(* Title: HOL/Quotient_Examples/Quotient_Message.thy
Author: Christian Urban
Message datatype, based on an older version by Larry Paulson.
*)
theory Quotient_Message
imports Main "HOL-Library.Quotient_Syntax"
begin
subsection\<open>Defining the Free Algebra\<close>
datatype
freemsg = NONCE nat
| MPAIR freemsg freemsg
| CRYPT nat freemsg
| DECRYPT nat freemsg
inductive
msgrel::"freemsg \ freemsg \ bool" (infixl "\" 50)
where
CD: "CRYPT K (DECRYPT K X) \ X"
| DC: "DECRYPT K (CRYPT K X) \ X"
| NONCE: "NONCE N \ NONCE N"
| MPAIR: "\X \ X'; Y \ Y'\ \ MPAIR X Y \ MPAIR X' Y'"
| CRYPT: "X \ X' \ CRYPT K X \ CRYPT K X'"
| DECRYPT: "X \ X' \ DECRYPT K X \ DECRYPT K X'"
| SYM: "X \ Y \ Y \ X"
| TRANS: "\X \ Y; Y \ Z\ \ X \ Z"
lemmas msgrel.intros[intro]
text\<open>Proving that it is an equivalence relation\<close>
lemma msgrel_refl: "X \ X"
by (induct X) (auto intro: msgrel.intros)
theorem equiv_msgrel: "equivp msgrel"
proof (rule equivpI)
show "reflp msgrel" by (rule reflpI) (simp add: msgrel_refl)
show "symp msgrel" by (rule sympI) (blast intro: msgrel.SYM)
show "transp msgrel" by (rule transpI) (blast intro: msgrel.TRANS)
qed
subsection\<open>Some Functions on the Free Algebra\<close>
subsubsection\<open>The Set of Nonces\<close>
primrec
freenonces :: "freemsg \ nat set"
where
"freenonces (NONCE N) = {N}"
| "freenonces (MPAIR X Y) = freenonces X \ freenonces Y"
| "freenonces (CRYPT K X) = freenonces X"
| "freenonces (DECRYPT K X) = freenonces X"
theorem msgrel_imp_eq_freenonces:
assumes a: "U \ V"
shows "freenonces U = freenonces V"
using a by (induct) (auto)
subsubsection\<open>The Left Projection\<close>
text\<open>A function to return the left part of the top pair in a message. It will
be lifted to the initial algebra, to serve as an example of that process.\<close>
primrec
freeleft :: "freemsg \ freemsg"
where
"freeleft (NONCE N) = NONCE N"
| "freeleft (MPAIR X Y) = X"
| "freeleft (CRYPT K X) = freeleft X"
| "freeleft (DECRYPT K X) = freeleft X"
text\<open>This theorem lets us prove that the left function respects the
equivalence relation. It also helps us prove that MPair
(the abstract constructor) is injective\<close>
lemma msgrel_imp_eqv_freeleft_aux:
shows "freeleft U \ freeleft U"
by (fact msgrel_refl)
theorem msgrel_imp_eqv_freeleft:
assumes a: "U \ V"
shows "freeleft U \ freeleft V"
using a
by (induct) (auto intro: msgrel_imp_eqv_freeleft_aux)
subsubsection\<open>The Right Projection\<close>
text\<open>A function to return the right part of the top pair in a message.\<close>
primrec
freeright :: "freemsg \ freemsg"
where
"freeright (NONCE N) = NONCE N"
| "freeright (MPAIR X Y) = Y"
| "freeright (CRYPT K X) = freeright X"
| "freeright (DECRYPT K X) = freeright X"
text\<open>This theorem lets us prove that the right function respects the
equivalence relation. It also helps us prove that MPair
(the abstract constructor) is injective\<close>
lemma msgrel_imp_eqv_freeright_aux:
shows "freeright U \ freeright U"
by (fact msgrel_refl)
theorem msgrel_imp_eqv_freeright:
assumes a: "U \ V"
shows "freeright U \ freeright V"
using a
by (induct) (auto intro: msgrel_imp_eqv_freeright_aux)
subsubsection\<open>The Discriminator for Constructors\<close>
text\<open>A function to distinguish nonces, mpairs and encryptions\<close>
primrec
freediscrim :: "freemsg \ int"
where
"freediscrim (NONCE N) = 0"
| "freediscrim (MPAIR X Y) = 1"
| "freediscrim (CRYPT K X) = freediscrim X + 2"
| "freediscrim (DECRYPT K X) = freediscrim X - 2"
text\<open>This theorem helps us prove \<^term>\<open>Nonce N \<noteq> MPair X Y\<close>\<close>
theorem msgrel_imp_eq_freediscrim:
assumes a: "U \ V"
shows "freediscrim U = freediscrim V"
using a by (induct) (auto)
subsection\<open>The Initial Algebra: A Quotiented Message Type\<close>
quotient_type msg = freemsg / msgrel
by (rule equiv_msgrel)
text\<open>The abstract message constructors\<close>
quotient_definition
"Nonce :: nat \ msg"
is
"NONCE"
done
quotient_definition
"MPair :: msg \ msg \ msg"
is
"MPAIR"
by (rule MPAIR)
quotient_definition
"Crypt :: nat \ msg \ msg"
is
"CRYPT"
by (rule CRYPT)
quotient_definition
"Decrypt :: nat \ msg \ msg"
is
"DECRYPT"
by (rule DECRYPT)
text\<open>Establishing these two equations is the point of the whole exercise\<close>
theorem CD_eq [simp]:
shows "Crypt K (Decrypt K X) = X"
by (lifting CD)
theorem DC_eq [simp]:
shows "Decrypt K (Crypt K X) = X"
by (lifting DC)
subsection\<open>The Abstract Function to Return the Set of Nonces\<close>
quotient_definition
"nonces:: msg \ nat set"
is
"freenonces"
by (rule msgrel_imp_eq_freenonces)
text\<open>Now prove the four equations for \<^term>\<open>nonces\<close>\<close>
lemma nonces_Nonce [simp]:
shows "nonces (Nonce N) = {N}"
by (lifting freenonces.simps(1))
lemma nonces_MPair [simp]:
shows "nonces (MPair X Y) = nonces X \ nonces Y"
by (lifting freenonces.simps(2))
lemma nonces_Crypt [simp]:
shows "nonces (Crypt K X) = nonces X"
by (lifting freenonces.simps(3))
lemma nonces_Decrypt [simp]:
shows "nonces (Decrypt K X) = nonces X"
by (lifting freenonces.simps(4))
subsection\<open>The Abstract Function to Return the Left Part\<close>
quotient_definition
"left:: msg \ msg"
is
"freeleft"
by (rule msgrel_imp_eqv_freeleft)
lemma left_Nonce [simp]:
shows "left (Nonce N) = Nonce N"
by (lifting freeleft.simps(1))
lemma left_MPair [simp]:
shows "left (MPair X Y) = X"
by (lifting freeleft.simps(2))
lemma left_Crypt [simp]:
shows "left (Crypt K X) = left X"
by (lifting freeleft.simps(3))
lemma left_Decrypt [simp]:
shows "left (Decrypt K X) = left X"
by (lifting freeleft.simps(4))
subsection\<open>The Abstract Function to Return the Right Part\<close>
quotient_definition
"right:: msg \ msg"
is
"freeright"
by (rule msgrel_imp_eqv_freeright)
text\<open>Now prove the four equations for \<^term>\<open>right\<close>\<close>
lemma right_Nonce [simp]:
shows "right (Nonce N) = Nonce N"
by (lifting freeright.simps(1))
lemma right_MPair [simp]:
shows "right (MPair X Y) = Y"
by (lifting freeright.simps(2))
lemma right_Crypt [simp]:
shows "right (Crypt K X) = right X"
by (lifting freeright.simps(3))
lemma right_Decrypt [simp]:
shows "right (Decrypt K X) = right X"
by (lifting freeright.simps(4))
subsection\<open>Injectivity Properties of Some Constructors\<close>
text\<open>Can also be proved using the function \<^term>\<open>nonces\<close>\<close>
lemma Nonce_Nonce_eq [iff]:
shows "(Nonce m = Nonce n) = (m = n)"
proof
assume "Nonce m = Nonce n"
then show "m = n"
by (descending) (drule msgrel_imp_eq_freenonces, simp)
next
assume "m = n"
then show "Nonce m = Nonce n" by simp
qed
lemma MPair_imp_eq_left:
assumes eq: "MPair X Y = MPair X' Y'"
shows "X = X'"
using eq
by (descending) (drule msgrel_imp_eqv_freeleft, simp)
lemma MPair_imp_eq_right:
shows "MPair X Y = MPair X' Y' \ Y = Y'"
by (descending) (drule msgrel_imp_eqv_freeright, simp)
theorem MPair_MPair_eq [iff]:
shows "(MPair X Y = MPair X' Y') = (X=X' & Y=Y')"
by (blast dest: MPair_imp_eq_left MPair_imp_eq_right)
theorem Nonce_neq_MPair [iff]:
shows "Nonce N \ MPair X Y"
by (descending) (auto dest: msgrel_imp_eq_freediscrim)
text\<open>Example suggested by a referee\<close>
theorem Crypt_Nonce_neq_Nonce:
shows "Crypt K (Nonce M) \ Nonce N"
by (descending) (auto dest: msgrel_imp_eq_freediscrim)
text\<open>...and many similar results\<close>
theorem Crypt2_Nonce_neq_Nonce:
shows "Crypt K (Crypt K' (Nonce M)) \ Nonce N"
by (descending) (auto dest: msgrel_imp_eq_freediscrim)
theorem Crypt_Crypt_eq [iff]:
shows "(Crypt K X = Crypt K X') = (X=X')"
proof
assume "Crypt K X = Crypt K X'"
hence "Decrypt K (Crypt K X) = Decrypt K (Crypt K X')" by simp
thus "X = X'" by simp
next
assume "X = X'"
thus "Crypt K X = Crypt K X'" by simp
qed
theorem Decrypt_Decrypt_eq [iff]:
shows "(Decrypt K X = Decrypt K X') = (X=X')"
proof
assume "Decrypt K X = Decrypt K X'"
hence "Crypt K (Decrypt K X) = Crypt K (Decrypt K X')" by simp
thus "X = X'" by simp
next
assume "X = X'"
thus "Decrypt K X = Decrypt K X'" by simp
qed
lemma msg_induct [case_names Nonce MPair Crypt Decrypt, cases type: msg]:
assumes N: "\N. P (Nonce N)"
and M: "\X Y. \P X; P Y\ \ P (MPair X Y)"
and C: "\K X. P X \ P (Crypt K X)"
and D: "\K X. P X \ P (Decrypt K X)"
shows "P msg"
using N M C D
by (descending) (auto intro: freemsg.induct)
subsection\<open>The Abstract Discriminator\<close>
text\<open>However, as \<open>Crypt_Nonce_neq_Nonce\<close> above illustrates, we don't
need this function in order to prove discrimination theorems.\<close>
quotient_definition
"discrim:: msg \ int"
is
"freediscrim"
by (rule msgrel_imp_eq_freediscrim)
text\<open>Now prove the four equations for \<^term>\<open>discrim\<close>\<close>
lemma discrim_Nonce [simp]:
shows "discrim (Nonce N) = 0"
by (lifting freediscrim.simps(1))
lemma discrim_MPair [simp]:
shows "discrim (MPair X Y) = 1"
by (lifting freediscrim.simps(2))
lemma discrim_Crypt [simp]:
shows "discrim (Crypt K X) = discrim X + 2"
by (lifting freediscrim.simps(3))
lemma discrim_Decrypt [simp]:
shows "discrim (Decrypt K X) = discrim X - 2"
by (lifting freediscrim.simps(4))
end
¤ Dauer der Verarbeitung: 0.2 Sekunden
(vorverarbeitet)
¤
|
Haftungshinweis
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
|
