section "Security Type Systems"
theory Sec_Type_Expr imports Big_Step
begin
subsection "Security Levels and Expressions"
type_synonym level = nat
class sec =
fixes sec :: "'a => nat"
text{* The security/confidentiality level of each variable is globally fixed
for simplicity. For the sake of examples --- the general theory does not rely
on it! --- a variable of length @{text n} has security level @{text n}: *}
instantiation list :: (type)sec
begin
definition "sec(x :: 'a list) = length x"
instance ..
end
instantiation aexp :: sec
begin
fun sec_aexp :: "aexp => level" where
"sec (N n) = 0" |
"sec (V x) = sec x" |
"sec (Plus a⇩_{1} a⇩_{2}) = max (sec a⇩_{1}) (sec a⇩_{2})"
instance ..
end
instantiation bexp :: sec
begin
fun sec_bexp :: "bexp => level" where
"sec (Bc v) = 0" |
"sec (Not b) = sec b" |
"sec (And b⇩_{1} b⇩_{2}) = max (sec b⇩_{1}) (sec b⇩_{2})" |
"sec (Less a⇩_{1} a⇩_{2}) = max (sec a⇩_{1}) (sec a⇩_{2})"
instance ..
end
abbreviation eq_le :: "state => state => level => bool"
("(_ = _ '(≤ _'))" [51,51,0] 50) where
"s = s' (≤ l) == (∀ x. sec x ≤ l --> s x = s' x)"
abbreviation eq_less :: "state => state => level => bool"
("(_ = _ '(< _'))" [51,51,0] 50) where
"s = s' (< l) == (∀ x. sec x < l --> s x = s' x)"
lemma aval_eq_if_eq_le:
"[| s⇩_{1} = s⇩_{2} (≤ l); sec a ≤ l |] ==> aval a s⇩_{1} = aval a s⇩_{2}"
by (induct a) auto
lemma bval_eq_if_eq_le:
"[| s⇩_{1} = s⇩_{2} (≤ l); sec b ≤ l |] ==> bval b s⇩_{1} = bval b s⇩_{2}"
by (induct b) (auto simp add: aval_eq_if_eq_le)
end