src/HOL/Induct/QuoNestedDataType.thy
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```
(*  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 begin

subsection{*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.*}
consts  exprel :: "(freeExp * freeExp) set"

syntax
"_exprel" :: "[freeExp, freeExp] => bool"  (infixl "~~" 50)
syntax (xsymbols)
"_exprel" :: "[freeExp, freeExp] => bool"  (infixl "\<sim>" 50)
syntax (HTML output)
"_exprel" :: "[freeExp, freeExp] => bool"  (infixl "\<sim>" 50)
translations
"X \<sim> Y" == "(X,Y) \<in> exprel"

text{*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.*}
inductive "exprel"
intros
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{*Proving that it is an equivalence relation*}

lemma exprel_refl_conj: "X \<sim> X & (Xs,Xs) \<in> listrel(exprel)"
apply (induct X and Xs)
apply (blast intro: exprel.intros listrel.intros)+
done

lemmas exprel_refl = exprel_refl_conj [THEN conjunct1]
lemmas list_exprel_refl = exprel_refl_conj [THEN conjunct2]

theorem equiv_exprel: "equiv UNIV exprel"
proof (simp add: equiv_def, intro conjI)
show "reflexive exprel" by (simp add: refl_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)"
by (insert equiv_listrel [OF equiv_exprel], simp)

lemma FNCALL_Nil: "FNCALL F [] \<sim> FNCALL F []"
apply (rule exprel.intros)
apply (rule listrel.intros)
done

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{*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 will
be 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 concept
of 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 the
equivalence 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 (erule exprel.induct)
apply (erule_tac [4] listrel.induct)
apply (simp_all add: Un_assoc)
done

subsubsection{*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 (erule exprel.induct, auto)

text{*This function, which returns the function name, is used to
prove 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 (erule exprel.induct, simp_all add: listrel.intros)

text{*This function, which returns the list of function arguments, is used to
prove 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 (erule exprel.induct)
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]])
done

subsection{*The Initial Algebra: A Quotiented Message Type*}

typedef (Exp)  exp = "UNIV//exprel"
by (auto simp add: quotient_def)

text{*The abstract message constructors*}

constdefs
Var :: "nat \<Rightarrow> exp"
"Var N == Abs_Exp(exprel``{VAR N})"

Plus :: "[exp,exp] \<Rightarrow> exp"
"Plus X Y ==
Abs_Exp (\<Union>U \<in> Rep_Exp X. \<Union>V \<in> Rep_Exp Y. exprel``{PLUS U V})"

FnCall :: "[nat, exp list] \<Rightarrow> exp"
"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)
done

text{*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)
done

subsection{*Every list of abstract expressions can be expressed in terms of a
list of concrete expressions*}

constdefs Abs_ExpList :: "freeExp list => exp list"
"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 intro: exprel_refl)
done

lemma 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])
qed

text{*It is not clear what to do with FnCall: it's argument is an abstraction
of an @{typ "exp list"}. Is it just Nil or Cons? What seems to work best is to
regard an @{typ "exp list"} as a @{term "listrel exprel"} equivalence class*}

text{*This theorem is easily proved but never used. There's no obvious way
even 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])
qed

lemma listset_Rep_Exp_Abs_Exp:
"listset (map Rep_Exp (Abs_ExpList Us)) = listrel exprel `` {Us}";
by (induct_tac 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)
qed

text{*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*}

constdefs
vars :: "exp \<Rightarrow> nat set"
"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])
done

lemma 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])
done

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{*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)
done

theorem 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)
done

subsection{*Injectivity of @{term FnCall}*}

constdefs
fun :: "exp \<Rightarrow> nat"
"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])
done

constdefs
args :: "exp \<Rightarrow> exp list"
"args X == contents (\<Union>U \<in> Rep_Exp X. {Abs_ExpList (freeargs U)})"

text{*This result can probably be generalized to arbitrary equivalence
relations, but with little benefit here.*}
lemma Abs_ExpList_eq:
"(y, z) \<in> listrel exprel \<Longrightarrow> Abs_ExpList (y) = Abs_ExpList (z)"
by (erule listrel.induct, simp_all)

lemma 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])
done

lemma 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 simp
next
assume "F=F' & Xs=Xs'" thus "FnCall F Xs = FnCall F' Xs'" by simp
qed

subsection{*The Abstract Discriminator*}
text{*However, as @{text FnCall_Var_neq_Var} illustrates, we don't need this
function in order to prove discrimination theorems.*}

constdefs
discrim :: "exp \<Rightarrow> int"
"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])
done

lemma 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])
done

text{*The structural induction rule for the abstract type*}
theorem exp_induct:
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 & P2 list"
proof (cases exp, rule eq_Abs_ExpList [of list], clarify)
fix U Us
show "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
qed

end

```