Abstract definitions
Definitions can be marked as abstract, for the purpose of hiding implementation details, or to speed up type-checking of other parts. In essence, abstract definitions behave like postulates, thus, do not reduce/compute. For instance, proofs whose content does not matter could be marked abstract, to prevent Agda from unfolding them (which might slow down type-checking).
As a guiding principle, all the rules concerning abstract
are
designed to prevent the leaking of implementation details of abstract
definitions. Similar concepts of other programming language include
(non-representative sample):
UCSD Pascal’s and Java’s interfaces and ML’s signatures.
(Especially when abstract definitions are used in combination with modules.)
Synopsis
Declarations can be marked as abstract using the block keyword
abstract
.Outside of abstract blocks, abstract definitions do not reduce, they are treated as postulates, in particular:
Abstract functions never match, thus, do not reduce.
Abstract data types do not expose their constructors.
Abstract record types do not expose their fields nor constructor.
Other declarations cannot be abstract.
Inside abstract blocks, abstract definitions reduce while type checking definitions, but not while checking their type signatures. Otherwise, due to dependent types, one could leak implementation details (e.g. expose reduction behavior by using propositional equality).
Consequently information from checking the body of a definition cannot leak into its type signature, effectively disabling type inference for abstract definitions. This means that all abstract definitions need a complete type signature.
The reach of the
abstract
keyword block extends recursively to thewhere
-blocks of a function and the declarations inside of arecord
declaration, but not inside modules declared in an abstract block.
Examples
Integers can be implemented in various ways, e.g. as difference of two natural numbers:
module Integer where
abstract
ℤ : Set
ℤ = Nat × Nat
0ℤ : ℤ
0ℤ = 0 , 0
1ℤ : ℤ
1ℤ = 1 , 0
_+ℤ_ : (x y : ℤ) → ℤ
(p , n) +ℤ (p' , n') = (p + p') , (n + n')
_*ℤ_ : (x y : ℤ) → ℤ
(a , b) *ℤ (c , d) = ((a * c) + (b * d)) , ((a * d) + (b * c))
infixl 20 _+ℤ_
infixl 30 _*ℤ_
-ℤ_ : ℤ → ℤ
-ℤ (p , n) = (n , p)
_≡ℤ_ : (x y : ℤ) → Set
(p , n) ≡ℤ (p' , n') = (p + n') ≡ (p' + n)
infix 10 _≡ℤ_
private
postulate
+comm : ∀ n m → (n + m) ≡ (m + n)
invℤ : ∀ x → (x +ℤ (-ℤ x)) ≡ℤ 0ℤ
invℤ (p , n) rewrite +comm (p + n) 0 | +comm p n = refl
Using abstract
we do not give away the actual representation of
integers, nor the implementation of the operations. We can construct
them from 0ℤ
, 1ℤ
, _+ℤ_
, and -ℤ
, but only reason about
equality ≡ℤ
with the provided lemma invℤ
.
The following property shape-of-0ℤ
of the integer zero exposes the
representation of integers as pairs. As such, it is rejected by Agda:
when checking its type signature, proj₁ x
fails to type check
since x
is of abstract type ℤ
. Remember that the abstract
definition of ℤ
does not unfold in type signatures, even when in
an abstract block! To work around this we have to define aliases for
the projections functions:
-- A property about the representation of zero integers:
abstract
private
posZ : ℤ → Nat
posZ = proj₁
negZ : ℤ → Nat
negZ = proj₂
shape-of-0ℤ : ∀ (x : ℤ) (is0ℤ : x ≡ℤ 0ℤ) → posZ x ≡ negZ x
shape-of-0ℤ (p , n) refl rewrite +comm p 0 = refl
By requiring shape-of-0ℤ
to be private to type-check, leaking of
representation details is prevented.
Scope of abstraction
In child modules, when checking an abstract definition, the abstract definitions of the parent module are transparent:
module M1 where
abstract
x : Nat
x = 0
module M2 where
abstract
x-is-0 : x ≡ 0
x-is-0 = refl
Thus, child modules can see into the representation choices of their parent modules. However, parent modules cannot see like this into child modules, nor can sibling modules see through each others abstract definitions. An exception to this is anonymous modules, which share abstract scope with their parent module, allowing parent or sibling modules to see inside their abstract definitions.
The reach of the abstract
keyword does not extend into modules:
module Parent where
abstract
module Child where
y : Nat
y = 0
x : Nat
x = 0 -- to avoid "useless abstract" error
y-is-0 : Child.y ≡ 0
y-is-0 = refl
The declarations in module Child
are not abstract!
Abstract definitions with where-blocks
Definitions in a where
block of an abstract definition are abstract
as well. This means, they can see through the abstractions of their
uncles:
module Where where
abstract
x : Nat
x = 0
y : Nat
y = x
where
x≡y : x ≡ 0
x≡y = refl