Lambda Abstraction

Pattern matching lambda

Anonymous pattern matching functions can be defined using one of the two following syntaxes:

\ { p11 .. p1n -> e1 ; … ; pm1 .. pmn -> em }

\ where
  p11 .. p1n -> e1
  …
  pm1 .. pmn -> em

(where, as usual, \ and -> can be replaced by λ and →). Note that the where keyword introduces an indented block of clauses; if there is only one clause then it may be used inline.

Internally this is translated into a function definition of the following form:

extlam p11 .. p1n = e1
…
extlam pm1 .. pmn = em

where extlam is a fresh name. This means that anonymous pattern matching functions are generative. For instance, refl will not be accepted as an inhabitant of the type

(λ { true → true ; false → false }) ==
(λ { true → true ; false → false })

because this is equivalent to extlam1 ≡ extlam2 for some distinct fresh names extlam1 and extlam2. Currently the where and with constructions are not allowed in (the top-level clauses of) anonymous pattern matching functions.

Examples:

and : Bool → Bool → Bool
and = λ { true x → x ; false _ → false }

xor : Bool → Bool → Bool
xor = λ { true  true  → false
        ; false false → false
        ; _     _     → true
        }

eq : Bool → Bool → Bool
eq = λ where
  true  true  → true
  false false → true
  _ _ → false

fst : {A : Set} {B : A → Set} → Σ A B → A
fst = λ { (a , b) → a }

snd : {A : Set} {B : A → Set} (p : Σ A B) → B (fst p)
snd = λ { (a , b) → b }

swap : {A B : Set} → Σ A (λ _ → B) → Σ B (λ _ → A)
swap = λ where (a , b) → (b , a)

Regular pattern-matching lambdas are treated as non-erased function definitions. One can make a pattern-matching lambda erased by writing @0 or @erased after the lambda:

@0 _ : @0 Set → Set
_ = λ @0 { A → A }

@0 _ : @0 Set → Set
_ = λ @erased where
  A → A