Theory ElementaryPolicies

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section‹Elementary Policies›
theory 
  ElementaryPolicies
  imports 
    UPFCore
begin
text‹
  In this theory, we introduce the elementary policies of UPF that build the basis 
  for more complex policies. These complex policies, respectively, embedding of 
  well-known access control or security models, are build by composing the elementary 
  policies defined in this theory. 
›

subsection‹The Core Policy Combinators: Allow and Deny Everything›

definition
   deny_pfun    :: "('α ⇀'β) ⇒ ('α ↦ 'β)" ("AllD")
   where 
  "deny_pfun pf ≡ (λ x. case pf x of
                          ⌊y⌋ ⇒ ⌊deny (y)⌋
                         |⊥ ⇒ ⊥)"

definition
   allow_pfun    :: "('α ⇀'β) ⇒ ('α ↦ 'β)" ( "AllA")
   where 
  "allow_pfun pf ≡ (λ x. case pf x of
                          ⌊y⌋ ⇒ ⌊allow (y)⌋
                         |⊥ ⇒ ⊥)"

syntax
  "_allow_pfun"  :: "('α ⇀'β) ⇒ ('α ↦ 'β)" ("A⇩p")
translations
  "A⇩p f" ⇌ "AllA f"

syntax
  "_deny_pfun"  :: "('α ⇀'β) ⇒ ('α ↦ 'β)" ("D⇩p")
translations
  "D⇩p f" ⇌ "AllD f"

notation
   "deny_pfun"  (binder "∀D" 10) and
   "allow_pfun" (binder "∀A" 10)

lemma AllD_norm[simp]: "deny_pfun (id o (λx. ⌊x⌋)) = (∀Dx. ⌊x⌋)"
  by(simp add:id_def comp_def)
    
lemma AllD_norm2[simp]: "deny_pfun (Some o id) = (∀Dx. ⌊x⌋)"
  by(simp add:id_def comp_def)
    
lemma AllA_norm[simp]: "allow_pfun (id o Some) = (∀Ax. ⌊x⌋)"
  by(simp add:id_def comp_def)
    
lemma AllA_norm2[simp]: "allow_pfun (Some o id) = (∀Ax. ⌊x⌋)"
  by(simp add:id_def comp_def)
    
lemma AllA_apply[simp]: "(∀Ax. Some (P x)) x = ⌊allow (P x)⌋"
  by(simp add:allow_pfun_def)
    
lemma AllD_apply[simp]: "(∀Dx. Some (P x)) x = ⌊deny (P x)⌋"
  by(simp add:deny_pfun_def)

lemma neq_Allow_Deny: "pf ≠ ∅ ⟹ (deny_pfun pf) ≠ (allow_pfun pf)"
  apply (erule contrapos_nn)
  apply (rule ext)
  subgoal for x
    apply (drule_tac x=x in fun_cong)
    apply (auto simp: deny_pfun_def allow_pfun_def)
    apply (case_tac "pf x =  ⊥")
     apply (auto)
    done
  done

subsection‹Common Instances›

definition allow_all_fun :: "('α ⇒ 'β) ⇒ ('α ↦ 'β)" ("A⇩f")
  where "allow_all_fun f =  allow_pfun (Some o f)"

definition deny_all_fun :: "('α ⇒ 'β) ⇒ ('α ↦ 'β)" ("D⇩f")
  where "deny_all_fun f ≡ deny_pfun (Some o f)"

definition
   deny_all_id   :: "'α ↦ 'α" ("D⇩I") where 
  "deny_all_id  ≡ deny_pfun (id o Some)"

definition
   allow_all_id    :: "'α ↦ 'α" ("A⇩I") where
  "allow_all_id  ≡ allow_pfun (id o Some)"

definition 
  allow_all    :: "('α ↦ unit)"  ("A⇩U") where 
  "allow_all p  = ⌊allow ()⌋" 

definition 
  deny_all :: "('α ↦ unit)" ("D⇩U") where
  "deny_all p   = ⌊deny ()⌋"              

text‹... and resulting properties:›

lemma "A⇩I  ⨁ Map.empty  = A⇩I"
  by simp 
  
lemma "A⇩f f  ⨁ Map.empty  = A⇩f f"
  by simp 
  
lemma "allow_pfun Map.empty = Map.empty"
  apply (rule ext)
  apply (simp add: allow_pfun_def)
  done

lemma allow_left_cancel :"dom pf = UNIV ⟹ (allow_pfun pf) ⨁ x = (allow_pfun pf)" 
  apply (rule ext)+
  apply (auto simp: allow_pfun_def option.splits)
  done


lemma deny_left_cancel :"dom pf = UNIV ⟹ (deny_pfun pf) ⨁ x = (deny_pfun pf)"
  apply (rule ext)+
  by (auto simp: deny_pfun_def option.splits)

subsection‹Domain, Range, and Restrictions›

text‹
  Since policies are essentially maps, we inherit the basic definitions for 
  domain and range on  Maps: \\
  \verb+Map.dom_def+ :  @{thm Map.dom_def} \\
  whereas range is just an abrreviation for image:
  \begin{verbatim}
  abbreviation range :: "('a => 'b) => 'b set" 
  where -- "of function"  "range f == f ` UNIV"
  \end{verbatim}
  As a consequence, we inherit the following properties on
  policies:
  \begin{itemize}
  \item  \verb+Map.domD+ @{thm Map.domD}
  \item\verb+Map.domI+ @{thm Map.domI}
  \item\verb+Map.domIff+ @{thm Map.domIff}
  \item\verb+Map.dom_const+ @{thm Map.dom_const}
  \item\verb+Map.dom_def+ @{thm Map.dom_def}
  \item\verb+Map.dom_empty+ @{thm Map.dom_empty}
  \item\verb+Map.dom_eq_empty_conv+ @{thm Map.dom_eq_empty_conv}
  \item\verb+Map.dom_eq_singleton_conv+ @{thm Map.dom_eq_singleton_conv}
  \item\verb+Map.dom_fun_upd+ @{thm Map.dom_fun_upd}
  \item\verb+Map.dom_if+ @{thm Map.dom_if}
  \item\verb+Map.dom_map_add+ @{thm Map.dom_map_add}
  \end{itemize}
›

text‹
  However, some properties are specific to policy concepts: 
›
lemma sub_ran : "ran p  ⊆ Allow ∪ Deny"
  apply (auto simp: Allow_def Deny_def ran_def full_SetCompr_eq[symmetric])[1]
  subgoal for x a
    apply (case_tac "x")
     apply (simp_all)
    done
  done 
    
lemma dom_allow_pfun [simp]:"dom(allow_pfun f) = dom f"
  apply (auto simp: allow_pfun_def)
  subgoal for x y
    apply (case_tac "f x", simp_all)
    done
  done
    
lemma dom_allow_all: "dom(A⇩f f) = UNIV"
  by(auto simp: allow_all_fun_def o_def)

lemma dom_deny_pfun [simp]:"dom(deny_pfun f) = dom f"
  apply (auto simp: deny_pfun_def)[1]
  apply (case_tac "f x")
  apply (simp_all)
  done

lemma dom_deny_all: " dom(D⇩f f) = UNIV"
  by(auto simp: deny_all_fun_def o_def)

lemma ran_allow_pfun [simp]:"ran(allow_pfun f) = allow `(ran f)"
  apply (simp add: allow_pfun_def ran_def) 
  apply (rule set_eqI)
  apply (auto)[1]
  subgoal for x a 
    apply (case_tac "f a")
     apply (auto simp: image_def)[1]
     apply (auto simp: image_def)[1]
    done 
  subgoal for xa a
    apply (rule_tac x=a in exI)
    apply (simp)
    done
  done

lemma ran_allow_all: "ran(A⇩f id) = Allow"
  apply (simp add: allow_all_fun_def Allow_def o_def)
  apply (rule set_eqI)
  apply (auto simp: image_def ran_def)
  done
    
lemma ran_deny_pfun[simp]: "ran(deny_pfun f) = deny ` (ran f)"
  apply (simp add: deny_pfun_def ran_def)
  apply (rule set_eqI)
  apply (auto)[1] 
  subgoal for x a 
    apply (case_tac "f a")
     apply (auto simp: image_def)[1]
    apply (auto simp: image_def)[1]
    done
  subgoal for xa a
    apply (rule_tac x=a in exI)
    apply (simp)
    done
  done 
    
lemma ran_deny_all: "ran(D⇩f id) = Deny"
  apply (simp add: deny_all_fun_def Deny_def o_def)
  apply (rule set_eqI)
  apply (auto simp: image_def ran_def)
  done


text‹
  Reasoning over \verb+dom+ is most crucial since it paves the way for simplification and 
  reordering of policies composed by override (i.e. by the normal left-to-right rule composition
  method.
  \begin{itemize}
    \item \verb+Map.dom_map_add+ @{thm Map.dom_map_add}
    \item \verb+Map.inj_on_map_add_dom+ @{thm Map.inj_on_map_add_dom}
    \item \verb+Map.map_add_comm+ @{thm Map.map_add_comm}
    \item \verb+Map.map_add_dom_app_simps(1)+ @{thm Map.map_add_dom_app_simps(1)}
    \item \verb+Map.map_add_dom_app_simps(2)+ @{thm Map.map_add_dom_app_simps(2)}
    \item \verb+Map.map_add_dom_app_simps(3)+ @{thm Map.map_add_dom_app_simps(3)}
    \item \verb+Map.map_add_upd_left+ @{thm Map.map_add_upd_left}
  \end{itemize}
  The latter rule also applies to allow- and deny-override.
›

definition dom_restrict :: "['α set, 'α↦'β] ⇒ 'α↦'β" (infixr "◃" 55)
where     "S ◃ p ≡ (λx. if x ∈ S then p x else ⊥)"

lemma dom_dom_restrict[simp] : "dom(S ◃ p) = S ∩ dom p"
  apply (auto simp: dom_restrict_def)
  subgoal for x y
    apply (case_tac "x ∈ S")
     apply (simp_all)
    done 
  subgoal for x y 
    apply (case_tac "x ∈ S")
     apply (simp_all)
    done
  done 

lemma dom_restrict_idem[simp] : "(dom p) ◃ p = p"
  apply (rule ext) 
  apply (auto simp: dom_restrict_def
      dest: neq_commute[THEN iffD1,THEN not_None_eq [THEN iffD1]])
  done

lemma dom_restrict_inter[simp] : "T ◃ S ◃ p = T ∩ S ◃ p"
  apply (rule ext)
  apply (auto simp: dom_restrict_def
      dest: neq_commute[THEN iffD1,THEN not_None_eq [THEN iffD1]])
  done

definition ran_restrict :: "['α↦'β,'β decision set] ⇒ 'α ↦'β" (infixr "▹" 55)
where     "p ▹ S ≡ (λx. if p x ∈ (Some`S) then p x else ⊥)"

definition ran_restrict2 :: "['α↦'β,'β decision set] ⇒ 'α ↦'β" (infixr "▹2" 55)
where     "p ▹2 S ≡ (λx. if (the (p x)) ∈ (S) then p x else ⊥)"

lemma "ran_restrict = ran_restrict2"
  apply (rule ext)+
  apply (simp add: ran_restrict_def ran_restrict2_def)
  subgoal for x xa xb
    apply (case_tac "x xb")
     apply simp_all 
    apply (metis inj_Some inj_image_mem_iff)
    done
  done 


lemma ran_ran_restrict[simp] : "ran(p ▹ S) = S ∩ ran p"
  by(auto simp: ran_restrict_def image_def ran_def)
    
lemma ran_restrict_idem[simp] : "p ▹ (ran p) = p"
  apply (rule ext)
  apply (auto simp: ran_restrict_def image_def Ball_def ran_def)
  apply (erule contrapos_pp)
  apply (auto dest!: neq_commute[THEN iffD1,THEN not_None_eq [THEN iffD1]])
  done
    
lemma ran_restrict_inter[simp] : "(p ▹ S) ▹ T = p ▹ T ∩ S"
  apply (rule ext) 
  apply (auto simp: ran_restrict_def
      dest: neq_commute[THEN iffD1,THEN not_None_eq [THEN iffD1]])
  done
    
lemma ran_gen_A[simp] : "(∀Ax. ⌊P x⌋) ▹ Allow = (∀Ax. ⌊P x⌋)"
  apply (rule ext)
  apply (auto simp: Allow_def ran_restrict_def)
  done
    
lemma ran_gen_D[simp] : "(∀Dx. ⌊P x⌋) ▹ Deny = (∀Dx. ⌊P x⌋)"
  apply (rule ext)
  apply (auto simp: Deny_def ran_restrict_def)
  done

lemmas ElementaryPoliciesDefs = deny_pfun_def allow_pfun_def allow_all_fun_def deny_all_fun_def 
                                allow_all_id_def deny_all_id_def allow_all_def deny_all_def 
                                dom_restrict_def ran_restrict_def

end