# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f704_0_nth_LE, f303_0_createIntList_Return, f1_0_main_Load, f754_0_main_LE, f964_0_nth_LE, f673_0_createIntList_LE, f517_0_random_ArrayAccess, __init
• Transitions: (pre-variables and post-variables)  f303_0_createIntList_Return 1 f517_0_random_ArrayAccess: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ −1 ≤ _arg1P − 1 ∧ −1 ≤ _arg1 − 1 ∧ _arg1P ≤ _arg1 f1_0_main_Load 2 f517_0_random_ArrayAccess: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x1 = _x4 ∧ x2 = _x5 ∧ x3 = _x6 ∧ x4 = _x8 ∧ −1 ≤ _x4 − 1 ∧ 0 ≤ _x − 1 f517_0_random_ArrayAccess 3 f704_0_nth_LE: x1 = _x9 ∧ x2 = _x10 ∧ x3 = _x11 ∧ x4 = _x12 ∧ x1 = _x13 ∧ x2 = _x14 ∧ x3 = _x15 ∧ x4 = _x16 ∧ 0 ≤ _x17 − 1 ∧ −1 ≤ _x15 − 1 ∧ _x13 ≤ _x9 ∧ _x14 ≤ _x9 ∧ −1 ≤ _x9 − 1 ∧ −1 ≤ _x13 − 1 ∧ −1 ≤ _x14 − 1 ∧ _x17 + 1 = _x16 f704_0_nth_LE 4 f754_0_main_LE: x1 = _x18 ∧ x2 = _x19 ∧ x3 = _x20 ∧ x4 = _x21 ∧ x1 = _x22 ∧ x2 = _x23 ∧ x3 = _x24 ∧ x4 = _x25 ∧ _x21 = _x24 ∧ _x23 + 2 ≤ _x19 ∧ −1 ≤ _x22 − 1 ∧ 0 ≤ _x19 − 1 ∧ −1 ≤ _x18 − 1 ∧ _x20 ≤ 1 ∧ _x22 ≤ _x18 f704_0_nth_LE 5 f704_0_nth_LE: x1 = _x26 ∧ x2 = _x27 ∧ x3 = _x28 ∧ x4 = _x29 ∧ x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x4 = _x33 ∧ _x29 = _x33 ∧ _x28 − 1 = _x32 ∧ −1 ≤ _x31 − 1 ∧ −1 ≤ _x30 − 1 ∧ 0 ≤ _x27 − 1 ∧ −1 ≤ _x26 − 1 ∧ _x31 + 1 ≤ _x27 ∧ 1 ≤ _x28 − 1 ∧ _x30 ≤ _x26 f754_0_main_LE 6 f964_0_nth_LE: x1 = _x34 ∧ x2 = _x35 ∧ x3 = _x36 ∧ x4 = _x37 ∧ x1 = _x38 ∧ x2 = _x39 ∧ x3 = _x40 ∧ x4 = _x41 ∧ _x36 + 1 = _x41 ∧ −1 ≤ _x39 − 1 ∧ −1 ≤ _x38 − 1 ∧ 0 ≤ _x34 − 1 ∧ _x39 + 1 ≤ _x34 ∧ _x38 + 1 ≤ _x34 ∧ 0 ≤ _x35 − 1 ∧ 0 ≤ _x36 − 1 ∧ −1 ≤ _x40 − 1 f964_0_nth_LE 7 f754_0_main_LE: x1 = _x42 ∧ x2 = _x43 ∧ x3 = _x44 ∧ x4 = _x45 ∧ x1 = _x46 ∧ x2 = _x47 ∧ x3 = _x48 ∧ x4 = _x49 ∧ _x45 = _x48 ∧ _x47 + 2 ≤ _x43 ∧ −1 ≤ _x46 − 1 ∧ 0 ≤ _x43 − 1 ∧ −1 ≤ _x42 − 1 ∧ _x44 ≤ 1 ∧ _x46 ≤ _x42 f964_0_nth_LE 8 f964_0_nth_LE: x1 = _x50 ∧ x2 = _x51 ∧ x3 = _x52 ∧ x4 = _x53 ∧ x1 = _x54 ∧ x2 = _x55 ∧ x3 = _x56 ∧ x4 = _x57 ∧ _x53 = _x57 ∧ _x52 − 1 = _x56 ∧ −1 ≤ _x55 − 1 ∧ −1 ≤ _x54 − 1 ∧ 0 ≤ _x51 − 1 ∧ −1 ≤ _x50 − 1 ∧ _x55 + 1 ≤ _x51 ∧ 1 ≤ _x52 − 1 ∧ _x54 ≤ _x50 f964_0_nth_LE 9 f754_0_main_LE: x1 = _x58 ∧ x2 = _x59 ∧ x3 = _x60 ∧ x4 = _x61 ∧ x1 = _x62 ∧ x2 = _x63 ∧ x3 = _x64 ∧ x4 = _x65 ∧ _x61 = _x64 ∧ 0 = _x63 ∧ −1 ≤ _x62 − 1 ∧ −1 ≤ _x59 − 1 ∧ −1 ≤ _x58 − 1 ∧ _x60 ≤ 1 ∧ _x62 ≤ _x58 f964_0_nth_LE 10 f754_0_main_LE: x1 = _x66 ∧ x2 = _x67 ∧ x3 = _x68 ∧ x4 = _x69 ∧ x1 = _x70 ∧ x2 = _x71 ∧ x3 = _x72 ∧ x4 = _x73 ∧ _x69 = _x72 ∧ 0 = _x71 ∧ −1 ≤ _x70 − 1 ∧ −1 ≤ _x67 − 1 ∧ −1 ≤ _x66 − 1 ∧ 1 ≤ _x68 − 1 ∧ _x70 ≤ _x66 f1_0_main_Load 11 f673_0_createIntList_LE: x1 = _x74 ∧ x2 = _x75 ∧ x3 = _x76 ∧ x4 = _x77 ∧ x1 = _x78 ∧ x2 = _x79 ∧ x3 = _x80 ∧ x4 = _x81 ∧ 1 = _x79 ∧ 0 ≤ _x74 − 1 ∧ −1 ≤ _x78 − 1 ∧ −1 ≤ _x75 − 1 f673_0_createIntList_LE 12 f673_0_createIntList_LE: x1 = _x82 ∧ x2 = _x83 ∧ x3 = _x84 ∧ x4 = _x85 ∧ x1 = _x86 ∧ x2 = _x87 ∧ x3 = _x88 ∧ x4 = _x89 ∧ _x83 + 1 = _x87 ∧ _x82 − 1 = _x86 ∧ 0 ≤ _x83 − 1 ∧ 0 ≤ _x82 − 1 __init 13 f1_0_main_Load: x1 = _x90 ∧ x2 = _x91 ∧ x3 = _x92 ∧ x4 = _x93 ∧ x1 = _x94 ∧ x2 = _x95 ∧ x3 = _x96 ∧ x4 = _x97 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f704_0_nth_LE f704_0_nth_LE f704_0_nth_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f303_0_createIntList_Return f303_0_createIntList_Return f303_0_createIntList_Return: 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 f754_0_main_LE f754_0_main_LE f754_0_main_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f964_0_nth_LE f964_0_nth_LE f964_0_nth_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f673_0_createIntList_LE f673_0_createIntList_LE f673_0_createIntList_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f517_0_random_ArrayAccess f517_0_random_ArrayAccess f517_0_random_ArrayAccess: 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 3 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/3

Here we consider the SCC { f673_0_createIntList_LE }.

### 2.1.1 Transition Removal

We remove transition 12 using the following ranking functions, which are bounded by 0.

 f673_0_createIntList_LE: x1

### 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 { f704_0_nth_LE }.

### 2.2.1 Transition Removal

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

 f704_0_nth_LE: x3

### 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 { f754_0_main_LE, f964_0_nth_LE }.

### 2.3.1 Transition Removal

We remove transitions 6, 10, 9, 7 using the following ranking functions, which are bounded by 0.

 f754_0_main_LE: −1 + 2⋅x1 f964_0_nth_LE: 2⋅x1

### 2.3.2 Transition Removal

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

 f964_0_nth_LE: x2

### 2.3.3 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 % (11 real / 0 unknown / 0 assumptions / 11 total proof steps)