Complemented subsets #1805
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| # Measure theory | ||
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| ```agda | ||
| module measure-theory where | ||
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| open import measure-theory.apart-subtypes public | ||
| open import measure-theory.complemented-subtypes public | ||
| open import measure-theory.empty-complemented-subtypes public | ||
| open import measure-theory.full-complemented-subtypes public | ||
| open import measure-theory.intersections-complemented-subtypes public | ||
| open import measure-theory.large-poset-complemented-subtypes public | ||
| open import measure-theory.unions-complemented-subtypes public | ||
| ``` | ||
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| ## References | ||
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| Our setup for measure theory closely follows {{#cite Zeuner22}}. | ||
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| {{#bibliography}} |
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| # Apart subtypes | ||
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| ```agda | ||
| module measure-theory.apart-subtypes where | ||
| ``` | ||
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| <details><summary>Imports</summary> | ||
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| ```agda | ||
| open import foundation.apartness-relations | ||
| open import foundation.propositions | ||
| open import foundation.subtypes | ||
| open import foundation.universe-levels | ||
| ``` | ||
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| </details> | ||
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| ## Idea | ||
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| Two [subtypes](foundation.subtypes.md) `A` and `B` of a type `X` equipped with | ||
| an [apartness relation](foundation.apartness-relations.md) are | ||
| {{#concept "apart" Disambiguation="subtypes of a type equipped with an apartness relation" Agda=apart-subtype-Type-With-Apartness}} | ||
| if every one of their elements is apart. | ||
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| ## Definition | ||
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| ```agda | ||
| module _ | ||
| {l1 l2 l3 l4 : Level} | ||
| (X : Type-With-Apartness l1 l2) | ||
| (A : subtype l3 (type-Type-With-Apartness X)) | ||
| (B : subtype l4 (type-Type-With-Apartness X)) | ||
| where | ||
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| apart-prop-subtype-Type-With-Apartness : Prop (l1 ⊔ l2 ⊔ l3 ⊔ l4) | ||
| apart-prop-subtype-Type-With-Apartness = | ||
| Π-Prop | ||
| ( type-Type-With-Apartness X) | ||
| ( λ a → | ||
| Π-Prop | ||
| ( type-Type-With-Apartness X) | ||
| ( λ b → | ||
| A a ⇒ B b ⇒ rel-apart-Type-With-Apartness X a b)) | ||
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| apart-subtype-Type-With-Apartness : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) | ||
| apart-subtype-Type-With-Apartness = | ||
| type-Prop apart-prop-subtype-Type-With-Apartness | ||
| ``` |
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| # Complemented subtypes | ||||||
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| ```agda | ||||||
| module measure-theory.complemented-subtypes where | ||||||
| ``` | ||||||
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| <details><summary>Imports</summary> | ||||||
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| ```agda | ||||||
| open import foundation.apartness-relations | ||||||
| open import foundation.coproduct-types | ||||||
| open import foundation.dependent-pair-types | ||||||
| open import foundation.disjoint-subtypes | ||||||
| open import foundation.propositions | ||||||
| open import foundation.subtypes | ||||||
| open import foundation.universe-levels | ||||||
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| open import measure-theory.apart-subtypes | ||||||
| ``` | ||||||
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| </details> | ||||||
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| ## Idea | ||||||
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| A | ||||||
| {{#concept "complemented subtype" Disambiguation="of a type with apartness" Agda=complemented-subtype-Type-With-Apartness}} | ||||||
| of a type `X` equipped with an | ||||||
| [apartness relation](foundation.apartness-relations.md) is an ordered pair of | ||||||
| [subtypes](foundation-core.subtypes.md) `A` and `A'` of `X` that are | ||||||
| [apart](measure-theory.apart-subtypes.md). | ||||||
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| This definition follows {{#cite Zeuner22}}.. | ||||||
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| ## Definition | ||||||
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| ```agda | ||||||
| module _ | ||||||
| {l1 l2 : Level} | ||||||
| (X : Type-With-Apartness l1 l2) | ||||||
| where | ||||||
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| complemented-subtype-Type-With-Apartness : (l : Level) → UU (l1 ⊔ l2 ⊔ lsuc l) | ||||||
| complemented-subtype-Type-With-Apartness l = | ||||||
| Σ ( subtype l (type-Type-With-Apartness X)) | ||||||
| ( λ A → | ||||||
| Σ ( subtype l (type-Type-With-Apartness X)) | ||||||
| ( apart-subtype-Type-With-Apartness X A)) | ||||||
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| module _ | ||||||
| {l1 l2 l : Level} | ||||||
| (X : Type-With-Apartness l1 l2) | ||||||
| (cA@(A , A' , A#A') : complemented-subtype-Type-With-Apartness X l) | ||||||
| where | ||||||
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| subtype-complemented-subtype-Type-With-Apartness : | ||||||
| subtype l (type-Type-With-Apartness X) | ||||||
| subtype-complemented-subtype-Type-With-Apartness = A | ||||||
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| complement-subtype-complemented-subtype-Type-With-Apartness : | ||||||
| subtype l (type-Type-With-Apartness X) | ||||||
| complement-subtype-complemented-subtype-Type-With-Apartness = A' | ||||||
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| apart-subtype-complemented-subtype-Type-With-Apartness : | ||||||
| apart-subtype-Type-With-Apartness | ||||||
| ( X) | ||||||
| ( subtype-complemented-subtype-Type-With-Apartness) | ||||||
| ( complement-subtype-complemented-subtype-Type-With-Apartness) | ||||||
| apart-subtype-complemented-subtype-Type-With-Apartness = A#A' | ||||||
| ``` | ||||||
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| ## Properties | ||||||
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| ### The complement of a complemented subtype | ||||||
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| ```agda | ||||||
| module _ | ||||||
| {l1 l2 : Level} | ||||||
| (X : Type-With-Apartness l1 l2) | ||||||
| where | ||||||
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| complement-complemented-subtype-Type-With-Apartness : | ||||||
| {l : Level} → complemented-subtype-Type-With-Apartness X l → | ||||||
| complemented-subtype-Type-With-Apartness X l | ||||||
| complement-complemented-subtype-Type-With-Apartness (A , A' , A#A') = | ||||||
| ( A' , | ||||||
| A , | ||||||
| λ a' a a'∈A' a∈A → | ||||||
| symmetric-apart-Type-With-Apartness X a a' (A#A' a a' a∈A a'∈A')) | ||||||
| ``` | ||||||
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| ### The subtype and complement subtype associated with a complemented subtype are disjoint | ||||||
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| ```agda | ||||||
| module _ | ||||||
| {l1 l2 l : Level} | ||||||
| (X : Type-With-Apartness l1 l2) | ||||||
| (cA@(A , A' , A#A') : complemented-subtype-Type-With-Apartness X l) | ||||||
| where | ||||||
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| abstract | ||||||
| disjoint-complement-subtype-complemented-subtype-Type-With-Apartness : | ||||||
| disjoint-subtype | ||||||
| ( subtype-complemented-subtype-Type-With-Apartness X cA) | ||||||
| ( complement-subtype-complemented-subtype-Type-With-Apartness X cA) | ||||||
| disjoint-complement-subtype-complemented-subtype-Type-With-Apartness | ||||||
| x (x∈A , x∈A') = | ||||||
| antirefl-apart-Type-With-Apartness X x (A#A' x x x∈A x∈A') | ||||||
| ``` | ||||||
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| ### The total subtype associated with a complemented subtype | ||||||
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| ```agda | ||||||
| module _ | ||||||
| {l1 l2 l : Level} | ||||||
| (X : Type-With-Apartness l1 l2) | ||||||
| (cA@(A , A' , A#A') : complemented-subtype-Type-With-Apartness X l) | ||||||
| where | ||||||
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| is-in-total-subtype-complemented-subtype-Type-With-Apartness : | ||||||
| type-Type-With-Apartness X → UU l | ||||||
| is-in-total-subtype-complemented-subtype-Type-With-Apartness x = | ||||||
| is-in-subtype A x + is-in-subtype A' x | ||||||
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| abstract | ||||||
| is-prop-is-in-total-subtype-complemented-subtype-Type-With-Apartness : | ||||||
| (x : type-Type-With-Apartness X) → | ||||||
| is-prop (is-in-total-subtype-complemented-subtype-Type-With-Apartness x) | ||||||
| is-prop-is-in-total-subtype-complemented-subtype-Type-With-Apartness x = | ||||||
| is-prop-coproduct | ||||||
| ( ev-pair | ||||||
| ( disjoint-complement-subtype-complemented-subtype-Type-With-Apartness | ||||||
| ( X) | ||||||
| ( cA) | ||||||
| ( x))) | ||||||
| ( is-prop-is-in-subtype A x) | ||||||
| ( is-prop-is-in-subtype A' x) | ||||||
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| total-subtype-complemented-subtype-Type-With-Apartness : | ||||||
| subtype l (type-Type-With-Apartness X) | ||||||
| total-subtype-complemented-subtype-Type-With-Apartness x = | ||||||
| ( is-in-total-subtype-complemented-subtype-Type-With-Apartness x , | ||||||
| is-prop-is-in-total-subtype-complemented-subtype-Type-With-Apartness x) | ||||||
| ``` | ||||||
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| ## References | ||||||
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| {{#bibliography}} | ||||||
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I find this choice of terminology a bit confusing, since it is certainly not a subtype with a complement, but a subtype equipped with a subtype of its complement. Maybe you can say something more about how these objects should be interpreted and what justifies the name?
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It should also be noted that this concept is not to be confused with "decidable subtypes", i.e., subtypes$X \subseteq A$ such that $(X \cup {(A\setminus X)}) = A$ , which appear in the literature under the name "complemented subtype".