# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f182_0_f_LE, f135_0_f_LE, f1_0_main_Load, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f135_0_f_LE: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ _arg2 = _arg2P ∧ _arg2 = _arg1P ∧ −1 ≤ _arg2 − 1 ∧ 0 ≤ _arg1 − 1 f135_0_f_LE 2 f182_0_f_LE: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x1 = _x3 ∧ x2 = _x4 ∧ x3 = _x5 ∧ _x1 = _x5 ∧ 2 = _x4 ∧ _x = _x3 ∧ 0 ≤ _x − 1 f182_0_f_LE 3 f135_0_f_LE: x1 = _x6 ∧ x2 = _x7 ∧ x3 = _x8 ∧ x1 = _x9 ∧ x2 = _x10 ∧ x3 = _x11 ∧ _x8 − 1 = _x10 ∧ _x6 − _x7 = _x9 ∧ 0 ≤ _x6 − 1 ∧ 0 ≤ _x7 − 1 f182_0_f_LE 4 f182_0_f_LE: x1 = _x12 ∧ x2 = _x13 ∧ x3 = _x14 ∧ x1 = _x15 ∧ x2 = _x16 ∧ x3 = _x17 ∧ _x14 − 1 = _x17 ∧ _x13 − 1 = _x16 ∧ _x12 = _x15 ∧ 0 ≤ _x12 − 1 ∧ 0 ≤ _x13 − 1 f182_0_f_LE 5 f182_0_f_LE: x1 = _x18 ∧ x2 = _x19 ∧ x3 = _x20 ∧ x1 = _x21 ∧ x2 = _x22 ∧ x3 = _x23 ∧ _x19 − 1 = _x22 ∧ _x18 = _x21 ∧ 0 ≤ _x18 − 1 ∧ 0 ≤ _x19 − 1 __init 6 f1_0_main_Load: x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x1 = _x27 ∧ x2 = _x28 ∧ x3 = _x29 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f182_0_f_LE f182_0_f_LE f182_0_f_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f135_0_f_LE f135_0_f_LE f135_0_f_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f1_0_main_Load f1_0_main_Load f1_0_main_Load: 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 { f182_0_f_LE, f135_0_f_LE }.

### 2.1.1 Transition Removal

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

 f135_0_f_LE: 3 + 3⋅x1 f182_0_f_LE: 3⋅x1 + 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)