Theory List_Msg

(*  Title:      HOL/Auth/Guard/List_Msg.thy
    Author:     Frederic Blanqui, University of Cambridge Computer Laboratory
    Copyright   2001  University of Cambridge

section‹Lists of Messages and Lists of Agents›

theory List_Msg imports Extensions begin

subsection‹Implementation of Lists by Messages›

subsubsection‹nil is represented by any message which is not a pair›

abbreviation (input)
  cons :: "msg => msg => msg" where
  "cons x l == x,l"

subsubsection‹induction principle›

lemma lmsg_induct: "!!x. not_MPair x  P x; !!x l. P l  P (cons x l)
 P l"
by (induct l) auto


primrec head :: "msg => msg" where
"head (cons x l) = x"


primrec tail :: "msg => msg" where
"tail (cons x l) = l"


fun len :: "msg => nat" where
"len (cons x l) = Suc (len l)" |
"len other = 0"

lemma len_not_empty: "n < len l  x l'. l = cons x l'"
by (cases l) auto


fun isin :: "msg * msg => bool" where
"isin (x, cons y l) = (x=y | isin (x,l))" |
"isin (x, other) = False"

subsubsection‹delete an element›

fun del :: "msg * msg => msg" where
"del (x, cons y l) = (if x=y then l else cons y (del (x,l)))" |
"del (x, other) = other"

lemma notin_del [simp]: "~ isin (x,l)  del (x,l) = l"
by (induct l) auto

lemma isin_del [rule_format]: "isin (y, del (x,l)) --> isin (y,l)"
by (induct l) auto


fun app :: "msg * msg => msg" where
"app (cons x l, l') = cons x (app (l,l'))" |
"app (other, l') = l'"

lemma isin_app [iff]: "isin (x, app(l,l')) = (isin (x,l) | isin (x,l'))"
by (induct l) auto


fun repl :: "msg * nat * msg => msg" where
"repl (cons x l, Suc i, x') = cons x (repl (l,i,x'))" |
"repl (cons x l, 0, x') = cons x' l" |
"repl (other, i, M') = other"

subsubsection‹ith element›

fun ith :: "msg * nat => msg" where
"ith (cons x l, Suc i) = ith (l,i)" |
"ith (cons x l, 0) = x" |
"ith (other, i) = other"

lemma ith_head: "0 < len l  ith (l,0) = head l"
by (cases l) auto


fun ins :: "msg * nat * msg => msg" where
"ins (cons x l, Suc i, y) = cons x (ins (l,i,y))" |
"ins (l, 0, y) = cons y l"

lemma ins_head [simp]: "ins (l,0,y) = cons y l"
by (cases l) auto


fun trunc :: "msg * nat => msg" where
"trunc (l,0) = l" |
"trunc (cons x l, Suc i) = trunc (l,i)"

lemma trunc_zero [simp]: "trunc (l,0) = l"
by (cases l) auto

subsection‹Agent Lists›

subsubsection‹set of well-formed agent-list messages›

  nil :: msg where
  "nil == Number 0"

inductive_set agl :: "msg set"
  Nil[intro]: "nil  agl"
| Cons[intro]: "A  agent; I  agl  cons (Agent A) I  agl"

subsubsection‹basic facts about agent lists›

lemma del_in_agl [intro]: "I  agl  del (a,I)  agl"
by (erule agl.induct, auto)

lemma app_in_agl [intro]: "I  agl; J  agl  app (I,J)  agl"
by (erule agl.induct, auto)

lemma no_Key_in_agl: "I  agl  Key K  parts {I}"
by (erule agl.induct, auto)

lemma no_Nonce_in_agl: "I  agl  Nonce n  parts {I}"
by (erule agl.induct, auto)

lemma no_Key_in_appdel: "I  agl; J  agl 
Key K  parts {app (J, del (Agent B, I))}"
by (rule no_Key_in_agl, auto)

lemma no_Nonce_in_appdel: "I  agl; J  agl 
Nonce n  parts {app (J, del (Agent B, I))}"
by (rule no_Nonce_in_agl, auto)

lemma no_Crypt_in_agl: "I  agl  Crypt K X  parts {I}"
by (erule agl.induct, auto)

lemma no_Crypt_in_appdel: "I  agl; J  agl 
Crypt K X  parts {app (J, del (Agent B,I))}"
by (rule no_Crypt_in_agl, auto)