Decidability of dependent pair types

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

Created on 2022-02-14.
Last modified on 2024-02-06.

module foundation.decidable-dependent-pair-types where
open import foundation.decidable-types
open import foundation.dependent-pair-types
open import foundation.maybe
open import foundation.type-arithmetic-coproduct-types
open import foundation.type-arithmetic-unit-type
open import foundation.universe-levels

open import foundation-core.coproduct-types
open import foundation-core.equivalences
open import foundation-core.function-types
open import foundation-core.functoriality-dependent-pair-types


We describe conditions under which dependent sums are decidable.

is-decidable-Σ-coproduct :
  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (C : A + B  UU l3) 
  is-decidable (Σ A (C  inl))  is-decidable (Σ B (C  inr)) 
  is-decidable (Σ (A + B) C)
is-decidable-Σ-coproduct {l1} {l2} {l3} {A} {B} C dA dB =
    ( right-distributive-Σ-coproduct A B C)
    ( is-decidable-coproduct dA dB)

is-decidable-Σ-Maybe :
  {l1 l2 : Level} {A : UU l1} {B : Maybe A  UU l2} 
  is-decidable (Σ A (B  unit-Maybe))  is-decidable (B exception-Maybe) 
  is-decidable (Σ (Maybe A) B)
is-decidable-Σ-Maybe {l1} {l2} {A} {B} dA de =
  is-decidable-Σ-coproduct B dA
    ( is-decidable-equiv
      ( left-unit-law-Σ (B  inr))
      ( de))

is-decidable-Σ-equiv :
  {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : A  UU l3} {D : B  UU l4}
  (e : A  B) (f : (x : A)  C x  D (map-equiv e x)) 
  is-decidable (Σ A C)  is-decidable (Σ B D)
is-decidable-Σ-equiv {D = D} e f =
  is-decidable-equiv' (equiv-Σ D e f)

is-decidable-Σ-equiv' :
  {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : A  UU l3} {D : B  UU l4}
  (e : A  B) (f : (x : A)  C x  D (map-equiv e x)) 
  is-decidable (Σ B D)  is-decidable (Σ A C)
is-decidable-Σ-equiv' {D = D} e f =
  is-decidable-equiv (equiv-Σ D e f)

Recent changes