diff --git a/MIXFIX-OPERATORS.md b/MIXFIX-OPERATORS.md index 4ec18bd8cc..c0e723b26e 100644 --- a/MIXFIX-OPERATORS.md +++ b/MIXFIX-OPERATORS.md @@ -163,22 +163,22 @@ Below, we outline a list of general rules when assigning associativities. ## Full table of assigned precedences -| Precedence level | Operators | -| ---------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -| 50 | Unary nonparametric operators. This class is currently empty | -| 45 | Exponentiative arithmetic operators | -| 40 | Multiplicative arithmetic operators | -| 36 | Subtractive arithmetic operators | -| 35 | Additive arithmetic operators | -| 30 | Relational arithmetic operators like`_≤-ℕ_` and `_<-ℕ_` | -| 25 | Parametric unary operators. This class is currently empty | -| 20 | Parametric exponentiative operators. This class is currently empty | -| 17 | Left homotopy whiskering `_·l_` | -| 16 | Right homotopy whiskering `_·r_` | -| 15 | Parametric multiplicative operators like `_×_`,`_×∗_`, `_∧_`, `_∧∗_`, function composition operators like `_∘_`,`_∘∗_`, `_∘e_`, and `_∘iff_`, concatenation operators like `_∙_` and `_∙h_` | -| 10 | Parametric additive operators like `_+_`, `_∨_`, `_∨∗_`, list concatenation, monadic bind operators for the type checking monad | -| 6 | Parametric relational operators like `_＝_`, `_~_`, `_≃_`, and `_↔_`, elementhood relations, subtype relations | -| 5 | Directed function type-like formation operators, e.g. `_⇒_`, `_→∗_`, `_↠_`, `_↪_`, `_↪ᵈ_`, and `_⊆_` | -| 3 | The pairing operators `_,_` and `_,ω_` | -| 0-1 | Reasoning syntaxes | -| -∞ | Function type formation `_→_` | +| Precedence level | Operators | +| ---------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | +| 50 | Unary nonparametric operators. This class is currently empty | +| 45 | Exponentiative arithmetic operators | +| 40 | Multiplicative arithmetic operators | +| 36 | Subtractive arithmetic operators | +| 35 | Additive arithmetic operators | +| 30 | Relational arithmetic operators like`_≤-ℕ_` and `_<-ℕ_` | +| 25 | Parametric unary operators like `¬_` and `¬¬_` | +| 20 | Parametric exponentiative operators. This class is currently empty | +| 17 | Left homotopy whiskering `_·l_` | +| 16 | Right homotopy whiskering `_·r_` | +| 15 | Parametric multiplicative operators like `_×_`,`_×∗_`, `_∧_`, `_∧∗_`, `_*_`, function composition operators like `_∘_`,`_∘∗_`, `_∘e_`, and `_∘iff_`, concatenation operators like `_∙_` and `_∙h_` | +| 10 | Parametric additive operators like `_+_`, `_∨_`, `_∨∗_`, list concatenation, monadic bind operators for the type checking monad | +| 6 | Parametric relational operators like `_＝_`, `_~_`, `_≃_`, `_⇔_`, and `_↔_`, elementhood relations, subtype relations | +| 5 | Directed function type-like formation operators, e.g. `_⇒_`, `_↪_`, `_→∗_`, `_↠_`, `_↪ᵈ_`, and `_⊆_` | +| 3 | The pairing operators `_,_` and `_,ω_` | +| 0-1 | Reasoning syntaxes | +| -∞ | Function type formation `_→_` | diff --git a/src/category-theory/categories.lagda.md b/src/category-theory/categories.lagda.md index e015106c69..810689aba3 100644 --- a/src/category-theory/categories.lagda.md +++ b/src/category-theory/categories.lagda.md @@ -18,6 +18,7 @@ open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.propositions open import foundation.sets open import foundation.strictly-involutive-identity-types @@ -312,7 +313,7 @@ module _ is-equiv-is-category-is-surjective-iso-eq-Preunivalent-Category : is-equiv is-category-is-surjective-iso-eq-Preunivalent-Category is-equiv-is-category-is-surjective-iso-eq-Preunivalent-Category = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-is-surjective-iso-eq-Precategory ( precategory-Preunivalent-Category C)) ( is-prop-is-category-Precategory (precategory-Preunivalent-Category C)) @@ -321,7 +322,7 @@ module _ is-equiv-is-surjective-iso-eq-is-category-Preunivalent-Category : is-equiv is-surjective-iso-eq-is-category-Preunivalent-Category is-equiv-is-surjective-iso-eq-is-category-Preunivalent-Category = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-is-category-Precategory (precategory-Preunivalent-Category C)) ( is-prop-is-surjective-iso-eq-Precategory ( precategory-Preunivalent-Category C)) diff --git a/src/category-theory/coproducts-in-precategories.lagda.md b/src/category-theory/coproducts-in-precategories.lagda.md index 1ef104ed59..cc72f57907 100644 --- a/src/category-theory/coproducts-in-precategories.lagda.md +++ b/src/category-theory/coproducts-in-precategories.lagda.md @@ -16,7 +16,7 @@ open import foundation.dependent-pair-types open import foundation.identity-types open import foundation.iterated-dependent-product-types open import foundation.propositions -open import foundation.unique-existence +open import foundation.uniqueness-quantification open import foundation.universe-levels ``` @@ -39,7 +39,7 @@ module _ (z : obj-Precategory C) (f : hom-Precategory C x z) → (g : hom-Precategory C y z) → - ∃! + uniquely-exists-structure ( hom-Precategory C p z) ( λ h → ( comp-hom-Precategory C h l ＝ f) × diff --git a/src/category-theory/dependent-products-of-large-precategories.lagda.md b/src/category-theory/dependent-products-of-large-precategories.lagda.md index 83d18e5945..eacb31a0b5 100644 --- a/src/category-theory/dependent-products-of-large-precategories.lagda.md +++ b/src/category-theory/dependent-products-of-large-precategories.lagda.md @@ -14,6 +14,7 @@ open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.propositions open import foundation.sets open import foundation.strictly-involutive-identity-types @@ -215,7 +216,7 @@ module _ (f : hom-Π-Large-Precategory I C x y) → is-equiv (is-fiberwise-iso-is-iso-Π-Large-Precategory f) is-equiv-is-fiberwise-iso-is-iso-Π-Large-Precategory f = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-is-iso-Large-Precategory (Π-Large-Precategory I C) f) ( is-prop-Π (λ i → is-prop-is-iso-Large-Precategory (C i) (f i))) ( is-iso-is-fiberwise-iso-Π-Large-Precategory f) @@ -233,7 +234,7 @@ module _ (f : hom-Π-Large-Precategory I C x y) → is-equiv (is-iso-is-fiberwise-iso-Π-Large-Precategory f) is-equiv-is-iso-is-fiberwise-iso-Π-Large-Precategory f = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-Π (λ i → is-prop-is-iso-Large-Precategory (C i) (f i))) ( is-prop-is-iso-Large-Precategory (Π-Large-Precategory I C) f) ( is-fiberwise-iso-is-iso-Π-Large-Precategory f) diff --git a/src/category-theory/dependent-products-of-precategories.lagda.md b/src/category-theory/dependent-products-of-precategories.lagda.md index 6e7448dc99..4ee2a1858a 100644 --- a/src/category-theory/dependent-products-of-precategories.lagda.md +++ b/src/category-theory/dependent-products-of-precategories.lagda.md @@ -15,6 +15,7 @@ open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.propositions open import foundation.sets open import foundation.strictly-involutive-identity-types @@ -183,7 +184,7 @@ module _ (f : hom-Π-Precategory I C x y) → is-equiv (is-fiberwise-iso-is-iso-Π-Precategory f) is-equiv-is-fiberwise-iso-is-iso-Π-Precategory f = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-is-iso-Precategory (Π-Precategory I C) f) ( is-prop-Π (λ i → is-prop-is-iso-Precategory (C i) (f i))) ( is-iso-is-fiberwise-iso-Π-Precategory f) @@ -201,7 +202,7 @@ module _ (f : hom-Π-Precategory I C x y) → is-equiv (is-iso-is-fiberwise-iso-Π-Precategory f) is-equiv-is-iso-is-fiberwise-iso-Π-Precategory f = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-Π (λ i → is-prop-is-iso-Precategory (C i) (f i))) ( is-prop-is-iso-Precategory (Π-Precategory I C) f) ( is-fiberwise-iso-is-iso-Π-Precategory f) diff --git a/src/category-theory/essentially-surjective-functors-precategories.lagda.md b/src/category-theory/essentially-surjective-functors-precategories.lagda.md index ef49c3c337..b33efd03a0 100644 --- a/src/category-theory/essentially-surjective-functors-precategories.lagda.md +++ b/src/category-theory/essentially-surjective-functors-precategories.lagda.md @@ -43,14 +43,15 @@ module _ Π-Prop ( obj-Precategory D) ( λ y → - ∃-Prop + exists-structure-Prop ( obj-Precategory C) ( λ x → iso-Precategory D (obj-functor-Precategory C D F x) y)) is-essentially-surjective-functor-Precategory : UU (l1 ⊔ l3 ⊔ l4) is-essentially-surjective-functor-Precategory = ( y : obj-Precategory D) → - ∃ ( obj-Precategory C) + exists-structure + ( obj-Precategory C) ( λ x → iso-Precategory D (obj-functor-Precategory C D F x) y) is-prop-is-essentially-surjective-functor-Precategory : diff --git a/src/category-theory/exponential-objects-precategories.lagda.md b/src/category-theory/exponential-objects-precategories.lagda.md index 941b98f1d7..e2fad05451 100644 --- a/src/category-theory/exponential-objects-precategories.lagda.md +++ b/src/category-theory/exponential-objects-precategories.lagda.md @@ -13,7 +13,7 @@ open import category-theory.products-in-precategories open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.identity-types -open import foundation.unique-existence +open import foundation.uniqueness-quantification open import foundation.universe-levels ``` @@ -47,7 +47,7 @@ module _ is-exponential-obj-Precategory x y e ev = (z : obj-Precategory C) (f : hom-Precategory C (object-product-obj-Precategory C p z x) y) → - ∃! + uniquely-exists-structure ( hom-Precategory C z e) ( λ g → comp-hom-Precategory C ev diff --git a/src/category-theory/faithful-maps-precategories.lagda.md b/src/category-theory/faithful-maps-precategories.lagda.md index 40b26f802d..69cc0cd483 100644 --- a/src/category-theory/faithful-maps-precategories.lagda.md +++ b/src/category-theory/faithful-maps-precategories.lagda.md @@ -16,7 +16,9 @@ open import foundation.equivalences open import foundation.function-types open import foundation.injective-maps open import foundation.iterated-dependent-product-types +open import foundation.logical-equivalences open import foundation.propositions +open import foundation.sets open import foundation.universe-levels ``` @@ -156,7 +158,7 @@ module _ is-equiv-is-injective-hom-is-faithful-map-Precategory : is-equiv is-injective-hom-is-faithful-map-Precategory is-equiv-is-injective-hom-is-faithful-map-Precategory = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-is-faithful-map-Precategory C D F) ( is-prop-is-injective-hom-map-Precategory C D F) ( is-faithful-is-injective-hom-map-Precategory) @@ -164,7 +166,7 @@ module _ is-equiv-is-faithful-is-injective-hom-map-Precategory : is-equiv is-faithful-is-injective-hom-map-Precategory is-equiv-is-faithful-is-injective-hom-map-Precategory = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-is-injective-hom-map-Precategory C D F) ( is-prop-is-faithful-map-Precategory C D F) ( is-injective-hom-is-faithful-map-Precategory) diff --git a/src/category-theory/isomorphisms-in-categories.lagda.md b/src/category-theory/isomorphisms-in-categories.lagda.md index 5efbcdb841..a0f12490f9 100644 --- a/src/category-theory/isomorphisms-in-categories.lagda.md +++ b/src/category-theory/isomorphisms-in-categories.lagda.md @@ -212,7 +212,7 @@ module _ ### The type of isomorphisms form a set The type of isomorphisms between objects `x y : A` is a -[subtype](foundation-core.subtypes.md) of the [foundation-core.sets.md) +[subtype](foundation-core.subtypes.md) of the [set](foundation-core.sets.md) `hom x y` since being an isomorphism is a proposition. ```agda diff --git a/src/category-theory/isomorphisms-in-precategories.lagda.md b/src/category-theory/isomorphisms-in-precategories.lagda.md index 4e7be2a906..31a63e5d4f 100644 --- a/src/category-theory/isomorphisms-in-precategories.lagda.md +++ b/src/category-theory/isomorphisms-in-precategories.lagda.md @@ -248,7 +248,7 @@ module _ ### The type of isomorphisms form a set The type of isomorphisms between objects `x y : A` is a -[subtype](foundation-core.subtypes.md) of the [foundation-core.sets.md) +[subtype](foundation-core.subtypes.md) of the [set](foundation-core.sets.md) `hom x y` since being an isomorphism is a proposition. ```agda diff --git a/src/category-theory/natural-numbers-object-precategories.lagda.md b/src/category-theory/natural-numbers-object-precategories.lagda.md index b55eb9b842..33cad6e351 100644 --- a/src/category-theory/natural-numbers-object-precategories.lagda.md +++ b/src/category-theory/natural-numbers-object-precategories.lagda.md @@ -14,7 +14,7 @@ open import foundation.action-on-identifications-functions open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.identity-types -open import foundation.unique-existence +open import foundation.uniqueness-quantification open import foundation.universe-levels ``` @@ -43,7 +43,7 @@ module _ (x : obj-Precategory C) (q : hom-Precategory C t x) (f : hom-Precategory C x x) → - ∃! + uniquely-exists-structure ( hom-Precategory C n x) ( λ u → ( comp-hom-Precategory C u z ＝ q) × diff --git a/src/category-theory/precategory-of-functors.lagda.md b/src/category-theory/precategory-of-functors.lagda.md index 9130ba3fe0..d0dd6873e4 100644 --- a/src/category-theory/precategory-of-functors.lagda.md +++ b/src/category-theory/precategory-of-functors.lagda.md @@ -20,6 +20,7 @@ open import foundation.equivalences open import foundation.function-extensionality open import foundation.functoriality-dependent-pair-types open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.propositions open import foundation.strictly-involutive-identity-types open import foundation.universe-levels @@ -199,7 +200,7 @@ module _ (f : natural-transformation-Precategory C D F G) → is-equiv (is-iso-functor-is-natural-isomorphism-Precategory f) is-equiv-is-iso-functor-is-natural-isomorphism-Precategory f = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-is-natural-isomorphism-Precategory C D F G f) ( is-prop-is-iso-Precategory ( functor-precategory-Precategory C D) {F} {G} f) @@ -209,7 +210,7 @@ module _ (f : natural-transformation-Precategory C D F G) → is-equiv (is-natural-isomorphism-is-iso-functor-Precategory f) is-equiv-is-natural-isomorphism-is-iso-functor-Precategory f = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-is-iso-Precategory ( functor-precategory-Precategory C D) {F} {G} f) ( is-prop-is-natural-isomorphism-Precategory C D F G f) diff --git a/src/category-theory/precategory-of-maps-precategories.lagda.md b/src/category-theory/precategory-of-maps-precategories.lagda.md index bb248ce3b4..8fc3c449a5 100644 --- a/src/category-theory/precategory-of-maps-precategories.lagda.md +++ b/src/category-theory/precategory-of-maps-precategories.lagda.md @@ -20,6 +20,7 @@ open import foundation.equivalences open import foundation.function-extensionality open import foundation.functoriality-dependent-pair-types open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.propositions open import foundation.strictly-involutive-identity-types open import foundation.universe-levels @@ -199,7 +200,7 @@ module _ (f : natural-transformation-map-Precategory C D F G) → is-equiv (is-iso-map-is-natural-isomorphism-map-Precategory f) is-equiv-is-iso-map-is-natural-isomorphism-map-Precategory f = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-is-natural-isomorphism-map-Precategory C D F G f) ( is-prop-is-iso-Precategory ( map-precategory-Precategory C D) {F} {G} f) @@ -209,7 +210,7 @@ module _ (f : natural-transformation-map-Precategory C D F G) → is-equiv (is-natural-isomorphism-map-is-iso-map-Precategory f) is-equiv-is-natural-isomorphism-map-is-iso-map-Precategory f = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-is-iso-Precategory ( map-precategory-Precategory C D) {F} {G} f) ( is-prop-is-natural-isomorphism-map-Precategory C D F G f) diff --git a/src/category-theory/products-in-precategories.lagda.md b/src/category-theory/products-in-precategories.lagda.md index bb27e473b9..78e0668af4 100644 --- a/src/category-theory/products-in-precategories.lagda.md +++ b/src/category-theory/products-in-precategories.lagda.md @@ -16,7 +16,7 @@ open import foundation.dependent-pair-types open import foundation.identity-types open import foundation.iterated-dependent-product-types open import foundation.propositions -open import foundation.unique-existence +open import foundation.uniqueness-quantification open import foundation.universe-levels ``` @@ -52,8 +52,11 @@ module _ (z : obj-Precategory C) (f : hom-Precategory C z x) → (g : hom-Precategory C z y) → - (∃! (hom-Precategory C z p) λ h → - (comp-hom-Precategory C l h ＝ f) × (comp-hom-Precategory C r h ＝ g)) + ( uniquely-exists-structure + ( hom-Precategory C z p) + ( λ h → + ( comp-hom-Precategory C l h ＝ f) × + ( comp-hom-Precategory C r h ＝ g))) product-obj-Precategory : obj-Precategory C → obj-Precategory C → UU (l1 ⊔ l2) product-obj-Precategory x y = diff --git a/src/category-theory/pullbacks-in-precategories.lagda.md b/src/category-theory/pullbacks-in-precategories.lagda.md index 1d18cc1147..aaefb61ec6 100644 --- a/src/category-theory/pullbacks-in-precategories.lagda.md +++ b/src/category-theory/pullbacks-in-precategories.lagda.md @@ -16,7 +16,7 @@ open import foundation.dependent-pair-types open import foundation.identity-types open import foundation.iterated-dependent-product-types open import foundation.propositions -open import foundation.unique-existence +open import foundation.uniqueness-quantification open import foundation.universe-levels ``` @@ -62,7 +62,7 @@ module _ (p₁' : hom-Precategory C w' y) → (p₂' : hom-Precategory C w' z) → comp-hom-Precategory C f p₁' ＝ comp-hom-Precategory C g p₂' → - ∃! + uniquely-exists-structure ( hom-Precategory C w' w) ( λ h → ( comp-hom-Precategory C p₁ h ＝ p₁') × diff --git a/src/category-theory/replete-subprecategories.lagda.md b/src/category-theory/replete-subprecategories.lagda.md index 2f89bd6296..50d39bd9b3 100644 --- a/src/category-theory/replete-subprecategories.lagda.md +++ b/src/category-theory/replete-subprecategories.lagda.md @@ -18,6 +18,7 @@ open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.functoriality-dependent-pair-types open import foundation.iterated-dependent-product-types +open import foundation.logical-equivalences open import foundation.propositions open import foundation.subsingleton-induction open import foundation.subtypes @@ -216,7 +217,7 @@ module _ is-equiv-is-iso-is-iso-base-is-replete-Subprecategory : is-equiv is-iso-is-iso-base-is-replete-Subprecategory is-equiv-is-iso-is-iso-base-is-replete-Subprecategory = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-is-iso-Precategory C (inclusion-hom-Subprecategory C P x y f)) ( is-prop-is-iso-Subprecategory C P f) ( is-iso-base-is-iso-Subprecategory C P) diff --git a/src/category-theory/representing-arrow-category.lagda.md b/src/category-theory/representing-arrow-category.lagda.md index ff5fd7b484..dbdfec5962 100644 --- a/src/category-theory/representing-arrow-category.lagda.md +++ b/src/category-theory/representing-arrow-category.lagda.md @@ -15,6 +15,7 @@ open import foundation.booleans open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.propositions open import foundation.sets open import foundation.subtypes @@ -118,7 +119,7 @@ representing-arrow-Precategory = is-category-representing-arrow-Category : is-category-Precategory representing-arrow-Precategory is-category-representing-arrow-Category true true = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-set-bool true true) ( is-prop-type-subtype ( is-iso-prop-Precategory representing-arrow-Precategory {true} {true}) @@ -133,7 +134,7 @@ is-category-representing-arrow-Category false true = ( iso-eq-Precategory representing-arrow-Precategory false true) ( hom-inv-iso-Precategory representing-arrow-Precategory) is-category-representing-arrow-Category false false = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-set-bool false false) ( is-prop-type-subtype ( is-iso-prop-Precategory representing-arrow-Precategory {false} {false}) diff --git a/src/category-theory/slice-precategories.lagda.md b/src/category-theory/slice-precategories.lagda.md index 00eaee9ffd..ccef207457 100644 --- a/src/category-theory/slice-precategories.lagda.md +++ b/src/category-theory/slice-precategories.lagda.md @@ -22,6 +22,7 @@ open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.propositions open import foundation.sets open import foundation.strictly-involutive-identity-types @@ -263,7 +264,11 @@ module _ eq-hom-Slice-Precategory C A _ _ (pr2 (pr2 (pr1 (ϕ Z h₁ h₂ β₂)))) q : - ∀ k → + (k : + hom-Precategory + ( Slice-Precategory C A) + ( Z , comp-hom-Precategory C f h₁) + ( W , p)) → is-prop ( ( comp-hom-Precategory (Slice-Precategory C A) (p₁ , α₁) k ＝ (h₁ , refl)) × @@ -275,7 +280,11 @@ module _ ( is-set-hom-Slice-Precategory C A _ _ _ _) σ : - ∀ k → + (k : + hom-Precategory + ( Slice-Precategory C A) + ( Z , comp-hom-Precategory C f h₁) + ( W , p)) → ( ( comp-hom-Precategory ( Slice-Precategory C A) ( p₁ , α₁) @@ -331,7 +340,7 @@ module _ ( p₂' , α'))))) q : - ∀ k' → + (k' : hom-Precategory C W' W) → is-prop (( comp-hom-Precategory C p₁ k' ＝ p₁') × ( comp-hom-Precategory C p₂ k' ＝ p₂')) @@ -363,7 +372,7 @@ module _ is-product-obj-Precategory (Slice-Precategory C A) (X , f) (Y , g) (W , p) (p₁ , α₁) (p₂ , α₂) equiv-is-pullback-is-product-Slice-Precategory = - equiv-prop + equiv-iff-is-prop ( is-prop-is-pullback-obj-Precategory C A X Y f g W p₁ p₂ α) ( is-prop-is-product-obj-Precategory (Slice-Precategory C A) (X , f) (Y , g) (W , p) (p₁ , α₁) (p₂ , α₂)) diff --git a/src/commutative-algebra/intersections-radical-ideals-commutative-rings.lagda.md b/src/commutative-algebra/intersections-radical-ideals-commutative-rings.lagda.md index 850554b720..1bf031cb1c 100644 --- a/src/commutative-algebra/intersections-radical-ideals-commutative-rings.lagda.md +++ b/src/commutative-algebra/intersections-radical-ideals-commutative-rings.lagda.md @@ -177,7 +177,7 @@ module _ ( intersection-ideal-Commutative-Ring A I J) ( x)) ( λ (m , K') → - intro-∃ + intro-exists ( add-ℕ n m) ( ( is-closed-under-eq-ideal-Commutative-Ring A I ( is-closed-under-right-multiplication-ideal-Commutative-Ring @@ -209,7 +209,8 @@ module _ ( ideal-radical-of-ideal-Commutative-Ring A I) ( ideal-radical-of-ideal-Commutative-Ring A J) ( x)) - ( λ (n , H' , K') → (intro-∃ n H' , intro-∃ n K')) + ( λ (n , H' , K') → + ( intro-exists n H' , intro-exists n K')) preserves-intersection-radical-of-ideal-Commutative-Ring : ( intersection-ideal-Commutative-Ring A diff --git a/src/commutative-algebra/nilradical-commutative-rings.lagda.md b/src/commutative-algebra/nilradical-commutative-rings.lagda.md index 8acddc3744..e8cae03fe9 100644 --- a/src/commutative-algebra/nilradical-commutative-rings.lagda.md +++ b/src/commutative-algebra/nilradical-commutative-rings.lagda.md @@ -49,7 +49,7 @@ contains-zero-nilradical-Commutative-Ring : {l : Level} (A : Commutative-Ring l) → contains-zero-subset-Commutative-Ring A ( subset-nilradical-Commutative-Ring A) -contains-zero-nilradical-Commutative-Ring A = intro-∃ 1 refl +contains-zero-nilradical-Commutative-Ring A = intro-exists 1 refl ``` ### The nilradical is closed under addition diff --git a/src/commutative-algebra/nilradicals-commutative-semirings.lagda.md b/src/commutative-algebra/nilradicals-commutative-semirings.lagda.md index f60df9ffde..994c83bc29 100644 --- a/src/commutative-algebra/nilradicals-commutative-semirings.lagda.md +++ b/src/commutative-algebra/nilradicals-commutative-semirings.lagda.md @@ -42,7 +42,7 @@ contains-zero-nilradical-Commutative-Semiring : {l : Level} (A : Commutative-Semiring l) → contains-zero-subset-Commutative-Semiring A ( subset-nilradical-Commutative-Semiring A) -contains-zero-nilradical-Commutative-Semiring A = intro-∃ 1 refl +contains-zero-nilradical-Commutative-Semiring A = intro-exists 1 refl ``` ### The nilradical is closed under addition diff --git a/src/commutative-algebra/prime-ideals-commutative-rings.lagda.md b/src/commutative-algebra/prime-ideals-commutative-rings.lagda.md index 317ca4828a..de9a327c13 100644 --- a/src/commutative-algebra/prime-ideals-commutative-rings.lagda.md +++ b/src/commutative-algebra/prime-ideals-commutative-rings.lagda.md @@ -164,11 +164,10 @@ is-radical-prime-ideal-Commutative-Ring R P x zero-ℕ p = ( p) ( x) is-radical-prime-ideal-Commutative-Ring R P x (succ-ℕ n) p = - elim-disjunction-Prop - ( subset-prime-ideal-Commutative-Ring R P (power-Commutative-Ring R n x)) + elim-disjunction ( subset-prime-ideal-Commutative-Ring R P x) - ( subset-prime-ideal-Commutative-Ring R P x) - ( is-radical-prime-ideal-Commutative-Ring R P x n , id) + ( is-radical-prime-ideal-Commutative-Ring R P x n) + ( id) ( is-prime-ideal-ideal-prime-ideal-Commutative-Ring R P ( power-Commutative-Ring R n x) ( x) diff --git a/src/commutative-algebra/products-radical-ideals-commutative-rings.lagda.md b/src/commutative-algebra/products-radical-ideals-commutative-rings.lagda.md index 6177f5690e..f112058f6d 100644 --- a/src/commutative-algebra/products-radical-ideals-commutative-rings.lagda.md +++ b/src/commutative-algebra/products-radical-ideals-commutative-rings.lagda.md @@ -182,7 +182,7 @@ module _ ( J)) ( mul-Commutative-Ring A x y)) ( λ (n , p) → - intro-∃ + intro-exists ( succ-ℕ n) ( is-closed-under-eq-ideal-Commutative-Ring' A ( product-ideal-Commutative-Ring A diff --git a/src/commutative-algebra/radicals-of-ideals-commutative-rings.lagda.md b/src/commutative-algebra/radicals-of-ideals-commutative-rings.lagda.md index 40c4fe1491..13a2f12ae2 100644 --- a/src/commutative-algebra/radicals-of-ideals-commutative-rings.lagda.md +++ b/src/commutative-algebra/radicals-of-ideals-commutative-rings.lagda.md @@ -82,8 +82,7 @@ module _ subset-radical-of-ideal-Commutative-Ring : type-Commutative-Ring A → Prop l2 subset-radical-of-ideal-Commutative-Ring f = - ∃-Prop ℕ - ( λ n → is-in-ideal-Commutative-Ring A I (power-Commutative-Ring A n f)) + ∃ ℕ (λ n → subset-ideal-Commutative-Ring A I (power-Commutative-Ring A n f)) is-in-radical-of-ideal-Commutative-Ring : type-Commutative-Ring A → UU l2 is-in-radical-of-ideal-Commutative-Ring = @@ -93,7 +92,7 @@ module _ (f : type-Commutative-Ring A) → is-in-ideal-Commutative-Ring A I f → is-in-radical-of-ideal-Commutative-Ring f - contains-ideal-radical-of-ideal-Commutative-Ring f H = intro-∃ 1 H + contains-ideal-radical-of-ideal-Commutative-Ring f H = intro-exists 1 H contains-zero-radical-of-ideal-Commutative-Ring : is-in-radical-of-ideal-Commutative-Ring (zero-Commutative-Ring A) @@ -113,7 +112,7 @@ module _ ( subset-radical-of-ideal-Commutative-Ring ( add-Commutative-Ring A x y)) ( λ (m , q) → - intro-∃ + intro-exists ( n +ℕ m) ( is-closed-under-eq-ideal-Commutative-Ring' A I ( is-closed-under-addition-ideal-Commutative-Ring A I @@ -138,7 +137,7 @@ module _ apply-universal-property-trunc-Prop H ( subset-radical-of-ideal-Commutative-Ring (neg-Commutative-Ring A x)) ( λ (n , p) → - intro-∃ n + intro-exists n ( is-closed-under-eq-ideal-Commutative-Ring' A I ( is-closed-under-left-multiplication-ideal-Commutative-Ring A I _ ( power-Commutative-Ring A n x) @@ -152,7 +151,7 @@ module _ apply-universal-property-trunc-Prop H ( subset-radical-of-ideal-Commutative-Ring (mul-Commutative-Ring A x y)) ( λ (n , p) → - intro-∃ n + intro-exists n ( is-closed-under-eq-ideal-Commutative-Ring' A I ( is-closed-under-right-multiplication-ideal-Commutative-Ring A I ( power-Commutative-Ring A n x) @@ -167,7 +166,7 @@ module _ apply-universal-property-trunc-Prop H ( subset-radical-of-ideal-Commutative-Ring (mul-Commutative-Ring A x y)) ( λ (n , p) → - intro-∃ n + intro-exists n ( is-closed-under-eq-ideal-Commutative-Ring' A I ( is-closed-under-left-multiplication-ideal-Commutative-Ring A I ( power-Commutative-Ring A n x) @@ -190,7 +189,7 @@ module _ apply-universal-property-trunc-Prop H ( subset-radical-of-ideal-Commutative-Ring x) ( λ (m , K) → - intro-∃ + intro-exists ( mul-ℕ n m) ( is-closed-under-eq-ideal-Commutative-Ring' A I K ( power-mul-Commutative-Ring A n m))) diff --git a/src/commutative-algebra/zariski-topology.lagda.md b/src/commutative-algebra/zariski-topology.lagda.md index 3b325ea92e..b3bbbec25c 100644 --- a/src/commutative-algebra/zariski-topology.lagda.md +++ b/src/commutative-algebra/zariski-topology.lagda.md @@ -42,10 +42,9 @@ is-closed-subset-zariski-topology-Commutative-Ring : (U : subtype (l1 ⊔ l2 ⊔ l3) (prime-ideal-Commutative-Ring l2 A)) → Prop (lsuc l1 ⊔ lsuc l2 ⊔ lsuc l3) is-closed-subset-zariski-topology-Commutative-Ring {l1} {l2} {l3} A U = - ∃-Prop + exists-structure-Prop ( subtype l3 (type-Commutative-Ring A)) - ( λ V → - standard-closed-subset-zariski-topology-Commutative-Ring A V ＝ U) + ( λ V → standard-closed-subset-zariski-topology-Commutative-Ring A V ＝ U) closed-subset-zariski-topology-Commutative-Ring : {l1 l2 : Level} (l3 : Level) (A : Commutative-Ring l1) → diff --git a/src/elementary-number-theory/addition-integers.lagda.md b/src/elementary-number-theory/addition-integers.lagda.md index 51c6422199..0074401613 100644 --- a/src/elementary-number-theory/addition-integers.lagda.md +++ b/src/elementary-number-theory/addition-integers.lagda.md @@ -26,6 +26,7 @@ open import foundation.function-types open import foundation.identity-types open import foundation.injective-maps open import foundation.interchange-law +open import foundation.sets open import foundation.unit-type ``` diff --git a/src/elementary-number-theory/addition-natural-numbers.lagda.md b/src/elementary-number-theory/addition-natural-numbers.lagda.md index 403385d149..47633f70b0 100644 --- a/src/elementary-number-theory/addition-natural-numbers.lagda.md +++ b/src/elementary-number-theory/addition-natural-numbers.lagda.md @@ -21,6 +21,7 @@ open import foundation.identity-types open import foundation.injective-maps open import foundation.interchange-law open import foundation.negated-equality +open import foundation.sets ``` diff --git a/src/elementary-number-theory/finitary-natural-numbers.lagda.md b/src/elementary-number-theory/finitary-natural-numbers.lagda.md index f97fed792d..57acf9d9c0 100644 --- a/src/elementary-number-theory/finitary-natural-numbers.lagda.md +++ b/src/elementary-number-theory/finitary-natural-numbers.lagda.md @@ -21,6 +21,7 @@ open import foundation.empty-types open import foundation.function-types open import foundation.identity-types open import foundation.injective-maps +open import foundation.sets open import foundation.universe-levels open import univalent-combinatorics.standard-finite-types diff --git a/src/elementary-number-theory/inequality-rational-numbers.lagda.md b/src/elementary-number-theory/inequality-rational-numbers.lagda.md index c4b8e44389..3b96f951fd 100644 --- a/src/elementary-number-theory/inequality-rational-numbers.lagda.md +++ b/src/elementary-number-theory/inequality-rational-numbers.lagda.md @@ -21,6 +21,7 @@ open import elementary-number-theory.positive-integers open import elementary-number-theory.rational-numbers open import foundation.cartesian-product-types +open import foundation.conjunction open import foundation.coproduct-types open import foundation.decidable-propositions open import foundation.dependent-pair-types diff --git a/src/elementary-number-theory/initial-segments-natural-numbers.lagda.md b/src/elementary-number-theory/initial-segments-natural-numbers.lagda.md index d6d10e541f..9823b2c9a1 100644 --- a/src/elementary-number-theory/initial-segments-natural-numbers.lagda.md +++ b/src/elementary-number-theory/initial-segments-natural-numbers.lagda.md @@ -38,7 +38,7 @@ holds for every `n : ℕ`. ```agda is-initial-segment-subset-ℕ-Prop : {l : Level} (P : subtype l ℕ) → Prop l is-initial-segment-subset-ℕ-Prop P = - Π-Prop ℕ (λ n → implication-Prop (P (succ-ℕ n)) (P n)) + Π-Prop ℕ (λ n → hom-Prop (P (succ-ℕ n)) (P n)) is-initial-segment-subset-ℕ : {l : Level} (P : subtype l ℕ) → UU l is-initial-segment-subset-ℕ P = type-Prop (is-initial-segment-subset-ℕ-Prop P) diff --git a/src/elementary-number-theory/modular-arithmetic-standard-finite-types.lagda.md b/src/elementary-number-theory/modular-arithmetic-standard-finite-types.lagda.md index 9b643eebd3..fe5615c83e 100644 --- a/src/elementary-number-theory/modular-arithmetic-standard-finite-types.lagda.md +++ b/src/elementary-number-theory/modular-arithmetic-standard-finite-types.lagda.md @@ -26,6 +26,7 @@ open import foundation.equivalences open import foundation.function-types open import foundation.identity-types open import foundation.injective-maps +open import foundation.sets open import foundation.split-surjective-maps open import foundation.surjective-maps open import foundation.universe-levels diff --git a/src/elementary-number-theory/multiplication-integers.lagda.md b/src/elementary-number-theory/multiplication-integers.lagda.md index 8d9326a47b..5306775315 100644 --- a/src/elementary-number-theory/multiplication-integers.lagda.md +++ b/src/elementary-number-theory/multiplication-integers.lagda.md @@ -29,6 +29,7 @@ open import foundation.function-types open import foundation.identity-types open import foundation.injective-maps open import foundation.interchange-law +open import foundation.sets open import foundation.transport-along-identifications open import foundation.type-arithmetic-empty-type open import foundation.unit-type diff --git a/src/elementary-number-theory/multiplication-natural-numbers.lagda.md b/src/elementary-number-theory/multiplication-natural-numbers.lagda.md index 43f53a469c..8e9ffa25d0 100644 --- a/src/elementary-number-theory/multiplication-natural-numbers.lagda.md +++ b/src/elementary-number-theory/multiplication-natural-numbers.lagda.md @@ -19,6 +19,7 @@ open import foundation.identity-types open import foundation.injective-maps open import foundation.interchange-law open import foundation.negated-equality +open import foundation.sets ``` diff --git a/src/elementary-number-theory/nonzero-integers.lagda.md b/src/elementary-number-theory/nonzero-integers.lagda.md index d5b2dcd318..2d7dd2e4a9 100644 --- a/src/elementary-number-theory/nonzero-integers.lagda.md +++ b/src/elementary-number-theory/nonzero-integers.lagda.md @@ -38,7 +38,7 @@ holds. ```agda is-nonzero-prop-ℤ : ℤ → Prop lzero -is-nonzero-prop-ℤ k = neg-Prop' (k ＝ zero-ℤ) +is-nonzero-prop-ℤ k = neg-type-Prop (k ＝ zero-ℤ) is-nonzero-ℤ : ℤ → UU lzero is-nonzero-ℤ k = type-Prop (is-nonzero-prop-ℤ k) diff --git a/src/elementary-number-theory/pisano-periods.lagda.md b/src/elementary-number-theory/pisano-periods.lagda.md index d3d99b98ee..f765a59fb3 100644 --- a/src/elementary-number-theory/pisano-periods.lagda.md +++ b/src/elementary-number-theory/pisano-periods.lagda.md @@ -25,6 +25,7 @@ open import foundation.equivalences open import foundation.identity-types open import foundation.injective-maps open import foundation.repetitions-sequences +open import foundation.sets open import foundation.universe-levels open import univalent-combinatorics.cartesian-product-types diff --git a/src/elementary-number-theory/prime-numbers.lagda.md b/src/elementary-number-theory/prime-numbers.lagda.md index a7543fbf13..767f0d0529 100644 --- a/src/elementary-number-theory/prime-numbers.lagda.md +++ b/src/elementary-number-theory/prime-numbers.lagda.md @@ -157,7 +157,7 @@ has-unique-proper-divisor-is-prime-ℕ n H = fundamental-theorem-id' ( λ x p → pr2 (H x) (inv p)) ( λ x → - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-set-ℕ 1 x) ( is-prop-is-proper-divisor-ℕ n x) ( λ p → inv (pr1 (H x) p))) diff --git a/src/elementary-number-theory/standard-cyclic-rings.lagda.md b/src/elementary-number-theory/standard-cyclic-rings.lagda.md index 692df2b922..60ffeb3ce6 100644 --- a/src/elementary-number-theory/standard-cyclic-rings.lagda.md +++ b/src/elementary-number-theory/standard-cyclic-rings.lagda.md @@ -179,7 +179,7 @@ is-generating-element-one-ℤ-Mod n = is-cyclic-ℤ-Mod-Group : ( n : ℕ) → is-cyclic-Group (ℤ-Mod-Group n) is-cyclic-ℤ-Mod-Group n = - intro-∃ + intro-exists ( one-ℤ-Mod n) ( is-generating-element-one-ℤ-Mod n) diff --git a/src/elementary-number-theory/strict-inequality-integer-fractions.lagda.md b/src/elementary-number-theory/strict-inequality-integer-fractions.lagda.md index 0801560c95..966dad6794 100644 --- a/src/elementary-number-theory/strict-inequality-integer-fractions.lagda.md +++ b/src/elementary-number-theory/strict-inequality-integer-fractions.lagda.md @@ -26,6 +26,7 @@ open import elementary-number-theory.strict-inequality-integers open import foundation.action-on-identifications-functions open import foundation.cartesian-product-types +open import foundation.conjunction open import foundation.coproduct-types open import foundation.decidable-propositions open import foundation.dependent-pair-types @@ -292,9 +293,9 @@ module _ where dense-le-fraction-ℤ : - ∃ fraction-ℤ (λ r → le-fraction-ℤ x r × le-fraction-ℤ r y) + exists fraction-ℤ (λ r → le-fraction-ℤ-Prop x r ∧ le-fraction-ℤ-Prop r y) dense-le-fraction-ℤ = - intro-∃ + intro-exists ( mediant-fraction-ℤ x y) ( le-left-mediant-fraction-ℤ x y H , le-right-mediant-fraction-ℤ x y H) ``` @@ -307,7 +308,8 @@ module _ where located-le-fraction-ℤ : - le-fraction-ℤ y z → (le-fraction-ℤ-Prop y x) ∨ (le-fraction-ℤ-Prop x z) + le-fraction-ℤ y z → + type-disjunction-Prop (le-fraction-ℤ-Prop y x) (le-fraction-ℤ-Prop x z) located-le-fraction-ℤ H = unit-trunc-Prop ( map-coproduct diff --git a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md index 0f342d811e..e6c91bbdf5 100644 --- a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md +++ b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md @@ -27,6 +27,7 @@ open import elementary-number-theory.strict-inequality-integers open import foundation.binary-relations open import foundation.cartesian-product-types +open import foundation.conjunction open import foundation.coproduct-types open import foundation.decidable-propositions open import foundation.dependent-pair-types @@ -197,28 +198,30 @@ module _ (x : ℚ) where - left-∃-le-ℚ : ∃ ℚ (λ q → le-ℚ q x) - left-∃-le-ℚ = intro-∃ - ( rational-fraction-ℤ frac) - ( preserves-le-left-rational-fraction-ℤ x frac - ( le-fraction-le-numerator-fraction-ℤ - ( frac) - ( fraction-ℚ x) - ( refl) - ( le-pred-ℤ (numerator-ℚ x)))) + exists-lesser-ℚ : exists ℚ (λ q → le-ℚ-Prop q x) + exists-lesser-ℚ = + intro-exists + ( rational-fraction-ℤ frac) + ( preserves-le-left-rational-fraction-ℤ x frac + ( le-fraction-le-numerator-fraction-ℤ + ( frac) + ( fraction-ℚ x) + ( refl) + ( le-pred-ℤ (numerator-ℚ x)))) where frac : fraction-ℤ frac = (pred-ℤ (numerator-ℚ x) , positive-denominator-ℚ x) - right-∃-le-ℚ : ∃ ℚ (λ r → le-ℚ x r) - right-∃-le-ℚ = intro-∃ - ( rational-fraction-ℤ frac) - ( preserves-le-right-rational-fraction-ℤ x frac - ( le-fraction-le-numerator-fraction-ℤ - ( fraction-ℚ x) - ( frac) - ( refl) - ( le-succ-ℤ (numerator-ℚ x)))) + exists-greater-ℚ : exists ℚ (λ r → le-ℚ-Prop x r) + exists-greater-ℚ = + intro-exists + ( rational-fraction-ℤ frac) + ( preserves-le-right-rational-fraction-ℤ x frac + ( le-fraction-le-numerator-fraction-ℤ + ( fraction-ℚ x) + ( frac) + ( refl) + ( le-succ-ℤ (numerator-ℚ x)))) where frac : fraction-ℤ frac = (succ-ℤ (numerator-ℚ x) , positive-denominator-ℚ x) @@ -282,9 +285,9 @@ module _ (x y : ℚ) (H : le-ℚ x y) where - dense-le-ℚ : ∃ ℚ (λ r → le-ℚ x r × le-ℚ r y) + dense-le-ℚ : exists ℚ (λ r → le-ℚ-Prop x r ∧ le-ℚ-Prop r y) dense-le-ℚ = - intro-∃ + intro-exists ( mediant-ℚ x y) ( le-left-mediant-ℚ x y H , le-right-mediant-ℚ x y H) ``` @@ -292,7 +295,8 @@ module _ ### Strict inequality on the rational numbers is located ```agda -located-le-ℚ : (x y z : ℚ) → le-ℚ y z → (le-ℚ-Prop y x) ∨ (le-ℚ-Prop x z) +located-le-ℚ : + (x y z : ℚ) → le-ℚ y z → type-disjunction-Prop (le-ℚ-Prop y x) (le-ℚ-Prop x z) located-le-ℚ x y z H = unit-trunc-Prop ( map-coproduct diff --git a/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md b/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md index fd45692c37..a422c722f0 100644 --- a/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md +++ b/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md @@ -55,13 +55,13 @@ numbers. ### For any decidable family `P` over `Fin n`, if `P x` doesn't hold for all `x` then there exists an `x` for which `P x` is false ```agda -exists-not-not-forall-Fin : +exists-not-not-for-all-Fin : {l : Level} (k : ℕ) {P : Fin k → UU l} → (is-decidable-fam P) → ¬ ((x : Fin k) → P x) → Σ (Fin k) (λ x → ¬ (P x)) -exists-not-not-forall-Fin {l} zero-ℕ d H = ex-falso (H ind-empty) -exists-not-not-forall-Fin {l} (succ-ℕ k) {P} d H with d (inr star) +exists-not-not-for-all-Fin {l} zero-ℕ d H = ex-falso (H ind-empty) +exists-not-not-for-all-Fin {l} (succ-ℕ k) {P} d H with d (inr star) ... | inl p = - T ( exists-not-not-forall-Fin k + T ( exists-not-not-for-all-Fin k ( λ x → d (inl x)) ( λ f → H (ind-coproduct P f (ind-unit p)))) where @@ -69,13 +69,13 @@ exists-not-not-forall-Fin {l} (succ-ℕ k) {P} d H with d (inr star) T z = pair (inl (pr1 z)) (pr2 z) ... | inr f = pair (inr star) f -exists-not-not-forall-count : +exists-not-not-for-all-count : {l1 l2 : Level} {X : UU l1} (P : X → UU l2) → (is-decidable-fam P) → count X → ¬ ((x : X) → P x) → Σ X (λ x → ¬ (P x)) -exists-not-not-forall-count {l1} {l2} {X} P p e = +exists-not-not-for-all-count {l1} {l2} {X} P p e = ( g) ∘ - ( ( exists-not-not-forall-Fin + ( ( exists-not-not-for-all-Fin ( number-of-elements-count e) ( p ∘ map-equiv-count e)) ∘ f) where @@ -180,7 +180,7 @@ well-ordering-principle-Σ-Fin (succ-ℕ k) {P} d (pair (inr star) p) well-ordering-principle-∃-Fin : {l : Level} (k : ℕ) (P : decidable-subtype l (Fin k)) → - ∃ (Fin k) (is-in-decidable-subtype P) → + exists (Fin k) (subtype-decidable-subtype P) → minimal-element-Fin k (is-in-decidable-subtype P) well-ordering-principle-∃-Fin k P H = apply-universal-property-trunc-Prop H diff --git a/src/finite-algebra/semisimple-commutative-finite-rings.lagda.md b/src/finite-algebra/semisimple-commutative-finite-rings.lagda.md index 2f4184550c..ddfea4c796 100644 --- a/src/finite-algebra/semisimple-commutative-finite-rings.lagda.md +++ b/src/finite-algebra/semisimple-commutative-finite-rings.lagda.md @@ -44,8 +44,7 @@ is-semisimple-Commutative-Ring-𝔽 l2 R = exists ( ℕ) ( λ n → - exists-Prop - ( Fin n → Field-𝔽 l2) + ∃ ( Fin n → Field-𝔽 l2) ( λ A → trunc-Prop ( hom-Commutative-Ring-𝔽 diff --git a/src/finite-group-theory/orbits-permutations.lagda.md b/src/finite-group-theory/orbits-permutations.lagda.md index 525f6e0278..4b629bb9c1 100644 --- a/src/finite-group-theory/orbits-permutations.lagda.md +++ b/src/finite-group-theory/orbits-permutations.lagda.md @@ -799,13 +799,17 @@ module _ (pair k ( (equal-iterate-transposition-other-orbits k) ∙ ( p))))) - ( is-equiv-is-prop is-prop-type-trunc-Prop is-prop-type-trunc-Prop - ( λ P' → apply-universal-property-trunc-Prop P' - ( prop-equivalence-relation (same-orbits-permutation-count g) x y) - ( λ (pair k p) → unit-trunc-Prop - ( pair k - ( (inv (equal-iterate-transposition-other-orbits k)) ∙ - ( p)))))) + ( is-equiv-has-converse-is-prop + ( is-prop-type-trunc-Prop) + ( is-prop-type-trunc-Prop) + ( λ P' → + apply-universal-property-trunc-Prop P' + ( prop-equivalence-relation (same-orbits-permutation-count g) x y) + ( λ (pair k p) → + unit-trunc-Prop + ( ( k) , + ( (inv (equal-iterate-transposition-other-orbits k)) ∙ + ( p)))))) where equal-iterate-transposition-other-orbits : (k : ℕ) → @@ -1124,7 +1128,7 @@ module _ apply-universal-property-trunc-Prop T ( coproduct-sim-equivalence-relation-a-b-Prop g P x) (λ pa → lemma2 g (pair (pr1 pa) (inl (pr2 pa))))) - ( is-equiv-is-prop is-prop-type-trunc-Prop + ( is-equiv-has-converse-is-prop is-prop-type-trunc-Prop ( is-prop-type-Prop ( coproduct-sim-equivalence-relation-a-b-Prop g P x)) ( λ where diff --git a/src/finite-group-theory/transpositions.lagda.md b/src/finite-group-theory/transpositions.lagda.md index faa40ddb52..51817489c5 100644 --- a/src/finite-group-theory/transpositions.lagda.md +++ b/src/finite-group-theory/transpositions.lagda.md @@ -290,7 +290,7 @@ module _ element-is-not-identity-map-transposition : Σ X (λ x → map-transposition Y x ≠ x) element-is-not-identity-map-transposition = - exists-not-not-forall-count + exists-not-not-for-all-count ( λ z → Id (map-transposition Y z) z) ( λ x → has-decidable-equality-count eX (map-transposition Y x) x) ( eX) diff --git a/src/foundation-core/coproduct-types.lagda.md b/src/foundation-core/coproduct-types.lagda.md index 2c6b56433a..02461897b7 100644 --- a/src/foundation-core/coproduct-types.lagda.md +++ b/src/foundation-core/coproduct-types.lagda.md @@ -14,8 +14,8 @@ 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`. +The {{#concept "coproduct" Disambiguation="of types"}} of two types `A` and `B` +can be thought of as the disjoint union of `A` and `B`. ## Definition diff --git a/src/foundation-core/decidable-propositions.lagda.md b/src/foundation-core/decidable-propositions.lagda.md index cd694d7194..7134f32b4b 100644 --- a/src/foundation-core/decidable-propositions.lagda.md +++ b/src/foundation-core/decidable-propositions.lagda.md @@ -28,8 +28,9 @@ open import foundation-core.subtypes ## Idea -A **decidable proposition** is a [proposition](foundation-core.propositions.md) -that has a [decidable](foundation.decidable-types.md) underlying type. +A {{#concept "decidable proposition" Agda=is-decidable-Prop}} is a +[proposition](foundation-core.propositions.md) that has a +[decidable](foundation.decidable-types.md) underlying type. ## Definition @@ -52,7 +53,7 @@ pr2 (is-decidable-Prop P) = is-prop-is-decidable (is-prop-type-Prop P) is-prop-is-decidable-prop : {l : Level} (X : UU l) → is-prop (is-decidable-prop X) is-prop-is-decidable-prop X = - is-prop-is-inhabited + is-prop-has-element ( λ H → is-prop-product ( is-prop-is-prop X) @@ -121,6 +122,38 @@ pr1 unit-Decidable-Prop = unit pr2 unit-Decidable-Prop = is-decidable-prop-unit ``` +### The product of two decidable propositions is a decidable proposition + +```agda +module _ + {l1 l2 : Level} (P : Decidable-Prop l1) (Q : Decidable-Prop l2) + where + + type-product-Decidable-Prop : UU (l1 ⊔ l2) + type-product-Decidable-Prop = + type-product-Prop (prop-Decidable-Prop P) (prop-Decidable-Prop Q) + + is-prop-product-Decidable-Prop : is-prop type-product-Decidable-Prop + is-prop-product-Decidable-Prop = + is-prop-product-Prop (prop-Decidable-Prop P) (prop-Decidable-Prop Q) + + is-decidable-product-Decidable-Prop : is-decidable type-product-Decidable-Prop + is-decidable-product-Decidable-Prop = + is-decidable-product + ( is-decidable-Decidable-Prop P) + ( is-decidable-Decidable-Prop Q) + + is-decidable-prop-product-Decidable-Prop : + is-decidable-prop type-product-Decidable-Prop + pr1 is-decidable-prop-product-Decidable-Prop = is-prop-product-Decidable-Prop + pr2 is-decidable-prop-product-Decidable-Prop = + is-decidable-product-Decidable-Prop + + product-Decidable-Prop : Decidable-Prop (l1 ⊔ l2) + pr1 product-Decidable-Prop = type-product-Decidable-Prop + pr2 product-Decidable-Prop = is-decidable-prop-product-Decidable-Prop +``` + ### Decidability of a propositional truncation ```agda @@ -151,7 +184,7 @@ is-merely-decidable-is-decidable-trunc-Prop : is-decidable (type-trunc-Prop A) → is-merely-decidable A is-merely-decidable-is-decidable-trunc-Prop A (inl x) = apply-universal-property-trunc-Prop x - ( is-merely-Decidable-Prop A) + ( is-merely-decidable-Prop A) ( unit-trunc-Prop ∘ inl) is-merely-decidable-is-decidable-trunc-Prop A (inr f) = unit-trunc-Prop (inr (f ∘ unit-trunc-Prop)) diff --git a/src/foundation-core/injective-maps.lagda.md b/src/foundation-core/injective-maps.lagda.md index a47703cb60..ea67f3fc16 100644 --- a/src/foundation-core/injective-maps.lagda.md +++ b/src/foundation-core/injective-maps.lagda.md @@ -17,11 +17,8 @@ open import foundation-core.equivalences open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.propositional-maps -open import foundation-core.propositions open import foundation-core.retractions open import foundation-core.sections -open import foundation-core.sets ``` @@ -148,63 +145,6 @@ module _ is-injective-emb e {x} {y} = map-inv-is-equiv (is-emb-map-emb e x y) ``` -### Any injective map between sets is an embedding - -```agda -abstract - is-emb-is-injective' : - {l1 l2 : Level} {A : UU l1} (is-set-A : is-set A) - {B : UU l2} (is-set-B : is-set B) (f : A → B) → - is-injective f → is-emb f - is-emb-is-injective' is-set-A is-set-B f is-injective-f x y = - is-equiv-is-prop - ( is-set-A x y) - ( is-set-B (f x) (f y)) - ( is-injective-f) - - is-set-is-injective : - {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} → - is-set B → is-injective f → is-set A - is-set-is-injective {f = f} H I = - is-set-prop-in-id - ( λ x y → f x ＝ f y) - ( λ x y → H (f x) (f y)) - ( λ x → refl) - ( λ x y → I) - - is-emb-is-injective : - {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} → - is-set B → is-injective f → is-emb f - is-emb-is-injective {f = f} H I = - is-emb-is-injective' (is-set-is-injective H I) H f I - - is-prop-map-is-injective : - {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} → - is-set B → is-injective f → is-prop-map f - is-prop-map-is-injective {f = f} H I = - is-prop-map-is-emb (is-emb-is-injective H I) -``` - -### For a map between sets, being injective is a property - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} - where - - is-prop-is-injective : - is-set A → (f : A → B) → is-prop (is-injective f) - is-prop-is-injective H f = - is-prop-implicit-Π - ( λ x → - is-prop-implicit-Π - ( λ y → is-prop-function-type (H x y))) - - is-injective-Prop : is-set A → (A → B) → Prop (l1 ⊔ l2) - pr1 (is-injective-Prop H f) = is-injective f - pr2 (is-injective-Prop H f) = is-prop-is-injective H f -``` - ### Any map out of a contractible type is injective ```agda diff --git a/src/foundation-core/negation.lagda.md b/src/foundation-core/negation.lagda.md index e565a46dfc..7021a1e464 100644 --- a/src/foundation-core/negation.lagda.md +++ b/src/foundation-core/negation.lagda.md @@ -16,14 +16,17 @@ open import foundation-core.empty-types ## Idea -The Curry-Howard interpretation of negation in type theory is the interpretation -of the proposition `P ⇒ ⊥` using propositions as types. Thus, the negation of a -type `A` is the type `A → empty`. +The Curry-Howard interpretation of _negation_ in type theory is the +interpretation of the proposition `P ⇒ ⊥` using propositions as types. Thus, the +{{#concept "negation" Disambiguation="of a type" WDID=Q190558 WD="logical negation" Agda=¬_}} +of a type `A` is the type `A → empty`. ## Definition ```agda -¬ : {l : Level} → UU l → UU l +infix 25 ¬_ + +¬_ : {l : Level} → UU l → UU l ¬ A = A → empty map-neg : @@ -31,3 +34,8 @@ map-neg : (P → Q) → (¬ Q → ¬ P) map-neg f nq p = nq (f p) ``` + +## External links + +- [negation](https://ncatlab.org/nlab/show/negation) at $n$Lab +- [Negation](https://en.wikipedia.org/wiki/Negation) at Wikipedia diff --git a/src/foundation-core/propositions.lagda.md b/src/foundation-core/propositions.lagda.md index 2afe3a74bd..041ce6881d 100644 --- a/src/foundation-core/propositions.lagda.md +++ b/src/foundation-core/propositions.lagda.md @@ -25,16 +25,24 @@ open import foundation-core.transport-along-identifications ## Idea -A type is considered to be a proposition if its identity types are contractible. -This condition is equivalent to the condition that it has up to identification -at most one element. +A type is a {{#concept "proposition" Agda=is-prop}} if its +[identity types](foundation-core.identity-types.md) are +[contractible](foundation-core.contractible-types.md). This condition is +[equivalent](foundation-core.equivalences.md) to the condition that it has up to +identification at most one element. -## Definition +## Definitions + +### The predicate of being a proposition ```agda is-prop : {l : Level} (A : UU l) → UU l is-prop A = (x y : A) → is-contr (x ＝ y) +``` +### The type of propositions + +```agda Prop : (l : Level) → UU (lsuc l) Prop l = Σ (UU l) is-prop @@ -56,20 +64,18 @@ module _ We prove here only that any contractible type is a proposition. The fact that the empty type and the unit type are propositions can be found in -```text -foundation.empty-types -foundation.unit-type -``` +- [`foundation.empty-types`](foundation.empty-types.md), and +- [`foundation.unit-type`](foundation.unit-type.md). ## Properties -### To show that a type is a proposition, we may assume it is inhabited +### To show that a type is a proposition we may assume it has an element ```agda abstract - is-prop-is-inhabited : + is-prop-has-element : {l1 : Level} {X : UU l1} → (X → is-prop X) → is-prop X - is-prop-is-inhabited f x y = f x x y + is-prop-has-element f x y = f x x y ``` ### Equivalent characterizations of propositions @@ -122,29 +128,6 @@ module _ eq-is-proof-irrelevant = eq-is-prop' ∘ is-prop-is-proof-irrelevant ``` -### A map between propositions is an equivalence if there is a map in the reverse direction - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} - where - - abstract - is-equiv-is-prop : - is-prop A → is-prop B → {f : A → B} → (B → A) → is-equiv f - is-equiv-is-prop is-prop-A is-prop-B {f} g = - is-equiv-is-invertible - ( g) - ( λ y → eq-is-prop is-prop-B) - ( λ x → eq-is-prop is-prop-A) - - abstract - equiv-prop : is-prop A → is-prop B → (A → B) → (B → A) → A ≃ B - pr1 (equiv-prop is-prop-A is-prop-B f g) = f - pr2 (equiv-prop is-prop-A is-prop-B f g) = - is-equiv-is-prop is-prop-A is-prop-B g -``` - ### Propositions are closed under equivalences ```agda @@ -192,8 +175,7 @@ abstract ( λ p → K (pr1 y) (tr _ p (pr2 x)) (pr2 y))) Σ-Prop : - {l1 l2 : Level} (P : Prop l1) (Q : type-Prop P → Prop l2) → - Prop (l1 ⊔ l2) + {l1 l2 : Level} (P : Prop l1) (Q : type-Prop P → Prop l2) → Prop (l1 ⊔ l2) pr1 (Σ-Prop P Q) = Σ (type-Prop P) (λ p → type-Prop (Q p)) pr2 (Σ-Prop P Q) = is-prop-Σ @@ -210,10 +192,19 @@ abstract is-prop A → is-prop B → is-prop (A × B) is-prop-product H K = is-prop-Σ H (λ x → K) -product-Prop : {l1 l2 : Level} → Prop l1 → Prop l2 → Prop (l1 ⊔ l2) -pr1 (product-Prop P Q) = type-Prop P × type-Prop Q -pr2 (product-Prop P Q) = - is-prop-product (is-prop-type-Prop P) (is-prop-type-Prop Q) +module _ + {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) + where + + type-product-Prop : UU (l1 ⊔ l2) + type-product-Prop = type-Prop P × type-Prop Q + + is-prop-product-Prop : is-prop type-product-Prop + is-prop-product-Prop = + is-prop-product (is-prop-type-Prop P) (is-prop-type-Prop Q) + + product-Prop : Prop (l1 ⊔ l2) + product-Prop = (type-product-Prop , is-prop-product-Prop) ``` ### Products of families of propositions are propositions @@ -234,15 +225,15 @@ module _ type-Π-Prop : UU (l1 ⊔ l2) type-Π-Prop = (x : A) → type-Prop (P x) - is-prop-type-Π-Prop : is-prop type-Π-Prop - is-prop-type-Π-Prop = is-prop-Π (λ x → is-prop-type-Prop (P x)) + is-prop-Π-Prop : is-prop type-Π-Prop + is-prop-Π-Prop = is-prop-Π (λ x → is-prop-type-Prop (P x)) Π-Prop : Prop (l1 ⊔ l2) pr1 Π-Prop = type-Π-Prop - pr2 Π-Prop = is-prop-type-Π-Prop + pr2 Π-Prop = is-prop-Π-Prop ``` -We repeat the above for implicit Π-types. +We now repeat the above for implicit Π-types. ```agda abstract @@ -262,13 +253,13 @@ module _ type-implicit-Π-Prop : UU (l1 ⊔ l2) type-implicit-Π-Prop = {x : A} → type-Prop (P x) - is-prop-type-implicit-Π-Prop : is-prop type-implicit-Π-Prop - is-prop-type-implicit-Π-Prop = + is-prop-implicit-Π-Prop : is-prop type-implicit-Π-Prop + is-prop-implicit-Π-Prop = is-prop-implicit-Π (λ x → is-prop-type-Prop (P x)) implicit-Π-Prop : Prop (l1 ⊔ l2) pr1 implicit-Π-Prop = type-implicit-Π-Prop - pr2 implicit-Π-Prop = is-prop-type-implicit-Π-Prop + pr2 implicit-Π-Prop = is-prop-implicit-Π-Prop ``` ### The type of functions into a proposition is a proposition @@ -284,41 +275,33 @@ type-function-Prop : {l1 l2 : Level} → UU l1 → Prop l2 → UU (l1 ⊔ l2) type-function-Prop A P = A → type-Prop P -is-prop-type-function-Prop : - {l1 l2 : Level} (A : UU l1) (P : Prop l2) → +is-prop-function-Prop : + {l1 l2 : Level} {A : UU l1} (P : Prop l2) → is-prop (type-function-Prop A P) -is-prop-type-function-Prop A P = +is-prop-function-Prop P = is-prop-function-type (is-prop-type-Prop P) function-Prop : {l1 l2 : Level} → UU l1 → Prop l2 → Prop (l1 ⊔ l2) pr1 (function-Prop A P) = type-function-Prop A P -pr2 (function-Prop A P) = is-prop-type-function-Prop A P +pr2 (function-Prop A P) = is-prop-function-Prop P type-hom-Prop : {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) → UU (l1 ⊔ l2) type-hom-Prop P = type-function-Prop (type-Prop P) -is-prop-type-hom-Prop : +is-prop-hom-Prop : {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) → is-prop (type-hom-Prop P Q) -is-prop-type-hom-Prop P = is-prop-type-function-Prop (type-Prop P) +is-prop-hom-Prop P = is-prop-function-Prop hom-Prop : {l1 l2 : Level} → Prop l1 → Prop l2 → Prop (l1 ⊔ l2) pr1 (hom-Prop P Q) = type-hom-Prop P Q -pr2 (hom-Prop P Q) = is-prop-type-hom-Prop P Q - -implication-Prop : - {l1 l2 : Level} → Prop l1 → Prop l2 → Prop (l1 ⊔ l2) -implication-Prop = hom-Prop - -type-implication-Prop : - {l1 l2 : Level} → Prop l1 → Prop l2 → UU (l1 ⊔ l2) -type-implication-Prop = type-hom-Prop +pr2 (hom-Prop P Q) = is-prop-hom-Prop P Q infixr 5 _⇒_ -_⇒_ = type-implication-Prop +_⇒_ = hom-Prop ``` ### The type of equivalences between two propositions is a proposition @@ -386,3 +369,61 @@ is-prop-Prop : {l : Level} (A : UU l) → Prop l pr1 (is-prop-Prop A) = is-prop A pr2 (is-prop-Prop A) = is-prop-is-prop A ``` + +## See also + +### Operations on propositions + +There is a wide range of operations on propositions due to the rich structure of +intuitionistic logic. Below we give a structured overview of a notable selection +of such operations and their notation in the library. + +The list is split into two sections, the first consists of operations that +generalize to arbitrary types and even sufficiently nice +[subuniverses](foundation.subuniverses.md), such as +$n$-[types](foundation-core.truncated-types.md). + +| Name | Operator on types | Operator on propositions/subtypes | +| ----------------------------------------------------------- | ----------------- | --------------------------------- | +| [Dependent sum](foundation.dependent-pair-types.md) | `Σ` | `Σ-Prop` | +| [Dependent product](foundation.dependent-function-types.md) | `Π` | `Π-Prop` | +| [Functions](foundation-core.function-types.md) | `→` | `⇒` | +| [Logical equivalence](foundation.logical-equivalences.md) | `↔` | `⇔` | +| [Product](foundation-core.cartesian-product-types.md) | `×` | `product-Prop` | +| [Join](synthetic-homotopy-theory.joins-of-types.md) | `*` | `join-Prop` | +| [Exclusive sum](foundation.exclusive-sum.md) | `exclusive-sum` | `exclusive-sum-Prop` | +| [Coproduct](foundation-core.coproduct-types.md) | `+` | _N/A_ | + +Note that for many operations in the second section, there is an equivalent +operation on propositions in the first. + +| Name | Operator on types | Operator on propositions/subtypes | +| ---------------------------------------------------------------------------- | --------------------------- | ---------------------------------------- | +| [Initial object](foundation-core.empty-types.md) | `empty` | `empty-Prop` | +| [Terminal object](foundation.unit-type.md) | `unit` | `unit-Prop` | +| [Existential quantification](foundation.existential-quantification.md) | `exists-structure` | `∃` | +| [Unique existential quantification](foundation.uniqueness-quantification.md) | `uniquely-exists-structure` | `∃!` | +| [Universal quantification](foundation.universal-quantification.md) | | `∀'` (equivalent to `Π-Prop`) | +| [Conjunction](foundation.conjunction.md) | | `∧` (equivalent to `product-Prop`) | +| [Disjunction](foundation.disjunction.md) | `disjunction-type` | `∨` (equivalent to `join-Prop`) | +| [Exclusive disjunction](foundation.exclusive-disjunction.md) | `xor-type` | `⊻` (equivalent to `exclusive-sum-Prop`) | +| [Negation](foundation.negation.md) | `¬` | `¬'` | +| [Double negation](foundation.double-negation.md) | `¬¬` | `¬¬'` | + +We can also organize these operations by indexed and binary variants, giving us +the following table: + +| Name | Binary | Indexed | +| ---------------------- | ------ | ------- | +| Product | `×` | `Π` | +| Conjunction | `∧` | `∀'` | +| Constructive existence | `+` | `Σ` | +| Existence | `∨` | `∃` | +| Unique existence | `⊻` | `∃!` | + +### Table of files about propositional logic + +The following table gives an overview of basic constructions in propositional +logic and related considerations. + +{{#include tables/propositional-logic.md}} diff --git a/src/foundation-core/sets.lagda.md b/src/foundation-core/sets.lagda.md index 603e8413ec..27390c8d90 100644 --- a/src/foundation-core/sets.lagda.md +++ b/src/foundation-core/sets.lagda.md @@ -12,8 +12,10 @@ open import foundation.fundamental-theorem-of-identity-types open import foundation.universe-levels open import foundation-core.contractible-types +open import foundation-core.embeddings open import foundation-core.equivalences open import foundation-core.identity-types +open import foundation-core.injective-maps open import foundation-core.propositions open import foundation-core.truncated-types open import foundation-core.truncation-levels @@ -46,8 +48,7 @@ module _ is-set-type-Set = pr2 X Id-Prop : (x y : type-Set) → Prop l - pr1 (Id-Prop x y) = (x ＝ y) - pr2 (Id-Prop x y) = is-set-type-Set x y + Id-Prop x y = (x ＝ y , is-set-type-Set x y) ``` ## Properties @@ -166,3 +167,21 @@ module _ pr2 equiv-set-Set = is-set-equiv-is-set (is-set-type-Set A) (is-set-type-Set B) ``` + +### If a type injects into a set, then it is a set + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + abstract + is-set-is-injective : + {f : A → B} → is-set B → is-injective f → is-set A + is-set-is-injective {f} H I = + is-set-prop-in-id + ( λ x y → f x ＝ f y) + ( λ x y → H (f x) (f y)) + ( λ x → refl) + ( λ x y → I) +``` diff --git a/src/foundation-core/subtypes.lagda.md b/src/foundation-core/subtypes.lagda.md index a6a943e0c6..3aa30a0277 100644 --- a/src/foundation-core/subtypes.lagda.md +++ b/src/foundation-core/subtypes.lagda.md @@ -316,7 +316,11 @@ equiv-type-subtype : pr1 (equiv-type-subtype is-subtype-P is-subtype-Q f g) = tot f pr2 (equiv-type-subtype is-subtype-P is-subtype-Q f g) = is-equiv-tot-is-fiberwise-equiv {f = f} - ( λ x → is-equiv-is-prop (is-subtype-P x) (is-subtype-Q x) (g x)) + ( λ x → + is-equiv-has-converse-is-prop + ( is-subtype-P x) + ( is-subtype-Q x) + ( g x)) ``` ### Equivalences of subtypes @@ -342,7 +346,11 @@ abstract is-equiv f → ((x : A) → (Q (f x)) → P x) → is-equiv (map-Σ Q f g) is-equiv-subtype-is-equiv {Q = Q} is-subtype-P is-subtype-Q f g is-equiv-f h = is-equiv-map-Σ Q is-equiv-f - ( λ x → is-equiv-is-prop (is-subtype-P x) (is-subtype-Q (f x)) (h x)) + ( λ x → + is-equiv-has-converse-is-prop + ( is-subtype-P x) + ( is-subtype-Q (f x)) + ( h x)) abstract is-equiv-subtype-is-equiv' : @@ -356,6 +364,9 @@ abstract is-equiv-subtype-is-equiv' {P = P} {Q} is-subtype-P is-subtype-Q f g is-equiv-f h = is-equiv-map-Σ Q is-equiv-f - ( λ x → is-equiv-is-prop (is-subtype-P x) (is-subtype-Q (f x)) - ( (tr P (is-retraction-map-inv-is-equiv is-equiv-f x)) ∘ (h (f x)))) + ( λ x → + is-equiv-has-converse-is-prop + ( is-subtype-P x) + ( is-subtype-Q (f x)) + ( (tr P (is-retraction-map-inv-is-equiv is-equiv-f x)) ∘ (h (f x)))) ``` diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md index b4ebc64e2e..ecda4a6b37 100644 --- a/src/foundation.lagda.md +++ b/src/foundation.lagda.md @@ -54,6 +54,7 @@ open import foundation.category-of-sets public open import foundation.choice-of-representatives-equivalence-relation public open import foundation.codiagonal-maps-of-types public open import foundation.coherently-invertible-maps public +open import foundation.coinhabited-pairs-of-types public open import foundation.commuting-cubes-of-maps public open import foundation.commuting-hexagons-of-identifications public open import foundation.commuting-pentagons-of-identifications public @@ -164,6 +165,7 @@ open import foundation.equivalences-spans public open import foundation.equivalences-spans-families-of-types public open import foundation.evaluation-functions public open import foundation.exclusive-disjunction public +open import foundation.exclusive-sum public open import foundation.existential-quantification public open import foundation.exponents-set-quotients public open import foundation.extensions-types public @@ -203,6 +205,7 @@ open import foundation.homotopies public open import foundation.homotopies-morphisms-arrows public open import foundation.homotopy-algebra public open import foundation.homotopy-induction public +open import foundation.homotopy-preorder-of-types public open import foundation.identity-systems public open import foundation.identity-truncated-types public open import foundation.identity-types public @@ -247,6 +250,8 @@ open import foundation.maybe public open import foundation.mere-embeddings public open import foundation.mere-equality public open import foundation.mere-equivalences public +open import foundation.mere-functions public +open import foundation.mere-logical-equivalences public open import foundation.monomorphisms public open import foundation.morphisms-arrows public open import foundation.morphisms-binary-relations public @@ -409,8 +414,8 @@ open import foundation.type-arithmetic-unit-type public open import foundation.type-duality public open import foundation.type-theoretic-principle-of-choice public open import foundation.unions-subtypes public -open import foundation.unique-existence public open import foundation.uniqueness-image public +open import foundation.uniqueness-quantification public open import foundation.uniqueness-set-quotients public open import foundation.uniqueness-set-truncations public open import foundation.uniqueness-truncation public @@ -441,6 +446,7 @@ open import foundation.universal-property-set-quotients public open import foundation.universal-property-set-truncation public open import foundation.universal-property-truncation public open import foundation.universal-property-unit-type public +open import foundation.universal-quantification public open import foundation.universe-levels public open import foundation.unordered-pairs public open import foundation.unordered-pairs-of-types public diff --git a/src/foundation/apartness-relations.lagda.md b/src/foundation/apartness-relations.lagda.md index 973e7c3afe..baf80f0f6b 100644 --- a/src/foundation/apartness-relations.lagda.md +++ b/src/foundation/apartness-relations.lagda.md @@ -12,6 +12,7 @@ open import foundation.dependent-pair-types open import foundation.disjunction open import foundation.existential-quantification open import foundation.propositional-truncations +open import foundation.universal-quantification open import foundation.universe-levels open import foundation-core.cartesian-product-types @@ -53,9 +54,12 @@ module _ is-consistent : UU (l1 ⊔ l2) is-consistent = (a b : A) → (a ＝ b) → ¬ (type-Prop (R a b)) + is-cotransitive-Prop : Prop (l1 ⊔ l2) + is-cotransitive-Prop = + ∀' A (λ a → ∀' A (λ b → ∀' A (λ c → R a b ⇒ (R a c ∨ R b c)))) + is-cotransitive : UU (l1 ⊔ l2) - is-cotransitive = - (a b c : A) → type-hom-Prop (R a b) (disjunction-Prop (R a c) (R b c)) + is-cotransitive = type-Prop is-cotransitive-Prop is-apartness-relation : UU (l1 ⊔ l2) is-apartness-relation = @@ -150,7 +154,7 @@ module _ rel-apart-function-into-Type-With-Apartness : Relation-Prop (l1 ⊔ l3) (X → type-Type-With-Apartness Y) rel-apart-function-into-Type-With-Apartness f g = - ∃-Prop X (λ x → apart-Type-With-Apartness Y (f x) (g x)) + ∃ X (λ x → rel-apart-Type-With-Apartness Y (f x) (g x)) apart-function-into-Type-With-Apartness : Relation (l1 ⊔ l3) (X → type-Type-With-Apartness Y) @@ -202,16 +206,8 @@ module _ ( rel-apart-function-into-Type-With-Apartness X Y f h) ( rel-apart-function-into-Type-With-Apartness X Y g h)) ( λ where - ( inl b) → - inl-disjunction-Prop - ( rel-apart-function-into-Type-With-Apartness X Y f h) - ( rel-apart-function-into-Type-With-Apartness X Y g h) - ( unit-trunc-Prop (x , b)) - ( inr b) → - inr-disjunction-Prop - ( rel-apart-function-into-Type-With-Apartness X Y f h) - ( rel-apart-function-into-Type-With-Apartness X Y g h) - ( unit-trunc-Prop (x , b)))) + ( inl b) → inl-disjunction (intro-exists x b) + ( inr b) → inr-disjunction (intro-exists x b))) exp-Type-With-Apartness : Type-With-Apartness (l1 ⊔ l2) (l1 ⊔ l3) pr1 exp-Type-With-Apartness = X → type-Type-With-Apartness Y diff --git a/src/foundation/axiom-of-choice.lagda.md b/src/foundation/axiom-of-choice.lagda.md index 1d28e42414..a7bd0fea94 100644 --- a/src/foundation/axiom-of-choice.lagda.md +++ b/src/foundation/axiom-of-choice.lagda.md @@ -10,6 +10,7 @@ module foundation.axiom-of-choice where open import foundation.dependent-pair-types open import foundation.function-extensionality open import foundation.functoriality-propositional-truncation +open import foundation.inhabited-types open import foundation.postcomposition-functions open import foundation.projective-types open import foundation.propositional-truncations @@ -17,7 +18,6 @@ open import foundation.sections open import foundation.split-surjective-maps open import foundation.surjective-maps open import foundation.universe-levels -open import foundation.whiskering-homotopies-composition open import foundation-core.equivalences open import foundation-core.fibers-of-maps @@ -32,8 +32,10 @@ open import foundation-core.sets ## Idea -The axiom of choice asserts that for every family of inhabited types indexed by -a set, the type of sections of that family is inhabited. +The {{#concept "axiom of choice" Agda=AC-0}} asserts that for every family of +[inhabited](foundation.inhabited-types.md) types `B` indexed by a +[set](foundation-core.sets.md) `A`, the type of sections of that family +`(x : A) → B x` is inhabited. ## Definition @@ -43,8 +45,8 @@ a set, the type of sections of that family is inhabited. AC-Set : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) AC-Set l1 l2 = (A : Set l1) (B : type-Set A → Set l2) → - ((x : type-Set A) → type-trunc-Prop (type-Set (B x))) → - type-trunc-Prop ((x : type-Set A) → type-Set (B x)) + ((x : type-Set A) → is-inhabited (type-Set (B x))) → + is-inhabited ((x : type-Set A) → type-Set (B x)) ``` ### The axiom of choice @@ -53,8 +55,8 @@ AC-Set l1 l2 = AC-0 : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) AC-0 l1 l2 = (A : Set l1) (B : type-Set A → UU l2) → - ((x : type-Set A) → type-trunc-Prop (B x)) → - type-trunc-Prop ((x : type-Set A) → B x) + ((x : type-Set A) → is-inhabited (B x)) → + is-inhabited ((x : type-Set A) → B x) ``` ## Properties diff --git a/src/foundation/booleans.lagda.md b/src/foundation/booleans.lagda.md index 844bfeeb29..704c90a92d 100644 --- a/src/foundation/booleans.lagda.md +++ b/src/foundation/booleans.lagda.md @@ -8,6 +8,7 @@ module foundation.booleans where ```agda open import foundation.dependent-pair-types +open import foundation.involutions open import foundation.negated-equality open import foundation.raising-universe-levels open import foundation.unit-type @@ -23,6 +24,7 @@ open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.negation open import foundation-core.propositions +open import foundation-core.sections open import foundation-core.sets open import univalent-combinatorics.finite-types @@ -237,9 +239,9 @@ neq-neg-bool false () ### Boolean negation is an involution ```agda -neg-neg-bool : (neg-bool ∘ neg-bool) ~ id -neg-neg-bool true = refl -neg-neg-bool false = refl +is-involution-neg-bool : is-involution neg-bool +is-involution-neg-bool true = refl +is-involution-neg-bool false = refl ``` ### Boolean negation is an equivalence @@ -247,10 +249,7 @@ neg-neg-bool false = refl ```agda abstract is-equiv-neg-bool : is-equiv neg-bool - pr1 (pr1 is-equiv-neg-bool) = neg-bool - pr2 (pr1 is-equiv-neg-bool) = neg-neg-bool - pr1 (pr2 is-equiv-neg-bool) = neg-bool - pr2 (pr2 is-equiv-neg-bool) = neg-neg-bool + is-equiv-neg-bool = is-equiv-is-involution is-involution-neg-bool equiv-neg-bool : bool ≃ bool pr1 equiv-neg-bool = neg-bool @@ -261,8 +260,12 @@ pr2 equiv-neg-bool = is-equiv-neg-bool ```agda abstract - not-equiv-const : + no-section-const-bool : (b : bool) → ¬ (section (const bool bool b)) + no-section-const-bool true (g , G) = neq-true-false-bool (G false) + no-section-const-bool false (g , G) = neq-false-true-bool (G true) + +abstract + is-not-equiv-const-bool : (b : bool) → ¬ (is-equiv (const bool bool b)) - not-equiv-const true ((g , G) , _) = neq-true-false-bool (G false) - not-equiv-const false ((g , G) , _) = neq-false-true-bool (G true) + is-not-equiv-const-bool b e = no-section-const-bool b (section-is-equiv e) ``` diff --git a/src/foundation/coherently-invertible-maps.lagda.md b/src/foundation/coherently-invertible-maps.lagda.md index 4a590a17f1..e2e34ab431 100644 --- a/src/foundation/coherently-invertible-maps.lagda.md +++ b/src/foundation/coherently-invertible-maps.lagda.md @@ -13,6 +13,7 @@ open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.type-arithmetic-dependent-pair-types open import foundation.universe-levels open import foundation.whiskering-higher-homotopies-composition @@ -111,7 +112,7 @@ module _ is-equiv-is-coherently-invertible-is-equiv : is-equiv (is-coherently-invertible-is-equiv {f = f}) is-equiv-is-coherently-invertible-is-equiv = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-property-is-equiv f) ( is-prop-is-coherently-invertible) ( is-equiv-is-coherently-invertible) diff --git a/src/foundation/coinhabited-pairs-of-types.lagda.md b/src/foundation/coinhabited-pairs-of-types.lagda.md new file mode 100644 index 0000000000..bb9be37fb9 --- /dev/null +++ b/src/foundation/coinhabited-pairs-of-types.lagda.md @@ -0,0 +1,104 @@ +# Coinhabited pairs of types + +```agda +module foundation.coinhabited-pairs-of-types where +``` + +
Imports + +```agda +open import foundation.inhabited-types +open import foundation.logical-equivalences +open import foundation.propositional-truncations +open import foundation.universe-levels + +open import foundation-core.propositions +``` + +
+ +## Idea + +Two types `A` and `B` are said to be +{{#concept "coinhabited" Agda=is-coinhabited}} if `A` is +[inhabited](foundation.inhabited-types.md) +[if and only if](foundation.logical-equivalences.md) `B` is. + +## Definitions + +### The predicate of being coinhabited + +```agda +module _ + {l1 l2 : Level} (A : UU l1) (B : UU l2) + where + + is-coinhabited-Prop : Prop (l1 ⊔ l2) + is-coinhabited-Prop = iff-Prop (is-inhabited-Prop A) (is-inhabited-Prop B) + + is-coinhabited : UU (l1 ⊔ l2) + is-coinhabited = type-Prop is-coinhabited-Prop +``` + +### Forward and backward implications of coinhabited types + +```agda +module _ + {l1 l2 : Level} (A : UU l1) (B : UU l2) + where + + forward-implication-is-coinhabited : + is-coinhabited A B → is-inhabited A → is-inhabited B + forward-implication-is-coinhabited = forward-implication + + forward-implication-is-coinhabited' : is-coinhabited A B → A → is-inhabited B + forward-implication-is-coinhabited' e a = + forward-implication e (unit-trunc-Prop a) + + backward-implication-is-coinhabited : + is-coinhabited A B → is-inhabited B → is-inhabited A + backward-implication-is-coinhabited = backward-implication + + backward-implication-is-coinhabited' : is-coinhabited A B → B → is-inhabited A + backward-implication-is-coinhabited' e b = + backward-implication e (unit-trunc-Prop b) +``` + +### Every type is coinhabited with itself + +```agda +module _ + {l : Level} (A : UU l) + where + + is-reflexive-is-coinhabited : is-coinhabited A A + is-reflexive-is-coinhabited = id-iff +``` + +### Coinhabitedness is a transitive relation + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} + where + + is-transitive-is-coinhabited : + is-coinhabited B C → is-coinhabited A B → is-coinhabited A C + is-transitive-is-coinhabited = _∘iff_ +``` + +### Coinhabitedness is a symmetric relation + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-symmetric-is-coinhabited : is-coinhabited A B → is-coinhabited B A + is-symmetric-is-coinhabited = inv-iff +``` + +## See also + +- [Mere logical equivalence of types](foundation.mere-logical-equivalences.md) + is a related but stronger notion than coinhabitedness. diff --git a/src/foundation/conjunction.lagda.md b/src/foundation/conjunction.lagda.md index 23f062e227..54c4cc00c9 100644 --- a/src/foundation/conjunction.lagda.md +++ b/src/foundation/conjunction.lagda.md @@ -1,4 +1,4 @@ -# Conjunction of propositions +# Conjunction ```agda module foundation.conjunction where @@ -10,11 +10,14 @@ module foundation.conjunction where open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.logical-equivalences +open import foundation.propositional-truncations +open import foundation.universal-property-cartesian-product-types open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.decidable-propositions open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.propositions ``` @@ -22,77 +25,204 @@ open import foundation-core.propositions ## Idea -The **conjunction** of two [propositions](foundation-core.propositions.md) `P` -and `Q` is the proposition that both `P` and `Q` hold. +The +{{#concept "conjunction" Disambiguation="of propositions" WDID=Q191081 WD="logical conjunction" Agda=conjunction-Prop}} +`P ∧ Q` of two [propositions](foundation-core.propositions.md) `P` and `Q` is +the proposition that both `P` and `Q` hold and is thus defined by the +[cartesian product](foundation-core.cartesian-product-types.md) of their +underlying types -## Definition +```text + P ∧ Q := P × Q +``` + +The conjunction of two propositions satisfies the universal property of the +[meet](order-theory.greatest-lower-bounds-large-posets.md) in the +[locale of propositions](foundation.large-locale-of-propositions.md). This means +that any span of propositions over `P` and `Q` factor (uniquely) through the +conjunction + +```text + R + / ∶ \ + / ∶ \ + ∨ ∨ ∨ + P <--- P ∧ Q ---> Q. +``` + +In other words, we have a +[logical equivalence](foundation.logical-equivalences.md) + +```text + (R → P) ∧ (R → Q) ↔ (R → P ∧ Q) +``` + +for every proposition `R`. In fact, `R` can be any type. + +The +{{#concept "recursion principle" Disambiguation"of the conjunction of propositions" Agda=elimination-principle-conjunction-Prop}} +of the conjunction of propositions states that to define a function `P ∧ Q → R` +into a proposition `R`, or indeed any type, is equivalent to defining a function +`P → Q → R`. + +## Definitions + +### The conjunction ```agda -conjunction-Prop = product-Prop +module _ + {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) + where + + conjunction-Prop : Prop (l1 ⊔ l2) + conjunction-Prop = product-Prop P Q -type-conjunction-Prop : {l1 l2 : Level} → Prop l1 → Prop l2 → UU (l1 ⊔ l2) -type-conjunction-Prop P Q = type-Prop (conjunction-Prop P Q) + type-conjunction-Prop : UU (l1 ⊔ l2) + type-conjunction-Prop = type-Prop conjunction-Prop -abstract - is-prop-type-conjunction-Prop : - {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) → - is-prop (type-conjunction-Prop P Q) - is-prop-type-conjunction-Prop P Q = is-prop-type-Prop (conjunction-Prop P Q) + is-prop-conjunction-Prop : + is-prop type-conjunction-Prop + is-prop-conjunction-Prop = is-prop-type-Prop conjunction-Prop -infixr 15 _∧_ -_∧_ = type-conjunction-Prop + infixr 15 _∧_ + _∧_ : Prop (l1 ⊔ l2) + _∧_ = conjunction-Prop ``` **Note**: The symbol used for the conjunction `_∧_` is the -[logical and](https://codepoints.net/U+2227) `∧` (agda-input: `\wedge` `\and`). +[logical and](https://codepoints.net/U+2227) `∧` (agda-input: `\wedge` or +`\and`). + +### The conjunction of decidable propositions ```agda -conjunction-Decidable-Prop : - {l1 l2 : Level} → Decidable-Prop l1 → Decidable-Prop l2 → - Decidable-Prop (l1 ⊔ l2) -pr1 (conjunction-Decidable-Prop P Q) = - type-conjunction-Prop (prop-Decidable-Prop P) (prop-Decidable-Prop Q) -pr1 (pr2 (conjunction-Decidable-Prop P Q)) = - is-prop-type-conjunction-Prop (prop-Decidable-Prop P) (prop-Decidable-Prop Q) -pr2 (pr2 (conjunction-Decidable-Prop P Q)) = - is-decidable-product - ( is-decidable-Decidable-Prop P) - ( is-decidable-Decidable-Prop Q) +module _ + {l1 l2 : Level} (P : Decidable-Prop l1) (Q : Decidable-Prop l2) + where + + is-decidable-conjunction-Decidable-Prop : + is-decidable + ( type-conjunction-Prop (prop-Decidable-Prop P) (prop-Decidable-Prop Q)) + is-decidable-conjunction-Decidable-Prop = + is-decidable-product + ( is-decidable-Decidable-Prop P) + ( is-decidable-Decidable-Prop Q) + + conjunction-Decidable-Prop : Decidable-Prop (l1 ⊔ l2) + conjunction-Decidable-Prop = + ( type-conjunction-Prop (prop-Decidable-Prop P) (prop-Decidable-Prop Q) , + is-prop-conjunction-Prop (prop-Decidable-Prop P) (prop-Decidable-Prop Q) , + is-decidable-conjunction-Decidable-Prop) ``` -## Properties +### The introduction rule and projections for the conjunction of propositions -### Introduction rule for conjunction +While we define the introduction rule and projections for the conjunction below, +we advice users to use the standard pairing and projection functions for the +cartesian product types: `pair`, `pr1` and `pr2`. ```agda -intro-conjunction-Prop : - {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) → - type-Prop P → type-Prop Q → type-conjunction-Prop P Q -pr1 (intro-conjunction-Prop P Q p q) = p -pr2 (intro-conjunction-Prop P Q p q) = q +module _ + {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) + where + + intro-conjunction-Prop : type-Prop P → type-Prop Q → type-conjunction-Prop P Q + intro-conjunction-Prop = pair + + pr1-conjunction-Prop : type-conjunction-Prop P Q → type-Prop P + pr1-conjunction-Prop = pr1 + + pr2-conjunction-Prop : type-conjunction-Prop P Q → type-Prop Q + pr2-conjunction-Prop = pr2 ``` -### The universal property of conjunction +### The elimination principle of the conjunction ```agda -iff-universal-property-conjunction-Prop : +module _ {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) - {l3 : Level} (R : Prop l3) → - ( type-hom-Prop R P × type-hom-Prop R Q) ↔ - ( type-hom-Prop R (conjunction-Prop P Q)) -pr1 (pr1 (iff-universal-property-conjunction-Prop P Q R) (f , g) r) = f r -pr2 (pr1 (iff-universal-property-conjunction-Prop P Q R) (f , g) r) = g r -pr1 (pr2 (iff-universal-property-conjunction-Prop P Q R) h) r = pr1 (h r) -pr2 (pr2 (iff-universal-property-conjunction-Prop P Q R) h) r = pr2 (h r) - -equiv-universal-property-conjunction-Prop : + where + + ev-conjunction-Prop : + {l : Level} (R : Prop l) → type-Prop (((P ∧ Q) ⇒ R) ⇒ P ⇒ Q ⇒ R) + ev-conjunction-Prop R = ev-pair + + elimination-principle-conjunction-Prop : UUω + elimination-principle-conjunction-Prop = + {l : Level} (R : Prop l) → has-converse (ev-conjunction-Prop R) +``` + +## Properties + +### The conjunction satisfies the recursion principle of the conjunction + +```agda +module _ {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) - {l3 : Level} (R : Prop l3) → - ( type-hom-Prop R P × type-hom-Prop R Q) ≃ - ( type-hom-Prop R (conjunction-Prop P Q)) -equiv-universal-property-conjunction-Prop P Q R = - equiv-iff' - ( conjunction-Prop (hom-Prop R P) (hom-Prop R Q)) - ( hom-Prop R (conjunction-Prop P Q)) - ( iff-universal-property-conjunction-Prop P Q R) + where + + elim-conjunction-Prop : elimination-principle-conjunction-Prop P Q + elim-conjunction-Prop R f (p , q) = f p q ``` + +### The conjunction is the meet in the locale of propositions + +```agda +module _ + {l1 l2 l3 : Level} (P : Prop l1) (Q : Prop l2) (C : UU l3) + where + + map-distributive-conjunction-Prop : + type-conjunction-Prop (function-Prop C P) (function-Prop C Q) → + C → type-conjunction-Prop P Q + map-distributive-conjunction-Prop (f , g) y = (f y , g y) + + map-inv-distributive-conjunction-Prop : + (C → type-conjunction-Prop P Q) → + type-conjunction-Prop (function-Prop C P) (function-Prop C Q) + map-inv-distributive-conjunction-Prop f = (pr1 ∘ f , pr2 ∘ f) + + is-equiv-map-distributive-conjunction-Prop : + is-equiv map-distributive-conjunction-Prop + is-equiv-map-distributive-conjunction-Prop = + is-equiv-has-converse + ( conjunction-Prop (function-Prop C P) (function-Prop C Q)) + ( function-Prop C (conjunction-Prop P Q)) + ( map-inv-distributive-conjunction-Prop) + + distributive-conjunction-Prop : + type-conjunction-Prop (function-Prop C P) (function-Prop C Q) ≃ + (C → type-conjunction-Prop P Q) + distributive-conjunction-Prop = + ( map-distributive-conjunction-Prop , + is-equiv-map-distributive-conjunction-Prop) + +module _ + {l1 l2 l3 : Level} (P : Prop l1) (Q : Prop l2) (R : Prop l3) + where + + is-greatest-binary-lower-bound-conjunction-Prop : + type-Prop (((R ⇒ P) ∧ (R ⇒ Q)) ⇔ (R ⇒ P ∧ Q)) + is-greatest-binary-lower-bound-conjunction-Prop = + ( map-distributive-conjunction-Prop P Q (type-Prop R) , + map-inv-distributive-conjunction-Prop P Q (type-Prop R)) +``` + +## See also + +- The indexed counterpart to conjunction is + [universal quantification](foundation.universal-quantification.md). + +## Table of files about propositional logic + +The following table gives an overview of basic constructions in propositional +logic and related considerations. + +{{#include tables/propositional-logic.md}} + +## External links + +- [logical conjunction](https://ncatlab.org/nlab/show/logical+conjunction) at + $n$Lab +- [Logical conjunction](https://en.wikipedia.org/wiki/Logical_conjunction) at + Wikipedia diff --git a/src/foundation/contractible-types.lagda.md b/src/foundation/contractible-types.lagda.md index e489774da7..5d78e164f1 100644 --- a/src/foundation/contractible-types.lagda.md +++ b/src/foundation/contractible-types.lagda.md @@ -12,6 +12,7 @@ open import foundation-core.contractible-types public open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.logical-equivalences open import foundation.subuniverses open import foundation.unit-type open import foundation.universe-levels @@ -123,7 +124,7 @@ is-closed-under-is-contr-subuniverse P = equiv-is-contr-equiv : {l1 l2 : Level} {A : UU l1} {B : UU l2} → A ≃ B → is-contr A ≃ is-contr B equiv-is-contr-equiv {A = A} {B = B} e = - equiv-prop + equiv-iff-is-prop ( is-property-is-contr) ( is-property-is-contr) ( is-contr-retract-of A diff --git a/src/foundation/coproduct-types.lagda.md b/src/foundation/coproduct-types.lagda.md index a64a9b8d9b..14b74377c5 100644 --- a/src/foundation/coproduct-types.lagda.md +++ b/src/foundation/coproduct-types.lagda.md @@ -214,7 +214,8 @@ module _ coproduct-Prop : {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) → (type-Prop P → ¬ (type-Prop Q)) → Prop (l1 ⊔ l2) -pr1 (coproduct-Prop P Q H) = type-Prop P + type-Prop Q +pr1 (coproduct-Prop P Q H) = + type-Prop P + type-Prop Q pr2 (coproduct-Prop P Q H) = is-prop-coproduct H (is-prop-type-Prop P) (is-prop-type-Prop Q) ``` diff --git a/src/foundation/decidable-embeddings.lagda.md b/src/foundation/decidable-embeddings.lagda.md index 4bc7896cbb..d18d5f6df8 100644 --- a/src/foundation/decidable-embeddings.lagda.md +++ b/src/foundation/decidable-embeddings.lagda.md @@ -17,6 +17,7 @@ open import foundation.functoriality-cartesian-product-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopy-induction open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.propositional-maps open import foundation.structured-type-duality open import foundation.subtype-identity-principle @@ -219,7 +220,7 @@ abstract {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) → is-prop (is-decidable-emb f) is-prop-is-decidable-emb f = - is-prop-is-inhabited + is-prop-has-element ( λ H → is-prop-product ( is-property-is-emb f) @@ -322,7 +323,7 @@ equiv-precomp-decidable-emb-equiv e C = ( is-decidable-emb) ( equiv-precomp e C) ( λ g → - equiv-prop + equiv-iff-is-prop ( is-prop-is-decidable-emb g) ( is-prop-is-decidable-emb (g ∘ map-equiv e)) ( is-decidable-emb-comp (is-decidable-emb-is-equiv (pr2 e))) diff --git a/src/foundation/decidable-equality.lagda.md b/src/foundation/decidable-equality.lagda.md index 5dbe76a61d..39b6f711fc 100644 --- a/src/foundation/decidable-equality.lagda.md +++ b/src/foundation/decidable-equality.lagda.md @@ -12,9 +12,11 @@ open import foundation.coproduct-types open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.double-negation +open import foundation.injective-maps open import foundation.negation open import foundation.retracts-of-types open import foundation.sections +open import foundation.sets open import foundation.type-arithmetic-dependent-pair-types open import foundation.unit-type open import foundation.universe-levels @@ -25,9 +27,7 @@ open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps open import foundation-core.identity-types -open import foundation-core.injective-maps open import foundation-core.propositions -open import foundation-core.sets open import foundation-core.transport-along-identifications ``` @@ -223,7 +223,7 @@ abstract is-prop-has-decidable-equality : {l1 : Level} {X : UU l1} → is-prop (has-decidable-equality X) is-prop-has-decidable-equality {l1} {X} = - is-prop-is-inhabited + is-prop-has-element ( λ d → is-prop-Π ( λ x → diff --git a/src/foundation/decidable-equivalence-relations.lagda.md b/src/foundation/decidable-equivalence-relations.lagda.md index 9addc2138d..948b84936d 100644 --- a/src/foundation/decidable-equivalence-relations.lagda.md +++ b/src/foundation/decidable-equivalence-relations.lagda.md @@ -128,7 +128,7 @@ module _ sim-Decidable-equivalence-relation x y ≃ sim-Decidable-equivalence-relation y x equiv-symmetric-Decidable-equivalence-relation {x} {y} = - equiv-prop + equiv-iff-is-prop ( is-prop-sim-Decidable-equivalence-relation x y) ( is-prop-sim-Decidable-equivalence-relation y x) ( symmetric-Decidable-equivalence-relation x y) @@ -169,7 +169,9 @@ module _ is-equivalence-class-Decidable-equivalence-relation : decidable-subtype l2 X → UU (l1 ⊔ lsuc l2) is-equivalence-class-Decidable-equivalence-relation P = - ∃ X (λ x → P ＝ decidable-relation-Decidable-equivalence-relation R x) + exists-structure + ( X) + ( λ x → P ＝ decidable-relation-Decidable-equivalence-relation R x) equivalence-class-Decidable-equivalence-relation : UU (l1 ⊔ lsuc l2) equivalence-class-Decidable-equivalence-relation = @@ -180,7 +182,7 @@ module _ pr1 (class-Decidable-equivalence-relation x) = decidable-relation-Decidable-equivalence-relation R x pr2 (class-Decidable-equivalence-relation x) = - intro-∃ x refl + intro-exists x refl emb-equivalence-class-Decidable-equivalence-relation : equivalence-class-Decidable-equivalence-relation ↪ decidable-subtype l2 X diff --git a/src/foundation/decidable-propositions.lagda.md b/src/foundation/decidable-propositions.lagda.md index 02de050cce..877d8c5626 100644 --- a/src/foundation/decidable-propositions.lagda.md +++ b/src/foundation/decidable-propositions.lagda.md @@ -142,8 +142,8 @@ equiv-universes-Decidable-Prop l l' = iff-universes-Decidable-Prop : (l l' : Level) (P : Decidable-Prop l) → - ( prop-Decidable-Prop P ⇔ - prop-Decidable-Prop (map-equiv (equiv-universes-Decidable-Prop l l') P)) + ( type-Decidable-Prop P) ↔ + ( type-Decidable-Prop (map-equiv (equiv-universes-Decidable-Prop l l') P)) pr1 (iff-universes-Decidable-Prop l l' P) p = map-inv-equiv ( compute-equiv-bool-Decidable-Prop diff --git a/src/foundation/decidable-subtypes.lagda.md b/src/foundation/decidable-subtypes.lagda.md index 20472c465b..b8c8bce746 100644 --- a/src/foundation/decidable-subtypes.lagda.md +++ b/src/foundation/decidable-subtypes.lagda.md @@ -157,12 +157,12 @@ module _ iff-universes-decidable-subtype : (l l' : Level) (S : decidable-subtype l X) → ( (x : X) → - prop-Decidable-Prop (S x) ⇔ - prop-Decidable-Prop + type-Decidable-Prop (S x) ↔ + type-Decidable-Prop ( map-equiv (equiv-universes-decidable-subtype l l') S x)) iff-universes-decidable-subtype l l' S x = tr - ( λ P → prop-Decidable-Prop (S x) ⇔ prop-Decidable-Prop P) + ( λ P → type-Decidable-Prop (S x) ↔ type-Decidable-Prop P) ( inv ( compute-map-equiv-Π ( λ _ → Decidable-Prop l') @@ -246,7 +246,7 @@ module _ (P ＝ Q) ≃ has-same-elements-decidable-subtype Q extensionality-decidable-subtype = extensionality-Π P - ( λ x Q → prop-Decidable-Prop (P x) ⇔ prop-Decidable-Prop Q) + ( λ x Q → (type-Decidable-Prop (P x)) ↔ (type-Decidable-Prop Q)) ( λ x Q → extensionality-Decidable-Prop (P x) Q) has-same-elements-eq-decidable-subtype : diff --git a/src/foundation/decidable-types.lagda.md b/src/foundation/decidable-types.lagda.md index 4ffe535d77..8073a1ae80 100644 --- a/src/foundation/decidable-types.lagda.md +++ b/src/foundation/decidable-types.lagda.md @@ -61,13 +61,15 @@ is-inhabited-or-empty A = type-trunc-Prop A + is-empty A ### Merely decidable types -A type `A` is said to be merely decidable if it comes equipped with an element -of `type-trunc-Prop (is-decidable A)`. +A type `A` is said to be +{{#concept "merely decidable" Agda=is-merely-decidable}} if it comes equipped +with an element of `║ is-decidable A ║₋₁`, or equivalently, the +[disjunction](foundation.disjunction.md) `A ∨ ¬ A` holds. ```agda -is-merely-Decidable-Prop : +is-merely-decidable-Prop : {l : Level} → UU l → Prop l -is-merely-Decidable-Prop A = trunc-Prop (is-decidable A) +is-merely-decidable-Prop A = trunc-Prop (is-decidable A) is-merely-decidable : {l : Level} → UU l → UU l is-merely-decidable A = type-trunc-Prop (is-decidable A) @@ -208,8 +210,7 @@ double-negation-elim-is-decidable (inr x) p = ex-falso (p x) ```agda double-negation-is-decidable : {l : Level} {P : UU l} → ¬¬ (is-decidable P) double-negation-is-decidable {P = P} f = - map-neg (inr {A = P} {B = ¬ P}) f - ( map-neg (inl {A = P} {B = ¬ P}) f) + map-neg (inr {A = P} {B = ¬ P}) f (map-neg (inl {A = P} {B = ¬ P}) f) ``` ### Decidable types have ε-operators @@ -236,9 +237,9 @@ idempotent-is-decidable P (inr np) = inr (λ p → np (inl p)) ```agda abstract - is-prop-is-inhabited-or-empty : + is-property-is-inhabited-or-empty : {l1 : Level} (A : UU l1) → is-prop (is-inhabited-or-empty A) - is-prop-is-inhabited-or-empty A = + is-property-is-inhabited-or-empty A = is-prop-coproduct ( λ t → apply-universal-property-trunc-Prop t empty-Prop) ( is-prop-type-trunc-Prop) @@ -246,7 +247,7 @@ abstract is-inhabited-or-empty-Prop : {l1 : Level} → UU l1 → Prop l1 pr1 (is-inhabited-or-empty-Prop A) = is-inhabited-or-empty A -pr2 (is-inhabited-or-empty-Prop A) = is-prop-is-inhabited-or-empty A +pr2 (is-inhabited-or-empty-Prop A) = is-property-is-inhabited-or-empty A ``` ### Any inhabited type is a fixed point for `is-decidable` diff --git a/src/foundation/discrete-types.lagda.md b/src/foundation/discrete-types.lagda.md index ae29cad8b8..c9c1243aa0 100644 --- a/src/foundation/discrete-types.lagda.md +++ b/src/foundation/discrete-types.lagda.md @@ -39,7 +39,7 @@ module _ where rel-apart-Discrete-Type : Relation-Prop l (type-Discrete-Type X) - rel-apart-Discrete-Type x y = neg-Prop' (x ＝ y) + rel-apart-Discrete-Type x y = neg-type-Prop (x ＝ y) apart-Discrete-Type : (x y : type-Discrete-Type X) → UU l apart-Discrete-Type x y = type-Prop (rel-apart-Discrete-Type x y) diff --git a/src/foundation/disjunction.lagda.md b/src/foundation/disjunction.lagda.md index 8cc5a61022..21a764be49 100644 --- a/src/foundation/disjunction.lagda.md +++ b/src/foundation/disjunction.lagda.md @@ -1,4 +1,4 @@ -# Disjunction of propositions +# Disjunction ```agda module foundation.disjunction where @@ -7,12 +7,15 @@ module foundation.disjunction where
Imports ```agda -open import foundation.conjunction open import foundation.decidable-types open import foundation.dependent-pair-types +open import foundation.functoriality-coproduct-types +open import foundation.inhabited-types +open import foundation.logical-equivalences open import foundation.propositional-truncations open import foundation.universe-levels +open import foundation-core.cartesian-product-types open import foundation-core.coproduct-types open import foundation-core.decidable-propositions open import foundation-core.empty-types @@ -25,96 +28,214 @@ open import foundation-core.propositions ## Idea -The disjunction of two propositions `P` and `Q` is the proposition that `P` -holds or `Q` holds. +The +{{#concept "disjunction" Disambiguation="of propositions" WDID=Q1651704 WD="logical disjunction" Agda=disjunction-Prop}} +of two [propositions](foundation-core.propositions.md) `P` and `Q` is the +proposition that `P` holds or `Q` holds, and is defined as +[propositional truncation](foundation.propositional-truncations.md) of the +[coproduct](foundation-core.coproduct-types.md) of their underlying types -## Definition +```text + P ∨ Q := ║ P + Q ║₋₁ +``` + +The +{{#concept "universal property" Disambiguation="of the disjunction" Agda=universal-property-disjunction-Prop}} +of the disjunction states that, for every proposition `R`, the evaluation map + +```text + ev : ((P ∨ Q) → R) → ((P → R) ∧ (Q → R)) +``` + +is a [logical equivalence](foundation.logical-equivalences.md), and thus the +disjunction is the least upper bound of `P` and `Q` in the +[locale of propositions](foundation.large-locale-of-propositions.md): there is a +pair of logical implications `P → R` and `Q → R`, if and only if `P ∨ Q` implies +`R` + +```text +P ---> P ∨ Q <--- Q + \ ∶ / + \ ∶ / + ∨ ∨ ∨ + R. +``` + +## Definitions + +### The disjunction of arbitrary types + +Given arbitrary types `A` and `B`, the truncation + +```text + ║ A + B ║₋₁ +``` + +satisfies the universal property of + +```text + ║ A ║₋₁ ∨ ║ B ║₋₁ +``` + +and is thus equivalent to it. Therefore, we may reasonably call this +construction the +{{#concept "disjunction" Disambiguation="of types" Agda=disjunction-type-Prop}} +of types. It is important to keep in mind that this is not a generalization of +the concept but rather a conflation, and should be read as the statement _`A` or +`B` is (merely) [inhabited](foundation.inhabited-types.md)_. + +Because propositions are a special case of types, and Agda can generally infer +types for us, we will continue to conflate the two in our formalizations for the +benefit that we have to specify the propositions in our code less often. For +instance, even though the introduction rules for disjunction are phrased in +terms of disjunction of types, they equally apply to disjunction of +propositions. ```agda -disjunction-Prop : {l1 l2 : Level} → Prop l1 → Prop l2 → Prop (l1 ⊔ l2) -disjunction-Prop P Q = trunc-Prop (type-Prop P + type-Prop Q) +module _ + {l1 l2 : Level} (A : UU l1) (B : UU l2) + where -type-disjunction-Prop : {l1 l2 : Level} → Prop l1 → Prop l2 → UU (l1 ⊔ l2) -type-disjunction-Prop P Q = type-Prop (disjunction-Prop P Q) + disjunction-type-Prop : Prop (l1 ⊔ l2) + disjunction-type-Prop = trunc-Prop (A + B) -abstract - is-prop-type-disjunction-Prop : - {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) → - is-prop (type-disjunction-Prop P Q) - is-prop-type-disjunction-Prop P Q = is-prop-type-Prop (disjunction-Prop P Q) + disjunction-type : UU (l1 ⊔ l2) + disjunction-type = type-Prop disjunction-type-Prop -infixr 10 _∨_ -_∨_ = type-disjunction-Prop + is-prop-disjunction-type : is-prop disjunction-type + is-prop-disjunction-type = is-prop-type-Prop disjunction-type-Prop ``` -**Note**: The symbol used for the disjunction `_∨_` is the +### The disjunction + +```agda +module _ + {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) + where + + disjunction-Prop : Prop (l1 ⊔ l2) + disjunction-Prop = disjunction-type-Prop (type-Prop P) (type-Prop Q) + + type-disjunction-Prop : UU (l1 ⊔ l2) + type-disjunction-Prop = type-Prop disjunction-Prop + + abstract + is-prop-disjunction-Prop : is-prop type-disjunction-Prop + is-prop-disjunction-Prop = is-prop-type-Prop disjunction-Prop + + infixr 10 _∨_ + _∨_ : Prop (l1 ⊔ l2) + _∨_ = disjunction-Prop +``` + +**Notation.** The symbol used for the disjunction `_∨_` is the [logical or](https://codepoints.net/U+2228) `∨` (agda-input: `\vee` `\or`), and not the [latin small letter v](https://codepoints.net/U+0076) `v`. +### The introduction rules for the disjunction + ```agda -disjunction-Decidable-Prop : - {l1 l2 : Level} → - Decidable-Prop l1 → Decidable-Prop l2 → Decidable-Prop (l1 ⊔ l2) -pr1 (disjunction-Decidable-Prop P Q) = - type-disjunction-Prop (prop-Decidable-Prop P) (prop-Decidable-Prop Q) -pr1 (pr2 (disjunction-Decidable-Prop P Q)) = - is-prop-type-disjunction-Prop (prop-Decidable-Prop P) (prop-Decidable-Prop Q) -pr2 (pr2 (disjunction-Decidable-Prop P Q)) = - is-decidable-trunc-Prop-is-merely-decidable - ( type-Decidable-Prop P + type-Decidable-Prop Q) - ( unit-trunc-Prop - ( is-decidable-coproduct - ( is-decidable-Decidable-Prop P) - ( is-decidable-Decidable-Prop Q))) +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + inl-disjunction : A → disjunction-type A B + inl-disjunction = unit-trunc-Prop ∘ inl + + inr-disjunction : B → disjunction-type A B + inr-disjunction = unit-trunc-Prop ∘ inr +``` + +**Note.** Even though the introduction rules are formalized in terms of +disjunction of types, it equally applies to disjunction of propositions. This is +because propositions are a special case of types. The benefit of this approach +is that Agda can infer types for us, but not generally propositions. + +### The universal property of disjunctions + +```agda +ev-disjunction : + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} → + (disjunction-type A B → C) → (A → C) × (B → C) +pr1 (ev-disjunction h) = h ∘ inl-disjunction +pr2 (ev-disjunction h) = h ∘ inr-disjunction + +universal-property-disjunction-type : + {l1 l2 l3 : Level} → UU l1 → UU l2 → Prop l3 → UUω +universal-property-disjunction-type A B S = + {l : Level} (R : Prop l) → + (type-Prop S → type-Prop R) ↔ ((A → type-Prop R) × (B → type-Prop R)) + +universal-property-disjunction-Prop : + {l1 l2 l3 : Level} → Prop l1 → Prop l2 → Prop l3 → UUω +universal-property-disjunction-Prop P Q = + universal-property-disjunction-type (type-Prop P) (type-Prop Q) ``` ## Properties -### The introduction rules for disjunction +### The disjunction satisfies the universal property of disjunctions ```agda -inl-disjunction-Prop : - {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) → - type-hom-Prop P (disjunction-Prop P Q) -inl-disjunction-Prop P Q = unit-trunc-Prop ∘ inl +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + elim-disjunction' : + {l : Level} (R : Prop l) → + (A → type-Prop R) × (B → type-Prop R) → + disjunction-type A B → type-Prop R + elim-disjunction' R (f , g) = + map-universal-property-trunc-Prop R (rec-coproduct f g) -inr-disjunction-Prop : - {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) → - type-hom-Prop Q (disjunction-Prop P Q) -inr-disjunction-Prop P Q = unit-trunc-Prop ∘ inr + up-disjunction : + universal-property-disjunction-type A B (disjunction-type-Prop A B) + up-disjunction R = ev-disjunction , elim-disjunction' R ``` -### The elimination rule and universal property of disjunction +### The elimination principle of disjunctions ```agda -ev-disjunction-Prop : - {l1 l2 l3 : Level} (P : Prop l1) (Q : Prop l2) (R : Prop l3) → - type-hom-Prop - ( hom-Prop (disjunction-Prop P Q) R) - ( conjunction-Prop (hom-Prop P R) (hom-Prop Q R)) -pr1 (ev-disjunction-Prop P Q R h) = h ∘ (inl-disjunction-Prop P Q) -pr2 (ev-disjunction-Prop P Q R h) = h ∘ (inr-disjunction-Prop P Q) - -elim-disjunction-Prop : - {l1 l2 l3 : Level} (P : Prop l1) (Q : Prop l2) (R : Prop l3) → - type-hom-Prop - ( conjunction-Prop (hom-Prop P R) (hom-Prop Q R)) - ( hom-Prop (disjunction-Prop P Q) R) -elim-disjunction-Prop P Q R (pair f g) = - map-universal-property-trunc-Prop R (rec-coproduct f g) - -abstract - is-equiv-ev-disjunction-Prop : - {l1 l2 l3 : Level} (P : Prop l1) (Q : Prop l2) (R : Prop l3) → - is-equiv (ev-disjunction-Prop P Q R) - is-equiv-ev-disjunction-Prop P Q R = - is-equiv-is-prop - ( is-prop-type-Prop (hom-Prop (disjunction-Prop P Q) R)) - ( is-prop-type-Prop (conjunction-Prop (hom-Prop P R) (hom-Prop Q R))) - ( elim-disjunction-Prop P Q R) -``` - -### The unit laws for disjunction +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (R : Prop l3) + where + + elim-disjunction : + (A → type-Prop R) → (B → type-Prop R) → + disjunction-type A B → type-Prop R + elim-disjunction f g = elim-disjunction' R (f , g) +``` + +### Propositions that satisfy the universal property of a disjunction are equivalent to the disjunction + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (Q : Prop l3) + (up-Q : universal-property-disjunction-type A B Q) + where + + forward-implication-iff-universal-property-disjunction : + disjunction-type A B → type-Prop Q + forward-implication-iff-universal-property-disjunction = + elim-disjunction Q + ( pr1 (forward-implication (up-Q Q) id)) + ( pr2 (forward-implication (up-Q Q) id)) + + backward-implication-iff-universal-property-disjunction : + type-Prop Q → disjunction-type A B + backward-implication-iff-universal-property-disjunction = + backward-implication + ( up-Q (disjunction-type-Prop A B)) + ( inl-disjunction , inr-disjunction) + + iff-universal-property-disjunction : + disjunction-type A B ↔ type-Prop Q + iff-universal-property-disjunction = + ( forward-implication-iff-universal-property-disjunction , + backward-implication-iff-universal-property-disjunction) +``` + +### The unit laws for the disjunction ```agda module _ @@ -124,10 +245,110 @@ module _ map-left-unit-law-disjunction-is-empty-Prop : is-empty (type-Prop P) → type-disjunction-Prop P Q → type-Prop Q map-left-unit-law-disjunction-is-empty-Prop f = - elim-disjunction-Prop P Q Q (ex-falso ∘ f , id) + elim-disjunction Q (ex-falso ∘ f) id map-right-unit-law-disjunction-is-empty-Prop : is-empty (type-Prop Q) → type-disjunction-Prop P Q → type-Prop P map-right-unit-law-disjunction-is-empty-Prop f = - elim-disjunction-Prop P Q P (id , ex-falso ∘ f) + elim-disjunction P id (ex-falso ∘ f) +``` + +### The unit laws for the disjunction of types + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + map-left-unit-law-disjunction-is-empty-type : + is-empty A → disjunction-type A B → is-inhabited B + map-left-unit-law-disjunction-is-empty-type f = + elim-disjunction (is-inhabited-Prop B) (ex-falso ∘ f) unit-trunc-Prop + + map-right-unit-law-disjunction-is-empty-type : + is-empty B → disjunction-type A B → is-inhabited A + map-right-unit-law-disjunction-is-empty-type f = + elim-disjunction (is-inhabited-Prop A) unit-trunc-Prop (ex-falso ∘ f) +``` + +### The disjunction of arbitrary types is the disjunction of inhabitedness propositions + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + universal-property-disjunction-trunc : + universal-property-disjunction-type A B + ( disjunction-Prop (trunc-Prop A) (trunc-Prop B)) + universal-property-disjunction-trunc R = + ( λ f → + ( f ∘ inl-disjunction ∘ unit-trunc-Prop , + f ∘ inr-disjunction ∘ unit-trunc-Prop)) , + ( λ (f , g) → + rec-trunc-Prop R + ( rec-coproduct (rec-trunc-Prop R f) (rec-trunc-Prop R g))) + + iff-compute-disjunction-trunc : + disjunction-type A B ↔ type-disjunction-Prop (trunc-Prop A) (trunc-Prop B) + iff-compute-disjunction-trunc = + iff-universal-property-disjunction + ( disjunction-Prop (trunc-Prop A) (trunc-Prop B)) + ( universal-property-disjunction-trunc) +``` + +### The disjunction preserves decidability + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-decidable-disjunction : + is-decidable A → is-decidable B → is-decidable (disjunction-type A B) + is-decidable-disjunction is-decidable-A is-decidable-B = + is-decidable-trunc-Prop-is-merely-decidable + ( A + B) + ( unit-trunc-Prop (is-decidable-coproduct is-decidable-A is-decidable-B)) + +module _ + {l1 l2 : Level} (P : Decidable-Prop l1) (Q : Decidable-Prop l2) + where + + type-disjunction-Decidable-Prop : UU (l1 ⊔ l2) + type-disjunction-Decidable-Prop = + type-disjunction-Prop (prop-Decidable-Prop P) (prop-Decidable-Prop Q) + + is-prop-disjunction-Decidable-Prop : + is-prop type-disjunction-Decidable-Prop + is-prop-disjunction-Decidable-Prop = + is-prop-disjunction-Prop + ( prop-Decidable-Prop P) + ( prop-Decidable-Prop Q) + + disjunction-Decidable-Prop : Decidable-Prop (l1 ⊔ l2) + disjunction-Decidable-Prop = + ( type-disjunction-Decidable-Prop , + is-prop-disjunction-Decidable-Prop , + is-decidable-disjunction + ( is-decidable-Decidable-Prop P) + ( is-decidable-Decidable-Prop Q)) ``` + +## See also + +- The indexed counterpart to disjunction is + [existential quantification](foundation.existential-quantification.md). + +## Table of files about propositional logic + +The following table gives an overview of basic constructions in propositional +logic and related considerations. + +{{#include tables/propositional-logic.md}} + +## External links + +- [disjunction](https://ncatlab.org/nlab/show/disjunction) at $n$Lab +- [Logical disjunction](https://en.wikipedia.org/wiki/Logical_disjunction) at + Wikipedia diff --git a/src/foundation/double-negation-modality.lagda.md b/src/foundation/double-negation-modality.lagda.md index 1e406c460d..d313b548d3 100644 --- a/src/foundation/double-negation-modality.lagda.md +++ b/src/foundation/double-negation-modality.lagda.md @@ -33,7 +33,7 @@ The [double negation](foundation.double-negation.md) operation `¬¬` is a ```agda operator-double-negation-modality : (l : Level) → operator-modality l l -operator-double-negation-modality _ = ¬¬ +operator-double-negation-modality _ = ¬¬_ unit-double-negation-modality : {l : Level} → unit-modality (operator-double-negation-modality l) @@ -57,7 +57,7 @@ is-uniquely-eliminating-modality-double-negation-modality {l} {A} P = double-negation-extend ( λ (a : A) → tr - ( ¬¬ ∘ P) + ( ¬¬_ ∘ P) ( eq-is-prop is-prop-double-negation) ( f a)) ( z)) diff --git a/src/foundation/double-negation.lagda.md b/src/foundation/double-negation.lagda.md index 70aecd6fc0..4529283ca7 100644 --- a/src/foundation/double-negation.lagda.md +++ b/src/foundation/double-negation.lagda.md @@ -26,10 +26,12 @@ open import foundation-core.propositions We define double negation and triple negation ```agda -¬¬ : {l : Level} → UU l → UU l +infix 25 ¬¬_ ¬¬¬_ + +¬¬_ : {l : Level} → UU l → UU l ¬¬ P = ¬ (¬ P) -¬¬¬ : {l : Level} → UU l → UU l +¬¬¬_ : {l : Level} → UU l → UU l ¬¬¬ P = ¬ (¬ (¬ P)) ``` @@ -41,7 +43,7 @@ intro-double-negation : {l : Level} {P : UU l} → P → ¬¬ P intro-double-negation p f = f p map-double-negation : - {l1 l2 : Level} {P : UU l1} {Q : UU l2} → (P → Q) → (¬¬ P → ¬¬ Q) + {l1 l2 : Level} {P : UU l1} {Q : UU l2} → (P → Q) → ¬¬ P → ¬¬ Q map-double-negation f = map-neg (map-neg f) ``` @@ -50,17 +52,22 @@ map-double-negation f = map-neg (map-neg f) ### The double negation of a type is a proposition ```agda -double-negation-Prop' : +double-negation-type-Prop : {l : Level} (A : UU l) → Prop l -double-negation-Prop' A = neg-Prop' (¬ A) +double-negation-type-Prop A = neg-type-Prop (¬ A) double-negation-Prop : {l : Level} (P : Prop l) → Prop l -double-negation-Prop P = double-negation-Prop' (type-Prop P) +double-negation-Prop P = double-negation-type-Prop (type-Prop P) is-prop-double-negation : {l : Level} {A : UU l} → is-prop (¬¬ A) is-prop-double-negation = is-prop-neg + +infix 25 ¬¬'_ + +¬¬'_ : {l : Level} (P : Prop l) → Prop l +¬¬'_ = double-negation-Prop ``` ### Double negations of classical laws @@ -109,10 +116,10 @@ double-negation-elim-exp {P = P} {Q = Q} f p = ( ¬ Q) ( map-double-negation (λ (g : P → ¬¬ Q) → g p) f) -double-negation-elim-forall : +double-negation-elim-for-all : {l1 l2 : Level} {P : UU l1} {Q : P → UU l2} → ¬¬ ((p : P) → ¬¬ (Q p)) → (p : P) → ¬¬ (Q p) -double-negation-elim-forall {P = P} {Q = Q} f p = +double-negation-elim-for-all {P = P} {Q = Q} f p = double-negation-elim-neg ( ¬ (Q p)) ( map-double-negation (λ (g : (u : P) → ¬¬ (Q u)) → g p) f) @@ -137,7 +144,7 @@ abstract double-negation-double-negation-type-trunc-Prop A = double-negation-extend ( map-universal-property-trunc-Prop - ( double-negation-Prop' A) + ( double-negation-type-Prop A) ( intro-double-negation)) abstract @@ -146,3 +153,10 @@ abstract double-negation-type-trunc-Prop-double-negation = map-double-negation unit-trunc-Prop ``` + +## Table of files about propositional logic + +The following table gives an overview of basic constructions in propositional +logic and related considerations. + +{{#include tables/propositional-logic.md}} diff --git a/src/foundation/double-powersets.lagda.md b/src/foundation/double-powersets.lagda.md index 9788978ec3..45d84375c0 100644 --- a/src/foundation/double-powersets.lagda.md +++ b/src/foundation/double-powersets.lagda.md @@ -81,7 +81,7 @@ union-double-powerset : {l1 l2 l3 : Level} {A : UU l1} → double-powerset l2 l3 A → powerset (l1 ⊔ lsuc l2 ⊔ l3) A union-double-powerset F a = - ∃-Prop (type-subtype F) (λ X → is-in-subtype (inclusion-subtype F X) a) + ∃ (type-subtype F) (λ X → inclusion-subtype F X a) module _ {l1 l2 l3 : Level} {A : UU l1} (F : double-powerset l2 l3 A) @@ -92,7 +92,7 @@ module _ inclusion-union-double-powerset : (X : type-subtype F) → inclusion-subtype F X ⊆ union-double-powerset F - inclusion-union-double-powerset X a = intro-∃ X + inclusion-union-double-powerset X a = intro-exists X universal-property-union-double-powerset : {l : Level} (P : powerset l A) diff --git a/src/foundation/dubuc-penon-compact-types.lagda.md b/src/foundation/dubuc-penon-compact-types.lagda.md index bfb43239fc..7a80fcc588 100644 --- a/src/foundation/dubuc-penon-compact-types.lagda.md +++ b/src/foundation/dubuc-penon-compact-types.lagda.md @@ -8,6 +8,7 @@ module foundation.dubuc-penon-compact-types where ```agda open import foundation.disjunction +open import foundation.universal-quantification open import foundation.universe-levels open import foundation-core.propositions @@ -18,9 +19,12 @@ open import foundation-core.subtypes ## Idea -A type is said to be Dubuc-Penon compact if for every proposition `P` and every -subtype `Q` of `X` such that `P ∨ Q x` holds for all `x`, then either `P` is -true or `Q` contains every element of `X`. +A type is said to be +{{#concept "Dubuc-Penon compact" Agda=is-dubuc-penon-compact}} if for every +[proposition](foundation-core.propositions.md) `P` and every +[subtype](foundation-core.subtypes.md) `Q` of `X` such that the +[disjunction](foundation.disjunction.md) `P ∨ Q x` holds for all `x`, then +either `P` is true or `Q` contains every element of `X`. ## Definition @@ -28,15 +32,12 @@ true or `Q` contains every element of `X`. is-dubuc-penon-compact-Prop : {l : Level} (l1 l2 : Level) → UU l → Prop (l ⊔ lsuc l1 ⊔ lsuc l2) is-dubuc-penon-compact-Prop l1 l2 X = - Π-Prop + ∀' ( Prop l1) ( λ P → - Π-Prop + ∀' ( subtype l2 X) - ( λ Q → - function-Prop - ( (x : X) → type-disjunction-Prop P (Q x)) - ( disjunction-Prop P (Π-Prop X Q)))) + ( λ Q → (∀' X (λ x → P ∨ Q x)) ⇒ (P ∨ (∀' X Q)))) is-dubuc-penon-compact : {l : Level} (l1 l2 : Level) → UU l → UU (l ⊔ lsuc l1 ⊔ lsuc l2) diff --git a/src/foundation/epimorphisms-with-respect-to-sets.lagda.md b/src/foundation/epimorphisms-with-respect-to-sets.lagda.md index 01a52b2c70..f34760598e 100644 --- a/src/foundation/epimorphisms-with-respect-to-sets.lagda.md +++ b/src/foundation/epimorphisms-with-respect-to-sets.lagda.md @@ -12,6 +12,7 @@ open import foundation.dependent-pair-types open import foundation.existential-quantification open import foundation.function-extensionality open import foundation.identity-types +open import foundation.injective-maps open import foundation.propositional-extensionality open import foundation.propositional-truncations open import foundation.sets @@ -24,7 +25,6 @@ open import foundation-core.equivalences open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies -open import foundation-core.injective-maps open import foundation-core.precomposition-functions open import foundation-core.propositional-maps open import foundation-core.propositions @@ -100,7 +100,7 @@ abstract g : B → Prop (l1 ⊔ l2) g y = raise-unit-Prop (l1 ⊔ l2) h : B → Prop (l1 ⊔ l2) - h y = ∃-Prop A (λ x → f x ＝ y) + h y = exists-structure-Prop A (λ x → f x ＝ y) ``` ### There is at most one extension of a map into a set along a surjection diff --git a/src/foundation/equivalence-classes.lagda.md b/src/foundation/equivalence-classes.lagda.md index 2a8551e680..71954bae57 100644 --- a/src/foundation/equivalence-classes.lagda.md +++ b/src/foundation/equivalence-classes.lagda.md @@ -7,6 +7,7 @@ module foundation.equivalence-classes where
Imports ```agda +open import foundation.conjunction open import foundation.dependent-pair-types open import foundation.effective-maps-equivalence-relations open import foundation.existential-quantification @@ -56,7 +57,7 @@ module _ is-equivalence-class-Prop : subtype l2 A → Prop (l1 ⊔ l2) is-equivalence-class-Prop P = - ∃-Prop A (λ x → has-same-elements-subtype P (prop-equivalence-relation R x)) + ∃ A (λ x → has-same-elements-subtype-Prop P (prop-equivalence-relation R x)) is-equivalence-class : subtype l2 A → UU (l1 ⊔ l2) is-equivalence-class P = type-Prop (is-equivalence-class-Prop P) @@ -246,8 +247,9 @@ module _ share-common-element-equivalence-class-Prop : (C D : equivalence-class R) → Prop (l1 ⊔ l2) share-common-element-equivalence-class-Prop C D = - ∃-Prop A - ( λ x → is-in-equivalence-class R C x × is-in-equivalence-class R D x) + ∃ ( A) + ( λ x → + is-in-equivalence-class-Prop R C x ∧ is-in-equivalence-class-Prop R D x) share-common-element-equivalence-class : (C D : equivalence-class R) → UU (l1 ⊔ l2) diff --git a/src/foundation/equivalences.lagda.md b/src/foundation/equivalences.lagda.md index 7ffbfa28a8..3b62d1bde4 100644 --- a/src/foundation/equivalences.lagda.md +++ b/src/foundation/equivalences.lagda.md @@ -15,6 +15,7 @@ open import foundation.dependent-pair-types open import foundation.equivalence-extensionality open import foundation.function-extensionality open import foundation.functoriality-fibers-of-maps +open import foundation.logical-equivalences open import foundation.transposition-identifications-along-equivalences open import foundation.truncated-maps open import foundation.universal-property-equivalences @@ -188,7 +189,7 @@ module _ ( H : coherence-triangle-maps h (map-equiv e) f) → is-equiv f ≃ is-equiv h equiv-is-equiv-right-map-triangle {f} e h H = - equiv-prop + equiv-iff-is-prop ( is-property-is-equiv f) ( is-property-is-equiv h) ( λ is-equiv-f → @@ -215,7 +216,7 @@ module _ ( H : coherence-triangle-maps h f (map-equiv e)) → is-equiv f ≃ is-equiv h equiv-is-equiv-top-map-triangle e {f} h H = - equiv-prop + equiv-iff-is-prop ( is-property-is-equiv f) ( is-property-is-equiv h) ( is-equiv-left-map-triangle h f (map-equiv e) H (is-equiv-map-equiv e)) @@ -380,7 +381,7 @@ module _ { f g : A → B} → (f ~ g) → is-equiv f ≃ is-equiv g equiv-is-equiv-htpy {f} {g} H = - equiv-prop + equiv-iff-is-prop ( is-property-is-equiv f) ( is-property-is-equiv g) ( is-equiv-htpy f (inv-htpy H)) diff --git a/src/foundation/exclusive-disjunction.lagda.md b/src/foundation/exclusive-disjunction.lagda.md index 78aacaf3cd..5fb3e08ee3 100644 --- a/src/foundation/exclusive-disjunction.lagda.md +++ b/src/foundation/exclusive-disjunction.lagda.md @@ -1,4 +1,4 @@ -# Exclusive disjunction of propositions +# Exclusive disjunctions ```agda module foundation.exclusive-disjunction where @@ -7,61 +7,88 @@ module foundation.exclusive-disjunction where
Imports ```agda -open import foundation.conjunction 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.exclusive-sum open import foundation.functoriality-coproduct-types -open import foundation.negation -open import foundation.propositional-extensionality -open import foundation.symmetric-operations +open import foundation.propositional-truncations open import foundation.type-arithmetic-cartesian-product-types open import foundation.type-arithmetic-coproduct-types open import foundation.universal-property-coproduct-types open import foundation.universe-levels -open import foundation.unordered-pairs -open import foundation-core.cartesian-product-types -open import foundation-core.decidable-propositions open import foundation-core.embeddings -open import foundation-core.empty-types -open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.transport-along-identifications - -open import univalent-combinatorics.2-element-types -open import univalent-combinatorics.equality-finite-types -open import univalent-combinatorics.finite-types -open import univalent-combinatorics.standard-finite-types ```
## Idea -The exclusive disjunction of two propositions `P` and `Q` is the proposition -that `P` holds and `Q` doesn't hold or `P` doesn't hold and `Q` holds. +The +{{#concept "exclusive disjunction" Disambiguation="of propositions" WDID=Q498186 WD="exclusive or" Agda=xor-Prop}} +of two [propositions](foundation-core.propositions.md) `P` and `Q` is the +proposition that precisely one of `P` and `Q` holds, and is defined as the +proposition that the [coproduct](foundation-core.coproduct-types.md) of their +underlying types is [contractible](foundation-core.contractible-types.md) + +```text + P ⊻ Q := is-contr (P + Q) +``` + +It necessarily follows that precisely one of the two propositions hold, and the +other does not. This is captured by the +[exclusive sum](foundation.exclusive-sum.md). ## Definitions -### Exclusive disjunction of types +### The exclusive disjunction of arbitrary types + +The definition of exclusive sum is sometimes generalized to arbitrary types, +which we record here for completeness. + +The +{{#concept "exclusive disjunction" Disambiguation="of types" Agda=xor-type-Prop}} +of the types `A` and `B` is the proposition that their coproduct is contractible + +```text + A ⊻ B := is-contr (A + B). +``` + +Note that unlike the case for [disjunction](foundation.disjunction.md) and +[existential quantification](foundation.existential-quantification.md), but +analogous to the case of +[uniqueness quantification](foundation.uniqueness-quantification.md), the +exclusive disjunction of types does _not_ coincide with the exclusive +disjunction of the summands' +[propositional reflections](foundation.propositional-truncations.md): + +```text + A ⊻ B ≠ ║ A ║₋₁ ⊻ ║ B ║₋₁. +``` ```agda module _ {l1 l2 : Level} (A : UU l1) (B : UU l2) where - xor : UU (l1 ⊔ l2) - xor = (A × ¬ B) + (B × ¬ A) + xor-type-Prop : Prop (l1 ⊔ l2) + xor-type-Prop = is-contr-Prop (A + B) + + xor-type : UU (l1 ⊔ l2) + xor-type = type-Prop xor-type-Prop + + is-prop-xor-type : is-prop xor-type + is-prop-xor-type = is-prop-type-Prop xor-type-Prop ``` -### Exclusive disjunction of propositions +### The exclusive disjunction ```agda module _ @@ -69,119 +96,52 @@ module _ where xor-Prop : Prop (l1 ⊔ l2) - xor-Prop = - coproduct-Prop - ( conjunction-Prop P (neg-Prop Q)) - ( conjunction-Prop Q (neg-Prop P)) - ( λ p q → pr2 q (pr1 p)) + xor-Prop = xor-type-Prop (type-Prop P) (type-Prop Q) type-xor-Prop : UU (l1 ⊔ l2) type-xor-Prop = type-Prop xor-Prop - abstract - is-prop-type-xor-Prop : is-prop type-xor-Prop - is-prop-type-xor-Prop = is-prop-type-Prop xor-Prop -``` + is-prop-xor-Prop : is-prop type-xor-Prop + is-prop-xor-Prop = is-prop-type-Prop xor-Prop -### The symmetric operation of exclusive disjunction - -```agda -predicate-symmetric-xor : - {l : Level} (p : unordered-pair (UU l)) → type-unordered-pair p → UU l -predicate-symmetric-xor p x = - ( element-unordered-pair p x) × (¬ (other-element-unordered-pair p x)) - -symmetric-xor : {l : Level} → symmetric-operation (UU l) (UU l) -symmetric-xor p = Σ (type-unordered-pair p) (predicate-symmetric-xor p) + infixr 10 _⊻_ + _⊻_ : Prop (l1 ⊔ l2) + _⊻_ = xor-Prop ``` -### The symmetric operation of exclusive disjunction of propositions +**Notation.** The +[symbol used for exclusive disjunction](https://codepoints.net/U+22BB?lang=en) +`⊻` can be written with the escape sequence `\veebar`. -```agda -predicate-symmetric-xor-Prop : - {l : Level} (p : unordered-pair (Prop l)) → - type-unordered-pair p → UU l -predicate-symmetric-xor-Prop p = - predicate-symmetric-xor (map-unordered-pair type-Prop p) - -type-symmetric-xor-Prop : - {l : Level} → symmetric-operation (Prop l) (UU l) -type-symmetric-xor-Prop p = symmetric-xor (map-unordered-pair type-Prop p) - -all-elements-equal-type-symmetric-xor-Prop : - {l : Level} (p : unordered-pair (Prop l)) → - all-elements-equal (type-symmetric-xor-Prop p) -all-elements-equal-type-symmetric-xor-Prop (pair X P) x y = - cases-is-prop-type-symmetric-xor-Prop - ( has-decidable-equality-is-finite - ( is-finite-type-UU-Fin 2 X) - ( pr1 x) - ( pr1 y)) - where - cases-is-prop-type-symmetric-xor-Prop : - is-decidable (pr1 x ＝ pr1 y) → x ＝ y - cases-is-prop-type-symmetric-xor-Prop (inl p) = - eq-pair-Σ - ( p) - ( eq-is-prop - ( is-prop-product (is-prop-type-Prop (P (pr1 y))) is-prop-neg)) - cases-is-prop-type-symmetric-xor-Prop (inr np) = - ex-falso - ( tr - ( λ z → ¬ (type-Prop (P z))) - ( compute-swap-2-Element-Type X (pr1 x) (pr1 y) np) - ( pr2 (pr2 x)) - ( pr1 (pr2 y))) - -is-prop-type-symmetric-xor-Prop : - {l : Level} (p : unordered-pair (Prop l)) → - is-prop (type-symmetric-xor-Prop p) -is-prop-type-symmetric-xor-Prop p = - is-prop-all-elements-equal - ( all-elements-equal-type-symmetric-xor-Prop p) - -symmetric-xor-Prop : - {l : Level} → symmetric-operation (Prop l) (Prop l) -pr1 (symmetric-xor-Prop E) = type-symmetric-xor-Prop E -pr2 (symmetric-xor-Prop E) = is-prop-type-symmetric-xor-Prop E -``` +## Properties -### Second definition of exclusive disjunction +### The canonical map from the exclusive disjunction into the exclusive sum ```agda module _ - {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) + {l1 l2 : Level} {A : UU l1} {B : UU l2} where - xor-Prop' : Prop (l1 ⊔ l2) - xor-Prop' = is-contr-Prop (type-Prop P + type-Prop Q) - - type-xor-Prop' : UU (l1 ⊔ l2) - type-xor-Prop' = type-Prop xor-Prop' + map-exclusive-sum-xor : xor-type A B → exclusive-sum A B + map-exclusive-sum-xor (inl a , H) = + inl (a , (λ b → is-empty-eq-coproduct-inl-inr a b (H (inr b)))) + map-exclusive-sum-xor (inr b , H) = + inr (b , (λ a → is-empty-eq-coproduct-inr-inl b a (H (inl a)))) ``` -## Properties - -### The definitions `xor-Prop` and `xor-Prop'` are equivalent +### The exclusive disjunction of two propositions is equivalent to their exclusive sum ```agda module _ {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) where - map-equiv-xor-Prop : type-xor-Prop' P Q → type-xor-Prop P Q - map-equiv-xor-Prop (pair (inl p) H) = - inl (pair p (λ q → is-empty-eq-coproduct-inl-inr p q (H (inr q)))) - map-equiv-xor-Prop (pair (inr q) H) = - inr (pair q (λ p → is-empty-eq-coproduct-inr-inl q p (H (inl p)))) - - equiv-xor-Prop : type-xor-Prop' P Q ≃ type-xor-Prop P Q - equiv-xor-Prop = + equiv-exclusive-sum-xor-Prop : type-xor-Prop P Q ≃ type-exclusive-sum-Prop P Q + equiv-exclusive-sum-xor-Prop = ( equiv-coproduct ( equiv-tot ( λ p → - ( ( equiv-Π-equiv-family - ( λ q → compute-eq-coproduct-inl-inr p q)) ∘e + ( ( equiv-Π-equiv-family (compute-eq-coproduct-inl-inr p)) ∘e ( left-unit-law-product-is-contr ( is-contr-Π ( λ p' → @@ -189,11 +149,10 @@ module _ ( p ＝ p') ( equiv-ap-emb (emb-inl (type-Prop P) (type-Prop Q))) ( is-prop-type-Prop P p p'))))) ∘e - ( equiv-dependent-universal-property-coproduct (λ x → inl p ＝ x)))) + ( equiv-dependent-universal-property-coproduct (inl p ＝_)))) ( equiv-tot ( λ q → - ( ( equiv-Π-equiv-family - ( λ p → compute-eq-coproduct-inr-inl q p)) ∘e + ( ( equiv-Π-equiv-family (compute-eq-coproduct-inr-inl q)) ∘e ( right-unit-law-product-is-contr ( is-contr-Π ( λ q' → @@ -201,211 +160,28 @@ module _ ( q ＝ q') ( equiv-ap-emb (emb-inr (type-Prop P) (type-Prop Q))) ( is-prop-type-Prop Q q q'))))) ∘e - ( equiv-dependent-universal-property-coproduct - ( λ x → inr q ＝ x))))) ∘e + ( equiv-dependent-universal-property-coproduct (inr q ＝_))))) ∘e ( right-distributive-Σ-coproduct ( type-Prop P) ( type-Prop Q) ( λ x → (y : type-Prop P + type-Prop Q) → x ＝ y)) ``` -### The symmetric exclusive disjunction at a standard unordered pair +## See also -```agda -module _ - {l : Level} {A B : UU l} - where +- The indexed counterpart to exclusive disjunction is + [unique existence](foundation.uniqueness-quantification.md). - xor-symmetric-xor : - symmetric-xor (standard-unordered-pair A B) → xor A B - xor-symmetric-xor (pair (inl (inr _)) (pair p nq)) = - inl - ( pair p - ( tr - ( λ t → ¬ (element-unordered-pair (standard-unordered-pair A B) t)) - ( compute-swap-Fin-two-ℕ (zero-Fin 1)) - ( nq))) - xor-symmetric-xor (pair (inr _) (pair q np)) = - inr - ( pair - ( q) - ( tr - ( λ t → ¬ (element-unordered-pair (standard-unordered-pair A B) t)) - ( compute-swap-Fin-two-ℕ (one-Fin 1)) - ( np))) - - symmetric-xor-xor : - xor A B → symmetric-xor (standard-unordered-pair A B) - pr1 (symmetric-xor-xor (inl (pair a nb))) = (zero-Fin 1) - pr1 (pr2 (symmetric-xor-xor (inl (pair a nb)))) = a - pr2 (pr2 (symmetric-xor-xor (inl (pair a nb)))) = - tr - ( λ t → ¬ (element-unordered-pair (standard-unordered-pair A B) t)) - ( inv (compute-swap-Fin-two-ℕ (zero-Fin 1))) - ( nb) - pr1 (symmetric-xor-xor (inr (pair na b))) = (one-Fin 1) - pr1 (pr2 (symmetric-xor-xor (inr (pair b na)))) = b - pr2 (pr2 (symmetric-xor-xor (inr (pair b na)))) = - tr - ( λ t → ¬ (element-unordered-pair (standard-unordered-pair A B) t)) - ( inv (compute-swap-Fin-two-ℕ (one-Fin 1))) - ( na) - -{- - eq-equiv-Prop - ( ( ( equiv-coproduct - ( ( ( left-unit-law-coproduct (type-Prop (conjunction-Prop P (neg-Prop Q)))) ∘e - ( equiv-coproduct - ( left-absorption-Σ - ( λ x → - ( type-Prop - ( pr2 (standard-unordered-pair P Q) (inl (inl x)))) × - ( ¬ ( type-Prop - ( pr2 - ( standard-unordered-pair P Q) - ( map-swap-2-Element-Type - ( pr1 (standard-unordered-pair P Q)) - ( inl (inl x)))))))) - ( ( equiv-product - ( id-equiv) - ( tr - ( λ b → - ( ¬ (type-Prop (pr2 (standard-unordered-pair P Q) b))) ≃ - ( ¬ (type-Prop Q))) - ( inv (compute-swap-Fin-two-ℕ (zero-Fin ?))) - ( id-equiv))) ∘e - ( left-unit-law-Σ - ( λ y → - ( type-Prop - ( pr2 (standard-unordered-pair P Q) (inl (inr y)))) × - ( ¬ ( type-Prop - ( pr2 - ( standard-unordered-pair P Q) - ( map-swap-2-Element-Type - ( pr1 (standard-unordered-pair P Q)) - ( inl (inr y))))))))))) ∘e - ( ( right-distributive-Σ-coproduct - ( Fin 0) - ( unit) - ( λ x → - ( type-Prop (pr2 (standard-unordered-pair P Q) (inl x))) × - ( ¬ ( type-Prop - ( pr2 - ( standard-unordered-pair P Q) - ( map-swap-2-Element-Type - ( pr1 (standard-unordered-pair P Q)) (inl x))))))))) - ( ( equiv-product - ( tr - ( λ b → - ¬ (type-Prop (pr2 (standard-unordered-pair P Q) b)) ≃ - ¬ (type-Prop P)) - ( inv (compute-swap-Fin-two-ℕ (one-Fin ?))) - ( id-equiv)) - ( id-equiv)) ∘e - ( ( commutative-product) ∘e - ( left-unit-law-Σ - ( λ y → - ( type-Prop (pr2 (standard-unordered-pair P Q) (inr y))) × - ( ¬ ( type-Prop - ( pr2 - ( standard-unordered-pair P Q) - ( map-swap-2-Element-Type - ( pr1 (standard-unordered-pair P Q)) - ( inr y)))))))))) ∘e - ( right-distributive-Σ-coproduct - ( Fin 1) - ( unit) - ( λ x → - ( type-Prop (pr2 (standard-unordered-pair P Q) x)) × - ( ¬ ( type-Prop - ( pr2 - ( standard-unordered-pair P Q) - ( map-swap-2-Element-Type - ( pr1 (standard-unordered-pair P Q)) - ( x))))))))) --} -``` +## Table of files about propositional logic -```agda -module _ - {l : Level} (P Q : Prop l) - where +The following table gives an overview of basic constructions in propositional +logic and related considerations. - xor-symmetric-xor-Prop : - type-hom-Prop - ( symmetric-xor-Prop (standard-unordered-pair P Q)) - ( xor-Prop P Q) - xor-symmetric-xor-Prop (pair (inl (inr _)) (pair p nq)) = - inl - ( pair p - ( tr - ( λ t → - ¬ ( type-Prop - ( element-unordered-pair (standard-unordered-pair P Q) t))) - ( compute-swap-Fin-two-ℕ (zero-Fin 1)) - ( nq))) - xor-symmetric-xor-Prop (pair (inr _) (pair q np)) = - inr - ( pair q - ( tr - ( λ t → - ¬ ( type-Prop - ( element-unordered-pair (standard-unordered-pair P Q) t))) - ( compute-swap-Fin-two-ℕ (one-Fin 1)) - ( np))) - - symmetric-xor-xor-Prop : - type-hom-Prop - ( xor-Prop P Q) - ( symmetric-xor-Prop (standard-unordered-pair P Q)) - pr1 (symmetric-xor-xor-Prop (inl (pair p nq))) = (zero-Fin 1) - pr1 (pr2 (symmetric-xor-xor-Prop (inl (pair p nq)))) = p - pr2 (pr2 (symmetric-xor-xor-Prop (inl (pair p nq)))) = - tr - ( λ t → - ¬ (type-Prop (element-unordered-pair (standard-unordered-pair P Q) t))) - ( inv (compute-swap-Fin-two-ℕ (zero-Fin 1))) - ( nq) - pr1 (symmetric-xor-xor-Prop (inr (pair q np))) = (one-Fin 1) - pr1 (pr2 (symmetric-xor-xor-Prop (inr (pair q np)))) = q - pr2 (pr2 (symmetric-xor-xor-Prop (inr (pair q np)))) = - tr - ( λ t → - ¬ (type-Prop (element-unordered-pair (standard-unordered-pair P Q) t))) - ( inv (compute-swap-Fin-two-ℕ (one-Fin 1))) - ( np) - -eq-commmutative-xor-xor : - {l : Level} (P Q : Prop l) → - symmetric-xor-Prop (standard-unordered-pair P Q) ＝ xor-Prop P Q -eq-commmutative-xor-xor P Q = - eq-iff (xor-symmetric-xor-Prop P Q) (symmetric-xor-xor-Prop P Q) -``` +{{#include tables/propositional-logic.md}} -### Exclusive disjunction of decidable propositions +## External links -```agda -is-decidable-xor : - {l1 l2 : Level} {A : UU l1} {B : UU l2} → - is-decidable A → is-decidable B → is-decidable (xor A B) -is-decidable-xor d e = - is-decidable-coproduct - ( is-decidable-product d (is-decidable-neg e)) - ( is-decidable-product e (is-decidable-neg d)) - -xor-Decidable-Prop : - {l1 l2 : Level} → Decidable-Prop l1 → Decidable-Prop l2 → - Decidable-Prop (l1 ⊔ l2) -pr1 (xor-Decidable-Prop P Q) = - type-xor-Prop (prop-Decidable-Prop P) (prop-Decidable-Prop Q) -pr1 (pr2 (xor-Decidable-Prop P Q)) = - is-prop-type-xor-Prop (prop-Decidable-Prop P) (prop-Decidable-Prop Q) -pr2 (pr2 (xor-Decidable-Prop P Q)) = - is-decidable-coproduct - ( is-decidable-product - ( is-decidable-Decidable-Prop P) - ( is-decidable-neg (is-decidable-Decidable-Prop Q))) - ( is-decidable-product - ( is-decidable-Decidable-Prop Q) - ( is-decidable-neg (is-decidable-Decidable-Prop P))) -``` +- [exclusive disjunction](https://ncatlab.org/nlab/show/exclusive+disjunction) + at $n$Lab +- [Exclusive disjunction](https://simple.wikipedia.org/wiki/Exclusive_disjunction) + at Wikipedia diff --git a/src/foundation/exclusive-sum.lagda.md b/src/foundation/exclusive-sum.lagda.md new file mode 100644 index 0000000000..fc12a94489 --- /dev/null +++ b/src/foundation/exclusive-sum.lagda.md @@ -0,0 +1,395 @@ +# Exclusive sums + +```agda +module foundation.exclusive-sum where +``` + +
Imports + +```agda +open import foundation.conjunction +open import foundation.coproduct-types +open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.negation +open import foundation.propositional-extensionality +open import foundation.symmetric-operations +open import foundation.universal-quantification +open import foundation.universe-levels +open import foundation.unordered-pairs + +open import foundation-core.cartesian-product-types +open import foundation-core.decidable-propositions +open import foundation-core.embeddings +open import foundation-core.empty-types +open import foundation-core.equality-dependent-pair-types +open import foundation-core.identity-types +open import foundation-core.propositions +open import foundation-core.transport-along-identifications + +open import univalent-combinatorics.2-element-types +open import univalent-combinatorics.equality-finite-types +open import univalent-combinatorics.finite-types +open import univalent-combinatorics.standard-finite-types +``` + +
+ +## Idea + +The {{#concept "exclusive sum" Disambiguation="of types" Agda=exclusive-sum}} of +two types `A` and `B`, is the type consisting of + +- elements of `A` together with a proof that `B` is + [empty](foundation-core.empty-types.md), or +- elements of `B` together with a proof that `A` is empty. + +The +{{#concept "exclusive sum" Disambiguation="of propositions" Agda=exclusive-sum-Prop}} +of [propositions](foundation-core.propositions.md) `P` and `Q` is likewise the +[proposition](foundation-core.propositions.md) that `P` holds and `Q` does not +hold, or `P` does not hold and `Q` holds. The exclusive sum of two propositions +is equivalent to the +[exclusive disjunction](foundation.exclusive-disjunction.md), which is shown +there. + +## Definitions + +### The exclusive sum of types + +```agda +module _ + {l1 l2 : Level} (A : UU l1) (B : UU l2) + where + + exclusive-sum : UU (l1 ⊔ l2) + exclusive-sum = (A × ¬ B) + (B × ¬ A) +``` + +### The exclusive sum of propositions + +```agda +module _ + {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) + where + + exclusive-sum-Prop : Prop (l1 ⊔ l2) + exclusive-sum-Prop = + coproduct-Prop + ( P ∧ (¬' Q)) + ( Q ∧ (¬' P)) + ( λ p q → pr2 q (pr1 p)) + + type-exclusive-sum-Prop : UU (l1 ⊔ l2) + type-exclusive-sum-Prop = type-Prop exclusive-sum-Prop + + abstract + is-prop-type-exclusive-sum-Prop : is-prop type-exclusive-sum-Prop + is-prop-type-exclusive-sum-Prop = is-prop-type-Prop exclusive-sum-Prop +``` + +### The canonical inclusion of the exclusive sum into the coproduct + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + map-coproduct-exclusive-sum : exclusive-sum A B → A + B + map-coproduct-exclusive-sum (inl (a , _)) = inl a + map-coproduct-exclusive-sum (inr (b , _)) = inr b +``` + +### The symmetric operation of exclusive sum + +```agda +predicate-symmetric-exclusive-sum : + {l : Level} (p : unordered-pair (UU l)) → type-unordered-pair p → UU l +predicate-symmetric-exclusive-sum p x = + ( element-unordered-pair p x) × (¬ (other-element-unordered-pair p x)) + +symmetric-exclusive-sum : {l : Level} → symmetric-operation (UU l) (UU l) +symmetric-exclusive-sum p = + Σ (type-unordered-pair p) (predicate-symmetric-exclusive-sum p) +``` + +### The symmetric operation of the exclusive sum of propositions + +```agda +predicate-symmetric-exclusive-sum-Prop : + {l : Level} (p : unordered-pair (Prop l)) → + type-unordered-pair p → UU l +predicate-symmetric-exclusive-sum-Prop p = + predicate-symmetric-exclusive-sum (map-unordered-pair type-Prop p) + +type-symmetric-exclusive-sum-Prop : + {l : Level} → symmetric-operation (Prop l) (UU l) +type-symmetric-exclusive-sum-Prop p = + symmetric-exclusive-sum (map-unordered-pair type-Prop p) + +all-elements-equal-type-symmetric-exclusive-sum-Prop : + {l : Level} (p : unordered-pair (Prop l)) → + all-elements-equal (type-symmetric-exclusive-sum-Prop p) +all-elements-equal-type-symmetric-exclusive-sum-Prop (X , P) x y = + cases-is-prop-type-symmetric-exclusive-sum-Prop + ( has-decidable-equality-is-finite + ( is-finite-type-UU-Fin 2 X) + ( pr1 x) + ( pr1 y)) + where + cases-is-prop-type-symmetric-exclusive-sum-Prop : + is-decidable (pr1 x ＝ pr1 y) → x ＝ y + cases-is-prop-type-symmetric-exclusive-sum-Prop (inl p) = + eq-pair-Σ + ( p) + ( eq-is-prop + ( is-prop-product-Prop + ( P (pr1 y)) + ( neg-type-Prop + ( other-element-unordered-pair + ( map-unordered-pair (type-Prop) (X , P)) + ( pr1 y))))) + cases-is-prop-type-symmetric-exclusive-sum-Prop (inr np) = + ex-falso + ( tr + ( λ z → ¬ (type-Prop (P z))) + ( compute-swap-2-Element-Type X (pr1 x) (pr1 y) np) + ( pr2 (pr2 x)) + ( pr1 (pr2 y))) + +is-prop-type-symmetric-exclusive-sum-Prop : + {l : Level} (p : unordered-pair (Prop l)) → + is-prop (type-symmetric-exclusive-sum-Prop p) +is-prop-type-symmetric-exclusive-sum-Prop p = + is-prop-all-elements-equal + ( all-elements-equal-type-symmetric-exclusive-sum-Prop p) + +symmetric-exclusive-sum-Prop : + {l : Level} → symmetric-operation (Prop l) (Prop l) +pr1 (symmetric-exclusive-sum-Prop E) = type-symmetric-exclusive-sum-Prop E +pr2 (symmetric-exclusive-sum-Prop E) = + is-prop-type-symmetric-exclusive-sum-Prop E +``` + +## Properties + +### The symmetric exclusive sum at a standard unordered pair + +```agda +module _ + {l : Level} {A B : UU l} + where + + exclusive-sum-symmetric-exclusive-sum : + symmetric-exclusive-sum (standard-unordered-pair A B) → exclusive-sum A B + exclusive-sum-symmetric-exclusive-sum (pair (inl (inr _)) (p , nq)) = + inl + ( pair p + ( tr + ( λ t → ¬ (element-unordered-pair (standard-unordered-pair A B) t)) + ( compute-swap-Fin-two-ℕ (zero-Fin 1)) + ( nq))) + exclusive-sum-symmetric-exclusive-sum (pair (inr _) (q , np)) = + inr + ( pair + ( q) + ( tr + ( λ t → ¬ (element-unordered-pair (standard-unordered-pair A B) t)) + ( compute-swap-Fin-two-ℕ (one-Fin 1)) + ( np))) + + symmetric-exclusive-sum-exclusive-sum : + exclusive-sum A B → symmetric-exclusive-sum (standard-unordered-pair A B) + pr1 (symmetric-exclusive-sum-exclusive-sum (inl (a , nb))) = (zero-Fin 1) + pr1 (pr2 (symmetric-exclusive-sum-exclusive-sum (inl (a , nb)))) = a + pr2 (pr2 (symmetric-exclusive-sum-exclusive-sum (inl (a , nb)))) = + tr + ( λ t → ¬ (element-unordered-pair (standard-unordered-pair A B) t)) + ( inv (compute-swap-Fin-two-ℕ (zero-Fin 1))) + ( nb) + pr1 (symmetric-exclusive-sum-exclusive-sum (inr (na , b))) = (one-Fin 1) + pr1 (pr2 (symmetric-exclusive-sum-exclusive-sum (inr (b , na)))) = b + pr2 (pr2 (symmetric-exclusive-sum-exclusive-sum (inr (b , na)))) = + tr + ( λ t → ¬ (element-unordered-pair (standard-unordered-pair A B) t)) + ( inv (compute-swap-Fin-two-ℕ (one-Fin 1))) + ( na) +``` + +```text + eq-equiv-Prop + ( ( ( equiv-coproduct + ( ( ( left-unit-law-coproduct (type-Prop (P ∧ (¬' Q)))) ∘e + ( equiv-coproduct + ( left-absorption-Σ + ( λ x → + ( type-Prop + ( pr2 (standard-unordered-pair P Q) (inl (inl x)))) × + ( ¬ ( type-Prop + ( pr2 + ( standard-unordered-pair P Q) + ( map-swap-2-Element-Type + ( pr1 (standard-unordered-pair P Q)) + ( inl (inl x)))))))) + ( ( equiv-product + ( id-equiv) + ( tr + ( λ b → + ( ¬ (type-Prop (pr2 (standard-unordered-pair P Q) b))) ≃ + ( ¬ (type-Prop Q))) + ( inv (compute-swap-Fin-two-ℕ (zero-Fin ?))) + ( id-equiv))) ∘e + ( left-unit-law-Σ + ( λ y → + ( type-Prop + ( pr2 (standard-unordered-pair P Q) (inl (inr y)))) × + ( ¬ ( type-Prop + ( pr2 + ( standard-unordered-pair P Q) + ( map-swap-2-Element-Type + ( pr1 (standard-unordered-pair P Q)) + ( inl (inr y))))))))))) ∘e + ( ( right-distributive-Σ-coproduct + ( Fin 0) + ( unit) + ( λ x → + ( type-Prop (pr2 (standard-unordered-pair P Q) (inl x))) × + ( ¬ ( type-Prop + ( pr2 + ( standard-unordered-pair P Q) + ( map-swap-2-Element-Type + ( pr1 (standard-unordered-pair P Q)) (inl x))))))))) + ( ( equiv-product + ( tr + ( λ b → + ¬ (type-Prop (pr2 (standard-unordered-pair P Q) b)) ≃ + ¬ (type-Prop P)) + ( inv (compute-swap-Fin-two-ℕ (one-Fin ?))) + ( id-equiv)) + ( id-equiv)) ∘e + ( ( commutative-product) ∘e + ( left-unit-law-Σ + ( λ y → + ( type-Prop (pr2 (standard-unordered-pair P Q) (inr y))) × + ( ¬ ( type-Prop + ( pr2 + ( standard-unordered-pair P Q) + ( map-swap-2-Element-Type + ( pr1 (standard-unordered-pair P Q)) + ( inr y)))))))))) ∘e + ( right-distributive-Σ-coproduct + ( Fin 1) + ( unit) + ( λ x → + ( type-Prop (pr2 (standard-unordered-pair P Q) x)) × + ( ¬ ( type-Prop + ( pr2 + ( standard-unordered-pair P Q) + ( map-swap-2-Element-Type + ( pr1 (standard-unordered-pair P Q)) + ( x))))))))) +``` + +```agda +module _ + {l : Level} (P Q : Prop l) + where + + exclusive-sum-symmetric-exclusive-sum-Prop : + type-hom-Prop + ( symmetric-exclusive-sum-Prop (standard-unordered-pair P Q)) + ( exclusive-sum-Prop P Q) + exclusive-sum-symmetric-exclusive-sum-Prop (pair (inl (inr _)) (p , nq)) = + inl + ( pair p + ( tr + ( λ t → + ¬ ( type-Prop + ( element-unordered-pair (standard-unordered-pair P Q) t))) + ( compute-swap-Fin-two-ℕ (zero-Fin 1)) + ( nq))) + exclusive-sum-symmetric-exclusive-sum-Prop (pair (inr _) (q , np)) = + inr + ( pair q + ( tr + ( λ t → + ¬ ( type-Prop + ( element-unordered-pair (standard-unordered-pair P Q) t))) + ( compute-swap-Fin-two-ℕ (one-Fin 1)) + ( np))) + + symmetric-exclusive-sum-exclusive-sum-Prop : + type-hom-Prop + ( exclusive-sum-Prop P Q) + ( symmetric-exclusive-sum-Prop (standard-unordered-pair P Q)) + pr1 (symmetric-exclusive-sum-exclusive-sum-Prop (inl (p , nq))) = zero-Fin 1 + pr1 (pr2 (symmetric-exclusive-sum-exclusive-sum-Prop (inl (p , nq)))) = p + pr2 (pr2 (symmetric-exclusive-sum-exclusive-sum-Prop (inl (p , nq)))) = + tr + ( λ t → + ¬ (type-Prop (element-unordered-pair (standard-unordered-pair P Q) t))) + ( inv (compute-swap-Fin-two-ℕ (zero-Fin 1))) + ( nq) + pr1 (symmetric-exclusive-sum-exclusive-sum-Prop (inr (q , np))) = one-Fin 1 + pr1 (pr2 (symmetric-exclusive-sum-exclusive-sum-Prop (inr (q , np)))) = q + pr2 (pr2 (symmetric-exclusive-sum-exclusive-sum-Prop (inr (q , np)))) = + tr + ( λ t → + ¬ (type-Prop (element-unordered-pair (standard-unordered-pair P Q) t))) + ( inv (compute-swap-Fin-two-ℕ (one-Fin 1))) + ( np) + +eq-commmutative-exclusive-sum-exclusive-sum : + {l : Level} (P Q : Prop l) → + symmetric-exclusive-sum-Prop (standard-unordered-pair P Q) ＝ + exclusive-sum-Prop P Q +eq-commmutative-exclusive-sum-exclusive-sum P Q = + eq-iff + ( exclusive-sum-symmetric-exclusive-sum-Prop P Q) + ( symmetric-exclusive-sum-exclusive-sum-Prop P Q) +``` + +### The exclusive sum of decidable types is decidable + +```agda +is-decidable-exclusive-sum : + {l1 l2 : Level} {A : UU l1} {B : UU l2} → + is-decidable A → is-decidable B → is-decidable (exclusive-sum A B) +is-decidable-exclusive-sum d e = + is-decidable-coproduct + ( is-decidable-product d (is-decidable-neg e)) + ( is-decidable-product e (is-decidable-neg d)) + +module _ + {l1 l2 : Level} (P : Decidable-Prop l1) (Q : Decidable-Prop l2) + where + + is-decidable-exclusive-sum-Decidable-Prop : + is-decidable + ( type-exclusive-sum-Prop (prop-Decidable-Prop P) (prop-Decidable-Prop Q)) + is-decidable-exclusive-sum-Decidable-Prop = + is-decidable-coproduct + ( is-decidable-product + ( is-decidable-Decidable-Prop P) + ( is-decidable-neg (is-decidable-Decidable-Prop Q))) + ( is-decidable-product + ( is-decidable-Decidable-Prop Q) + ( is-decidable-neg (is-decidable-Decidable-Prop P))) + + exclusive-sum-Decidable-Prop : Decidable-Prop (l1 ⊔ l2) + pr1 exclusive-sum-Decidable-Prop = + type-exclusive-sum-Prop (prop-Decidable-Prop P) (prop-Decidable-Prop Q) + pr1 (pr2 exclusive-sum-Decidable-Prop) = + is-prop-type-exclusive-sum-Prop + ( prop-Decidable-Prop P) + ( prop-Decidable-Prop Q) + pr2 (pr2 exclusive-sum-Decidable-Prop) = + is-decidable-exclusive-sum-Decidable-Prop +``` + +## Table of files about propositional logic + +The following table gives an overview of basic constructions in propositional +logic and related considerations. + +{{#include tables/propositional-logic.md}} diff --git a/src/foundation/existential-quantification.lagda.md b/src/foundation/existential-quantification.lagda.md index a749f4825c..8fdf7327a3 100644 --- a/src/foundation/existential-quantification.lagda.md +++ b/src/foundation/existential-quantification.lagda.md @@ -9,12 +9,16 @@ module foundation.existential-quantification where ```agda open import foundation.conjunction open import foundation.dependent-pair-types +open import foundation.functoriality-propositional-truncation open import foundation.logical-equivalences open import foundation.propositional-extensionality open import foundation.propositional-truncations open import foundation.universe-levels +open import foundation-core.cartesian-product-types open import foundation-core.equivalences +open import foundation-core.function-types +open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.propositions ``` @@ -23,122 +27,322 @@ open import foundation-core.propositions ## Idea -Given a family of propositions `P` over `A`, the existential quantification of -`P` over `A` is the proposition `∃ A P` asserting that there is an element -`a : A` such that `P a` holds. We use the propositional truncation to define the -existential quantification, because the Curry-Howard interpretation of the -existential quantification as `Σ A P` does not guarantee that existential -quantifications are interpreted as propositions. +Given a family of [propositions](foundation-core.propositions.md) `P` over `A`, +the +{{#concept "existential quantification" Disambiguation="on a subtype" WDID=Q773483 WD="existential quantification" Agda=exists}} +of `P` over `A` is the proposition `∃ A P` asserting that there is an element +`a : A` such that `P a` holds. We use the +[propositional truncation](foundation.propositional-truncations.md) to define +the existential quantification, -## Definition +```text + ∃ (x : A), (P x) := ║ Σ (x : A), (P x) ║₋₁ +``` + +because the Curry-Howard interpretation of the existential quantification as +`Σ A P` does not guarantee that existential quantifications are interpreted as +propositions. + +The +{{#concept "universal property" Disambiguation="of existential quantification" Agda=universal-property-exists}} +of existential quantification states that it is the +[least upper bound](order-theory.least-upper-bounds-large-posets.md) on the +family of propositions `P` in the +[locale of propositions](foundation.large-locale-of-propositions.md), by which +we mean that for every proposition `Q` we have the +[logical equivalence](foundation.logical-equivalences.md) + +```text + (∀ (x : A), (P x ⇒ Q)) ⇔ ((∃ (x : A), (P x)) ⇒ Q). +``` + +## Definitions + +### Existence of structure + +Given a [structure](foundation.structure.md) `B : A → 𝒰` on a type `A`, the +propositional truncation + +```text + ║ Σ (x : A), (B x) ║₋₁ +``` + +satisfies the universal property of the existential quantification + +```text + ∃ (x : A), ║ B x ║₋₁ +``` + +and is thus equivalent to it. Therefore, we may reasonably call this +construction the +{{#concept "existential quantification" Disambiguation="structure" Agda=exists-structure-Prop}} +of structure. It is important to keep in mind that this is not a generalization +of the concept but rather a conflation, and should be read as the statement _the +type of elements `x : A` equipped with `y : B x` is +[inhabited](foundation.inhabited-types.md)_. + +Existence of structure is a widely occurring notion in univalent mathematics. +For instance, the condition that an element `y : B` is in the +[image](foundation.images.md) of a map `f : A → B` is formulated using existence +of structure: The element `y` is in the image of `f` if the type of `x : A` +equipped with an identification `f x = y` is inhabited. -### Existential quantification for families of propositions +Because subtypes are a special case of structure, and Agda can generally infer +structures for us, we will continue to conflate the two in our formalizations +for the benefit that we have to specify the subtype in our code less often. For +instance, even though the introduction rule for existential quantification +`intro-exists` is phrased in terms of existential quantification on structures, +it equally applies to existential quantification on subtypes. ```agda -exists-Prop : - {l1 l2 : Level} (A : UU l1) (P : A → Prop l2) → Prop (l1 ⊔ l2) -exists-Prop {l1} {l2} A P = trunc-Prop (Σ A (λ x → type-Prop (P x))) +module _ + {l1 l2 : Level} (A : UU l1) (B : A → UU l2) + where + + exists-structure-Prop : Prop (l1 ⊔ l2) + exists-structure-Prop = trunc-Prop (Σ A B) -exists : - {l1 l2 : Level} (A : UU l1) (P : A → Prop l2) → UU (l1 ⊔ l2) -exists A P = type-Prop (exists-Prop A P) + exists-structure : UU (l1 ⊔ l2) + exists-structure = type-Prop exists-structure-Prop -abstract - is-prop-exists : - {l1 l2 : Level} (A : UU l1) (P : A → Prop l2) → is-prop (exists A P) - is-prop-exists A P = is-prop-type-Prop (exists-Prop A P) + is-prop-exists-structure : is-prop exists-structure + is-prop-exists-structure = is-prop-type-Prop exists-structure-Prop ``` -### Existential quantification of arbitrary type families +### Existential quantification ```agda -∃-Prop : - {l1 l2 : Level} (A : UU l1) (B : A → UU l2) → Prop (l1 ⊔ l2) -∃-Prop A B = trunc-Prop (Σ A B) +module _ + {l1 l2 : Level} (A : UU l1) (P : A → Prop l2) + where + + exists-Prop : Prop (l1 ⊔ l2) + exists-Prop = exists-structure-Prop A (type-Prop ∘ P) -∃ : - {l1 l2 : Level} (A : UU l1) (B : A → UU l2) → UU (l1 ⊔ l2) -∃ A B = type-Prop (∃-Prop A B) + exists : UU (l1 ⊔ l2) + exists = type-Prop exists-Prop -is-prop-∃ : - {l1 l2 : Level} (A : UU l1) (B : A → UU l2) → is-prop (∃ A B) -is-prop-∃ A B = is-prop-type-Prop (∃-Prop A B) + abstract + is-prop-exists : is-prop exists + is-prop-exists = is-prop-type-Prop exists-Prop + + ∃ : Prop (l1 ⊔ l2) + ∃ = exists-Prop +``` + +### The introduction rule for existential quantification + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + intro-exists : (a : A) (b : B a) → exists-structure A B + intro-exists a b = unit-trunc-Prop (a , b) +``` + +**Note.** Even though the introduction rule is formalized in terms of +existential quantification on structures, it equally applies to existential +quantification on subtypes. This is because subtypes are a special case of +structure. The benefit of this approach is that Agda can infer structures for +us, but not generally subtypes. + +### The universal property of existential quantification + +```agda +module _ + {l1 l2 l3 : Level} (A : UU l1) (B : A → UU l2) (S : Prop l3) + where + + universal-property-exists-structure : UUω + universal-property-exists-structure = + {l : Level} (Q : Prop l) → + (type-Prop S → type-Prop Q) ↔ ((x : A) → B x → type-Prop Q) + +module _ + {l1 l2 l3 : Level} (A : UU l1) (P : A → Prop l2) (S : Prop l3) + where + + universal-property-exists : UUω + universal-property-exists = + universal-property-exists-structure A (type-Prop ∘ P) S ``` ## Properties -### The introduction rule for existential quantification +### The elimination rule of existential quantification + +The +{{#concept "universal property" Disambiguation="of existential quantification"}} +of existential quantification states `∃ A P` is the least upper bound on the +predicate `P` in the locale of propositions. ```agda -intro-exists : - {l1 l2 : Level} {A : UU l1} (P : A → Prop l2) → - (x : A) → type-Prop (P x) → exists A P -intro-exists P x p = unit-trunc-Prop (pair x p) +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} + where + + ev-intro-exists : + {C : UU l3} → (exists-structure A B → C) → (x : A) → B x → C + ev-intro-exists H x p = H (intro-exists x p) -intro-∃ : - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (a : A) (b : B a) → - ∃ A B -intro-∃ a b = unit-trunc-Prop (pair a b) + elim-exists : + (Q : Prop l3) → + ((x : A) → B x → type-Prop Q) → (exists-structure A B → type-Prop Q) + elim-exists Q f = map-universal-property-trunc-Prop Q (ind-Σ f) + + abstract + is-equiv-ev-intro-exists : + (Q : Prop l3) → is-equiv (ev-intro-exists {type-Prop Q}) + is-equiv-ev-intro-exists Q = + is-equiv-has-converse + ( function-Prop (exists-structure A B) Q) + ( Π-Prop A (λ x → function-Prop (B x) Q)) + ( elim-exists Q) ``` -### The elimination rule and the universal property of existential quantification +### The existential quantification satisfies the universal property of existential quantification ```agda -ev-intro-exists-Prop : - {l1 l2 l3 : Level} {A : UU l1} (P : A → Prop l2) (Q : Prop l3) → - type-hom-Prop (exists-Prop A P) Q → (x : A) → type-hom-Prop (P x) Q -ev-intro-exists-Prop P Q H x p = H (intro-exists P x p) - -elim-exists-Prop : - {l1 l2 l3 : Level} {A : UU l1} (P : A → Prop l2) (Q : Prop l3) → - ((x : A) → type-hom-Prop (P x) Q) → type-hom-Prop (exists-Prop A P) Q -elim-exists-Prop P Q f = - map-universal-property-trunc-Prop Q (ind-Σ f) - -abstract - is-equiv-ev-intro-exists-Prop : - {l1 l2 l3 : Level} (A : UU l1) (P : A → Prop l2) (Q : Prop l3) → - is-equiv (ev-intro-exists-Prop P Q) - is-equiv-ev-intro-exists-Prop A P Q = - is-equiv-is-prop - ( is-prop-type-hom-Prop (exists-Prop A P) Q) - ( is-prop-Π ((λ x → is-prop-type-hom-Prop (P x) Q))) - ( elim-exists-Prop P Q) - -is-least-upper-bound-exists-Prop : - {l1 l2 l3 : Level} {A : UU l1} (P : A → Prop l2) (Q : Prop l3) → - ((a : A) → type-hom-Prop (P a) Q) ↔ type-hom-Prop (exists-Prop A P) Q -pr1 (is-least-upper-bound-exists-Prop P Q) = elim-exists-Prop P Q -pr2 (is-least-upper-bound-exists-Prop P Q) h a p = h (intro-∃ a p) -``` - -### Conjunction distributes over existential quatification +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + up-exists : + universal-property-exists-structure A B (exists-structure-Prop A B) + up-exists Q = (ev-intro-exists , elim-exists Q) +``` + +### Propositions that satisfy the universal property of a existential quantification are equivalent to the existential quantification ```agda module _ - {l1 l2 l3 : Level} (P : Prop l1) {A : UU l2} (Q : A → Prop l3) + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} (Q : Prop l3) + (up-Q : universal-property-exists-structure A B Q) + where + + forward-implication-iff-universal-property-exists : + exists-structure A B → type-Prop Q + forward-implication-iff-universal-property-exists = + elim-exists Q (forward-implication (up-Q Q) id) + + backward-implication-iff-universal-property-exists : + type-Prop Q → exists-structure A B + backward-implication-iff-universal-property-exists = + backward-implication (up-Q (exists-structure-Prop A B)) intro-exists + + iff-universal-property-exists : + exists-structure A B ↔ type-Prop Q + iff-universal-property-exists = + ( forward-implication-iff-universal-property-exists , + backward-implication-iff-universal-property-exists) +``` + +### Existential quantification of structure is the same as existential quantification over its propositional reflection + +We proceed by showing that the latter satisfies the universal property of the +former. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + universal-property-exists-structure-exists-trunc-Prop : + universal-property-exists-structure A B (exists-Prop A (trunc-Prop ∘ B)) + universal-property-exists-structure-exists-trunc-Prop Q = + ( λ f a b → f (unit-trunc-Prop (a , unit-trunc-Prop b))) , + ( λ f → rec-trunc-Prop Q (λ (a , |b|) → rec-trunc-Prop Q (f a) |b|)) + + compute-exists-trunc-Prop : exists-structure A B ↔ exists A (trunc-Prop ∘ B) + compute-exists-trunc-Prop = + iff-universal-property-exists + ( exists-Prop A (trunc-Prop ∘ B)) + ( universal-property-exists-structure-exists-trunc-Prop) +``` + +### Taking the cartesian product with a proposition distributes over existential quantification of structures + +```agda +module _ + {l1 l2 l3 : Level} (P : Prop l1) {A : UU l2} {B : A → UU l3} where - iff-distributive-conjunction-exists-Prop : - ( conjunction-Prop P (exists-Prop A Q)) ⇔ - ( exists-Prop A (λ a → conjunction-Prop P (Q a))) - pr1 iff-distributive-conjunction-exists-Prop (p , e) = - elim-exists-Prop Q - ( exists-Prop A (λ a → conjunction-Prop P (Q a))) - ( λ x q → intro-∃ x (p , q)) + map-distributive-product-exists-structure : + type-Prop P × exists-structure A B → + exists-structure A (λ x → type-Prop P × B x) + map-distributive-product-exists-structure (p , e) = + elim-exists + ( exists-structure-Prop A (λ x → type-Prop P × B x)) + ( λ x q → intro-exists x (p , q)) ( e) - pr2 iff-distributive-conjunction-exists-Prop = - elim-exists-Prop - ( λ x → conjunction-Prop P (Q x)) - ( conjunction-Prop P (exists-Prop A Q)) - ( λ x (p , q) → (p , intro-∃ x q)) - - distributive-conjunction-exists-Prop : - conjunction-Prop P (exists-Prop A Q) ＝ - exists-Prop A (λ a → conjunction-Prop P (Q a)) - distributive-conjunction-exists-Prop = + + map-inv-distributive-product-exists-structure : + exists-structure A (λ x → type-Prop P × B x) → + type-Prop P × exists-structure A B + map-inv-distributive-product-exists-structure = + elim-exists + ( P ∧ exists-structure-Prop A B) + ( λ x (p , q) → (p , intro-exists x q)) + + iff-distributive-product-exists-structure : + ( type-Prop P × exists-structure A B) ↔ + ( exists-structure A (λ x → type-Prop P × B x)) + iff-distributive-product-exists-structure = + ( map-distributive-product-exists-structure , + map-inv-distributive-product-exists-structure) + + eq-distributive-product-exists-structure : + P ∧ exists-structure-Prop A B ＝ + exists-structure-Prop A (λ x → type-Prop P × B x) + eq-distributive-product-exists-structure = eq-iff' - ( conjunction-Prop P (exists-Prop A Q)) - ( exists-Prop A (λ a → conjunction-Prop P (Q a))) - ( iff-distributive-conjunction-exists-Prop) + ( P ∧ exists-structure-Prop A B) + ( exists-structure-Prop A (λ x → type-Prop P × B x)) + ( iff-distributive-product-exists-structure) ``` + +### Conjunction distributes over existential quantification + +```agda +module _ + {l1 l2 l3 : Level} (P : Prop l1) {A : UU l2} (Q : A → Prop l3) + where + + map-distributive-conjunction-exists : + type-Prop (P ∧ (∃ A Q) ⇒ ∃ A (λ x → P ∧ Q x)) + map-distributive-conjunction-exists = + map-distributive-product-exists-structure P + + map-inv-distributive-conjunction-exists : + type-Prop (∃ A (λ x → P ∧ Q x) ⇒ P ∧ (∃ A Q)) + map-inv-distributive-conjunction-exists = + map-inv-distributive-product-exists-structure P + + iff-distributive-conjunction-exists : + type-Prop (P ∧ ∃ A Q ⇔ ∃ A (λ x → P ∧ Q x)) + iff-distributive-conjunction-exists = + iff-distributive-product-exists-structure P + + eq-distributive-conjunction-exists : + P ∧ (∃ A Q) ＝ ∃ A (λ x → P ∧ Q x) + eq-distributive-conjunction-exists = + eq-distributive-product-exists-structure P +``` + +## See also + +- Existential quantification is the indexed counterpart to + [disjunction](foundation.disjunction.md) + +## Table of files about propositional logic + +The following table gives an overview of basic constructions in propositional +logic and related considerations. + +{{#include tables/propositional-logic.md}} + +## External links + +- [existential quantifier](https://ncatlab.org/nlab/show/existential+quantifier) + at $n$Lab diff --git a/src/foundation/fibered-equivalences.lagda.md b/src/foundation/fibered-equivalences.lagda.md index 9b4a35d2c4..2b90a61064 100644 --- a/src/foundation/fibered-equivalences.lagda.md +++ b/src/foundation/fibered-equivalences.lagda.md @@ -11,6 +11,7 @@ open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalences open import foundation.fibered-maps +open import foundation.logical-equivalences open import foundation.pullbacks open import foundation.slice open import foundation.universe-levels @@ -333,7 +334,7 @@ module _ is-pullback (pr1 ihH) g (cone-fibered-map f g ihH) ≃ is-fibered-equiv-fibered-map f g ihH equiv-is-fibered-equiv-is-pullback is-equiv-i = - equiv-prop + equiv-iff-is-prop ( is-prop-is-pullback (pr1 ihH) g (cone-fibered-map f g ihH)) ( is-prop-is-fibered-equiv-fibered-map f g ihH) ( is-fibered-equiv-is-pullback is-equiv-i) diff --git a/src/foundation/functoriality-propositional-truncation.lagda.md b/src/foundation/functoriality-propositional-truncation.lagda.md index 8df3a04a2a..1c895736c6 100644 --- a/src/foundation/functoriality-propositional-truncation.lagda.md +++ b/src/foundation/functoriality-propositional-truncation.lagda.md @@ -10,6 +10,7 @@ module foundation.functoriality-propositional-truncation where open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.logical-equivalences open import foundation.propositional-truncations open import foundation.universe-levels open import foundation.whiskering-homotopies-composition @@ -116,7 +117,7 @@ abstract (A ≃ B) → (type-trunc-Prop A ≃ type-trunc-Prop B) pr1 (equiv-trunc-Prop e) = map-equiv-trunc-Prop e pr2 (equiv-trunc-Prop e) = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-type-trunc-Prop) ( is-prop-type-trunc-Prop) ( map-inv-equiv-trunc-Prop e) diff --git a/src/foundation/functoriality-set-truncation.lagda.md b/src/foundation/functoriality-set-truncation.lagda.md index 136c3f4933..c0539827f8 100644 --- a/src/foundation/functoriality-set-truncation.lagda.md +++ b/src/foundation/functoriality-set-truncation.lagda.md @@ -11,8 +11,10 @@ open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.functoriality-truncation open import foundation.images +open import foundation.injective-maps open import foundation.propositional-truncations open import foundation.set-truncations +open import foundation.sets open import foundation.slice open import foundation.surjective-maps open import foundation.uniqueness-image @@ -30,9 +32,7 @@ open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.injective-maps open import foundation-core.propositions -open import foundation-core.sets open import foundation-core.truncation-levels ``` diff --git a/src/foundation/homotopy-preorder-of-types.lagda.md b/src/foundation/homotopy-preorder-of-types.lagda.md new file mode 100644 index 0000000000..c6c1e1902e --- /dev/null +++ b/src/foundation/homotopy-preorder-of-types.lagda.md @@ -0,0 +1,60 @@ +# The homotopy preorder of types + +```agda +module + foundation.homotopy-preorder-of-types + where +``` + +
Imports + +```agda +open import foundation.dependent-pair-types +open import foundation.empty-types +open import foundation.identity-types +open import foundation.mere-functions +open import foundation.propositional-truncations +open import foundation.propositions +open import foundation.sets +open import foundation.universe-levels + +open import order-theory.large-preorders +open import order-theory.posets +open import order-theory.preorders +``` + +
+ +## Idea + +The {{#concept "homotopy preorder of types" Agda=Homotopy-Type-Large-Preorder}} +is the [(large) preorder](order-theory.large-preorders.md) whose objects are +types, and whose ordering relation is defined by +[mere functions](foundation.mere-functions.md), i.e. by the +[propositional truncation](foundation.propositional-truncations.md) of the +function types: + +```text + A ≤ B := ║(A → B)║₋₁. +``` + +## Definitions + +### The large homotopy preorder of types + +```agda +Homotopy-Type-Large-Preorder : Large-Preorder lsuc (_⊔_) +Homotopy-Type-Large-Preorder = + λ where + .type-Large-Preorder l → UU l + .leq-prop-Large-Preorder → prop-mere-function + .refl-leq-Large-Preorder → refl-mere-function + .transitive-leq-Large-Preorder X Y Z → transitive-mere-function +``` + +### The small homotopy preorder of types + +```agda +Homotopy-Type-Preorder : (l : Level) → Preorder (lsuc l) l +Homotopy-Type-Preorder = preorder-Large-Preorder Homotopy-Type-Large-Preorder +``` diff --git a/src/foundation/impredicative-encodings.lagda.md b/src/foundation/impredicative-encodings.lagda.md index 274f2edc0e..dbf1136729 100644 --- a/src/foundation/impredicative-encodings.lagda.md +++ b/src/foundation/impredicative-encodings.lagda.md @@ -10,15 +10,18 @@ module foundation.impredicative-encodings where open import foundation.conjunction open import foundation.dependent-pair-types open import foundation.disjunction +open import foundation.empty-types open import foundation.existential-quantification +open import foundation.homotopies open import foundation.logical-equivalences open import foundation.negation open import foundation.propositional-truncations +open import foundation.universal-quantification open import foundation.universe-levels open import foundation-core.coproduct-types -open import foundation-core.empty-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.sets @@ -28,278 +31,278 @@ open import foundation-core.sets ## Idea -By quantifying over all propositions in a universe, we can define all the -logical operations. The idea is to use the fact that any proposition `P` is -equivalent to the proposition `(Q : Prop l) → (P ⇒ Q) ⇒ Q`, which can be thought -of as the least proposition `Q` containing `P`. +By universally quantifying over all +[propositions](foundation-core.propositions.md) in a universe, we can define all +the logical operations. The idea is to use the fact that any proposition `P` is +[equivalent](foundation.logical-equivalences.md) to the proposition +`(Q : Prop l) → (P ⇒ Q) ⇒ Q`, which can be thought of as the least proposition +`Q` containing `P`. ### The impredicative encoding of the propositional truncation ```agda -impredicative-trunc-Prop : - {l : Level} → UU l → Prop (lsuc l) -impredicative-trunc-Prop {l} A = - Π-Prop - ( Prop l) - ( λ Q → function-Prop (A → type-Prop Q) Q) - -type-impredicative-trunc-Prop : - {l : Level} → UU l → UU (lsuc l) -type-impredicative-trunc-Prop {l} A = - type-Prop (impredicative-trunc-Prop A) - -map-impredicative-trunc-Prop : - {l : Level} (A : UU l) → - type-trunc-Prop A → type-impredicative-trunc-Prop A -map-impredicative-trunc-Prop {l} A = - map-universal-property-trunc-Prop - ( impredicative-trunc-Prop A) - ( λ x Q f → f x) - -map-inv-impredicative-trunc-Prop : - {l : Level} (A : UU l) → - type-impredicative-trunc-Prop A → type-trunc-Prop A -map-inv-impredicative-trunc-Prop A H = - H (trunc-Prop A) unit-trunc-Prop - -equiv-impredicative-trunc-Prop : - {l : Level} (A : UU l) → - type-trunc-Prop A ≃ type-impredicative-trunc-Prop A -equiv-impredicative-trunc-Prop A = - equiv-iff - ( trunc-Prop A) - ( impredicative-trunc-Prop A) - ( map-impredicative-trunc-Prop A) - ( map-inv-impredicative-trunc-Prop A) +module _ + {l : Level} (A : UU l) + where + + impredicative-trunc-Prop : Prop (lsuc l) + impredicative-trunc-Prop = + ∀' (Prop l) (λ Q → function-Prop (A → type-Prop Q) Q) + + type-impredicative-trunc-Prop : UU (lsuc l) + type-impredicative-trunc-Prop = + type-Prop impredicative-trunc-Prop + + map-impredicative-trunc-Prop : + type-trunc-Prop A → type-impredicative-trunc-Prop + map-impredicative-trunc-Prop = + map-universal-property-trunc-Prop + ( impredicative-trunc-Prop) + ( λ x Q f → f x) + + map-inv-impredicative-trunc-Prop : + type-impredicative-trunc-Prop → type-trunc-Prop A + map-inv-impredicative-trunc-Prop H = + H (trunc-Prop A) unit-trunc-Prop + + equiv-impredicative-trunc-Prop : + type-trunc-Prop A ≃ type-impredicative-trunc-Prop + equiv-impredicative-trunc-Prop = + equiv-iff + ( trunc-Prop A) + ( impredicative-trunc-Prop) + ( map-impredicative-trunc-Prop) + ( map-inv-impredicative-trunc-Prop) ``` ### The impredicative encoding of conjunction ```agda -impredicative-conjunction-Prop : - {l1 l2 : Level} → Prop l1 → Prop l2 → Prop (lsuc (l1 ⊔ l2)) -impredicative-conjunction-Prop {l1} {l2} P1 P2 = - Π-Prop - ( Prop (l1 ⊔ l2)) - ( λ Q → function-Prop (type-Prop P1 → (type-Prop P2 → type-Prop Q)) Q) - -type-impredicative-conjunction-Prop : - {l1 l2 : Level} → Prop l1 → Prop l2 → UU (lsuc (l1 ⊔ l2)) -type-impredicative-conjunction-Prop P1 P2 = - type-Prop (impredicative-conjunction-Prop P1 P2) - -map-impredicative-conjunction-Prop : - {l1 l2 : Level} (P1 : Prop l1) (P2 : Prop l2) → - type-conjunction-Prop P1 P2 → type-impredicative-conjunction-Prop P1 P2 -map-impredicative-conjunction-Prop {l1} {l2} P1 P2 (pair p1 p2) Q f = - f p1 p2 - -map-inv-impredicative-conjunction-Prop : - {l1 l2 : Level} (P1 : Prop l1) (P2 : Prop l2) → - type-impredicative-conjunction-Prop P1 P2 → type-conjunction-Prop P1 P2 -map-inv-impredicative-conjunction-Prop P1 P2 H = - H (conjunction-Prop P1 P2) (λ p1 p2 → pair p1 p2) - -equiv-impredicative-conjunction-Prop : - {l1 l2 : Level} (P1 : Prop l1) (P2 : Prop l2) → - type-conjunction-Prop P1 P2 ≃ type-impredicative-conjunction-Prop P1 P2 -equiv-impredicative-conjunction-Prop P1 P2 = - equiv-iff - ( conjunction-Prop P1 P2) - ( impredicative-conjunction-Prop P1 P2) - ( map-impredicative-conjunction-Prop P1 P2) - ( map-inv-impredicative-conjunction-Prop P1 P2) +module _ + {l1 l2 : Level} (P1 : Prop l1) (P2 : Prop l2) + where + + impredicative-conjunction-Prop : Prop (lsuc (l1 ⊔ l2)) + impredicative-conjunction-Prop = + ∀' (Prop (l1 ⊔ l2)) (λ Q → (P1 ⇒ P2 ⇒ Q) ⇒ Q) + + type-impredicative-conjunction-Prop : UU (lsuc (l1 ⊔ l2)) + type-impredicative-conjunction-Prop = + type-Prop impredicative-conjunction-Prop + + map-product-impredicative-conjunction-Prop : + type-conjunction-Prop P1 P2 → type-impredicative-conjunction-Prop + map-product-impredicative-conjunction-Prop (p1 , p2) Q f = f p1 p2 + + map-inv-product-impredicative-conjunction-Prop : + type-impredicative-conjunction-Prop → type-conjunction-Prop P1 P2 + map-inv-product-impredicative-conjunction-Prop H = H (P1 ∧ P2) pair + + equiv-product-impredicative-conjunction-Prop : + type-conjunction-Prop P1 P2 ≃ type-impredicative-conjunction-Prop + equiv-product-impredicative-conjunction-Prop = + equiv-iff + ( P1 ∧ P2) + ( impredicative-conjunction-Prop) + ( map-product-impredicative-conjunction-Prop) + ( map-inv-product-impredicative-conjunction-Prop) ``` ### The impredicative encoding of disjunction ```agda -impredicative-disjunction-Prop : - {l1 l2 : Level} → Prop l1 → Prop l2 → Prop (lsuc (l1 ⊔ l2)) -impredicative-disjunction-Prop {l1} {l2} P1 P2 = - Π-Prop - ( Prop (l1 ⊔ l2)) - ( λ Q → - function-Prop - ( type-implication-Prop P1 Q) - ( function-Prop (type-implication-Prop P2 Q) Q)) - -type-impredicative-disjunction-Prop : - {l1 l2 : Level} → Prop l1 → Prop l2 → UU (lsuc (l1 ⊔ l2)) -type-impredicative-disjunction-Prop P1 P2 = - type-Prop (impredicative-disjunction-Prop P1 P2) - -map-impredicative-disjunction-Prop : - {l1 l2 : Level} (P1 : Prop l1) (P2 : Prop l2) → - type-disjunction-Prop P1 P2 → type-impredicative-disjunction-Prop P1 P2 -map-impredicative-disjunction-Prop {l1} {l2} P1 P2 = - map-universal-property-trunc-Prop - ( impredicative-disjunction-Prop P1 P2) - ( rec-coproduct - ( λ x Q f1 f2 → f1 x) - ( λ y Q f1 f2 → f2 y)) - -map-inv-impredicative-disjunction-Prop : - {l1 l2 : Level} (P1 : Prop l1) (P2 : Prop l2) → - type-impredicative-disjunction-Prop P1 P2 → type-disjunction-Prop P1 P2 -map-inv-impredicative-disjunction-Prop P1 P2 H = - H ( disjunction-Prop P1 P2) - ( inl-disjunction-Prop P1 P2) - ( inr-disjunction-Prop P1 P2) - -equiv-impredicative-disjunction-Prop : - {l1 l2 : Level} (P1 : Prop l1) (P2 : Prop l2) → - type-disjunction-Prop P1 P2 ≃ type-impredicative-disjunction-Prop P1 P2 -equiv-impredicative-disjunction-Prop P1 P2 = - equiv-iff - ( disjunction-Prop P1 P2) - ( impredicative-disjunction-Prop P1 P2) - ( map-impredicative-disjunction-Prop P1 P2) - ( map-inv-impredicative-disjunction-Prop P1 P2) +module _ + {l1 l2 : Level} (P1 : Prop l1) (P2 : Prop l2) + where + + impredicative-disjunction-Prop : Prop (lsuc (l1 ⊔ l2)) + impredicative-disjunction-Prop = + ∀' (Prop (l1 ⊔ l2)) (λ Q → (P1 ⇒ Q) ⇒ (P2 ⇒ Q) ⇒ Q) + + type-impredicative-disjunction-Prop : UU (lsuc (l1 ⊔ l2)) + type-impredicative-disjunction-Prop = + type-Prop impredicative-disjunction-Prop + + map-impredicative-disjunction-Prop : + type-disjunction-Prop P1 P2 → type-impredicative-disjunction-Prop + map-impredicative-disjunction-Prop = + map-universal-property-trunc-Prop + ( impredicative-disjunction-Prop) + ( rec-coproduct (λ x Q f1 f2 → f1 x) (λ y Q f1 f2 → f2 y)) + + map-inv-impredicative-disjunction-Prop : + type-impredicative-disjunction-Prop → type-disjunction-Prop P1 P2 + map-inv-impredicative-disjunction-Prop H = + H (P1 ∨ P2) (inl-disjunction) (inr-disjunction) + + equiv-impredicative-disjunction-Prop : + type-disjunction-Prop P1 P2 ≃ type-impredicative-disjunction-Prop + equiv-impredicative-disjunction-Prop = + equiv-iff + ( P1 ∨ P2) + ( impredicative-disjunction-Prop) + ( map-impredicative-disjunction-Prop) + ( map-inv-impredicative-disjunction-Prop) +``` + +### The impredicative encoding of the empty type + +```agda +impredicative-empty-Prop : Prop (lsuc lzero) +impredicative-empty-Prop = ∀' (Prop lzero) (λ P → P) + +type-impredicative-empty-Prop : UU (lsuc lzero) +type-impredicative-empty-Prop = type-Prop impredicative-empty-Prop + +is-empty-impredicative-empty-Prop : + is-empty type-impredicative-empty-Prop +is-empty-impredicative-empty-Prop e = e empty-Prop + +equiv-impredicative-empty-Prop : + empty ≃ type-impredicative-empty-Prop +equiv-impredicative-empty-Prop = + equiv-is-empty id is-empty-impredicative-empty-Prop ``` ### The impredicative encoding of negation ```agda -impredicative-neg-Prop : - {l : Level} → UU l → Prop (lsuc l) -impredicative-neg-Prop {l} A = - Π-Prop (Prop l) (λ Q → function-Prop A Q) - -type-impredicative-neg-Prop : - {l : Level} → UU l → UU (lsuc l) -type-impredicative-neg-Prop A = - type-Prop (impredicative-neg-Prop A) - -map-impredicative-neg-Prop : - {l : Level} (A : UU l) → - ¬ A → type-impredicative-neg-Prop A -map-impredicative-neg-Prop A f Q a = ex-falso (f a) - -map-inv-impredicative-neg-Prop : - {l : Level} (A : UU l) → - type-impredicative-neg-Prop A → ¬ A -map-inv-impredicative-neg-Prop A H a = H (neg-Prop' A) a a - -equiv-impredicative-neg-Prop : - {l : Level} (A : UU l) → - ¬ A ≃ type-impredicative-neg-Prop A -equiv-impredicative-neg-Prop A = - equiv-iff - ( neg-Prop' A) - ( impredicative-neg-Prop A) - ( map-impredicative-neg-Prop A) - ( map-inv-impredicative-neg-Prop A) +module _ + {l : Level} (A : UU l) + where + + impredicative-neg-Prop : Prop (lsuc l) + impredicative-neg-Prop = + Π-Prop (Prop l) (λ Q → function-Prop A Q) + + type-impredicative-neg-Prop : UU (lsuc l) + type-impredicative-neg-Prop = type-Prop impredicative-neg-Prop + + map-impredicative-neg-Prop : ¬ A → type-impredicative-neg-Prop + map-impredicative-neg-Prop f Q a = ex-falso (f a) + + map-inv-impredicative-neg-Prop : type-impredicative-neg-Prop → ¬ A + map-inv-impredicative-neg-Prop H a = H (neg-type-Prop A) a a + + equiv-impredicative-neg-Prop : ¬ A ≃ type-impredicative-neg-Prop + equiv-impredicative-neg-Prop = + equiv-iff + ( neg-type-Prop A) + ( impredicative-neg-Prop) + ( map-impredicative-neg-Prop) + ( map-inv-impredicative-neg-Prop) ``` ### The impredicative encoding of existential quantification ```agda -impredicative-exists-Prop : - {l1 l2 : Level} {A : UU l1} (P : A → Prop l2) → Prop (lsuc (l1 ⊔ l2)) -impredicative-exists-Prop {l1} {l2} {A} P = - Π-Prop - ( Prop (l1 ⊔ l2)) - ( λ Q → function-Prop ((x : A) → type-Prop (P x) → type-Prop Q) Q) - -type-impredicative-exists-Prop : - {l1 l2 : Level} {A : UU l1} (P : A → Prop l2) → UU (lsuc (l1 ⊔ l2)) -type-impredicative-exists-Prop P = - type-Prop (impredicative-exists-Prop P) - -map-impredicative-exists-Prop : - {l1 l2 : Level} {A : UU l1} (P : A → Prop l2) → - exists A P → type-impredicative-exists-Prop P -map-impredicative-exists-Prop {l1} {l2} {A} P = - map-universal-property-trunc-Prop - ( impredicative-exists-Prop P) - ( ind-Σ (λ x y Q h → h x y)) - -map-inv-impredicative-exists-Prop : - {l1 l2 : Level} {A : UU l1} (P : A → Prop l2) → - type-impredicative-exists-Prop P → exists A P -map-inv-impredicative-exists-Prop {A = A} P H = - H ( exists-Prop A P) - ( λ x y → unit-trunc-Prop (pair x y)) - -equiv-impredicative-exists-Prop : - {l1 l2 : Level} {A : UU l1} (P : A → Prop l2) → - exists A P ≃ type-impredicative-exists-Prop P -equiv-impredicative-exists-Prop {A = A} P = - equiv-iff - ( exists-Prop A P) - ( impredicative-exists-Prop P) - ( map-impredicative-exists-Prop P) - ( map-inv-impredicative-exists-Prop P) +module _ + {l1 l2 : Level} {A : UU l1} (P : A → Prop l2) + where + + impredicative-exists-Prop : Prop (lsuc (l1 ⊔ l2)) + impredicative-exists-Prop = + ∀' (Prop (l1 ⊔ l2)) (λ Q → (∀' A (λ x → P x ⇒ Q)) ⇒ Q) + + type-impredicative-exists-Prop : UU (lsuc (l1 ⊔ l2)) + type-impredicative-exists-Prop = + type-Prop impredicative-exists-Prop + + map-impredicative-exists-Prop : + exists A P → type-impredicative-exists-Prop + map-impredicative-exists-Prop = + map-universal-property-trunc-Prop + ( impredicative-exists-Prop) + ( ind-Σ (λ x y Q h → h x y)) + + map-inv-impredicative-exists-Prop : + type-impredicative-exists-Prop → exists A P + map-inv-impredicative-exists-Prop H = + H (∃ A P) (λ x y → unit-trunc-Prop (x , y)) + + equiv-impredicative-exists-Prop : + exists A P ≃ type-impredicative-exists-Prop + equiv-impredicative-exists-Prop = + equiv-iff + ( ∃ A P) + ( impredicative-exists-Prop) + ( map-impredicative-exists-Prop) + ( map-inv-impredicative-exists-Prop) ``` ### The impredicative encoding of the based identity type of a set ```agda -impredicative-based-id-Prop : - {l : Level} (A : Set l) (a x : type-Set A) → Prop (lsuc l) -impredicative-based-id-Prop {l} A a x = - Π-Prop (type-Set A → Prop l) (λ Q → hom-Prop (Q a) (Q x)) - -type-impredicative-based-id-Prop : - {l : Level} (A : Set l) (a x : type-Set A) → UU (lsuc l) -type-impredicative-based-id-Prop A a x = - type-Prop (impredicative-based-id-Prop A a x) - -map-impredicative-based-id-Prop : - {l : Level} (A : Set l) (a x : type-Set A) → - a ＝ x → type-impredicative-based-id-Prop A a x -map-impredicative-based-id-Prop A a .a refl Q p = p - -map-inv-impredicative-based-id-Prop : - {l : Level} (A : Set l) (a x : type-Set A) → - type-impredicative-based-id-Prop A a x → a ＝ x -map-inv-impredicative-based-id-Prop A a x H = - H (λ x → pair (a ＝ x) (is-set-type-Set A a x)) refl - -equiv-impredicative-based-id-Prop : - {l : Level} (A : Set l) (a x : type-Set A) → - (a ＝ x) ≃ type-impredicative-based-id-Prop A a x -equiv-impredicative-based-id-Prop A a x = - equiv-iff - ( pair (a ＝ x) (is-set-type-Set A a x)) - ( impredicative-based-id-Prop A a x) - ( map-impredicative-based-id-Prop A a x) - ( map-inv-impredicative-based-id-Prop A a x) +module _ + {l : Level} (A : Set l) (a x : type-Set A) + where + + impredicative-based-id-Prop : Prop (lsuc l) + impredicative-based-id-Prop = ∀' (type-Set A → Prop l) (λ Q → Q a ⇒ Q x) + + type-impredicative-based-id-Prop : UU (lsuc l) + type-impredicative-based-id-Prop = type-Prop impredicative-based-id-Prop + + map-impredicative-based-id-Prop : + a ＝ x → type-impredicative-based-id-Prop + map-impredicative-based-id-Prop refl Q p = p + + map-inv-impredicative-based-id-Prop : + type-impredicative-based-id-Prop → a ＝ x + map-inv-impredicative-based-id-Prop H = H (Id-Prop A a) refl + + equiv-impredicative-based-id-Prop : + (a ＝ x) ≃ type-impredicative-based-id-Prop + equiv-impredicative-based-id-Prop = + equiv-iff + ( Id-Prop A a x) + ( impredicative-based-id-Prop) + ( map-impredicative-based-id-Prop) + ( map-inv-impredicative-based-id-Prop) ``` ### The impredicative encoding of Martin-Löf's identity type of a set ```agda -impredicative-id-Prop : - {l : Level} (A : Set l) (x y : type-Set A) → Prop (lsuc l) -impredicative-id-Prop {l} A x y = - Π-Prop (type-Set A → type-Set A → Prop l) - (λ Q → function-Prop ((a : type-Set A) → type-Prop (Q a a)) (Q x y)) - -type-impredicative-id-Prop : - {l : Level} (A : Set l) (x y : type-Set A) → UU (lsuc l) -type-impredicative-id-Prop A x y = - type-Prop (impredicative-id-Prop A x y) - -map-impredicative-id-Prop : - {l : Level} (A : Set l) (x y : type-Set A) → - x ＝ y → type-impredicative-id-Prop A x y -map-impredicative-id-Prop A x .x refl Q r = r x - -map-inv-impredicative-id-Prop : - {l : Level} (A : Set l) (x y : type-Set A) → - type-impredicative-id-Prop A x y → x ＝ y -map-inv-impredicative-id-Prop A x y H = - H (λ a b → pair (a ＝ b) (is-set-type-Set A a b)) (λ a → refl) - -equiv-impredicative-id-Prop : - {l : Level} (A : Set l) (x y : type-Set A) → - (x ＝ y) ≃ type-impredicative-id-Prop A x y -equiv-impredicative-id-Prop A x y = - equiv-iff - ( pair (x ＝ y) (is-set-type-Set A x y)) - ( impredicative-id-Prop A x y) - ( map-impredicative-id-Prop A x y) - ( map-inv-impredicative-id-Prop A x y) +module _ + {l : Level} (A : Set l) (x y : type-Set A) + where + + impredicative-id-Prop : Prop (lsuc l) + impredicative-id-Prop = + ∀' + ( type-Set A → type-Set A → Prop l) + ( λ Q → (∀' (type-Set A) (λ a → Q a a)) ⇒ Q x y) + + type-impredicative-id-Prop : UU (lsuc l) + type-impredicative-id-Prop = type-Prop impredicative-id-Prop + + map-impredicative-id-Prop : + x ＝ y → type-impredicative-id-Prop + map-impredicative-id-Prop refl Q r = r x + + map-inv-impredicative-id-Prop : + type-impredicative-id-Prop → x ＝ y + map-inv-impredicative-id-Prop H = H (Id-Prop A) (refl-htpy) + + equiv-impredicative-id-Prop : + (x ＝ y) ≃ type-impredicative-id-Prop + equiv-impredicative-id-Prop = + equiv-iff + ( Id-Prop A x y) + ( impredicative-id-Prop) + ( map-impredicative-id-Prop) + ( map-inv-impredicative-id-Prop) ``` + +## Table of files about propositional logic + +The following table gives an overview of basic constructions in propositional +logic and related considerations. + +{{#include tables/propositional-logic.md}} + +## External links + +- [Constructing coproduct types and boolean types from universes](https://mathoverflow.net/questions/457904/constructing-coproduct-types-and-boolean-types-from-universes) + at mathoverflow diff --git a/src/foundation/impredicative-universes.lagda.md b/src/foundation/impredicative-universes.lagda.md index 8699e535dc..c1f2ed3f59 100644 --- a/src/foundation/impredicative-universes.lagda.md +++ b/src/foundation/impredicative-universes.lagda.md @@ -17,7 +17,8 @@ open import foundation-core.small-types ## Idea -A universe `U` is impredicative if the type of propositions in `U` is `U`-small. +A universe `𝒰` is {{#concept "impredicative"}} if the type of +[propositions](foundation-core.propositions.md) in `𝒰` is `𝒰`-small. ## Definition @@ -25,3 +26,7 @@ A universe `U` is impredicative if the type of propositions in `U` is `U`-small. is-impredicative-UU : (l : Level) → UU (lsuc l) is-impredicative-UU l = is-small l (Prop l) ``` + +## See also + +- [Impredicative encodings of the logical operations](foundation.impredicative-encodings.md) diff --git a/src/foundation/inhabited-types.lagda.md b/src/foundation/inhabited-types.lagda.md index 6360251748..874d306572 100644 --- a/src/foundation/inhabited-types.lagda.md +++ b/src/foundation/inhabited-types.lagda.md @@ -46,7 +46,7 @@ is-inhabited-Prop X = trunc-Prop X is-inhabited : {l : Level} → UU l → UU l is-inhabited X = type-Prop (is-inhabited-Prop X) -is-property-is-inhabited : {l : Level} → (X : UU l) → is-prop (is-inhabited X) +is-property-is-inhabited : {l : Level} (X : UU l) → is-prop (is-inhabited X) is-property-is-inhabited X = is-prop-type-Prop (is-inhabited-Prop X) Inhabited-Type : (l : Level) → UU (lsuc l) diff --git a/src/foundation/injective-maps.lagda.md b/src/foundation/injective-maps.lagda.md index 5eaf79f193..38991f22b2 100644 --- a/src/foundation/injective-maps.lagda.md +++ b/src/foundation/injective-maps.lagda.md @@ -9,10 +9,17 @@ open import foundation-core.injective-maps public
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.logical-equivalences open import foundation.universe-levels +open import foundation-core.embeddings open import foundation-core.empty-types +open import foundation-core.identity-types open import foundation-core.negation +open import foundation-core.propositional-maps +open import foundation-core.propositions +open import foundation-core.sets ```
@@ -49,3 +56,51 @@ is-injective-is-empty : is-empty A → is-injective f is-injective-is-empty f is-empty-A {x} = ex-falso (is-empty-A x) ``` + +### Any injective map between sets is an embedding + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + abstract + is-emb-is-injective' : + (is-set-A : is-set A) (is-set-B : is-set B) (f : A → B) → + is-injective f → is-emb f + is-emb-is-injective' is-set-A is-set-B f is-injective-f x y = + is-equiv-has-converse-is-prop + ( is-set-A x y) + ( is-set-B (f x) (f y)) + ( is-injective-f) + + is-emb-is-injective : + {f : A → B} → is-set B → is-injective f → is-emb f + is-emb-is-injective {f} H I = + is-emb-is-injective' (is-set-is-injective H I) H f I + + is-prop-map-is-injective : + {f : A → B} → is-set B → is-injective f → is-prop-map f + is-prop-map-is-injective {f} H I = + is-prop-map-is-emb (is-emb-is-injective H I) +``` + +### For a map between sets, being injective is a property + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-prop-is-injective : + is-set A → (f : A → B) → is-prop (is-injective f) + is-prop-is-injective H f = + is-prop-implicit-Π + ( λ x → + is-prop-implicit-Π + ( λ y → is-prop-function-type (H x y))) + + is-injective-Prop : is-set A → (A → B) → Prop (l1 ⊔ l2) + pr1 (is-injective-Prop H f) = is-injective f + pr2 (is-injective-Prop H f) = is-prop-is-injective H f +``` diff --git a/src/foundation/intersections-subtypes.lagda.md b/src/foundation/intersections-subtypes.lagda.md index 33b3583dbb..3ff9cc3f3b 100644 --- a/src/foundation/intersections-subtypes.lagda.md +++ b/src/foundation/intersections-subtypes.lagda.md @@ -14,6 +14,7 @@ open import foundation.large-locale-of-subtypes open import foundation.powersets open import foundation.universe-levels +open import foundation-core.decidable-propositions open import foundation-core.propositions open import foundation-core.subtypes diff --git a/src/foundation/isolated-elements.lagda.md b/src/foundation/isolated-elements.lagda.md index c42a2f870c..ebf38b94f7 100644 --- a/src/foundation/isolated-elements.lagda.md +++ b/src/foundation/isolated-elements.lagda.md @@ -17,9 +17,11 @@ open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.fundamental-theorem-of-identity-types open import foundation.identity-types +open import foundation.injective-maps open import foundation.maybe open import foundation.negated-equality open import foundation.negation +open import foundation.sets open import foundation.type-arithmetic-unit-type open import foundation.unit-type open import foundation.universe-levels @@ -32,9 +34,7 @@ open import foundation-core.equivalences open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies -open import foundation-core.injective-maps open import foundation-core.propositions -open import foundation-core.sets open import foundation-core.subtypes open import foundation-core.torsorial-type-families open import foundation-core.transport-along-identifications @@ -214,7 +214,7 @@ module _ is-prop-is-isolated : {l1 : Level} {A : UU l1} (a : A) → is-prop (is-isolated a) is-prop-is-isolated a = - is-prop-is-inhabited + is-prop-has-element ( λ H → is-prop-Π (is-prop-is-decidable ∘ is-prop-eq-isolated-element a H)) is-isolated-Prop : @@ -257,7 +257,7 @@ module _ ( is-emb-inclusion-isolated-element A) ( is-emb-is-injective ( is-set-isolated-element A) - ( λ {star} {star} p → refl)) + ( λ p → refl)) emb-point-isolated-element : unit ↪ A pr1 emb-point-isolated-element = point-isolated-element diff --git a/src/foundation/large-locale-of-propositions.lagda.md b/src/foundation/large-locale-of-propositions.lagda.md index 233f214be3..816cd18c3d 100644 --- a/src/foundation/large-locale-of-propositions.lagda.md +++ b/src/foundation/large-locale-of-propositions.lagda.md @@ -9,8 +9,10 @@ module foundation.large-locale-of-propositions where ```agda open import foundation.conjunction open import foundation.existential-quantification +open import foundation.logical-equivalences open import foundation.propositional-extensionality open import foundation.unit-type +open import foundation.universal-property-cartesian-product-types open import foundation.universe-levels open import foundation-core.function-types @@ -30,8 +32,17 @@ open import order-theory.top-elements-large-posets ## Idea -The types of [propositions](foundation-core.propositions.md) `Prop l` combined -form a [large locale](order-theory.large-locales.md). +The [large locale](order-theory.large-locales.md) of +[propositions](foundation-core.propositions.md) consists of all the propositions +of any [universe level](foundation.universe-levels.md) and is ordered by the +implications between them. [Conjunction](foundation.conjunction.md) gives this +[large poset](order-theory.large-posets.md) the structure of a +[large meet-semilattice](order-theory.large-meet-semilattices.md), and +[existential quantification](foundation.existential-quantification.md) gives it +the structure of a [large suplattice](order-theory.large-suplattices.md). + +**Note.** The collection of all propositions is large because we do not assume +[propositional resizing](foundation.propositional-resizing.md). ## Definitions @@ -60,8 +71,8 @@ has-meets-Prop-Large-Locale : has-meets-Large-Poset Prop-Large-Poset meet-has-meets-Large-Poset has-meets-Prop-Large-Locale = conjunction-Prop is-greatest-binary-lower-bound-meet-has-meets-Large-Poset - has-meets-Prop-Large-Locale = - iff-universal-property-conjunction-Prop + has-meets-Prop-Large-Locale P Q R = + is-greatest-binary-lower-bound-conjunction-Prop P Q R ``` ### The largest element in the large poset of propositions @@ -96,10 +107,10 @@ is-large-suplattice-Prop-Large-Locale : is-large-suplattice-Large-Poset lzero Prop-Large-Poset sup-has-least-upper-bound-family-of-elements-Large-Poset ( is-large-suplattice-Prop-Large-Locale {I = I} P) = - exists-Prop I P + ∃ I P is-least-upper-bound-sup-has-least-upper-bound-family-of-elements-Large-Poset - ( is-large-suplattice-Prop-Large-Locale {I = I} P) = - is-least-upper-bound-exists-Prop P + ( is-large-suplattice-Prop-Large-Locale {I = I} P) R = + inv-iff (up-exists R) ``` ### The large frame of propositions @@ -113,7 +124,7 @@ is-large-meet-semilattice-Large-Frame Prop-Large-Frame = is-large-suplattice-Large-Frame Prop-Large-Frame = is-large-suplattice-Prop-Large-Locale distributive-meet-sup-Large-Frame Prop-Large-Frame = - distributive-conjunction-exists-Prop + eq-distributive-conjunction-exists ``` ### The large locale of propositions @@ -122,3 +133,7 @@ distributive-meet-sup-Large-Frame Prop-Large-Frame = Prop-Large-Locale : Large-Locale lsuc (_⊔_) lzero Prop-Large-Locale = Prop-Large-Frame ``` + +## See also + +- [Propositional resizing](foundation.propositional-resizing.md) diff --git a/src/foundation/law-of-excluded-middle.lagda.md b/src/foundation/law-of-excluded-middle.lagda.md index d28bb30393..d82d58a594 100644 --- a/src/foundation/law-of-excluded-middle.lagda.md +++ b/src/foundation/law-of-excluded-middle.lagda.md @@ -22,7 +22,7 @@ open import univalent-combinatorics.2-element-types ## Idea -The **law of excluded middle** asserts that any +The {{#concept "law of excluded middle" Agda=LEM}} asserts that any [proposition](foundation-core.propositions.md) `P` is [decidable](foundation.decidable-types.md). diff --git a/src/foundation/lawveres-fixed-point-theorem.lagda.md b/src/foundation/lawveres-fixed-point-theorem.lagda.md index 00323ca065..bbe7fbd83e 100644 --- a/src/foundation/lawveres-fixed-point-theorem.lagda.md +++ b/src/foundation/lawveres-fixed-point-theorem.lagda.md @@ -21,22 +21,26 @@ open import foundation-core.identity-types ## Idea -Lawvere's fixed point theorem asserts that if there is a surjective map +{{#concept "Lawvere's fixed point theorem" Agda=fixed-point-theorem-Lawvere}} +asserts that if there is a [surjective map](foundation.surjective-maps.md) `A → (A → B)`, then any map `B → B` must have a fixed point. ## Theorem ```agda -abstract - fixed-point-theorem-Lawvere : - {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → A → B} → - is-surjective f → (h : B → B) → ∃ B (λ b → h b ＝ b) - fixed-point-theorem-Lawvere {A = A} {B} {f} H h = - apply-universal-property-trunc-Prop - ( H g) - ( ∃-Prop B (λ b → h b ＝ b)) - ( λ p → intro-∃ (f (pr1 p) (pr1 p)) (inv (htpy-eq (pr2 p) (pr1 p)))) - where - g : A → B - g a = h (f a a) +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → A → B} + where + + abstract + fixed-point-theorem-Lawvere : + is-surjective f → (h : B → B) → exists-structure B (λ b → h b ＝ b) + fixed-point-theorem-Lawvere H h = + apply-universal-property-trunc-Prop + ( H g) + ( exists-structure-Prop B (λ b → h b ＝ b)) + ( λ (x , p) → intro-exists (f x x) (inv (htpy-eq p x))) + where + g : A → B + g a = h (f a a) ``` diff --git a/src/foundation/limited-principle-of-omniscience.lagda.md b/src/foundation/limited-principle-of-omniscience.lagda.md index bd00f84e57..5b6d52ed2e 100644 --- a/src/foundation/limited-principle-of-omniscience.lagda.md +++ b/src/foundation/limited-principle-of-omniscience.lagda.md @@ -11,6 +11,7 @@ open import elementary-number-theory.natural-numbers open import foundation.disjunction open import foundation.existential-quantification +open import foundation.universal-quantification open import foundation.universe-levels open import foundation-core.identity-types @@ -24,17 +25,19 @@ open import univalent-combinatorics.standard-finite-types ## Statement -The **limited principle of omniscience** (LPO) asserts that for every sequence -`f : ℕ → Fin 2` there either exists an `n` such that `f n ＝ 1` or for all `n` -we have `f n ＝ 0`. +The +{{#concept "limited principle of omniscience" WDID=Q6549544 WD="limited principle of omniscience" Agda=LPO}} +(LPO) asserts that for every [sequence](foundation.sequences.md) `f : ℕ → Fin 2` +there either [exists](foundation.existential-quantification.md) an `n` such that +`f n ＝ 1` or for all `n` we have `f n ＝ 0`. ```agda LPO : UU lzero LPO = (f : ℕ → Fin 2) → type-disjunction-Prop - ( ∃-Prop ℕ (λ n → f n ＝ one-Fin 1)) - ( Π-Prop ℕ (λ n → Id-Prop (Fin-Set 2) (f n) (zero-Fin 1))) + ( ∃ ℕ (λ n → Id-Prop (Fin-Set 2) (f n) (one-Fin 1))) + ( ∀' ℕ (λ n → Id-Prop (Fin-Set 2) (f n) (zero-Fin 1))) ``` ## See also diff --git a/src/foundation/logical-equivalences.lagda.md b/src/foundation/logical-equivalences.lagda.md index 8c72c70a96..b03f4e4a5b 100644 --- a/src/foundation/logical-equivalences.lagda.md +++ b/src/foundation/logical-equivalences.lagda.md @@ -32,20 +32,44 @@ open import foundation-core.torsorial-type-families ## Idea -**Logical equivalences** between two types `A` and `B` consist of a map `A → B` -and a map `B → A`. The type of logical equivalences between types is the -Curry-Howard interpretation of logical equivalences between +{{#concept "Logical equivalences" Agda=}} between two types `A` and `B` consist +of a map `A → B` and a map `B → A`. The type of logical equivalences between +types is the Curry-Howard interpretation of logical equivalences between [propositions](foundation-core.propositions.md). ## Definition +### The structure on a map of being a logical equivalence + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + has-converse : (A → B) → UU (l1 ⊔ l2) + has-converse f = B → A + + is-prop-has-converse : + is-prop A → (f : A → B) → is-prop (has-converse f) + is-prop-has-converse is-prop-A f = is-prop-function-type is-prop-A + +has-converse-Prop : + {l1 l2 : Level} (A : Prop l1) {B : UU l2} → (type-Prop A → B) → Prop (l1 ⊔ l2) +has-converse-Prop A f = + ( has-converse f , + is-prop-has-converse (is-prop-type-Prop A) f) +``` + ### Logical equivalences between types ```agda +iff : {l1 l2 : Level} → UU l1 → UU l2 → UU (l1 ⊔ l2) +iff A B = Σ (A → B) has-converse + infix 6 _↔_ _↔_ : {l1 l2 : Level} → UU l1 → UU l2 → UU (l1 ⊔ l2) -A ↔ B = (A → B) × (B → A) +_↔_ = iff module _ {l1 l2 : Level} {A : UU l1} {B : UU l2} (H : A ↔ B) @@ -80,8 +104,15 @@ module _ infix 6 _⇔_ - _⇔_ : UU (l1 ⊔ l2) - _⇔_ = type-iff-Prop + _⇔_ : Prop (l1 ⊔ l2) + _⇔_ = iff-Prop +``` + +### The identity logical equivalence + +```agda +id-iff : {l1 : Level} {A : UU l1} → A ↔ A +id-iff = (id , id) ``` ### Composition of logical equivalences @@ -136,13 +167,44 @@ module _ ### Logical equivalences between propositions induce equivalences ```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + abstract + is-equiv-has-converse-is-prop : + is-prop A → is-prop B → {f : A → B} → (B → A) → is-equiv f + is-equiv-has-converse-is-prop is-prop-A is-prop-B {f} g = + is-equiv-is-invertible + ( g) + ( λ y → eq-is-prop is-prop-B) + ( λ x → eq-is-prop is-prop-A) + + abstract + equiv-iff-is-prop : is-prop A → is-prop B → (A → B) → (B → A) → A ≃ B + pr1 (equiv-iff-is-prop is-prop-A is-prop-B f g) = f + pr2 (equiv-iff-is-prop is-prop-A is-prop-B f g) = + is-equiv-has-converse-is-prop is-prop-A is-prop-B g + module _ {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) where - equiv-iff' : (P ⇔ Q) → (type-Prop P ≃ type-Prop Q) - pr1 (equiv-iff' t) = pr1 t - pr2 (equiv-iff' t) = is-equiv-is-prop (pr2 P) (pr2 Q) (pr2 t) + abstract + is-equiv-has-converse : + {f : type-Prop P → type-Prop Q} → (type-Prop Q → type-Prop P) → is-equiv f + is-equiv-has-converse = + is-equiv-has-converse-is-prop + ( is-prop-type-Prop P) + ( is-prop-type-Prop Q) + + equiv-iff' : type-Prop (P ⇔ Q) → (type-Prop P ≃ type-Prop Q) + pr1 (equiv-iff' t) = forward-implication t + pr2 (equiv-iff' t) = + is-equiv-has-converse-is-prop + ( is-prop-type-Prop P) + ( is-prop-type-Prop Q) + ( backward-implication t) equiv-iff : (type-Prop P → type-Prop Q) → (type-Prop Q → type-Prop P) → @@ -176,7 +238,7 @@ compute-fiber-iff-equiv {A = A} {B} (f , g) = ### Two equal propositions are logically equivalent ```agda -iff-eq : {l1 : Level} {P Q : Prop l1} → P ＝ Q → P ⇔ Q +iff-eq : {l1 : Level} {P Q : Prop l1} → P ＝ Q → type-Prop (P ⇔ Q) pr1 (iff-eq refl) = id pr2 (iff-eq refl) = id ``` @@ -189,14 +251,14 @@ abstract {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) → is-equiv (equiv-iff' P Q) is-equiv-equiv-iff P Q = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-iff-Prop P Q) ( is-prop-type-equiv-Prop P Q) ( iff-equiv) equiv-equiv-iff : {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) → - (P ⇔ Q) ≃ (type-Prop P ≃ type-Prop Q) + (type-Prop P ↔ type-Prop Q) ≃ (type-Prop P ≃ type-Prop Q) pr1 (equiv-equiv-iff P Q) = equiv-iff' P Q pr2 (equiv-equiv-iff P Q) = is-equiv-equiv-iff P Q ``` diff --git a/src/foundation/mere-functions.lagda.md b/src/foundation/mere-functions.lagda.md new file mode 100644 index 0000000000..4597357ccd --- /dev/null +++ b/src/foundation/mere-functions.lagda.md @@ -0,0 +1,101 @@ +# Mere functions + +```agda +module foundation.mere-functions where +``` + +
Imports + +```agda +open import foundation.propositional-truncations +open import foundation.universe-levels + +open import foundation-core.function-types +open import foundation-core.propositions +``` + +
+ +## Idea + +The type of +{{#concept "mere functions" Disambiguation="of types" Agda=mere-function}} from +`A` to `B` is the +[propositional truncation](foundation.propositional-truncations.md) of the type +of maps from `A` to `B`. + +```text + mere-function A B := ║(A → B)║₋₁ +``` + +## Definitions + +### Mere functions between types + +```agda +module _ + {l1 l2 : Level} (A : UU l1) (B : UU l2) + where + + prop-mere-function : Prop (l1 ⊔ l2) + prop-mere-function = trunc-Prop (A → B) + + mere-function : UU (l1 ⊔ l2) + mere-function = type-Prop prop-mere-function + + is-prop-mere-function : is-prop mere-function + is-prop-mere-function = is-prop-type-Prop prop-mere-function +``` + +### The evaluation map on mere functions + +If we have a mere function from `A` to `B` and `A` is inhabited, then `B` is +inhabited. + +```agda +module _ + {l1 l2 : Level} (A : UU l1) (B : UU l2) + where + + ev-mere-function' : (mere-function A B) → A → ║ B ║₋₁ + ev-mere-function' |f| a = + rec-trunc-Prop (trunc-Prop B) (λ f → unit-trunc-Prop (f a)) |f| + + ev-mere-function : (mere-function A B) → ║ A ║₋₁ → ║ B ║₋₁ + ev-mere-function |f| |a| = + rec-trunc-Prop (trunc-Prop B) (ev-mere-function' |f|) (|a|) +``` + +### Mere functions form a reflexive relation + +```agda +module _ + {l : Level} (A : UU l) + where + + refl-mere-function : mere-function A A + refl-mere-function = unit-trunc-Prop id +``` + +### Mere functions form a transitive relation + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} + where + + transitive-mere-function : + mere-function B C → mere-function A B → mere-function A C + transitive-mere-function |g| = + rec-trunc-Prop + ( prop-mere-function A C) + ( λ f → + rec-trunc-Prop + ( prop-mere-function A C) + ( λ g → unit-trunc-Prop (g ∘ f)) + ( |g|)) +``` + +## See also + +- [Mere logical equivalences](foundation.mere-logical-equivalences.md) diff --git a/src/foundation/mere-logical-equivalences.lagda.md b/src/foundation/mere-logical-equivalences.lagda.md new file mode 100644 index 0000000000..5bf9ebe781 --- /dev/null +++ b/src/foundation/mere-logical-equivalences.lagda.md @@ -0,0 +1,181 @@ +# Mere logical equivalences + +```agda +module foundation.mere-logical-equivalences where +``` + +
Imports + +```agda +open import foundation.conjunction +open import foundation.dependent-pair-types +open import foundation.inhabited-types +open import foundation.logical-equivalences +open import foundation.mere-functions +open import foundation.propositional-truncations +open import foundation.universe-levels + +open import foundation-core.cartesian-product-types +open import foundation-core.equivalences +open import foundation-core.function-types +open import foundation-core.propositions +``` + +
+ +## Idea + +Two types `A` and `B` are +{{#concept "merely logically equivalent" Disambiguation="types" Agda=mere-iff}} +if the type of [logical equivalences](foundation.logical-equivalences.md) +between `A` and `B` is [inhabited](foundation.inhabited-types.md). + +```text + mere-iff A B := ║(A ↔ B)║₋₁ +``` + +## Definitions + +### Mere logical equivalence of types + +```agda +module _ + {l1 l2 : Level} (A : UU l1) (B : UU l2) + where + + prop-mere-iff : Prop (l1 ⊔ l2) + prop-mere-iff = trunc-Prop (A ↔ B) + + mere-iff : UU (l1 ⊔ l2) + mere-iff = type-Prop prop-mere-iff + + is-prop-mere-iff : is-prop mere-iff + is-prop-mere-iff = is-prop-type-Prop prop-mere-iff +``` + +## Properties + +### Mere logical equivalence is a reflexive relation + +```agda +module _ + {l : Level} (A : UU l) + where + + refl-mere-iff : mere-iff A A + refl-mere-iff = unit-trunc-Prop id-iff +``` + +### Mere logical equivalence is a transitive relation + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} + where + + transitive-mere-iff : mere-iff B C → mere-iff A B → mere-iff A C + transitive-mere-iff |g| = + rec-trunc-Prop + ( prop-mere-iff A C) + ( λ f → + rec-trunc-Prop + ( prop-mere-iff A C) + ( λ g → unit-trunc-Prop (g ∘iff f)) + ( |g|)) +``` + +### Mere logical equivalence is a symmetric relation + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + symmetric-mere-iff : mere-iff A B → mere-iff B A + symmetric-mere-iff = + rec-trunc-Prop (prop-mere-iff B A) (unit-trunc-Prop ∘ inv-iff) +``` + +### Merely logically equivalent types are coinhabited + +If `A` and `B` are merely logically equivalent then they are +[coinhabited](foundation.coinhabited-pairs-of-types.md). + +```agda +module _ + {l1 l2 : Level} (A : UU l1) (B : UU l2) + where + + ev-forward-mere-iff' : mere-iff A B → A → is-inhabited B + ev-forward-mere-iff' |f| a = + rec-trunc-Prop + ( trunc-Prop B) + ( λ f → unit-trunc-Prop (forward-implication f a)) + ( |f|) + + ev-forward-mere-iff : mere-iff A B → is-inhabited A → is-inhabited B + ev-forward-mere-iff |f| = + rec-trunc-Prop (trunc-Prop B) (ev-forward-mere-iff' |f|) + + ev-backward-mere-iff' : mere-iff A B → B → is-inhabited A + ev-backward-mere-iff' |f| b = + rec-trunc-Prop + ( trunc-Prop A) + ( λ f → unit-trunc-Prop (backward-implication f b)) + ( |f|) + + ev-backward-mere-iff : mere-iff A B → is-inhabited B → is-inhabited A + ev-backward-mere-iff |f| = + rec-trunc-Prop (trunc-Prop A) (ev-backward-mere-iff' |f|) + + is-coinhabited-mere-iff : mere-iff A B → is-inhabited A ↔ is-inhabited B + is-coinhabited-mere-iff |f| = + ( ev-forward-mere-iff |f| , ev-backward-mere-iff |f|) +``` + +### Merely logically equivalent types have bidirectional mere functions + +If `A` and `B` are merely logically equivalent then we have a mere function from +`A` to `B` and a mere function from `B` to `A`. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + forward-mere-function-mere-iff : mere-iff A B → mere-function A B + forward-mere-function-mere-iff = + rec-trunc-Prop + ( prop-mere-function A B) + ( unit-trunc-Prop ∘ forward-implication) + + backward-mere-function-mere-iff : mere-iff A B → mere-function B A + backward-mere-function-mere-iff = + rec-trunc-Prop + ( prop-mere-function B A) + ( unit-trunc-Prop ∘ backward-implication) +``` + +### Mere logical equivalence is equivalent to having bidirectional mere functions + +For all types we have the equivalence + +```text + (mere-iff A B) ≃ (mere-function A B × mere-function B A). +``` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + compute-mere-iff : + mere-iff A B ≃ mere-function A B × mere-function B A + compute-mere-iff = equiv-product-trunc-Prop (A → B) (B → A) +``` + +## See also + +- [Mere functions](foundation.mere-functions.md) +- [Coinhabitedness](foundation.coinhabited-pairs-of-types.md) is a related but + weaker notion than mere logical equivalence. diff --git a/src/foundation/negation.lagda.md b/src/foundation/negation.lagda.md index aa4a880ec6..adbbb19004 100644 --- a/src/foundation/negation.lagda.md +++ b/src/foundation/negation.lagda.md @@ -34,12 +34,19 @@ type `A` is the type `A → empty`. is-prop-neg : {l : Level} {A : UU l} → is-prop (¬ A) is-prop-neg {A = A} = is-prop-function-type is-prop-empty -neg-Prop' : {l1 : Level} → UU l1 → Prop l1 -pr1 (neg-Prop' A) = ¬ A -pr2 (neg-Prop' A) = is-prop-neg +neg-type-Prop : {l1 : Level} → UU l1 → Prop l1 +neg-type-Prop A = ¬ A , is-prop-neg neg-Prop : {l1 : Level} → Prop l1 → Prop l1 -neg-Prop P = neg-Prop' (type-Prop P) +neg-Prop P = neg-type-Prop (type-Prop P) + +type-neg-Prop : {l1 : Level} → Prop l1 → UU l1 +type-neg-Prop P = type-Prop (neg-Prop P) + +infix 25 ¬'_ + +¬'_ : {l1 : Level} → Prop l1 → Prop l1 +¬'_ = neg-Prop ``` ### Reductio ad absurdum @@ -57,8 +64,8 @@ equiv-neg : (X ≃ Y) → (¬ X ≃ ¬ Y) equiv-neg {l1} {l2} {X} {Y} e = equiv-iff' - ( neg-Prop' X) - ( neg-Prop' Y) + ( neg-type-Prop X) + ( neg-type-Prop Y) ( pair (map-neg (map-inv-equiv e)) (map-neg (map-equiv e))) ``` @@ -66,14 +73,24 @@ equiv-neg {l1} {l2} {X} {Y} e = ```agda no-fixed-points-neg : - {l : Level} (A : UU l) → ¬ (A ↔ (¬ A)) -no-fixed-points-neg A (pair f g) = + {l : Level} (A : UU l) → ¬ (A ↔ ¬ A) +no-fixed-points-neg A (f , g) = ( λ (h : ¬ A) → h (g h)) (λ (a : A) → f a a) -``` -```agda abstract no-fixed-points-neg-Prop : - {l1 : Level} (P : Prop l1) → ¬ (P ⇔ neg-Prop P) + {l : Level} (P : Prop l) → ¬ (type-Prop P ↔ ¬ (type-Prop P)) no-fixed-points-neg-Prop P = no-fixed-points-neg (type-Prop P) ``` + +## Table of files about propositional logic + +The following table gives an overview of basic constructions in propositional +logic and related considerations. + +{{#include tables/propositional-logic.md}} + +## External links + +- [negation](https://ncatlab.org/nlab/show/negation) at $n$Lab +- [Negation](https://en.wikipedia.org/wiki/Negation) at Wikipedia diff --git a/src/foundation/partitions.lagda.md b/src/foundation/partitions.lagda.md index 414d0edeca..11658b834b 100644 --- a/src/foundation/partitions.lagda.md +++ b/src/foundation/partitions.lagda.md @@ -8,6 +8,7 @@ module foundation.partitions where ```agda open import foundation.action-on-identifications-functions +open import foundation.conjunction open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.embeddings @@ -512,9 +513,7 @@ module _ share-common-element-block-partition-Prop : (C : block-partition P) → Prop (l1 ⊔ l2) share-common-element-block-partition-Prop C = - ∃-Prop A - ( λ a → - is-in-block-partition P B a × is-in-block-partition P C a) + ∃ A (λ a → subtype-block-partition P B a ∧ subtype-block-partition P C a) share-common-element-block-partition : (C : block-partition P) → UU (l1 ⊔ l2) diff --git a/src/foundation/path-split-maps.lagda.md b/src/foundation/path-split-maps.lagda.md index 857a1071b5..0255429411 100644 --- a/src/foundation/path-split-maps.lagda.md +++ b/src/foundation/path-split-maps.lagda.md @@ -12,6 +12,7 @@ open import foundation-core.path-split-maps public open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.iterated-dependent-product-types +open import foundation.logical-equivalences open import foundation.universe-levels open import foundation-core.contractible-types @@ -47,7 +48,7 @@ module _ is-equiv-is-path-split-is-equiv : (f : A → B) → is-equiv (is-path-split-is-equiv f) is-equiv-is-path-split-is-equiv f = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-property-is-equiv f) ( is-prop-is-path-split f) ( is-equiv-is-path-split f) @@ -60,7 +61,7 @@ module _ is-equiv-is-equiv-is-path-split : (f : A → B) → is-equiv (is-equiv-is-path-split f) is-equiv-is-equiv-is-path-split f = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-is-path-split f) ( is-property-is-equiv f) ( is-path-split-is-equiv f) diff --git a/src/foundation/preunivalent-type-families.lagda.md b/src/foundation/preunivalent-type-families.lagda.md index 6a4a7e9f2a..0a71d755dd 100644 --- a/src/foundation/preunivalent-type-families.lagda.md +++ b/src/foundation/preunivalent-type-families.lagda.md @@ -14,16 +14,16 @@ open import foundation.embeddings open import foundation.equivalence-injective-type-families open import foundation.faithful-maps open import foundation.function-types +open import foundation.injective-maps open import foundation.preunivalence open import foundation.retractions +open import foundation.sets open import foundation.subuniverses open import foundation.transport-along-identifications open import foundation.universe-levels open import foundation-core.equivalences open import foundation-core.identity-types -open import foundation-core.injective-maps -open import foundation-core.sets open import foundation-core.univalence ``` diff --git a/src/foundation/propositional-extensionality.lagda.md b/src/foundation/propositional-extensionality.lagda.md index 4a65faa25d..a92f6d6176 100644 --- a/src/foundation/propositional-extensionality.lagda.md +++ b/src/foundation/propositional-extensionality.lagda.md @@ -37,8 +37,15 @@ open import foundation-core.torsorial-type-families ## Idea -Propositional extensionality characterizes identifications of propositions. It -asserts that for any two propositions `P` and `Q`, we have `Id P Q ≃ (P ⇔ Q)`. +{{#concept "Propositional extensionality" Agda=propositional-extensionality}} +characterizes [identifications](foundation-core.identity-types.md) of +[propositions](foundation-core.propositions.md). It asserts that for any two +propositions `P` and `Q`, we have `(P ＝ Q) ≃ (P ⇔ Q)`. + +**Note.** While we derive propositional extensionality from the +[univalence axiom](foundation-core.univalence.md), it is a strictly weaker +principle, meaning that the principle of propositional extensionality does not +imply univalence. ## Properties @@ -51,7 +58,7 @@ module _ abstract is-torsorial-iff : - (P : Prop l1) → is-torsorial (λ (Q : Prop l1) → P ⇔ Q) + (P : Prop l1) → is-torsorial (λ (Q : Prop l1) → type-Prop P ↔ type-Prop Q) is-torsorial-iff P = is-contr-equiv ( Σ (Prop l1) (λ Q → type-Prop P ≃ type-Prop Q)) @@ -66,16 +73,14 @@ module _ abstract is-equiv-iff-eq : (P Q : Prop l1) → is-equiv (iff-eq {l1} {P} {Q}) is-equiv-iff-eq P = - fundamental-theorem-id - ( is-torsorial-iff P) - ( λ Q → iff-eq {P = P} {Q}) + fundamental-theorem-id (is-torsorial-iff P) (λ Q → iff-eq {P = P} {Q}) propositional-extensionality : - (P Q : Prop l1) → (P ＝ Q) ≃ (P ⇔ Q) + (P Q : Prop l1) → (P ＝ Q) ≃ (type-Prop P ↔ type-Prop Q) pr1 (propositional-extensionality P Q) = iff-eq pr2 (propositional-extensionality P Q) = is-equiv-iff-eq P Q - eq-iff' : (P Q : Prop l1) → P ⇔ Q → P ＝ Q + eq-iff' : (P Q : Prop l1) → type-Prop P ↔ type-Prop Q → P ＝ Q eq-iff' P Q = map-inv-is-equiv (is-equiv-iff-eq P Q) eq-iff : @@ -84,7 +89,7 @@ module _ eq-iff {P} {Q} f g = eq-iff' P Q (pair f g) eq-equiv-Prop : - {P Q : Prop l1} → (type-Prop P ≃ type-Prop Q) → P ＝ Q + {P Q : Prop l1} → type-Prop P ≃ type-Prop Q → P ＝ Q eq-equiv-Prop e = eq-iff (map-equiv e) (map-inv-equiv e) @@ -93,11 +98,10 @@ module _ equiv-eq-Prop {P} refl = id-equiv is-torsorial-equiv-Prop : - (P : Prop l1) → - is-torsorial (λ Q → type-Prop P ≃ type-Prop Q) + (P : Prop l1) → is-torsorial (λ Q → type-Prop P ≃ type-Prop Q) is-torsorial-equiv-Prop P = is-contr-equiv' - ( Σ (Prop l1) (λ Q → P ⇔ Q)) + ( Σ (Prop l1) (λ Q → type-Prop P ↔ type-Prop Q)) ( equiv-tot (equiv-equiv-iff P)) ( is-torsorial-iff P) ``` @@ -106,7 +110,7 @@ module _ ```agda is-set-type-Prop : {l : Level} → is-set (Prop l) -is-set-type-Prop {l} P Q = +is-set-type-Prop P Q = is-prop-equiv ( propositional-extensionality P Q) ( is-prop-iff-Prop P Q) @@ -120,7 +124,7 @@ pr2 (Prop-Set l) = is-set-type-Prop ```agda is-univalent-type-Prop : {l : Level} → is-univalent (type-Prop {l}) -is-univalent-type-Prop {l} P = +is-univalent-type-Prop P = fundamental-theorem-id ( is-torsorial-equiv-Prop P) ( λ Q → equiv-tr type-Prop) @@ -134,7 +138,7 @@ abstract {l1 : Level} → is-torsorial (λ (P : Prop l1) → type-Prop P) is-torsorial-true-Prop {l1} = is-contr-equiv - ( Σ (Prop l1) (λ P → raise-unit-Prop l1 ⇔ P)) + ( Σ (Prop l1) (λ P → raise-unit l1 ↔ type-Prop P)) ( equiv-tot ( λ P → inv-equiv @@ -144,7 +148,7 @@ abstract ( type-Prop P)) ∘e ( right-unit-law-product-is-contr ( is-contr-Π - ( λ x → + ( λ _ → is-proof-irrelevant-is-prop ( is-prop-raise-unit) ( raise-star))))))) @@ -156,10 +160,10 @@ abstract ```agda abstract is-torsorial-false-Prop : - {l1 : Level} → is-torsorial (λ (P : Prop l1) → type-Prop (neg-Prop P)) + {l1 : Level} → is-torsorial (λ (P : Prop l1) → ¬ (type-Prop P)) is-torsorial-false-Prop {l1} = is-contr-equiv - ( Σ (Prop l1) (λ P → raise-empty-Prop l1 ⇔ P)) + ( Σ (Prop l1) (λ P → raise-empty l1 ↔ type-Prop P)) ( equiv-tot ( λ P → inv-equiv @@ -172,3 +176,15 @@ abstract ( type-Prop P)))))) ( is-torsorial-iff (raise-empty-Prop l1)) ``` + +## Table of files about propositional logic + +The following table gives an overview of basic constructions in propositional +logic and related considerations. + +{{#include tables/propositional-logic.md}} + +## External links + +- [propositional extensionality](https://ncatlab.org/nlab/show/propositional+extensionality) + at $n$Lab. diff --git a/src/foundation/propositional-resizing.lagda.md b/src/foundation/propositional-resizing.lagda.md index b3be0891b7..46d48494aa 100644 --- a/src/foundation/propositional-resizing.lagda.md +++ b/src/foundation/propositional-resizing.lagda.md @@ -18,10 +18,13 @@ open import foundation-core.subtypes ## Idea -We say that there is propositional resizing for propositions of universe levels -`l1` and `l2` if there is a type `Ω : UU l1` equipped with a subtype `Q` such -that for each proposition `P` of universe level `l2` there is an element `u : Ω` -such that `Q u ≃ P`. +We say that a universe `𝒱` satisfies `𝒰`-small +{{#concept "propositional resizing"}} if there is a type `Ω` in `𝒰` +[equipped](foundation.structure.md) with a +[subtype](foundation-core.subtypes.md) `Q` such that for each proposition `P` in +`𝒱` there is an element `u : Ω` such that `Q u ≃ P`. Such a type `Ω` is called +an `𝒰`-small {{#concept "classifier" Disambiguation="of small subobjects"}} of +`𝒱`-small subobjects. ## Definition @@ -31,3 +34,14 @@ propositional-resizing l1 l2 = Σ ( Σ (UU l1) (subtype l1)) ( λ Ω → (P : Prop l2) → Σ (pr1 Ω) (λ u → type-equiv-Prop (pr2 Ω u) P)) ``` + +## See also + +- [The large locale of propositions](foundation.large-locale-of-propositions.md) + +## Table of files about propositional logic + +The following table gives an overview of basic constructions in propositional +logic and related considerations. + +{{#include tables/propositional-logic.md}} diff --git a/src/foundation/propositional-truncations.lagda.md b/src/foundation/propositional-truncations.lagda.md index 8f12d80321..1181cc2572 100644 --- a/src/foundation/propositional-truncations.lagda.md +++ b/src/foundation/propositional-truncations.lagda.md @@ -10,6 +10,7 @@ module foundation.propositional-truncations where open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.functoriality-cartesian-product-types +open import foundation.logical-equivalences open import foundation.propositions open import foundation.truncations open import foundation.universal-property-propositional-truncation @@ -20,7 +21,6 @@ open import foundation-core.contractible-types open import foundation-core.coproduct-types open import foundation-core.equivalences open import foundation-core.function-types -open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.precomposition-dependent-functions open import foundation-core.precomposition-functions @@ -59,8 +59,16 @@ all-elements-equal-type-trunc-Prop {l} {A} = trunc-Prop : {l : Level} → UU l → Prop l trunc-Prop = trunc neg-one-𝕋 + +║_║₋₁ : {l : Level} → UU l → UU l +║_║₋₁ = type-trunc-Prop ``` +**Notation.** The [box drawings double vertical](https://codepoints.net/U+2551) +symbol `║` in the propositional truncation notation `║_║₋₁` can be inserted with +`agda-input` using the escape sequence `\--=` and selecting the second item in +the list. + ## Properties ### The condition in the induction principle is a property @@ -245,7 +253,7 @@ module _ abstract is-equiv-map-idempotent-trunc-Prop : is-equiv map-idempotent-trunc-Prop is-equiv-map-idempotent-trunc-Prop = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-type-trunc-Prop) ( is-prop-type-trunc-Prop) ( unit-trunc-Prop) @@ -259,7 +267,7 @@ module _ is-equiv-map-inv-idempotent-trunc-Prop : is-equiv (unit-trunc-Prop {A = type-trunc-Prop A}) is-equiv-map-inv-idempotent-trunc-Prop = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-type-trunc-Prop) ( is-prop-type-trunc-Prop) ( map-idempotent-trunc-Prop) @@ -356,8 +364,8 @@ abstract is-equiv-map-distributive-trunc-product-Prop : {l1 l2 : Level} {A : UU l1} {B : UU l2} → is-equiv (map-distributive-trunc-product-Prop {A = A} {B = B}) - is-equiv-map-distributive-trunc-product-Prop {l1} {l2} {A} {B} = - is-equiv-is-prop + is-equiv-map-distributive-trunc-product-Prop = + is-equiv-has-converse-is-prop ( is-prop-type-trunc-Prop) ( is-prop-product is-prop-type-trunc-Prop is-prop-type-trunc-Prop) ( map-inv-distributive-trunc-product-Prop) @@ -373,8 +381,8 @@ abstract is-equiv-map-inv-distributive-trunc-product-Prop : {l1 l2 : Level} {A : UU l1} {B : UU l2} → is-equiv (map-inv-distributive-trunc-product-Prop {A = A} {B = B}) - is-equiv-map-inv-distributive-trunc-product-Prop {l1} {l2} {A} {B} = - is-equiv-is-prop + is-equiv-map-inv-distributive-trunc-product-Prop = + is-equiv-has-converse-is-prop ( is-prop-product is-prop-type-trunc-Prop is-prop-type-trunc-Prop) ( is-prop-type-trunc-Prop) ( map-distributive-trunc-product-Prop) @@ -411,7 +419,7 @@ module _ is-equiv-map-trunc-Prop-diagonal-coproduct : is-equiv map-trunc-Prop-diagonal-coproduct is-equiv-map-trunc-Prop-diagonal-coproduct = - is-equiv-is-prop + is-equiv-has-converse-is-prop is-prop-type-trunc-Prop is-prop-type-trunc-Prop map-inv-trunc-Prop-diagonal-coproduct @@ -419,21 +427,32 @@ module _ is-equiv-map-inv-trunc-Prop-diagonal-coproduct : is-equiv map-inv-trunc-Prop-diagonal-coproduct is-equiv-map-inv-trunc-Prop-diagonal-coproduct = - is-equiv-is-prop + is-equiv-has-converse-is-prop is-prop-type-trunc-Prop is-prop-type-trunc-Prop map-trunc-Prop-diagonal-coproduct equiv-trunc-Prop-diagonal-coproduct : - (type-trunc-Prop (A + A)) ≃ type-trunc-Prop A + type-trunc-Prop (A + A) ≃ type-trunc-Prop A pr1 equiv-trunc-Prop-diagonal-coproduct = map-trunc-Prop-diagonal-coproduct pr2 equiv-trunc-Prop-diagonal-coproduct = is-equiv-map-trunc-Prop-diagonal-coproduct inv-equiv-trunc-Prop-diagonal-coproduct : - (type-trunc-Prop A) ≃ type-trunc-Prop (A + A) + type-trunc-Prop A ≃ type-trunc-Prop (A + A) pr1 inv-equiv-trunc-Prop-diagonal-coproduct = map-inv-trunc-Prop-diagonal-coproduct pr2 inv-equiv-trunc-Prop-diagonal-coproduct = is-equiv-map-inv-trunc-Prop-diagonal-coproduct ``` + +## Table of files about propositional logic + +The following table gives an overview of basic constructions in propositional +logic and related considerations. + +{{#include tables/propositional-logic.md}} + +## External links + +- [bracket type](https://ncatlab.org/nlab/show/bracket+type) at $n$Lab diff --git a/src/foundation/propositions.lagda.md b/src/foundation/propositions.lagda.md index e0f0a8094c..5a9092e007 100644 --- a/src/foundation/propositions.lagda.md +++ b/src/foundation/propositions.lagda.md @@ -11,6 +11,7 @@ open import foundation-core.propositions public ```agda open import foundation.contractible-types open import foundation.dependent-pair-types +open import foundation.logical-equivalences open import foundation.retracts-of-types open import foundation.universe-levels @@ -69,9 +70,67 @@ abstract equiv-is-prop-equiv : {l1 l2 : Level} {A : UU l1} {B : UU l2} → A ≃ B → is-prop A ≃ is-prop B equiv-is-prop-equiv {A = A} {B = B} e = - equiv-prop - (is-prop-is-prop A) - (is-prop-is-prop B) - (is-prop-equiv' e) - (is-prop-equiv e) + equiv-iff-is-prop + ( is-prop-is-prop A) + ( is-prop-is-prop B) + ( is-prop-equiv' e) + ( is-prop-equiv e) ``` + +## See also + +### Operations on propositions + +There is a wide range of operations on propositions due to the rich structure of +intuitionistic logic. Below we give a structured overview of a notable selection +of such operations and their notation in the library. + +The list is split into two sections, the first consists of operations that +generalize to arbitrary types and even sufficiently nice +[subuniverses](foundation.subuniverses.md), such as +$n$-[types](foundation-core.truncated-types.md). + +| Name | Operator on types | Operator on propositions/subtypes | +| ----------------------------------------------------------- | ----------------- | --------------------------------- | +| [Dependent sum](foundation.dependent-pair-types.md) | `Σ` | `Σ-Prop` | +| [Dependent product](foundation.dependent-function-types.md) | `Π` | `Π-Prop` | +| [Functions](foundation-core.function-types.md) | `→` | `⇒` | +| [Logical equivalence](foundation.logical-equivalences.md) | `↔` | `⇔` | +| [Product](foundation-core.cartesian-product-types.md) | `×` | `product-Prop` | +| [Join](synthetic-homotopy-theory.joins-of-types.md) | `*` | `join-Prop` | +| [Exclusive sum](foundation.exclusive-sum.md) | `exclusive-sum` | `exclusive-sum-Prop` | +| [Coproduct](foundation-core.coproduct-types.md) | `+` | _N/A_ | + +Note that for many operations in the second section, there is an equivalent +operation on propositions in the first. + +| Name | Operator on types | Operator on propositions/subtypes | +| ---------------------------------------------------------------------------- | --------------------------- | ---------------------------------------- | +| [Initial object](foundation-core.empty-types.md) | `empty` | `empty-Prop` | +| [Terminal object](foundation.unit-type.md) | `unit` | `unit-Prop` | +| [Existential quantification](foundation.existential-quantification.md) | `exists-structure` | `∃` | +| [Unique existential quantification](foundation.uniqueness-quantification.md) | `uniquely-exists-structure` | `∃!` | +| [Universal quantification](foundation.universal-quantification.md) | | `∀'` (equivalent to `Π-Prop`) | +| [Conjunction](foundation.conjunction.md) | | `∧` (equivalent to `product-Prop`) | +| [Disjunction](foundation.disjunction.md) | `disjunction-type` | `∨` (equivalent to `join-Prop`) | +| [Exclusive disjunction](foundation.exclusive-disjunction.md) | `xor-type` | `⊻` (equivalent to `exclusive-sum-Prop`) | +| [Negation](foundation.negation.md) | `¬` | `¬'` | +| [Double negation](foundation.double-negation.md) | `¬¬` | `¬¬'` | + +We can also organize these operations by indexed and binary variants, giving us +the following table: + +| Name | Binary | Indexed | +| ---------------------- | ------ | ------- | +| Product | `×` | `Π` | +| Conjunction | `∧` | `∀'` | +| Constructive existence | `+` | `Σ` | +| Existence | `∨` | `∃` | +| Unique existence | `⊻` | `∃!` | + +### Table of files about propositional logic + +The following table gives an overview of basic constructions in propositional +logic and related considerations. + +{{#include tables/propositional-logic.md}} diff --git a/src/foundation/retractions.lagda.md b/src/foundation/retractions.lagda.md index 43253b5f8c..fe2da2d349 100644 --- a/src/foundation/retractions.lagda.md +++ b/src/foundation/retractions.lagda.md @@ -9,7 +9,6 @@ open import foundation-core.retractions public
Imports ```agda -open import foundation.action-on-identifications-functions open import foundation.coslice open import foundation.dependent-pair-types open import foundation.retracts-of-types @@ -20,7 +19,6 @@ open import foundation-core.equivalences open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.injective-maps ```
diff --git a/src/foundation/set-presented-types.lagda.md b/src/foundation/set-presented-types.lagda.md index ee7d3b9858..e6f75f27db 100644 --- a/src/foundation/set-presented-types.lagda.md +++ b/src/foundation/set-presented-types.lagda.md @@ -7,11 +7,11 @@ module foundation.set-presented-types where
Imports ```agda +open import foundation.equivalences open import foundation.existential-quantification open import foundation.set-truncations open import foundation.universe-levels -open import foundation-core.equivalences open import foundation-core.function-types open import foundation-core.propositions open import foundation-core.sets @@ -21,12 +21,18 @@ open import foundation-core.sets ## Idea -A type `A` is said to be set presented if there exists a map `f : X → A` from a -set `X` such that the composite `X → A → type-trunc-Set A` is an equivalence. +A type `A` is said to be +{{#concept "set presented" Agda=has-set-presentation-Prop}} if there +[exists](foundation.existential-quantification.md) a map `f : X → A` from a +[set](foundation-core.sets.md) `X` such that the composite +`X → A → type-trunc-Set A` is an [equivalence](foundation.equivalences.md). ```agda -has-set-presentation-Prop : - {l1 l2 : Level} (A : Set l1) (B : UU l2) → Prop (l1 ⊔ l2) -has-set-presentation-Prop A B = - ∃-Prop (type-Set A → B) (λ f → is-equiv (unit-trunc-Set ∘ f)) +module _ + {l1 l2 : Level} (A : Set l1) (B : UU l2) + where + + has-set-presentation-Prop : Prop (l1 ⊔ l2) + has-set-presentation-Prop = + ∃ (type-Set A → B) (λ f → is-equiv-Prop (unit-trunc-Set ∘ f)) ``` diff --git a/src/foundation/set-truncations.lagda.md b/src/foundation/set-truncations.lagda.md index df3386684a..154cc2d2b5 100644 --- a/src/foundation/set-truncations.lagda.md +++ b/src/foundation/set-truncations.lagda.md @@ -49,30 +49,38 @@ open import foundation-core.truncation-levels ## Idea -The **set truncation** of a type `A` is a map `η : A → trunc-Set A` that -satisfies +The {{#concept "set truncation" Agda=trunc-Set}} of a type `A` is a map +`η : A → trunc-Set A` that satisfies [the universal property of set truncations](foundation.universal-property-set-truncation.md). -## Definition +## Definitions ```agda +trunc-Set : {l : Level} → UU l → Set l +trunc-Set = trunc zero-𝕋 + type-trunc-Set : {l : Level} → UU l → UU l type-trunc-Set = type-trunc zero-𝕋 is-set-type-trunc-Set : {l : Level} {A : UU l} → is-set (type-trunc-Set A) is-set-type-trunc-Set = is-trunc-type-trunc -trunc-Set : {l : Level} → UU l → Set l -trunc-Set = trunc zero-𝕋 - unit-trunc-Set : {l : Level} {A : UU l} → A → type-trunc-Set A unit-trunc-Set = unit-trunc is-set-truncation-trunc-Set : {l1 : Level} (A : UU l1) → is-set-truncation (trunc-Set A) unit-trunc-Set is-set-truncation-trunc-Set A = is-truncation-trunc + +║_║₀ : {l : Level} → UU l → UU l +║_║₀ = type-trunc-Set ``` +**Notation.** The [box drawings double vertical](https://codepoints.net/U+2551) +symbol `║` in the set truncation notation `║_║₀` can be inserted with +`agda-input` using the escape sequence `\--=` and selecting the second item in +the list. + ## Properties ### The dependent universal property of set truncations diff --git a/src/foundation/sets.lagda.md b/src/foundation/sets.lagda.md index 15ad33b747..32edfb667b 100644 --- a/src/foundation/sets.lagda.md +++ b/src/foundation/sets.lagda.md @@ -11,6 +11,7 @@ open import foundation-core.sets public ```agda open import foundation.contractible-types open import foundation.dependent-pair-types +open import foundation.logical-equivalences open import foundation.subuniverses open import foundation.truncated-types open import foundation.univalent-type-families @@ -21,6 +22,7 @@ open import foundation-core.cartesian-product-types open import foundation-core.embeddings open import foundation-core.equivalences open import foundation-core.identity-types +open import foundation-core.injective-maps open import foundation-core.precomposition-functions open import foundation-core.propositional-maps open import foundation-core.propositions diff --git a/src/foundation/slice.lagda.md b/src/foundation/slice.lagda.md index 146aa27710..0a41b6dd82 100644 --- a/src/foundation/slice.lagda.md +++ b/src/foundation/slice.lagda.md @@ -14,6 +14,7 @@ open import foundation.function-extensionality open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.homotopy-induction +open import foundation.logical-equivalences open import foundation.structure-identity-principle open import foundation.type-arithmetic-dependent-pair-types open import foundation.univalence @@ -246,11 +247,12 @@ module _ (h : hom-slice f g) → is-equiv (pr1 h) ≃ is-fiberwise-equiv (map-equiv (equiv-fiberwise-hom-hom-slice f g) h) - α h = equiv-prop - ( is-property-is-equiv _) - ( is-prop-Π (λ _ → is-property-is-equiv _)) - ( is-fiberwise-equiv-fiberwise-equiv-equiv-slice h) - ( is-equiv-hom-slice-is-fiberwise-equiv-fiberwise-hom-hom-slice h) + α h = + equiv-iff-is-prop + ( is-property-is-equiv _) + ( is-prop-Π (λ _ → is-property-is-equiv _)) + ( is-fiberwise-equiv-fiberwise-equiv-equiv-slice h) + ( is-equiv-hom-slice-is-fiberwise-equiv-fiberwise-hom-hom-slice h) equiv-equiv-slice-fiberwise-equiv : fiberwise-equiv (fiber f) (fiber g) ≃ equiv-slice f g diff --git a/src/foundation/subsingleton-induction.lagda.md b/src/foundation/subsingleton-induction.lagda.md index 1e6b1f0c7d..7007417be8 100644 --- a/src/foundation/subsingleton-induction.lagda.md +++ b/src/foundation/subsingleton-induction.lagda.md @@ -21,11 +21,12 @@ open import foundation-core.sections ## Idea -**Subsingleton induction** on a type `A` is a slight variant of -[singleton induction](foundation.singleton-induction.md) which in turn is a -principle analogous to induction for the [unit type](foundation.unit-type.md). -Subsingleton induction uses the observation that a type equipped with an element -is [contractible](foundation-core.contractible-types.md) if and only if it is a +{{#concept "Subsingleton induction" Agda=ind-subsingleton}} on a type `A` is a +slight variant of [singleton induction](foundation.singleton-induction.md) which +in turn is a principle analogous to induction for the +[unit type](foundation.unit-type.md). Subsingleton induction uses the +observation that a type equipped with an element is +[contractible](foundation-core.contractible-types.md) if and only if it is a [proposition](foundation-core.propositions.md). Subsingleton induction states that given a type family `B` over `A`, to @@ -95,3 +96,10 @@ abstract {l1 : Level} (A : UU l1) → ({l2 : Level} → is-subsingleton l2 A) → is-prop A is-prop-is-subsingleton A S = is-prop-ind-subsingleton A (pr1 ∘ S) ``` + +## Table of files about propositional logic + +The following table gives an overview of basic constructions in propositional +logic and related considerations. + +{{#include tables/propositional-logic.md}} diff --git a/src/foundation/subterminal-types.lagda.md b/src/foundation/subterminal-types.lagda.md index 317b7132b6..85ab7093da 100644 --- a/src/foundation/subterminal-types.lagda.md +++ b/src/foundation/subterminal-types.lagda.md @@ -23,8 +23,10 @@ open import foundation-core.propositions ## Idea -A type is said to be subterminal if it embeds into the unit type. A type is -subterminal if and only if it is a proposition. +A type is said to be {{#concept "subterminal" Agda=is-subterminal}} if it +[embeds](foundation-core.embeddings.md) into the +[unit type](foundation.unit-type.md). A type is subterminal if and only if it is +a [proposition](foundation-core.propositions.md). ## Definition @@ -82,3 +84,10 @@ module _ is-proof-irrelevant-is-subterminal H = is-proof-irrelevant-all-elements-equal (eq-is-subterminal H) ``` + +## Table of files about propositional logic + +The following table gives an overview of basic constructions in propositional +logic and related considerations. + +{{#include tables/propositional-logic.md}} diff --git a/src/foundation/subtypes.lagda.md b/src/foundation/subtypes.lagda.md index 33ae55ad1a..b181a9a10d 100644 --- a/src/foundation/subtypes.lagda.md +++ b/src/foundation/subtypes.lagda.md @@ -147,7 +147,7 @@ module _ {l2 l3 : Level} (P : subtype l2 A) (Q : subtype l3 A) → P ⊆ Q → Q ⊆ P → (x : A) → is-in-subtype P x ≃ is-in-subtype Q x equiv-antisymmetric-leq-subtype P Q H K x = - equiv-prop + equiv-iff-is-prop ( is-prop-is-in-subtype P x) ( is-prop-is-in-subtype Q x) ( H x) diff --git a/src/foundation/symmetric-difference.lagda.md b/src/foundation/symmetric-difference.lagda.md index 806c2bbce1..13308cf30a 100644 --- a/src/foundation/symmetric-difference.lagda.md +++ b/src/foundation/symmetric-difference.lagda.md @@ -10,8 +10,7 @@ module foundation.symmetric-difference where open import foundation.decidable-subtypes open import foundation.decidable-types open import foundation.dependent-pair-types -open import foundation.exclusive-disjunction -open import foundation.identity-types hiding (inv) +open import foundation.exclusive-sum open import foundation.intersections-subtypes open import foundation.universe-levels @@ -19,6 +18,7 @@ open import foundation-core.coproduct-types open import foundation-core.decidable-propositions open import foundation-core.equivalences open import foundation-core.function-types +open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes open import foundation-core.transport-along-identifications @@ -41,12 +41,14 @@ module _ symmetric-difference-subtype : subtype l1 X → subtype l2 X → subtype (l1 ⊔ l2) X - symmetric-difference-subtype P Q x = xor-Prop (P x) (Q x) + symmetric-difference-subtype P Q x = + exclusive-sum-Prop (P x) (Q x) symmetric-difference-decidable-subtype : decidable-subtype l1 X → decidable-subtype l2 X → decidable-subtype (l1 ⊔ l2) X - symmetric-difference-decidable-subtype P Q x = xor-Decidable-Prop (P x) (Q x) + symmetric-difference-decidable-subtype P Q x = + exclusive-sum-Decidable-Prop (P x) (Q x) ``` ## Properties diff --git a/src/foundation/truncated-types.lagda.md b/src/foundation/truncated-types.lagda.md index 3adb80b12a..d2248f9dab 100644 --- a/src/foundation/truncated-types.lagda.md +++ b/src/foundation/truncated-types.lagda.md @@ -13,6 +13,7 @@ open import elementary-number-theory.natural-numbers open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.logical-equivalences open import foundation.subtype-identity-principle open import foundation.truncation-levels open import foundation.univalence @@ -104,7 +105,7 @@ module _ equiv-is-trunc-equiv : A ≃ B → is-trunc k A ≃ is-trunc k B equiv-is-trunc-equiv e = - equiv-prop + equiv-iff-is-prop ( is-prop-is-trunc k A) ( is-prop-is-trunc k B) ( is-trunc-equiv' k A e) diff --git a/src/foundation/unions-subtypes.lagda.md b/src/foundation/unions-subtypes.lagda.md index d726dd99da..f1bd0aee75 100644 --- a/src/foundation/unions-subtypes.lagda.md +++ b/src/foundation/unions-subtypes.lagda.md @@ -37,7 +37,7 @@ module _ where union-subtype : subtype l1 X → subtype l2 X → subtype (l1 ⊔ l2) X - union-subtype P Q x = disjunction-Prop (P x) (Q x) + union-subtype P Q x = (P x) ∨ (Q x) ``` ### Unions of decidable subtypes diff --git a/src/foundation/unique-existence.lagda.md b/src/foundation/unique-existence.lagda.md deleted file mode 100644 index a60d5ee118..0000000000 --- a/src/foundation/unique-existence.lagda.md +++ /dev/null @@ -1,26 +0,0 @@ -# Unique existence - -```agda -module foundation.unique-existence where -``` - -
Imports - -```agda -open import foundation.universe-levels - -open import foundation-core.torsorial-type-families -``` - -
- -## Idea - -Unique existence `∃! A B` is defined as `Σ A B` being contractible. - -## Definition - -```agda -∃! : {l1 l2 : Level} → (A : UU l1) → (A → UU l2) → UU (l1 ⊔ l2) -∃! A B = is-torsorial B -``` diff --git a/src/foundation/uniqueness-quantification.lagda.md b/src/foundation/uniqueness-quantification.lagda.md new file mode 100644 index 0000000000..dbbce2a023 --- /dev/null +++ b/src/foundation/uniqueness-quantification.lagda.md @@ -0,0 +1,133 @@ +# Uniqueness quantification + +```agda +module foundation.uniqueness-quantification where +``` + +
Imports + +```agda +open import foundation.dependent-pair-types +open import foundation.torsorial-type-families +open import foundation.universe-levels + +open import foundation-core.contractible-types +open import foundation-core.function-types +open import foundation-core.identity-types +open import foundation-core.propositions +``` + +
+ +## Idea + +Given a predicate `P : A → Prop` we say there +{{#concept "uniquely exists" Disambiguation="in a subtype" WDID=Q2502253 WD="uniqueness quantification" Agda=∃!}} +_an `x : A` satisfying `P`_, if the [subtype](foundation-core.subtypes.md) +`Σ (x : A), (P x)` is [contractible](foundation-core.contractible-types.md). + +More generally, given a [structure](foundation.structure.md) `B : A → 𝒰` we say +there +{{#concept "uniquely exists" Disambiguation="structure" Agda=uniquely-exists-structure}} +_an `x : A` and a `y : B x`_, if the +[total type](foundation.dependent-pair-types.md) `Σ (x : A), (B x)` is +contractible. + +Note that the unique existence of structure is defined in the exact same way as +the concept of +[torsorial type families](foundation-core.torsorial-type-families.md). Whether +to use the concept of unique existence of a structure or the concept of +torsorial type family is dependent on the context. Torsoriality is used often to +emphasize the relation of the type family to the identity type, whereas +uniqueness of structure is used to emphasize the uniqueness of elements equipped +with further structure. For example, we tend to use unique existence in +combination with universal properties, in order to conclude that a certain map +equipped with some homotopies exists uniquely. + +As a special case of uniqueness quantification, we recover +[exclusive disjunction](foundation.exclusive-disjunction.md) when the indexing +type is a [2-element type](univalent-combinatorics.2-element-types.md). +Concretely, we have the equivalence + +```text + ∃! (t : bool), (P t) ≐ is-contr (Σ (t : bool), (P t)) + ≃ is-contr ((P false) + (P true)) + ≐ P false ⊻ P true. +``` + +## Definitions + +### Unique existence of structure + +```agda +module _ + {l1 l2 : Level} (A : UU l1) (B : A → UU l2) + where + + uniquely-exists-structure-Prop : Prop (l1 ⊔ l2) + uniquely-exists-structure-Prop = is-torsorial-Prop B + + uniquely-exists-structure : UU (l1 ⊔ l2) + uniquely-exists-structure = type-Prop uniquely-exists-structure-Prop + + is-prop-uniquely-exists-structure : is-prop uniquely-exists-structure + is-prop-uniquely-exists-structure = + is-prop-type-Prop uniquely-exists-structure-Prop +``` + +### Unique existence in a subtype + +```agda +module _ + {l1 l2 : Level} (A : UU l1) (P : A → Prop l2) + where + + uniquely-exists-Prop : Prop (l1 ⊔ l2) + uniquely-exists-Prop = uniquely-exists-structure-Prop A (type-Prop ∘ P) + + uniquely-exists : UU (l1 ⊔ l2) + uniquely-exists = type-Prop uniquely-exists-Prop + + is-prop-uniquely-exists : is-prop uniquely-exists + is-prop-uniquely-exists = is-prop-type-Prop uniquely-exists-Prop + + ∃! : Prop (l1 ⊔ l2) + ∃! = uniquely-exists-Prop +``` + +### Components of unique existence + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + pair-uniquely-exists : uniquely-exists-structure A B → Σ A B + pair-uniquely-exists = center + + pr1-uniquely-exists : uniquely-exists-structure A B → A + pr1-uniquely-exists = pr1 ∘ pair-uniquely-exists + + pr2-uniquely-exists : + (p : uniquely-exists-structure A B) → B (pr1-uniquely-exists p) + pr2-uniquely-exists = pr2 ∘ pair-uniquely-exists + + contraction-uniquely-exists : + (p : uniquely-exists-structure A B) → + (q : Σ A B) → pair-uniquely-exists p ＝ q + contraction-uniquely-exists = contraction +``` + +## See also + +- Unique existence is the indexed counterpart to + [exclusive disjunction](foundation.exclusive-disjunction.md). +- A different name for _unique existence_ is + [torsoriality](foundation.torsorial-type-families.md). + +## External links + +- [uniqueness quantifier](https://ncatlab.org/nlab/show/uniqueness+quantifier) + at $n$Lab +- [Uniqueness quantification](https://en.wikipedia.org/wiki/Uniqueness_quantification) + at Wikipedia diff --git a/src/foundation/universal-property-cartesian-product-types.lagda.md b/src/foundation/universal-property-cartesian-product-types.lagda.md index e2ff4ee524..fe91b41474 100644 --- a/src/foundation/universal-property-cartesian-product-types.lagda.md +++ b/src/foundation/universal-property-cartesian-product-types.lagda.md @@ -9,6 +9,7 @@ module foundation.universal-property-cartesian-product-types where ```agda open import foundation.cones-over-cospan-diagrams open import foundation.dependent-pair-types +open import foundation.logical-equivalences open import foundation.standard-pullbacks open import foundation.unit-type open import foundation.universal-property-dependent-pair-types @@ -74,6 +75,10 @@ module _ pr1 (map-inv-up-product (f , g) x) = f x pr2 (map-inv-up-product (f , g) x) = g x + iff-up-product : + ((x : X) → A x × B x) ↔ ((x : X) → A x) × ((x : X) → B x) + iff-up-product = (map-up-product , map-inv-up-product) + up-product : is-equiv map-up-product up-product = is-equiv-is-invertible (map-inv-up-product) (refl-htpy) (refl-htpy) diff --git a/src/foundation/universal-property-image.lagda.md b/src/foundation/universal-property-image.lagda.md index 1e1636519d..0d8de3ea76 100644 --- a/src/foundation/universal-property-image.lagda.md +++ b/src/foundation/universal-property-image.lagda.md @@ -104,7 +104,7 @@ abstract { B : UU l3} (i : B ↪ X) (q : hom-slice f (map-emb i)) → is-image' f i q → is-image f i q is-image-is-image' f i q up' C j = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-hom-slice (map-emb i) j) ( is-prop-hom-slice f j) ( up' C j) diff --git a/src/foundation/universal-property-propositional-truncation.lagda.md b/src/foundation/universal-property-propositional-truncation.lagda.md index bd74f3e7d0..61fbe14f5b 100644 --- a/src/foundation/universal-property-propositional-truncation.lagda.md +++ b/src/foundation/universal-property-propositional-truncation.lagda.md @@ -11,6 +11,7 @@ open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality open import foundation.functoriality-cartesian-product-types +open import foundation.logical-equivalences open import foundation.precomposition-functions-into-subuniverses open import foundation.subtype-identity-principle open import foundation.unit-type @@ -165,7 +166,7 @@ abstract extension-property-propositional-truncation P f → is-propositional-truncation P f is-propositional-truncation-extension-property P f up-P Q = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-Π (λ x → is-prop-type-Prop Q)) ( is-prop-Π (λ x → is-prop-type-Prop Q)) ( up-P Q) @@ -183,7 +184,7 @@ abstract is-propositional-truncation P' f' → is-equiv h is-equiv-is-ptruncation-is-ptruncation P P' f f' h H is-ptr-P is-ptr-P' = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-type-Prop P) ( is-prop-type-Prop P') ( map-inv-is-equiv (is-ptr-P' P) f) @@ -278,7 +279,7 @@ abstract {l1 l2 : Level} {A : UU l1} (P : Prop l2) (f : A → type-Prop P) → (g : type-Prop P → A) → is-propositional-truncation P f is-propositional-truncation-has-section {A = A} P f g Q = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-function-type (is-prop-type-Prop Q)) ( is-prop-function-type (is-prop-type-Prop Q)) ( λ h → h ∘ g) @@ -358,7 +359,7 @@ abstract ( is-equiv-comp ( λ h a a' → h a (f' a')) ( λ h a p' → h (f a) p') - ( is-ptr-f (pair (type-hom-Prop P' Q) (is-prop-type-hom-Prop P' Q))) + ( is-ptr-f (pair (type-hom-Prop P' Q) (is-prop-hom-Prop P' Q))) ( is-equiv-map-Π-is-fiberwise-equiv ( λ a → is-ptr-f' Q))) ``` diff --git a/src/foundation/universal-property-set-quotients.lagda.md b/src/foundation/universal-property-set-quotients.lagda.md index a4b7eb03fc..5e8ef52d96 100644 --- a/src/foundation/universal-property-set-quotients.lagda.md +++ b/src/foundation/universal-property-set-quotients.lagda.md @@ -20,7 +20,9 @@ open import foundation.function-extensionality open import foundation.fundamental-theorem-of-identity-types open import foundation.identity-types open import foundation.images +open import foundation.injective-maps open import foundation.locally-small-types +open import foundation.logical-equivalences open import foundation.propositional-extensionality open import foundation.propositional-truncations open import foundation.reflecting-maps-equivalence-relations @@ -43,7 +45,6 @@ open import foundation-core.fibers-of-maps open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies -open import foundation-core.injective-maps open import foundation-core.propositional-maps open import foundation-core.propositions open import foundation-core.small-types @@ -527,7 +528,7 @@ module _ P-Prop : (b : type-Set B) (x : type-Set X) → Prop (l1 ⊔ l3 ⊔ l) P-Prop b x = - ∃-Prop A + exists-structure-Prop A ( λ a → ( map-reflecting-map-equivalence-relation R f a ＝ x) × ( map-reflecting-map-equivalence-relation R q a ＝ b)) @@ -838,7 +839,7 @@ module _ pr1 equiv-equalizes-equal-values-extension-along-surjection-Set = equalizes-equal-values-extension-along-surjection-Set pr2 equiv-equalizes-equal-values-extension-along-surjection-Set = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-extension-along-surjection-Set f C g) ( is-prop-equalizes-equal-values-surjection-Set) ( extension-equalizes-equal-values-surjection-Set) diff --git a/src/foundation/universal-quantification.lagda.md b/src/foundation/universal-quantification.lagda.md new file mode 100644 index 0000000000..e4b0bd024f --- /dev/null +++ b/src/foundation/universal-quantification.lagda.md @@ -0,0 +1,151 @@ +# Universal quantification + +```agda +module foundation.universal-quantification where +``` + +
Imports + +```agda +open import foundation.dependent-pair-types +open import foundation.evaluation-functions +open import foundation.logical-equivalences +open import foundation.propositional-truncations +open import foundation.universe-levels + +open import foundation-core.equivalences +open import foundation-core.function-types +open import foundation-core.propositions +``` + +
+ +## Idea + +Given a type `A` and a [subtype](foundation-core.subtypes.md) `P : A → Prop`, +the +{{#concept "universal quantification" Disambiguation="on a subtype" WDID=Q126695 WD="universal quantification"}} + +```text + ∀ (x : A), (P x) +``` + +is the [proposition](foundation-core.propositions.md) that there exists a proof +of `P x` for every `x` in `A`. + +The +{{#concept "universal property" Disambiguation="of universal quantification" Agda=universal-property-for-all}} +of universal quantification states that it is the +[greatest lower bound](order-theory.greatest-lower-bounds-large-posets.md) on +the family of propositions `P` in the +[locale of propositions](foundation.large-locale-of-propositions.md), by which +we mean that for every proposition `Q` we have the +[logical equivalence](foundation.logical-equivalences.md) + +```text + (∀ (a : A), (R → P a)) ↔ (R → ∀ (a : A), (P a)) +``` + +**Notation.** Because of syntactic limitations of the Agda language, we cannot +use `∀` for the universal quantification in formalizations, and instead use +`∀'`. + +## Definitions + +### Universal quantification + +```agda +module _ + {l1 l2 : Level} (A : UU l1) (P : A → Prop l2) + where + + for-all-Prop : Prop (l1 ⊔ l2) + for-all-Prop = Π-Prop A P + + type-for-all-Prop : UU (l1 ⊔ l2) + type-for-all-Prop = type-Prop for-all-Prop + + is-prop-for-all-Prop : is-prop type-for-all-Prop + is-prop-for-all-Prop = is-prop-type-Prop for-all-Prop + + for-all : UU (l1 ⊔ l2) + for-all = type-for-all-Prop + + ∀' : Prop (l1 ⊔ l2) + ∀' = for-all-Prop +``` + +### The universal property of universal quantification + +The +{{#concept "universal property" Disambiguation="of universal quantification" Agda=universal-property-for-all}} +of the universal quantification `∀ (a : A), (P a)` states that for every +proposition `R`, the canonical map + +```text + (∀ (a : A), (R → P a)) → (R → ∀ (a : A), (P a)) +``` + +is a [logical equivalence](foundation.logical-equivalences.md). Indeed, this +holds for any type `R`. + +```agda +module _ + {l1 l2 : Level} (A : UU l1) (P : A → Prop l2) + where + + universal-property-for-all : {l3 : Level} (S : Prop l3) → UUω + universal-property-for-all S = + {l : Level} (R : Prop l) → + type-Prop ((∀' A (λ a → R ⇒ P a)) ⇔ (R ⇒ S)) + + ev-for-all : + {l : Level} {B : UU l} → + for-all A (λ a → function-Prop B (P a)) → B → for-all A P + ev-for-all f r a = f a r +``` + +## Properties + +### Universal quantification satisfies its universal property + +```agda +module _ + {l1 l2 : Level} (A : UU l1) (P : A → Prop l2) + where + + map-up-for-all : + {l : Level} {B : UU l} → + (B → for-all A P) → for-all A (λ a → function-Prop B (P a)) + map-up-for-all f a r = f r a + + is-equiv-ev-for-all : + {l : Level} {B : UU l} → is-equiv (ev-for-all A P {B = B}) + is-equiv-ev-for-all {B = B} = + is-equiv-has-converse + ( ∀' A (λ a → function-Prop B (P a))) + ( function-Prop B (∀' A P)) + ( map-up-for-all) + + up-for-all : universal-property-for-all A P (∀' A P) + up-for-all R = (ev-for-all A P , map-up-for-all) +``` + +## See also + +- Universal quantification is the indexed counterpart to + [conjunction](foundation.conjunction.md). + +## Table of files about propositional logic + +The following table gives an overview of basic constructions in propositional +logic and related considerations. + +{{#include tables/propositional-logic.md}} + +## External links + +- [universal quantifier](https://ncatlab.org/nlab/show/universal+quantifier) at + $n$Lab +- [Universal quantification](https://en.wikipedia.org/wiki/Universal_quantification) + at Wikipedia diff --git a/src/foundation/unordered-pairs.lagda.md b/src/foundation/unordered-pairs.lagda.md index fcc7e11206..3a6a00c0ad 100644 --- a/src/foundation/unordered-pairs.lagda.md +++ b/src/foundation/unordered-pairs.lagda.md @@ -107,7 +107,9 @@ module _ is-in-unordered-pair-Prop : {l : Level} {A : UU l} (p : unordered-pair A) (a : A) → Prop l is-in-unordered-pair-Prop p a = - ∃-Prop (type-unordered-pair p) (λ x → element-unordered-pair p x ＝ a) + exists-structure-Prop + ( type-unordered-pair p) + ( λ x → element-unordered-pair p x ＝ a) is-in-unordered-pair : {l : Level} {A : UU l} (p : unordered-pair A) (a : A) → UU l diff --git a/src/foundation/weak-limited-principle-of-omniscience.lagda.md b/src/foundation/weak-limited-principle-of-omniscience.lagda.md index ff5da85ae7..b4a805bc24 100644 --- a/src/foundation/weak-limited-principle-of-omniscience.lagda.md +++ b/src/foundation/weak-limited-principle-of-omniscience.lagda.md @@ -11,6 +11,7 @@ open import elementary-number-theory.natural-numbers open import foundation.disjunction open import foundation.negation +open import foundation.universal-quantification open import foundation.universe-levels open import foundation-core.propositions @@ -23,17 +24,23 @@ open import univalent-combinatorics.standard-finite-types ## Statement -The **Weak Limited Principle of Omniscience** asserts that for any sequence -`f : ℕ → Fin 2` either `f n ＝ 0` for all `n : ℕ` or not. In particular, it is a -restricted form of the law of excluded middle. +The {{#concept "Weak Limited Principle of Omniscience"}} asserts that for any +[sequence](foundation.sequences.md) `f : ℕ → Fin 2` either `f n ＝ 0` for all +`n : ℕ` or not. In particular, it is a restricted form of the +[law of excluded middle](foundation.law-of-excluded-middle.md). ```agda +WLPO-Prop : Prop lzero +WLPO-Prop = + ∀' + ( ℕ → Fin 2) + ( λ f → + disjunction-Prop + ( ∀' ℕ (λ n → Id-Prop (Fin-Set 2) (f n) (zero-Fin 1))) + ( ¬' (∀' ℕ (λ n → Id-Prop (Fin-Set 2) (f n) (zero-Fin 1))))) + WLPO : UU lzero -WLPO = - (f : ℕ → Fin 2) → - type-disjunction-Prop - ( Π-Prop ℕ (λ n → Id-Prop (Fin-Set 2) (f n) (zero-Fin 1))) - ( neg-Prop (Π-Prop ℕ (λ n → Id-Prop (Fin-Set 2) (f n) (zero-Fin 1)))) +WLPO = type-Prop WLPO-Prop ``` ## See also diff --git a/src/group-theory/cayleys-theorem.lagda.md b/src/group-theory/cayleys-theorem.lagda.md index 8a1459ba57..6b7594ab67 100644 --- a/src/group-theory/cayleys-theorem.lagda.md +++ b/src/group-theory/cayleys-theorem.lagda.md @@ -12,6 +12,7 @@ open import foundation.embeddings open import foundation.equivalence-extensionality open import foundation.identity-types open import foundation.injective-maps +open import foundation.sets open import foundation.universe-levels open import group-theory.embeddings-groups diff --git a/src/group-theory/commutator-subgroups.lagda.md b/src/group-theory/commutator-subgroups.lagda.md index 9309698b09..2316c49c7e 100644 --- a/src/group-theory/commutator-subgroups.lagda.md +++ b/src/group-theory/commutator-subgroups.lagda.md @@ -129,7 +129,7 @@ module _ ( generating-subset-commutator-subgroup-Group (conjugation-Group G x y)) ( λ where ( (y , z) , refl) → - intro-∃ + intro-exists ( conjugation-Group G x y , conjugation-Group G x z) ( inv (distributive-conjugation-commutator-Group G x y z))) diff --git a/src/group-theory/cyclic-groups.lagda.md b/src/group-theory/cyclic-groups.lagda.md index b51af13775..08c862b1fc 100644 --- a/src/group-theory/cyclic-groups.lagda.md +++ b/src/group-theory/cyclic-groups.lagda.md @@ -51,9 +51,7 @@ module _ is-cyclic-prop-Group : Prop l1 is-cyclic-prop-Group = - ∃-Prop - ( type-Group G) - ( is-generating-element-Group G) + ∃ (type-Group G) (is-generating-element-prop-Group G) is-cyclic-Group : UU l1 is-cyclic-Group = type-Prop is-cyclic-prop-Group @@ -136,13 +134,17 @@ module _ is-cyclic-has-generating-element-Group : is-cyclic-Group G → - {l : Level} → ∃ (type-Group G) (λ g → is-emb-ev-element-hom-Group' G g l) - is-cyclic-has-generating-element-Group H = + {l : Level} → + exists-structure + ( type-Group G) + ( λ g → is-emb-ev-element-hom-Group' G g l) + is-cyclic-has-generating-element-Group H {l} = apply-universal-property-trunc-Prop H - ( ∃-Prop (type-Group G) (λ g → is-emb-ev-element-hom-Group' G g _)) + ( exists-structure-Prop + ( type-Group G) + ( λ g → is-emb-ev-element-hom-Group' G g l)) ( λ (g , u) → - intro-∃ g - ( is-emb-ev-element-is-generating-element-Group G g u)) + intro-exists g (is-emb-ev-element-is-generating-element-Group G g u)) ``` ## See also diff --git a/src/group-theory/elements-of-finite-order-groups.lagda.md b/src/group-theory/elements-of-finite-order-groups.lagda.md index 083748e2e4..de05cf476b 100644 --- a/src/group-theory/elements-of-finite-order-groups.lagda.md +++ b/src/group-theory/elements-of-finite-order-groups.lagda.md @@ -47,10 +47,9 @@ module _ has-finite-order-element-prop-Group : Prop l1 has-finite-order-element-prop-Group = - ∃-Prop - ( nonzero-ℤ) + ∃ ( nonzero-ℤ) ( λ k → - has-same-elements-Subgroup ℤ-Group + has-same-elements-prop-Subgroup ℤ-Group ( subgroup-element-Group ℤ-Group (int-nonzero-ℤ k)) ( subgroup-order-element-Group G x)) ``` diff --git a/src/group-theory/integer-powers-of-elements-groups.lagda.md b/src/group-theory/integer-powers-of-elements-groups.lagda.md index 1e105e8922..47de6edf13 100644 --- a/src/group-theory/integer-powers-of-elements-groups.lagda.md +++ b/src/group-theory/integer-powers-of-elements-groups.lagda.md @@ -78,7 +78,7 @@ module _ is-integer-power-of-element-prop-Group : (x y : type-Group G) → Prop l is-integer-power-of-element-prop-Group x y = - ∃-Prop ℤ (λ k → integer-power-Group G k x ＝ y) + exists-structure-Prop ℤ (λ k → integer-power-Group G k x ＝ y) is-integer-power-of-element-Group : (x y : type-Group G) → UU l diff --git a/src/group-theory/isomorphisms-group-actions.lagda.md b/src/group-theory/isomorphisms-group-actions.lagda.md index 9337d23248..49635eee09 100644 --- a/src/group-theory/isomorphisms-group-actions.lagda.md +++ b/src/group-theory/isomorphisms-group-actions.lagda.md @@ -16,6 +16,7 @@ open import foundation.equivalences open import foundation.function-extensionality open import foundation.functoriality-dependent-pair-types open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.propositions open import foundation.subtypes open import foundation.torsorial-type-families @@ -196,7 +197,7 @@ module _ is-equiv-is-equiv-hom-is-iso-action-Group : is-equiv is-equiv-hom-is-iso-action-Group is-equiv-is-equiv-hom-is-iso-action-Group = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-is-iso-Large-Precategory ( action-Group-Large-Precategory G) {X = X} {Y = Y} f) ( is-property-is-equiv (map-hom-action-Group G X Y f)) @@ -205,7 +206,7 @@ module _ is-equiv-is-iso-is-equiv-hom-action-Group : is-equiv is-iso-is-equiv-hom-action-Group is-equiv-is-iso-is-equiv-hom-action-Group = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-property-is-equiv (map-hom-action-Group G X Y f)) ( is-prop-is-iso-Large-Precategory ( action-Group-Large-Precategory G) {X = X} {Y = Y} f) diff --git a/src/group-theory/nontrivial-groups.lagda.md b/src/group-theory/nontrivial-groups.lagda.md index 83a4e7c5a2..725296c895 100644 --- a/src/group-theory/nontrivial-groups.lagda.md +++ b/src/group-theory/nontrivial-groups.lagda.md @@ -49,7 +49,7 @@ module _ is-nontrivial-prop-Group : Prop l1 is-nontrivial-prop-Group = - ∃-Prop (type-Group G) (λ g → unit-Group G ≠ g) + exists-structure-Prop (type-Group G) (λ g → unit-Group G ≠ g) is-nontrivial-Group : UU l1 is-nontrivial-Group = @@ -70,7 +70,7 @@ module _ is-not-trivial-prop-Group : Prop l1 is-not-trivial-prop-Group = - neg-Prop' ((x : type-Group G) → unit-Group G ＝ x) + neg-type-Prop ((x : type-Group G) → unit-Group G ＝ x) is-not-trivial-Group : UU l1 is-not-trivial-Group = @@ -138,10 +138,7 @@ module _ ( f) ( forward-implication ( iff-eq (ap (λ T → subset-Subgroup G T x) α)) - ( inr-disjunction-Prop - ( Id-Prop (set-Group G) _ _) - ( P) - ( p)))) + ( inr-disjunction p))) ( λ q → map-left-unit-law-disjunction-is-empty-Prop ( Id-Prop (set-Group G) _ _) @@ -149,10 +146,7 @@ module _ ( f) ( backward-implication ( iff-eq (ap (λ T → subset-Subgroup G T x) α)) - ( inr-disjunction-Prop - ( Id-Prop (set-Group G) _ _) - ( Q) - ( q)))))) + ( inr-disjunction q))))) ``` ### If the map `subgroup-Prop G : Prop lzero → Subgroup l1 G` is an embedding, then `G` is not a trivial group diff --git a/src/group-theory/normal-closures-subgroups.lagda.md b/src/group-theory/normal-closures-subgroups.lagda.md index b978b2b76a..704d72a7b2 100644 --- a/src/group-theory/normal-closures-subgroups.lagda.md +++ b/src/group-theory/normal-closures-subgroups.lagda.md @@ -75,7 +75,7 @@ module _ generating-subset-normal-closure-Subgroup : subset-Group (l1 ⊔ l2) G generating-subset-normal-closure-Subgroup x = - ∃-Prop + exists-structure-Prop ( type-Group G) ( λ y → fiber (conjugation-Group G y ∘ inclusion-Subgroup G H) x) diff --git a/src/group-theory/powers-of-elements-monoids.lagda.md b/src/group-theory/powers-of-elements-monoids.lagda.md index 4cc4d89acd..66c4e4f41b 100644 --- a/src/group-theory/powers-of-elements-monoids.lagda.md +++ b/src/group-theory/powers-of-elements-monoids.lagda.md @@ -58,7 +58,7 @@ module _ is-power-of-element-prop-Monoid : (x y : type-Monoid M) → Prop l is-power-of-element-prop-Monoid x y = - ∃-Prop ℕ (λ n → power-Monoid M n x ＝ y) + exists-structure-Prop ℕ (λ n → power-Monoid M n x ＝ y) is-power-of-element-Monoid : (x y : type-Monoid M) → UU l is-power-of-element-Monoid x y = diff --git a/src/group-theory/quotient-groups.lagda.md b/src/group-theory/quotient-groups.lagda.md index f4c13fd08f..0f695c97ba 100644 --- a/src/group-theory/quotient-groups.lagda.md +++ b/src/group-theory/quotient-groups.lagda.md @@ -19,6 +19,7 @@ open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.functoriality-set-quotients open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.propositions open import foundation.reflecting-maps-equivalence-relations open import foundation.set-quotients @@ -584,7 +585,7 @@ module _ ( equivalence-relation-congruence-Normal-Subgroup G N) ( set-Group H)) ( λ f → - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-preserves-mul-Group (quotient-Group G N) H f) ( is-prop-preserves-mul-Group G H ( f ∘ map-quotient-hom-Group G N)) diff --git a/src/group-theory/subgroups-concrete-groups.lagda.md b/src/group-theory/subgroups-concrete-groups.lagda.md index 187b71f58b..eb86e36730 100644 --- a/src/group-theory/subgroups-concrete-groups.lagda.md +++ b/src/group-theory/subgroups-concrete-groups.lagda.md @@ -244,7 +244,7 @@ is-set-subgroup-Concrete-Group G X Y = ( transitive-action-subgroup-Concrete-Group G Y) ( pr1 e) ( pr1 f) - ( intro-∃ + ( intro-exists ( unit-coset-subgroup-Concrete-Group G X) ( pr2 e ∙ inv (pr2 f))))))) ``` diff --git a/src/group-theory/subgroups.lagda.md b/src/group-theory/subgroups.lagda.md index 3b2d7daa93..98631e94e6 100644 --- a/src/group-theory/subgroups.lagda.md +++ b/src/group-theory/subgroups.lagda.md @@ -671,11 +671,7 @@ module _ contains-unit-subgroup-Prop : contains-unit-subset-Group G subset-subgroup-Prop - contains-unit-subgroup-Prop = - inl-disjunction-Prop - ( Id-Prop (set-Group G) (unit-Group G) (unit-Group G)) - ( P) - ( refl) + contains-unit-subgroup-Prop = inl-disjunction refl is-closed-under-multiplication-subgroup-Prop' : (x y : type-Group G) → diff --git a/src/group-theory/substitution-functor-group-actions.lagda.md b/src/group-theory/substitution-functor-group-actions.lagda.md index ef24eda041..d8d73e7e03 100644 --- a/src/group-theory/substitution-functor-group-actions.lagda.md +++ b/src/group-theory/substitution-functor-group-actions.lagda.md @@ -135,7 +135,7 @@ module _ ( equivalence-relation-obj-left-adjoint-subst-action-Group X) ( h , x) ( h' , x') = - ∃-Prop + exists-structure-Prop ( type-Group G) ( λ g → ( Id (mul-Group H (map-hom-Group G H f g) h) h') × @@ -143,7 +143,7 @@ module _ pr1 ( pr2 (equivalence-relation-obj-left-adjoint-subst-action-Group X)) ( h , x) = - intro-∃ + intro-exists ( unit-Group G) ( pair ( ( ap (mul-Group' H h) (preserves-unit-hom-Group G H f)) ∙ @@ -162,7 +162,7 @@ module _ ( h' , x') ( h , x)) ( λ (g , p , q) → - intro-∃ + intro-exists ( inv-Group G g) ( ( ( ap ( mul-Group' H h') @@ -191,7 +191,7 @@ module _ ( h , x) ( h'' , x'')) ( λ (g' , p' , q') → - intro-∃ + intro-exists ( mul-Group G g' g) ( ( ( ap ( mul-Group' H h) diff --git a/src/group-theory/torsion-elements-groups.lagda.md b/src/group-theory/torsion-elements-groups.lagda.md index 5fbdd3842e..9b7a129185 100644 --- a/src/group-theory/torsion-elements-groups.lagda.md +++ b/src/group-theory/torsion-elements-groups.lagda.md @@ -54,7 +54,7 @@ module _ is-torsion-element-prop-Group : Prop l1 is-torsion-element-prop-Group = - ∃-Prop + exists-structure-Prop ( nonzero-ℤ) ( λ k → integer-power-Group G (int-nonzero-ℤ k) x ＝ unit-Group G) @@ -88,7 +88,7 @@ module _ is-torsion-element-unit-Group : is-torsion-element-Group G (unit-Group G) is-torsion-element-unit-Group = - intro-∃ + intro-exists ( one-nonzero-ℤ) ( integer-power-unit-Group G one-ℤ) diff --git a/src/group-theory/torsion-free-groups.lagda.md b/src/group-theory/torsion-free-groups.lagda.md index 4269e473a5..f207fffb76 100644 --- a/src/group-theory/torsion-free-groups.lagda.md +++ b/src/group-theory/torsion-free-groups.lagda.md @@ -17,6 +17,7 @@ open import foundation.equivalences open import foundation.existential-quantification open import foundation.fundamental-theorem-of-identity-types open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.propositions open import foundation.sets open import foundation.singleton-subtypes @@ -149,7 +150,7 @@ module _ is-torsion-free-has-unique-torsion-element-Group : has-unique-torsion-element-Group G → is-torsion-free-Group G is-torsion-free-has-unique-torsion-element-Group H x k p = - ap pr1 (eq-is-contr H {x , intro-∃ k p} {unit-torsion-element-Group G}) + ap pr1 (eq-is-contr H {x , intro-exists k p} {unit-torsion-element-Group G}) abstract has-unique-torsion-element-is-torsion-free-Group : @@ -158,11 +159,10 @@ module _ fundamental-theorem-id' ( λ where x refl → is-torsion-element-unit-Group G) ( λ x → - is-equiv-is-prop - ( is-set-type-Group G _ _) + is-equiv-has-converse-is-prop + ( is-set-type-Group G (unit-Group G) x) ( is-prop-is-torsion-element-Group G x) - ( elim-exists-Prop - ( λ k → Id-Prop (set-Group G) _ _) - ( Id-Prop (set-Group G) _ _) - ( λ k p → inv (H x k p)))) + ( elim-exists + ( Id-Prop (set-Group G) (unit-Group G) x) + ( λ k p → inv (H x k p)))) ``` diff --git a/src/group-theory/transitive-concrete-group-actions.lagda.md b/src/group-theory/transitive-concrete-group-actions.lagda.md index 5b56c26696..f6af5d47b3 100644 --- a/src/group-theory/transitive-concrete-group-actions.lagda.md +++ b/src/group-theory/transitive-concrete-group-actions.lagda.md @@ -278,7 +278,8 @@ module _ ( f) htpy-exists-equiv-transitive-action-Concrete-Group : - ∃ ( type-transitive-action-Concrete-Group G X) + exists-structure + ( type-transitive-action-Concrete-Group G X) ( λ x → map-equiv-transitive-action-Concrete-Group G X Y e x ＝ map-equiv-transitive-action-Concrete-Group G X Y f x) → diff --git a/src/higher-group-theory/cyclic-higher-groups.lagda.md b/src/higher-group-theory/cyclic-higher-groups.lagda.md index 34d954d1f2..d9067fb971 100644 --- a/src/higher-group-theory/cyclic-higher-groups.lagda.md +++ b/src/higher-group-theory/cyclic-higher-groups.lagda.md @@ -60,7 +60,7 @@ module _ is-cyclic-prop-∞-Group : Prop (l1 ⊔ lsuc l2) is-cyclic-prop-∞-Group = - ∃-Prop + exists-structure-Prop ( type-∞-Group G) ( λ g → (H : ∞-Group l2) → is-emb (ev-element-∞-Group G H g)) diff --git a/src/linear-algebra/vectors.lagda.md b/src/linear-algebra/vectors.lagda.md index 0fd59958df..a870ba74c9 100644 --- a/src/linear-algebra/vectors.lagda.md +++ b/src/linear-algebra/vectors.lagda.md @@ -51,7 +51,7 @@ infixr 10 _∷_ data vec {l : Level} (A : UU l) : ℕ → UU l where empty-vec : vec A zero-ℕ - _∷_ : ∀ {n} → A → vec A n → vec A (succ-ℕ n) + _∷_ : {n : ℕ} → A → vec A n → vec A (succ-ℕ n) module _ {l : Level} {A : UU l} diff --git a/src/lists/sorted-vectors.lagda.md b/src/lists/sorted-vectors.lagda.md index 2d7921986a..dbb2353d4f 100644 --- a/src/lists/sorted-vectors.lagda.md +++ b/src/lists/sorted-vectors.lagda.md @@ -100,7 +100,7 @@ module _ ```agda is-least-element-vec-is-leq-head-sorted-vec : {n : ℕ} - (x : type-Decidable-Total-Order X) + (x : type-Decidable-Total-Order X) (v : vec (type-Decidable-Total-Order X) (succ-ℕ n)) → is-sorted-vec v → leq-Decidable-Total-Order X x (head-vec v) → is-least-element-vec x v @@ -109,16 +109,16 @@ module _ is-least-element-vec-is-leq-head-sorted-vec {succ-ℕ n} x (y ∷ v) s p = p , ( is-least-element-vec-is-leq-head-sorted-vec + ( x) + ( v) + ( is-sorted-tail-is-sorted-vec (y ∷ v) s) + ( transitive-leq-Decidable-Total-Order + ( X) ( x) - ( v) - ( is-sorted-tail-is-sorted-vec (y ∷ v) s) - ( transitive-leq-Decidable-Total-Order - ( X) - ( x) - ( y) - ( head-vec v) - ( is-leq-head-head-tail-is-sorted-vec (y ∷ v) s) - ( p))) + ( y) + ( head-vec v) + ( is-leq-head-head-tail-is-sorted-vec (y ∷ v) s) + ( p))) ``` ### An equivalent definition of being sorted @@ -187,7 +187,7 @@ module _ (fv : functional-vec (type-Decidable-Total-Order X) n) → Prop l2 is-least-element-functional-vec-Prop n x fv = - Π-Prop (Fin n) λ k → leq-Decidable-Total-Order-Prop X x (fv k) + Π-Prop (Fin n) (λ k → leq-Decidable-Total-Order-Prop X x (fv k)) is-least-element-functional-vec : (n : ℕ) @@ -217,10 +217,10 @@ module _ is-least-element-vec-is-least-element-functional-vec (succ-ℕ n) x fv p = (p (inr star)) , ( is-least-element-vec-is-least-element-functional-vec - ( n) - ( x) - ( tail-functional-vec n fv) - ( p ∘ inl)) + ( n) + ( x) + ( tail-functional-vec n fv) + ( p ∘ inl)) is-least-element-functional-vec-is-least-element-vec : (n : ℕ) @@ -256,9 +256,9 @@ module _ ( x) ( functional-vec-vec n v ∘ map-equiv a) ( is-least-element-permute-functional-vec - ( n) - ( x) - ( functional-vec-vec n v) - ( a) - ( is-least-element-functional-vec-is-least-element-vec n x v p)) + ( n) + ( x) + ( functional-vec-vec n v) + ( a) + ( is-least-element-functional-vec-is-least-element-vec n x v p)) ``` diff --git a/src/order-theory/directed-families.lagda.md b/src/order-theory/directed-families.lagda.md index ec0e563f2a..e82194883e 100644 --- a/src/order-theory/directed-families.lagda.md +++ b/src/order-theory/directed-families.lagda.md @@ -8,10 +8,12 @@ module order-theory.directed-families where ```agda open import foundation.cartesian-product-types +open import foundation.conjunction open import foundation.dependent-pair-types open import foundation.existential-quantification open import foundation.inhabited-types open import foundation.propositions +open import foundation.universal-quantification open import foundation.universe-levels open import order-theory.posets @@ -32,15 +34,15 @@ is-directed-family-Poset-Prop : {l1 l2 l3 : Level} (P : Poset l1 l2) (I : Inhabited-Type l3) (x : type-Inhabited-Type I → type-Poset P) → Prop (l2 ⊔ l3) is-directed-family-Poset-Prop P I x = - Π-Prop + ∀' ( type-Inhabited-Type I) ( λ i → - Π-Prop + ∀' ( type-Inhabited-Type I) ( λ j → - ∃-Prop - ( type-Inhabited-Type I) - ( λ k → leq-Poset P (x i) (x k) × leq-Poset P (x j) (x k)))) + ∃ ( type-Inhabited-Type I) + ( λ k → + leq-Poset-Prop P (x i) (x k) ∧ leq-Poset-Prop P (x j) (x k)))) is-directed-family-Poset : {l1 l2 l3 : Level} (P : Poset l1 l2) (I : Inhabited-Type l3) diff --git a/src/order-theory/galois-connections.lagda.md b/src/order-theory/galois-connections.lagda.md index 111bc3090a..166d4d22a0 100644 --- a/src/order-theory/galois-connections.lagda.md +++ b/src/order-theory/galois-connections.lagda.md @@ -390,7 +390,7 @@ module _ (G H : Galois-Connection P Q) → is-equiv (htpy-upper-adjoint-htpy-lower-adjoint-Galois-Connection G H) is-equiv-htpy-upper-adjoint-htpy-lower-adjoint-Galois-Connection G H = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-htpy-lower-adjoint-Galois-Connection G H) ( is-prop-htpy-upper-adjoint-Galois-Connection P Q G H) ( htpy-lower-adjoint-htpy-upper-adjoint-Galois-Connection G H) @@ -399,7 +399,7 @@ module _ (G H : Galois-Connection P Q) → is-equiv (htpy-lower-adjoint-htpy-upper-adjoint-Galois-Connection G H) is-equiv-htpy-lower-adjoint-htpy-upper-adjoint-Galois-Connection G H = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-htpy-upper-adjoint-Galois-Connection P Q G H) ( is-prop-htpy-lower-adjoint-Galois-Connection G H) ( htpy-upper-adjoint-htpy-lower-adjoint-Galois-Connection G H) diff --git a/src/order-theory/greatest-lower-bounds-posets.lagda.md b/src/order-theory/greatest-lower-bounds-posets.lagda.md index 8a1f391c47..c6afa32f62 100644 --- a/src/order-theory/greatest-lower-bounds-posets.lagda.md +++ b/src/order-theory/greatest-lower-bounds-posets.lagda.md @@ -76,8 +76,8 @@ module _ {x y : type-Poset P} → is-greatest-binary-lower-bound-Poset P a b x → is-binary-lower-bound-Poset P a b y → leq-Poset P y x - forward-implication-is-greatest-binary-lower-bound-Poset H = - forward-implication (H _) + forward-implication-is-greatest-binary-lower-bound-Poset {x} {y} H = + forward-implication (H y) backward-implication-is-greatest-binary-lower-bound-Poset : {x y : type-Poset P} → @@ -100,9 +100,9 @@ module _ {x : type-Poset P} → is-greatest-binary-lower-bound-Poset P a b x → is-binary-lower-bound-Poset P a b x - is-binary-lower-bound-is-greatest-binary-lower-bound-Poset H = + is-binary-lower-bound-is-greatest-binary-lower-bound-Poset {x} H = backward-implication-is-greatest-binary-lower-bound-Poset H - ( refl-leq-Poset P _) + ( refl-leq-Poset P x) leq-left-is-greatest-binary-lower-bound-Poset : {x : type-Poset P} → @@ -209,23 +209,25 @@ module _ {x y : type-Poset P} → is-greatest-lower-bound-family-of-elements-Poset P a x → ((i : I) → leq-Poset P y (a i)) → leq-Poset P y x - forward-implication-is-greatest-lower-bound-family-of-elements-Poset H = - forward-implication (H _) + forward-implication-is-greatest-lower-bound-family-of-elements-Poset + { x} {y} H = + forward-implication (H y) backward-implication-is-greatest-lower-bound-family-of-elements-Poset : {x y : type-Poset P} → is-greatest-lower-bound-family-of-elements-Poset P a x → leq-Poset P y x → (i : I) → leq-Poset P y (a i) - backward-implication-is-greatest-lower-bound-family-of-elements-Poset H = - backward-implication (H _) + backward-implication-is-greatest-lower-bound-family-of-elements-Poset + {x} {y} H = + backward-implication (H y) is-lower-bound-is-greatest-lower-bound-family-of-elements-Poset : {x : type-Poset P} → is-greatest-lower-bound-family-of-elements-Poset P a x → is-lower-bound-family-of-elements-Poset P a x - is-lower-bound-is-greatest-lower-bound-family-of-elements-Poset H = + is-lower-bound-is-greatest-lower-bound-family-of-elements-Poset {x} H = backward-implication-is-greatest-lower-bound-family-of-elements-Poset H - ( refl-leq-Poset P _) + ( refl-leq-Poset P x) ``` ### The proposition that a family of elements has a greatest lower bound @@ -280,12 +282,12 @@ module _ is-greatest-lower-bound-family-of-elements-Poset P a x → is-greatest-lower-bound-family-of-elements-Poset P a y → x ＝ y - eq-is-greatest-lower-bound-family-of-elements-Poset H K = + eq-is-greatest-lower-bound-family-of-elements-Poset {x} {y} H K = ap ( pr1) ( all-elements-equal-has-greatest-lower-bound-family-of-elements-Poset ( P) ( a) - ( _ , H) - ( _ , K)) + ( x , H) + ( y , K)) ``` diff --git a/src/order-theory/inhabited-finite-total-orders.lagda.md b/src/order-theory/inhabited-finite-total-orders.lagda.md index d9908e0043..1a8d6c8968 100644 --- a/src/order-theory/inhabited-finite-total-orders.lagda.md +++ b/src/order-theory/inhabited-finite-total-orders.lagda.md @@ -38,8 +38,8 @@ module _ is-inhabited-Total-Order-𝔽 : UU (l1 ⊔ l2) is-inhabited-Total-Order-𝔽 = is-finite-Poset (poset-Total-Order-𝔽 P) - is-prop-is-inhabited-Total-Order-𝔽 : is-prop is-inhabited-Total-Order-𝔽 - is-prop-is-inhabited-Total-Order-𝔽 = + is-property-is-inhabited-Total-Order-𝔽 : is-prop is-inhabited-Total-Order-𝔽 + is-property-is-inhabited-Total-Order-𝔽 = is-prop-is-finite-Poset (poset-Total-Order-𝔽 P) is-finite-type-is-inhabited-Total-Order-𝔽 : diff --git a/src/order-theory/large-posets.lagda.md b/src/order-theory/large-posets.lagda.md index 8f6800de89..ebe0411daf 100644 --- a/src/order-theory/large-posets.lagda.md +++ b/src/order-theory/large-posets.lagda.md @@ -16,6 +16,7 @@ open import category-theory.precategories open import foundation.dependent-pair-types open import foundation.identity-types open import foundation.large-binary-relations +open import foundation.logical-equivalences open import foundation.propositions open import foundation.sets open import foundation.universe-levels @@ -172,7 +173,7 @@ module _ is-large-category-Large-Poset : is-large-category-Large-Precategory large-precategory-Large-Poset is-large-category-Large-Poset {l} x y = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-set-type-Large-Poset P x y) ( is-prop-iso-is-prop-hom-Precategory ( precategory-Large-Poset l) diff --git a/src/order-theory/least-upper-bounds-posets.lagda.md b/src/order-theory/least-upper-bounds-posets.lagda.md index 9052b87c52..7bddf850d8 100644 --- a/src/order-theory/least-upper-bounds-posets.lagda.md +++ b/src/order-theory/least-upper-bounds-posets.lagda.md @@ -76,8 +76,8 @@ module _ {x y : type-Poset P} → is-least-binary-upper-bound-Poset P a b x → is-binary-upper-bound-Poset P a b y → leq-Poset P x y - forward-implication-is-least-binary-upper-bound-Poset H = - forward-implication (H _) + forward-implication-is-least-binary-upper-bound-Poset {x} {y} H = + forward-implication (H y) backward-implication-is-least-binary-upper-bound-Poset : {x y : type-Poset P} → @@ -100,9 +100,9 @@ module _ {x : type-Poset P} → is-least-binary-upper-bound-Poset P a b x → is-binary-upper-bound-Poset P a b x - is-binary-upper-bound-is-least-binary-upper-bound-Poset H = + is-binary-upper-bound-is-least-binary-upper-bound-Poset {x} H = backward-implication-is-least-binary-upper-bound-Poset H - ( refl-leq-Poset P _) + ( refl-leq-Poset P x) leq-left-is-least-binary-upper-bound-Poset : {x : type-Poset P} → @@ -209,23 +209,23 @@ module _ {x y : type-Poset P} → is-least-upper-bound-family-of-elements-Poset P a x → is-upper-bound-family-of-elements-Poset P a y → leq-Poset P x y - forward-implication-is-least-upper-bound-family-of-elements-Poset H = - forward-implication (H _) + forward-implication-is-least-upper-bound-family-of-elements-Poset {x} {y} H = + forward-implication (H y) backward-implication-is-least-upper-bound-family-of-elements-Poset : {x y : type-Poset P} → is-least-upper-bound-family-of-elements-Poset P a x → leq-Poset P x y → is-upper-bound-family-of-elements-Poset P a y - backward-implication-is-least-upper-bound-family-of-elements-Poset H = - backward-implication (H _) + backward-implication-is-least-upper-bound-family-of-elements-Poset {x} {y} H = + backward-implication (H y) is-upper-bound-is-least-upper-bound-family-of-elements-Poset : {x : type-Poset P} → is-least-upper-bound-family-of-elements-Poset P a x → is-upper-bound-family-of-elements-Poset P a x - is-upper-bound-is-least-upper-bound-family-of-elements-Poset H = + is-upper-bound-is-least-upper-bound-family-of-elements-Poset {x} H = backward-implication-is-least-upper-bound-family-of-elements-Poset H - ( refl-leq-Poset P _) + ( refl-leq-Poset P x) ``` ### The proposition that a family of elements has a least upper bound @@ -280,12 +280,12 @@ module _ is-least-upper-bound-family-of-elements-Poset P a x → is-least-upper-bound-family-of-elements-Poset P a y → x ＝ y - eq-is-least-upper-bound-family-of-elements-Poset H K = + eq-is-least-upper-bound-family-of-elements-Poset {x} {y} H K = ap ( pr1) ( all-elements-equal-has-least-upper-bound-family-of-elements-Poset ( P) ( a) - ( _ , H) - ( _ , K)) + ( x , H) + ( y , K)) ``` diff --git a/src/order-theory/posets.lagda.md b/src/order-theory/posets.lagda.md index 488d467a33..e3e5d5e91b 100644 --- a/src/order-theory/posets.lagda.md +++ b/src/order-theory/posets.lagda.md @@ -16,6 +16,7 @@ open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.propositions open import foundation.sets open import foundation.universe-levels @@ -155,7 +156,7 @@ module _ is-category-precategory-Poset : is-category-Precategory precategory-Poset is-category-precategory-Poset x y = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-set-type-Poset X x y) ( is-prop-iso-is-prop-hom-Precategory precategory-Poset ( is-prop-leq-Poset X x y)) diff --git a/src/order-theory/total-preorders.lagda.md b/src/order-theory/total-preorders.lagda.md index 1c744e1e93..fdc75e0459 100644 --- a/src/order-theory/total-preorders.lagda.md +++ b/src/order-theory/total-preorders.lagda.md @@ -36,7 +36,7 @@ module _ incident-Preorder-Prop : (x y : type-Preorder X) → Prop l2 incident-Preorder-Prop x y = - disjunction-Prop (leq-Preorder-Prop X x y) (leq-Preorder-Prop X y x) + (leq-Preorder-Prop X x y) ∨ (leq-Preorder-Prop X y x) incident-Preorder : (x y : type-Preorder X) → UU l2 incident-Preorder x y = type-Prop (incident-Preorder-Prop x y) diff --git a/src/organic-chemistry/alkanes.lagda.md b/src/organic-chemistry/alkanes.lagda.md index 807e7c425a..ae5bc42c05 100644 --- a/src/organic-chemistry/alkanes.lagda.md +++ b/src/organic-chemistry/alkanes.lagda.md @@ -24,5 +24,6 @@ not have any double or triple carbon-carbon bonds. ```agda is-alkane-hydrocarbon : {l1 l2 : Level} → hydrocarbon l1 l2 → UU (l1 ⊔ l2) -is-alkane-hydrocarbon H = ∀ c → is-saturated-carbon-hydrocarbon H c +is-alkane-hydrocarbon H = + (c : vertex-hydrocarbon H) → is-saturated-carbon-hydrocarbon H c ``` diff --git a/src/organic-chemistry/ethane.lagda.md b/src/organic-chemistry/ethane.lagda.md index c795c7937b..43f02201ec 100644 --- a/src/organic-chemistry/ethane.lagda.md +++ b/src/organic-chemistry/ethane.lagda.md @@ -22,6 +22,7 @@ open import foundation.identity-types open import foundation.injective-maps open import foundation.propositional-truncations open import foundation.propositions +open import foundation.sets open import foundation.torsorial-type-families open import foundation.transport-along-identifications open import foundation.unit-type diff --git a/src/orthogonal-factorization-systems/factorizations-of-maps-function-classes.lagda.md b/src/orthogonal-factorization-systems/factorizations-of-maps-function-classes.lagda.md index 0a65bb2a43..31bf354643 100644 --- a/src/orthogonal-factorization-systems/factorizations-of-maps-function-classes.lagda.md +++ b/src/orthogonal-factorization-systems/factorizations-of-maps-function-classes.lagda.md @@ -8,7 +8,6 @@ module orthogonal-factorization-systems.factorizations-of-maps-function-classes ```agda open import foundation.action-on-identifications-functions -open import foundation.conjunction open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences @@ -65,7 +64,7 @@ module _ is-function-class-factorization-Prop : Prop (lL ⊔ lR) is-function-class-factorization-Prop = - conjunction-Prop + product-Prop ( L (left-map-factorization F)) ( R (right-map-factorization F)) diff --git a/src/orthogonal-factorization-systems/factorizations-of-maps-global-function-classes.lagda.md b/src/orthogonal-factorization-systems/factorizations-of-maps-global-function-classes.lagda.md index d3c3e3b691..49d6546998 100644 --- a/src/orthogonal-factorization-systems/factorizations-of-maps-global-function-classes.lagda.md +++ b/src/orthogonal-factorization-systems/factorizations-of-maps-global-function-classes.lagda.md @@ -8,7 +8,6 @@ module orthogonal-factorization-systems.factorizations-of-maps-global-function-c ```agda open import foundation.action-on-identifications-functions -open import foundation.conjunction open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences diff --git a/src/orthogonal-factorization-systems/function-classes.lagda.md b/src/orthogonal-factorization-systems/function-classes.lagda.md index cacff136e9..db1a440376 100644 --- a/src/orthogonal-factorization-systems/function-classes.lagda.md +++ b/src/orthogonal-factorization-systems/function-classes.lagda.md @@ -16,6 +16,7 @@ open import foundation.equivalences open import foundation.function-types open import foundation.identity-types open import foundation.iterated-dependent-product-types +open import foundation.logical-equivalences open import foundation.propositions open import foundation.pullbacks open import foundation.subtypes @@ -233,7 +234,7 @@ module _ is-equiv-has-identity-maps-has-equivalences-function-class : is-equiv has-identity-maps-has-equivalences-function-class is-equiv-has-identity-maps-has-equivalences-function-class = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-has-equivalences-function-class P) ( is-prop-has-identity-maps-function-class P) ( has-equivalences-has-identity-maps-function-class) @@ -248,7 +249,7 @@ module _ is-equiv-has-equivalences-has-identity-maps-function-class : is-equiv has-equivalences-has-identity-maps-function-class is-equiv-has-equivalences-has-identity-maps-function-class = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-has-identity-maps-function-class P) ( is-prop-has-equivalences-function-class P) ( has-identity-maps-has-equivalences-function-class) diff --git a/src/orthogonal-factorization-systems/local-types.lagda.md b/src/orthogonal-factorization-systems/local-types.lagda.md index 07c0e3ca8c..38b13c18ae 100644 --- a/src/orthogonal-factorization-systems/local-types.lagda.md +++ b/src/orthogonal-factorization-systems/local-types.lagda.md @@ -21,6 +21,7 @@ open import foundation.function-types open import foundation.functoriality-dependent-function-types open import foundation.homotopies open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.postcomposition-functions open import foundation.precomposition-dependent-functions open import foundation.precomposition-functions @@ -276,7 +277,7 @@ module _ (((y : Y) → A (f y)) → ((x : X) → A x)) → is-local-dependent-type f A is-local-dependent-type-is-prop A is-prop-A = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-Π is-prop-A) ( is-prop-Π (is-prop-A ∘ f)) diff --git a/src/orthogonal-factorization-systems/null-types.lagda.md b/src/orthogonal-factorization-systems/null-types.lagda.md index 7b75cf5b95..f2cc5610e0 100644 --- a/src/orthogonal-factorization-systems/null-types.lagda.md +++ b/src/orthogonal-factorization-systems/null-types.lagda.md @@ -11,6 +11,7 @@ open import foundation.constant-maps open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.fibers-of-maps +open import foundation.logical-equivalences open import foundation.precomposition-functions open import foundation.propositions open import foundation.type-arithmetic-unit-type @@ -77,7 +78,7 @@ module _ equiv-is-local-is-null : is-null Y A ≃ is-local (λ y → star) A equiv-is-local-is-null = - equiv-prop + equiv-iff-is-prop ( is-property-is-equiv (const Y A)) ( is-property-is-equiv (precomp (λ y → star) A)) ( is-local-is-null) diff --git a/src/primitives/machine-integers.lagda.md b/src/primitives/machine-integers.lagda.md index 7a285b84d7..84db3aa726 100644 --- a/src/primitives/machine-integers.lagda.md +++ b/src/primitives/machine-integers.lagda.md @@ -32,5 +32,5 @@ primitive primWord64ToNat : Word64 → ℕ primWord64FromNat : ℕ → Word64 primWord64ToNatInjective : - ∀ a b → primWord64ToNat a ＝ primWord64ToNat b → a ＝ b + (a b : Word64) → primWord64ToNat a ＝ primWord64ToNat b → a ＝ b ``` diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md index 599a3c4926..0604f765e2 100644 --- a/src/real-numbers/dedekind-real-numbers.lagda.md +++ b/src/real-numbers/dedekind-real-numbers.lagda.md @@ -33,6 +33,7 @@ open import foundation.sets open import foundation.subtypes open import foundation.transport-along-identifications open import foundation.truncated-types +open import foundation.universal-quantification open import foundation.universe-levels open import foundation-core.truncation-levels @@ -74,29 +75,15 @@ module _ is-dedekind-cut-Prop : Prop (l1 ⊔ l2) is-dedekind-cut-Prop = - product-Prop - ( product-Prop (exists-Prop ℚ L) (exists-Prop ℚ U)) - ( product-Prop - ( product-Prop - ( Π-Prop ℚ - ( λ q → - iff-Prop - ( L q) - ( exists-Prop ℚ (λ r → product-Prop (le-ℚ-Prop q r) (L r))))) - ( Π-Prop ℚ - ( λ r → - iff-Prop - ( U r) - ( exists-Prop ℚ (λ q → product-Prop (le-ℚ-Prop q r) (U q)))))) - ( product-Prop - ( Π-Prop ℚ (λ q → neg-Prop (product-Prop (L q) (U q)))) - ( Π-Prop ℚ - ( λ q → - Π-Prop ℚ - ( λ r → - implication-Prop - ( le-ℚ-Prop q r) - ( disjunction-Prop (L q) (U r))))))) + conjunction-Prop + ( (∃ ℚ L) ∧ (∃ ℚ U)) + ( conjunction-Prop + ( conjunction-Prop + ( ∀' ℚ ( λ q → L q ⇔ ∃ ℚ (λ r → le-ℚ-Prop q r ∧ L r))) + ( ∀' ℚ ( λ r → U r ⇔ ∃ ℚ (λ q → le-ℚ-Prop q r ∧ U q)))) + ( conjunction-Prop + ( ∀' ℚ (λ q → ¬' (L q ∧ U q))) + ( ∀' ℚ (λ q → ∀' ℚ (λ r → le-ℚ-Prop q r ⇒ (L q ∨ U r)))))) is-dedekind-cut : UU (l1 ⊔ l2) is-dedekind-cut = type-Prop is-dedekind-cut-Prop @@ -141,13 +128,15 @@ module _ is-rounded-lower-cut-ℝ : (q : ℚ) → - is-in-lower-cut-ℝ q ↔ ∃ ℚ (λ r → (le-ℚ q r) × (is-in-lower-cut-ℝ r)) + is-in-lower-cut-ℝ q ↔ + exists ℚ (λ r → (le-ℚ-Prop q r) ∧ (lower-cut-ℝ r)) is-rounded-lower-cut-ℝ = pr1 (pr1 (pr2 is-dedekind-cut-cut-ℝ)) is-rounded-upper-cut-ℝ : (r : ℚ) → - is-in-upper-cut-ℝ r ↔ ∃ ℚ (λ q → (le-ℚ q r) × (is-in-upper-cut-ℝ q)) + is-in-upper-cut-ℝ r ↔ + exists ℚ (λ q → (le-ℚ-Prop q r) ∧ (upper-cut-ℝ q)) is-rounded-upper-cut-ℝ = pr2 (pr1 (pr2 is-dedekind-cut-cut-ℝ)) @@ -156,7 +145,8 @@ module _ pr1 (pr2 (pr2 is-dedekind-cut-cut-ℝ)) is-located-lower-upper-cut-ℝ : - (q r : ℚ) → le-ℚ q r → (lower-cut-ℝ q) ∨ (upper-cut-ℝ r) + (q r : ℚ) → le-ℚ q r → + type-disjunction-Prop (lower-cut-ℝ q) (upper-cut-ℝ r) is-located-lower-upper-cut-ℝ = pr2 (pr2 (pr2 is-dedekind-cut-cut-ℝ)) @@ -182,16 +172,15 @@ abstract is-set-Σ ( is-set-function-type (is-trunc-Truncated-Type neg-one-𝕋)) ( λ x → - ( is-set-Σ + is-set-Σ ( is-set-function-type (is-trunc-Truncated-Type neg-one-𝕋)) ( λ y → ( is-set-is-prop ( is-prop-type-Prop - ( is-dedekind-cut-Prop x y)))))) + ( is-dedekind-cut-Prop x y))))) ℝ-Set : (l : Level) → Set (lsuc l) -pr1 (ℝ-Set l) = ℝ l -pr2 (ℝ-Set l) = is-set-ℝ l +ℝ-Set l = ℝ l , is-set-ℝ l ``` ## Properties of lower/upper Dedekind cuts @@ -288,13 +277,11 @@ module _ is-lower-complement-upper-cut-ℝ-Prop : (p q : ℚ) → Prop l is-lower-complement-upper-cut-ℝ-Prop p q = - product-Prop - ( le-ℚ-Prop p q) - ( neg-Prop ( upper-cut-ℝ x q)) + ( le-ℚ-Prop p q) ∧ (¬' (upper-cut-ℝ x q)) lower-complement-upper-cut-ℝ : subtype l ℚ lower-complement-upper-cut-ℝ p = - exists-Prop ℚ (is-lower-complement-upper-cut-ℝ-Prop p) + ∃ ℚ (is-lower-complement-upper-cut-ℝ-Prop p) ``` ```agda @@ -305,8 +292,7 @@ module _ subset-lower-cut-lower-complement-upper-cut-ℝ : lower-complement-upper-cut-ℝ x ⊆ lower-cut-ℝ x subset-lower-cut-lower-complement-upper-cut-ℝ p = - elim-exists-Prop - ( is-lower-complement-upper-cut-ℝ-Prop x p) + elim-exists ( lower-cut-ℝ x p) ( λ q I → map-right-unit-law-disjunction-is-empty-Prop @@ -318,12 +304,10 @@ module _ subset-lower-complement-upper-cut-lower-cut-ℝ : lower-cut-ℝ x ⊆ lower-complement-upper-cut-ℝ x subset-lower-complement-upper-cut-lower-cut-ℝ p H = - elim-exists-Prop - ( λ q → product-Prop (le-ℚ-Prop p q) (lower-cut-ℝ x q)) + elim-exists ( lower-complement-upper-cut-ℝ x p) ( λ q I → intro-exists - ( is-lower-complement-upper-cut-ℝ-Prop x p) ( q) ( map-product ( id) @@ -350,13 +334,11 @@ module _ is-upper-complement-lower-cut-ℝ-Prop : (q p : ℚ) → Prop l is-upper-complement-lower-cut-ℝ-Prop q p = - product-Prop - ( le-ℚ-Prop p q) - ( neg-Prop ( lower-cut-ℝ x p)) + (le-ℚ-Prop p q) ∧ (¬' (lower-cut-ℝ x p)) upper-complement-lower-cut-ℝ : subtype l ℚ upper-complement-lower-cut-ℝ q = - exists-Prop ℚ (is-upper-complement-lower-cut-ℝ-Prop q) + ∃ ℚ (is-upper-complement-lower-cut-ℝ-Prop q) ``` ```agda @@ -367,8 +349,7 @@ module _ subset-upper-cut-upper-complement-lower-cut-ℝ : upper-complement-lower-cut-ℝ x ⊆ upper-cut-ℝ x subset-upper-cut-upper-complement-lower-cut-ℝ q = - elim-exists-Prop - ( is-upper-complement-lower-cut-ℝ-Prop x q) + elim-exists ( upper-cut-ℝ x q) ( λ p I → map-left-unit-law-disjunction-is-empty-Prop @@ -380,12 +361,10 @@ module _ subset-upper-complement-lower-cut-upper-cut-ℝ : upper-cut-ℝ x ⊆ upper-complement-lower-cut-ℝ x subset-upper-complement-lower-cut-upper-cut-ℝ q H = - elim-exists-Prop - ( λ p → product-Prop (le-ℚ-Prop p q) (upper-cut-ℝ x p)) + elim-exists ( upper-complement-lower-cut-ℝ x q) ( λ p I → intro-exists - ( is-upper-complement-lower-cut-ℝ-Prop x q) ( p) ( map-product ( id) @@ -418,10 +397,9 @@ module _ ( eq-lower-cut-lower-complement-upper-cut-ℝ x) ( eq-lower-cut-lower-complement-upper-cut-ℝ y) ( λ p → - elim-exists-Prop - ( is-lower-complement-upper-cut-ℝ-Prop x p) + elim-exists ( lower-complement-upper-cut-ℝ y p) - ( λ q → intro-∃ q ∘ tot (λ _ K → K ∘ H q))) + ( λ q → intro-exists q ∘ tot (λ _ K → K ∘ H q))) subset-upper-cut-lower-cut-ℝ : lower-cut-ℝ x ⊆ lower-cut-ℝ y → upper-cut-ℝ y ⊆ upper-cut-ℝ x @@ -431,10 +409,9 @@ module _ ( eq-upper-cut-upper-complement-lower-cut-ℝ y) ( eq-upper-cut-upper-complement-lower-cut-ℝ x) ( λ q → - elim-exists-Prop - ( is-upper-complement-lower-cut-ℝ-Prop y q) + elim-exists ( upper-complement-lower-cut-ℝ x q) - ( λ p → intro-∃ p ∘ tot (λ _ K → K ∘ H p))) + ( λ p → intro-exists p ∘ tot (λ _ K → K ∘ H p))) module _ {l : Level} (x y : ℝ l) diff --git a/src/real-numbers/rational-real-numbers.lagda.md b/src/real-numbers/rational-real-numbers.lagda.md index eb34a051e8..777cecf09b 100644 --- a/src/real-numbers/rational-real-numbers.lagda.md +++ b/src/real-numbers/rational-real-numbers.lagda.md @@ -13,6 +13,7 @@ open import elementary-number-theory.strict-inequality-rational-numbers open import foundation.action-on-identifications-functions open import foundation.cartesian-product-types +open import foundation.conjunction open import foundation.dependent-pair-types open import foundation.disjunction open import foundation.embeddings @@ -56,17 +57,15 @@ is-dedekind-cut-le-ℚ : (λ (q : ℚ) → le-ℚ-Prop q x) (λ (r : ℚ) → le-ℚ-Prop x r) is-dedekind-cut-le-ℚ x = - ( left-∃-le-ℚ x , right-∃-le-ℚ x) , + ( exists-lesser-ℚ x , exists-greater-ℚ x) , ( ( λ (q : ℚ) → dense-le-ℚ q x , - elim-exists-Prop - ( λ r → product-Prop ( le-ℚ-Prop q r) ( le-ℚ-Prop r x)) + elim-exists ( le-ℚ-Prop q x) ( λ r (H , H') → transitive-le-ℚ q r x H' H)) , ( λ (r : ℚ) → α x r ∘ dense-le-ℚ x r , - elim-exists-Prop - ( λ q → product-Prop ( le-ℚ-Prop q r) ( le-ℚ-Prop x q)) + elim-exists ( le-ℚ-Prop x r) ( λ q (H , H') → transitive-le-ℚ x q r H H'))) , ( λ (q : ℚ) (H , H') → asymmetric-le-ℚ q x H H') , @@ -74,20 +73,12 @@ is-dedekind-cut-le-ℚ x = where α : (a b : ℚ) → - ∃ ℚ (λ r → le-ℚ a r × le-ℚ r b) → - ∃ ℚ (λ r → le-ℚ r b × le-ℚ a r) + exists ℚ (λ r → le-ℚ-Prop a r ∧ le-ℚ-Prop r b) → + exists ℚ (λ r → le-ℚ-Prop r b ∧ le-ℚ-Prop a r) α a b = - elim-exists-Prop - ( ( λ r → - product-Prop - ( le-ℚ-Prop a r) - ( le-ℚ-Prop r b))) - ( exists-Prop ℚ - ( λ r → - product-Prop - ( le-ℚ-Prop r b) - ( le-ℚ-Prop a r))) - ( λ r ( p , q) → intro-∃ r ( q , p)) + elim-exists + ( ∃ ℚ (λ r → le-ℚ-Prop r b ∧ le-ℚ-Prop a r)) + ( λ r ( p , q) → intro-exists r ( q , p)) ``` ### The canonical map from `ℚ` to `ℝ` @@ -110,9 +101,7 @@ module _ is-rational-ℝ-Prop : Prop l is-rational-ℝ-Prop = - product-Prop - ( neg-Prop (lower-cut-ℝ x p)) - ( neg-Prop (upper-cut-ℝ x p)) + (¬' (lower-cut-ℝ x p)) ∧ (¬' (upper-cut-ℝ x p)) is-rational-ℝ : UU l is-rational-ℝ = type-Prop is-rational-ℝ-Prop @@ -130,21 +119,19 @@ all-eq-is-rational-ℝ x p q H H' = left-case : le-ℚ p q → p ＝ q left-case I = ex-falso - ( elim-disjunction-Prop - ( lower-cut-ℝ x p) - ( upper-cut-ℝ x q) + ( elim-disjunction ( empty-Prop) - ( pr1 H , pr2 H') + ( pr1 H) + ( pr2 H') ( is-located-lower-upper-cut-ℝ x p q I)) right-case : le-ℚ q p → p ＝ q right-case I = ex-falso - ( elim-disjunction-Prop - ( lower-cut-ℝ x q) - ( upper-cut-ℝ x p) + ( elim-disjunction ( empty-Prop) - ( pr1 H' , pr2 H) + ( pr1 H') + ( pr2 H) ( is-located-lower-upper-cut-ℝ x q p I)) is-prop-rational-real : {l : Level} (x : ℝ l) → is-prop (Σ ℚ (is-rational-ℝ x)) @@ -204,12 +191,12 @@ eq-real-rational-is-rational-ℝ x q H = ( lower-cut-ℝ x) ( λ r → pair - ( λ I → elim-disjunction-Prop - ( lower-cut-ℝ x r) - ( upper-cut-ℝ x q) - ( lower-cut-ℝ x r) - ( id , λ H' → ex-falso (pr2 H H')) - ( is-located-lower-upper-cut-ℝ x r q I)) + ( λ I → + elim-disjunction + ( lower-cut-ℝ x r) + ( id) + ( λ H' → ex-falso (pr2 H H')) + ( is-located-lower-upper-cut-ℝ x r q I)) ( trichotomy-le-ℚ r q ( λ I _ → I) ( λ E H' → ex-falso (pr1 (tr (is-rational-ℝ x) (inv E) H) H')) diff --git a/src/ring-theory/nilpotent-elements-rings.lagda.md b/src/ring-theory/nilpotent-elements-rings.lagda.md index 1c96f4d478..96f8512935 100644 --- a/src/ring-theory/nilpotent-elements-rings.lagda.md +++ b/src/ring-theory/nilpotent-elements-rings.lagda.md @@ -79,7 +79,7 @@ is-nilpotent-element-neg-Ring R {x} H = apply-universal-property-trunc-Prop H ( is-nilpotent-element-ring-Prop R (neg-Ring R x)) ( λ (n , p) → - intro-∃ n + intro-exists n ( ( power-neg-Ring R n x) ∙ ( ( ap (mul-Ring R (power-Ring R n (neg-one-Ring R))) p) ∙ ( right-zero-law-mul-Ring R (power-Ring R n (neg-one-Ring R)))))) diff --git a/src/ring-theory/nilpotent-elements-semirings.lagda.md b/src/ring-theory/nilpotent-elements-semirings.lagda.md index 190f920d60..434bdf7fb8 100644 --- a/src/ring-theory/nilpotent-elements-semirings.lagda.md +++ b/src/ring-theory/nilpotent-elements-semirings.lagda.md @@ -37,7 +37,7 @@ number `n` such that `x^n = 0`. is-nilpotent-element-semiring-Prop : {l : Level} (R : Semiring l) → type-Semiring R → Prop l is-nilpotent-element-semiring-Prop R x = - ∃-Prop ℕ (λ n → power-Semiring R n x ＝ zero-Semiring R) + exists-structure-Prop ℕ (λ n → power-Semiring R n x ＝ zero-Semiring R) is-nilpotent-element-Semiring : {l : Level} (R : Semiring l) → type-Semiring R → UU l @@ -59,7 +59,7 @@ is-prop-is-nilpotent-element-Semiring R x = is-nilpotent-zero-Semiring : {l : Level} (R : Semiring l) → is-nilpotent-element-Semiring R (zero-Semiring R) -is-nilpotent-zero-Semiring R = intro-∃ 1 refl +is-nilpotent-zero-Semiring R = intro-exists 1 refl ``` ### If `x` and `y` commute and are both nilpotent, then `x + y` is nilpotent @@ -77,7 +77,7 @@ is-nilpotent-add-Semiring R x y H f h = apply-universal-property-trunc-Prop h ( is-nilpotent-element-semiring-Prop R (add-Semiring R x y)) ( λ (m , q) → - intro-∃ + intro-exists ( n +ℕ m) ( ( is-linear-combination-power-add-Semiring R n m x y H) ∙ ( ( ap-add-Semiring R @@ -104,7 +104,7 @@ module _ apply-universal-property-trunc-Prop f ( is-nilpotent-element-semiring-Prop R (mul-Semiring R x y)) ( λ (n , p) → - intro-∃ n + intro-exists n ( ( distributive-power-mul-Semiring R n H) ∙ ( ( ap (mul-Semiring' R (power-Semiring R n y)) p) ∙ ( left-zero-law-mul-Semiring R diff --git a/src/set-theory/countable-sets.lagda.md b/src/set-theory/countable-sets.lagda.md index 71c87d6103..38263b07f1 100644 --- a/src/set-theory/countable-sets.lagda.md +++ b/src/set-theory/countable-sets.lagda.md @@ -71,7 +71,8 @@ module _ is-surjective-map-enumeration E = pr2 E is-countable-Prop : Prop l - is-countable-Prop = ∃-Prop (ℕ → Maybe (type-Set X)) is-surjective + is-countable-Prop = + ∃ (ℕ → Maybe (type-Set X)) (is-surjective-Prop) is-countable : UU l is-countable = type-Prop (is-countable-Prop) @@ -94,7 +95,7 @@ module _ is-countable-Prop' : Prop (lsuc l ⊔ l) is-countable-Prop' = - ∃-Prop + exists-structure-Prop ( decidable-subtype l ℕ) ( λ P → type-decidable-subtype P ↠ type-Set X) @@ -113,7 +114,7 @@ surjective map `f : ℕ → X`. Let us call the latter as "directly countable". ```agda is-directly-countable-Prop : {l : Level} → Set l → Prop l is-directly-countable-Prop X = - ∃-Prop (ℕ → type-Set X) is-surjective + ∃ (ℕ → type-Set X) (is-surjective-Prop) is-directly-countable : {l : Level} → Set l → UU l is-directly-countable X = type-Prop (is-directly-countable-Prop X) diff --git a/src/set-theory/cumulative-hierarchy.lagda.md b/src/set-theory/cumulative-hierarchy.lagda.md index b908befe92..78520e6978 100644 --- a/src/set-theory/cumulative-hierarchy.lagda.md +++ b/src/set-theory/cumulative-hierarchy.lagda.md @@ -51,7 +51,7 @@ has-smaller-image : {A : UU l1} {B : UU l2} {C : UU l3} → (A → C) → (B → C) → UU (l1 ⊔ l2 ⊔ l3) has-smaller-image {l1} {l2} {l3} {A} {B} {C} f g = - (a : A) → ∃ B (λ b → g b ＝ f a) + (a : A) → exists-structure B (λ b → g b ＝ f a) has-same-image : { l1 l2 l3 : Level} → @@ -124,9 +124,9 @@ module _ ( e : has-same-image f g) ( IH₁ : (a : A) → P (f a)) ( IH₂ : (b : B) → P (g b)) → - ( ( a : A) → ∃ B (λ b → Σ (f a ＝ g b) + ( ( a : A) → exists-structure B (λ b → Σ (f a ＝ g b) ( λ p → tr P p (IH₁ a) ＝ IH₂ b))) → - ( ( b : B) → ∃ A (λ a → + ( ( b : B) → exists-structure A (λ a → Σ (g b ＝ f a) (λ p → tr P p (IH₂ b) ＝ IH₁ a))) → tr P (set-ext-pseudo-cumulative-hierarchy V f g e) (ρ f IH₁) ＝ ρ g IH₂) → (x : type-pseudo-cumulative-hierarchy V) → P x @@ -145,9 +145,9 @@ module _ ( e : has-same-image f g) ( IH₁ : (a : A) → P (f a)) ( IH₂ : (b : B) → P (g b)) → - ( (a : A) → ∃ B (λ b → Σ (f a ＝ g b) + ( (a : A) → exists-structure B (λ b → Σ (f a ＝ g b) ( λ p → tr P p (IH₁ a) ＝ IH₂ b))) → - ( (b : B) → ∃ A (λ a → Σ (g b ＝ f a) + ( (b : B) → exists-structure A (λ a → Σ (g b ＝ f a) (λ p → tr P p (IH₂ b) ＝ IH₁ a))) → tr P (set-ext-pseudo-cumulative-hierarchy V f g e) (ρ f IH₁) ＝ ρ g IH₂) → { A : UU l1} (f : A → type-pseudo-cumulative-hierarchy V) @@ -192,7 +192,7 @@ module _ ```agda underlying-function-cumulative-hierarchy : (v : type-pseudo-cumulative-hierarchy V) → - ∃ ( UU l1) + exists-structure ( UU l1) ( λ A → Σ ( A → type-pseudo-cumulative-hierarchy V) ( λ f → set-pseudo-cumulative-hierarchy V f ＝ v)) @@ -251,8 +251,8 @@ module _ ( e : has-same-image f g) ( IH₁ : A → X) ( IH₂ : B → X) → - ( (a : A) → ∃ B (λ b → (f a ＝ g b) × (IH₁ a ＝ IH₂ b))) → - ( (b : B) → ∃ A (λ a → (g b ＝ f a) × (IH₂ b ＝ IH₁ a))) → + ( (a : A) → exists-structure B (λ b → (f a ＝ g b) × (IH₁ a ＝ IH₂ b))) → + ( (b : B) → exists-structure A (λ a → (g b ＝ f a) × (IH₂ b ＝ IH₁ a))) → ρ f IH₁ ＝ ρ g IH₂) → type-pseudo-cumulative-hierarchy V → X recursion-principle-cumulative-hierarchy {l2} X σ ρ τ = @@ -262,9 +262,9 @@ module _ { A B : UU l1} (f : A → pr1 V) (g : B → pr1 V) ( e : has-same-image f g) ( IH₁ : (a : A) → X) (IH₂ : (b : B) → X) → - ( (a : A) → ∃ B (λ b → Σ (f a ＝ g b) + ( (a : A) → exists-structure B (λ b → Σ (f a ＝ g b) (λ p → tr (λ _ → X) p (IH₁ a) ＝ IH₂ b))) → - ( (b : B) → ∃ A (λ a → Σ (g b ＝ f a) + ( (b : B) → exists-structure A (λ a → Σ (g b ＝ f a) (λ p → tr (λ _ → X) p (IH₂ b) ＝ IH₁ a))) → tr (λ _ → X) (pr2 (pr2 (pr2 V)) f g e) (ρ f IH₁) ＝ ρ g IH₂ τ' {A} {B} f g e IH₁ IH₂ hIH₁ hIH₂ = @@ -279,13 +279,17 @@ module _ set-pseudo-cumulative-hierarchy V f ＝ set-pseudo-cumulative-hierarchy V g path-f-g = set-ext-pseudo-cumulative-hierarchy V f g e - hIH₁' : (a : A) → ∃ B (λ b → Σ (f a ＝ g b) (λ _ → IH₁ a ＝ IH₂ b)) + hIH₁' : + (a : A) → + exists-structure B (λ b → Σ (f a ＝ g b) (λ _ → IH₁ a ＝ IH₂ b)) hIH₁' a = map-trunc-Prop ( λ (b , p , q) → ( b , p , (inv (tr-constant-type-family p _) ∙ q))) ( hIH₁ a) - hIH₂' : (b : B) → ∃ A (λ a → Σ (g b ＝ f a) (λ _ → IH₂ b ＝ IH₁ a)) + hIH₂' : + (b : B) → + exists-structure A (λ a → Σ (g b ＝ f a) (λ _ → IH₂ b ＝ IH₁ a)) hIH₂' b = map-trunc-Prop ( λ (a , p , q) → @@ -301,8 +305,8 @@ module _ ( e : has-same-image f g) ( IH₁ : A → X) ( IH₂ : B → X) → - ( ( a : A) → ∃ B (λ b → (f a ＝ g b) × (IH₁ a ＝ IH₂ b))) → - ( ( b : B) → ∃ A (λ a → (g b ＝ f a) × (IH₂ b ＝ IH₁ a))) → + ( ( a : A) → exists-structure B (λ b → (f a ＝ g b) × (IH₁ a ＝ IH₂ b))) → + ( ( b : B) → exists-structure A (λ a → (g b ＝ f a) × (IH₂ b ＝ IH₁ a))) → ρ f IH₁ ＝ ρ g IH₂) → {A : UU l1} → (f : A → type-pseudo-cumulative-hierarchy V) → (IH : A → X) → recursion-principle-cumulative-hierarchy X σ ρ τ @@ -315,9 +319,9 @@ module _ { A B : UU l1} (f : A → pr1 V) (g : B → pr1 V) ( e : has-same-image f g) ( IH₁ : (a : A) → X) (IH₂ : (b : B) → X) → - ( ( a : A) → ∃ B (λ b → Σ (f a ＝ g b) + ( ( a : A) → exists-structure B (λ b → Σ (f a ＝ g b) ( λ p → tr (λ _ → X) p (IH₁ a) ＝ IH₂ b))) → - ( ( b : B) → ∃ A (λ a → Σ (g b ＝ f a) + ( ( b : B) → exists-structure A (λ a → Σ (g b ＝ f a) ( λ p → tr (λ _ → X) p (IH₂ b) ＝ IH₁ a))) → tr (λ _ → X) (pr2 (pr2 (pr2 V)) f g e) (ρ f IH₁) ＝ ρ g IH₂ τ' {A} {B} f g e IH₁ IH₂ hIH₁ hIH₂ = @@ -332,13 +336,17 @@ module _ set-pseudo-cumulative-hierarchy V f ＝ set-pseudo-cumulative-hierarchy V g path-f-g = set-ext-pseudo-cumulative-hierarchy V f g e - hIH₁' : (a : A) → ∃ B (λ b → Σ (f a ＝ g b) (λ _ → IH₁ a ＝ IH₂ b)) + hIH₁' : + (a : A) → + exists-structure B (λ b → Σ (f a ＝ g b) (λ _ → IH₁ a ＝ IH₂ b)) hIH₁' a = map-trunc-Prop ( λ (b , p , q) → ( b , p , (inv (tr-constant-type-family p _) ∙ q))) ( hIH₁ a) - hIH₂' : (b : B) → ∃ A (λ a → Σ (g b ＝ f a) (λ _ → IH₂ b ＝ IH₁ a)) + hIH₂' : + (b : B) → + exists-structure A (λ a → Σ (g b ＝ f a) (λ _ → IH₂ b ＝ IH₁ a)) hIH₂' b = map-trunc-Prop ( λ (a , p , q) → @@ -359,7 +367,7 @@ A simplification of the recursion principle, when the codomain is `Prop l2`. ( e : has-smaller-image f g) ( IH₁ : A → Prop l2) ( IH₂ : B → Prop l2) → - ( (a : A) → ∃ B (λ b → (f a ＝ g b) × (IH₁ a ＝ IH₂ b))) → + ( (a : A) → exists-structure B (λ b → (f a ＝ g b) × (IH₁ a ＝ IH₂ b))) → type-Prop (ρ f IH₁) → type-Prop (ρ g IH₂)) → type-pseudo-cumulative-hierarchy V → Prop l2 prop-recursion-principle-cumulative-hierarchy {l2} ρ τ = @@ -371,8 +379,8 @@ A simplification of the recursion principle, when the codomain is `Prop l2`. ( g : B → type-pseudo-cumulative-hierarchy V) → ( e : has-same-image f g) ( IH₁ : A → Prop l2) (IH₂ : B → Prop l2) → - ( (a : A) → ∃ B (λ b → (f a ＝ g b) × (IH₁ a ＝ IH₂ b))) → - ( (b : B) → ∃ A (λ a → (g b ＝ f a) × (IH₂ b ＝ IH₁ a))) → + ( (a : A) → exists-structure B (λ b → (f a ＝ g b) × (IH₁ a ＝ IH₂ b))) → + ( (b : B) → exists-structure A (λ a → (g b ＝ f a) × (IH₂ b ＝ IH₁ a))) → ρ f IH₁ ＝ ρ g IH₂ τ' f g (e₁ , e₂) IH₁ IH₂ hIH₁ hIH₂ = eq-iff (τ f g e₁ IH₁ IH₂ hIH₁) (τ g f e₂ IH₂ IH₁ hIH₂) @@ -388,7 +396,7 @@ A simplification of the recursion principle, when the codomain is `Prop l2`. ( e : has-smaller-image f g) ( IH₁ : A → Prop l2) ( IH₂ : B → Prop l2) → - ( (a : A) → ∃ B (λ b → (f a ＝ g b) × (IH₁ a ＝ IH₂ b))) → + ( (a : A) → exists-structure B (λ b → (f a ＝ g b) × (IH₁ a ＝ IH₂ b))) → type-Prop (ρ f IH₁) → type-Prop (ρ g IH₂)) → { A : UU l1} → (f : A → type-pseudo-cumulative-hierarchy V) → ( IH : A → Prop l2) → @@ -403,8 +411,8 @@ A simplification of the recursion principle, when the codomain is `Prop l2`. ( g : B → type-pseudo-cumulative-hierarchy V) → ( e : has-same-image f g) ( IH₁ : A → Prop l2) (IH₂ : B → Prop l2) → - ( (a : A) → ∃ B (λ b → (f a ＝ g b) × (IH₁ a ＝ IH₂ b))) → - ( (b : B) → ∃ A (λ a → (g b ＝ f a) × (IH₂ b ＝ IH₁ a))) → + ( (a : A) → exists-structure B (λ b → (f a ＝ g b) × (IH₁ a ＝ IH₂ b))) → + ( (b : B) → exists-structure A (λ a → (g b ＝ f a) × (IH₂ b ＝ IH₁ a))) → ρ f IH₁ ＝ ρ g IH₂ τ' f g (e₁ , e₂) IH₁ IH₂ hIH₁ hIH₂ = eq-iff (τ f g e₁ IH₁ IH₂ hIH₁) (τ g f e₂ IH₂ IH₁ hIH₂) @@ -452,16 +460,18 @@ needed. Prop (lsuc l1) ∈-cumulative-hierarchy-Prop x = simple-prop-recursion-principle-cumulative-hierarchy - (λ {A} f → (∃ A (λ a → f a ＝ x)) , is-prop-∃ _ _) e + ( λ {A} f → exists-structure-Prop A (λ a → f a ＝ x)) + ( e) where e : {A B : UU l1} (f : A → type-pseudo-cumulative-hierarchy V) ( g : B → type-pseudo-cumulative-hierarchy V) → ( e : has-smaller-image f g) → - ( ∃ A (λ a → f a ＝ x) → ∃ B (λ b → g b ＝ x)) + ( exists-structure A (λ a → f a ＝ x) → + exists-structure B (λ b → g b ＝ x)) e {A} {B} f g s = map-universal-property-trunc-Prop - ( ∃-Prop B (λ b → g b ＝ x)) + ( exists-structure-Prop B (λ b → g b ＝ x)) ( λ ( a , p) → map-trunc-Prop (λ (b , q) → (b , q ∙ p)) (s a)) @@ -476,7 +486,7 @@ needed. ( x : type-pseudo-cumulative-hierarchy V) {A : UU l1} ( f : A → type-pseudo-cumulative-hierarchy V) → ( ∈-cumulative-hierarchy x (set-pseudo-cumulative-hierarchy V f)) ＝ - ∃ A (λ a → f a ＝ x) + exists-structure A (λ a → f a ＝ x) id-∈-cumulative-hierarchy x f = ap pr1 (compute-simple-prop-recursion-principle-cumulative-hierarchy _ _ f) @@ -485,7 +495,7 @@ needed. { A : UU l1} { f : A → type-pseudo-cumulative-hierarchy V} → ( ∈-cumulative-hierarchy x (set-pseudo-cumulative-hierarchy V f)) → - ∃ A (λ a → f a ＝ x) + exists-structure A (λ a → f a ＝ x) ∈-cumulative-hierarchy-mere-preimage {x} {A} {f} = tr id (id-∈-cumulative-hierarchy x f) @@ -493,7 +503,7 @@ needed. { x : type-pseudo-cumulative-hierarchy V} → { A : UU l1} { f : A → type-pseudo-cumulative-hierarchy V} → - ∃ A (λ a → f a ＝ x) → + exists-structure A (λ a → f a ＝ x) → ( ∈-cumulative-hierarchy x (set-pseudo-cumulative-hierarchy V f)) mere-preimage-∈-cumulative-hierarchy {x} {A} {f} = tr id (inv (id-∈-cumulative-hierarchy x f)) @@ -549,7 +559,8 @@ needed. ( set-pseudo-cumulative-hierarchy V g) has-smaller-image-⊆-cumulative-hierarchy {A} {B} f g s x m = mere-preimage-∈-cumulative-hierarchy - ( map-universal-property-trunc-Prop (∃-Prop B (λ b → g b ＝ x)) + ( map-universal-property-trunc-Prop + ( exists-structure-Prop B (λ b → g b ＝ x)) ( λ (a , p) → map-trunc-Prop (λ (b , q) → (b , q ∙ p)) (s a)) ( ∈-cumulative-hierarchy-mere-preimage m)) @@ -723,15 +734,17 @@ needed. ( x : type-pseudo-cumulative-hierarchy V) → ( r : type-pseudo-cumulative-hierarchy V → type-pseudo-cumulative-hierarchy V) → - ∃ ( type-pseudo-cumulative-hierarchy V) + exists-structure + ( type-pseudo-cumulative-hierarchy V) ( λ v → ( y : type-pseudo-cumulative-hierarchy V) → ∈-cumulative-hierarchy y v ↔ - ∃ ( type-pseudo-cumulative-hierarchy V) + exists-structure + ( type-pseudo-cumulative-hierarchy V) ( λ z → (∈-cumulative-hierarchy z x) × (y ＝ r z))) replacement-cumulative-hierarchy x r = map-universal-property-trunc-Prop - ( ∃-Prop (type-pseudo-cumulative-hierarchy V) _) + ( exists-structure-Prop (type-pseudo-cumulative-hierarchy V) _) ( λ where ( A , f , refl) → unit-trunc-Prop @@ -750,7 +763,7 @@ needed. ( λ H → mere-preimage-∈-cumulative-hierarchy ( map-universal-property-trunc-Prop - ( ∃-Prop A _) + ( exists-structure-Prop A _) ( λ where ( z , K , refl) → map-trunc-Prop diff --git a/src/structured-types/pointed-isomorphisms.lagda.md b/src/structured-types/pointed-isomorphisms.lagda.md index 8c97d7ac80..f38534a262 100644 --- a/src/structured-types/pointed-isomorphisms.lagda.md +++ b/src/structured-types/pointed-isomorphisms.lagda.md @@ -13,6 +13,7 @@ open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.propositions open import foundation.retractions open import foundation.sections @@ -379,7 +380,7 @@ module _ pr1 equiv-is-pointed-iso-is-pointed-equiv = is-pointed-iso-is-pointed-equiv f pr2 equiv-is-pointed-iso-is-pointed-equiv = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-property-is-equiv (map-pointed-map f)) ( is-prop-is-pointed-iso f) ( is-pointed-equiv-is-pointed-iso f) diff --git a/src/structured-types/small-pointed-types.lagda.md b/src/structured-types/small-pointed-types.lagda.md index 11bbfa1986..c1ee5d36fa 100644 --- a/src/structured-types/small-pointed-types.lagda.md +++ b/src/structured-types/small-pointed-types.lagda.md @@ -112,7 +112,7 @@ module _ is-prop-is-pointed-small-Pointed-Type : is-prop (is-pointed-small-Pointed-Type l2 X) is-prop-is-pointed-small-Pointed-Type = - is-prop-is-inhabited + is-prop-has-element ( λ (Y , e) → is-prop-equiv' ( equiv-tot (λ Z → equiv-comp-pointed-equiv' e)) diff --git a/src/synthetic-homotopy-theory/joins-of-types.lagda.md b/src/synthetic-homotopy-theory/joins-of-types.lagda.md index 6d1e096c00..36a860e028 100644 --- a/src/synthetic-homotopy-theory/joins-of-types.lagda.md +++ b/src/synthetic-homotopy-theory/joins-of-types.lagda.md @@ -17,6 +17,7 @@ open import foundation.equivalences open import foundation.function-types open import foundation.homotopies open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.propositions open import foundation.type-arithmetic-cartesian-product-types open import foundation.type-arithmetic-dependent-pair-types @@ -318,13 +319,13 @@ module _ where cocone-disjunction : cocone pr1 pr2 (type-disjunction-Prop A B) - pr1 cocone-disjunction = inl-disjunction-Prop A B - pr1 (pr2 cocone-disjunction) = inr-disjunction-Prop A B + pr1 cocone-disjunction = inl-disjunction + pr1 (pr2 cocone-disjunction) = inr-disjunction pr2 (pr2 cocone-disjunction) (a , b) = eq-is-prop' - ( is-prop-type-disjunction-Prop A B) - ( inl-disjunction-Prop A B a) - ( inr-disjunction-Prop A B b) + ( is-prop-disjunction-Prop A B) + ( inl-disjunction a) + ( inr-disjunction b) map-disjunction-join-Prop : type-join-Prop A B → type-disjunction-Prop A B map-disjunction-join-Prop = @@ -332,15 +333,16 @@ module _ map-join-disjunction-Prop : type-disjunction-Prop A B → type-join-Prop A B map-join-disjunction-Prop = - elim-disjunction-Prop A B + elim-disjunction ( join-Prop A B) - ( inl-join-Prop A B , inr-join-Prop A B) + ( inl-join-Prop A B) + ( inr-join-Prop A B) is-equiv-map-disjunction-join-Prop : is-equiv map-disjunction-join-Prop is-equiv-map-disjunction-join-Prop = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-type-join-Prop A B) - ( is-prop-type-disjunction-Prop A B) + ( is-prop-disjunction-Prop A B) ( map-join-disjunction-Prop) equiv-disjunction-join-Prop : @@ -350,8 +352,8 @@ module _ is-equiv-map-join-disjunction-Prop : is-equiv map-join-disjunction-Prop is-equiv-map-join-disjunction-Prop = - is-equiv-is-prop - ( is-prop-type-disjunction-Prop A B) + is-equiv-has-converse-is-prop + ( is-prop-disjunction-Prop A B) ( is-prop-type-join-Prop A B) ( map-disjunction-join-Prop) @@ -369,10 +371,10 @@ module _ ( cocone-join) ( cocone-disjunction) ( map-disjunction-join-Prop) - ( ( λ _ → eq-is-prop (is-prop-type-disjunction-Prop A B)) , - ( λ _ → eq-is-prop (is-prop-type-disjunction-Prop A B)) , + ( ( λ _ → eq-is-prop (is-prop-disjunction-Prop A B)) , + ( λ _ → eq-is-prop (is-prop-disjunction-Prop A B)) , ( λ (a , b) → eq-is-contr - ( is-prop-type-disjunction-Prop A B + ( is-prop-disjunction-Prop A B ( horizontal-map-cocone pr1 pr2 ( cocone-map pr1 pr2 ( cocone-join) diff --git a/src/trees/bounded-multisets.lagda.md b/src/trees/bounded-multisets.lagda.md index ada81ad1ab..ceda35ae57 100644 --- a/src/trees/bounded-multisets.lagda.md +++ b/src/trees/bounded-multisets.lagda.md @@ -12,6 +12,7 @@ open import elementary-number-theory.natural-numbers open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.existential-quantification +open import foundation.propositions open import foundation.universe-levels open import trees.empty-multisets @@ -51,6 +52,16 @@ module _ is-empty-𝕍 X is-of-natural-height-𝕍 (succ-ℕ n) (tree-𝕎 X Y) = (x : X) → is-of-natural-height-𝕍 n (Y x) + + is-prop-is-of-natural-height-𝕍 : + (n : ℕ) (X : 𝕍 l) → is-prop (is-of-natural-height-𝕍 n X) + is-prop-is-of-natural-height-𝕍 zero-ℕ = is-property-is-empty-𝕍 + is-prop-is-of-natural-height-𝕍 (succ-ℕ n) (tree-𝕎 X Y) = + is-prop-Π (λ x → is-prop-is-of-natural-height-𝕍 n (Y x)) + + is-of-natural-height-prop-𝕍 : ℕ → 𝕍 l → Prop l + is-of-natural-height-prop-𝕍 n X = + ( is-of-natural-height-𝕍 n X , is-prop-is-of-natural-height-𝕍 n X) ``` ### Explicitly bounded multisets @@ -80,7 +91,8 @@ data {n : ℕ} {X : UU l} (Y : X → Bounded-𝕍 l n) → Bounded-𝕍 l (succ-ℕ n) Bounded-𝕍' : (l : Level) → UU (lsuc l) -Bounded-𝕍' l = Σ (𝕍 l) (λ X → ∃ ℕ (λ n → is-of-natural-height-𝕍 n X)) +Bounded-𝕍' l = + Σ (𝕍 l) (λ X → exists ℕ (λ n → is-of-natural-height-prop-𝕍 n X)) ``` ## Properties diff --git a/src/trees/directed-trees.lagda.md b/src/trees/directed-trees.lagda.md index 6af472ebe5..0198184e27 100644 --- a/src/trees/directed-trees.lagda.md +++ b/src/trees/directed-trees.lagda.md @@ -178,7 +178,8 @@ module _ where is-proper-node-Directed-Tree-Prop : node-Directed-Tree T → Prop l1 - is-proper-node-Directed-Tree-Prop x = neg-Prop' (is-root-Directed-Tree T x) + is-proper-node-Directed-Tree-Prop x = + neg-type-Prop (is-root-Directed-Tree T x) is-proper-node-Directed-Tree : node-Directed-Tree T → UU l1 is-proper-node-Directed-Tree x = diff --git a/src/trees/empty-multisets.lagda.md b/src/trees/empty-multisets.lagda.md index 880a5c680b..d31b3e8a6b 100644 --- a/src/trees/empty-multisets.lagda.md +++ b/src/trees/empty-multisets.lagda.md @@ -10,6 +10,7 @@ module trees.empty-multisets where open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.identity-types +open import foundation.propositions open import foundation.universe-levels open import trees.elementhood-relation-w-types @@ -35,6 +36,12 @@ module _ is-empty-𝕍 : 𝕍 l → UU l is-empty-𝕍 (tree-𝕎 X Y) = is-empty X + + is-property-is-empty-𝕍 : (X : 𝕍 l) → is-prop (is-empty-𝕍 X) + is-property-is-empty-𝕍 (tree-𝕎 X Y) = is-property-is-empty + + is-empty-prop-𝕍 : 𝕍 l → Prop l + is-empty-prop-𝕍 X = is-empty-𝕍 X , is-property-is-empty-𝕍 X ``` ### The predicate of being a multiset with no elements diff --git a/src/trees/lower-types-w-types.lagda.md b/src/trees/lower-types-w-types.lagda.md index 0f2c8176a7..5e283be7bb 100644 --- a/src/trees/lower-types-w-types.lagda.md +++ b/src/trees/lower-types-w-types.lagda.md @@ -42,5 +42,5 @@ module _ upper-bound-rank-inclusion-lower-𝕎 : {x : 𝕎 A B} (y : lower-𝕎 x) → inclusion-lower-𝕎 y ≼-𝕎 x upper-bound-rank-inclusion-lower-𝕎 (lower-tree-𝕎 g) y = - intro-∃ y (upper-bound-rank-inclusion-lower-𝕎 (g y)) + intro-exists y (upper-bound-rank-inclusion-lower-𝕎 (g y)) ``` diff --git a/src/trees/ranks-of-elements-w-types.lagda.md b/src/trees/ranks-of-elements-w-types.lagda.md index bc8589c340..26e9e63740 100644 --- a/src/trees/ranks-of-elements-w-types.lagda.md +++ b/src/trees/ranks-of-elements-w-types.lagda.md @@ -42,7 +42,7 @@ module _ _≼-𝕎-Prop_ : 𝕎 A B → 𝕎 A B → Prop (l1 ⊔ l2) (tree-𝕎 x α) ≼-𝕎-Prop (tree-𝕎 y β) = - Π-Prop (B x) (λ b → exists-Prop (B y) (λ c → (α b) ≼-𝕎-Prop (β c))) + Π-Prop (B x) (λ b → ∃ (B y) (λ c → (α b) ≼-𝕎-Prop (β c))) _≼-𝕎_ : 𝕎 A B → 𝕎 A B → UU (l1 ⊔ l2) x ≼-𝕎 y = type-Prop (x ≼-𝕎-Prop y) @@ -63,7 +63,7 @@ module _ _≺-𝕎-Prop_ : 𝕎 A B → 𝕎 A B → Prop (l1 ⊔ l2) x ≺-𝕎-Prop y = - exists-Prop (Σ (𝕎 A B) (λ w → w ∈-𝕎 y)) (λ t → x ≼-𝕎-Prop (pr1 t)) + ∃ (Σ (𝕎 A B) (λ w → w ∈-𝕎 y)) (λ t → x ≼-𝕎-Prop (pr1 t)) _≺-𝕎_ : 𝕎 A B → 𝕎 A B → UU (l1 ⊔ l2) x ≺-𝕎 y = type-Prop (x ≺-𝕎-Prop y) @@ -93,10 +93,9 @@ module _ Π-Prop ( u ∈-𝕎 x) ( λ H → - exists-Prop - ( 𝕎 A B) + ∃ ( 𝕎 A B) ( λ v → - exists-Prop (v ∈-𝕎 y) (λ K → u ≼-𝕎-Prop v)))) + ∃ (v ∈-𝕎 y) (λ K → u ≼-𝕎-Prop v)))) _strong-≼-𝕎_ : 𝕎 A B → 𝕎 A B → UU (l1 ⊔ l2) x strong-≼-𝕎 y = type-Prop (x strong-≼-𝕎-Prop y) @@ -126,11 +125,11 @@ module _ transitive-≼-𝕎 {tree-𝕎 x α} {tree-𝕎 y β} {tree-𝕎 z γ} H K a = apply-universal-property-trunc-Prop ( H a) - ( exists-Prop (B z) (λ c → (α a) ≼-𝕎-Prop (γ c))) + ( ∃ (B z) (λ c → (α a) ≼-𝕎-Prop (γ c))) ( λ t → apply-universal-property-trunc-Prop ( K (pr1 t)) - ( exists-Prop (B z) (λ c → (α a) ≼-𝕎-Prop (γ c))) + ( ∃ (B z) (λ c → (α a) ≼-𝕎-Prop (γ c))) ( λ s → unit-trunc-Prop ( pair @@ -153,30 +152,23 @@ module _ strong-≼-≼-𝕎 : {x y : 𝕎 A B} → (x ≼-𝕎 y) → (x strong-≼-𝕎 y) strong-≼-≼-𝕎 {tree-𝕎 x α} {tree-𝕎 y β} H .(α b) (pair b refl) = apply-universal-property-trunc-Prop (H b) - ( exists-Prop - ( 𝕎 A B) - ( (λ v → exists-Prop (v ∈-𝕎 tree-𝕎 y β) (λ hv → (α b) ≼-𝕎-Prop v)))) + ( ∃ ( 𝕎 A B) + ( (λ v → ∃ (v ∈-𝕎 tree-𝕎 y β) (λ hv → (α b) ≼-𝕎-Prop v)))) ( f) where f : Σ (B y) (λ c → pr1 (α b ≼-𝕎-Prop β c)) → exists ( 𝕎 A B) - ( λ v → exists-Prop (v ∈-𝕎 tree-𝕎 y β) (λ hv → α b ≼-𝕎-Prop v)) + ( λ v → ∃ (v ∈-𝕎 tree-𝕎 y β) (λ hv → α b ≼-𝕎-Prop v)) f (pair c K) = - intro-exists - ( λ v → exists-Prop (v ∈-𝕎 tree-𝕎 y β) (λ hv → α b ≼-𝕎-Prop v)) - ( β c) - ( intro-exists - ( λ hβc → α b ≼-𝕎-Prop β c) - ( pair c refl) - ( K)) + intro-exists (β c) ( intro-exists (pair c refl) K) ≼-strong-≼-𝕎 : {x y : 𝕎 A B} → (x strong-≼-𝕎 y) → (x ≼-𝕎 y) ≼-strong-≼-𝕎 {tree-𝕎 x α} {tree-𝕎 y β} H b = apply-universal-property-trunc-Prop - ( H (α b) (pair b refl)) - ( exists-Prop (B y) (λ c → α b ≼-𝕎-Prop β c)) + ( H (α b) (b , refl)) + ( ∃ (B y) (λ c → α b ≼-𝕎-Prop β c)) ( f) where f : @@ -185,11 +177,13 @@ module _ exists (B y) (λ c → α b ≼-𝕎-Prop β c) f (pair v K) = apply-universal-property-trunc-Prop K - ( exists-Prop (B y) (λ c → α b ≼-𝕎-Prop β c)) + ( ∃ (B y) (λ c → α b ≼-𝕎-Prop β c)) ( g) where - g : (v ∈-𝕎 tree-𝕎 y β) × (α b ≼-𝕎 v) → ∃ (B y) (λ c → α b ≼-𝕎 β c) - g (pair (pair c p) M) = intro-∃ c (tr (λ t → α b ≼-𝕎 t) (inv p) M) + g : + (v ∈-𝕎 tree-𝕎 y β) × (α b ≼-𝕎 v) → + exists (B y) (λ c → α b ≼-𝕎-Prop β c) + g (pair (pair c p) M) = intro-exists c (tr (λ t → α b ≼-𝕎 t) (inv p) M) ``` ### If `x ∈ y` then the rank of `x` is at most the rank of `y` @@ -202,7 +196,6 @@ module _ ≼-∈-𝕎 : {x y : 𝕎 A B} → (x ∈-𝕎 y) → (x ≼-𝕎 y) ≼-∈-𝕎 {tree-𝕎 x α} {tree-𝕎 y β} (pair v p) u = intro-exists - ( λ z → α u ≼-𝕎-Prop β z) ( v) ( tr ( λ t → α u ≼-𝕎 t) @@ -307,7 +300,6 @@ module _ g : Σ (Σ (𝕎 A B) (λ w → w ∈-𝕎 z)) (λ t → y ≼-𝕎 pr1 t) → x ≺-𝕎 z g (pair (pair v P) Q) = intro-exists - ( λ t → x ≼-𝕎-Prop (pr1 t)) ( pair v P) ( transitive-≼-𝕎 {x = x} {w} {v} M ( transitive-≼-𝕎 {x = w} {y} {v} (≼-∈-𝕎 L) Q)) diff --git a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md index 1cc4c33057..31511a5324 100644 --- a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md +++ b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md @@ -34,6 +34,7 @@ open import foundation.negated-equality open import foundation.negation open import foundation.propositional-truncations open import foundation.propositions +open import foundation.sets open import foundation.subtypes open import foundation.transport-along-identifications open import foundation.type-arithmetic-coproduct-types diff --git a/src/univalent-combinatorics/2-element-subtypes.lagda.md b/src/univalent-combinatorics/2-element-subtypes.lagda.md index b86c98de5c..782ef754ab 100644 --- a/src/univalent-combinatorics/2-element-subtypes.lagda.md +++ b/src/univalent-combinatorics/2-element-subtypes.lagda.md @@ -254,7 +254,7 @@ module _ Σ ( type-2-Element-Type X) ( λ x → ¬ ((y : type-2-Element-Type X) → Id (f x) (f y))) first-element h = - exists-not-not-forall-count (λ z → (w : type-2-Element-Type X) → + exists-not-not-for-all-count (λ z → (w : type-2-Element-Type X) → Id (f z) (f w)) (λ z → {!!}) {!!} {!!} two-elements-different-image : diff --git a/src/univalent-combinatorics/2-element-types.lagda.md b/src/univalent-combinatorics/2-element-types.lagda.md index 00a6a7d5dd..84068ec248 100644 --- a/src/univalent-combinatorics/2-element-types.lagda.md +++ b/src/univalent-combinatorics/2-element-types.lagda.md @@ -526,7 +526,7 @@ abstract ( ¬ (pr1 X)) ( apply-universal-property-trunc-Prop ( pr2 X) - ( double-negation-Prop' (pr1 X)) + ( double-negation-type-Prop (pr1 X)) ( λ e → intro-double-negation {l} (map-equiv e (zero-Fin 1)))) ( d X)) ``` diff --git a/src/univalent-combinatorics/binomial-types.lagda.md b/src/univalent-combinatorics/binomial-types.lagda.md index bf406c5b33..67c3f5a8a2 100644 --- a/src/univalent-combinatorics/binomial-types.lagda.md +++ b/src/univalent-combinatorics/binomial-types.lagda.md @@ -271,7 +271,7 @@ abstract ( pair (raise-empty-Prop _) map-inv-raise)))) ∘e ( right-distributive-Σ-coproduct ( Σ (Prop _) type-Prop) - ( Σ (Prop _) (¬ ∘ type-Prop)) + ( Σ (Prop _) (¬_ ∘ type-Prop)) ( ind-coproduct _ ( λ Q → mere-equiv (Maybe B) ((Σ A _) + (type-Prop (pr1 Q)))) diff --git a/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md b/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md index d300a6adc9..ce45d38d86 100644 --- a/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md +++ b/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md @@ -52,8 +52,8 @@ Consider a type family `B` over `A`, and consider the following statements 3. The elements of `Σ A B` can be counted. If 1 holds, then 2 holds if and only if 3 holds. Furthermore if 2 and 3 hold, -then 1 holds if and only if the elements of `Σ A (¬ ∘ B)` can be counted, i.e., -if the elements in the complement of `B` can be counted. +then 1 holds if and only if the elements of `Σ (x : A), ¬ (B x)` can be counted, +i.e., if the elements in the complement of `B` can be counted. ## Proofs @@ -168,12 +168,12 @@ count-base-count-Σ b e f = ``` More generally, if `Σ A B` and each `B x` can be counted, then `A` can be -counted if and only if the type `Σ A (¬ ∘ B)` can be counted. However, to avoid -having to invoke function extensionality, we show that if `Σ A B` and each `B x` -can be counted, then `A` can be counted if and only if +counted if and only if the type `Σ (x : A), ¬ (B x)` can be counted. However, to +avoid having to invoke function extensionality, we show that if `Σ A B` and each +`B x` can be counted, then `A` can be counted if and only if ```text - count (Σ A (λ x → is-zero-ℕ (number-of-elements-count (f x)))), + count (Σ A (λ x → is-zero-ℕ (number-of-elements-count (f x)))), ``` where `f : (x : A) → count (B x)`. Thus, we have a precise characterization of diff --git a/src/univalent-combinatorics/dependent-pair-types.lagda.md b/src/univalent-combinatorics/dependent-pair-types.lagda.md index 4e25dec6af..932efc1d7a 100644 --- a/src/univalent-combinatorics/dependent-pair-types.lagda.md +++ b/src/univalent-combinatorics/dependent-pair-types.lagda.md @@ -155,7 +155,7 @@ abstract ( ( right-unit-law-Σ-is-contr ( λ x → is-proof-irrelevant-is-prop - ( is-prop-is-inhabited-or-empty (B x)) + ( is-property-is-inhabited-or-empty (B x)) ( is-inhabited-or-empty-is-finite (g x)))) ∘e ( inv-equiv ( left-distributive-Σ-coproduct A diff --git a/src/univalent-combinatorics/embeddings-standard-finite-types.lagda.md b/src/univalent-combinatorics/embeddings-standard-finite-types.lagda.md index 69b7455380..458ec1b433 100644 --- a/src/univalent-combinatorics/embeddings-standard-finite-types.lagda.md +++ b/src/univalent-combinatorics/embeddings-standard-finite-types.lagda.md @@ -18,6 +18,7 @@ open import foundation.embeddings open import foundation.empty-types open import foundation.identity-types open import foundation.injective-maps +open import foundation.sets open import foundation.unit-type open import foundation.universe-levels diff --git a/src/univalent-combinatorics/embeddings.lagda.md b/src/univalent-combinatorics/embeddings.lagda.md index 625f1cd7da..5ec8168e06 100644 --- a/src/univalent-combinatorics/embeddings.lagda.md +++ b/src/univalent-combinatorics/embeddings.lagda.md @@ -19,6 +19,7 @@ open import foundation.empty-types open import foundation.equivalences open import foundation.propositional-truncations open import foundation.propositions +open import foundation.sets open import foundation.unit-type open import foundation.universe-levels diff --git a/src/univalent-combinatorics/equality-standard-finite-types.lagda.md b/src/univalent-combinatorics/equality-standard-finite-types.lagda.md index d8bc06357e..2ed5c9e278 100644 --- a/src/univalent-combinatorics/equality-standard-finite-types.lagda.md +++ b/src/univalent-combinatorics/equality-standard-finite-types.lagda.md @@ -21,6 +21,7 @@ open import foundation.empty-types open import foundation.equivalences open import foundation.functoriality-coproduct-types open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.negated-equality open import foundation.negation open import foundation.propositions @@ -77,7 +78,7 @@ extensionality-Fin : (x ＝ y) ≃ (Eq-Fin k x y) pr1 (extensionality-Fin k x y) = Eq-Fin-eq k pr2 (extensionality-Fin k x y) = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-set-Fin k x y) ( is-prop-Eq-Fin k x y) ( eq-Eq-Fin k) diff --git a/src/univalent-combinatorics/fibers-of-maps.lagda.md b/src/univalent-combinatorics/fibers-of-maps.lagda.md index ad8da9ae05..914e7cabd5 100644 --- a/src/univalent-combinatorics/fibers-of-maps.lagda.md +++ b/src/univalent-combinatorics/fibers-of-maps.lagda.md @@ -18,6 +18,7 @@ open import foundation.equality-dependent-pair-types open import foundation.equivalences open import foundation.functoriality-dependent-pair-types open import foundation.identity-types +open import foundation.logical-equivalences open import foundation.propositional-truncations open import foundation.propositions open import foundation.sections @@ -146,7 +147,7 @@ equiv-is-finite-domain-is-finite-fiber : (B : 𝔽 l2) (f : A → (type-𝔽 B)) → ((b : type-𝔽 B) → is-finite (fiber f b)) ≃ is-finite A equiv-is-finite-domain-is-finite-fiber {A = A} B f = - equiv-prop + equiv-iff-is-prop ( is-prop-Π (λ b → is-prop-is-finite (fiber f b))) ( is-prop-is-finite A) ( λ P → diff --git a/src/univalent-combinatorics/finite-types.lagda.md b/src/univalent-combinatorics/finite-types.lagda.md index dcee18540b..5da3f667e3 100644 --- a/src/univalent-combinatorics/finite-types.lagda.md +++ b/src/univalent-combinatorics/finite-types.lagda.md @@ -25,6 +25,7 @@ open import foundation.functoriality-dependent-pair-types open import foundation.functoriality-propositional-truncation open import foundation.identity-types open import foundation.inhabited-types +open import foundation.logical-equivalences open import foundation.mere-equivalences open import foundation.propositional-truncations open import foundation.propositions @@ -510,7 +511,7 @@ compute-total-UU-Fin : {l : Level} → Σ ℕ (UU-Fin l) ≃ 𝔽 l compute-total-UU-Fin = ( equiv-tot ( λ X → - equiv-prop + equiv-iff-is-prop ( is-prop-has-finite-cardinality) ( is-prop-is-finite X) ( is-finite-has-finite-cardinality) @@ -697,7 +698,7 @@ abstract ( has-finite-cardinality-is-finite H) ( pair n Q)) pr2 (equiv-has-cardinality-id-number-of-elements-is-finite X H n) = - is-equiv-is-prop + is-equiv-has-converse-is-prop ( is-prop-type-trunc-Prop) ( is-set-ℕ (number-of-elements-is-finite H) n) ( λ p → diff --git a/src/univalent-combinatorics/inequality-types-with-counting.lagda.md b/src/univalent-combinatorics/inequality-types-with-counting.lagda.md index 0a0a2fcd3d..137bb6bd70 100644 --- a/src/univalent-combinatorics/inequality-types-with-counting.lagda.md +++ b/src/univalent-combinatorics/inequality-types-with-counting.lagda.md @@ -13,6 +13,7 @@ open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.identity-types open import foundation.injective-maps +open import foundation.sets open import foundation.universe-levels open import univalent-combinatorics.counting diff --git a/src/univalent-combinatorics/kuratowsky-finite-sets.lagda.md b/src/univalent-combinatorics/kuratowsky-finite-sets.lagda.md index dd9d41741b..adc3eb2069 100644 --- a/src/univalent-combinatorics/kuratowsky-finite-sets.lagda.md +++ b/src/univalent-combinatorics/kuratowsky-finite-sets.lagda.md @@ -37,7 +37,7 @@ the set-quotient of a standard finite type. ```agda is-kuratowsky-finite-set-Prop : {l : Level} → Set l → Prop l is-kuratowsky-finite-set-Prop X = - ∃-Prop ℕ (λ n → Fin n ↠ type-Set X) + exists-structure-Prop ℕ (λ n → Fin n ↠ type-Set X) is-kuratowsky-finite-set : {l : Level} → Set l → UU l is-kuratowsky-finite-set X = type-Prop (is-kuratowsky-finite-set-Prop X) diff --git a/src/univalent-combinatorics/pi-finite-types.lagda.md b/src/univalent-combinatorics/pi-finite-types.lagda.md index d530ac77ea..9f4a0b0633 100644 --- a/src/univalent-combinatorics/pi-finite-types.lagda.md +++ b/src/univalent-combinatorics/pi-finite-types.lagda.md @@ -143,9 +143,9 @@ mere-equiv-number-of-connected-components H = is-π-finite-Prop : {l : Level} (k : ℕ) → UU l → Prop l is-π-finite-Prop zero-ℕ X = has-finite-connected-components-Prop X is-π-finite-Prop (succ-ℕ k) X = - product-Prop ( is-π-finite-Prop zero-ℕ X) - ( Π-Prop X - ( λ x → Π-Prop X (λ y → is-π-finite-Prop k (Id x y)))) + product-Prop + ( is-π-finite-Prop zero-ℕ X) + ( Π-Prop X (λ x → Π-Prop X (λ y → is-π-finite-Prop k (Id x y)))) is-π-finite : {l : Level} (k : ℕ) → UU l → UU l is-π-finite k X = type-Prop (is-π-finite-Prop k X) diff --git a/src/univalent-combinatorics/pigeonhole-principle.lagda.md b/src/univalent-combinatorics/pigeonhole-principle.lagda.md index 97927d8ab5..8c52386070 100644 --- a/src/univalent-combinatorics/pigeonhole-principle.lagda.md +++ b/src/univalent-combinatorics/pigeonhole-principle.lagda.md @@ -24,6 +24,7 @@ open import foundation.negation open import foundation.pairs-of-distinct-elements open import foundation.propositional-truncations open import foundation.propositions +open import foundation.sets open import foundation.unit-type open import foundation.universe-levels open import foundation.whiskering-homotopies-composition diff --git a/src/univalent-combinatorics/repetitions-of-values.lagda.md b/src/univalent-combinatorics/repetitions-of-values.lagda.md index 9fe994bec5..c38d663bcd 100644 --- a/src/univalent-combinatorics/repetitions-of-values.lagda.md +++ b/src/univalent-combinatorics/repetitions-of-values.lagda.md @@ -18,6 +18,7 @@ open import foundation.identity-types open import foundation.injective-maps open import foundation.negated-equality open import foundation.negation +open import foundation.sets open import univalent-combinatorics.decidable-dependent-function-types open import univalent-combinatorics.decidable-propositions @@ -48,7 +49,7 @@ repetition-of-values-is-not-injective-Fin k l f N = where u : Σ (Fin k) (λ x → ¬ ((y : Fin k) → f x ＝ f y → x ＝ y)) u = - exists-not-not-forall-Fin k + exists-not-not-for-all-Fin k ( λ x → is-decidable-Π-Fin k ( λ y → @@ -61,7 +62,7 @@ repetition-of-values-is-not-injective-Fin k l f N = H : ¬ ((y : Fin k) → f x ＝ f y → x ＝ y) H = pr2 u v : Σ (Fin k) (λ y → ¬ (f x ＝ f y → x ＝ y)) - v = exists-not-not-forall-Fin k + v = exists-not-not-for-all-Fin k ( λ y → is-decidable-function-type ( has-decidable-equality-Fin l (f x) (f y)) @@ -72,7 +73,7 @@ repetition-of-values-is-not-injective-Fin k l f N = K : ¬ (f x ＝ f y → x ＝ y) K = pr2 v w : (f x ＝ f y) × (x ≠ y) - w = exists-not-not-forall-count + w = exists-not-not-for-all-count ( λ _ → Id x y) ( λ _ → has-decidable-equality-Fin k x y) diff --git a/src/univalent-combinatorics/retracts-of-finite-types.lagda.md b/src/univalent-combinatorics/retracts-of-finite-types.lagda.md index bc30866f9d..f12270336a 100644 --- a/src/univalent-combinatorics/retracts-of-finite-types.lagda.md +++ b/src/univalent-combinatorics/retracts-of-finite-types.lagda.md @@ -18,6 +18,7 @@ open import foundation.functoriality-propositional-truncation open import foundation.injective-maps open import foundation.retractions open import foundation.retracts-of-types +open import foundation.sets open import foundation.universe-levels open import univalent-combinatorics.counting diff --git a/src/univalent-combinatorics/sequences-finite-types.lagda.md b/src/univalent-combinatorics/sequences-finite-types.lagda.md index b9d113365b..e9cd8106af 100644 --- a/src/univalent-combinatorics/sequences-finite-types.lagda.md +++ b/src/univalent-combinatorics/sequences-finite-types.lagda.md @@ -24,6 +24,7 @@ open import foundation.pairs-of-distinct-elements open import foundation.repetitions-of-values open import foundation.repetitions-sequences open import foundation.sequences +open import foundation.sets open import foundation.universe-levels open import univalent-combinatorics.counting diff --git a/src/univalent-combinatorics/sigma-decompositions.lagda.md b/src/univalent-combinatorics/sigma-decompositions.lagda.md index 1d1a8ff667..f45a3477c8 100644 --- a/src/univalent-combinatorics/sigma-decompositions.lagda.md +++ b/src/univalent-combinatorics/sigma-decompositions.lagda.md @@ -18,6 +18,7 @@ open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types open import foundation.inhabited-types +open import foundation.logical-equivalences open import foundation.precomposition-functions open import foundation.propositions open import foundation.relaxed-sigma-decompositions @@ -437,7 +438,7 @@ module _ type-Prop (is-finite-displayed-Σ-Decomposition A (map-equiv equiv-displayed-fibered-Σ-Decomposition D)) equiv-is-finite-displayed-fibered-Σ-Decomposition = - equiv-prop + equiv-iff-is-prop ( is-prop-type-Prop (is-finite-fibered-Σ-Decomposition A D)) ( is-prop-type-Prop ( is-finite-displayed-Σ-Decomposition A diff --git a/src/univalent-combinatorics/skipping-element-standard-finite-types.lagda.md b/src/univalent-combinatorics/skipping-element-standard-finite-types.lagda.md index 1ab1535c34..3495723917 100644 --- a/src/univalent-combinatorics/skipping-element-standard-finite-types.lagda.md +++ b/src/univalent-combinatorics/skipping-element-standard-finite-types.lagda.md @@ -16,6 +16,7 @@ open import foundation.embeddings open import foundation.equality-coproduct-types open import foundation.identity-types open import foundation.injective-maps +open import foundation.sets open import foundation.unit-type open import univalent-combinatorics.standard-finite-types diff --git a/src/univalent-combinatorics/surjective-maps.lagda.md b/src/univalent-combinatorics/surjective-maps.lagda.md index f4950acb3d..231a339513 100644 --- a/src/univalent-combinatorics/surjective-maps.lagda.md +++ b/src/univalent-combinatorics/surjective-maps.lagda.md @@ -17,6 +17,7 @@ open import foundation.decidable-equality open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.logical-equivalences open import foundation.propositional-truncations open import foundation.propositions open import foundation.type-arithmetic-dependent-pair-types @@ -119,7 +120,7 @@ module _ is-finite X ≃ ( has-decidable-equality X × type-trunc-Prop (Σ ℕ (λ n → Fin n ↠ X))) is-finite-iff-∃-surjection-has-decidable-equality = - equiv-prop + equiv-iff-is-prop ( is-prop-is-finite X) ( is-prop-product is-prop-has-decidable-equality is-prop-type-trunc-Prop) ( λ fin-X → is-finite-if-∃-surjection-has-decidable-equality fin-X) diff --git a/tables/propositional-logic.md b/tables/propositional-logic.md new file mode 100644 index 0000000000..85f1cc601c --- /dev/null +++ b/tables/propositional-logic.md @@ -0,0 +1,26 @@ +| Concept | File | +| ---------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------- | +| Propositions (foundation-core) | [`foundation-core.propositions`](foundation-core.propositions.md) | +| Propositions (foundation) | [`foundation.propositions`](foundation.propositions.md) | +| Subterminal types | [`foundation.subterminal-types`](foundation.subterminal-types.md) | +| Subsingleton induction | [`foundation.subsingleton-induction`](foundation.subsingleton-induction.md) | +| Empty types (foundation-core) | [`foundation-core.empty-types`](foundation-core.empty-types.md) | +| Empty types (foundation) | [`foundation.empty-types`](foundation.empty-types.md) | +| Unit type | [`foundation.unit-type`](foundation.unit-type.md) | +| Logical equivalences | [`foundation.logical-equivalences`](foundation.logical-equivalences.md) | +| Propositional extensionality | [`foundation.propositional-extensionality`](foundation.propositional-extensionality.md) | +| Mere logical equivalences | [`foundation.mere-logical-equivalences`](foundation.mere-logical-equivalences.md) | +| Conjunction | [`foundation.conjunction`](foundation.conjunction.md) | +| Disjunction | [`foundation.disjunction`](foundation.disjunction.md) | +| Exclusive disjunction | [`foundation.exclusive-disjunction`](foundation.exclusive-disjunction.md) | +| Existential quantification | [`foundation.existential-quantification`](foundation.existential-quantification.md) | +| Uniqueness quantification | [`foundation.uniqueness-quantification`](foundation.uniqueness-quantification.md) | +| Universal quantification | [`foundation.universal-quantification`](foundation.universal-quantification.md) | +| Negation | [`foundation.negation`](foundation.negation.md) | +| Double negation | [`foundation.double-negation`](foundation.double-negation.md) | +| Propositional truncations | [`foundation.propositional-truncations`](foundation.propositional-truncations.md) | +| Universal property of propositional truncations | [`foundation.universal-property-propositional-truncation`](foundation.universal-property-propositional-truncation.md) | +| The induction principle of propositional truncations | [`foundation.induction-principle-propositional-truncation`](foundation.induction-principle-propositional-truncation.md) | +| Functoriality of propositional truncations | [`foundation.functoriality-propositional-truncations`](foundation.functoriality-propositional-truncation.md) | +| Propositional resizing | [`foundation.propositional-resizing`](foundation.propositional-resizing.md) | +| Impredicative encodings of the logical operations | [`foundation.impredicative-encodings`](foundation.impredicative-encodings.md) |