60 lines
1.6 KiB
Plaintext
60 lines
1.6 KiB
Plaintext
|
-- no pattern matching, instead a recursor is given based on your inductive definition
|
||
|
|
-- for example: the following
|
||
|
|
|
||
|
|
type List A
|
||
|
|
| nil : List A
|
||
|
|
| cons : A → List A → List A
|
||
|
|
|
||
|
|
-- will bring the following into scope
|
||
|
|
-- rec[List] : B → (A → B → B) → List A → B
|
||
|
|
|
||
|
|
-- map could then be written as
|
||
|
|
|
||
|
|
map : (A → B) → List A → List B
|
||
|
|
:= rec[List] nil (λx xs. cons (f x) xs)
|
||
|
|
|
||
|
|
-- one could then define the naturals as follows
|
||
|
|
|
||
|
|
type Nat
|
||
|
|
| zero : Nat
|
||
|
|
| succ : Nat → Nat
|
||
|
|
|
||
|
|
-- defining addition as
|
||
|
|
|
||
|
|
add : Nat → Nat → Nat
|
||
|
|
:= rec[Nat] (λx. x) (λn f. succ (f n))
|
||
|
|
|
||
|
|
-- since | rec[Nat] : B → (Nat → B → B) → Nat → B
|
||
|
|
-- which generalizes to | rec[Nat] : (Nat → Nat) → (Nat → (Nat → Nat) → (Nat → Nat)) → Nat → Nat → Nat
|
||
|
|
-- ^ adding 0 ^ recursively adding one more ^ resulting addition type
|
||
|
|
|
||
|
|
-- id is used to not add anything, the second function takes the last addition function and adds a layer
|
||
|
|
-- of succ onto it, this way it generates a function for adding the right amount.
|
||
|
|
|
||
|
|
-- multiplication is defined similairly
|
||
|
|
|
||
|
|
mul : Nat → Nat → Nat
|
||
|
|
:= rec[Nat] (λx. zero) (λn f. add n (f n))
|
||
|
|
|
||
|
|
-- here's an example of a simpler type
|
||
|
|
|
||
|
|
type Bool
|
||
|
|
| true : Bool
|
||
|
|
| false : Bool
|
||
|
|
|
||
|
|
not : Bool → Bool
|
||
|
|
:= rec[Bool] false true
|
||
|
|
|
||
|
|
-- now, let's look at a bit more interesting example
|
||
|
|
|
||
|
|
type Expr
|
||
|
|
| num : Nat
|
||
|
|
| add : Expr Expr
|
||
|
|
| mul : Expr Expr
|
||
|
|
|
||
|
|
-- this generates the following recursor
|
||
|
|
-- rec[Expr] : (Nat → B) → (B → B → B) → (B → B → B) → Expr → B
|
||
|
|
|
||
|
|
eval : Expr → Nat
|
||
|
|
:= rec[Expr] (λx. x) add mul
|