Theory IPv4

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subsection ‹Formalizing IPv4 Addresses›
theory 
  IPv4
  imports 
    NetworkCore
begin
text‹
  A theory describing IPv4 addresses with ports. The host address is a four-tuple of Integers, 
  the port number is a single Integer.
›

type_synonym 
  ipv4_ip = "(int × int × int × int)" 

type_synonym 
  port    =  "int"

type_synonym 
  ipv4    =  "(ipv4_ip × port)"

overloading src_port_ipv4 ≡ "src_port :: ('α::adr,'β) packet ⇒ 'γ::port" 
begin 
definition 
  "src_port_ipv4 (x::(ipv4,'β) packet) ≡ (snd o fst o snd) x"
end 

overloading dest_port_ipv4 ≡ "dest_port :: ('α::adr,'β) packet ⇒ 'γ::port"
begin
definition 
  "dest_port_ipv4  (x::(ipv4,'β) packet) ≡ (snd o fst o snd o snd) x"
end 

overloading subnet_of_ipv4 ≡ "subnet_of :: 'α::adr ⇒ 'α net"
begin 
definition
  "subnet_of_ipv4 (x::ipv4) ≡ {{(a,b::int). a =  fst x}}" 
end

definition subnet_of_ip :: "ipv4_ip ⇒ ipv4 net"
  where "subnet_of_ip ip = {{(a,b). (a = ip)}}"

lemma src_port: "src_port (a,(x::ipv4),d,e) = snd x"
  by (simp add: src_port_ipv4_def in_subnet)

lemma dest_port: "dest_port (a,d,(x::ipv4),e) = snd x"
  by (simp add: dest_port_ipv4_def in_subnet)


lemmas IPv4Lemmas = src_port dest_port src_port_ipv4_def dest_port_ipv4_def

end