rewrite www to non www doesn't work using two htaccess file

I have two .htaccess file.

in root:(this file redirect to public folder)

<IfModule mod_rewrite.c>
RewriteEngine On
RewriteRule ^$ /public/index.php/home/index [L]
RewriteRule ^(.*)/$ $1
RewriteRule ^(.*)$ /public/$1 [L]
</IfModule>

and in public folder I have:

# Disable directory browsing
Options All -Indexes

# ----------------------------------------------------------------------
# Rewrite engine
# ----------------------------------------------------------------------

# Turning on the rewrite engine is necessary for the following rules and features.
# FollowSymLinks must be enabled for this to work.
<IfModule mod_rewrite.c>
    Options +FollowSymlinks
    RewriteEngine On

    # If you installed CodeIgniter in a subfolder, you will need to
    # change the following line to match the subfolder you need.
    # http://httpd.apache.org/docs/current/mod/mod_rewrite.html#rewritebase
    # RewriteBase /

    # Redirect Trailing Slashes...
    RewriteCond %{REQUEST_FILENAME} !-d
    RewriteRule ^(.*)/$ /$1 [L,R=301]

    # Rewrite "www.example.com -> example.com"
    RewriteCond %{HTTPS} !=on
    RewriteCond %{HTTP_HOST} ^www\.(.+)$ [NC]
    RewriteRule ^ http://%1%{REQUEST_URI} [R=301,L]

    # Checks to see if the user is attempting to access a valid file,
    # such as an image or css document, if this isn't true it sends the
    # request to the front controller, index.php
    RewriteCond %{REQUEST_FILENAME} !-f
    RewriteCond %{REQUEST_FILENAME} !-d
    RewriteRule ^([\s\S]*)$ index.php/$1 [L,NC,QSA]

    # Ensure Authorization header is passed along
    RewriteCond %{HTTP:Authorization} .
    RewriteRule .* - [E=HTTP_AUTHORIZATION:%{HTTP:Authorization}]
</IfModule>

<IfModule !mod_rewrite.c>
    # If we don't have mod_rewrite installed, all 404's
    # can be sent to index.php, and everything works as normal.
    ErrorDocument 404 index.php
</IfModule>
# X-XSS-Protection
<IfModule mod_headers.c>
    Header set X-XSS-Protection "1; mode=block"
</IfModule>
# X-Frame-Options
<IfModule mod_headers.c>
    Header always append X-Frame-Options SAMEORIGIN
</IfModule>
# X-Content-Type nosniff
<IfModule mod_headers.c>
    Header set X-Content-Type-Options nosniff
</IfModule>
# Disable server signature start
    ServerSignature Off
# Disable server signature end

I add this code for redirect www to none www in public folder .htaccess file:

# Rewrite "www.example.com -> example.com"
RewriteCond %{HTTPS} !=on
RewriteCond %{HTTP_HOST} ^www\.(.+)$ [NC]
RewriteRule ^ http://%1%{REQUEST_URI} [R=301,L]

but in action, my www not redirect to none www. how do can I fix this problem?