theory Hotel_Example
imports Main
begin
datatype guest = Guest0 | Guest1
datatype key = Key0 | Key1 | Key2 | Key3
datatype room = Room0
type_synonym card = "key * key"
datatype event =
Check_in guest room card | Enter guest room card | Exit guest room
definition initk :: "room \<Rightarrow> key"
where "initk = (%r. Key0)"
declare initk_def[code_pred_def, code]
primrec owns :: "event list \<Rightarrow> room \<Rightarrow> guest option"
where
"owns [] r = None"
| "owns (e#s) r = (case e of
Check_in g r' c \<Rightarrow> if r' = r then Some g else owns s r |
Enter g r' c \<Rightarrow> owns s r |
Exit g r' \<Rightarrow> owns s r)"
primrec currk :: "event list \<Rightarrow> room \<Rightarrow> key"
where
"currk [] r = initk r"
| "currk (e#s) r = (let k = currk s r in
case e of Check_in g r' (k1, k2) \<Rightarrow> if r' = r then k2 else k
| Enter g r' c \<Rightarrow> k
| Exit g r \<Rightarrow> k)"
primrec issued :: "event list \<Rightarrow> key set"
where
"issued [] = range initk"
| "issued (e#s) = issued s \<union>
(case e of Check_in g r (k1, k2) \<Rightarrow> {k2} | Enter g r c \<Rightarrow> {} | Exit g r \<Rightarrow> {})"
primrec cards :: "event list \<Rightarrow> guest \<Rightarrow> card set"
where
"cards [] g = {}"
| "cards (e#s) g = (let C = cards s g in
case e of Check_in g' r c \<Rightarrow> if g' = g then insert c C
else C
| Enter g r c \<Rightarrow> C
| Exit g r \<Rightarrow> C)"
primrec roomk :: "event list \<Rightarrow> room \<Rightarrow> key"
where
"roomk [] r = initk r"
| "roomk (e#s) r = (let k = roomk s r in
case e of Check_in g r' c \<Rightarrow> k
| Enter g r' (x,y) \<Rightarrow> if r' = r (*& x = k*) then y else k
| Exit g r \<Rightarrow> k)"
primrec isin :: "event list \<Rightarrow> room \<Rightarrow> guest set"
where
"isin [] r = {}"
| "isin (e#s) r = (let G = isin s r in
case e of Check_in g r c \<Rightarrow> G
| Enter g r' c \<Rightarrow> if r' = r then {g} \<union> G else G
| Exit g r' \<Rightarrow> if r'=r then G - {g} else G)"
primrec hotel :: "event list \<Rightarrow> bool"
where
"hotel [] = True"
| "hotel (e # s) = (hotel s & (case e of
Check_in g r (k,k') \<Rightarrow> k = currk s r \<and> k' \<notin> issued s |
Enter g r (k,k') \<Rightarrow> (k,k') : cards s g & (roomk s r : {k, k'}) |
Exit g r \<Rightarrow> g : isin s r))"
definition no_Check_in :: "event list \<Rightarrow> room \<Rightarrow> bool" where(*>*)
[code del]: "no_Check_in s r \<equiv> \<not>(\<exists>g c. Check_in g r c \<in> set s)"
definition feels_safe :: "event list \<Rightarrow> room \<Rightarrow> bool"
where
"feels_safe s r = (\<exists>s\<^isub>1 s\<^isub>2 s\<^isub>3 g c c'.
s = s\<^isub>3 @ [Enter g r c] @ s\<^isub>2 @ [Check_in g r c'] @ s\<^isub>1 \<and>
no_Check_in (s\<^isub>3 @ s\<^isub>2) r \<and> isin (s\<^isub>2 @ [Check_in g r c] @ s\<^isub>1) r = {})"
section {* Some setup *}
lemma issued_nil: "issued [] = {Key0}"
by (auto simp add: initk_def)
lemmas issued_simps[code, code_pred_def] = issued_nil issued.simps(2)
end