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subsection \<open>IPv4 with Ports and Protocols\<close>
theory 
  IPv4_TCPUDP
  imports IPv4
begin

type_synonym 
  ipv4_TCPUDP    =  "(ipv4_ip \<times> port \<times> protocol)"
 
instance protocol :: adr ..

overloading src_port_ipv4_TCPUDP \<equiv> "src_port :: ('\<alpha>::adr,'\<beta>) packet \<Rightarrow> '\<gamma>::port" 
begin 
definition 
  "src_port_ipv4_TCPUDP (x::(ipv4_TCPUDP,'\<beta>) packet) \<equiv> (fst o snd o fst o snd) x"
end 

overloading dest_port_ipv4_TCPUDP \<equiv> "dest_port :: ('\<alpha>::adr,'\<beta>) packet \<Rightarrow> '\<gamma>::port"
begin
definition 
  "dest_port_ipv4_TCPUDP  (x::(ipv4_TCPUDP,'\<beta>) packet) \<equiv> (fst o snd o fst o snd o snd) x"
end 

overloading subnet_of_ipv4_TCPUDP \<equiv> "subnet_of :: '\<alpha>::adr \<Rightarrow> '\<alpha> net"
begin 
definition
  "subnet_of_ipv4_TCPUDP (x::ipv4_TCPUDP) \<equiv> {{(a,b). a = fst x}}::(ipv4_TCPUDP net)" 
end

overloading dest_protocol_ipv4_TCPUDP \<equiv> "dest_protocol :: ('\<alpha>::adr,'\<beta>) packet \<Rightarrow> protocol"
begin
definition 
  "dest_protocol_ipv4_TCPUDP (x::(ipv4_TCPUDP,'\<beta>) packet) \<equiv> (snd o snd o fst o snd o snd) x"
end

definition subnet_of_ip :: "ipv4_ip \<Rightarrow> ipv4_TCPUDP net"
  where "subnet_of_ip ip = {{(a,b). (a = ip)}}"

lemma src_port: "src_port (a,(x::ipv4_TCPUDP),d,e) = fst (snd x)"
  by (simp add: src_port_ipv4_TCPUDP_def in_subnet)

lemma dest_port: "dest_port (a,d,(x::ipv4_TCPUDP),e) = fst (snd x)"
  by (simp add: dest_port_ipv4_TCPUDP_def in_subnet)

lemmas Ipv4_TCPUDPLemmas = src_port dest_port src_port_ipv4_TCPUDP_def dest_port_ipv4_TCPUDP_def 
  dest_protocol_ipv4_TCPUDP_def subnet_of_ipv4_TCPUDP_def
end