# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f167_0_log_LT, f96_0_random_GT, f154_0_main_InvokeMethod, f1_0_main_Load, f167_0_log_LT', __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f96_0_random_GT: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ 0 = _arg2P ∧ 0 ≤ _arg1P − 1 ∧ 0 ≤ _arg1 − 1 ∧ 0 ≤ _arg2 − 1 ∧ _arg1P ≤ _arg1 f1_0_main_Load 2 f96_0_random_GT: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x1 = _x3 ∧ x2 = _x4 ∧ x3 = _x5 ∧ 0 ≤ _x3 − 1 ∧ 0 ≤ _x − 1 ∧ _x3 ≤ _x ∧ 0 ≤ _x1 − 1 ∧ −1 ≤ _x4 − 1 f96_0_random_GT 3 f154_0_main_InvokeMethod: x1 = _x6 ∧ x2 = _x8 ∧ x3 = _x9 ∧ x1 = _x10 ∧ x2 = _x12 ∧ x3 = _x13 ∧ _x10 ≤ _x6 ∧ 1 ≤ _x14 − 1 ∧ 0 ≤ _x6 − 1 ∧ 0 ≤ _x10 − 1 ∧ _x8 = _x12 ∧ 0 = _x13 f96_0_random_GT 4 f154_0_main_InvokeMethod: x1 = _x15 ∧ x2 = _x16 ∧ x3 = _x17 ∧ x1 = _x18 ∧ x2 = _x20 ∧ x3 = _x21 ∧ 1 ≤ _x22 − 1 ∧ −1 ≤ _x21 − 1 ∧ _x18 ≤ _x15 ∧ 0 ≤ _x15 − 1 ∧ 0 ≤ _x18 − 1 ∧ _x16 = _x20 f1_0_main_Load 5 f167_0_log_LT: x1 = _x23 ∧ x2 = _x24 ∧ x3 = _x25 ∧ x1 = _x26 ∧ x2 = _x27 ∧ x3 = _x29 ∧ 0 = _x29 ∧ 0 = _x27 ∧ 0 = _x26 ∧ 0 = _x24 ∧ 0 ≤ _x23 − 1 f96_0_random_GT 6 f167_0_log_LT: x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x1 = _x33 ∧ x2 = _x34 ∧ x3 = _x35 ∧ 0 = _x35 ∧ _x31 = _x34 ∧ 0 = _x33 ∧ 0 ≤ _x30 − 1 f154_0_main_InvokeMethod 7 f167_0_log_LT: x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x1 = _x39 ∧ x2 = _x40 ∧ x3 = _x41 ∧ 0 ≤ _x36 − 1 ∧ 1 ≤ _x42 − 1 ∧ _x38 = _x39 ∧ _x37 = _x40 ∧ _x38 = _x41 f167_0_log_LT 8 f167_0_log_LT': x1 = _x43 ∧ x2 = _x44 ∧ x3 = _x45 ∧ x1 = _x46 ∧ x2 = _x47 ∧ x3 = _x48 ∧ 1 ≤ _x44 − 1 ∧ 1 ≤ _x43 − 1 ∧ _x49 ≤ _x44 − 1 ∧ _x43 ≤ _x44 ∧ _x43 = _x45 ∧ _x43 = _x46 ∧ _x44 = _x47 ∧ _x43 = _x48 f167_0_log_LT' 9 f167_0_log_LT: x1 = _x50 ∧ x2 = _x51 ∧ x3 = _x52 ∧ x1 = _x53 ∧ x2 = _x54 ∧ x3 = _x55 ∧ _x50 = _x55 ∧ _x50 = _x53 ∧ _x50 = _x52 ∧ 0 ≤ _x51 − _x50⋅_x54 ∧ _x51 − _x50⋅_x54 ≤ _x50 − 1 ∧ _x54 ≤ _x51 − 1 ∧ _x50 ≤ _x51 ∧ 1 ≤ _x50 − 1 ∧ 1 ≤ _x51 − 1 __init 10 f1_0_main_Load: x1 = _x56 ∧ x2 = _x57 ∧ x3 = _x58 ∧ x1 = _x59 ∧ x2 = _x60 ∧ x3 = _x61 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f167_0_log_LT f167_0_log_LT f167_0_log_LT: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f96_0_random_GT f96_0_random_GT f96_0_random_GT: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f154_0_main_InvokeMethod f154_0_main_InvokeMethod f154_0_main_InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f167_0_log_LT' f167_0_log_LT' f167_0_log_LT': x1 = x1 ∧ x2 = x2 ∧ x3 = x3 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3
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 { f167_0_log_LT, f167_0_log_LT' }.

### 2.1.1 Transition Removal

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

 f167_0_log_LT: 2⋅x2 + 1 f167_0_log_LT': 2⋅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 % (4 real / 0 unknown / 0 assumptions / 4 total proof steps)