Your missing ; in the constructor area.
public function __construct() {
    parent::__construct();
    $this->load->model('user_model');    
}
If your using codeigniter 3 you must have the FIRST LETTER of class and file name as upper case example Welcome.php and class Welcome extends CI_Controller same goes with models and libararies.
And Controller removed ; from index public function index();
User.php
<?php if(!defined('BASEPATH')) die("No direct Script Access allowed"); 
class User extends CI_Controller { 
   public function __construct() {
       parent::__construct();
       $this->load->model('user_model');    
   }
   public function index() { 
       if ($this->session->userdata('user_name') == TRUE) { 
           $this->welcome(); 
       } else {
          // Just a idea make them go to error page or something
          // Up to you.
          $this->some_other_function(); 
       }   
   } 
   public function welcome() {
       // Example.
       $variable = $this->user_model->function();
       $this->load->view('some_view', $variable);
   }
   public function some_other_function() {
       $this->load->view('another_view');
   }
}
I would recommend reading through most of the User Guide Codeigniter 3
And Codeigniter Doc's Page Here http://www.codeigniter.com/docs