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 *               2008-2015 Achim D. Brucker, Germany
 *               2009-2016 Université Paris-Sud, France
 *               2015-2016 The University of Sheffield, UK
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subsection \<open>Elementary Firewall Policy Transformation Rules\<close> 
theory 
  ElementaryRules
  imports 
    FWNormalisationCore
begin
  
  
text\<open>
  This theory contains those elementary transformation rules which are presented in the ICST 
  2010 paper~\cite{brucker.ea:firewall:2010}. They are not used elsewhere.
\<close>
  
lemma elem1:
  "C (AllowPortFromTo x y p \<oplus> DenyAllFromTo x y) = C (DenyAllFromTo x y)"
  by (rule ext, auto simp: PLemmas)
    
    
lemma elem2:
  "C ((a \<oplus> b) \<oplus> c) = C (a \<oplus> (b \<oplus> c))"
  by (simp add: C.simps) 
    
lemma elem3:
  "C (AllowPortFromTo x y a \<oplus> AllowPortFromTo x y b) = 
  C (AllowPortFromTo x y b \<oplus> AllowPortFromTo x y a)"
  by (rule ext, auto simp: PLemmas)
    
lemma elem4:
  "C (a \<oplus> DenyAll) = C DenyAll"
  by (rule ext, auto simp: PLemmas)
    
lemma elem5:
  "C (DenyAllFromTo x y \<oplus> DenyAllFromTo u v) = C (DenyAllFromTo u v \<oplus> DenyAllFromTo x y)"
  by (rule ext, auto simp: PLemmas)
    
    
lemma elem6: 
  "dom (C a) \<inter> dom (C b) = {} \<Longrightarrow> C (a \<oplus> b) = C (b \<oplus> a)"
  by (rule ext, metis C.simps(4) map_add_comm)
    
end