# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f355_0_reverse_NULL, f385_0_reverse_FieldAccess, f1_0_main_Load, f172_0_createList_LE, f370_0_reverse_FieldAccess, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f172_0_createList_LE: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ −1 ≤ _x4 − 1 ∧ 0 ≤ _arg2 − 1 ∧ _arg1P − 1 ≤ _arg1 ∧ _arg2P − 1 ≤ _arg1 ∧ 0 ≤ _arg1 − 1 ∧ 1 ≤ _arg1P − 1 ∧ 1 ≤ _arg2P − 1 ∧ _x4 − 2 = _arg3P f172_0_createList_LE 2 f172_0_createList_LE: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x1 = _x5 ∧ x2 = _x6 ∧ x3 = _x7 ∧ x4 = _x8 ∧ _x2 − 1 = _x7 ∧ 2 ≤ _x6 − 1 ∧ 0 ≤ _x5 − 1 ∧ 0 ≤ _x1 − 1 ∧ 0 ≤ _x − 1 ∧ _x6 − 2 ≤ _x1 ∧ 0 ≤ _x2 − 1 ∧ _x5 ≤ _x f172_0_createList_LE 3 f355_0_reverse_NULL: x1 = _x9 ∧ x2 = _x10 ∧ x3 = _x11 ∧ x4 = _x12 ∧ x1 = _x13 ∧ x2 = _x14 ∧ x3 = _x15 ∧ x4 = _x16 ∧ 1 ≤ _x14 − 1 ∧ 1 ≤ _x13 − 1 ∧ 1 ≤ _x10 − 1 ∧ 1 ≤ _x9 − 1 ∧ _x14 ≤ _x10 ∧ _x14 ≤ _x9 ∧ _x13 ≤ _x10 ∧ _x11 ≤ 0 ∧ _x13 ≤ _x9 f172_0_createList_LE 4 f355_0_reverse_NULL: x1 = _x17 ∧ x2 = _x18 ∧ x3 = _x19 ∧ x4 = _x20 ∧ x1 = _x21 ∧ x2 = _x22 ∧ x3 = _x23 ∧ x4 = _x24 ∧ −1 ≤ _x22 − 1 ∧ 1 ≤ _x21 − 1 ∧ 0 ≤ _x18 − 1 ∧ 1 ≤ _x17 − 1 ∧ _x21 − 1 ≤ _x18 ∧ _x19 ≤ 0 ∧ _x21 ≤ _x17 f172_0_createList_LE 5 f355_0_reverse_NULL: x1 = _x25 ∧ x2 = _x26 ∧ x3 = _x27 ∧ x4 = _x28 ∧ x1 = _x29 ∧ x2 = _x30 ∧ x3 = _x31 ∧ x4 = _x32 ∧ 0 ≤ _x30 − 1 ∧ 1 ≤ _x29 − 1 ∧ 0 ≤ _x26 − 1 ∧ 1 ≤ _x25 − 1 ∧ _x29 − 1 ≤ _x26 ∧ _x27 ≤ 0 ∧ _x29 ≤ _x25 f172_0_createList_LE 6 f370_0_reverse_FieldAccess: x1 = _x33 ∧ x2 = _x34 ∧ x3 = _x35 ∧ x4 = _x36 ∧ x1 = _x37 ∧ x2 = _x38 ∧ x3 = _x39 ∧ x4 = _x40 ∧ 0 ≤ _x38 − 1 ∧ 1 ≤ _x37 − 1 ∧ 0 ≤ _x34 − 1 ∧ 1 ≤ _x33 − 1 ∧ _x37 − 1 ≤ _x34 ∧ _x35 ≤ 0 ∧ _x37 ≤ _x33 f355_0_reverse_NULL 7 f385_0_reverse_FieldAccess: x1 = _x41 ∧ x2 = _x42 ∧ x3 = _x43 ∧ x4 = _x44 ∧ x1 = _x45 ∧ x2 = _x46 ∧ x3 = _x47 ∧ x4 = _x48 ∧ −1 ≤ _x48 − 1 ∧ 0 ≤ _x47 − 1 ∧ 0 ≤ _x46 − 1 ∧ −1 ≤ _x45 − 1 ∧ 0 ≤ _x42 − 1 ∧ 0 ≤ _x41 − 1 ∧ _x48 + 1 ≤ _x42 ∧ _x47 ≤ _x41 ∧ _x46 ≤ _x42 ∧ _x45 + 1 ≤ _x42 f385_0_reverse_FieldAccess 8 f355_0_reverse_NULL: x1 = _x49 ∧ x2 = _x50 ∧ x3 = _x51 ∧ x4 = _x52 ∧ x1 = _x53 ∧ x2 = _x54 ∧ x3 = _x55 ∧ x4 = _x56 ∧ −1 ≤ _x54 − 1 ∧ 2 ≤ _x53 − 1 ∧ −1 ≤ _x52 − 1 ∧ 0 ≤ _x51 − 1 ∧ 0 ≤ _x50 − 1 ∧ −1 ≤ _x49 − 1 ∧ _x54 ≤ _x52 ∧ _x54 + 1 ≤ _x50 ∧ _x54 ≤ _x49 f385_0_reverse_FieldAccess 9 f355_0_reverse_NULL: x1 = _x57 ∧ x2 = _x58 ∧ x3 = _x59 ∧ x4 = _x60 ∧ x1 = _x61 ∧ x2 = _x62 ∧ x3 = _x63 ∧ x4 = _x64 ∧ −1 ≤ _x62 − 1 ∧ 2 ≤ _x61 − 1 ∧ −1 ≤ _x60 − 1 ∧ 0 ≤ _x59 − 1 ∧ 0 ≤ _x58 − 1 ∧ −1 ≤ _x57 − 1 ∧ _x62 ≤ _x60 ∧ _x62 + 1 ≤ _x59 ∧ _x62 + 1 ≤ _x58 ∧ _x62 ≤ _x57 ∧ _x61 − 3 ≤ _x60 ∧ _x61 − 2 ≤ _x59 ∧ _x61 − 2 ≤ _x58 ∧ _x61 − 3 ≤ _x57 f370_0_reverse_FieldAccess 10 f370_0_reverse_FieldAccess: x1 = _x67 ∧ x2 = _x68 ∧ x3 = _x69 ∧ x4 = _x70 ∧ x1 = _x71 ∧ x2 = _x72 ∧ x3 = _x75 ∧ x4 = _x76 ∧ _x69 = _x75 ∧ 0 ≤ _x72 − 1 ∧ 2 ≤ _x71 − 1 ∧ 2 ≤ _x68 − 1 ∧ 0 ≤ _x67 − 1 ∧ _x71 − 2 ≤ _x67 ∧ _x76 ≤ _x70 − 1 ∧ 0 ≤ _x70 − 1 ∧ 0 ≤ _x69 − 1 f370_0_reverse_FieldAccess 11 f385_0_reverse_FieldAccess: x1 = _x77 ∧ x2 = _x78 ∧ x3 = _x79 ∧ x4 = _x80 ∧ x1 = _x81 ∧ x2 = _x82 ∧ x3 = _x83 ∧ x4 = _x84 ∧ 0 ≤ _x85 − 1 ∧ −1 ≤ _x86 − 1 ∧ _x85 ≤ _x86 − 1 ∧ _x83 − 2 ≤ _x77 ∧ 0 ≤ _x77 − 1 ∧ 2 ≤ _x78 − 1 ∧ −1 ≤ _x81 − 1 ∧ 0 ≤ _x82 − 1 ∧ 2 ≤ _x83 − 1 ∧ −1 ≤ _x84 − 1 f370_0_reverse_FieldAccess 12 f355_0_reverse_NULL: x1 = _x87 ∧ x2 = _x88 ∧ x3 = _x89 ∧ x4 = _x90 ∧ x1 = _x91 ∧ x2 = _x92 ∧ x3 = _x93 ∧ x4 = _x94 ∧ −1 ≤ _x95 − 1 ∧ _x96 ≤ _x95 − 1 ∧ _x91 − 2 ≤ _x87 ∧ _x91 ≤ _x88 ∧ _x92 ≤ _x87 ∧ 0 ≤ _x87 − 1 ∧ 2 ≤ _x88 − 1 ∧ 2 ≤ _x91 − 1 ∧ 0 ≤ _x92 − 1 __init 13 f1_0_main_Load: x1 = _x97 ∧ x2 = _x98 ∧ x3 = _x99 ∧ x4 = _x100 ∧ x1 = _x101 ∧ x2 = _x102 ∧ x3 = _x103 ∧ x4 = _x104 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f355_0_reverse_NULL f355_0_reverse_NULL f355_0_reverse_NULL: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f385_0_reverse_FieldAccess f385_0_reverse_FieldAccess f385_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 f172_0_createList_LE f172_0_createList_LE f172_0_createList_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f370_0_reverse_FieldAccess f370_0_reverse_FieldAccess f370_0_reverse_FieldAccess: 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 { f172_0_createList_LE }.

### 2.1.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by 0.

 f172_0_createList_LE: 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 { f370_0_reverse_FieldAccess }.

### 2.2.1 Transition Removal

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

 f370_0_reverse_FieldAccess: x4

### 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 { f355_0_reverse_NULL, f385_0_reverse_FieldAccess }.

### 2.3.1 Transition Removal

We remove transitions 7, 9, 8 using the following ranking functions, which are bounded by 0.

 f355_0_reverse_NULL: −1 + 2⋅x2 f385_0_reverse_FieldAccess: 2⋅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)