# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f97_0_random_GT, f162_0_power_GT', f155_0_main_InvokeMethod, f1_0_main_Load, f213_0_power_NE, f213_0_power_NE', f162_0_power_GT, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f162_0_power_GT: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ 0 = _arg2P ∧ 0 = _arg1P ∧ 0 = _arg2 ∧ 0 ≤ _arg1 − 1 f1_0_main_Load 2 f97_0_random_GT: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x1 = _x4 ∧ x2 = _x5 ∧ x3 = _x6 ∧ x4 = _x7 ∧ 0 = _x5 ∧ 0 ≤ _x4 − 1 ∧ 0 ≤ _x − 1 ∧ 0 ≤ _x1 − 1 ∧ _x4 ≤ _x f1_0_main_Load 3 f97_0_random_GT: x1 = _x8 ∧ x2 = _x9 ∧ x3 = _x11 ∧ x4 = _x12 ∧ x1 = _x13 ∧ x2 = _x15 ∧ x3 = _x16 ∧ x4 = _x17 ∧ 0 ≤ _x13 − 1 ∧ 0 ≤ _x8 − 1 ∧ _x13 ≤ _x8 ∧ 0 ≤ _x9 − 1 ∧ −1 ≤ _x15 − 1 f97_0_random_GT 4 f162_0_power_GT: x1 = _x18 ∧ x2 = _x20 ∧ x3 = _x21 ∧ x4 = _x22 ∧ x1 = _x23 ∧ x2 = _x24 ∧ x3 = _x25 ∧ x4 = _x26 ∧ 0 = _x24 ∧ _x20 = _x23 ∧ 0 ≤ _x18 − 1 f97_0_random_GT 5 f155_0_main_InvokeMethod: x1 = _x27 ∧ x2 = _x28 ∧ x3 = _x29 ∧ x4 = _x30 ∧ x1 = _x31 ∧ x2 = _x32 ∧ x3 = _x33 ∧ x4 = _x34 ∧ _x31 ≤ _x27 ∧ 1 ≤ _x35 − 1 ∧ 0 ≤ _x27 − 1 ∧ 0 ≤ _x31 − 1 ∧ _x28 = _x32 ∧ 0 = _x33 f97_0_random_GT 6 f155_0_main_InvokeMethod: x1 = _x36 ∧ x2 = _x38 ∧ x3 = _x39 ∧ x4 = _x40 ∧ x1 = _x42 ∧ x2 = _x43 ∧ x3 = _x44 ∧ x4 = _x45 ∧ 1 ≤ _x46 − 1 ∧ −1 ≤ _x44 − 1 ∧ _x42 ≤ _x36 ∧ 0 ≤ _x36 − 1 ∧ 0 ≤ _x42 − 1 ∧ _x38 = _x43 f155_0_main_InvokeMethod 7 f162_0_power_GT: x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x4 = _x51 ∧ x1 = _x52 ∧ x2 = _x53 ∧ x3 = _x54 ∧ x4 = _x55 ∧ 0 ≤ _x48 − 1 ∧ 1 ≤ _x56 − 1 ∧ _x49 = _x52 ∧ _x50 = _x53 f162_0_power_GT 8 f162_0_power_GT': x1 = _x57 ∧ x2 = _x58 ∧ x3 = _x59 ∧ x4 = _x60 ∧ x1 = _x61 ∧ x2 = _x62 ∧ x3 = _x63 ∧ x4 = _x64 ∧ _x58 = _x62 ∧ _x57 = _x61 ∧ 1 ≤ _x58 − 1 ∧ _x57 ≤ 1 f162_0_power_GT' 9 f213_0_power_NE: x1 = _x65 ∧ x2 = _x66 ∧ x3 = _x67 ∧ x4 = _x68 ∧ x1 = _x69 ∧ x2 = _x70 ∧ x3 = _x71 ∧ x4 = _x72 ∧ 1 ≤ _x66 − 1 ∧ _x65 ≤ 1 ∧ _x66 − 2⋅_x73 ≤ 1 ∧ 0 ≤ _x66 − 2⋅_x73 ∧ _x65 = _x69 ∧ _x66 = _x70 ∧ _x66 − 2⋅_x73 = _x71 f162_0_power_GT 10 f162_0_power_GT': x1 = _x74 ∧ x2 = _x75 ∧ x3 = _x76 ∧ x4 = _x77 ∧ x1 = _x78 ∧ x2 = _x79 ∧ x3 = _x80 ∧ x4 = _x81 ∧ _x75 = _x79 ∧ _x74 = _x78 ∧ 1 ≤ _x75 − 1 ∧ 2 ≤ _x74 − 1 f162_0_power_GT' 11 f213_0_power_NE: x1 = _x82 ∧ x2 = _x83 ∧ x3 = _x84 ∧ x4 = _x85 ∧ x1 = _x86 ∧ x2 = _x87 ∧ x3 = _x88 ∧ x4 = _x89 ∧ 1 ≤ _x83 − 1 ∧ 2 ≤ _x82 − 1 ∧ _x83 − 2⋅_x90 ≤ 1 ∧ 0 ≤ _x83 − 2⋅_x90 ∧ _x82 = _x86 ∧ _x83 = _x87 ∧ _x83 − 2⋅_x90 = _x88 f213_0_power_NE 12 f213_0_power_NE': x1 = _x91 ∧ x2 = _x92 ∧ x3 = _x93 ∧ x4 = _x94 ∧ x1 = _x95 ∧ x2 = _x96 ∧ x3 = _x97 ∧ x4 = _x98 ∧ 1 ≤ _x92 − 1 ∧ 0 ≤ _x99 − 1 ∧ _x99 ≤ _x92 − 1 ∧ 0 = _x93 ∧ _x91 = _x95 ∧ _x92 = _x96 ∧ 0 = _x97 ∧ _x94 = _x98 f213_0_power_NE' 13 f162_0_power_GT: x1 = _x100 ∧ x2 = _x101 ∧ x3 = _x102 ∧ x4 = _x103 ∧ x1 = _x104 ∧ x2 = _x105 ∧ x3 = _x106 ∧ x4 = _x107 ∧ _x100 = _x104 ∧ 0 = _x102 ∧ 0 ≤ _x101 − 2⋅_x105 ∧ _x101 − 2⋅_x105 ≤ 1 ∧ 0 ≤ _x105 − 1 ∧ _x105 ≤ _x101 − 1 ∧ 1 ≤ _x101 − 1 f213_0_power_NE 14 f162_0_power_GT: x1 = _x108 ∧ x2 = _x109 ∧ x3 = _x110 ∧ x4 = _x111 ∧ x1 = _x112 ∧ x2 = _x113 ∧ x3 = _x114 ∧ x4 = _x115 ∧ _x109 − 1 = _x113 ∧ _x108 = _x112 ∧ 1 = _x110 ∧ _x109 − 1 ≤ _x109 − 1 ∧ 1 ≤ _x109 − 1 __init 15 f1_0_main_Load: x1 = _x116 ∧ x2 = _x117 ∧ x3 = _x118 ∧ x4 = _x119 ∧ x1 = _x120 ∧ x2 = _x121 ∧ x3 = _x122 ∧ x4 = _x123 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f97_0_random_GT f97_0_random_GT f97_0_random_GT: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f162_0_power_GT' f162_0_power_GT' f162_0_power_GT': x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f155_0_main_InvokeMethod f155_0_main_InvokeMethod f155_0_main_InvokeMethod: 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 f213_0_power_NE f213_0_power_NE f213_0_power_NE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f213_0_power_NE' f213_0_power_NE' f213_0_power_NE': x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f162_0_power_GT f162_0_power_GT f162_0_power_GT: 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 1 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/1

Here we consider the SCC { f162_0_power_GT', f213_0_power_NE, f213_0_power_NE', f162_0_power_GT }.

### 2.1.1 Transition Removal

We remove transitions 8, 13, 12, 9, 10, 14, 11 using the following ranking functions, which are bounded by 0.

 f162_0_power_GT: 4⋅x2 + 2 f162_0_power_GT': 4⋅x2 + 1 f213_0_power_NE': 2⋅x2 + 3 f213_0_power_NE: 4⋅x2

### 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)