# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f274_0_power_LE, f1_0_main_Load, f274_0_power_LE', __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f274_0_power_LE: x1 = _arg1 ∧ x2 = _arg2 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ −1 ≤ _arg2 − 1 ∧ 1 ≤ _arg1P − 1 ∧ _x3 ≤ 1 ∧ −1 ≤ _x3 − 1 ∧ 0 ≤ _arg1 − 1 f1_0_main_Load 2 f274_0_power_LE: x1 = _x ∧ x2 = _x1 ∧ x1 = _x2 ∧ x2 = _x4 ∧ −1 ≤ _x1 − 1 ∧ 1 ≤ _x2 − 1 ∧ 2 ≤ _x5 − 1 ∧ 0 ≤ _x − 1 f274_0_power_LE 3 f274_0_power_LE': x1 = _x6 ∧ x2 = _x8 ∧ x1 = _x9 ∧ x2 = _x10 ∧ _x6 − 2⋅_x13 = 0 ∧ _x14 ≤ _x6 − 1 ∧ 0 ≤ _x6 − 1 ∧ _x6 = _x9 f274_0_power_LE' 4 f274_0_power_LE: x1 = _x16 ∧ x2 = _x19 ∧ x1 = _x20 ∧ x2 = _x22 ∧ _x16 − 2⋅_x23 = 0 ∧ 0 ≤ _x16 − 1 ∧ _x20 ≤ _x16 − 1 ∧ 0 ≤ _x16 − 2⋅_x23 ∧ _x16 − 2⋅_x23 ≤ 1 ∧ _x16 − 2⋅_x20 ≤ 1 ∧ 0 ≤ _x16 − 2⋅_x20 f274_0_power_LE 5 f274_0_power_LE': x1 = _x24 ∧ x2 = _x25 ∧ x1 = _x26 ∧ x2 = _x27 ∧ _x24 − 2⋅_x28 = 1 ∧ _x29 ≤ _x24 − 1 ∧ 0 ≤ _x24 − 1 ∧ _x24 = _x26 f274_0_power_LE' 6 f274_0_power_LE: x1 = _x30 ∧ x2 = _x31 ∧ x1 = _x32 ∧ x2 = _x33 ∧ _x30 − 2⋅_x34 = 1 ∧ 0 ≤ _x30 − 1 ∧ _x32 ≤ _x30 − 1 ∧ 0 ≤ _x30 − 2⋅_x34 ∧ _x30 − 2⋅_x34 ≤ 1 ∧ _x30 − 2⋅_x32 ≤ 1 ∧ 0 ≤ _x30 − 2⋅_x32 __init 7 f1_0_main_Load: x1 = _x35 ∧ x2 = _x36 ∧ x1 = _x37 ∧ x2 = _x38 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f274_0_power_LE f274_0_power_LE f274_0_power_LE: x1 = x1 ∧ x2 = x2 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 f274_0_power_LE' f274_0_power_LE' f274_0_power_LE': x1 = x1 ∧ x2 = x2 __init __init __init: x1 = x1 ∧ x2 = x2
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 { f274_0_power_LE, f274_0_power_LE' }.

### 2.1.1 Transition Removal

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

 f274_0_power_LE: 1 + 2⋅x1 f274_0_power_LE': 2⋅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)