# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f361_0_fractale_LE, f361_0_fractale_LE', f425_0_fractale_InvokeMethod, f1_0_main_Load, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f361_0_fractale_LE: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x5 = _arg5 ∧ x6 = _arg6 ∧ x7 = _arg7 ∧ x8 = _arg8 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ x5 = _arg5P ∧ x6 = _arg6P ∧ x7 = _arg7P ∧ x8 = _arg8P ∧ 200 = _arg7P ∧ 200 = _arg6P ∧ 0 = _arg5P ∧ 0 = _arg4P ∧ _arg2 = _arg3P ∧ 201 ≤ _arg2P − 1 ∧ 1 ≤ _arg1P − 1 ∧ 0 ≤ _arg1 − 1 ∧ _arg2P − 201 ≤ _arg1 ∧ −1 ≤ _arg2 − 1 ∧ _arg1P − 1 ≤ _arg1 f361_0_fractale_LE 2 f361_0_fractale_LE': x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x7 = _x6 ∧ x8 = _x7 ∧ x1 = _x8 ∧ x2 = _x9 ∧ x3 = _x10 ∧ x4 = _x13 ∧ x5 = _x14 ∧ x6 = _x15 ∧ x7 = _x16 ∧ x8 = _x17 ∧ 0 ≤ _x2 − 1 ∧ _x2 − 1 ≤ _x2 − 1 ∧ _x18 ≤ _x ∧ 0 ≤ _x − 1 ∧ 0 ≤ _x1 − 1 ∧ 0 ≤ _x18 − 1 ∧ 0 ≤ _x19 − 1 ∧ _x3 + 2 ≤ _x ∧ _x4 + 2 ≤ _x ∧ _x6 + 2 ≤ _x1 ∧ _x5 + 2 ≤ _x1 ∧ _x = _x8 ∧ _x1 = _x9 ∧ _x2 = _x10 ∧ _x3 = _x13 ∧ _x4 = _x14 ∧ _x5 = _x15 ∧ _x6 = _x16 f361_0_fractale_LE' 3 f361_0_fractale_LE: x1 = _x20 ∧ x2 = _x23 ∧ x3 = _x24 ∧ x4 = _x25 ∧ x5 = _x26 ∧ x6 = _x27 ∧ x7 = _x28 ∧ x8 = _x29 ∧ x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x34 ∧ x4 = _x35 ∧ x5 = _x36 ∧ x6 = _x37 ∧ x7 = _x38 ∧ x8 = _x39 ∧ _x26 = _x36 ∧ _x25 = _x35 ∧ _x24 − 1 = _x34 ∧ 0 ≤ _x27 + _x28 + _x26 − _x25 − 2⋅_x38 ∧ _x27 + _x28 + _x26 − _x25 − 2⋅_x38 ≤ 1 ∧ _x25 + _x26 + _x27 − _x28 − 2⋅_x37 ≤ 1 ∧ 0 ≤ _x25 + _x26 + _x27 − _x28 − 2⋅_x37 ∧ _x28 + 2 ≤ _x23 ∧ _x27 + 2 ≤ _x23 ∧ _x26 + 2 ≤ _x20 ∧ _x25 + 2 ≤ _x20 ∧ 0 ≤ _x31 − 1 ∧ 0 ≤ _x30 − 1 ∧ 0 ≤ _x23 − 1 ∧ 0 ≤ _x20 − 1 ∧ _x30 ≤ _x20 ∧ _x24 − 1 ≤ _x24 − 1 ∧ 0 ≤ _x24 − 1 f361_0_fractale_LE 4 f361_0_fractale_LE': x1 = _x40 ∧ x2 = _x41 ∧ x3 = _x42 ∧ x4 = _x43 ∧ x5 = _x44 ∧ x6 = _x45 ∧ x7 = _x46 ∧ x8 = _x47 ∧ x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x4 = _x51 ∧ x5 = _x52 ∧ x6 = _x53 ∧ x7 = _x54 ∧ x8 = _x55 ∧ _x56 ≤ _x41 ∧ 0 ≤ _x40 − 1 ∧ 0 ≤ _x41 − 1 ∧ 0 ≤ _x57 − 1 ∧ 0 ≤ _x56 − 1 ∧ _x43 + 2 ≤ _x40 ∧ _x44 + 2 ≤ _x40 ∧ _x46 + 2 ≤ _x41 ∧ _x45 + 2 ≤ _x41 ∧ 1 = _x42 ∧ _x40 = _x48 ∧ _x41 = _x49 ∧ 1 = _x50 ∧ _x43 = _x51 ∧ _x44 = _x52 ∧ _x45 = _x53 ∧ _x46 = _x54 f361_0_fractale_LE' 5 f425_0_fractale_InvokeMethod: x1 = _x58 ∧ x2 = _x59 ∧ x3 = _x60 ∧ x4 = _x61 ∧ x5 = _x62 ∧ x6 = _x63 ∧ x7 = _x64 ∧ x8 = _x65 ∧ x1 = _x66 ∧ x2 = _x67 ∧ x3 = _x68 ∧ x4 = _x69 ∧ x5 = _x70 ∧ x6 = _x71 ∧ x7 = _x72 ∧ x8 = _x73 ∧ _x64 = _x73 ∧ _x63 = _x72 ∧ 0 = _x67 ∧ 1 = _x66 ∧ 1 = _x60 ∧ 0 ≤ _x61 + _x62 + _x63 − _x64 − 2⋅_x70 ∧ _x61 + _x62 + _x63 − _x64 − 2⋅_x70 ≤ 1 ∧ _x63 + _x64 + _x62 − _x61 − 2⋅_x71 ≤ 1 ∧ 0 ≤ _x63 + _x64 + _x62 − _x61 − 2⋅_x71 ∧ _x64 + 2 ≤ _x59 ∧ _x63 + 2 ≤ _x59 ∧ _x62 + 2 ≤ _x58 ∧ _x61 + 2 ≤ _x58 ∧ 0 ≤ _x69 − 1 ∧ 0 ≤ _x68 − 1 ∧ 0 ≤ _x59 − 1 ∧ 0 ≤ _x58 − 1 ∧ _x69 ≤ _x59 f361_0_fractale_LE 6 f361_0_fractale_LE': x1 = _x74 ∧ x2 = _x75 ∧ x3 = _x76 ∧ x4 = _x77 ∧ x5 = _x78 ∧ x6 = _x79 ∧ x7 = _x80 ∧ x8 = _x81 ∧ x1 = _x82 ∧ x2 = _x83 ∧ x3 = _x84 ∧ x4 = _x85 ∧ x5 = _x86 ∧ x6 = _x87 ∧ x7 = _x88 ∧ x8 = _x89 ∧ 0 ≤ _x76 − 1 ∧ _x76 − 1 ≤ _x76 − 1 ∧ _x90 ≤ _x75 ∧ 0 ≤ _x74 − 1 ∧ 0 ≤ _x75 − 1 ∧ 0 ≤ _x91 − 1 ∧ 0 ≤ _x90 − 1 ∧ _x77 + 2 ≤ _x74 ∧ _x78 + 2 ≤ _x74 ∧ _x80 + 2 ≤ _x75 ∧ _x79 + 2 ≤ _x75 ∧ _x74 = _x82 ∧ _x75 = _x83 ∧ _x76 = _x84 ∧ _x77 = _x85 ∧ _x78 = _x86 ∧ _x79 = _x87 ∧ _x80 = _x88 f361_0_fractale_LE' 7 f425_0_fractale_InvokeMethod: x1 = _x92 ∧ x2 = _x93 ∧ x3 = _x94 ∧ x4 = _x95 ∧ x5 = _x96 ∧ x6 = _x97 ∧ x7 = _x98 ∧ x8 = _x99 ∧ x1 = _x100 ∧ x2 = _x101 ∧ x3 = _x102 ∧ x4 = _x103 ∧ x5 = _x104 ∧ x6 = _x105 ∧ x7 = _x106 ∧ x8 = _x107 ∧ _x98 = _x107 ∧ _x97 = _x106 ∧ _x94 − 1 = _x101 ∧ _x94 = _x100 ∧ 0 ≤ _x97 + _x98 + _x96 − _x95 − 2⋅_x105 ∧ _x97 + _x98 + _x96 − _x95 − 2⋅_x105 ≤ 1 ∧ _x95 + _x96 + _x97 − _x98 − 2⋅_x104 ≤ 1 ∧ 0 ≤ _x95 + _x96 + _x97 − _x98 − 2⋅_x104 ∧ _x98 + 2 ≤ _x93 ∧ _x97 + 2 ≤ _x93 ∧ _x96 + 2 ≤ _x92 ∧ _x95 + 2 ≤ _x92 ∧ 0 ≤ _x103 − 1 ∧ 0 ≤ _x102 − 1 ∧ 0 ≤ _x93 − 1 ∧ 0 ≤ _x92 − 1 ∧ _x103 ≤ _x93 ∧ _x94 − 1 ≤ _x94 − 1 ∧ 0 ≤ _x94 − 1 f425_0_fractale_InvokeMethod 8 f361_0_fractale_LE: x1 = _x108 ∧ x2 = _x109 ∧ x3 = _x110 ∧ x4 = _x111 ∧ x5 = _x112 ∧ x6 = _x113 ∧ x7 = _x114 ∧ x8 = _x115 ∧ x1 = _x116 ∧ x2 = _x117 ∧ x3 = _x118 ∧ x4 = _x119 ∧ x5 = _x120 ∧ x6 = _x121 ∧ x7 = _x122 ∧ x8 = _x123 ∧ _x115 = _x122 ∧ _x114 = _x121 ∧ _x113 = _x120 ∧ _x112 = _x119 ∧ _x109 = _x118 ∧ _x114 + 2 ≤ _x111 ∧ _x115 + 2 ≤ _x111 ∧ _x113 + 2 ≤ _x110 ∧ _x112 + 2 ≤ _x110 ∧ 0 ≤ _x117 − 1 ∧ 0 ≤ _x116 − 1 ∧ 0 ≤ _x111 − 1 ∧ 0 ≤ _x110 − 1 ∧ _x117 ≤ _x111 ∧ _x116 ≤ _x110 ∧ 0 ≤ _x108 − 1 ∧ _x109 ≤ _x108 − 1 __init 9 f1_0_main_Load: x1 = _x124 ∧ x2 = _x125 ∧ x3 = _x126 ∧ x4 = _x127 ∧ x5 = _x128 ∧ x6 = _x129 ∧ x7 = _x130 ∧ x8 = _x131 ∧ x1 = _x132 ∧ x2 = _x133 ∧ x3 = _x134 ∧ x4 = _x135 ∧ x5 = _x136 ∧ x6 = _x137 ∧ x7 = _x138 ∧ x8 = _x139 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f361_0_fractale_LE f361_0_fractale_LE f361_0_fractale_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 f361_0_fractale_LE' f361_0_fractale_LE' f361_0_fractale_LE': x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 f425_0_fractale_InvokeMethod f425_0_fractale_InvokeMethod f425_0_fractale_InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
and for every transition t, a duplicate t is considered.

### 2 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/1

Here we consider the SCC { f361_0_fractale_LE, f361_0_fractale_LE', f425_0_fractale_InvokeMethod }.

### 2.1.1 Transition Removal

We remove transitions 2, 3, 4, 8, 7, 5, 6 using the following ranking functions, which are bounded by 0.

 f361_0_fractale_LE: 2 + 3⋅x3 f361_0_fractale_LE': 1 + 3⋅x3 f425_0_fractale_InvokeMethod: 3⋅x1

### 2.1.2 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 % (5 real / 0 unknown / 0 assumptions / 5 total proof steps)