src/HOL/Quotient_Examples/Quotient_Message.thy

+(*  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 Quotient_Syntax
+begin
+
+subsection{*Defining the Free Algebra*}
+
+datatype
+  freemsg = NONCE  nat
+        | MPAIR  freemsg freemsg
+        | CRYPT  nat freemsg
+        | DECRYPT  nat freemsg
+
+inductive
+  msgrel::"freemsg \<Rightarrow> freemsg \<Rightarrow> bool" (infixl "\<sim>" 50)
+where
+  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"
+
+lemmas msgrel.intros[intro]
+
+text{*Proving that it is an equivalence relation*}
+
+lemma msgrel_refl: "X \<sim> X"
+by (induct X, (blast intro: msgrel.intros)+)
+
+theorem equiv_msgrel: "equivp msgrel"
+proof (rule equivpI)
+  show "reflp msgrel" by (simp add: reflp_def msgrel_refl)
+  show "symp msgrel" by (simp add: symp_def, blast intro: msgrel.SYM)
+  show "transp msgrel" by (simp add: transp_def, blast intro: msgrel.TRANS)
+qed
+
+subsection{*Some Functions on the Free Algebra*}
+
+subsubsection{*The Set of Nonces*}
+
+fun
+  freenonces :: "freemsg \<Rightarrow> nat set"
+where
+  "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"
+
+theorem msgrel_imp_eq_freenonces:
+  assumes a: "U \<sim> V"
+  shows "freenonces U = freenonces V"
+  using a by (induct) (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.*}
+fun
+  freeleft :: "freemsg \<Rightarrow> 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{*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*}
+lemma msgrel_imp_eqv_freeleft_aux:
+  shows "freeleft U \<sim> freeleft U"
+  by (induct rule: freeleft.induct) (auto)
+
+theorem msgrel_imp_eqv_freeleft:
+  assumes a: "U \<sim> V"
+  shows "freeleft U \<sim> freeleft V"
+  using a
+  by (induct) (auto intro: msgrel_imp_eqv_freeleft_aux)
+
+subsubsection{*The Right Projection*}
+
+text{*A function to return the right part of the top pair in a message.*}
+fun
+  freeright :: "freemsg \<Rightarrow> 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{*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*}
+lemma msgrel_imp_eqv_freeright_aux:
+  shows "freeright U \<sim> freeright U"
+  by (induct rule: freeright.induct) (auto)
+
+theorem msgrel_imp_eqv_freeright:
+  assumes a: "U \<sim> V"
+  shows "freeright U \<sim> freeright V"
+  using a
+  by (induct) (auto intro: msgrel_imp_eqv_freeright_aux)
+
+subsubsection{*The Discriminator for Constructors*}
+
+text{*A function to distinguish nonces, mpairs and encryptions*}
+fun
+  freediscrim :: "freemsg \<Rightarrow> 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{*This theorem helps us prove @{term "Nonce N \<noteq> MPair X Y"}*}
+theorem msgrel_imp_eq_freediscrim:
+  assumes a: "U \<sim> V"
+  shows "freediscrim U = freediscrim V"
+  using a by (induct) (auto)
+
+subsection{*The Initial Algebra: A Quotiented Message Type*}
+
+quotient_type msg = freemsg / msgrel
+  by (rule equiv_msgrel)
+
+text{*The abstract message constructors*}
+
+quotient_definition
+  "Nonce :: nat \<Rightarrow> msg"
+is
+  "NONCE"
+
+quotient_definition
+  "MPair :: msg \<Rightarrow> msg \<Rightarrow> msg"
+is
+  "MPAIR"
+
+quotient_definition
+  "Crypt :: nat \<Rightarrow> msg \<Rightarrow> msg"
+is
+  "CRYPT"
+
+quotient_definition
+  "Decrypt :: nat \<Rightarrow> msg \<Rightarrow> msg"
+is
+  "DECRYPT"
+
+lemma [quot_respect]:
+  shows "(op = ===> op \<sim> ===> op \<sim>) CRYPT CRYPT"
+by (auto intro: CRYPT)
+
+lemma [quot_respect]:
+  shows "(op = ===> op \<sim> ===> op \<sim>) DECRYPT DECRYPT"
+by (auto intro: DECRYPT)
+
+text{*Establishing these two equations is the point of the whole exercise*}
+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{*The Abstract Function to Return the Set of Nonces*}
+
+quotient_definition
+   "nonces:: msg \<Rightarrow> nat set"
+is
+  "freenonces"
+
+text{*Now prove the four equations for @{term nonces}*}
+
+lemma [quot_respect]:
+  shows "(op \<sim> ===> op =) freenonces freenonces"
+  by (simp add: msgrel_imp_eq_freenonces)
+
+lemma [quot_respect]:
+  shows "(op = ===> op \<sim>) NONCE NONCE"
+  by (simp add: NONCE)
+
+lemma nonces_Nonce [simp]:
+  shows "nonces (Nonce N) = {N}"
+  by (lifting freenonces.simps(1))
+
+lemma [quot_respect]:
+  shows " (op \<sim> ===> op \<sim> ===> op \<sim>) MPAIR MPAIR"
+  by (simp add: MPAIR)
+
+lemma nonces_MPair [simp]:
+  shows "nonces (MPair X Y) = nonces X \<union> 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{*The Abstract Function to Return the Left Part*}
+
+quotient_definition
+  "left:: msg \<Rightarrow> msg"
+is
+  "freeleft"
+
+lemma [quot_respect]:
+  shows "(op \<sim> ===> op \<sim>) freeleft freeleft"
+  by (simp add: 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{*The Abstract Function to Return the Right Part*}
+
+quotient_definition
+  "right:: msg \<Rightarrow> msg"
+is
+  "freeright"
+
+text{*Now prove the four equations for @{term right}*}
+
+lemma [quot_respect]:
+  shows "(op \<sim> ===> op \<sim>) freeright freeright"
+  by (simp add: msgrel_imp_eqv_freeright)
+
+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{*Injectivity Properties of Some Constructors*}
+
+lemma NONCE_imp_eq:
+  shows "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]:
+  shows "(Nonce m = Nonce n) = (m = n)"
+proof
+  assume "Nonce m = Nonce n"
+  then show "m = n" by (lifting NONCE_imp_eq)
+next
+  assume "m = n"
+  then show "Nonce m = Nonce n" by simp
+qed
+
+lemma MPAIR_imp_eqv_left:
+  shows "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'"
+  using eq by (lifting MPAIR_imp_eqv_left)
+
+lemma MPAIR_imp_eqv_right:
+  shows "MPAIR X Y \<sim> MPAIR X' Y' \<Longrightarrow> Y \<sim> Y'"
+  by (drule msgrel_imp_eqv_freeright) (simp)
+
+lemma MPair_imp_eq_right:
+  shows "MPair X Y = MPair X' Y' \<Longrightarrow> Y = Y'"
+  by (lifting  MPAIR_imp_eqv_right)
+
+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)
+
+lemma NONCE_neqv_MPAIR:
+  shows "\<not>(NONCE m \<sim> MPAIR X Y)"
+  by (auto dest: msgrel_imp_eq_freediscrim)
+
+theorem Nonce_neq_MPair [iff]:
+  shows "Nonce N \<noteq> MPair X Y"
+  by (lifting NONCE_neqv_MPAIR)
+
+text{*Example suggested by a referee*}
+
+lemma CRYPT_NONCE_neq_NONCE:
+  shows "\<not>(CRYPT K (NONCE M) \<sim> NONCE N)"
+  by (auto dest: msgrel_imp_eq_freediscrim)
+
+theorem Crypt_Nonce_neq_Nonce:
+  shows "Crypt K (Nonce M) \<noteq> Nonce N"
+  by (lifting CRYPT_NONCE_neq_NONCE)
+
+text{*...and many similar results*}
+lemma CRYPT2_NONCE_neq_NONCE:
+  shows "\<not>(CRYPT K (CRYPT K' (NONCE M)) \<sim> NONCE N)"
+  by (auto dest: msgrel_imp_eq_freediscrim)
+
+theorem Crypt2_Nonce_neq_Nonce:
+  shows "Crypt K (Crypt K' (Nonce M)) \<noteq> Nonce N"
+  by (lifting CRYPT2_NONCE_neq_NONCE)
+
+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_aux:
+  shows "\<lbrakk>\<And>N. P (Nonce N);
+          \<And>X Y. \<lbrakk>P X; P Y\<rbrakk> \<Longrightarrow> P (MPair X Y);
+          \<And>K X. P X \<Longrightarrow> P (Crypt K X);
+          \<And>K X. P X \<Longrightarrow> P (Decrypt K X)\<rbrakk> \<Longrightarrow> P msg"
+  by (lifting freemsg.induct)
+
+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"
+  using N M C D by (rule msg_induct_aux)
+
+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.*}
+
+quotient_definition
+  "discrim:: msg \<Rightarrow> int"
+is
+  "freediscrim"
+
+text{*Now prove the four equations for @{term discrim}*}
+
+lemma [quot_respect]:
+  shows "(op \<sim> ===> op =) freediscrim freediscrim"
+  by (auto simp add: msgrel_imp_eq_freediscrim)
+
+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
+