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 * Copyright (c) 2005-2010 ETH Zurich, Switzerland
 *               2008-2015 Achim D. Brucker, Germany
 *               2009-2016 Université Paris-Sud, France
 *               2015-2016 The University of Sheffield, UK
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subsection {* Transformation Example 1 *}
theory 
  Transformation01
imports 
  "../../UPF-Firewall"
begin

definition
 FWLink :: "adr\<^sub>i\<^sub>p net" where
"FWLink = {{(a,b). a = 1}}"

definition
 any :: "adr\<^sub>i\<^sub>p net" where
"any = {{(a,b). a > 5}}"

definition
 i4:: "adr\<^sub>i\<^sub>p net" where
"i4 = {{(a,b). a = 2 }}" 

definition
 i27:: "adr\<^sub>i\<^sub>p net" where
"i27 = {{(a,b). a = 3 }}" 
 
definition
 eth_intern:: "adr\<^sub>i\<^sub>p net" where
"eth_intern = {{(a,b). a = 4 }}"
 
definition
 eth_private:: "adr\<^sub>i\<^sub>p net" where
"eth_private = {{(a,b). a = 5 }}"


definition
(* Mandatory: Global *)

 MG2 :: "(adr\<^sub>i\<^sub>p net,port) Combinators" where
"MG2 = AllowPortFromTo i27 any 1 \<oplus> 
       AllowPortFromTo i27 any 2 \<oplus>
       AllowPortFromTo i27 any 3"

definition
 MG3 :: "(adr\<^sub>i\<^sub>p net,port) Combinators" where
"MG3 = AllowPortFromTo any FWLink 1"

definition
 MG4 :: "(adr\<^sub>i\<^sub>p net,port) Combinators" where
"MG4 = AllowPortFromTo FWLink FWLink 4"
 
definition
 MG7 :: "(adr\<^sub>i\<^sub>p net,port) Combinators" where
"MG7 = AllowPortFromTo FWLink i4 6 \<oplus>
       AllowPortFromTo FWLink i4 7"

definition
 MG8 :: "(adr\<^sub>i\<^sub>p net,port) Combinators" where
"MG8 = AllowPortFromTo FWLink i4 6 \<oplus>
       AllowPortFromTo FWLink i4 7"

(* Default Global *)

definition
 DG3:: "(adr\<^sub>i\<^sub>p net,port) Combinators" where
"DG3 = AllowPortFromTo any any 7"

 

definition
"Policy = DenyAll \<oplus> MG8 \<oplus>  MG7 \<oplus> MG4 \<oplus> MG3 \<oplus> MG2 \<oplus> DG3"   
 
 

 lemmas PolicyLemmas =  Policy_def
 FWLink_def
 any_def
 
 i27_def
 i4_def
 
 eth_intern_def
 eth_private_def
 
 MG2_def MG3_def MG4_def MG7_def MG8_def  
 DG3_def
 

lemmas PolicyL =  MG2_def MG3_def MG4_def MG7_def MG8_def  
 DG3_def Policy_def

definition 
 not_in_same_net :: "(adr\<^sub>i\<^sub>p,DummyContent) packet \<Rightarrow> bool" where
"not_in_same_net x =  (((src x \<sqsubset> i27) \<longrightarrow> ( \<not> (dest x \<sqsubset> i27))) \<and>
                      ((src x \<sqsubset>  i4) \<longrightarrow> ( \<not> (dest x \<sqsubset> i4))) \<and>
                      ((src x \<sqsubset>  eth_intern) \<longrightarrow> ( \<not> (dest x \<sqsubset> eth_intern))) \<and>
                      ((src x \<sqsubset>  eth_private) \<longrightarrow> ( \<not> (dest x \<sqsubset> eth_private))))"
                   
consts fixID :: id
consts fixContent :: DummyContent

definition "fixElements p = (id p = fixID \<and> content p = fixContent)"

lemmas fixDefs = fixElements_def NetworkCore.id_def NetworkCore.content_def



lemma sets_distinct1: "(n::int) \<noteq> m \<Longrightarrow> {(a,b). a = n} \<noteq> {(a,b). a = m}"
apply auto
done

lemma sets_distinct2: "(m::int) \<noteq> n \<Longrightarrow> {(a,b). a = n} \<noteq> {(a,b). a = m}"
apply auto
done



lemma sets_distinct3: "{((a::int),(b::int)). a = n} \<noteq> {(a,b). a > n}"
apply auto
done


lemma sets_distinct4: "{((a::int),(b::int)). a > n} \<noteq> {(a,b). a = n}"
apply auto
done


lemma aux: "\<lbrakk>a \<in> c; a \<notin> d; c = d\<rbrakk> \<Longrightarrow> False"
apply auto
done


lemma sets_distinct5: "(s::int) < g \<Longrightarrow> {(a::int, b::int). a = s} \<noteq> {(a::int, b::int).  g < a}"
apply (auto simp: sets_distinct3)
apply (subgoal_tac "(s,4) \<in> {(a::int,b::int). a = (s)}")
apply (subgoal_tac "(s,4) \<notin> {(a::int,b::int). g < a}")
apply (erule aux)
apply assumption+
apply simp
by blast


lemma sets_distinct6: "(s::int) < g \<Longrightarrow> {(a::int, b::int).  g < a} \<noteq> {(a::int, b::int).  a = s}"
apply (rule not_sym)
apply (rule sets_distinct5)
by simp


lemma distinctNets: "FWLink \<noteq>  any \<and> FWLink \<noteq>  i4 \<and> FWLink \<noteq>  i27 \<and> FWLink \<noteq>  eth_intern \<and> FWLink \<noteq>   eth_private \<and> 
any \<noteq>  FWLink \<and> any \<noteq>  i4 \<and> any \<noteq>  i27 \<and> any \<noteq>  eth_intern \<and> any \<noteq>  eth_private \<and> i4 \<noteq>  FWLink \<and> 
i4 \<noteq>  any \<and> i4 \<noteq>  i27 \<and> i4 \<noteq>  eth_intern \<and> i4 \<noteq>  eth_private \<and> i27 \<noteq>  FWLink \<and> i27 \<noteq>  any \<and> 
i27 \<noteq>  i4 \<and> i27 \<noteq>  eth_intern \<and> i27 \<noteq>  eth_private \<and> eth_intern \<noteq>  FWLink \<and> eth_intern \<noteq>  any \<and> 
eth_intern \<noteq>  i4 \<and> eth_intern \<noteq>  i27 \<and> eth_intern \<noteq>  eth_private \<and> eth_private \<noteq>  FWLink \<and> 
eth_private \<noteq>  any \<and> eth_private \<noteq>  i4 \<and> eth_private \<noteq>  i27 \<and> eth_private \<noteq>  eth_intern"
apply (simp add: PolicyLemmas sets_distinct1 sets_distinct2 sets_distinct3 sets_distinct4 sets_distinct5 sets_distinct6)
done



lemma aux5: "\<lbrakk>x \<noteq> a; y\<noteq>b; (x \<noteq> y \<and> x \<noteq> b) \<or>  (a \<noteq> b \<and> a \<noteq> y)\<rbrakk> \<Longrightarrow> {x,a} \<noteq> {y,b}"
  apply auto
done


lemma aux2: "{a,b} = {b,a}"
  apply auto
done

(*
lemma noMT: "\<forall> x \<in> set (policy2list Policy). dom (C x) \<noteq> {}"
apply (simp add: PolicyLemmas)
apply (simp add: PLemmas PolicyLemmas)
by arith
*)


lemma ANDex: "allNetsDistinct (policy2list Policy)"
apply (simp add: PolicyL allNetsDistinct_def distinctNets)
apply (auto simp: PLemmas PolicyLemmas netsDistinct_def sets_distinct5 sets_distinct6)
done

(*
lemma count_the_rules: "(int (length(policy2list (list2FWpolicy(normalize Policy)))) = post) \<and> 
(int(length (policy2list Policy)) = pre) \<and>
 (int (length((normalize Policy))) = Partitions)"
apply (insert distinctNets noMT)
apply (simp add: normalize_def PolicyL bothNets_def aux5 aux2 Nets_List_def, thin_tac "?X",thin_tac "?S")
oops



lemma normedPolicy: "normalize Policy = X"
apply (insert distinctNets noMT)
apply (simp add: normalize_def PolicyL bothNets_def aux5 aux2 Nets_List_def, thin_tac "?X",thin_tac "?S")
oops
*)

fun (sequential) numberOfRules where 
 "numberOfRules (a\<oplus>b) = numberOfRules a + numberOfRules b"
 |"numberOfRules a = (1::int)" 




fun numberOfRulesList where 
 "numberOfRulesList  (x#xs) = ((numberOfRules x)#(numberOfRulesList xs)) "
 |"numberOfRulesList [] = []" 

(*
lemma "numberOfRulesList (normalize Policy) = X"
apply (insert distinctNets noMT)
apply (simp add: normalize_def PolicyL bothNets_def aux5 aux2 Nets_List_def, thin_tac "?X",thin_tac "?S")
oops
*)

lemma all_in_list: "all_in_list (policy2list Policy) (Nets_List Policy)"
apply (simp add: PolicyL)
apply (unfold Nets_List_def)
apply (unfold bothNets_def)
apply (insert distinctNets)
apply simp
done

lemmas normalizeUnfold =   normalize_def Policy_def Nets_List_def bothNets_def aux aux2 bothNets_def

(*
lemma noMT2: "\<forall> x \<in> set (policy2list Policy). dom (C x) \<noteq> {}"
apply (simp add: PLemmas  normalize_def bothNets_def
                 PolicyLemmas aux5 aux2 Nets_List_def ) 
by (metis zless_add1_eq)
*)


end