On the Representation of Datatyp es in Isab elle HOL
N V olker
FernUniversit at Hagen Germany
Abstract
Representation of datatyp es is a necessary prerequisite if one wants to proverat
her than p ostulate the characteristic theorems of datatyp es This pap er intro duces
two notions of representation functions for typ es and shows how representations
of comp osed typ es can b e calculated from representations of their constituents
Together with a representation of basic typ es due to Paulson this provides a
basis for the mechanization of datatyp es in Isab elle HOL
Intro duction
Datatyp es are imp ortant ingredients of many theories mo delling computations We will
b e concerned here with datatyp es which are generated from a numb er of elements and
functions the constructors This means that every element of the typ e can b e written as
a constructor term i e as an application of constructors to constructors Furthermore
our datatyp es will b e freely generated by the constructors i e the constructors are
distinct and injective This implies that every element of the datatyp e is denoted by
a unique constructor term As a consequence suchtyp es enjoy a structural induction
theorem and allow the de nition of functions by primitive recursion
In the categorical setting these typ es arise as initial elements in certain categories
of algebras Therefore one also sp eaks of initial algebras Other names used are
recursive or inductivetyp es without laws We will simply call them datatyp es
y destructor functions whose domain is The dual kind of typ es are characterized b
the carrier of the typ e An example for such a co datatyp e are in nite lists Although
parts of our discussion should carry over to co datatyp es we will not consider them
here
Currently the Isab elle HOL system provides a datatyp e de nition package which
hasanumb er of shortcomings
The prop erties of the new typ es are p ostulated axiomatically
The declaration of a typ e T maynotcontain applications of typ e op erators to
T This forbids de nitions like
a b Tree LEAF a j NODE b a b Tree List
There is no supp ort for mutually recursive datatyp es
We will outline in this pap er an approach which aims to prove the characteristic theo
rems of the new datatyp e This is achieved by representing typ es by sets of a certain
typ e whichwas intro duced byPaulson for exactly this purp ose Our main contri
bution is the derivation of representations of typ e expressions from representations of
its constituents This construction based on a generalization of the map function from
lists to an arbitrary datatyp e
The typ e a b Tree mo dels trees with arbitrary nite branching and elements of
a resp b in the leafs resp no des This example is inspired from a slightly simpler
datatyp e in That article discusses some basic questions concerning automatic
supp ort for datatyp es in another higher order pro of assistant namely the HOL system
We are currently working on an implementation along the lines suggested in this
pap er We stress that this pap er rep orts on work in progress and do es not claim to b e
the nal word on representations of datatyp es in higher order logic
Map for arbitrary datatyp es
Atseveral places of our exp osition we will make use of the fact that for every datatyp e
map whichistheT analogue of the wellknown function T one can de ne a function T
map on lists Since the de nition of the general mapping function is technically some
what involved we will illustrate it rst by a couple of examples Recall that lists can
b e de ned by the datatyp e declaration
a List j Cons a a List
For the typ e T List the function T map agrees with map i e wehave
List map a b a List b List
and
map f List
List map f Cons x xs Cons fx List map fxs
Intuitively List map preserves the structure given by the list constructors and Cons
but changes the values of those constructor arguments whose typ e in is a
For the datatyp e Tree de ned ab ove the typ e of the mapping function is
Tree map a b a b a a Tree b b Tree
map is de ned by primitive recursion The recursive o ccur Again the function Tree
rence of Tree within List is re ected by an application of List map to Tree map f f
map f f LEAF a LEAF f a Tree
Tree map f f NODE a ts NODE f a List map Tree map f f ts
For ary datatyp es i e unparameterized typ es suchasnat the mapping function
map f is the identity on that typ e Next we will showhow to de ne the function T
for the case of an arbitrary one parameter datatyp e aT with m constructors The
declaration of suchatyp e has the form
aT C T j j C T
m m
where T is the j th argumenttyp e of constructor C Of course the number of typ e
i j i
arguments can vary from constructor to constructor and can also b e zero Every T is
i j
atyp e expression build up from the typ e variable a recursive o ccurrences of aT and
previously de ned datatyp es Note that T is only allowed to o ccur with the parameter
a i e instantiations of T are not allowed
Let f b e some function from a typ e A to another typ e B Then the mapping of f
over T
map f AT BT T
preserves constructors For a constructor C with k arguments we therefore have
i
T map f C x x C h T x h T x
i i i k i i i i k i k
T On ary constructors the function T map f is the identity for certain functions h
i j
As indicated by the notation the function h T dep ends on the typ e T in
i j i j
Ty is de ned by induction on the structure of Ty as follows For a typ e expression Ty h
h a f
h aT T map f
h T T D D map h T h T
l l
D stands here for an arbitrary previously datatyp e of some arity l
For a general n ary datatyp e T the typ e of the mapping function is
T map a b a b a a T b b T
n n n n
The general de nition of T map f f follows exactly the same scheme as for the
n
T stays the same except that case n The de nition of the auxiliary functions h
i j
equation is replaced by setting
h a f
i i
for i n
Although it is not usual in higher order logic the pro duct and sum typ e themselves
can b e de ned as datatyp es
a b Inl a j Inr b
a b a b
Their mapping op erators are characterized by the following equations
map f g Inl a Inl fa Sum
map f g Inr b Inr gb Sum
Prod map f g a b fa gb
A treatment of the generalized mapping function in the categorical framework can b e found in
Another generic function
For the de nition of representing sets we will b e interested in computing the range of
T map f f It turns out that this can b e expressed in terms of another generic
n
function
set a set a set a a T set T
n n
set A A which can b e de ned for an arbitrary n ary datatyp e T Intuitively T
n
will consist of those elements of T which can b e generated from the sets A A
n
The meaning of generated will b e made precise using inductive de nitions
First let us consider the example of the typ e List The function
List set a set a List set
should takeasetA into the set of all lists with elements in A This set is characterized
by the following twointro duction rules
a A l List set A
List set A Cons a l List set A
Note how each rule corresp onds to the typing of one List constructor In fact wecan
deriveevery rule systematically from the typing rule of the corresp onding constructor
by simply replacing the name List by List set and a by A
For a general datatyp e a a T with m constructors we obtain analogously
n
m intro duction rules for T set A A from the typing rules of the constructors
n
set and a by A for i n by replacing the name T by T
i i
By the principle of inductive de nition a nite numb er of rules such as the ones
ab ove uniquely sp eci es a set c f chapter of and the literature cited there This
set is the intersection of all the sets which comply to those rules
Supp ort for inductive de nitions in higher order logic pro of systems has b een des
crib ed in and Both approaches implement inductively de ned sets as least xed
p oints
Our interest in T set stems from the equality
range T map f f T set range f range f
n n
set is obviously monotonic We note the fact that T
Representations of typ es
The principal means to add new typ es in Isab elle HOL without risking inconsistencies is
the subtype facility This function is similar to the HOL systems new type definition
and allows to de ne a new typ e T which is isomorphic to a nonempty subset S of an
existing typ e RT Isomorphic here means that there exists an injective function T
rep
from the new typ e T onto its representing set S In the subtype package the new
typ e is declared and constants are intro duced for the representation function T
rep
and an inverse abstraction function T The inverse relationship b etween these two
abs
functions and the fact that the representing set S is the image of the representation function are p ostulated as axioms RT T_rep T S
T_abs
Figure De nition of new typ es using subtype
The use of the subtype declaration is the only waytoavoid the intro duction of
non trivial axioms in the de nition of a datatyp e T in Isab elle HOL It implies the
following sub division of our task
Construct a representing set S
De ne the new typ e T byasubtype declaration This requires a pro of that S is
not empty
De ne the constructors by using the abstraction and representation functions
Generate and prove the characteristic theorems
In the following we will b e considering not just the representation of a single typ e but
of a whole class C of typ es Such a representation asso ciates every typ e T in C with a
representation function T Obviously we require the representation functions to b e
rep
injective
Because we will need to combine representations all the representation functions
should ideally have the same target typ e Rep However there is a problem here
Consider the family A of typ es de ned by
A a a
n n
Clearly a typ e T can only b e emb edded in another typ e T if all typ e variables of T
also o ccur in T This implies that it is not p ossible to emb ed all elements of A in one
typ e For a general class C the b est we can ask for is that a typ e T parameterized by
n di erenttyp es variables a a has a representation
n
T a a T a a Rep
rep n n
Of course rep eated typ e parameters only have to o ccur once in the representation typ e
In particular if all typ e parameters of a typ e are instantiated to one typ e parameter
we can exp ect a representation
T a a T aRep
rep
Lastly unparameterized typ es T suchasnat should b e representable by functions
T T a Rep rep
Our aim is now to nd representations of typ e expressions built from typ es for which
we already have a representation As an example consider the typ e
a b Tree List
which o ccurs in the de nition of typ e Tree Intuitively a representation of a list ts of
trees should b e formed by rst representing all elements of ts and then representing the
resulting list of representations This suggests a di erent notion of a representation of
a parameterized typ e T namely as a function which turns a T structure of represen
tations into a representation Formally this means that for a n parameter datatyp e T
we are lo oking for an injective function
T a Rep a Rep T a Rep
REP
z
n
Given such a function this would allow us to de ne
a b Tree List List List map Tree
rep REP rep
More generally a representation of a typ e expression could b e derived from represen
tations of its constituents by setting
T T T T T map T T
n REP n
rep rep
rep
The injectivity of this function follows from the easily proven fact that the mapping
map of injective functions over some typ e Ty is again injective Ty
Can we construct T from a given representation function T Instantiating
REP rep
gives us
T a Rep a Rep a Rep Rep
rep
If we compare this with the typ e in we note that the target typ e should b e a Rep
This suggests that we require an injective function
flat a Rep Rep a Rep Rep
The name is b orrowed from the function flat which attens a list of lists by concate
nation Assuming the existence of such a function Rep flat we can simply de ne the
desired function T by
REP
T Rep flat T
REP rep
The injectivityofT follows of course from the fact that the comp osition of injective
REP
functions is injective
Unfortunately there are typ es Rep whichwe mightwant to use as a representation
typ e but for whichnosuch injection Rep flat exists A prominent example is the
typ e of endomorphic functions
a Fun a a
Its attening function is of typ e
flat a a a a a a Fun
Now for arbitrary typ es T and T the HOL typ e T T contains all set theoretic
functions from T to T However it is wellknown in set theory that only one element
sets S allow an injection of typ e S S S This follows from the fact that for S
with a cardinality strictly greater than one the cardinalityofS S is always greater
than the cardinalityofS
flat of the typ e required ab ove The Hence there exists no injective function Fun
same argument also shows immediately that the function space op erator has no REP
representation This is in contrast to the fact that it is easy to construct a representation
Fun a b a b Fun
rep
After constructing T from T assumenowconversely that we are given a repre
REP rep
sentation function T of typ e Let
REP
In a a a
n i i n
b e the i th injection into the n fold sum Further assume the existence of an emb edding
a a Rep
Then we can de ne a representation function T of typ e by setting
rep
T T T map In In
rep REP n n n
The two kinds of representation functions are therefore equivalent in the sense that we
can calculate one from the other under suitable assumptions ab out the representation
typ e Rep Note that the two notions coincide for typ es without constants
The ab ove section describ es only some basic requirements of typ e representations
We are currently investigating the usefulness of further prop erties like monad axioms
for Rep c f
Representation of some basic typ es
For the derivation of representing sets for datatyp es we require at least representations
ofanumb er of basic typ es These typ es are the p olymorphic typ e a the unit typ e
unit and the sum and pro duct typ es The necessity to represent these typ es stems from
the fact that semantically the bars and commas in a datatyp e declaration corresp ond
to the sum resp pro duct of typ es The constructor names are simply the tags of the
sum An empty argument list corresp onds to the unit typ e The representation of a
is necessary for the treatmentoftyp e variables
A construction of representations for these four typ es has b een given byL Paulson
in It was implemented by him in the theory Univ which is part of the standard
Isab elle HOL library Rather than listing this theory here we will just summarize
some of its main features Details can b e found in the cited pap er resp in the theory
les
In Univ thy the typ e used for representation is called item It corresp onds to
the typ e Rep of the previous section We will not go into the de nition of this typ e
itself For our discussion here the main p oint of the typ e item is the existence of three
functions
Leaf a a item
Numb nat a item
a item a item a item infix
which are injective and havedisjoint ranges Hence these functions generate a subset of
a item which is isomorphic to a datatyp e with these three functions as constructors
Function Leaf corresp onds to the function from ab ove It provides a represen
tation of a More generally n di erenttyp e variables a a are represented by
n
setting
a Leaf In
i n i
rep
for i n
Function Numb provides a representation of natural numb ers In particular this can
b e used to de ne a representation of the unit typ e by
unit Numb
REP
Using the bijection
fab f fst ab snd ab
between two parameter functions and functions with pairs as parameters we can trans
form op erator into a REP representation of the pro duct typ e
Prod a item a item a item
REP
Prod ab fst ab snd ab
REP
For the de nition of a representation of injections
In A Numb A
In A Numb A
are de ned Since these functions have disjoint images one obtains the following REP
representation of the sum typ e
Sum a item a item a item
REP
Sum sum case In In
REP
For the representation of typ e expressions it is imp ortantthatwe can de ne an injec
tion
item flat a item item a item
Its e ect on the subset of a item spanned by the three functions Leaf Numb and is
given by the primitive recursive equalities
flat Leaf a a item
item flat Numb n Numb n
flat M N item flat M item flatN item
For the derivation of map functions on datatyp es an item map function is useful Again
we do not give its de nition but only note the following equalities
map f Leaf a Leaf fa item
item map f Numb n Numb n
item map f M N item map fM item map fN
The two functions item flat and item map are at the time of this writing not part of
the standard Isab elle distribution
Because of the injectivity of representation functions the range of T can serveas
rep
a representingsetforatyp e T For the following range calculations note the equality
range f g f range g
It uses the image op erator de ned by
S
f A a A ffag
The range of a general typ e instantiation can b e computed as follows
range T T T
n
rep
f equation g
T range T T T map T
REP REP n
rep rep
f equations g
T T set range T range T
REP n
rep rep
This expression can b e evaluated further for concrete T In the case of the sum typ e
wehave
S S
set A B a A b B f a b g Prod
which implies
range A B
rep
f derivation ab ove g
set range A range B Prod Prod
rep rep REP
f equations simpli cations g
S S
a range A b range B fa b g
rep rep
f de nition of below g
range A range B
rep rep
where the binary op erator is de ned by
S S
X Y a A b B fa b g
Analogously we can derive
range A B range A range B
rep rep
rep
where the binary op erator is given by
X Y In X In Y
Note that the two op erators and are obviously monotonic
The representing set of a datatyp e
Wewillnow explain how the little representation theory of the preceding sections can
b e used to construct representing sets for datatyp es Instead of showing this abstractly
we will demonstrate it for the case of the typ e Tree and trust that the reader can infer
the general pro cedure Recall the de nition of Tree
a b Tree LEAF a j NODE b a b Tree List
As mentioned b efore the bars and the commas in a datatyp e declaration corresp ond
to sum and pro duct of typ es This means that we can interpret this typ e declaration
as an isomorphism of typ es
a b Tree a b a b Tree List
This isomorphism suggests that the range of the representation function of b oth sides
should b e the same i e
a b a b Tree List range a b Tree range
rep rep
Let
R range a b Tree
rep
Using the representation calculus develop ed in the previous sections equation can
b e reduced to
R range Leaf Inl
range Leaf Inr List List set R
REP
Because of the monotonicity of the o ccurring op erations on set the abstraction of the
right hand side over R is a monotonous function Hence Tarski s xed p oint theorem
implies the existence and uniqueness of a smallest solution R to this equality c f
or the theory Lfp in the standard Isab elle HOL distribution This set is used as the
representing set of a b Tree
In general the construction of the representing set requires that we already have
representation functions of all the typ es o ccurring in the de nition of a datatyp e T
except of course for T itself
Instead of the de nition of R as a least xed p oint an equivalentcharacterization
using inductive sets is also p ossible In our example we get the following two rules
rs List set R
In Leaf Inl a R In Leaf Inr b List rs
REP
In general if there are more than two constructors it is necessary to distinguish them
by suitable combinations of In and In
De nition of typ e and constructors
According to our work plan from section wecannow declare the typ e Tree bya
subtype de nition using R as representing set The non emptiness of R follows easily
from equality For a general datatyp e T it requires at least one constructor whose
argumenttyp es do not contain recursive o ccurrences of T
From the subset declaration of Tree we obtain a representation function Tree
rep
and an inverse abstraction function Tree The typing of Tree agrees with the
abs rep
general typing Per construction the ranges of the representing functions of the
twotyp es in are equal Note that nothing is said ab out the equality of the repre
sentation functions themselves
The constructors of a new typ e T are obtained by
representing the arguments
combining the results using and In In and
abstracting into T
In the case of typ e Tree this results in
LEAF a a b Tree
NODE b a b Tree a b Tree
LEAF a Tree In Leaf Inl a
abs
NODE b ts Tree In Leaf Inr a
abs
List List map Tree ts
REP rep
ary constructors are represented using Numb i e the representation of the only
element of the unit typ e
Theorems
For a datatyp e T the induction theorem expresses the wellknown principle of structural
induction over typ e T For its pro of note that the inductivecharacterization of the
representing set gives us an induction theorem for that set The induction theorem for
T is obtained directly by lifting this theorem to T using T For our example typ e
abs
it reads as follows
a P LEAF a
set f t Ptg P NODE b ts b ts ts List
t Pt
Primitive recursive functions are the unique functions solving certain functional equa
lities In our example this means that for a pair of functions
h a c
h b a b Tree List c List c
there exists a unique function f satisfying
f LEAF a h a
map fts f NODE b ts h b ts List
Theorems justifying primitive recursion can b e proven by an inductive construction of
the graph of the function c f
In a di erent notion of primitive recursion is suggested In the case of typ e
Tree for example it is required that given
h a c
h b a b Tree List d c
h d
h a b Tree a b Tree List c d d
there exists a unique pair of functions f and g satisfying
f LEAF a h a
f NODE b ts h b ts gts
g h
f Cons t ts h t ts ft gts
For the typ e Tree wehaveproven that this agrees with our notion of primitive recursive
functions It seems that this also holds for arbitrary datatyp es but wehave not proven
this formally yet
Concluding remarks
Wehave presented an approach to the representation of datatyp es which applies also to
datatyp es T with recursive o ccurrences of T inside of typ e expressions Our approachis
based on two generic functions T map and T set which can b e de ned for an arbitrary
datatyp e T The use of these functions has led us also to a di erentformulation of the
induction theorem and of primitive recursive functions than usual
The discussion of representations in the rst four sections applies also to co datatyp es
and to mutual recursivetyp es By suitable adjustments of the material in the last sec
tions an extension of our approach to these typ es should b e p ossible
Acknowledgements
Iwould like to thank M Cieliebak for his comments
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