# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f200_0_random_ArrayAccess, f387_0_length_NULL, f314_0_appendNewList_LE, f1_0_main_Load, f165_0_appendNewList_LE, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f165_0_appendNewList_LE: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x5 = _arg5 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ x5 = _arg5P ∧ 0 = _arg5P ∧ 3 ≤ _arg3P − 1 ∧ 5 ≤ _arg2P − 1 ∧ 0 ≤ _arg1P − 1 ∧ 0 ≤ _arg1 − 1 ∧ _arg3P − 3 ≤ _arg1 ∧ _arg2P − 5 ≤ _arg1 ∧ _arg1P ≤ _arg1 ∧ −1 ≤ _arg4P − 1 ∧ 0 ≤ _arg2 − 1 f165_0_appendNewList_LE 2 f200_0_random_ArrayAccess: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x1 = _x5 ∧ x2 = _x6 ∧ x3 = _x7 ∧ x4 = _x8 ∧ x5 = _x9 ∧ _x3 ≤ 1 ∧ 0 ≤ _x4 − 1 ∧ _x4 ≤ _x10 − 1 ∧ _x4 ≤ _x11 − 1 ∧ 0 ≤ _x − 1 ∧ 4 ≤ _x1 − 1 ∧ 2 ≤ _x2 − 1 ∧ 4 ≤ _x5 − 1 f165_0_appendNewList_LE 3 f200_0_random_ArrayAccess: x1 = _x12 ∧ x2 = _x15 ∧ x3 = _x16 ∧ x4 = _x17 ∧ x5 = _x18 ∧ x1 = _x19 ∧ x2 = _x20 ∧ x3 = _x21 ∧ x4 = _x22 ∧ x5 = _x23 ∧ 6 ≤ _x19 − 1 ∧ 3 ≤ _x16 − 1 ∧ 5 ≤ _x15 − 1 ∧ 0 ≤ _x12 − 1 ∧ _x19 − 3 ≤ _x16 ∧ _x19 − 1 ≤ _x15 ∧ _x17 ≤ 1 ∧ _x19 − 6 ≤ _x12 f165_0_appendNewList_LE 4 f165_0_appendNewList_LE: x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x4 = _x27 ∧ x5 = _x28 ∧ x1 = _x29 ∧ x2 = _x30 ∧ x3 = _x31 ∧ x4 = _x32 ∧ x5 = _x33 ∧ _x27 − 1 = _x32 ∧ 2 ≤ _x31 − 1 ∧ 4 ≤ _x30 − 1 ∧ 0 ≤ _x29 − 1 ∧ 2 ≤ _x26 − 1 ∧ 4 ≤ _x25 − 1 ∧ 0 ≤ _x24 − 1 ∧ _x29 + 2 ≤ _x26 ∧ _x29 + 4 ≤ _x25 ∧ _x29 ≤ _x24 ∧ 1 ≤ _x27 − 1 ∧ 0 ≤ _x28 − 1 ∧ _x28 ≤ _x33 − 1 f165_0_appendNewList_LE 5 f165_0_appendNewList_LE: x1 = _x34 ∧ x2 = _x35 ∧ x3 = _x36 ∧ x4 = _x37 ∧ x5 = _x38 ∧ x1 = _x39 ∧ x2 = _x40 ∧ x3 = _x41 ∧ x4 = _x42 ∧ x5 = _x43 ∧ 1 = _x43 ∧ _x37 − 1 = _x42 ∧ 5 ≤ _x41 − 1 ∧ 7 ≤ _x40 − 1 ∧ 0 ≤ _x39 − 1 ∧ 3 ≤ _x36 − 1 ∧ 5 ≤ _x35 − 1 ∧ 0 ≤ _x34 − 1 ∧ _x41 − 2 ≤ _x36 ∧ _x41 ≤ _x35 ∧ _x41 − 5 ≤ _x34 ∧ _x40 − 4 ≤ _x36 ∧ _x40 − 2 ≤ _x35 ∧ _x40 − 7 ≤ _x34 ∧ _x39 + 3 ≤ _x36 ∧ _x39 + 5 ≤ _x35 ∧ 1 ≤ _x37 − 1 ∧ _x39 ≤ _x34 f200_0_random_ArrayAccess 6 f314_0_appendNewList_LE: x1 = _x44 ∧ x2 = _x45 ∧ x3 = _x46 ∧ x4 = _x47 ∧ x5 = _x48 ∧ x1 = _x49 ∧ x2 = _x50 ∧ x3 = _x51 ∧ x4 = _x52 ∧ x5 = _x53 ∧ 0 = _x51 ∧ 3 ≤ _x49 − 1 ∧ 4 ≤ _x44 − 1 ∧ _x49 + 1 ≤ _x44 ∧ −1 ≤ _x50 − 1 ∧ 1 ≤ _x45 − 1 f314_0_appendNewList_LE 7 f314_0_appendNewList_LE: x1 = _x54 ∧ x2 = _x55 ∧ x3 = _x56 ∧ x4 = _x57 ∧ x5 = _x58 ∧ x1 = _x59 ∧ x2 = _x60 ∧ x3 = _x61 ∧ x4 = _x62 ∧ x5 = _x63 ∧ _x55 − 1 = _x60 ∧ 2 ≤ _x59 − 1 ∧ 2 ≤ _x54 − 1 ∧ 1 ≤ _x55 − 1 ∧ 0 ≤ _x56 − 1 ∧ _x56 ≤ _x61 − 1 f314_0_appendNewList_LE 8 f314_0_appendNewList_LE: x1 = _x64 ∧ x2 = _x65 ∧ x3 = _x66 ∧ x4 = _x67 ∧ x5 = _x68 ∧ x1 = _x69 ∧ x2 = _x70 ∧ x3 = _x71 ∧ x4 = _x72 ∧ x5 = _x73 ∧ 1 = _x71 ∧ _x65 − 1 = _x70 ∧ 5 ≤ _x69 − 1 ∧ 3 ≤ _x64 − 1 ∧ 1 ≤ _x65 − 1 ∧ _x69 − 2 ≤ _x64 f314_0_appendNewList_LE 9 f387_0_length_NULL: x1 = _x74 ∧ x2 = _x75 ∧ x3 = _x76 ∧ x4 = _x77 ∧ x5 = _x78 ∧ x1 = _x79 ∧ x2 = _x80 ∧ x3 = _x81 ∧ x4 = _x82 ∧ x5 = _x83 ∧ 0 ≤ _x79 − 1 ∧ 2 ≤ _x74 − 1 ∧ _x75 ≤ 1 ∧ _x79 + 2 ≤ _x74 f387_0_length_NULL 10 f387_0_length_NULL: x1 = _x84 ∧ x2 = _x85 ∧ x3 = _x86 ∧ x4 = _x87 ∧ x5 = _x88 ∧ x1 = _x89 ∧ x2 = _x90 ∧ x3 = _x91 ∧ x4 = _x92 ∧ x5 = _x93 ∧ −1 ≤ _x89 − 1 ∧ 0 ≤ _x84 − 1 ∧ _x89 + 1 ≤ _x84 __init 11 f1_0_main_Load: x1 = _x94 ∧ x2 = _x95 ∧ x3 = _x96 ∧ x4 = _x97 ∧ x5 = _x98 ∧ x1 = _x99 ∧ x2 = _x100 ∧ x3 = _x101 ∧ x4 = _x102 ∧ x5 = _x103 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f200_0_random_ArrayAccess f200_0_random_ArrayAccess f200_0_random_ArrayAccess: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 f387_0_length_NULL f387_0_length_NULL f387_0_length_NULL: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 f314_0_appendNewList_LE f314_0_appendNewList_LE f314_0_appendNewList_LE: 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 f165_0_appendNewList_LE f165_0_appendNewList_LE f165_0_appendNewList_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 __init __init __init: 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 3 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/3

Here we consider the SCC { f165_0_appendNewList_LE }.

### 2.1.1 Transition Removal

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

 f165_0_appendNewList_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 2/3

Here we consider the SCC { f314_0_appendNewList_LE }.

### 2.2.1 Transition Removal

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

 f314_0_appendNewList_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 3/3

Here we consider the SCC { f387_0_length_NULL }.

### 2.3.1 Transition Removal

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

 f387_0_length_NULL: x1

### 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)