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