# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f626_0_main_GE, f60_0_power_GT, f753_0_main_InvokeMethod, f168_0_odd_NE, f116_0_even_NE, f60_0_power_GT', f1_0_main_ConstantStackPush, f675_0_main_InvokeMethod, f712_0_main_ArrayAccess, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_ConstantStackPush 1 f626_0_main_GE: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ _arg2 = _arg3P ∧ 0 = _arg2P ∧ 0 ≤ _arg1P − 1 ∧ 0 ≤ _arg1 − 1 ∧ −1 ≤ _arg2 − 1 ∧ _arg1P ≤ _arg1 f626_0_main_GE 2 f675_0_main_InvokeMethod: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x1 = _x4 ∧ x2 = _x5 ∧ x3 = _x6 ∧ x4 = _x7 ∧ _x1 ≤ _x2 − 1 ∧ 0 ≤ _x2 − 1 ∧ −1 ≤ _x9 − 1 ∧ _x4 ≤ _x ∧ 0 ≤ _x − 1 ∧ 0 ≤ _x4 − 1 ∧ _x1 = _x5 ∧ _x2 = _x7 f626_0_main_GE 3 f60_0_power_GT: x1 = _x10 ∧ x2 = _x11 ∧ x3 = _x12 ∧ x4 = _x13 ∧ x1 = _x14 ∧ x2 = _x15 ∧ x3 = _x16 ∧ x4 = _x17 ∧ _x12 = _x14 ∧ 0 ≤ _x10 − 1 ∧ −1 ≤ _x15 − 1 ∧ 0 ≤ _x12 − 1 ∧ _x11 ≤ _x12 − 1 f626_0_main_GE 4 f675_0_main_InvokeMethod: x1 = _x18 ∧ x2 = _x19 ∧ x3 = _x20 ∧ x4 = _x22 ∧ x1 = _x23 ∧ x2 = _x24 ∧ x3 = _x25 ∧ x4 = _x26 ∧ _x20 = _x26 ∧ 1 = _x25 ∧ _x19 = _x24 ∧ 0 ≤ _x23 − 1 ∧ 0 ≤ _x18 − 1 ∧ _x23 ≤ _x18 ∧ 0 ≤ _x20 − 1 ∧ _x19 ≤ _x20 − 1 f626_0_main_GE 5 f675_0_main_InvokeMethod: x1 = _x27 ∧ x2 = _x28 ∧ x3 = _x29 ∧ x4 = _x30 ∧ x1 = _x31 ∧ x2 = _x32 ∧ x3 = _x33 ∧ x4 = _x34 ∧ _x28 ≤ _x29 − 1 ∧ 0 ≤ _x29 − 1 ∧ −1 ≤ _x35 − 1 ∧ _x31 ≤ _x27 ∧ 0 ≤ _x27 − 1 ∧ 0 ≤ _x31 − 1 ∧ _x28 = _x32 ∧ _x29 = _x33 ∧ _x29 = _x34 f675_0_main_InvokeMethod 6 f712_0_main_ArrayAccess: x1 = _x37 ∧ x2 = _x38 ∧ x3 = _x39 ∧ x4 = _x40 ∧ x1 = _x41 ∧ x2 = _x42 ∧ x3 = _x43 ∧ x4 = _x44 ∧ _x40 = _x43 ∧ _x38 = _x42 ∧ 0 ≤ _x41 − 1 ∧ 0 ≤ _x37 − 1 ∧ _x41 ≤ _x37 ∧ 0 ≤ _x39 − 1 ∧ 0 ≤ _x40 − 1 f712_0_main_ArrayAccess 7 f60_0_power_GT: x1 = _x46 ∧ x2 = _x47 ∧ x3 = _x48 ∧ x4 = _x49 ∧ x1 = _x50 ∧ x2 = _x51 ∧ x3 = _x52 ∧ x4 = _x53 ∧ _x48 = _x50 ∧ 0 ≤ _x46 − 1 ∧ _x47 ≤ _x48 − 1 ∧ −1 ≤ _x51 − 1 ∧ 0 ≤ _x48 − 1 f712_0_main_ArrayAccess 8 f753_0_main_InvokeMethod: x1 = _x54 ∧ x2 = _x55 ∧ x3 = _x56 ∧ x4 = _x57 ∧ x1 = _x58 ∧ x2 = _x59 ∧ x3 = _x60 ∧ x4 = _x61 ∧ 0 ≤ _x56 − 1 ∧ −1 ≤ _x62 − 1 ∧ _x55 ≤ _x56 − 1 ∧ _x58 ≤ _x54 ∧ 0 ≤ _x54 − 1 ∧ 0 ≤ _x58 − 1 ∧ _x55 = _x59 ∧ _x56 = _x61 f712_0_main_ArrayAccess 9 f753_0_main_InvokeMethod: x1 = _x63 ∧ x2 = _x64 ∧ x3 = _x65 ∧ x4 = _x66 ∧ x1 = _x67 ∧ x2 = _x68 ∧ x3 = _x70 ∧ x4 = _x71 ∧ _x65 = _x71 ∧ 1 = _x70 ∧ _x64 = _x68 ∧ 0 ≤ _x67 − 1 ∧ 0 ≤ _x63 − 1 ∧ _x67 ≤ _x63 ∧ 0 ≤ _x65 − 1 ∧ _x64 ≤ _x65 − 1 f712_0_main_ArrayAccess 10 f753_0_main_InvokeMethod: x1 = _x72 ∧ x2 = _x73 ∧ x3 = _x74 ∧ x4 = _x75 ∧ x1 = _x76 ∧ x2 = _x77 ∧ x3 = _x78 ∧ x4 = _x79 ∧ 0 ≤ _x74 − 1 ∧ −1 ≤ _x80 − 1 ∧ _x73 ≤ _x74 − 1 ∧ _x76 ≤ _x72 ∧ 0 ≤ _x72 − 1 ∧ 0 ≤ _x76 − 1 ∧ _x73 = _x77 ∧ _x74 = _x78 ∧ _x74 = _x79 f753_0_main_InvokeMethod 11 f626_0_main_GE: x1 = _x81 ∧ x2 = _x82 ∧ x3 = _x83 ∧ x4 = _x84 ∧ x1 = _x85 ∧ x2 = _x86 ∧ x3 = _x87 ∧ x4 = _x88 ∧ _x84 = _x87 ∧ _x82 + 1 = _x86 ∧ 1 = _x83 ∧ 0 ≤ _x85 − 1 ∧ 0 ≤ _x81 − 1 ∧ 0 ≤ _x84 − 1 ∧ _x85 ≤ _x81 f753_0_main_InvokeMethod 12 f626_0_main_GE: x1 = _x89 ∧ x2 = _x90 ∧ x3 = _x91 ∧ x4 = _x92 ∧ x1 = _x93 ∧ x2 = _x94 ∧ x3 = _x95 ∧ x4 = _x96 ∧ _x92 = _x95 ∧ _x90 + 1 = _x94 ∧ 0 ≤ _x93 − 1 ∧ 0 ≤ _x89 − 1 ∧ _x93 ≤ _x89 ∧ 0 ≤ _x92 − 1 ∧ 0 ≤ _x91 − 1 f60_0_power_GT 13 f60_0_power_GT': x1 = _x97 ∧ x2 = _x98 ∧ x3 = _x99 ∧ x4 = _x100 ∧ x1 = _x101 ∧ x2 = _x102 ∧ x3 = _x103 ∧ x4 = _x104 ∧ 0 ≤ _x97 − 1 ∧ 1 ≤ _x98 − 1 ∧ 1 ≤ _x97⋅_x97 ∧ 0 ≤ _x105 − 1 ∧ _x105 ≤ _x98 − 1 ∧ _x97 = _x101 ∧ _x98 = _x102 f60_0_power_GT' 14 f60_0_power_GT: x1 = _x106 ∧ x2 = _x107 ∧ x3 = _x108 ∧ x4 = _x109 ∧ x1 = _x110 ∧ x2 = _x111 ∧ x3 = _x112 ∧ x4 = _x113 ∧ _x106⋅_x106 = _x110 ∧ 0 ≤ _x107 − 2⋅_x111 ∧ _x107 − 2⋅_x111 ≤ 1 ∧ 0 ≤ _x111 − 1 ∧ _x111 ≤ _x107 − 1 ∧ 1 ≤ _x106⋅_x106 ∧ 1 ≤ _x107 − 1 ∧ 0 ≤ _x106 − 1 f675_0_main_InvokeMethod 15 f116_0_even_NE: x1 = _x114 ∧ x2 = _x115 ∧ x3 = _x116 ∧ x4 = _x117 ∧ x1 = _x118 ∧ x2 = _x119 ∧ x3 = _x120 ∧ x4 = _x121 ∧ _x116 = _x118 ∧ 0 ≤ _x114 − 1 ∧ 0 ≤ _x117 − 1 ∧ 0 ≤ _x116 − 1 f753_0_main_InvokeMethod 16 f168_0_odd_NE: x1 = _x122 ∧ x2 = _x123 ∧ x3 = _x124 ∧ x4 = _x125 ∧ x1 = _x126 ∧ x2 = _x127 ∧ x3 = _x128 ∧ x4 = _x129 ∧ _x124 = _x126 ∧ 0 ≤ _x122 − 1 ∧ 0 ≤ _x125 − 1 ∧ 0 ≤ _x124 − 1 f116_0_even_NE 17 f168_0_odd_NE: x1 = _x130 ∧ x2 = _x131 ∧ x3 = _x132 ∧ x4 = _x133 ∧ x1 = _x134 ∧ x2 = _x135 ∧ x3 = _x136 ∧ x4 = _x137 ∧ _x130 − 1 = _x134 ∧ _x130 − 1 ≤ _x130 − 1 ∧ 1 ≤ _x130 − 1 f168_0_odd_NE 18 f116_0_even_NE: x1 = _x138 ∧ x2 = _x139 ∧ x3 = _x140 ∧ x4 = _x141 ∧ x1 = _x142 ∧ x2 = _x143 ∧ x3 = _x144 ∧ x4 = _x145 ∧ _x138 − 1 = _x142 ∧ _x138 − 1 ≤ _x138 − 1 ∧ 1 ≤ _x138 − 1 __init 19 f1_0_main_ConstantStackPush: x1 = _x146 ∧ x2 = _x147 ∧ x3 = _x148 ∧ x4 = _x149 ∧ x1 = _x150 ∧ x2 = _x151 ∧ x3 = _x152 ∧ x4 = _x153 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f626_0_main_GE f626_0_main_GE f626_0_main_GE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f60_0_power_GT f60_0_power_GT f60_0_power_GT: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f753_0_main_InvokeMethod f753_0_main_InvokeMethod f753_0_main_InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f168_0_odd_NE f168_0_odd_NE f168_0_odd_NE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f116_0_even_NE f116_0_even_NE f116_0_even_NE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f60_0_power_GT' f60_0_power_GT' f60_0_power_GT': x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f1_0_main_ConstantStackPush f1_0_main_ConstantStackPush f1_0_main_ConstantStackPush: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f675_0_main_InvokeMethod f675_0_main_InvokeMethod f675_0_main_InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f712_0_main_ArrayAccess f712_0_main_ArrayAccess f712_0_main_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 { f626_0_main_GE, f753_0_main_InvokeMethod, f675_0_main_InvokeMethod, f712_0_main_ArrayAccess }.

### 2.1.1 Transition Removal

We remove transitions 8, 10, 9 using the following ranking functions, which are bounded by 0.

 f626_0_main_GE: −1 − 2⋅x2 + 2⋅x3 f675_0_main_InvokeMethod: −1 − 2⋅x2 + 2⋅x4 f753_0_main_InvokeMethod: −3 − 2⋅x2 + 2⋅x4 f712_0_main_ArrayAccess: −1 − 2⋅x2 + 2⋅x3

### 2.1.2 Transition Removal

We remove transitions 2, 11, 6, 5, 4, 12 using the following ranking functions, which are bounded by 0.

 f626_0_main_GE: 2 f675_0_main_InvokeMethod: 1 f753_0_main_InvokeMethod: 3 f712_0_main_ArrayAccess: 0

### 2.1.3 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 { f168_0_odd_NE, f116_0_even_NE }.

### 2.2.1 Transition Removal

We remove transitions 17, 18 using the following ranking functions, which are bounded by 0.

 f116_0_even_NE: x1 f168_0_odd_NE: x1

### 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 { f60_0_power_GT, f60_0_power_GT' }.

### 2.3.1 Transition Removal

We remove transitions 13, 14 using the following ranking functions, which are bounded by 0.

 f60_0_power_GT: 2⋅x2 f60_0_power_GT': x2 + 1

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