤ _x107 − 1 ∧ −1 ≤ _x108 − 1 f473_0_reverse_FieldAccess 15 f458_0_reverse_NULL: x1 = _x111 ∧ x2 = _x112 ∧ x3 = _x113 ∧ x4 = _x114 ∧ x1 = _x115 ∧ x2 = _x116 ∧ x3 = _x117 ∧ x4 = _x118 ∧ −1 ≤ _x119 − 1 ∧ _x120 ≤ _x119 − 1 ∧ _x115 − 2 ≤ _x111 ∧ _x115 ≤ _x112 ∧ _x116 ≤ _x111 ∧ 0 ≤ _x111 − 1 ∧ 2 ≤ _x112 − 1 ∧ 2 ≤ _x115 − 1 ∧ 0 ≤ _x116 − 1 __init 16 f1_0_main_Load: x1 = _x121 ∧ x2 = _x122 ∧ x3 = _x123 ∧ x4 = _x124 ∧ x1 = _x125 ∧ x2 = _x126 ∧ x3 = _x127 ∧ x4 = _x128 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f458_0_reverse_NULL f458_0_reverse_NULL f458_0_reverse_NULL: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f488_0_reverse_FieldAccess f488_0_reverse_FieldAccess f488_0_reverse_FieldAccess: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f266_0_createList_LE f266_0_createList_LE f266_0_createList_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f473_0_reverse_FieldAccess f473_0_reverse_FieldAccess f473_0_reverse_FieldAccess: 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 f231_0_createList_LE f231_0_createList_LE f231_0_createList_LE: 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 4 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/4

Here we consider the SCC { f231_0_createList_LE }.

### 2.1.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 0.

 f231_0_createList_LE: 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 2/4

Here we consider the SCC { f266_0_createList_LE }.

### 2.2.1 Transition Removal

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

 f266_0_createList_LE: x2

### 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 3/4

Here we consider the SCC { f473_0_reverse_FieldAccess }.

### 2.3.1 Transition Removal

We remove transition 13 using the following ranking functions, which are bounded by 0.

 f473_0_reverse_FieldAccess: x4

### 2.3.2 Trivial Cooperation Program

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

### 2.4 SCC Subproblem 4/4

Here we consider the SCC { f458_0_reverse_NULL, f488_0_reverse_FieldAccess }.

### 2.4.1 Transition Removal

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

 f458_0_reverse_NULL: −1 + 2⋅x2 f488_0_reverse_FieldAccess: 2⋅x1

### 2.4.2 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 % (12 real / 0 unknown / 0 assumptions / 12 total proof steps)