(* Title: HOL/Induct/QuoNestedDataType ID: $Id$ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 2004 University of Cambridge*)header{*Quotienting a Free Algebra Involving Nested Recursion*}theory QuoNestedDataType imports Main beginsubsection{*Defining the Free Algebra*}text{*Messages with encryption and decryption as free constructors.*}datatype freeExp = VAR nat | PLUS freeExp freeExp | FNCALL nat "freeExp list"text{*The equivalence relation, which makes PLUS associative.*}text{*The first rule is the desired equation. The next three rulesmake the equations applicable to subterms. The last two rules are symmetryand transitivity.*}inductive_set exprel :: "(freeExp * freeExp) set" and exp_rel :: "[freeExp, freeExp] => bool" (infixl "\<sim>" 50) where "X \<sim> Y == (X,Y) \<in> exprel" | ASSOC: "PLUS X (PLUS Y Z) \<sim> PLUS (PLUS X Y) Z" | VAR: "VAR N \<sim> VAR N" | PLUS: "\<lbrakk>X \<sim> X'; Y \<sim> Y'\<rbrakk> \<Longrightarrow> PLUS X Y \<sim> PLUS X' Y'" | FNCALL: "(Xs,Xs') \<in> listrel exprel \<Longrightarrow> FNCALL F Xs \<sim> FNCALL F Xs'" | SYM: "X \<sim> Y \<Longrightarrow> Y \<sim> X" | TRANS: "\<lbrakk>X \<sim> Y; Y \<sim> Z\<rbrakk> \<Longrightarrow> X \<sim> Z" monos listrel_monotext{*Proving that it is an equivalence relation*}lemma exprel_refl: "X \<sim> X" and list_exprel_refl: "(Xs,Xs) \<in> listrel(exprel)" by (induct X and Xs) (blast intro: exprel.intros listrel.intros)+theorem equiv_exprel: "equiv UNIV exprel"proof - have "reflexive exprel" by (simp add: refl_def exprel_refl) moreover have "sym exprel" by (simp add: sym_def, blast intro: exprel.SYM) moreover have "trans exprel" by (simp add: trans_def, blast intro: exprel.TRANS) ultimately show ?thesis by (simp add: equiv_def)qedtheorem equiv_list_exprel: "equiv UNIV (listrel exprel)" using equiv_listrel [OF equiv_exprel] by simplemma FNCALL_Nil: "FNCALL F [] \<sim> FNCALL F []"apply (rule exprel.intros) apply (rule listrel.intros) donelemma FNCALL_Cons: "\<lbrakk>X \<sim> X'; (Xs,Xs') \<in> listrel(exprel)\<rbrakk> \<Longrightarrow> FNCALL F (X#Xs) \<sim> FNCALL F (X'#Xs')"by (blast intro: exprel.intros listrel.intros) subsection{*Some Functions on the Free Algebra*}subsubsection{*The Set of Variables*}text{*A function to return the set of variables present in a message. It willbe lifted to the initial algrebra, to serve as an example of that process.Note that the "free" refers to the free datatype rather than to the conceptof a free variable.*}consts freevars :: "freeExp \<Rightarrow> nat set" freevars_list :: "freeExp list \<Rightarrow> nat set"primrec "freevars (VAR N) = {N}" "freevars (PLUS X Y) = freevars X \<union> freevars Y" "freevars (FNCALL F Xs) = freevars_list Xs" "freevars_list [] = {}" "freevars_list (X # Xs) = freevars X \<union> freevars_list Xs"text{*This theorem lets us prove that the vars function respects theequivalence relation. It also helps us prove that Variable (the abstract constructor) is injective*}theorem exprel_imp_eq_freevars: "U \<sim> V \<Longrightarrow> freevars U = freevars V"apply (induct set: exprel) apply (erule_tac [4] listrel.induct) apply (simp_all add: Un_assoc)donesubsubsection{*Functions for Freeness*}text{*A discriminator function to distinguish vars, sums and function calls*}consts freediscrim :: "freeExp \<Rightarrow> int"primrec "freediscrim (VAR N) = 0" "freediscrim (PLUS X Y) = 1" "freediscrim (FNCALL F Xs) = 2"theorem exprel_imp_eq_freediscrim: "U \<sim> V \<Longrightarrow> freediscrim U = freediscrim V" by (induct set: exprel) autotext{*This function, which returns the function name, is used toprove part of the injectivity property for FnCall.*}consts freefun :: "freeExp \<Rightarrow> nat"primrec "freefun (VAR N) = 0" "freefun (PLUS X Y) = 0" "freefun (FNCALL F Xs) = F"theorem exprel_imp_eq_freefun: "U \<sim> V \<Longrightarrow> freefun U = freefun V" by (induct set: exprel) (simp_all add: listrel.intros)text{*This function, which returns the list of function arguments, is used toprove part of the injectivity property for FnCall.*}consts freeargs :: "freeExp \<Rightarrow> freeExp list"primrec "freeargs (VAR N) = []" "freeargs (PLUS X Y) = []" "freeargs (FNCALL F Xs) = Xs"theorem exprel_imp_eqv_freeargs: "U \<sim> V \<Longrightarrow> (freeargs U, freeargs V) \<in> listrel exprel"apply (induct set: exprel)apply (erule_tac [4] listrel.induct) apply (simp_all add: listrel.intros)apply (blast intro: symD [OF equiv.sym [OF equiv_list_exprel]])apply (blast intro: transD [OF equiv.trans [OF equiv_list_exprel]])donesubsection{*The Initial Algebra: A Quotiented Message Type*}typedef (Exp) exp = "UNIV//exprel" by (auto simp add: quotient_def)text{*The abstract message constructors*}definition Var :: "nat \<Rightarrow> exp" where "Var N = Abs_Exp(exprel``{VAR N})"definition Plus :: "[exp,exp] \<Rightarrow> exp" where "Plus X Y = Abs_Exp (\<Union>U \<in> Rep_Exp X. \<Union>V \<in> Rep_Exp Y. exprel``{PLUS U V})"definition FnCall :: "[nat, exp list] \<Rightarrow> exp" where "FnCall F Xs = Abs_Exp (\<Union>Us \<in> listset (map Rep_Exp Xs). exprel `` {FNCALL F Us})"text{*Reduces equality of equivalence classes to the @{term exprel} relation: @{term "(exprel `` {x} = exprel `` {y}) = ((x,y) \<in> exprel)"} *}lemmas equiv_exprel_iff = eq_equiv_class_iff [OF equiv_exprel UNIV_I UNIV_I]declare equiv_exprel_iff [simp]text{*All equivalence classes belong to set of representatives*}lemma [simp]: "exprel``{U} \<in> Exp"by (auto simp add: Exp_def quotient_def intro: exprel_refl)lemma inj_on_Abs_Exp: "inj_on Abs_Exp Exp"apply (rule inj_on_inverseI)apply (erule Abs_Exp_inverse)donetext{*Reduces equality on abstractions to equality on representatives*}declare inj_on_Abs_Exp [THEN inj_on_iff, simp]declare Abs_Exp_inverse [simp]text{*Case analysis on the representation of a exp as an equivalence class.*}lemma eq_Abs_Exp [case_names Abs_Exp, cases type: exp]: "(!!U. z = Abs_Exp(exprel``{U}) ==> P) ==> P"apply (rule Rep_Exp [of z, unfolded Exp_def, THEN quotientE])apply (drule arg_cong [where f=Abs_Exp])apply (auto simp add: Rep_Exp_inverse intro: exprel_refl)donesubsection{*Every list of abstract expressions can be expressed in terms of a list of concrete expressions*}definition Abs_ExpList :: "freeExp list => exp list" where "Abs_ExpList Xs = map (%U. Abs_Exp(exprel``{U})) Xs"lemma Abs_ExpList_Nil [simp]: "Abs_ExpList [] == []"by (simp add: Abs_ExpList_def)lemma Abs_ExpList_Cons [simp]: "Abs_ExpList (X#Xs) == Abs_Exp (exprel``{X}) # Abs_ExpList Xs"by (simp add: Abs_ExpList_def)lemma ExpList_rep: "\<exists>Us. z = Abs_ExpList Us"apply (induct z)apply (rule_tac [2] z=a in eq_Abs_Exp)apply (auto simp add: Abs_ExpList_def Cons_eq_map_conv intro: exprel_refl)donelemma eq_Abs_ExpList [case_names Abs_ExpList]: "(!!Us. z = Abs_ExpList Us ==> P) ==> P"by (rule exE [OF ExpList_rep], blast) subsubsection{*Characteristic Equations for the Abstract Constructors*}lemma Plus: "Plus (Abs_Exp(exprel``{U})) (Abs_Exp(exprel``{V})) = Abs_Exp (exprel``{PLUS U V})"proof - have "(\<lambda>U V. exprel `` {PLUS U V}) respects2 exprel" by (simp add: congruent2_def exprel.PLUS) thus ?thesis by (simp add: Plus_def UN_equiv_class2 [OF equiv_exprel equiv_exprel])qedtext{*It is not clear what to do with FnCall: it's argument is an abstractionof an @{typ "exp list"}. Is it just Nil or Cons? What seems to work best is toregard an @{typ "exp list"} as a @{term "listrel exprel"} equivalence class*}text{*This theorem is easily proved but never used. There's no obvious wayeven to state the analogous result, @{text FnCall_Cons}.*}lemma FnCall_Nil: "FnCall F [] = Abs_Exp (exprel``{FNCALL F []})" by (simp add: FnCall_def)lemma FnCall_respects: "(\<lambda>Us. exprel `` {FNCALL F Us}) respects (listrel exprel)" by (simp add: congruent_def exprel.FNCALL)lemma FnCall_sing: "FnCall F [Abs_Exp(exprel``{U})] = Abs_Exp (exprel``{FNCALL F [U]})"proof - have "(\<lambda>U. exprel `` {FNCALL F [U]}) respects exprel" by (simp add: congruent_def FNCALL_Cons listrel.intros) thus ?thesis by (simp add: FnCall_def UN_equiv_class [OF equiv_exprel])qedlemma listset_Rep_Exp_Abs_Exp: "listset (map Rep_Exp (Abs_ExpList Us)) = listrel exprel `` {Us}"; by (induct Us) (simp_all add: listrel_Cons Abs_ExpList_def)lemma FnCall: "FnCall F (Abs_ExpList Us) = Abs_Exp (exprel``{FNCALL F Us})"proof - have "(\<lambda>Us. exprel `` {FNCALL F Us}) respects (listrel exprel)" by (simp add: congruent_def exprel.FNCALL) thus ?thesis by (simp add: FnCall_def UN_equiv_class [OF equiv_list_exprel] listset_Rep_Exp_Abs_Exp)qedtext{*Establishing this equation is the point of the whole exercise*}theorem Plus_assoc: "Plus X (Plus Y Z) = Plus (Plus X Y) Z"by (cases X, cases Y, cases Z, simp add: Plus exprel.ASSOC)subsection{*The Abstract Function to Return the Set of Variables*}definition vars :: "exp \<Rightarrow> nat set" where "vars X = (\<Union>U \<in> Rep_Exp X. freevars U)"lemma vars_respects: "freevars respects exprel"by (simp add: congruent_def exprel_imp_eq_freevars) text{*The extension of the function @{term vars} to lists*}consts vars_list :: "exp list \<Rightarrow> nat set"primrec "vars_list [] = {}" "vars_list(E#Es) = vars E \<union> vars_list Es"text{*Now prove the three equations for @{term vars}*}lemma vars_Variable [simp]: "vars (Var N) = {N}"by (simp add: vars_def Var_def UN_equiv_class [OF equiv_exprel vars_respects]) lemma vars_Plus [simp]: "vars (Plus X Y) = vars X \<union> vars Y"apply (cases X, cases Y) apply (simp add: vars_def Plus UN_equiv_class [OF equiv_exprel vars_respects]) donelemma vars_FnCall [simp]: "vars (FnCall F Xs) = vars_list Xs"apply (cases Xs rule: eq_Abs_ExpList) apply (simp add: FnCall)apply (induct_tac Us) apply (simp_all add: vars_def UN_equiv_class [OF equiv_exprel vars_respects])donelemma vars_FnCall_Nil: "vars (FnCall F Nil) = {}" by simplemma vars_FnCall_Cons: "vars (FnCall F (X#Xs)) = vars X \<union> vars_list Xs"by simpsubsection{*Injectivity Properties of Some Constructors*}lemma VAR_imp_eq: "VAR m \<sim> VAR n \<Longrightarrow> m = n"by (drule exprel_imp_eq_freevars, simp)text{*Can also be proved using the function @{term vars}*}lemma Var_Var_eq [iff]: "(Var m = Var n) = (m = n)"by (auto simp add: Var_def exprel_refl dest: VAR_imp_eq)lemma VAR_neqv_PLUS: "VAR m \<sim> PLUS X Y \<Longrightarrow> False"by (drule exprel_imp_eq_freediscrim, simp)theorem Var_neq_Plus [iff]: "Var N \<noteq> Plus X Y"apply (cases X, cases Y) apply (simp add: Var_def Plus) apply (blast dest: VAR_neqv_PLUS) donetheorem Var_neq_FnCall [iff]: "Var N \<noteq> FnCall F Xs"apply (cases Xs rule: eq_Abs_ExpList) apply (auto simp add: FnCall Var_def)apply (drule exprel_imp_eq_freediscrim, simp)donesubsection{*Injectivity of @{term FnCall}*}definition "fun" :: "exp \<Rightarrow> nat" where "fun X = contents (\<Union>U \<in> Rep_Exp X. {freefun U})"lemma fun_respects: "(%U. {freefun U}) respects exprel"by (simp add: congruent_def exprel_imp_eq_freefun) lemma fun_FnCall [simp]: "fun (FnCall F Xs) = F"apply (cases Xs rule: eq_Abs_ExpList) apply (simp add: FnCall fun_def UN_equiv_class [OF equiv_exprel fun_respects])donedefinition args :: "exp \<Rightarrow> exp list" where "args X = contents (\<Union>U \<in> Rep_Exp X. {Abs_ExpList (freeargs U)})"text{*This result can probably be generalized to arbitrary equivalencerelations, but with little benefit here.*}lemma Abs_ExpList_eq: "(y, z) \<in> listrel exprel \<Longrightarrow> Abs_ExpList (y) = Abs_ExpList (z)" by (induct set: listrel) simp_alllemma args_respects: "(%U. {Abs_ExpList (freeargs U)}) respects exprel"by (simp add: congruent_def Abs_ExpList_eq exprel_imp_eqv_freeargs) lemma args_FnCall [simp]: "args (FnCall F Xs) = Xs"apply (cases Xs rule: eq_Abs_ExpList) apply (simp add: FnCall args_def UN_equiv_class [OF equiv_exprel args_respects])donelemma FnCall_FnCall_eq [iff]: "(FnCall F Xs = FnCall F' Xs') = (F=F' & Xs=Xs')" proof assume "FnCall F Xs = FnCall F' Xs'" hence "fun (FnCall F Xs) = fun (FnCall F' Xs')" and "args (FnCall F Xs) = args (FnCall F' Xs')" by auto thus "F=F' & Xs=Xs'" by simpnext assume "F=F' & Xs=Xs'" thus "FnCall F Xs = FnCall F' Xs'" by simpqedsubsection{*The Abstract Discriminator*}text{*However, as @{text FnCall_Var_neq_Var} illustrates, we don't need thisfunction in order to prove discrimination theorems.*}definition discrim :: "exp \<Rightarrow> int" where "discrim X = contents (\<Union>U \<in> Rep_Exp X. {freediscrim U})"lemma discrim_respects: "(\<lambda>U. {freediscrim U}) respects exprel"by (simp add: congruent_def exprel_imp_eq_freediscrim) text{*Now prove the four equations for @{term discrim}*}lemma discrim_Var [simp]: "discrim (Var N) = 0"by (simp add: discrim_def Var_def UN_equiv_class [OF equiv_exprel discrim_respects]) lemma discrim_Plus [simp]: "discrim (Plus X Y) = 1"apply (cases X, cases Y) apply (simp add: discrim_def Plus UN_equiv_class [OF equiv_exprel discrim_respects]) donelemma discrim_FnCall [simp]: "discrim (FnCall F Xs) = 2"apply (rule_tac z=Xs in eq_Abs_ExpList) apply (simp add: discrim_def FnCall UN_equiv_class [OF equiv_exprel discrim_respects]) donetext{*The structural induction rule for the abstract type*}theorem exp_inducts: assumes V: "\<And>nat. P1 (Var nat)" and P: "\<And>exp1 exp2. \<lbrakk>P1 exp1; P1 exp2\<rbrakk> \<Longrightarrow> P1 (Plus exp1 exp2)" and F: "\<And>nat list. P2 list \<Longrightarrow> P1 (FnCall nat list)" and Nil: "P2 []" and Cons: "\<And>exp list. \<lbrakk>P1 exp; P2 list\<rbrakk> \<Longrightarrow> P2 (exp # list)" shows "P1 exp" and "P2 list"proof - obtain U where exp: "exp = (Abs_Exp (exprel `` {U}))" by (cases exp) obtain Us where list: "list = Abs_ExpList Us" by (rule eq_Abs_ExpList) have "P1 (Abs_Exp (exprel `` {U}))" and "P2 (Abs_ExpList Us)" proof (induct U and Us) case (VAR nat) with V show ?case by (simp add: Var_def) next case (PLUS X Y) with P [of "Abs_Exp (exprel `` {X})" "Abs_Exp (exprel `` {Y})"] show ?case by (simp add: Plus) next case (FNCALL nat list) with F [of "Abs_ExpList list"] show ?case by (simp add: FnCall) next case Nil_freeExp with Nil show ?case by simp next case Cons_freeExp with Cons show ?case by simp qed with exp and list show "P1 exp" and "P2 list" by (simp_all only:)qedend