(* Title: HOL/Induct/QuoNestedDataType.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 2004 University of Cambridge
*)
section\<open>Quotienting a Free Algebra Involving Nested Recursion\<close>
text \<open>This is the development promised in Lawrence Paulson's paper ``Defining functions on equivalence classes''
\emph{ACM Transactions on Computational Logic} \textbf{7}:40 (2006), 658--675,
illustrating bare-bones quotient constructions. Any comparison using lifting and transfer
should be done in a separate theory.\<close>
theory QuoNestedDataType imports Main begin
subsection\<open>Defining the Free Algebra\<close>
text\<open>Messages with encryption and decryption as free constructors.\<close>
datatype
freeExp = VAR nat
| PLUS freeExp freeExp
| FNCALL nat "freeExp list"
datatype_compat freeExp
text\<open>The equivalence relation, which makes PLUS associative.\<close>
text\<open>The first rule is the desired equation. The next three rules
make the equations applicable to subterms. The last two rules are symmetry
and transitivity.\<close>
inductive_set
exprel :: "(freeExp * freeExp) set"
and exp_rel :: "[freeExp, freeExp] => bool" (infixl "\<sim>" 50)
where
"X \<sim> Y \<equiv> (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_mono
text\<open>Proving that it is an equivalence relation\<close>
lemma exprel_refl: "X \<sim> X"
and list_exprel_refl: "(Xs,Xs) \<in> listrel(exprel)"
by (induct X and Xs rule: compat_freeExp.induct compat_freeExp_list.induct)
(blast intro: exprel.intros listrel.intros)+
theorem equiv_exprel: "equiv UNIV exprel"
proof (rule equivI)
show "refl exprel" by (simp add: refl_on_def exprel_refl)
show "sym exprel" by (simp add: sym_def, blast intro: exprel.SYM)
show "trans exprel" by (simp add: trans_def, blast intro: exprel.TRANS)
qed
theorem equiv_list_exprel: "equiv UNIV (listrel exprel)"
using equiv_listrel [OF equiv_exprel] by simp
lemma 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\<open>Some Functions on the Free Algebra\<close>
subsubsection\<open>The Set of Variables\<close>
text\<open>A function to return the set of variables present in a message. It will
be lifted to the initial algebra, to serve as an example of that process.
Note that the "free" refers to the free datatype rather than to the concept
of a free variable.\<close>
primrec freevars :: "freeExp \<Rightarrow> nat set" and freevars_list :: "freeExp list \<Rightarrow> nat set"
where
"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\<open>This theorem lets us prove that the vars function respects the
equivalence relation. It also helps us prove that Variable
(the abstract constructor) is injective\<close>
theorem exprel_imp_eq_freevars: "U \<sim> V \<Longrightarrow> freevars U = freevars V"
proof (induct set: exprel)
case (FNCALL Xs Xs' F)
then show ?case
by (induct rule: listrel.induct) auto
qed (simp_all add: Un_assoc)
subsubsection\<open>Functions for Freeness\<close>
text\<open>A discriminator function to distinguish vars, sums and function calls\<close>
primrec freediscrim :: "freeExp \<Rightarrow> int" where
"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) auto
text\<open>This function, which returns the function name, is used to
prove part of the injectivity property for FnCall.\<close>
primrec freefun :: "freeExp \<Rightarrow> nat" where
"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\<open>This function, which returns the list of function arguments, is used to
prove part of the injectivity property for FnCall.\<close>
primrec freeargs :: "freeExp \<Rightarrow> freeExp list" where
"freeargs (VAR N) = []"
| "freeargs (PLUS X Y) = []"
| "freeargs (FNCALL F Xs) = Xs"
theorem exprel_imp_eqv_freeargs:
assumes "U \<sim> V"
shows "(freeargs U, freeargs V) \<in> listrel exprel"
using assms
proof induction
case (FNCALL Xs Xs' F)
then show ?case
by (simp add: listrel_iff_nth)
next
case (SYM X Y)
then show ?case
by (meson equivE equiv_list_exprel symD)
next
case (TRANS X Y Z)
then show ?case
by (meson equivE equiv_list_exprel transD)
qed (use listrel.simps in auto)
subsection\<open>The Initial Algebra: A Quotiented Message Type\<close>
definition "Exp = UNIV//exprel"
typedef exp = Exp
morphisms Rep_Exp Abs_Exp
unfolding Exp_def by (auto simp add: quotient_def)
text\<open>The abstract message constructors\<close>
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\<open>Reduces equality of equivalence classes to the \<^term>\<open>exprel\<close> relation:
\<^term>\<open>(exprel``{x} = exprel``{y}) = ((x,y) \<in> exprel)\<close>\<close>
lemmas equiv_exprel_iff = eq_equiv_class_iff [OF equiv_exprel UNIV_I UNIV_I]
declare equiv_exprel_iff [simp]
text\<open>All equivalence classes belong to set of representatives\<close>
lemma exprel_in_Exp [simp]: "exprel``{U} \<in> Exp"
by (simp add: Exp_def quotientI)
lemma inj_on_Abs_Exp: "inj_on Abs_Exp Exp"
by (meson Abs_Exp_inject inj_onI)
text\<open>Reduces equality on abstractions to equality on representatives\<close>
declare inj_on_Abs_Exp [THEN inj_on_eq_iff, simp]
declare Abs_Exp_inverse [simp]
text\<open>Case analysis on the representation of a exp as an equivalence class.\<close>
lemma eq_Abs_Exp [case_names Abs_Exp, cases type: exp]:
"(\<And>U. z = Abs_Exp (exprel``{U}) \<Longrightarrow> P) \<Longrightarrow> P"
by (metis Abs_Exp_cases Exp_def quotientE)
subsection\<open>Every list of abstract expressions can be expressed in terms of a
list of concrete expressions\<close>
definition
Abs_ExpList :: "freeExp list => exp list" where
"Abs_ExpList Xs \<equiv> map (\<lambda>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"
by (smt (verit, del_insts) Abs_ExpList_def eq_Abs_Exp ex_map_conv)
subsubsection\<open>Characteristic Equations for the Abstract Constructors\<close>
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 (auto simp add: congruent2_def exprel.PLUS)
thus ?thesis
by (simp add: Plus_def UN_equiv_class2 [OF equiv_exprel equiv_exprel])
qed
text\<open>It is not clear what to do with FnCall: it's argument is an abstraction
of an \<^typ>\<open>exp list\<close>. Is it just Nil or Cons? What seems to work best is to
regard an \<^typ>\<open>exp list\<close> as a \<^term>\<open>listrel exprel\<close> equivalence class\<close>
text\<open>This theorem is easily proved but never used. There's no obvious way
even to state the analogous result, \<open>FnCall_Cons\<close>.\<close>
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 (auto 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 (auto simp add: congruent_def FNCALL_Cons listrel.intros)
thus ?thesis
by (simp add: FnCall_def UN_equiv_class [OF equiv_exprel])
qed
lemma 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 (auto 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)
qed
text\<open>Establishing this equation is the point of the whole exercise\<close>
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\<open>The Abstract Function to Return the Set of Variables\<close>
definition
vars :: "exp \<Rightarrow> nat set" where "vars X \<equiv> (\<Union>U \<in> Rep_Exp X. freevars U)"
lemma vars_respects: "freevars respects exprel"
by (auto simp add: congruent_def exprel_imp_eq_freevars)
text\<open>The extension of the function \<^term>\<open>vars\<close> to lists\<close>
primrec vars_list :: "exp list \<Rightarrow> nat set" where
"vars_list [] = {}"
| "vars_list(E#Es) = vars E \<union> vars_list Es"
text\<open>Now prove the three equations for \<^term>\<open>vars\<close>\<close>
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"
proof -
have "\<And>U V. \<lbrakk>X = Abs_Exp (exprel``{U}); Y = Abs_Exp (exprel``{V})\<rbrakk>
\<Longrightarrow> vars (Plus X Y) = vars X \<union> vars Y"
by (simp add: vars_def Plus UN_equiv_class [OF equiv_exprel vars_respects])
then show ?thesis
by (meson eq_Abs_Exp)
qed
lemma vars_FnCall [simp]: "vars (FnCall F Xs) = vars_list Xs"
proof -
have "vars (Abs_Exp (exprel``{FNCALL F Us})) = vars_list (Abs_ExpList Us)" for Us
by (induct Us) (auto simp: vars_def UN_equiv_class [OF equiv_exprel vars_respects])
then show ?thesis
by (metis ExpList_rep FnCall)
qed
lemma vars_FnCall_Nil: "vars (FnCall F Nil) = {}"
by simp
lemma vars_FnCall_Cons: "vars (FnCall F (X#Xs)) = vars X \<union> vars_list Xs"
by simp
subsection\<open>Injectivity Properties of Some Constructors\<close>
lemma VAR_imp_eq: "VAR m \<sim> VAR n \<Longrightarrow> m = n"
by (drule exprel_imp_eq_freevars, simp)
text\<open>Can also be proved using the function \<^term>\<open>vars\<close>\<close>
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"
using exprel_imp_eq_freediscrim by force
theorem Var_neq_Plus [iff]: "Var N \<noteq> Plus X Y"
proof -
have "\<And>U V. \<lbrakk>X = Abs_Exp (exprel``{U}); Y = Abs_Exp (exprel``{V})\<rbrakk> \<Longrightarrow> Var N \<noteq> Plus X Y"
using Plus VAR_neqv_PLUS Var_def by force
then show ?thesis
by (meson eq_Abs_Exp)
qed
theorem Var_neq_FnCall [iff]: "Var N \<noteq> FnCall F Xs"
proof -
have "\<And>Us. Var N \<noteq> FnCall F (Abs_ExpList Us)"
using FnCall Var_def exprel_imp_eq_freediscrim by fastforce
then show ?thesis
by (metis ExpList_rep)
qed
subsection\<open>Injectivity of \<^term>\<open>FnCall\<close>\<close>
definition
"fun" :: "exp \<Rightarrow> nat"
where "fun X \<equiv> the_elem (\<Union>U \<in> Rep_Exp X. {freefun U})"
lemma fun_respects: "(\<lambda>U. {freefun U}) respects exprel"
by (auto simp add: congruent_def exprel_imp_eq_freefun)
lemma fun_FnCall [simp]: "fun (FnCall F Xs) = F"
proof -
have "\<And>Us. fun (FnCall F (Abs_ExpList Us)) = F"
using FnCall UN_equiv_class [OF equiv_exprel] fun_def fun_respects by fastforce
then show ?thesis
by (metis ExpList_rep)
qed
definition
args :: "exp \<Rightarrow> exp list" where
"args X = the_elem (\<Union>U \<in> Rep_Exp X. {Abs_ExpList (freeargs U)})"
text\<open>This result can probably be generalized to arbitrary equivalence
relations, but with little benefit here.\<close>
lemma Abs_ExpList_eq:
"(y, z) \<in> listrel exprel \<Longrightarrow> Abs_ExpList (y) = Abs_ExpList (z)"
by (induct set: listrel) simp_all
lemma args_respects: "(\<lambda>U. {Abs_ExpList (freeargs U)}) respects exprel"
by (auto simp add: congruent_def Abs_ExpList_eq exprel_imp_eqv_freeargs)
lemma args_FnCall [simp]: "args (FnCall F Xs) = Xs"
proof -
have "\<And>Us. Xs = Abs_ExpList Us \<Longrightarrow> args (FnCall F Xs) = Xs"
by (simp add: FnCall args_def UN_equiv_class [OF equiv_exprel args_respects])
then show ?thesis
by (metis ExpList_rep)
qed
lemma FnCall_FnCall_eq [iff]: "(FnCall F Xs = FnCall F' Xs') \<longleftrightarrow> (F=F' \<and> Xs=Xs')"
by (metis args_FnCall fun_FnCall)
subsection\<open>The Abstract Discriminator\<close>
text\<open>However, as \<open>FnCall_Var_neq_Var\<close> illustrates, we don't need this
function in order to prove discrimination theorems.\<close>
definition
discrim :: "exp \<Rightarrow> int" where
"discrim X = the_elem (\<Union>U \<in> Rep_Exp X. {freediscrim U})"
lemma discrim_respects: "(\<lambda>U. {freediscrim U}) respects exprel"
by (auto simp add: congruent_def exprel_imp_eq_freediscrim)
text\<open>Now prove the four equations for \<^term>\<open>discrim\<close>\<close>
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"
proof -
have "\<And>U V. \<lbrakk>X = Abs_Exp (exprel``{U}); Y = Abs_Exp (exprel``{V})\<rbrakk> \<Longrightarrow> discrim (Plus X Y) = 1"
by (simp add: discrim_def Plus UN_equiv_class [OF equiv_exprel discrim_respects])
then show ?thesis
by (meson eq_Abs_Exp)
qed
lemma discrim_FnCall [simp]: "discrim (FnCall F Xs) = 2"
proof -
have "discrim (FnCall F (Abs_ExpList Us)) = 2" for Us
by (simp add: discrim_def FnCall UN_equiv_class [OF equiv_exprel discrim_respects])
then show ?thesis
by (metis ExpList_rep)
qed
text\<open>The structural induction rule for the abstract type\<close>
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 (metis ExpList_rep)
have "P1 (Abs_Exp (exprel``{U}))" and "P2 (Abs_ExpList Us)"
proof (induct U and Us rule: compat_freeExp.induct compat_freeExp_list.induct)
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:)
qed
end