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* Proof that embeddings are monotone over sums Also cleaned up several aspects of things. * Embeddings are monotone over products * Define cardinal exponentiation * Minor formatting clean-up * Removal of some excessive imports * Renamed annihiliation by 0 to be less vague * Renamed Cantor's Theorem results * Cleaned up order property definitions * Replace renaming * Require preordering for order properties * Remove extra level variable that's unused * Removed redundant implicit level term * Replace type families with embeddings as subtypes More consistent with the idea of a "subset" and also makes reasoning easier by keeping the sigma types out of the type signatures. * Cleaned up types for embeddings in order props * Added function in Embedding to simplify constructs Sigma-types where the type family is always a prop embed into the first argument's type * Minor clean-up * Better respecting camel case * Name-cleaning and removing unused imports * Reintroduce erroneously-removed newline * Renamed loset to toset, strict loset to loset Also required losets to be weakly linear * Fixed paths in top-level file * Defined induced relations over embeddings * Better capitalization * Renamed one more detail * Moved order properties into their own folders * Reduced imports for cardinals * Removed unused apartness import * Changed capitalization back for conversions * Proved induced relations preserve order properties * Loosened embedding to mere function for bounds * Fixed whitespace * Fix rebase issues * Resolve PR comments
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| Original file line number | Diff line number | Diff line change |
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| {-# OPTIONS --safe #-} | ||
| module Cubical.Data.Cardinality where | ||
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| open import Cubical.Data.Cardinality.Base public | ||
| open import Cubical.Data.Cardinality.Properties public |
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| Original file line number | Diff line number | Diff line change |
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| {- | ||
| This file contains: | ||
| - Treatment of set truncation as cardinality | ||
| as per the HoTT book, section 10.2 | ||
| -} | ||
| {-# OPTIONS --safe #-} | ||
| module Cubical.Data.Cardinality.Base where | ||
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| open import Cubical.HITs.SetTruncation as ∥₂ | ||
| open import Cubical.Foundations.Prelude | ||
| open import Cubical.Foundations.HLevels | ||
| open import Cubical.Data.Empty | ||
| open import Cubical.Data.Sigma | ||
| open import Cubical.Data.Sum | ||
| open import Cubical.Data.Unit | ||
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| private | ||
| variable | ||
| ℓ : Level | ||
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| -- First, we define a cardinal as the set truncation of Set | ||
| Card : Type (ℓ-suc ℓ) | ||
| Card {ℓ} = ∥ hSet ℓ ∥₂ | ||
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| -- Verify that it's a set | ||
| isSetCard : isSet (Card {ℓ}) | ||
| isSetCard = isSetSetTrunc | ||
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| -- Set truncation of a set is its cardinality | ||
| card : hSet ℓ → Card {ℓ} | ||
| card = ∣_∣₂ | ||
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| -- Some special cardinalities | ||
| 𝟘 : Card {ℓ} | ||
| 𝟘 = card (⊥* , isProp→isSet isProp⊥*) | ||
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| 𝟙 : Card {ℓ} | ||
| 𝟙 = card (Unit* , isSetUnit*) | ||
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| -- Now we define some arithmetic | ||
| _+_ : Card {ℓ} → Card {ℓ} → Card {ℓ} | ||
| _+_ = ∥₂.rec2 isSetCard λ (A , isSetA) (B , isSetB) | ||
| → card ((A ⊎ B) , isSet⊎ isSetA isSetB) | ||
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| _·_ : Card {ℓ} → Card {ℓ} → Card {ℓ} | ||
| _·_ = ∥₂.rec2 isSetCard λ (A , isSetA) (B , isSetB) | ||
| → card ((A × B) , isSet× isSetA isSetB) | ||
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| _^_ : Card {ℓ} → Card {ℓ} → Card {ℓ} | ||
| _^_ = ∥₂.rec2 isSetCard λ (A , isSetA) (B , _) | ||
| → card ((B → A) , isSet→ isSetA) |
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| Original file line number | Diff line number | Diff line change |
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| @@ -0,0 +1,194 @@ | ||
| {- | ||
| This file contains: | ||
| - Properties of cardinality | ||
| - Preordering of cardinalities | ||
| -} | ||
| {-# OPTIONS --safe #-} | ||
| module Cubical.Data.Cardinality.Properties where | ||
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| open import Cubical.HITs.SetTruncation as ∥₂ | ||
| open import Cubical.Data.Cardinality.Base | ||
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| open import Cubical.Algebra.CommSemiring | ||
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| open import Cubical.Foundations.Prelude | ||
| open import Cubical.Foundations.Isomorphism | ||
| open import Cubical.Foundations.HLevels | ||
| open import Cubical.Foundations.Structure | ||
| open import Cubical.Functions.Embedding | ||
| open import Cubical.Data.Empty as ⊥ | ||
| open import Cubical.Data.Sigma | ||
| open import Cubical.Data.Sum as ⊎ | ||
| open import Cubical.Data.Unit | ||
| open import Cubical.HITs.PropositionalTruncation as ∥₁ | ||
| open import Cubical.Relation.Binary.Base | ||
| open import Cubical.Relation.Binary.Order.Preorder.Base | ||
| open import Cubical.Relation.Binary.Order.Properties | ||
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| private | ||
| variable | ||
| ℓ : Level | ||
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| -- Cardinality is a commutative semiring | ||
| module _ where | ||
| private | ||
| +Assoc : (A B C : Card {ℓ}) → A + (B + C) ≡ (A + B) + C | ||
| +Assoc = ∥₂.elim3 (λ _ _ _ → isProp→isSet (isSetCard _ _)) | ||
| λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) | ||
| (sym (isoToPath ⊎-assoc-Iso))) | ||
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| ·Assoc : (A B C : Card {ℓ}) → A · (B · C) ≡ (A · B) · C | ||
| ·Assoc = ∥₂.elim3 (λ _ _ _ → isProp→isSet (isSetCard _ _)) | ||
| λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) | ||
| (sym (isoToPath Σ-assoc-Iso))) | ||
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| +IdR𝟘 : (A : Card {ℓ}) → A + 𝟘 ≡ A | ||
| +IdR𝟘 = ∥₂.elim (λ _ → isProp→isSet (isSetCard _ _)) | ||
| λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) | ||
| (isoToPath ⊎-IdR-⊥*-Iso)) | ||
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| +IdL𝟘 : (A : Card {ℓ}) → 𝟘 + A ≡ A | ||
| +IdL𝟘 = ∥₂.elim (λ _ → isProp→isSet (isSetCard _ _)) | ||
| λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) | ||
| (isoToPath ⊎-IdL-⊥*-Iso)) | ||
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| ·IdR𝟙 : (A : Card {ℓ}) → A · 𝟙 ≡ A | ||
| ·IdR𝟙 = ∥₂.elim (λ _ → isProp→isSet (isSetCard _ _)) | ||
| λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) | ||
| (isoToPath rUnit*×Iso)) | ||
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| ·IdL𝟙 : (A : Card {ℓ}) → 𝟙 · A ≡ A | ||
| ·IdL𝟙 = ∥₂.elim (λ _ → isProp→isSet (isSetCard _ _)) | ||
| λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) | ||
| (isoToPath lUnit*×Iso)) | ||
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| +Comm : (A B : Card {ℓ}) → (A + B) ≡ (B + A) | ||
| +Comm = ∥₂.elim2 (λ _ _ → isProp→isSet (isSetCard _ _)) | ||
| λ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) | ||
| (isoToPath ⊎-swap-Iso)) | ||
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| ·Comm : (A B : Card {ℓ}) → (A · B) ≡ (B · A) | ||
| ·Comm = ∥₂.elim2 (λ _ _ → isProp→isSet (isSetCard _ _)) | ||
| λ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) | ||
| (isoToPath Σ-swap-Iso)) | ||
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| ·LDist+ : (A B C : Card {ℓ}) → A · (B + C) ≡ (A · B) + (A · C) | ||
| ·LDist+ = ∥₂.elim3 (λ _ _ _ → isProp→isSet (isSetCard _ _)) | ||
| λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) | ||
| (isoToPath ×DistL⊎Iso)) | ||
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| AnnihilL : (A : Card {ℓ}) → 𝟘 · A ≡ 𝟘 | ||
| AnnihilL = ∥₂.elim (λ _ → isProp→isSet (isSetCard _ _)) | ||
| λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) | ||
| (isoToPath (ΣEmpty*Iso λ _ → _))) | ||
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| isCardCommSemiring : IsCommSemiring {ℓ-suc ℓ} 𝟘 𝟙 _+_ _·_ | ||
| isCardCommSemiring = makeIsCommSemiring isSetCard +Assoc +IdR𝟘 +Comm ·Assoc ·IdR𝟙 ·LDist+ AnnihilL ·Comm | ||
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| -- Exponentiation is also well-behaved | ||
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| ^AnnihilR𝟘 : (A : Card {ℓ}) → A ^ 𝟘 ≡ 𝟙 | ||
| ^AnnihilR𝟘 = ∥₂.elim (λ _ → isProp→isSet (isSetCard _ _)) | ||
| λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) | ||
| (isoToPath (iso⊥ _))) | ||
| where iso⊥ : ∀ A → Iso (⊥* → A) Unit* | ||
| Iso.fun (iso⊥ A) _ = tt* | ||
| Iso.inv (iso⊥ A) _ () | ||
| Iso.rightInv (iso⊥ A) _ = refl | ||
| Iso.leftInv (iso⊥ A) _ i () | ||
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| ^IdR𝟙 : (A : Card {ℓ}) → A ^ 𝟙 ≡ A | ||
| ^IdR𝟙 = ∥₂.elim (λ _ → isProp→isSet (isSetCard _ _)) | ||
| λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) | ||
| (isoToPath (iso⊤ _))) | ||
| where iso⊤ : ∀ A → Iso (Unit* → A) A | ||
| Iso.fun (iso⊤ _) f = f tt* | ||
| Iso.inv (iso⊤ _) a _ = a | ||
| Iso.rightInv (iso⊤ _) _ = refl | ||
| Iso.leftInv (iso⊤ _) _ = refl | ||
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| ^AnnihilL𝟙 : (A : Card {ℓ}) → 𝟙 ^ A ≡ 𝟙 | ||
| ^AnnihilL𝟙 = ∥₂.elim (λ _ → isProp→isSet (isSetCard _ _)) | ||
| λ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) | ||
| (isoToPath (iso⊤ _))) | ||
| where iso⊤ : ∀ A → Iso (A → Unit*) Unit* | ||
| Iso.fun (iso⊤ _) _ = tt* | ||
| Iso.inv (iso⊤ _) _ _ = tt* | ||
| Iso.rightInv (iso⊤ _) _ = refl | ||
| Iso.leftInv (iso⊤ _) _ = refl | ||
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| ^LDist+ : (A B C : Card {ℓ}) → A ^ (B + C) ≡ (A ^ B) · (A ^ C) | ||
| ^LDist+ = ∥₂.elim3 (λ _ _ _ → isProp→isSet (isSetCard _ _)) | ||
| λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) | ||
| (isoToPath Π⊎Iso)) | ||
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| ^Assoc· : (A B C : Card {ℓ}) → A ^ (B · C) ≡ (A ^ B) ^ C | ||
| ^Assoc· = ∥₂.elim3 (λ _ _ _ → isProp→isSet (isSetCard _ _)) | ||
| λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) | ||
| (isoToPath (is _ _ _))) | ||
| where is : ∀ A B C → Iso (B × C → A) (C → B → A) | ||
| is A B C = (B × C → A) Iso⟨ domIso Σ-swap-Iso ⟩ | ||
| (C × B → A) Iso⟨ curryIso ⟩ | ||
| (C → B → A) ∎Iso | ||
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| ^RDist· : (A B C : Card {ℓ}) → (A · B) ^ C ≡ (A ^ C) · (B ^ C) | ||
| ^RDist· = ∥₂.elim3 (λ _ _ _ → isProp→isSet (isSetCard _ _)) | ||
| λ _ _ _ → cong ∣_∣₂ (Σ≡Prop (λ _ → isPropIsSet) | ||
| (isoToPath Σ-Π-Iso)) | ||
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| -- With basic arithmetic done, we can now define an ordering over cardinals | ||
| module _ where | ||
| private | ||
| _≲'_ : Card {ℓ} → Card {ℓ} → hProp ℓ | ||
| _≲'_ = ∥₂.rec2 isSetHProp λ (A , _) (B , _) → ∥ A ↪ B ∥₁ , isPropPropTrunc | ||
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| _≲_ : Rel (Card {ℓ}) (Card {ℓ}) ℓ | ||
| A ≲ B = ⟨ A ≲' B ⟩ | ||
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| isPreorder≲ : IsPreorder {ℓ-suc ℓ} _≲_ | ||
| isPreorder≲ | ||
| = ispreorder isSetCard prop reflexive transitive | ||
| where prop : BinaryRelation.isPropValued _≲_ | ||
| prop a b = str (a ≲' b) | ||
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| reflexive : BinaryRelation.isRefl _≲_ | ||
| reflexive = ∥₂.elim (λ A → isProp→isSet (prop A A)) | ||
| (λ (A , _) → ∣ id↪ A ∣₁) | ||
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| transitive : BinaryRelation.isTrans _≲_ | ||
| transitive = ∥₂.elim3 (λ x _ z → isSetΠ2 | ||
| λ _ _ → isProp→isSet | ||
| (prop x z)) | ||
| (λ (A , _) (B , _) (C , _) | ||
| → ∥₁.map2 λ A↪B B↪C | ||
| → compEmbedding | ||
| B↪C | ||
| A↪B) | ||
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| isLeast𝟘 : ∀{ℓ} → isLeast isPreorder≲ (Card {ℓ} , id↪ (Card {ℓ})) (𝟘 {ℓ}) | ||
| isLeast𝟘 = ∥₂.elim (λ x → isProp→isSet (IsPreorder.is-prop-valued | ||
| isPreorder≲ 𝟘 x)) | ||
| (λ _ → ∣ ⊥.rec* , (λ ()) ∣₁) | ||
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| -- Our arithmetic behaves as expected over our preordering | ||
| +Monotone≲ : (A B C D : Card {ℓ}) → A ≲ C → B ≲ D → (A + B) ≲ (C + D) | ||
| +Monotone≲ | ||
| = ∥₂.elim4 (λ w x y z → isSetΠ2 λ _ _ → isProp→isSet (IsPreorder.is-prop-valued | ||
| isPreorder≲ | ||
| (w + x) | ||
| (y + z))) | ||
| λ (A , _) (B , _) (C , _) (D , _) | ||
| → ∥₁.map2 λ A↪C B↪D → ⊎Monotone↪ A↪C B↪D | ||
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| ·Monotone≲ : (A B C D : Card {ℓ}) → A ≲ C → B ≲ D → (A · B) ≲ (C · D) | ||
| ·Monotone≲ | ||
| = ∥₂.elim4 (λ w x y z → isSetΠ2 λ _ _ → isProp→isSet (IsPreorder.is-prop-valued | ||
| isPreorder≲ | ||
| (w · x) | ||
| (y · z))) | ||
| λ (A , _) (B , _) (C , _) (D , _) | ||
| → ∥₁.map2 λ A↪C B↪D → ×Monotone↪ A↪C B↪D |
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