(* Title: HOL/Induct/QuoDataType
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 2004 University of Cambridge
*)
header{*Defining an Initial Algebra by Quotienting a Free Algebra*}
theory QuoDataType imports Main begin
subsection{*Defining the Free Algebra*}
text{*Messages with encryption and decryption as free constructors.*}
datatype
freemsg = NONCE nat
| MPAIR freemsg freemsg
| CRYPT nat freemsg
| DECRYPT nat freemsg
text{*The equivalence relation, which makes encryption and decryption inverses
provided the keys are the same.*}
consts msgrel :: "(freemsg * freemsg) set"
abbreviation
msg_rel :: "[freemsg, freemsg] => bool" (infixl "~~" 50)
"X ~~ Y == (X,Y) \<in> msgrel"
const_syntax (xsymbols)
msg_rel (infixl "\<sim>" 50)
const_syntax (HTML output)
msg_rel (infixl "\<sim>" 50)
text{*The first two rules are the desired equations. The next four rules
make the equations applicable to subterms. The last two rules are symmetry
and transitivity.*}
inductive "msgrel"
intros
CD: "CRYPT K (DECRYPT K X) \<sim> X"
DC: "DECRYPT K (CRYPT K X) \<sim> X"
NONCE: "NONCE N \<sim> NONCE N"
MPAIR: "\<lbrakk>X \<sim> X'; Y \<sim> Y'\<rbrakk> \<Longrightarrow> MPAIR X Y \<sim> MPAIR X' Y'"
CRYPT: "X \<sim> X' \<Longrightarrow> CRYPT K X \<sim> CRYPT K X'"
DECRYPT: "X \<sim> X' \<Longrightarrow> DECRYPT K X \<sim> DECRYPT K X'"
SYM: "X \<sim> Y \<Longrightarrow> Y \<sim> X"
TRANS: "\<lbrakk>X \<sim> Y; Y \<sim> Z\<rbrakk> \<Longrightarrow> X \<sim> Z"
text{*Proving that it is an equivalence relation*}
lemma msgrel_refl: "X \<sim> X"
by (induct X) (blast intro: msgrel.intros)+
theorem equiv_msgrel: "equiv UNIV msgrel"
proof -
have "reflexive msgrel" by (simp add: refl_def msgrel_refl)
moreover have "sym msgrel" by (simp add: sym_def, blast intro: msgrel.SYM)
moreover have "trans msgrel" by (simp add: trans_def, blast intro: msgrel.TRANS)
ultimately show ?thesis by (simp add: equiv_def)
qed
subsection{*Some Functions on the Free Algebra*}
subsubsection{*The Set of Nonces*}
text{*A function to return the set of nonces present in a message. It will
be lifted to the initial algrebra, to serve as an example of that process.*}
consts
freenonces :: "freemsg \<Rightarrow> nat set"
primrec
"freenonces (NONCE N) = {N}"
"freenonces (MPAIR X Y) = freenonces X \<union> freenonces Y"
"freenonces (CRYPT K X) = freenonces X"
"freenonces (DECRYPT K X) = freenonces X"
text{*This theorem lets us prove that the nonces function respects the
equivalence relation. It also helps us prove that Nonce
(the abstract constructor) is injective*}
theorem msgrel_imp_eq_freenonces: "U \<sim> V \<Longrightarrow> freenonces U = freenonces V"
by (induct set: msgrel) auto
subsubsection{*The Left Projection*}
text{*A function to return the left part of the top pair in a message. It will
be lifted to the initial algrebra, to serve as an example of that process.*}
consts freeleft :: "freemsg \<Rightarrow> freemsg"
primrec
"freeleft (NONCE N) = NONCE N"
"freeleft (MPAIR X Y) = X"
"freeleft (CRYPT K X) = freeleft X"
"freeleft (DECRYPT K X) = freeleft X"
text{*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*}
theorem msgrel_imp_eqv_freeleft:
"U \<sim> V \<Longrightarrow> freeleft U \<sim> freeleft V"
by (induct set: msgrel) (auto intro: msgrel.intros)
subsubsection{*The Right Projection*}
text{*A function to return the right part of the top pair in a message.*}
consts freeright :: "freemsg \<Rightarrow> freemsg"
primrec
"freeright (NONCE N) = NONCE N"
"freeright (MPAIR X Y) = Y"
"freeright (CRYPT K X) = freeright X"
"freeright (DECRYPT K X) = freeright X"
text{*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*}
theorem msgrel_imp_eqv_freeright:
"U \<sim> V \<Longrightarrow> freeright U \<sim> freeright V"
by (induct set: msgrel) (auto intro: msgrel.intros)
subsubsection{*The Discriminator for Constructors*}
text{*A function to distinguish nonces, mpairs and encryptions*}
consts freediscrim :: "freemsg \<Rightarrow> int"
primrec
"freediscrim (NONCE N) = 0"
"freediscrim (MPAIR X Y) = 1"
"freediscrim (CRYPT K X) = freediscrim X + 2"
"freediscrim (DECRYPT K X) = freediscrim X - 2"
text{*This theorem helps us prove @{term "Nonce N \<noteq> MPair X Y"}*}
theorem msgrel_imp_eq_freediscrim:
"U \<sim> V \<Longrightarrow> freediscrim U = freediscrim V"
by (induct set: msgrel) auto
subsection{*The Initial Algebra: A Quotiented Message Type*}
typedef (Msg) msg = "UNIV//msgrel"
by (auto simp add: quotient_def)
text{*The abstract message constructors*}
definition
Nonce :: "nat \<Rightarrow> msg"
"Nonce N = Abs_Msg(msgrel``{NONCE N})"
MPair :: "[msg,msg] \<Rightarrow> msg"
"MPair X Y =
Abs_Msg (\<Union>U \<in> Rep_Msg X. \<Union>V \<in> Rep_Msg Y. msgrel``{MPAIR U V})"
Crypt :: "[nat,msg] \<Rightarrow> msg"
"Crypt K X =
Abs_Msg (\<Union>U \<in> Rep_Msg X. msgrel``{CRYPT K U})"
Decrypt :: "[nat,msg] \<Rightarrow> msg"
"Decrypt K X =
Abs_Msg (\<Union>U \<in> Rep_Msg X. msgrel``{DECRYPT K U})"
text{*Reduces equality of equivalence classes to the @{term msgrel} relation:
@{term "(msgrel `` {x} = msgrel `` {y}) = ((x,y) \<in> msgrel)"} *}
lemmas equiv_msgrel_iff = eq_equiv_class_iff [OF equiv_msgrel UNIV_I UNIV_I]
declare equiv_msgrel_iff [simp]
text{*All equivalence classes belong to set of representatives*}
lemma [simp]: "msgrel``{U} \<in> Msg"
by (auto simp add: Msg_def quotient_def intro: msgrel_refl)
lemma inj_on_Abs_Msg: "inj_on Abs_Msg Msg"
apply (rule inj_on_inverseI)
apply (erule Abs_Msg_inverse)
done
text{*Reduces equality on abstractions to equality on representatives*}
declare inj_on_Abs_Msg [THEN inj_on_iff, simp]
declare Abs_Msg_inverse [simp]
subsubsection{*Characteristic Equations for the Abstract Constructors*}
lemma MPair: "MPair (Abs_Msg(msgrel``{U})) (Abs_Msg(msgrel``{V})) =
Abs_Msg (msgrel``{MPAIR U V})"
proof -
have "(\<lambda>U V. msgrel `` {MPAIR U V}) respects2 msgrel"
by (simp add: congruent2_def msgrel.MPAIR)
thus ?thesis
by (simp add: MPair_def UN_equiv_class2 [OF equiv_msgrel equiv_msgrel])
qed
lemma Crypt: "Crypt K (Abs_Msg(msgrel``{U})) = Abs_Msg (msgrel``{CRYPT K U})"
proof -
have "(\<lambda>U. msgrel `` {CRYPT K U}) respects msgrel"
by (simp add: congruent_def msgrel.CRYPT)
thus ?thesis
by (simp add: Crypt_def UN_equiv_class [OF equiv_msgrel])
qed
lemma Decrypt:
"Decrypt K (Abs_Msg(msgrel``{U})) = Abs_Msg (msgrel``{DECRYPT K U})"
proof -
have "(\<lambda>U. msgrel `` {DECRYPT K U}) respects msgrel"
by (simp add: congruent_def msgrel.DECRYPT)
thus ?thesis
by (simp add: Decrypt_def UN_equiv_class [OF equiv_msgrel])
qed
text{*Case analysis on the representation of a msg as an equivalence class.*}
lemma eq_Abs_Msg [case_names Abs_Msg, cases type: msg]:
"(!!U. z = Abs_Msg(msgrel``{U}) ==> P) ==> P"
apply (rule Rep_Msg [of z, unfolded Msg_def, THEN quotientE])
apply (drule arg_cong [where f=Abs_Msg])
apply (auto simp add: Rep_Msg_inverse intro: msgrel_refl)
done
text{*Establishing these two equations is the point of the whole exercise*}
theorem CD_eq [simp]: "Crypt K (Decrypt K X) = X"
by (cases X, simp add: Crypt Decrypt CD)
theorem DC_eq [simp]: "Decrypt K (Crypt K X) = X"
by (cases X, simp add: Crypt Decrypt DC)
subsection{*The Abstract Function to Return the Set of Nonces*}
definition
nonces :: "msg \<Rightarrow> nat set"
"nonces X = (\<Union>U \<in> Rep_Msg X. freenonces U)"
lemma nonces_congruent: "freenonces respects msgrel"
by (simp add: congruent_def msgrel_imp_eq_freenonces)
text{*Now prove the four equations for @{term nonces}*}
lemma nonces_Nonce [simp]: "nonces (Nonce N) = {N}"
by (simp add: nonces_def Nonce_def
UN_equiv_class [OF equiv_msgrel nonces_congruent])
lemma nonces_MPair [simp]: "nonces (MPair X Y) = nonces X \<union> nonces Y"
apply (cases X, cases Y)
apply (simp add: nonces_def MPair
UN_equiv_class [OF equiv_msgrel nonces_congruent])
done
lemma nonces_Crypt [simp]: "nonces (Crypt K X) = nonces X"
apply (cases X)
apply (simp add: nonces_def Crypt
UN_equiv_class [OF equiv_msgrel nonces_congruent])
done
lemma nonces_Decrypt [simp]: "nonces (Decrypt K X) = nonces X"
apply (cases X)
apply (simp add: nonces_def Decrypt
UN_equiv_class [OF equiv_msgrel nonces_congruent])
done
subsection{*The Abstract Function to Return the Left Part*}
definition
left :: "msg \<Rightarrow> msg"
"left X = Abs_Msg (\<Union>U \<in> Rep_Msg X. msgrel``{freeleft U})"
lemma left_congruent: "(\<lambda>U. msgrel `` {freeleft U}) respects msgrel"
by (simp add: congruent_def msgrel_imp_eqv_freeleft)
text{*Now prove the four equations for @{term left}*}
lemma left_Nonce [simp]: "left (Nonce N) = Nonce N"
by (simp add: left_def Nonce_def
UN_equiv_class [OF equiv_msgrel left_congruent])
lemma left_MPair [simp]: "left (MPair X Y) = X"
apply (cases X, cases Y)
apply (simp add: left_def MPair
UN_equiv_class [OF equiv_msgrel left_congruent])
done
lemma left_Crypt [simp]: "left (Crypt K X) = left X"
apply (cases X)
apply (simp add: left_def Crypt
UN_equiv_class [OF equiv_msgrel left_congruent])
done
lemma left_Decrypt [simp]: "left (Decrypt K X) = left X"
apply (cases X)
apply (simp add: left_def Decrypt
UN_equiv_class [OF equiv_msgrel left_congruent])
done
subsection{*The Abstract Function to Return the Right Part*}
definition
right :: "msg \<Rightarrow> msg"
"right X = Abs_Msg (\<Union>U \<in> Rep_Msg X. msgrel``{freeright U})"
lemma right_congruent: "(\<lambda>U. msgrel `` {freeright U}) respects msgrel"
by (simp add: congruent_def msgrel_imp_eqv_freeright)
text{*Now prove the four equations for @{term right}*}
lemma right_Nonce [simp]: "right (Nonce N) = Nonce N"
by (simp add: right_def Nonce_def
UN_equiv_class [OF equiv_msgrel right_congruent])
lemma right_MPair [simp]: "right (MPair X Y) = Y"
apply (cases X, cases Y)
apply (simp add: right_def MPair
UN_equiv_class [OF equiv_msgrel right_congruent])
done
lemma right_Crypt [simp]: "right (Crypt K X) = right X"
apply (cases X)
apply (simp add: right_def Crypt
UN_equiv_class [OF equiv_msgrel right_congruent])
done
lemma right_Decrypt [simp]: "right (Decrypt K X) = right X"
apply (cases X)
apply (simp add: right_def Decrypt
UN_equiv_class [OF equiv_msgrel right_congruent])
done
subsection{*Injectivity Properties of Some Constructors*}
lemma NONCE_imp_eq: "NONCE m \<sim> NONCE n \<Longrightarrow> m = n"
by (drule msgrel_imp_eq_freenonces, simp)
text{*Can also be proved using the function @{term nonces}*}
lemma Nonce_Nonce_eq [iff]: "(Nonce m = Nonce n) = (m = n)"
by (auto simp add: Nonce_def msgrel_refl dest: NONCE_imp_eq)
lemma MPAIR_imp_eqv_left: "MPAIR X Y \<sim> MPAIR X' Y' \<Longrightarrow> X \<sim> X'"
by (drule msgrel_imp_eqv_freeleft, simp)
lemma MPair_imp_eq_left:
assumes eq: "MPair X Y = MPair X' Y'" shows "X = X'"
proof -
from eq
have "left (MPair X Y) = left (MPair X' Y')" by simp
thus ?thesis by simp
qed
lemma MPAIR_imp_eqv_right: "MPAIR X Y \<sim> MPAIR X' Y' \<Longrightarrow> Y \<sim> Y'"
by (drule msgrel_imp_eqv_freeright, simp)
lemma MPair_imp_eq_right: "MPair X Y = MPair X' Y' \<Longrightarrow> Y = Y'"
apply (cases X, cases X', cases Y, cases Y')
apply (simp add: MPair)
apply (erule MPAIR_imp_eqv_right)
done
theorem MPair_MPair_eq [iff]: "(MPair X Y = MPair X' Y') = (X=X' & Y=Y')"
by (blast dest: MPair_imp_eq_left MPair_imp_eq_right)
lemma NONCE_neqv_MPAIR: "NONCE m \<sim> MPAIR X Y \<Longrightarrow> False"
by (drule msgrel_imp_eq_freediscrim, simp)
theorem Nonce_neq_MPair [iff]: "Nonce N \<noteq> MPair X Y"
apply (cases X, cases Y)
apply (simp add: Nonce_def MPair)
apply (blast dest: NONCE_neqv_MPAIR)
done
text{*Example suggested by a referee*}
theorem Crypt_Nonce_neq_Nonce: "Crypt K (Nonce M) \<noteq> Nonce N"
by (auto simp add: Nonce_def Crypt dest: msgrel_imp_eq_freediscrim)
text{*...and many similar results*}
theorem Crypt2_Nonce_neq_Nonce: "Crypt K (Crypt K' (Nonce M)) \<noteq> Nonce N"
by (auto simp add: Nonce_def Crypt dest: msgrel_imp_eq_freediscrim)
theorem Crypt_Crypt_eq [iff]: "(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]: "(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: "\<And>N. P (Nonce N)"
and M: "\<And>X Y. \<lbrakk>P X; P Y\<rbrakk> \<Longrightarrow> P (MPair X Y)"
and C: "\<And>K X. P X \<Longrightarrow> P (Crypt K X)"
and D: "\<And>K X. P X \<Longrightarrow> P (Decrypt K X)"
shows "P msg"
proof (cases msg)
case (Abs_Msg U)
have "P (Abs_Msg (msgrel `` {U}))"
proof (induct U)
case (NONCE N)
with N show ?case by (simp add: Nonce_def)
next
case (MPAIR X Y)
with M [of "Abs_Msg (msgrel `` {X})" "Abs_Msg (msgrel `` {Y})"]
show ?case by (simp add: MPair)
next
case (CRYPT K X)
with C [of "Abs_Msg (msgrel `` {X})"]
show ?case by (simp add: Crypt)
next
case (DECRYPT K X)
with D [of "Abs_Msg (msgrel `` {X})"]
show ?case by (simp add: Decrypt)
qed
with Abs_Msg show ?thesis by (simp only:)
qed
subsection{*The Abstract Discriminator*}
text{*However, as @{text Crypt_Nonce_neq_Nonce} above illustrates, we don't
need this function in order to prove discrimination theorems.*}
definition
discrim :: "msg \<Rightarrow> int"
"discrim X = contents (\<Union>U \<in> Rep_Msg X. {freediscrim U})"
lemma discrim_congruent: "(\<lambda>U. {freediscrim U}) respects msgrel"
by (simp add: congruent_def msgrel_imp_eq_freediscrim)
text{*Now prove the four equations for @{term discrim}*}
lemma discrim_Nonce [simp]: "discrim (Nonce N) = 0"
by (simp add: discrim_def Nonce_def
UN_equiv_class [OF equiv_msgrel discrim_congruent])
lemma discrim_MPair [simp]: "discrim (MPair X Y) = 1"
apply (cases X, cases Y)
apply (simp add: discrim_def MPair
UN_equiv_class [OF equiv_msgrel discrim_congruent])
done
lemma discrim_Crypt [simp]: "discrim (Crypt K X) = discrim X + 2"
apply (cases X)
apply (simp add: discrim_def Crypt
UN_equiv_class [OF equiv_msgrel discrim_congruent])
done
lemma discrim_Decrypt [simp]: "discrim (Decrypt K X) = discrim X - 2"
apply (cases X)
apply (simp add: discrim_def Decrypt
UN_equiv_class [OF equiv_msgrel discrim_congruent])
done
end