Theory DatatypePort

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subsection ‹Datatype Addresses with Ports›
theory 
  DatatypePort
  imports 
    NetworkCore
begin

text‹
  A theory describing a network consisting of three subnetworks, including port numbers modelled 
  as Integers. Hosts within a network are not distinguished.
›

datatype DatatypeAddress = dmz_adr | intranet_adr | internet_adr

type_synonym  
  port = int
type_synonym  
  DatatypePort =  "(DatatypeAddress × port)"

instance DatatypeAddress :: adr ..

definition
  dmz::"DatatypePort net" where
  "dmz = {{(a,b). a = dmz_adr}}"
definition
  intranet::"DatatypePort net" where
  "intranet = {{(a,b). a = intranet_adr}}"
definition
  internet::"DatatypePort net" where
  "internet = {{(a,b). a = internet_adr}}"

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

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

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

lemma src_port : "src_port ((a,x,d,e)::(DatatypePort,'β) packet) = snd x"
  by (simp add: src_port_datatype_def in_subnet)

lemma dest_port : "dest_port ((a,d,x,e)::(DatatypePort,'β) packet) = snd x"
  by (simp add: dest_port_datatype_def in_subnet)
    
lemmas DatatypePortLemmas = src_port dest_port src_port_datatype_def dest_port_datatype_def 

end