Theory IntegerPort

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subsection‹Integer Addresses with Ports›
theory 
  IntegerPort
  imports 
    NetworkCore
begin

text‹
  A theory describing addresses which are modelled as a pair of Integers - the first being the 
  host address, the second the port number.
›

type_synonym 
  address = int

type_synonym 
  port = int

type_synonym 
  adr⇩i⇩p = "address × port" 

overloading src_port_int ≡ "src_port :: ('α::adr,'β) packet ⇒ 'γ::port" 
begin 
definition 
  "src_port_int (x::(adr⇩i⇩p,'β) packet)  ≡ (snd o fst  o snd) x"
end 

overloading dest_port_int ≡ "dest_port :: ('α::adr,'β) packet ⇒ 'γ::port"
begin
definition 
  "dest_port_int (x::(adr⇩i⇩p,'β) packet) ≡ (snd o fst o  snd o snd) x"
end 

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

lemma src_port: "src_port (a,x::adr⇩i⇩p,d,e) = snd x"
  by (simp add: src_port_int_def in_subnet)

lemma dest_port: "dest_port (a,d,x::adr⇩i⇩p,e) = snd x"
  by (simp add: dest_port_int_def in_subnet)

lemmas adr⇩i⇩pLemmas = src_port dest_port src_port_int_def dest_port_int_def

end