Coproduct types

Content created by Fredrik Bakke, Jonathan Prieto-Cubides, Egbert Rijke and Elisabeth Stenholm.

Created on 2023-01-20.
Last modified on 2024-04-11.

module foundation-core.coproduct-types where
Imports 
open import foundation.universe-levels

Idea

The coproduct of two types A and B can be thought of as the disjoint union of A and B.

Definition

Coproduct types

infixr 10 _+_

data _+_ {l1 l2 : Level} (A : UU l1) (B : UU l2) : UU (l1 ⊔ l2)
  where
  inl : A → A + B
  inr : B → A + B

The induction principle for coproduct types

ind-coproduct :
  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (C : A + B → UU l3) →
  ((x : A) → C (inl x)) → ((y : B) → C (inr y)) →
  (t : A + B) → C t
ind-coproduct C f g (inl x) = f x
ind-coproduct C f g (inr x) = g x

The recursion principle for coproduct types

rec-coproduct :
  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} →
  (A → C) → (B → C) → (A + B) → C
rec-coproduct {C = C} = ind-coproduct (λ _ → C)

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