-
Notifications
You must be signed in to change notification settings - Fork 74
Commit
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
add module
foundation.uniformly-decidable-type-families
- Loading branch information
1 parent
4f8b9c6
commit efaaa3a
Showing
6 changed files
with
213 additions
and
30 deletions.
There are no files selected for viewing
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
175 changes: 175 additions & 0 deletions
175
src/foundation/uniformly-decidable-type-families.lagda.md
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,175 @@ | ||
| # Uniformly decidable type families | ||
|
|
||
| ```agda | ||
| module foundation.uniformly-decidable-type-families where | ||
| ``` | ||
|
|
||
| <details><summary>Imports</summary> | ||
|
|
||
| ```agda | ||
| open import foundation.contractible-types | ||
| open import foundation.coproduct-types | ||
| open import foundation.decidable-types | ||
| open import foundation.dependent-pair-types | ||
| open import foundation.equality-coproduct-types | ||
| open import foundation.inhabited-types | ||
| open import foundation.negation | ||
| open import foundation.propositional-truncations | ||
| open import foundation.propositions | ||
| open import foundation.truncated-types | ||
| open import foundation.truncation-levels | ||
| open import foundation.type-arithmetic-empty-type | ||
| open import foundation.universe-levels | ||
|
|
||
| open import foundation-core.cartesian-product-types | ||
| open import foundation-core.contractible-maps | ||
| open import foundation-core.empty-types | ||
| open import foundation-core.equivalences | ||
| open import foundation-core.function-types | ||
| open import foundation-core.homotopies | ||
| open import foundation-core.identity-types | ||
| ``` | ||
|
|
||
| </details> | ||
|
|
||
| ## Idea | ||
|
|
||
| A type family `B : A → 𝒰` is | ||
| {{#concept "uniformly decidable" Agda=is-uniformly-decidable-type-family}} if | ||
| there either is an element of every fiber `B x`, or every fiber is | ||
| [empty](foundation-core.empty-types.md). | ||
|
|
||
| ## Definitions | ||
|
|
||
| ### The predicate of being uniformly decidable | ||
|
|
||
| ```agda | ||
| is-uniformly-decidable-family : | ||
| {l1 l2 : Level} {A : UU l1} → (A → UU l2) → UU (l1 ⊔ l2) | ||
| is-uniformly-decidable-family {A = A} B = | ||
| ((x : A) → B x) + ((x : A) → ¬ (B x)) | ||
| ``` | ||
|
|
||
| ## Properties | ||
|
|
||
| ### The fibers of a uniformly decidable type family are decidable | ||
|
|
||
| ```agda | ||
| is-decidable-is-uniformly-decidable-family : | ||
| {l1 l2 : Level} {A : UU l1} {B : A → UU l2} → | ||
| is-uniformly-decidable-family B → | ||
| (x : A) → is-decidable (B x) | ||
| is-decidable-is-uniformly-decidable-family (inl f) x = inl (f x) | ||
| is-decidable-is-uniformly-decidable-family (inr g) x = inr (g x) | ||
| ``` | ||
|
|
||
| ### The uniform decidability predicate on a family of truncated types | ||
|
|
||
| ```agda | ||
| module _ | ||
| {l1 l2 : Level} {A : UU l1} {B : A → UU l2} | ||
| where | ||
|
|
||
| is-contr-is-uniformly-decidable-family-is-inhabited-base'' : | ||
| is-contr ((x : A) → (B x)) → | ||
| A → | ||
| is-contr (is-uniformly-decidable-family B) | ||
| is-contr-is-uniformly-decidable-family-is-inhabited-base'' H a = | ||
| is-contr-equiv | ||
| ( (x : A) → B x) | ||
| ( right-unit-law-coproduct-is-empty | ||
| ( (x : A) → B x) | ||
| ( (x : A) → ¬ B x) | ||
| ( λ f → f a (center H a))) | ||
| ( H) | ||
|
|
||
| is-contr-is-uniformly-decidable-family-is-inhabited-base' : | ||
| ((x : A) → is-contr (B x)) → | ||
| A → | ||
| is-contr (is-uniformly-decidable-family B) | ||
| is-contr-is-uniformly-decidable-family-is-inhabited-base' H = | ||
| is-contr-is-uniformly-decidable-family-is-inhabited-base'' (is-contr-Π H) | ||
|
|
||
| is-contr-is-uniformly-decidable-family-is-inhabited-base : | ||
| ((x : A) → is-contr (B x)) → | ||
| is-inhabited A → | ||
| is-contr (is-uniformly-decidable-family B) | ||
| is-contr-is-uniformly-decidable-family-is-inhabited-base H = | ||
| rec-trunc-Prop | ||
| ( is-contr-Prop (is-uniformly-decidable-family B)) | ||
| ( is-contr-is-uniformly-decidable-family-is-inhabited-base' H) | ||
| ``` | ||
|
|
||
| ### The uniform decidability predicate on a family of propositions | ||
|
|
||
| ```agda | ||
| module _ | ||
| {l1 l2 : Level} {A : UU l1} {B : A → UU l2} | ||
| where | ||
|
|
||
| is-prop-is-uniformly-decidable-family-is-inhabited-base'' : | ||
| is-prop ((x : A) → (B x)) → | ||
| A → | ||
| is-prop (is-uniformly-decidable-family B) | ||
| is-prop-is-uniformly-decidable-family-is-inhabited-base'' H a = | ||
| is-prop-coproduct | ||
| ( λ f nf → nf a (f a)) | ||
| ( H) | ||
| ( is-prop-Π (λ x → is-prop-neg)) | ||
|
|
||
| is-prop-is-uniformly-decidable-family-is-inhabited-base' : | ||
| ((x : A) → is-prop (B x)) → | ||
| A → | ||
| is-prop (is-uniformly-decidable-family B) | ||
| is-prop-is-uniformly-decidable-family-is-inhabited-base' H = | ||
| is-prop-is-uniformly-decidable-family-is-inhabited-base'' (is-prop-Π H) | ||
|
|
||
| is-prop-is-uniformly-decidable-family-is-inhabited-base : | ||
| ((x : A) → is-prop (B x)) → | ||
| is-inhabited A → | ||
| is-prop (is-uniformly-decidable-family B) | ||
| is-prop-is-uniformly-decidable-family-is-inhabited-base H = | ||
| rec-trunc-Prop | ||
| ( is-prop-Prop (is-uniformly-decidable-family B)) | ||
| ( is-prop-is-uniformly-decidable-family-is-inhabited-base' H) | ||
| ``` | ||
|
|
||
| ### The uniform decidability predicate on a family of truncated types | ||
|
|
||
| ```agda | ||
| module _ | ||
| {l1 l2 : Level} {A : UU l1} {B : A → UU l2} | ||
| where | ||
|
|
||
| is-trunc-succ-succ-is-uniformly-decidable-family' : | ||
| (k : 𝕋) → | ||
| is-trunc (succ-𝕋 (succ-𝕋 k)) ((x : A) → (B x)) → | ||
| is-trunc (succ-𝕋 (succ-𝕋 k)) (is-uniformly-decidable-family B) | ||
| is-trunc-succ-succ-is-uniformly-decidable-family' k H = | ||
| is-trunc-coproduct k | ||
| ( H) | ||
| ( is-trunc-is-prop (succ-𝕋 k) (is-prop-Π (λ x → is-prop-neg))) | ||
|
|
||
| is-trunc-succ-succ-is-uniformly-decidable-family : | ||
| (k : 𝕋) → | ||
| ((x : A) → is-trunc (succ-𝕋 (succ-𝕋 k)) (B x)) → | ||
| is-trunc (succ-𝕋 (succ-𝕋 k)) (is-uniformly-decidable-family B) | ||
| is-trunc-succ-succ-is-uniformly-decidable-family k H = | ||
| is-trunc-succ-succ-is-uniformly-decidable-family' k | ||
| ( is-trunc-Π (succ-𝕋 (succ-𝕋 k)) H) | ||
|
|
||
| is-trunc-is-uniformly-decidable-family-is-inhabited-base : | ||
| (k : 𝕋) → | ||
| ((x : A) → is-trunc k (B x)) → | ||
| is-inhabited A → | ||
| is-trunc k (is-uniformly-decidable-family B) | ||
| is-trunc-is-uniformly-decidable-family-is-inhabited-base | ||
| neg-two-𝕋 = | ||
| is-contr-is-uniformly-decidable-family-is-inhabited-base | ||
| is-trunc-is-uniformly-decidable-family-is-inhabited-base | ||
| ( succ-𝕋 neg-two-𝕋) = | ||
| is-prop-is-uniformly-decidable-family-is-inhabited-base | ||
| is-trunc-is-uniformly-decidable-family-is-inhabited-base | ||
| ( succ-𝕋 (succ-𝕋 k)) H _ = | ||
| is-trunc-succ-succ-is-uniformly-decidable-family k H | ||
| ``` |