diff --git a/references.bib b/references.bib index aef4fe98c3..2c7232900e 100644 --- a/references.bib +++ b/references.bib @@ -285,6 +285,25 @@ @article{EKMS93 keywords = {axiality,closure operation,Galois connection,interior operation,polarity} } +@article{Esc08, + author = {Escardó, Martín Hötzel}, + title = {Exhaustible sets in higher-type computation}, + journal = {Logical Methods in Computer Science}, + volume = {4}, + year = {2008}, + month = {8}, + number = {3}, + issue = {3}, + publisher = {Centre pour la Communication Scientifique Directe (CCSD)}, + pages = {3:3, 37}, + issn = {1860-5974}, + doi = {10.2168/LMCS-4(3:3)2008}, + eprint = {0808.0441}, + eprinttype = {arxiv}, + eprintclass = {cs}, + primaryclass = {cs.LO} +} + @online{Esc19DefinitionsEquivalence, title = {Definitions of Equivalence Satisfying Judgmental/Strict Groupoid Laws?}, author = {Escardó, Martín Hötzel}, @@ -501,6 +520,22 @@ @online{MR23 keywords = {20B30 03B15,Mathematics - Group Theory,Mathematics - Logic} } +@article{Esc21b, + author = {Escardó, Martín Hötzel}, + title = {Injective types in univalent mathematics}, + journal = {Mathematical Structures in Computer Science}, + volume = {31}, + year = {2021}, + number = {1}, + pages = {89--111}, + issn = {0960-1295,1469-8072}, + doi = {10.1017/S0960129520000225}, + eprint = {1903.01211}, + archiveprefix = {arXiv}, + primaryclass = {math.CT}, + url = {https://martinescardo.github.io/TypeTopology/InjectiveTypes.Article.html} +} + @book{MRR88, title = {A {{Course}} in {{Constructive Algebra}}}, author = {Mines, Ray and Richman, Fred and Ruitenburg, Wim}, diff --git a/src/foundation-core/decidable-propositions.lagda.md b/src/foundation-core/decidable-propositions.lagda.md index 3f6eb2e2ce..eb48a26b7d 100644 --- a/src/foundation-core/decidable-propositions.lagda.md +++ b/src/foundation-core/decidable-propositions.lagda.md @@ -13,10 +13,12 @@ open import foundation.dependent-pair-types open import foundation.double-negation open import foundation.negation open import foundation.propositional-truncations +open import foundation.transport-along-identifications open import foundation.unit-type open import foundation.universe-levels open import foundation-core.cartesian-product-types +open import foundation-core.contractible-types open import foundation-core.empty-types open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types @@ -32,14 +34,11 @@ 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 +## Definitions -### The subtype of decidable propositions +### The property of a proposition of being decidable ```agda -is-decidable-prop : {l : Level} → UU l → UU l -is-decidable-prop A = is-prop A × is-decidable A - is-prop-is-decidable : {l : Level} {A : UU l} → is-prop A → is-prop (is-decidable A) is-prop-is-decidable is-prop-A = @@ -50,6 +49,16 @@ is-decidable-Prop : pr1 (is-decidable-Prop P) = is-decidable (type-Prop P) pr2 (is-decidable-Prop P) = is-prop-is-decidable (is-prop-type-Prop P) +is-decidable-type-Prop : {l : Level} → Prop l → UU l +is-decidable-type-Prop P = is-decidable (type-Prop P) +``` + +### The subuniverse of decidable propositions + +```agda +is-decidable-prop : {l : Level} → UU l → UU l +is-decidable-prop A = is-prop A × is-decidable A + is-prop-is-decidable-prop : {l : Level} (X : UU l) → is-prop (is-decidable-prop X) is-prop-is-decidable-prop X = @@ -63,6 +72,16 @@ is-decidable-prop-Prop : {l : Level} (A : UU l) → Prop l pr1 (is-decidable-prop-Prop A) = is-decidable-prop A pr2 (is-decidable-prop-Prop A) = is-prop-is-decidable-prop A + +module _ + {l : Level} {A : UU l} (H : is-decidable-prop A) + where + + is-prop-type-is-decidable-prop : is-prop A + is-prop-type-is-decidable-prop = pr1 H + + is-decidable-type-is-decidable-prop : is-decidable A + is-decidable-type-is-decidable-prop = pr2 H ``` ### Decidable propositions @@ -93,7 +112,8 @@ module _ is-decidable-prop-type-Decidable-Prop = pr2 P is-decidable-prop-Decidable-Prop : Prop l - pr1 is-decidable-prop-Decidable-Prop = is-decidable type-Decidable-Prop + pr1 is-decidable-prop-Decidable-Prop = + is-decidable type-Decidable-Prop pr2 is-decidable-prop-Decidable-Prop = is-prop-is-decidable is-prop-type-Decidable-Prop ``` @@ -110,6 +130,14 @@ pr1 empty-Decidable-Prop = empty pr2 empty-Decidable-Prop = is-decidable-prop-empty ``` +### Empty types are decidable propositions + +```agda +is-decidable-prop-is-empty : + {l : Level} {A : UU l} → is-empty A → is-decidable-prop A +is-decidable-prop-is-empty H = is-prop-is-empty H , inr H +``` + ### The unit type is a decidable proposition ```agda @@ -122,6 +150,14 @@ pr1 unit-Decidable-Prop = unit pr2 unit-Decidable-Prop = is-decidable-prop-unit ``` +### Contractible types are decidable propositions + +```agda +is-decidable-prop-is-contr : + {l : Level} {A : UU l} → is-contr A → is-decidable-prop A +is-decidable-prop-is-contr H = is-prop-is-contr H , inl (center H) +``` + ### The product of two decidable propositions is a decidable proposition ```agda @@ -154,6 +190,42 @@ module _ pr2 product-Decidable-Prop = is-decidable-prop-product-Decidable-Prop ``` +### The dependent sum of a family of decidable propositions over a decidable proposition + +```agda +module _ + {l1 l2 : Level} {P : UU l1} {Q : P → UU l2} + (H : is-decidable-prop P) (K : (x : P) → is-decidable-prop (Q x)) + where + + is-prop-is-decidable-prop-Σ : is-prop (Σ P Q) + is-prop-is-decidable-prop-Σ = + is-prop-Σ + ( is-prop-type-is-decidable-prop H) + ( is-prop-type-is-decidable-prop ∘ K) + + is-decidable-is-decidable-prop-Σ : is-decidable (Σ P Q) + is-decidable-is-decidable-prop-Σ = + rec-coproduct + ( λ x → + rec-coproduct + ( λ y → inl (x , y)) + ( λ ny → + inr + ( λ xy → + ny + ( tr Q + ( eq-is-prop (is-prop-type-is-decidable-prop H)) + ( pr2 xy)))) + ( is-decidable-type-is-decidable-prop (K x))) + ( λ nx → inr (λ xy → nx (pr1 xy))) + ( is-decidable-type-is-decidable-prop H) + + is-decidable-prop-Σ : is-decidable-prop (Σ P Q) + is-decidable-prop-Σ = + ( is-prop-is-decidable-prop-Σ , is-decidable-is-decidable-prop-Σ) +``` + ### The negation operation on decidable propositions ```agda diff --git a/src/foundation-core/embeddings.lagda.md b/src/foundation-core/embeddings.lagda.md index 3eac0587bb..6fe6dabc72 100644 --- a/src/foundation-core/embeddings.lagda.md +++ b/src/foundation-core/embeddings.lagda.md @@ -30,7 +30,9 @@ better behaved homotopically than being an equivalence is a [property](foundation-core.propositions.md) under [function extensionality](foundation.function-extensionality.md). -## Definition +## Definitions + +### The predicate on maps of being embeddings ```agda module _ @@ -48,7 +50,11 @@ module _ inv-equiv-ap-is-emb : {f : A → B} (e : is-emb f) {x y : A} → (f x ＝ f y) ≃ (x ＝ y) inv-equiv-ap-is-emb e = inv-equiv (equiv-ap-is-emb e) +``` +### The type of embeddings + +```agda infix 5 _↪_ _↪_ : {l1 l2 : Level} → UU l1 → UU l2 → UU (l1 ⊔ l2) @@ -73,6 +79,29 @@ module _ inv-equiv-ap-emb e = inv-equiv (equiv-ap-emb e) ``` +### The type of embeddings into a type + +```agda +Emb : (l1 : Level) {l2 : Level} (B : UU l2) → UU (lsuc l1 ⊔ l2) +Emb l1 B = Σ (UU l1) (λ A → A ↪ B) + +module _ + {l1 l2 : Level} {B : UU l2} (f : Emb l1 B) + where + + type-Emb : UU l1 + type-Emb = pr1 f + + emb-Emb : type-Emb ↪ B + emb-Emb = pr2 f + + map-Emb : type-Emb → B + map-Emb = map-emb emb-Emb + + is-emb-map-Emb : is-emb map-Emb + is-emb-map-Emb = is-emb-map-emb emb-Emb +``` + ## Examples ### The identity map is an embedding diff --git a/src/foundation-core/equality-dependent-pair-types.lagda.md b/src/foundation-core/equality-dependent-pair-types.lagda.md index 6b6374433d..9dacb2ce72 100644 --- a/src/foundation-core/equality-dependent-pair-types.lagda.md +++ b/src/foundation-core/equality-dependent-pair-types.lagda.md @@ -45,11 +45,18 @@ module _ ```agda refl-Eq-Σ : (s : Σ A B) → Eq-Σ s s - pr1 (refl-Eq-Σ (pair a b)) = refl - pr2 (refl-Eq-Σ (pair a b)) = refl + refl-Eq-Σ s = refl , refl + + eq-base-eq-pair : {s t : Σ A B} → s ＝ t → pr1 s ＝ pr1 t + eq-base-eq-pair = ap pr1 + + dependent-identification-eq-pair : + {s t : Σ A B} (p : s ＝ t) → + dependent-identification B (eq-base-eq-pair p) (pr2 s) (pr2 t) + dependent-identification-eq-pair {s} p = tr-ap pr1 (λ x _ → pr2 x) p (pr1 s) pair-eq-Σ : {s t : Σ A B} → s ＝ t → Eq-Σ s t - pair-eq-Σ {s} refl = refl-Eq-Σ s + pair-eq-Σ p = eq-base-eq-pair p , dependent-identification-eq-pair p eq-pair-eq-base : {x y : A} {s : B x} (p : x ＝ y) → (x , s) ＝ (y , tr B p s) @@ -78,11 +85,11 @@ module _ is-retraction-pair-eq-Σ : (s t : Σ A B) → pair-eq-Σ {s} {t} ∘ eq-pair-Σ' {s} {t} ~ id {A = Eq-Σ s t} - is-retraction-pair-eq-Σ (pair x y) (pair .x .y) (pair refl refl) = refl + is-retraction-pair-eq-Σ (x , y) (.x , .y) (refl , refl) = refl is-section-pair-eq-Σ : - (s t : Σ A B) → ((eq-pair-Σ' {s} {t}) ∘ (pair-eq-Σ {s} {t})) ~ id - is-section-pair-eq-Σ (pair x y) .(pair x y) refl = refl + (s t : Σ A B) → eq-pair-Σ' {s} {t} ∘ pair-eq-Σ {s} {t} ~ id + is-section-pair-eq-Σ (x , y) .(x , y) refl = refl abstract is-equiv-eq-pair-Σ : (s t : Σ A B) → is-equiv (eq-pair-Σ' {s} {t}) @@ -110,14 +117,6 @@ module _ η-pair : (t : Σ A B) → (pair (pr1 t) (pr2 t)) ＝ t η-pair t = refl - - eq-base-eq-pair-Σ : {s t : Σ A B} → (s ＝ t) → (pr1 s ＝ pr1 t) - eq-base-eq-pair-Σ p = pr1 (pair-eq-Σ p) - - dependent-eq-family-eq-pair-Σ : - {s t : Σ A B} → (p : s ＝ t) → - dependent-identification B (eq-base-eq-pair-Σ p) (pr2 s) (pr2 t) - dependent-eq-family-eq-pair-Σ p = pr2 (pair-eq-Σ p) ``` ### Lifting equality to the total space @@ -128,7 +127,7 @@ module _ where lift-eq-Σ : - {x y : A} (p : x ＝ y) (b : B x) → (pair x b) ＝ (pair y (tr B p b)) + {x y : A} (p : x ＝ y) (b : B x) → (x , b) ＝ (y , tr B p b) lift-eq-Σ refl b = refl ``` @@ -148,7 +147,7 @@ tr-Σ C refl z = refl ```agda tr-eq-pair-Σ : {l1 l2 l3 : Level} {A : UU l1} {a0 a1 : A} - {B : A → UU l2} {b0 : B a0} {b1 : B a1} (C : (Σ A B) → UU l3) + {B : A → UU l2} {b0 : B a0} {b1 : B a1} (C : Σ A B → UU l3) (p : a0 ＝ a1) (q : dependent-identification B p b0 b1) (u : C (a0 , b0)) → tr C (eq-pair-Σ p q) u ＝ tr (λ x → C (a1 , x)) q (tr C (eq-pair-Σ p refl) u) diff --git a/src/foundation-core/fibers-of-maps.lagda.md b/src/foundation-core/fibers-of-maps.lagda.md index 066a033e3c..72320d9a12 100644 --- a/src/foundation-core/fibers-of-maps.lagda.md +++ b/src/foundation-core/fibers-of-maps.lagda.md @@ -9,12 +9,14 @@ module foundation-core.fibers-of-maps where ```agda open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types +open import foundation.strictly-right-unital-concatenation-identifications open import foundation.universe-levels 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.postcomposition-functions open import foundation-core.retractions open import foundation-core.sections open import foundation-core.transport-along-identifications @@ -373,6 +375,76 @@ module _ pr2 inv-compute-fiber-comp = is-equiv-map-inv-compute-fiber-comp ``` +### Fibers of homotopic maps are equivalent + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + {f g : A → B} (H : g ~ f) (y : B) + where + + map-equiv-fiber-htpy : fiber f y → fiber g y + map-equiv-fiber-htpy (x , p) = x , H x ∙ᵣ p + + map-inv-equiv-fiber-htpy : fiber g y → fiber f y + map-inv-equiv-fiber-htpy (x , p) = x , inv (H x) ∙ᵣ p + + is-section-map-inv-equiv-fiber-htpy : + is-section map-equiv-fiber-htpy map-inv-equiv-fiber-htpy + is-section-map-inv-equiv-fiber-htpy (x , refl) = + eq-Eq-fiber g (g x) refl (inv (right-inv-right-strict-concat (H x))) + + is-retraction-map-inv-equiv-fiber-htpy : + is-retraction map-equiv-fiber-htpy map-inv-equiv-fiber-htpy + is-retraction-map-inv-equiv-fiber-htpy (x , refl) = + eq-Eq-fiber f (f x) refl (inv (left-inv-right-strict-concat (H x))) + + is-equiv-map-equiv-fiber-htpy : is-equiv map-equiv-fiber-htpy + is-equiv-map-equiv-fiber-htpy = + is-equiv-is-invertible + map-inv-equiv-fiber-htpy + is-section-map-inv-equiv-fiber-htpy + is-retraction-map-inv-equiv-fiber-htpy + + equiv-fiber-htpy : fiber f y ≃ fiber g y + equiv-fiber-htpy = map-equiv-fiber-htpy , is-equiv-map-equiv-fiber-htpy +``` + +We repeat the construction for `fiber'`. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + {f g : A → B} (H : g ~ f) (y : B) + where + + map-equiv-fiber-htpy' : fiber' f y → fiber' g y + map-equiv-fiber-htpy' (x , p) = (x , p ∙ inv (H x)) + + map-inv-equiv-fiber-htpy' : fiber' g y → fiber' f y + map-inv-equiv-fiber-htpy' (x , p) = (x , p ∙ H x) + + is-section-map-inv-equiv-fiber-htpy' : + is-section map-equiv-fiber-htpy' map-inv-equiv-fiber-htpy' + is-section-map-inv-equiv-fiber-htpy' (x , p) = + ap (pair x) (is-retraction-inv-concat' (H x) p) + + is-retraction-map-inv-equiv-fiber-htpy' : + is-retraction map-equiv-fiber-htpy' map-inv-equiv-fiber-htpy' + is-retraction-map-inv-equiv-fiber-htpy' (x , p) = + ap (pair x) (is-section-inv-concat' (H x) p) + + is-equiv-map-equiv-fiber-htpy' : is-equiv map-equiv-fiber-htpy' + is-equiv-map-equiv-fiber-htpy' = + is-equiv-is-invertible + map-inv-equiv-fiber-htpy' + is-section-map-inv-equiv-fiber-htpy' + is-retraction-map-inv-equiv-fiber-htpy' + + equiv-fiber-htpy' : fiber' f y ≃ fiber' g y + equiv-fiber-htpy' = map-equiv-fiber-htpy' , is-equiv-map-equiv-fiber-htpy' +``` + ## Table of files about fibers of maps The following table lists files that are about fibers of maps as a general diff --git a/src/foundation-core/path-split-maps.lagda.md b/src/foundation-core/path-split-maps.lagda.md index 2d4e324279..ef41e36e50 100644 --- a/src/foundation-core/path-split-maps.lagda.md +++ b/src/foundation-core/path-split-maps.lagda.md @@ -23,8 +23,8 @@ open import foundation-core.sections ## Idea -A map `f : A → B` is said to be **path split** if it has a -[section](foundation-core.sections.md) and its +A map `f : A → B` is said to be {{#concept "path split" Agda=is-path-split}} if +it has a [section](foundation-core.sections.md) and its [action on identifications](foundation.action-on-identifications-functions.md) `x ＝ y → f x ＝ f y` has a section for each `x y : A`. By the [fundamental theorem of identity types](foundation.fundamental-theorem-of-identity-types.md), diff --git a/src/foundation-core/propositions.lagda.md b/src/foundation-core/propositions.lagda.md index 005680a831..5aeafc52d4 100644 --- a/src/foundation-core/propositions.lagda.md +++ b/src/foundation-core/propositions.lagda.md @@ -126,6 +126,16 @@ module _ abstract eq-is-proof-irrelevant : is-proof-irrelevant A → all-elements-equal A eq-is-proof-irrelevant = eq-is-prop' ∘ is-prop-is-proof-irrelevant + +abstract + eq-type-Prop : {l : Level} (P : Prop l) → {x y : type-Prop P} → x ＝ y + eq-type-Prop P = eq-is-prop (is-prop-type-Prop P) + +abstract + is-proof-irrelevant-type-Prop : + {l : Level} (P : Prop l) → is-proof-irrelevant (type-Prop P) + is-proof-irrelevant-type-Prop P = + is-proof-irrelevant-is-prop (is-prop-type-Prop P) ``` ### Propositions are closed under equivalences diff --git a/src/foundation-core/transport-along-identifications.lagda.md b/src/foundation-core/transport-along-identifications.lagda.md index 911b7b0255..49b511e7e0 100644 --- a/src/foundation-core/transport-along-identifications.lagda.md +++ b/src/foundation-core/transport-along-identifications.lagda.md @@ -39,6 +39,47 @@ tr B refl b = b ## Properties +### The dependent action of a family of maps over a map on identifications in the base type + +```agda +tr-ap : + {l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} {C : UU l3} {D : C → UU l4} + (f : A → C) (g : (x : A) → B x → D (f x)) + {x y : A} (p : x ＝ y) (z : B x) → + tr D (ap f p) (g x z) ＝ g y (tr B p z) +tr-ap f g refl z = refl +``` + +### Every family of maps preserves transport + +```agda +module _ + {l1 l2 l3 : Level} {I : UU l1} {A : I → UU l2} {B : I → UU l3} + (f : (i : I) → A i → B i) + where + + preserves-tr : + {i j : I} (p : i ＝ j) (x : A i) → + f j (tr A p x) ＝ tr B p (f i x) + preserves-tr refl x = refl + + compute-preserves-tr : + {i j : I} (p : i ＝ j) (x : A i) → + preserves-tr p x ＝ + inv (tr-ap id f p x) ∙ ap (λ q → tr B q (f i x)) (ap-id p) + compute-preserves-tr refl x = refl +``` + +### Substitution law for transport + +```agda +substitution-law-tr : + {l1 l2 l3 : Level} {X : UU l1} {A : UU l2} (B : A → UU l3) (f : X → A) + {x y : X} (p : x ＝ y) {x' : B (f x)} → + tr B (ap f p) x' ＝ tr (B ∘ f) p x' +substitution-law-tr B f p {x'} = tr-ap f (λ _ → id) p x' +``` + ### Transport preserves concatenation of identifications ```agda @@ -70,34 +111,12 @@ module _ eq-transpose-tr' refl q = q ``` -### Every family of maps preserves transport - -```agda -preserves-tr : - {l1 l2 l3 : Level} {I : UU l1} {A : I → UU l2} {B : I → UU l3} - (f : (i : I) → A i → B i) - {i j : I} (p : i ＝ j) (x : A i) → - f j (tr A p x) ＝ tr B p (f i x) -preserves-tr f refl x = refl -``` - -### Transporting along the action on identifications of a function - -```agda -tr-ap : - {l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} {C : UU l3} {D : C → UU l4} - (f : A → C) (g : (x : A) → B x → D (f x)) - {x y : A} (p : x ＝ y) (z : B x) → - tr D (ap f p) (g x z) ＝ g y (tr B p z) -tr-ap f g refl z = refl -``` - ### Computing maps out of identity types as transports ```agda module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {a : A} - (f : (x : A) → (a ＝ x) → B x) + (f : (x : A) → a ＝ x → B x) where compute-map-out-of-identity-type : @@ -110,7 +129,7 @@ module _ ```agda tr-Id-left : {l : Level} {A : UU l} {a b c : A} (q : b ＝ c) (p : b ＝ a) → - tr (_＝ a) q p ＝ (inv q ∙ p) + tr (_＝ a) q p ＝ inv q ∙ p tr-Id-left refl p = refl ``` @@ -119,16 +138,6 @@ tr-Id-left refl p = refl ```agda tr-Id-right : {l : Level} {A : UU l} {a b c : A} (q : b ＝ c) (p : a ＝ b) → - tr (a ＝_) q p ＝ (p ∙ q) + tr (a ＝_) q p ＝ p ∙ q tr-Id-right refl p = inv right-unit ``` - -### Substitution law for transport - -```agda -substitution-law-tr : - {l1 l2 l3 : Level} {X : UU l1} {A : UU l2} (B : A → UU l3) (f : X → A) - {x y : X} (p : x ＝ y) {x' : B (f x)} → - tr B (ap f p) x' ＝ tr (B ∘ f) p x' -substitution-law-tr B f refl = refl -``` diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md index 0460bc24cd..4e72fe1c04 100644 --- a/src/foundation.lagda.md +++ b/src/foundation.lagda.md @@ -129,6 +129,7 @@ open import foundation.descent-coproduct-types public open import foundation.descent-dependent-pair-types public open import foundation.descent-empty-types public open import foundation.descent-equivalences public +open import foundation.diaconescus-theorem public open import foundation.diagonal-maps-cartesian-products-of-types public open import foundation.diagonal-maps-of-types public open import foundation.diagonal-span-diagrams public @@ -264,6 +265,7 @@ open import foundation.lawveres-fixed-point-theorem public open import foundation.lesser-limited-principle-of-omniscience public open import foundation.lifts-types public open import foundation.limited-principle-of-omniscience public +open import foundation.locale-of-propositions public open import foundation.locally-small-types public open import foundation.logical-equivalences public open import foundation.maps-in-global-subuniverses public @@ -311,6 +313,7 @@ open import foundation.partitions public open import foundation.path-algebra public open import foundation.path-cosplit-maps public open import foundation.path-split-maps public +open import foundation.path-split-type-families public open import foundation.perfect-images public open import foundation.permutations-spans-families-of-types public open import foundation.pi-decompositions public @@ -324,7 +327,6 @@ open import foundation.precomposition-dependent-functions public open import foundation.precomposition-functions public open import foundation.precomposition-functions-into-subuniverses public open import foundation.precomposition-type-families public -open import foundation.preimages-of-subtypes public open import foundation.preunivalence public open import foundation.preunivalent-type-families public open import foundation.principle-of-omniscience public @@ -443,6 +445,7 @@ open import foundation.type-arithmetic-standard-pullbacks public 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.uniformly-decidable-type-families public open import foundation.unions-subtypes public open import foundation.uniqueness-image public open import foundation.uniqueness-quantification public diff --git a/src/foundation/axiom-of-choice.lagda.md b/src/foundation/axiom-of-choice.lagda.md index 319776bde7..26774379b7 100644 --- a/src/foundation/axiom-of-choice.lagda.md +++ b/src/foundation/axiom-of-choice.lagda.md @@ -32,31 +32,53 @@ open import foundation-core.sets ## Idea -The {{#concept "axiom of choice" WD="axiom of choice" WDID=Q179692 Agda=AC-0}} +The {{#concept "axiom of choice" WD="axiom of choice" WDID=Q179692 Agda=AC0}} 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 +### Instances of choice + +```agda +instance-choice : {l1 l2 : Level} (A : UU l1) → (A → UU l2) → UU (l1 ⊔ l2) +instance-choice A B = + ((x : A) → is-inhabited (B x)) → is-inhabited ((x : A) → B x) +``` + +Note that the above statement, were it true for all indexing types `A`, is +inconsistent with [univalence](foundation.univalence.md). For a concrete +counterexample see Lemma 3.8.5 in {{#cite UF13}}. + ### The axiom of choice restricted to sets ```agda -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) → is-inhabited (type-Set (B x))) → - is-inhabited ((x : type-Set A) → type-Set (B x)) +instance-choice-Set : + {l1 l2 : Level} (A : Set l1) → (type-Set A → Set l2) → UU (l1 ⊔ l2) +instance-choice-Set A B = instance-choice (type-Set A) (type-Set ∘ B) + +level-AC-Set : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) +level-AC-Set l1 l2 = + (A : Set l1) (B : type-Set A → Set l2) → instance-choice-Set A B + +AC-Set : UUω +AC-Set = {l1 l2 : Level} → level-AC-Set l1 l2 ``` ### The axiom of choice ```agda -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) → is-inhabited (B x)) → - is-inhabited ((x : type-Set A) → B x) +instance-choice₀ : + {l1 l2 : Level} (A : Set l1) → (type-Set A → UU l2) → UU (l1 ⊔ l2) +instance-choice₀ A = instance-choice (type-Set A) + +level-AC0 : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) +level-AC0 l1 l2 = + (A : Set l1) (B : type-Set A → UU l2) → instance-choice₀ A B + +AC0 : UUω +AC0 = {l1 l2 : Level} → level-AC0 l1 l2 ``` ## Properties @@ -64,10 +86,10 @@ AC-0 l1 l2 = ### Every type is set-projective if and only if the axiom of choice holds ```agda -is-set-projective-AC-0 : - {l1 l2 l3 : Level} → AC-0 l2 (l1 ⊔ l2) → +is-set-projective-AC0 : + {l1 l2 l3 : Level} → level-AC0 l2 (l1 ⊔ l2) → (X : UU l3) → is-set-projective l1 l2 X -is-set-projective-AC-0 ac X A B f h = +is-set-projective-AC0 ac X A B f h = map-trunc-Prop ( ( map-Σ ( λ g → ((map-surjection f) ∘ g) ＝ h) @@ -76,11 +98,11 @@ is-set-projective-AC-0 ac X A B f h = ( section-is-split-surjective (map-surjection f))) ( ac B (fiber (map-surjection f)) (is-surjective-map-surjection f)) -AC-0-is-set-projective : +AC0-is-set-projective : {l1 l2 : Level} → ({l : Level} (X : UU l) → is-set-projective (l1 ⊔ l2) l1 X) → - AC-0 l1 l2 -AC-0-is-set-projective H A B K = + level-AC0 l1 l2 +AC0-is-set-projective H A B K = map-trunc-Prop ( map-equiv (equiv-Π-section-pr1 {B = B}) ∘ tot (λ g → htpy-eq)) ( H ( type-Set A) @@ -89,3 +111,12 @@ AC-0-is-set-projective H A B K = ( pr1 , (λ a → map-trunc-Prop (map-inv-fiber-pr1 B a) (K a))) ( id)) ``` + +## See also + +- [Diaconescu's theorem](foundation.diaconescus-theorem.md), which states that + the axiom of choice implies the law of excluded middle. + +## References + +{{#bibliography}} diff --git a/src/foundation/binary-transport.lagda.md b/src/foundation/binary-transport.lagda.md index 5ce9abd175..1bf6c1e593 100644 --- a/src/foundation/binary-transport.lagda.md +++ b/src/foundation/binary-transport.lagda.md @@ -37,7 +37,7 @@ module _ where binary-tr : (p : x ＝ x') (q : y ＝ y') → C x y → C x' y' - binary-tr p q c = tr (C _) q (tr (λ u → C u _) p c) + binary-tr p q c = tr (C x') q (tr (λ u → C u y) p c) is-equiv-binary-tr : (p : x ＝ x') (q : y ＝ y') → is-equiv (binary-tr p q) is-equiv-binary-tr refl refl = is-equiv-id diff --git a/src/foundation/booleans.lagda.md b/src/foundation/booleans.lagda.md index 82e07d5dfc..4c8195a978 100644 --- a/src/foundation/booleans.lagda.md +++ b/src/foundation/booleans.lagda.md @@ -7,7 +7,10 @@ module foundation.booleans where
Imports ```agda +open import foundation.decidable-equality +open import foundation.decidable-types open import foundation.dependent-pair-types +open import foundation.discrete-types open import foundation.involutions open import foundation.negated-equality open import foundation.raising-universe-levels @@ -193,18 +196,30 @@ abstract ( λ x y → eq-Eq-bool) bool-Set : Set lzero -pr1 bool-Set = bool -pr2 bool-Set = is-set-bool +bool-Set = bool , is-set-bool +``` + +### The booleans have decidable equality + +```agda +has-decidable-equality-bool : has-decidable-equality bool +has-decidable-equality-bool true true = inl refl +has-decidable-equality-bool true false = inr neq-true-false-bool +has-decidable-equality-bool false true = inr neq-false-true-bool +has-decidable-equality-bool false false = inl refl + +bool-Discrete-Type : Discrete-Type lzero +bool-Discrete-Type = bool , has-decidable-equality-bool ``` ### The "is true" predicate on booleans ```agda is-true : bool → UU lzero -is-true = Eq-bool true +is-true = _＝ true is-prop-is-true : (b : bool) → is-prop (is-true b) -is-prop-is-true = is-prop-Eq-bool true +is-prop-is-true b = is-set-bool b true is-true-Prop : bool → Prop lzero is-true-Prop b = is-true b , is-prop-is-true b @@ -214,15 +229,27 @@ is-true-Prop b = is-true b , is-prop-is-true b ```agda is-false : bool → UU lzero -is-false = Eq-bool false +is-false = _＝ false is-prop-is-false : (b : bool) → is-prop (is-false b) -is-prop-is-false = is-prop-Eq-bool false +is-prop-is-false b = is-set-bool b false is-false-Prop : bool → Prop lzero is-false-Prop b = is-false b , is-prop-is-false b ``` +### A boolean cannot be both true and false + +```agda +not-is-false-is-true : (x : bool) → is-true x → ¬ (is-false x) +not-is-false-is-true true t () +not-is-false-is-true false () f + +not-is-true-is-false : (x : bool) → is-false x → ¬ (is-true x) +not-is-true-is-false true () f +not-is-true-is-false false t () +``` + ### The type of booleans is equivalent to `Fin 2` ```agda diff --git a/src/foundation/cantors-theorem.lagda.md b/src/foundation/cantors-theorem.lagda.md index a00b09caeb..96593b0ef3 100644 --- a/src/foundation/cantors-theorem.lagda.md +++ b/src/foundation/cantors-theorem.lagda.md @@ -11,6 +11,7 @@ open import foundation.action-on-identifications-functions open import foundation.decidable-propositions open import foundation.decidable-subtypes open import foundation.dependent-pair-types +open import foundation.double-negation-stable-propositions open import foundation.function-extensionality open import foundation.logical-equivalences open import foundation.negation @@ -22,6 +23,10 @@ open import foundation.universe-levels open import foundation-core.empty-types open import foundation-core.fibers-of-maps open import foundation-core.propositions + +open import logic.de-morgan-propositions +open import logic.de-morgan-subtypes +open import logic.double-negation-stable-subtypes ```
@@ -85,6 +90,9 @@ module _ ### Cantor's theorem for the set of decidable subtypes +**Statement.** There is no surjective map from a type `X` to its set of +decidable subtypes `𝒫ᵈ(X)`. + ```agda module _ {l1 l2 : Level} {X : UU l1} (f : X → decidable-subtype l2 X) @@ -110,6 +118,68 @@ module _ ( not-in-image-map-theorem-decidable-Cantor) ``` +### Cantor's theorem for the set of double negation stable subtypes + +**Statement.** There is no surjective map from a type `X` to its set of double +negation stable subtypes `𝒫^¬¬(X)`. + +```agda +module _ + {l1 l2 : Level} {X : UU l1} (f : X → double-negation-stable-subtype l2 X) + where + + map-theorem-double-negation-stable-Cantor : + double-negation-stable-subtype l2 X + map-theorem-double-negation-stable-Cantor x = + neg-Double-Negation-Stable-Prop (f x x) + + abstract + not-in-image-map-theorem-double-negation-stable-Cantor : + ¬ (fiber f map-theorem-double-negation-stable-Cantor) + not-in-image-map-theorem-double-negation-stable-Cantor (x , α) = + no-fixed-points-neg-Double-Negation-Stable-Prop + ( f x x) + ( iff-eq (ap prop-Double-Negation-Stable-Prop (htpy-eq α x))) + + abstract + theorem-double-negation-stable-Cantor : ¬ (is-surjective f) + theorem-double-negation-stable-Cantor H = + apply-universal-property-trunc-Prop + ( H map-theorem-double-negation-stable-Cantor) + ( empty-Prop) + ( not-in-image-map-theorem-double-negation-stable-Cantor) +``` + +### Cantor's theorem for the set of De Morgan subtypes + +**Statement.** There is no surjective map from a type `X` to its set of De +Morgan subtypes `𝒫ᵈᵐ(X)`. + +```agda +module _ + {l1 l2 : Level} {X : UU l1} (f : X → de-morgan-subtype l2 X) + where + + map-theorem-de-morgan-Cantor : de-morgan-subtype l2 X + map-theorem-de-morgan-Cantor x = neg-De-Morgan-Prop (f x x) + + abstract + not-in-image-map-theorem-de-morgan-Cantor : + ¬ (fiber f map-theorem-de-morgan-Cantor) + not-in-image-map-theorem-de-morgan-Cantor (x , α) = + no-fixed-points-neg-De-Morgan-Prop + ( f x x) + ( iff-eq (ap prop-De-Morgan-Prop (htpy-eq α x))) + + abstract + theorem-de-morgan-Cantor : ¬ (is-surjective f) + theorem-de-morgan-Cantor H = + apply-universal-property-trunc-Prop + ( H map-theorem-de-morgan-Cantor) + ( empty-Prop) + ( not-in-image-map-theorem-de-morgan-Cantor) +``` + ## References A proof of Cantor's theorem first appeared in {{#cite Cantor1890/91}} where it diff --git a/src/foundation/complements-subtypes.lagda.md b/src/foundation/complements-subtypes.lagda.md index f105addb16..df47f07a3b 100644 --- a/src/foundation/complements-subtypes.lagda.md +++ b/src/foundation/complements-subtypes.lagda.md @@ -9,22 +9,37 @@ module foundation.complements-subtypes where ```agda open import foundation.decidable-propositions open import foundation.decidable-subtypes +open import foundation.double-negation-stable-propositions open import foundation.full-subtypes open import foundation.negation +open import foundation.postcomposition-functions +open import foundation.powersets open import foundation.propositional-truncations open import foundation.unions-subtypes open import foundation.universe-levels open import foundation-core.function-types open import foundation-core.subtypes + +open import logic.double-negation-stable-subtypes + +open import order-theory.large-posets +open import order-theory.opposite-large-posets +open import order-theory.order-preserving-maps-large-posets +open import order-theory.order-preserving-maps-large-preorders +open import order-theory.order-preserving-maps-posets +open import order-theory.order-preserving-maps-preorders +open import order-theory.posets ``` ## Idea -The **complement** of a [subtype](foundation-core.subtypes.md) `P` of `A` -consists of the elements that are not in `P`. +The +{{#concept "complement" Disambiguation="of a subtype" Agda=complement-subtype}} +of a [subtype](foundation-core.subtypes.md) `P ⊆ A` consists of the elements +that are [not](foundation-core.negation.md) in `P`. ## Definition @@ -36,45 +51,29 @@ complement-subtype : complement-subtype P x = neg-Prop (P x) ``` -### Complements of decidable subtypes +## Properties + +### Complements of subtypes are double negation stable ```agda -complement-decidable-subtype : - {l1 l2 : Level} {A : UU l1} → decidable-subtype l2 A → decidable-subtype l2 A -complement-decidable-subtype P x = neg-Decidable-Prop (P x) +complement-double-negation-stable-subtype' : + {l1 l2 : Level} {A : UU l1} → + subtype l2 A → double-negation-stable-subtype l2 A +complement-double-negation-stable-subtype' P x = + neg-type-Double-Negation-Stable-Prop (is-in-subtype P x) ``` -## Properties - -### The union of a subtype `P` with its complement is the full subtype if and only if `P` is a decidable subtype +### Taking complements gives a contravariant endooperator on the powerset posets ```agda -module _ - {l1 l2 : Level} {A : UU l1} - where - - is-full-union-subtype-complement-subtype : - (P : subtype l2 A) → is-decidable-subtype P → - is-full-subtype (union-subtype P (complement-subtype P)) - is-full-union-subtype-complement-subtype P d x = - unit-trunc-Prop (d x) - - is-decidable-subtype-is-full-union-subtype-complement-subtype : - (P : subtype l2 A) → - is-full-subtype (union-subtype P (complement-subtype P)) → - is-decidable-subtype P - is-decidable-subtype-is-full-union-subtype-complement-subtype P H x = - apply-universal-property-trunc-Prop - ( H x) - ( is-decidable-Prop (P x)) - ( id) - - is-full-union-subtype-complement-decidable-subtype : - (P : decidable-subtype l2 A) → - is-full-decidable-subtype - ( union-decidable-subtype P (complement-decidable-subtype P)) - is-full-union-subtype-complement-decidable-subtype P = - is-full-union-subtype-complement-subtype - ( subtype-decidable-subtype P) - ( is-decidable-decidable-subtype P) +neg-hom-powerset : + {l1 : Level} {A : UU l1} → + hom-Large-Poset + ( λ l → l) + ( powerset-Large-Poset A) + ( opposite-Large-Poset (powerset-Large-Poset A)) +neg-hom-powerset = + make-hom-Large-Preorder + ( λ P x → neg-Prop (P x)) + ( λ P Q f x → map-neg (f x)) ``` diff --git a/src/foundation/coproduct-decompositions-subuniverse.lagda.md b/src/foundation/coproduct-decompositions-subuniverse.lagda.md index 1662dfbdc9..3d8495b9a1 100644 --- a/src/foundation/coproduct-decompositions-subuniverse.lagda.md +++ b/src/foundation/coproduct-decompositions-subuniverse.lagda.md @@ -503,7 +503,7 @@ module _ eq-pair-Σ ( eq-pair-Σ ( eq-equiv (equiv-is-empty is-empty-raise-empty (pr2 x))) - ( eq-is-prop (is-prop-type-Prop (P _)))) + ( eq-type-Prop (P _))) ( eq-is-prop is-property-is-empty))) ( ( raise-empty l1 , C1) , is-empty-raise-empty)) ∘e ( ( inv-associative-Σ _ _ _) ∘e diff --git a/src/foundation/coproduct-types.lagda.md b/src/foundation/coproduct-types.lagda.md index 14b74377c5..aaadc73d63 100644 --- a/src/foundation/coproduct-types.lagda.md +++ b/src/foundation/coproduct-types.lagda.md @@ -176,11 +176,17 @@ module _ {l1 l2 : Level} {A : UU l1} {B : UU l2} where + noncontractibility-coproduct-is-contr' : + is-contr A → is-contr B → noncontractibility' (A + B) 1 + noncontractibility-coproduct-is-contr' HA HB = + inl (center HA) , inr (center HB) , neq-inl-inr + abstract is-not-contractible-coproduct-is-contr : is-contr A → is-contr B → is-not-contractible (A + B) - is-not-contractible-coproduct-is-contr HA HB HAB = - neq-inl-inr {x = center HA} {y = center HB} (eq-is-contr HAB) + is-not-contractible-coproduct-is-contr HA HB = + is-not-contractible-noncontractibility + ( 1 , noncontractibility-coproduct-is-contr' HA HB) ``` ### Coproducts of mutually exclusive propositions are propositions diff --git a/src/foundation/decidable-dependent-function-types.lagda.md b/src/foundation/decidable-dependent-function-types.lagda.md index f0045ab035..c63c6ddeea 100644 --- a/src/foundation/decidable-dependent-function-types.lagda.md +++ b/src/foundation/decidable-dependent-function-types.lagda.md @@ -10,26 +10,51 @@ module foundation.decidable-dependent-function-types where open import foundation.decidable-types open import foundation.functoriality-dependent-function-types open import foundation.maybe +open import foundation.uniformly-decidable-type-families open import foundation.universal-property-coproduct-types open import foundation.universal-property-maybe 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.negation ``` ## Idea -We describe conditions under which dependent products are decidable. +We describe conditions under which +[dependent function types](foundation.dependent-function-types.md) are +[decidable](foundation.decidable-types.md). + +## Properties + +### Decidability of dependent products of uniformly decidable families + +```agda +is-decidable-Π-uniformly-decidable-family : + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} → + is-decidable A → + is-uniformly-decidable-family B → + is-decidable ((a : A) → (B a)) +is-decidable-Π-uniformly-decidable-family (inl a) (inl b) = + inl b +is-decidable-Π-uniformly-decidable-family (inl a) (inr b) = + inr (λ f → b a (f a)) +is-decidable-Π-uniformly-decidable-family (inr na) _ = + inl (ex-falso ∘ na) +``` ### Decidablitilty of dependent products over coproducts ```agda is-decidable-Π-coproduct : {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : A + B → UU l3} → - is-decidable ((a : A) → C (inl a)) → is-decidable ((b : B) → C (inr b)) → + is-decidable ((a : A) → C (inl a)) → + is-decidable ((b : B) → C (inr b)) → is-decidable ((x : A + B) → C x) is-decidable-Π-coproduct {C = C} dA dB = is-decidable-equiv @@ -42,7 +67,8 @@ is-decidable-Π-coproduct {C = C} dA dB = ```agda is-decidable-Π-Maybe : {l1 l2 : Level} {A : UU l1} {B : Maybe A → UU l2} → - is-decidable ((x : A) → B (unit-Maybe x)) → is-decidable (B exception-Maybe) → + is-decidable ((x : A) → B (unit-Maybe x)) → + is-decidable (B exception-Maybe) → is-decidable ((x : Maybe A) → B x) is-decidable-Π-Maybe {B = B} du de = is-decidable-equiv @@ -53,15 +79,16 @@ is-decidable-Π-Maybe {B = B} du de = ### Decidability of dependent products over an equivalence ```agda -is-decidable-Π-equiv : +module _ {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : A → UU l3} {D : B → UU l4} - (e : A ≃ B) (f : (x : A) → C x ≃ D (map-equiv e x)) → - is-decidable ((x : A) → C x) → is-decidable ((y : B) → D y) -is-decidable-Π-equiv {D = D} e f = is-decidable-equiv' (equiv-Π D e f) + (e : A ≃ B) (f : (x : A) → C x ≃ D (map-equiv e x)) + where -is-decidable-Π-equiv' : - {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : A → UU l3} {D : B → UU l4} - (e : A ≃ B) (f : (x : A) → C x ≃ D (map-equiv e x)) → - is-decidable ((y : B) → D y) → is-decidable ((x : A) → C x) -is-decidable-Π-equiv' {D = D} e f = is-decidable-equiv (equiv-Π D e f) + is-decidable-Π-equiv : + is-decidable ((x : A) → C x) → is-decidable ((y : B) → D y) + is-decidable-Π-equiv = is-decidable-equiv' (equiv-Π D e f) + + is-decidable-Π-equiv' : + is-decidable ((y : B) → D y) → is-decidable ((x : A) → C x) + is-decidable-Π-equiv' = is-decidable-equiv (equiv-Π D e f) ``` diff --git a/src/foundation/decidable-dependent-pair-types.lagda.md b/src/foundation/decidable-dependent-pair-types.lagda.md index 1632117634..4af4b5e568 100644 --- a/src/foundation/decidable-dependent-pair-types.lagda.md +++ b/src/foundation/decidable-dependent-pair-types.lagda.md @@ -12,51 +12,78 @@ open import foundation.dependent-pair-types open import foundation.maybe open import foundation.type-arithmetic-coproduct-types open import foundation.type-arithmetic-unit-type +open import foundation.uniformly-decidable-type-families open import foundation.universe-levels open import foundation-core.coproduct-types open import foundation-core.equivalences open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types +open import foundation-core.negation ``` ## Idea -We describe conditions under which dependent sums are decidable. +We describe conditions under which +[dependent sums](foundation.dependent-pair-types.md) are +[decidable](foundation.decidable-types.md) + +## Properites + +### Dependent sums of a uniformly decidable family of types + +```agda +is-decidable-Σ-uniformly-decidable-family : + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} → + is-decidable A → + is-uniformly-decidable-family B → + is-decidable (Σ A B) +is-decidable-Σ-uniformly-decidable-family (inl a) (inl b) = + inl (a , b a) +is-decidable-Σ-uniformly-decidable-family (inl a) (inr b) = + inr (λ x → b (pr1 x) (pr2 x)) +is-decidable-Σ-uniformly-decidable-family (inr a) _ = + inr (λ x → a (pr1 x)) +``` + +### Decidability of dependent sums over coproducts ```agda is-decidable-Σ-coproduct : {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (C : A + B → UU l3) → is-decidable (Σ A (C ∘ inl)) → is-decidable (Σ B (C ∘ inr)) → is-decidable (Σ (A + B) C) -is-decidable-Σ-coproduct {l1} {l2} {l3} {A} {B} C dA dB = +is-decidable-Σ-coproduct {A = A} {B} C dA dB = is-decidable-equiv ( right-distributive-Σ-coproduct A B C) ( is-decidable-coproduct dA dB) +``` +### Decidability of dependent sums over `Maybe` + +```agda is-decidable-Σ-Maybe : {l1 l2 : Level} {A : UU l1} {B : Maybe A → UU l2} → is-decidable (Σ A (B ∘ unit-Maybe)) → is-decidable (B exception-Maybe) → is-decidable (Σ (Maybe A) B) -is-decidable-Σ-Maybe {l1} {l2} {A} {B} dA de = +is-decidable-Σ-Maybe {A = A} {B} dA de = is-decidable-Σ-coproduct B dA - ( is-decidable-equiv - ( left-unit-law-Σ (B ∘ inr)) - ( de)) + ( is-decidable-equiv (left-unit-law-Σ (B ∘ inr)) de) +``` -is-decidable-Σ-equiv : - {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : A → UU l3} {D : B → UU l4} - (e : A ≃ B) (f : (x : A) → C x ≃ D (map-equiv e x)) → - is-decidable (Σ A C) → is-decidable (Σ B D) -is-decidable-Σ-equiv {D = D} e f = - is-decidable-equiv' (equiv-Σ D e f) +### Decidability of dependent sums over equivalences -is-decidable-Σ-equiv' : +```agda +module _ {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : A → UU l3} {D : B → UU l4} - (e : A ≃ B) (f : (x : A) → C x ≃ D (map-equiv e x)) → - is-decidable (Σ B D) → is-decidable (Σ A C) -is-decidable-Σ-equiv' {D = D} e f = - is-decidable-equiv (equiv-Σ D e f) + (e : A ≃ B) (f : (x : A) → C x ≃ D (map-equiv e x)) + where + + is-decidable-Σ-equiv : is-decidable (Σ A C) → is-decidable (Σ B D) + is-decidable-Σ-equiv = is-decidable-equiv' (equiv-Σ D e f) + + is-decidable-Σ-equiv' : is-decidable (Σ B D) → is-decidable (Σ A C) + is-decidable-Σ-equiv' = is-decidable-equiv (equiv-Σ D e f) ``` diff --git a/src/foundation/decidable-embeddings.lagda.md b/src/foundation/decidable-embeddings.lagda.md index fc452b45c0..f4780285bd 100644 --- a/src/foundation/decidable-embeddings.lagda.md +++ b/src/foundation/decidable-embeddings.lagda.md @@ -14,6 +14,7 @@ open import foundation.decidable-propositions open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.embeddings +open import foundation.fibers-of-maps open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-coproduct-types open import foundation.fundamental-theorem-of-identity-types @@ -21,9 +22,11 @@ open import foundation.homotopy-induction open import foundation.identity-types open import foundation.logical-equivalences open import foundation.propositional-maps +open import foundation.propositions open import foundation.retracts-of-maps open import foundation.subtype-identity-principle open import foundation.type-arithmetic-dependent-pair-types +open import foundation.unit-type open import foundation.universal-property-equivalences open import foundation.universe-levels @@ -31,12 +34,10 @@ open import foundation-core.cartesian-product-types open import foundation-core.coproduct-types open import foundation-core.empty-types open import foundation-core.equivalences -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.propositions open import foundation-core.torsorial-type-families ``` @@ -64,88 +65,87 @@ is-decidable-emb : {l1 l2 : Level} {X : UU l1} {Y : UU l2} → (X → Y) → UU (l1 ⊔ l2) is-decidable-emb f = is-emb f × is-decidable-map f -abstract - is-emb-is-decidable-emb : - {l1 l2 : Level} {X : UU l1} {Y : UU l2} {f : X → Y} → - is-decidable-emb f → is-emb f - is-emb-is-decidable-emb = pr1 +is-emb-is-decidable-emb : + {l1 l2 : Level} {X : UU l1} {Y : UU l2} {f : X → Y} → + is-decidable-emb f → is-emb f +is-emb-is-decidable-emb = pr1 is-decidable-map-is-decidable-emb : {l1 l2 : Level} {X : UU l1} {Y : UU l2} {f : X → Y} → is-decidable-emb f → is-decidable-map f is-decidable-map-is-decidable-emb = pr2 + +is-injective-is-decidable-emb : + {l1 l2 : Level} {X : UU l1} {Y : UU l2} {f : X → Y} → + is-decidable-emb f → is-injective f +is-injective-is-decidable-emb = is-injective-is-emb ∘ is-emb-is-decidable-emb ``` ### Decidably propositional maps ```agda -is-decidable-prop-map : - {l1 l2 : Level} {X : UU l1} {Y : UU l2} → (X → Y) → UU (l1 ⊔ l2) -is-decidable-prop-map {Y = Y} f = (y : Y) → is-decidable-prop (fiber f y) +module _ + {l1 l2 : Level} {X : UU l1} {Y : UU l2} + where -abstract - is-prop-map-is-decidable-prop-map : - {l1 l2 : Level} {X : UU l1} {Y : UU l2} {f : X → Y} → - is-decidable-prop-map f → is-prop-map f - is-prop-map-is-decidable-prop-map H y = pr1 (H y) + is-decidable-prop-map : (X → Y) → UU (l1 ⊔ l2) + is-decidable-prop-map f = (y : Y) → is-decidable-prop (fiber f y) -is-decidable-map-is-decidable-prop-map : - {l1 l2 : Level} {X : UU l1} {Y : UU l2} {f : X → Y} → - is-decidable-prop-map f → is-decidable-map f -is-decidable-map-is-decidable-prop-map H y = pr2 (H y) + is-prop-is-decidable-prop-map : + (f : X → Y) → is-prop (is-decidable-prop-map f) + is-prop-is-decidable-prop-map f = + is-prop-Π (λ y → is-prop-is-decidable-prop (fiber f y)) + + is-decidable-prop-map-Prop : (X → Y) → Prop (l1 ⊔ l2) + is-decidable-prop-map-Prop f = + ( is-decidable-prop-map f , is-prop-is-decidable-prop-map f) + + abstract + is-prop-map-is-decidable-prop-map : + {f : X → Y} → is-decidable-prop-map f → is-prop-map f + is-prop-map-is-decidable-prop-map H y = pr1 (H y) + + is-decidable-map-is-decidable-prop-map : + {f : X → Y} → is-decidable-prop-map f → is-decidable-map f + is-decidable-map-is-decidable-prop-map H y = pr2 (H y) ``` ### The type of decidable embeddings ```agda infix 5 _↪ᵈ_ -_↪ᵈ_ : - {l1 l2 : Level} (X : UU l1) (Y : UU l2) → UU (l1 ⊔ l2) +_↪ᵈ_ : {l1 l2 : Level} (X : UU l1) (Y : UU l2) → UU (l1 ⊔ l2) X ↪ᵈ Y = Σ (X → Y) is-decidable-emb -map-decidable-emb : - {l1 l2 : Level} {X : UU l1} {Y : UU l2} → X ↪ᵈ Y → X → Y -map-decidable-emb e = pr1 e +decidable-emb : {l1 l2 : Level} (X : UU l1) (Y : UU l2) → UU (l1 ⊔ l2) +decidable-emb = _↪ᵈ_ + +module _ + {l1 l2 : Level} {X : UU l1} {Y : UU l2} (e : X ↪ᵈ Y) + where + + map-decidable-emb : X → Y + map-decidable-emb = pr1 e -abstract is-decidable-emb-map-decidable-emb : - {l1 l2 : Level} {X : UU l1} {Y : UU l2} (e : X ↪ᵈ Y) → - is-decidable-emb (map-decidable-emb e) - is-decidable-emb-map-decidable-emb e = pr2 e + is-decidable-emb map-decidable-emb + is-decidable-emb-map-decidable-emb = pr2 e -abstract - is-emb-map-decidable-emb : - {l1 l2 : Level} {X : UU l1} {Y : UU l2} (e : X ↪ᵈ Y) → - is-emb (map-decidable-emb e) - is-emb-map-decidable-emb e = - is-emb-is-decidable-emb (is-decidable-emb-map-decidable-emb e) + is-emb-map-decidable-emb : is-emb map-decidable-emb + is-emb-map-decidable-emb = + is-emb-is-decidable-emb is-decidable-emb-map-decidable-emb -abstract is-decidable-map-map-decidable-emb : - {l1 l2 : Level} {X : UU l1} {Y : UU l2} (e : X ↪ᵈ Y) → - is-decidable-map (map-decidable-emb e) - is-decidable-map-map-decidable-emb e = - is-decidable-map-is-decidable-emb (is-decidable-emb-map-decidable-emb e) - -emb-decidable-emb : - {l1 l2 : Level} {X : UU l1} {Y : UU l2} → X ↪ᵈ Y → X ↪ Y -pr1 (emb-decidable-emb e) = map-decidable-emb e -pr2 (emb-decidable-emb e) = is-emb-map-decidable-emb e + is-decidable-map map-decidable-emb + is-decidable-map-map-decidable-emb = + is-decidable-map-is-decidable-emb is-decidable-emb-map-decidable-emb + + emb-decidable-emb : X ↪ Y + emb-decidable-emb = map-decidable-emb , is-emb-map-decidable-emb ``` ## Properties -### Being a decidably propositional map is a proposition - -```agda -abstract - is-prop-is-decidable-prop-map : - {l1 l2 : Level} {X : UU l1} {Y : UU l2} (f : X → Y) → - is-prop (is-decidable-prop-map f) - is-prop-is-decidable-prop-map f = - is-prop-Π (λ y → is-prop-is-decidable-prop (fiber f y)) -``` - ### Any map of which the fibers are decidable propositions is a decidable embedding ```agda @@ -175,6 +175,26 @@ module _ is-decidable-map-is-decidable-emb H y ``` +### The first projection map of a dependent sum of decidable propositions is a decidable embedding + +```agda +module _ + {l1 l2 : Level} {A : UU l1} (Q : A → Decidable-Prop l2) + where + + is-decidable-prop-map-pr1 : + is-decidable-prop-map (pr1 {B = type-Decidable-Prop ∘ Q}) + is-decidable-prop-map-pr1 y = + is-decidable-prop-equiv + ( equiv-fiber-pr1 (type-Decidable-Prop ∘ Q) y) + ( is-decidable-prop-type-Decidable-Prop (Q y)) + + is-decidable-emb-pr1 : + is-decidable-emb (pr1 {B = type-Decidable-Prop ∘ Q}) + is-decidable-emb-pr1 = + is-decidable-emb-is-decidable-prop-map is-decidable-prop-map-pr1 +``` + ### Equivalences are decidable embeddings ```agda @@ -189,13 +209,16 @@ abstract ### Identity maps are decidable embeddings ```agda -abstract - is-decidable-emb-id : - {l1 : Level} {A : UU l1} → is-decidable-emb (id {A = A}) - is-decidable-emb-id = (is-emb-id , is-decidable-map-id) +is-decidable-emb-id : + {l : Level} {A : UU l} → is-decidable-emb (id {A = A}) +is-decidable-emb-id = (is-emb-id , is-decidable-map-id) -decidable-emb-id : {l1 : Level} {A : UU l1} → A ↪ᵈ A +decidable-emb-id : {l : Level} {A : UU l} → A ↪ᵈ A decidable-emb-id = (id , is-decidable-emb-id) + +is-decidable-prop-map-id : + {l : Level} {A : UU l} → is-decidable-prop-map (id {A = A}) +is-decidable-prop-map-id y = is-decidable-prop-is-contr (is-torsorial-Id' y) ``` ### Being a decidable embedding is a property @@ -214,6 +237,18 @@ abstract ( λ y → is-prop-is-decidable (is-prop-map-is-emb (pr1 H) y)))) ``` +### Decidable embeddings are closed under homotopies + +```agda +abstract + is-decidable-emb-htpy : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f g : A → B} → + f ~ g → is-decidable-emb g → is-decidable-emb f + is-decidable-emb-htpy {f = f} {g} H K = + ( is-emb-htpy H (is-emb-is-decidable-emb K) , + is-decidable-map-htpy H (is-decidable-map-is-decidable-emb K)) +``` + ### Decidable embeddings are closed under composition ```agda @@ -225,18 +260,10 @@ module _ abstract is-decidable-map-comp-is-decidable-emb' : is-decidable-emb g → is-decidable-map f → is-decidable-map (g ∘ f) - is-decidable-map-comp-is-decidable-emb' K H x = - rec-coproduct - ( λ u → - is-decidable-equiv - ( ( left-unit-law-Σ-is-contr - ( is-proof-irrelevant-is-prop - ( is-prop-map-is-emb (is-emb-is-decidable-emb K) x) u) - ( u)) ∘e - ( compute-fiber-comp g f x)) - ( H (pr1 u))) - ( λ α → inr (λ t → α (f (pr1 t) , pr2 t))) - ( is-decidable-map-is-decidable-emb K x) + is-decidable-map-comp-is-decidable-emb' K = + is-decidable-map-comp + ( is-injective-is-decidable-emb K) + ( is-decidable-map-is-decidable-emb K) is-decidable-map-comp-is-decidable-emb : is-decidable-emb g → is-decidable-emb f → is-decidable-map (g ∘ f) @@ -250,6 +277,29 @@ module _ is-decidable-emb-comp K H = ( is-emb-comp _ _ (pr1 K) (pr1 H) , is-decidable-map-comp-is-decidable-emb K H) + + abstract + is-decidable-prop-map-comp : + is-decidable-prop-map g → + is-decidable-prop-map f → + is-decidable-prop-map (g ∘ f) + is-decidable-prop-map-comp K H = + is-decidable-prop-map-is-decidable-emb + ( is-decidable-emb-comp + ( is-decidable-emb-is-decidable-prop-map K) + ( is-decidable-emb-is-decidable-prop-map H)) + +comp-decidable-emb : + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} → + B ↪ᵈ C → A ↪ᵈ B → A ↪ᵈ C +comp-decidable-emb (g , G) (f , F) = + ( g ∘ f , is-decidable-emb-comp G F) + +infixr 15 _∘ᵈ_ +_∘ᵈ_ : + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} → + B ↪ᵈ C → A ↪ᵈ B → A ↪ᵈ C +_∘ᵈ_ = comp-decidable-emb ``` ### Left cancellation for decidable embeddings @@ -273,18 +323,6 @@ module _ is-decidable-emb-right-factor' GH (is-emb-is-decidable-emb G) ``` -### Decidable embeddings are closed under homotopies - -```agda -abstract - is-decidable-emb-htpy : - {l1 l2 : Level} {A : UU l1} {B : UU l2} {f g : A → B} → - f ~ g → is-decidable-emb g → is-decidable-emb f - is-decidable-emb-htpy {f = f} {g} H K = - ( is-emb-htpy H (is-emb-is-decidable-emb K) , - is-decidable-map-htpy H (is-decidable-map-is-decidable-emb K)) -``` - ### In a commuting triangle of maps, if the top and right maps are decidable embeddings so is the left map ```agda @@ -360,9 +398,9 @@ abstract ( htpy-eq-decidable-emb f) eq-htpy-decidable-emb : - {l1 l2 : Level} {A : UU l1} {B : UU l2} {f g : A ↪ᵈ B} → + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f g : A ↪ᵈ B) → htpy-decidable-emb f g → f ＝ g -eq-htpy-decidable-emb {f = f} {g} = +eq-htpy-decidable-emb f g = map-inv-is-equiv (is-equiv-htpy-eq-decidable-emb f g) ``` @@ -561,3 +599,45 @@ module _ ( is-decidable-prop-map-retract-map R ( is-decidable-prop-map-is-decidable-emb G)) ``` + +### A type is a decidable proposition if and only if its terminal map is a decidable embedding + +```agda +module _ + {l : Level} {A : UU l} + where + + is-decidable-prop-is-decidable-emb-terminal-map : + is-decidable-emb (terminal-map A) → is-decidable-prop A + is-decidable-prop-is-decidable-emb-terminal-map H = + is-decidable-prop-equiv' + ( equiv-fiber-terminal-map star) + ( is-decidable-prop-map-is-decidable-emb H star) + + is-decidable-emb-terminal-map-is-decidable-prop : + is-decidable-prop A → is-decidable-emb (terminal-map A) + is-decidable-emb-terminal-map-is-decidable-prop H = + is-decidable-emb-is-decidable-prop-map + ( λ y → is-decidable-prop-equiv (equiv-fiber-terminal-map y) H) +``` + +### If a dependent sum of propositions over a proposition is decidable, then the family is a family of decidable propositions + +```agda +module _ + {l1 l2 : Level} (P : Prop l1) (Q : type-Prop P → Prop l2) + where + + is-decidable-prop-family-is-decidable-Σ : + is-decidable (Σ (type-Prop P) (type-Prop ∘ Q)) → + (p : type-Prop P) → is-decidable (type-Prop (Q p)) + is-decidable-prop-family-is-decidable-Σ H p = + is-decidable-equiv' + ( equiv-fiber-pr1 (type-Prop ∘ Q) p) + ( is-decidable-map-is-decidable-emb + ( is-decidable-emb-right-factor' + ( is-decidable-emb-terminal-map-is-decidable-prop + ( is-prop-Σ (is-prop-type-Prop P) (is-prop-type-Prop ∘ Q) , H)) + ( is-emb-terminal-map-is-prop (is-prop-type-Prop P))) + ( p)) +``` diff --git a/src/foundation/decidable-maps.lagda.md b/src/foundation/decidable-maps.lagda.md index 893f11da73..926253c0dc 100644 --- a/src/foundation/decidable-maps.lagda.md +++ b/src/foundation/decidable-maps.lagda.md @@ -29,17 +29,22 @@ open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.injective-maps open import foundation-core.retractions +open import foundation-core.sections ``` -## Definition +## Idea A [map](foundation-core.function-types.md) is said to be {{#concept "decidable" Disambiguation="map of types" Agda=is-decidable-map}} if its [fibers](foundation-core.fibers-of-maps.md) are [decidable types](foundation.decidable-types.md). +## Definition + +### The structure on a map of decidability + ```agda module _ {l1 l2 : Level} {A : UU l1} {B : UU l2} @@ -49,6 +54,28 @@ module _ is-decidable-map f = (y : B) → is-decidable (fiber f y) ``` +### The type of decidable maps + +```agda +infix 5 _→ᵈ_ + +_→ᵈ_ : {l1 l2 : Level} (A : UU l1) (B : UU l2) → UU (l1 ⊔ l2) +A →ᵈ B = Σ (A → B) (is-decidable-map) + +decidable-map : {l1 l2 : Level} (A : UU l1) (B : UU l2) → UU (l1 ⊔ l2) +decidable-map = _→ᵈ_ + +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A →ᵈ B) + where + + map-decidable-map : A → B + map-decidable-map = pr1 f + + is-decidable-decidable-map : is-decidable-map map-decidable-map + is-decidable-decidable-map = pr2 f +``` + ## Properties ### Decidable maps are closed under homotopy @@ -64,10 +91,36 @@ abstract ( K b) ``` +### Composition of decidable maps + +The composite `g ∘ f` of two decidable maps is decidable if `g` is injective. + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} + {g : B → C} {f : A → B} + where + + abstract + is-decidable-map-comp : + is-injective g → + is-decidable-map g → + is-decidable-map f → + is-decidable-map (g ∘ f) + is-decidable-map-comp H G F x = + rec-coproduct + ( λ u → + is-decidable-iff + ( λ v → (pr1 v) , ap g (pr2 v) ∙ pr2 u) + ( λ w → pr1 w , H (pr2 w ∙ inv (pr2 u))) + ( F (pr1 u))) + ( λ α → inr (λ t → α (f (pr1 t) , pr2 t))) + ( G x) +``` + ### Left cancellation for decidable maps -If a composite `g ∘ f` is decidable and the left factor `g` is injective, then -the right factor `f` is decidable. +If a composite `g ∘ f` is decidable and `g` is injective then `f` is decidable. ```agda module _ @@ -77,9 +130,11 @@ module _ abstract is-decidable-map-right-factor' : is-decidable-map (g ∘ f) → is-injective g → is-decidable-map f - is-decidable-map-right-factor' GF G y with (GF (g y)) - ... | inl q = inl (pr1 q , G (pr2 q)) - ... | inr q = inr (λ x → q ((pr1 x) , ap g (pr2 x))) + is-decidable-map-right-factor' GF G y = + rec-coproduct + ( λ q → inl (pr1 q , G (pr2 q))) + ( λ q → inr (λ x → q ((pr1 x) , ap g (pr2 x)))) + ( GF (g y)) ``` ### Retracts into types with decidable equality are decidable @@ -91,10 +146,19 @@ is-decidable-map-retraction : is-decidable-map-retraction d i (r , R) b = is-decidable-iff ( λ (p : i (r b) ＝ b) → r b , p) - ( λ t → ap (i ∘ r) (inv (pr2 t)) ∙ (ap i (R (pr1 t)) ∙ pr2 t)) + ( λ t → ap (i ∘ r) (inv (pr2 t)) ∙ ap i (R (pr1 t)) ∙ pr2 t) ( d (i (r b)) b) ``` +### Maps with sections are decidable + +```agda +is-decidable-map-section : + {l1 l2 : Level} {A : UU l1} {B : UU l2} → + (i : A → B) → section i → is-decidable-map i +is-decidable-map-section i (s , S) b = inl (s b , S b) +``` + ### Any map out of the empty type is decidable ```agda diff --git a/src/foundation/decidable-propositions.lagda.md b/src/foundation/decidable-propositions.lagda.md index 4315eab9c8..c6c5bfb814 100644 --- a/src/foundation/decidable-propositions.lagda.md +++ b/src/foundation/decidable-propositions.lagda.md @@ -115,8 +115,7 @@ module _ is-retraction-map-inv-equiv-bool-Decidable-Prop' equiv-bool-Decidable-Prop' : - ((Σ (Prop l) type-Prop) + (Σ (Prop l) (λ Q → ¬ (type-Prop Q)))) ≃ - bool + ((Σ (Prop l) type-Prop) + (Σ (Prop l) (λ Q → ¬ (type-Prop Q)))) ≃ bool pr1 equiv-bool-Decidable-Prop' = map-equiv-bool-Decidable-Prop' pr2 equiv-bool-Decidable-Prop' = is-equiv-map-equiv-bool-Decidable-Prop' diff --git a/src/foundation/decidable-subtypes.lagda.md b/src/foundation/decidable-subtypes.lagda.md index 9a162ba9f7..0bd5ecec7f 100644 --- a/src/foundation/decidable-subtypes.lagda.md +++ b/src/foundation/decidable-subtypes.lagda.md @@ -36,6 +36,8 @@ open import foundation-core.propositions open import foundation-core.transport-along-identifications open import foundation-core.truncated-types open import foundation-core.truncation-levels + +open import logic.double-negation-stable-subtypes ``` @@ -227,6 +229,16 @@ module _ ( iff-universes-Decidable-Prop l l' (S x)) ``` +### Decidable subtypes are double negation stable + +```agda +is-double-negation-stable-decicable-subtype : + {l1 l2 : Level} {A : UU l1} (P : decidable-subtype l2 A) → + is-double-negation-stable-subtype (subtype-decidable-subtype P) +is-double-negation-stable-decicable-subtype P x = + double-negation-elim-is-decidable (is-decidable-decidable-subtype P x) +``` + ### A decidable subtype of a `k+1`-truncated type is `k+1`-truncated ```agda diff --git a/src/foundation/decidable-types.lagda.md b/src/foundation/decidable-types.lagda.md index 2f34bdc31f..93bdd1707e 100644 --- a/src/foundation/decidable-types.lagda.md +++ b/src/foundation/decidable-types.lagda.md @@ -7,24 +7,27 @@ module foundation.decidable-types where
Imports ```agda +open import foundation.action-on-identifications-functions open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.double-negation open import foundation.empty-types +open import foundation.equivalences open import foundation.hilberts-epsilon-operators open import foundation.logical-equivalences open import foundation.negation open import foundation.propositional-truncations open import foundation.raising-universe-levels +open import foundation.retracts-of-types open import foundation.type-arithmetic-empty-type open import foundation.unit-type 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 -open import foundation-core.retracts-of-types +open import foundation-core.retractions +open import foundation-core.sections ```
@@ -154,36 +157,6 @@ is-decidable-function-type' (inl a) d with d a is-decidable-function-type' (inr na) d = inl (ex-falso ∘ na) ``` -### Dependent sums of a uniformly decidable family of types over a decidable base is decidable - -```agda -is-decidable-Σ-uniformly-decidable-family : - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} → - is-decidable A → (((a : A) → B a) + ((a : A) → ¬ B a)) → is-decidable (Σ A B) -is-decidable-Σ-uniformly-decidable-family (inl a) (inl b) = - inl (a , b a) -is-decidable-Σ-uniformly-decidable-family (inl a) (inr b) = - inr (λ x → b (pr1 x) (pr2 x)) -is-decidable-Σ-uniformly-decidable-family (inr a) _ = - inr (λ x → a (pr1 x)) -``` - -### Dependent products of uniformly decidable families over decidable bases are decidable - -```agda -is-decidable-Π-uniformly-decidable-family : - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} → - is-decidable A → - (((a : A) → B a) + ((a : A) → ¬ (B a))) → - is-decidable ((a : A) → (B a)) -is-decidable-Π-uniformly-decidable-family (inl a) (inl b) = - inl b -is-decidable-Π-uniformly-decidable-family (inl a) (inr b) = - inr (λ f → b a (f a)) -is-decidable-Π-uniformly-decidable-family (inr na) _ = - inl (ex-falso ∘ na) -``` - ### The negation of a decidable type is decidable ```agda @@ -202,7 +175,18 @@ module _ is-decidable-iff : (A → B) → (B → A) → is-decidable A → is-decidable B is-decidable-iff f g (inl a) = inl (f a) - is-decidable-iff f g (inr na) = inr (λ b → na (g b)) + is-decidable-iff f g (inr na) = inr (na ∘ g) + + is-decidable-iff' : + A ↔ B → is-decidable A → is-decidable B + is-decidable-iff' (f , g) = is-decidable-iff f g + +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + iff-is-decidable : A ↔ B → is-decidable A ↔ is-decidable B + iff-is-decidable e = is-decidable-iff' e , is-decidable-iff' (inv-iff e) ``` ### Decidable types are closed under retracts @@ -214,26 +198,70 @@ module _ is-decidable-retract-of : A retract-of B → is-decidable B → is-decidable A - is-decidable-retract-of (pair i (pair r H)) (inl b) = inl (r b) - is-decidable-retract-of (pair i (pair r H)) (inr f) = inr (f ∘ i) + is-decidable-retract-of R = is-decidable-iff' (iff-retract' R) + + is-decidable-retract-of' : + A retract-of B → is-decidable A → is-decidable B + is-decidable-retract-of' R = is-decidable-iff' (inv-iff (iff-retract' R)) ``` ### Decidable types are closed under equivalences ```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + is-decidable-is-equiv : {f : A → B} → is-equiv f → is-decidable B → is-decidable A - is-decidable-is-equiv {f} (pair (pair g G) (pair h H)) = - is-decidable-retract-of (pair f (pair h H)) + is-decidable-is-equiv {f} H = + is-decidable-retract-of (retract-equiv (f , H)) is-decidable-equiv : - (e : A ≃ B) → is-decidable B → is-decidable A + A ≃ B → is-decidable B → is-decidable A is-decidable-equiv e = is-decidable-iff (map-inv-equiv e) (map-equiv e) -is-decidable-equiv' : - {l1 l2 : Level} {A : UU l1} {B : UU l2} (e : A ≃ B) → - is-decidable A → is-decidable B -is-decidable-equiv' e = is-decidable-equiv (inv-equiv e) + is-decidable-equiv' : + A ≃ B → is-decidable A → is-decidable B + is-decidable-equiv' e = is-decidable-iff (map-equiv e) (map-inv-equiv e) +``` + +### Equivalent types have equivalent decidability predicates + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (e : A ≃ B) + where + + map-equiv-is-decidable : is-decidable A → is-decidable B + map-equiv-is-decidable = is-decidable-equiv' e + + map-inv-equiv-is-decidable : is-decidable B → is-decidable A + map-inv-equiv-is-decidable = is-decidable-equiv e + + is-section-map-inv-equiv-is-decidable : + is-section map-equiv-is-decidable map-inv-equiv-is-decidable + is-section-map-inv-equiv-is-decidable (inl x) = + ap inl (is-section-map-inv-equiv e x) + is-section-map-inv-equiv-is-decidable (inr x) = + ap inr eq-neg + + is-retraction-map-inv-equiv-is-decidable : + is-retraction map-equiv-is-decidable map-inv-equiv-is-decidable + is-retraction-map-inv-equiv-is-decidable (inl x) = + ap inl (is-retraction-map-inv-equiv e x) + is-retraction-map-inv-equiv-is-decidable (inr x) = + ap inr eq-neg + + is-equiv-map-equiv-is-decidable : is-equiv map-equiv-is-decidable + is-equiv-map-equiv-is-decidable = + is-equiv-is-invertible + map-inv-equiv-is-decidable + is-section-map-inv-equiv-is-decidable + is-retraction-map-inv-equiv-is-decidable + + equiv-is-decidable : is-decidable A ≃ is-decidable B + equiv-is-decidable = map-equiv-is-decidable , is-equiv-map-equiv-is-decidable ``` ### Decidability implies double negation elimination diff --git a/src/foundation/diaconescus-theorem.lagda.md b/src/foundation/diaconescus-theorem.lagda.md new file mode 100644 index 0000000000..adcacce8cc --- /dev/null +++ b/src/foundation/diaconescus-theorem.lagda.md @@ -0,0 +1,91 @@ +# Diaconescu's theorem + +```agda +module foundation.diaconescus-theorem where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.axiom-of-choice +open import foundation.booleans +open import foundation.decidable-propositions +open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.law-of-excluded-middle +open import foundation.logical-equivalences +open import foundation.propositional-truncations +open import foundation.propositions +open import foundation.universe-levels + +open import foundation-core.fibers-of-maps +open import foundation-core.function-types +open import foundation-core.identity-types + +open import synthetic-homotopy-theory.suspensions-of-propositions +open import synthetic-homotopy-theory.suspensions-of-types +``` + +
+ +## Idea + +The [axiom of choice](foundation.axiom-of-choice.md) implies the +[law of excluded middle](foundation.law-of-excluded-middle.md). This is often +referred to as +{{#concept "Diaconescu's theorem" WD="Diaconescu's theorem" WDID=Q3527059 Agda=theorem-Diaconescu}}. + +## Theorem + +We follow the proof given under Theorem 10.1.14 in {{#cite UF13}}. + +**Proof.** Given a [proposition](foundation-core.propositions.md) `P`, then its +[suspension](synthetic-homotopy-theory.suspensions-of-propositions.md) `ΣP` is a +[set](foundation-core.sets.md) with the property that `(N ＝ S) ≃ ΣP`, where `N` +and `S` are the _poles_ of `ΣP`. There is a surjection from the +[booleans](foundation.booleans.md) onto the suspension `f : bool ↠ ΣP` such that +`f true ≐ N` and `f false ≐ S`. Its +[fiber family](foundation-core.fibers-of-maps.md) is in other words an +[inhabited](foundation.inhabited-types.md) family over `ΣP`. Applying the axiom +of choice to this family, we obtain a +[mere](foundation.propositional-truncations.md) +[section](foundation-core.sections.md) `s` of `f` which thus exhibits `P` as a +[logical equivalent](foundation.logical-equivalences.md) to `f⁻¹ N ＝ f⁻¹ S`. +The latter is an [equation](foundation-core.identity-types.md) of booleans, and +the booleans have [decidable equality](foundation.decidable-equality.md) so `P` +must also be [decidable](foundation.decidable-propositions.md). + +```agda +instance-theorem-Diaconescu : + {l : Level} (P : Prop l) → + instance-choice₀ + ( suspension-set-Prop P) + ( fiber map-surjection-bool-suspension) → + is-decidable-type-Prop P +instance-theorem-Diaconescu P ac-P = + rec-trunc-Prop + ( is-decidable-Prop P) + ( λ s → + is-decidable-iff' + ( ( iff-equiv (compute-eq-north-south-suspension-Prop P)) ∘iff + ( ( λ p → + ( inv (pr2 (s north-suspension))) ∙ + ( ap map-surjection-bool-suspension p) ∙ + ( pr2 (s south-suspension))) , + ( ap (pr1 ∘ s)))) + ( has-decidable-equality-bool + ( pr1 (s north-suspension)) + ( pr1 (s south-suspension)))) + ( ac-P is-surjective-map-surjection-bool-suspension) + +theorem-Diaconescu : + {l : Level} → level-AC0 l l → LEM l +theorem-Diaconescu ac P = + instance-theorem-Diaconescu P + ( ac (suspension-set-Prop P) (fiber map-surjection-bool-suspension)) +``` + +## References + +{{#bibliography}} diff --git a/src/foundation/discrete-relaxed-sigma-decompositions.lagda.md b/src/foundation/discrete-relaxed-sigma-decompositions.lagda.md index e77ad9f312..76cdac43a7 100644 --- a/src/foundation/discrete-relaxed-sigma-decompositions.lagda.md +++ b/src/foundation/discrete-relaxed-sigma-decompositions.lagda.md @@ -85,13 +85,7 @@ module _ ( right-unit-law-Σ-is-contr is-discrete ∘e matching-correspondence-Relaxed-Σ-Decomposition D)) pr1 (pr2 equiv-discrete-is-discrete-Relaxed-Σ-Decomposition) x = - ( map-equiv (compute-raise-unit l4) ∘ - terminal-map (cotype-Relaxed-Σ-Decomposition D x) , - is-equiv-comp - ( map-equiv (compute-raise-unit l4)) - ( terminal-map (cotype-Relaxed-Σ-Decomposition D x)) - ( is-equiv-terminal-map-is-contr (is-discrete x)) - ( is-equiv-map-equiv ( compute-raise-unit l4))) + equiv-raise-unit-is-contr (is-discrete x) pr2 (pr2 equiv-discrete-is-discrete-Relaxed-Σ-Decomposition) a = eq-pair-Σ ( ap ( λ f → map-equiv f a) diff --git a/src/foundation/discrete-sigma-decompositions.lagda.md b/src/foundation/discrete-sigma-decompositions.lagda.md index 19316103ef..3ed376b244 100644 --- a/src/foundation/discrete-sigma-decompositions.lagda.md +++ b/src/foundation/discrete-sigma-decompositions.lagda.md @@ -87,13 +87,7 @@ module _ ( right-unit-law-Σ-is-contr is-discrete ∘e matching-correspondence-Σ-Decomposition D)) pr1 (pr2 equiv-discrete-is-discrete-Σ-Decomposition) x = - ( map-equiv (compute-raise-unit l4) ∘ - terminal-map (cotype-Σ-Decomposition D x) , - is-equiv-comp - ( map-equiv (compute-raise-unit l4)) - ( terminal-map (cotype-Σ-Decomposition D x)) - ( is-equiv-terminal-map-is-contr (is-discrete x)) - ( is-equiv-map-equiv ( compute-raise-unit l4))) + equiv-raise-unit-is-contr (is-discrete x) pr2 (pr2 equiv-discrete-is-discrete-Σ-Decomposition) a = eq-pair-Σ ( ap ( λ f → map-equiv f a) diff --git a/src/foundation/double-negation-modality.lagda.md b/src/foundation/double-negation-modality.lagda.md index a3158f9e97..89f91c3ee0 100644 --- a/src/foundation/double-negation-modality.lagda.md +++ b/src/foundation/double-negation-modality.lagda.md @@ -19,6 +19,8 @@ open import foundation.universe-levels open import foundation-core.function-types open import foundation-core.transport-along-identifications +open import logic.double-negation-elimination + open import orthogonal-factorization-systems.continuation-modalities open import orthogonal-factorization-systems.large-lawvere-tierney-topologies open import orthogonal-factorization-systems.lawvere-tierney-topologies diff --git a/src/foundation/double-negation-stable-propositions.lagda.md b/src/foundation/double-negation-stable-propositions.lagda.md index 760c158524..0ba325bf24 100644 --- a/src/foundation/double-negation-stable-propositions.lagda.md +++ b/src/foundation/double-negation-stable-propositions.lagda.md @@ -7,14 +7,36 @@ module foundation.double-negation-stable-propositions where
Imports ```agda +open import foundation.cartesian-product-types +open import foundation.conjunction +open import foundation.coproduct-types +open import foundation.decidable-propositions +open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.disjunction open import foundation.double-negation +open import foundation.embeddings open import foundation.empty-types +open import foundation.equivalences +open import foundation.existential-quantification +open import foundation.logical-equivalences open import foundation.negation +open import foundation.propositional-extensionality +open import foundation.propositions +open import foundation.sets +open import foundation.subtypes +open import foundation.transport-along-identifications +open import foundation.type-arithmetic-dependent-pair-types open import foundation.unit-type +open import foundation.universal-quantification open import foundation.universe-levels +open import foundation-core.contractible-types open import foundation-core.function-types -open import foundation-core.propositions +open import foundation-core.identity-types +open import foundation-core.retracts-of-types + +open import logic.double-negation-elimination ```
@@ -52,32 +74,327 @@ module _ is-prop-type-Prop is-double-negation-stable-Prop ``` +### The predicate on a type of being a double negation stable proposition + +```agda +is-double-negation-stable-prop : {l : Level} → UU l → UU l +is-double-negation-stable-prop X = (is-prop X) × (¬¬ X → X) + +is-prop-is-double-negation-stable-prop : + {l : Level} (X : UU l) → is-prop (is-double-negation-stable-prop X) +is-prop-is-double-negation-stable-prop X = + is-prop-Σ + ( is-prop-is-prop X) + ( λ is-prop-X → is-prop-is-double-negation-stable (X , is-prop-X)) + +is-double-negation-stable-prop-Prop : {l : Level} → UU l → Prop l +is-double-negation-stable-prop-Prop X = + ( is-double-negation-stable-prop X , is-prop-is-double-negation-stable-prop X) + +module _ + {l : Level} {A : UU l} (H : is-double-negation-stable-prop A) + where + + is-prop-type-is-double-negation-stable-prop : is-prop A + is-prop-type-is-double-negation-stable-prop = pr1 H + + has-double-negation-elim-is-double-negation-stable-prop : + has-double-negation-elim A + has-double-negation-elim-is-double-negation-stable-prop = pr2 H +``` + +### The subuniverse of double negation stable propositions + +```agda +Double-Negation-Stable-Prop : (l : Level) → UU (lsuc l) +Double-Negation-Stable-Prop l = Σ (UU l) (is-double-negation-stable-prop) + +module _ + {l : Level} (P : Double-Negation-Stable-Prop l) + where + + type-Double-Negation-Stable-Prop : UU l + type-Double-Negation-Stable-Prop = pr1 P + + is-double-negation-stable-prop-type-Double-Negation-Stable-Prop : + is-double-negation-stable-prop type-Double-Negation-Stable-Prop + is-double-negation-stable-prop-type-Double-Negation-Stable-Prop = pr2 P + + is-prop-type-Double-Negation-Stable-Prop : + is-prop type-Double-Negation-Stable-Prop + is-prop-type-Double-Negation-Stable-Prop = + is-prop-type-is-double-negation-stable-prop + ( is-double-negation-stable-prop-type-Double-Negation-Stable-Prop) + + prop-Double-Negation-Stable-Prop : Prop l + prop-Double-Negation-Stable-Prop = + ( type-Double-Negation-Stable-Prop , + is-prop-type-Double-Negation-Stable-Prop) + + has-double-negation-elim-type-Double-Negation-Stable-Prop : + has-double-negation-elim type-Double-Negation-Stable-Prop + has-double-negation-elim-type-Double-Negation-Stable-Prop = + has-double-negation-elim-is-double-negation-stable-prop + ( is-double-negation-stable-prop-type-Double-Negation-Stable-Prop) +``` + ## Properties +### The forgetful map from double negation stable propositions to propositions is an embedding + +```agda +is-emb-prop-Double-Negation-Stable-Prop : + {l : Level} → is-emb (prop-Double-Negation-Stable-Prop {l}) +is-emb-prop-Double-Negation-Stable-Prop = + is-emb-tot + ( λ X → + is-emb-inclusion-subtype + ( λ H → + has-double-negation-elim X , is-prop-has-double-negation-elim H)) + +emb-prop-Double-Negation-Stable-Prop : + {l : Level} → Double-Negation-Stable-Prop l ↪ Prop l +emb-prop-Double-Negation-Stable-Prop = + ( prop-Double-Negation-Stable-Prop , is-emb-prop-Double-Negation-Stable-Prop) +``` + +### The subuniverse of double negation stable propositions is a set + +```agda +is-set-Double-Negation-Stable-Prop : + {l : Level} → is-set (Double-Negation-Stable-Prop l) +is-set-Double-Negation-Stable-Prop {l} = + is-set-emb emb-prop-Double-Negation-Stable-Prop is-set-type-Prop + +set-Double-Negation-Stable-Prop : (l : Level) → Set (lsuc l) +set-Double-Negation-Stable-Prop l = + ( Double-Negation-Stable-Prop l , is-set-Double-Negation-Stable-Prop) +``` + +### Extensionality of double negation stable propositions + +```agda +module _ + {l : Level} (P Q : Double-Negation-Stable-Prop l) + where + + extensionality-Double-Negation-Stable-Prop : + ( P ＝ Q) ≃ + ( type-Double-Negation-Stable-Prop P ↔ type-Double-Negation-Stable-Prop Q) + extensionality-Double-Negation-Stable-Prop = + ( propositional-extensionality + ( prop-Double-Negation-Stable-Prop P) + ( prop-Double-Negation-Stable-Prop Q)) ∘e + ( equiv-ap-emb emb-prop-Double-Negation-Stable-Prop) + + iff-eq-Double-Negation-Stable-Prop : + P ＝ Q → + type-Double-Negation-Stable-Prop P ↔ type-Double-Negation-Stable-Prop Q + iff-eq-Double-Negation-Stable-Prop = + map-equiv extensionality-Double-Negation-Stable-Prop + + eq-iff-Double-Negation-Stable-Prop : + (type-Double-Negation-Stable-Prop P → type-Double-Negation-Stable-Prop Q) → + (type-Double-Negation-Stable-Prop Q → type-Double-Negation-Stable-Prop P) → + P ＝ Q + eq-iff-Double-Negation-Stable-Prop f g = + map-inv-equiv extensionality-Double-Negation-Stable-Prop (pair f g) +``` + +### Double negation stable propositions are closed under retracts + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-double-negation-stable-prop-retract : + A retract-of B → + is-double-negation-stable-prop B → + is-double-negation-stable-prop A + is-double-negation-stable-prop-retract e H = + ( is-prop-retract-of e + ( is-prop-type-is-double-negation-stable-prop H)) , + ( has-double-negation-elim-retract e + ( has-double-negation-elim-is-double-negation-stable-prop H)) +``` + +### Double negation stable propositions are closed under equivalences + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-double-negation-stable-prop-equiv : + A ≃ B → is-double-negation-stable-prop B → is-double-negation-stable-prop A + is-double-negation-stable-prop-equiv e = + is-double-negation-stable-prop-retract (retract-equiv e) + + is-double-negation-stable-prop-equiv' : + B ≃ A → is-double-negation-stable-prop B → is-double-negation-stable-prop A + is-double-negation-stable-prop-equiv' e = + is-double-negation-stable-prop-retract (retract-inv-equiv e) +``` + ### The empty proposition is double negation stable ```agda -is-double-negation-stable-empty : is-double-negation-stable empty-Prop -is-double-negation-stable-empty e = e id +empty-Double-Negation-Stable-Prop : Double-Negation-Stable-Prop lzero +empty-Double-Negation-Stable-Prop = + ( empty , is-prop-empty , double-negation-elim-empty) ``` ### The unit proposition is double negation stable ```agda -is-double-negation-stable-unit : is-double-negation-stable unit-Prop -is-double-negation-stable-unit _ = star +unit-Double-Negation-Stable-Prop : Double-Negation-Stable-Prop lzero +unit-Double-Negation-Stable-Prop = + ( unit , is-prop-unit , double-negation-elim-unit) +``` + +```agda +is-double-negation-stable-prop-is-contr : + {l : Level} {A : UU l} → is-contr A → is-double-negation-stable-prop A +is-double-negation-stable-prop-is-contr H = + (is-prop-is-contr H , double-negation-elim-is-contr H) +``` + +### Decidable propositions are double negation stable + +```agda +double-negation-stable-prop-Decidable-Prop : + {l : Level} → Decidable-Prop l → Double-Negation-Stable-Prop l +double-negation-stable-prop-Decidable-Prop (A , H , d) = + ( A , H , double-negation-elim-is-decidable d) +``` + +### Negations of types are double negation stable propositions + +```agda +neg-type-Double-Negation-Stable-Prop : + {l : Level} → UU l → Double-Negation-Stable-Prop l +neg-type-Double-Negation-Stable-Prop A = + ( ¬ A , is-prop-neg , double-negation-elim-neg A) + +neg-Double-Negation-Stable-Prop : + {l : Level} → Double-Negation-Stable-Prop l → Double-Negation-Stable-Prop l +neg-Double-Negation-Stable-Prop P = + neg-type-Double-Negation-Stable-Prop (type-Double-Negation-Stable-Prop P) +``` + +### Universal quantification over double negation stable propositions is double negation stable + +```agda +is-double-negation-stable-prop-Π : + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} → + ((a : A) → is-double-negation-stable-prop (B a)) → + is-double-negation-stable-prop ((a : A) → B a) +is-double-negation-stable-prop-Π b = + ( is-prop-Π (is-prop-type-is-double-negation-stable-prop ∘ b)) , + ( double-negation-elim-for-all + ( has-double-negation-elim-is-double-negation-stable-prop ∘ b)) + +Π-Double-Negation-Stable-Prop : + {l1 l2 : Level} + (A : UU l1) (B : A → Double-Negation-Stable-Prop l2) → + Double-Negation-Stable-Prop (l1 ⊔ l2) +Π-Double-Negation-Stable-Prop A B = + ( (a : A) → type-Double-Negation-Stable-Prop (B a)) , + ( is-double-negation-stable-prop-Π + ( is-double-negation-stable-prop-type-Double-Negation-Stable-Prop ∘ B)) +``` + +### Implication into double negation stable propositions is double negation stable + +```agda +is-double-negation-stable-prop-exp : + {l1 l2 : Level} {A : UU l1} {B : UU l2} → + is-double-negation-stable-prop B → + is-double-negation-stable-prop (A → B) +is-double-negation-stable-prop-exp b = + is-double-negation-stable-prop-Π (λ _ → b) + +exp-Double-Negation-Stable-Prop : + {l1 l2 : Level} + (A : UU l1) (B : Double-Negation-Stable-Prop l2) → + Double-Negation-Stable-Prop (l1 ⊔ l2) +exp-Double-Negation-Stable-Prop A B = Π-Double-Negation-Stable-Prop A (λ _ → B) +``` + +### Dependent sums of double negation stable propositions over double negation stable propositions are double negation stable + +```agda +is-double-negation-stable-prop-Σ : + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} → + is-double-negation-stable-prop A → + ((a : A) → is-double-negation-stable-prop (B a)) → + is-double-negation-stable-prop (Σ A B) +is-double-negation-stable-prop-Σ a b = + ( is-prop-Σ + ( is-prop-type-is-double-negation-stable-prop a) + ( is-prop-type-is-double-negation-stable-prop ∘ b)) , + ( double-negation-elim-Σ-is-prop-base + ( is-prop-type-is-double-negation-stable-prop a) + ( has-double-negation-elim-is-double-negation-stable-prop a) + ( has-double-negation-elim-is-double-negation-stable-prop ∘ b)) + +Σ-Double-Negation-Stable-Prop : + {l1 l2 : Level} + (A : Double-Negation-Stable-Prop l1) + (B : type-Double-Negation-Stable-Prop A → Double-Negation-Stable-Prop l2) → + Double-Negation-Stable-Prop (l1 ⊔ l2) +Σ-Double-Negation-Stable-Prop A B = + ( Σ ( type-Double-Negation-Stable-Prop A) + ( type-Double-Negation-Stable-Prop ∘ B)) , + ( is-double-negation-stable-prop-Σ + ( is-double-negation-stable-prop-type-Double-Negation-Stable-Prop A) + ( is-double-negation-stable-prop-type-Double-Negation-Stable-Prop ∘ B)) +``` + +### The conjunction of two double negation stable propositions is double negation stable + +```agda +is-double-negation-stable-prop-product : + {l1 l2 : Level} {A : UU l1} {B : UU l2} → + is-double-negation-stable-prop A → + is-double-negation-stable-prop B → + is-double-negation-stable-prop (A × B) +is-double-negation-stable-prop-product a b = + ( is-prop-product + ( is-prop-type-is-double-negation-stable-prop a) + ( is-prop-type-is-double-negation-stable-prop b)) , + ( double-negation-elim-product + ( has-double-negation-elim-is-double-negation-stable-prop a) + ( has-double-negation-elim-is-double-negation-stable-prop b)) + +product-Double-Negation-Stable-Prop : + {l1 l2 : Level} → + Double-Negation-Stable-Prop l1 → + Double-Negation-Stable-Prop l2 → + Double-Negation-Stable-Prop (l1 ⊔ l2) +product-Double-Negation-Stable-Prop A B = + ( ( type-Double-Negation-Stable-Prop A) × + ( type-Double-Negation-Stable-Prop B)) , + ( is-double-negation-stable-prop-product + ( is-double-negation-stable-prop-type-Double-Negation-Stable-Prop A) + ( is-double-negation-stable-prop-type-Double-Negation-Stable-Prop B)) ``` -### The negation of a type is double negation stable +### Negation has no fixed points on double negation stable propositions ```agda -is-double-negation-stable-neg : - {l : Level} (A : UU l) → is-double-negation-stable (neg-type-Prop A) -is-double-negation-stable-neg = double-negation-elim-neg +abstract + no-fixed-points-neg-Double-Negation-Stable-Prop : + {l : Level} (P : Double-Negation-Stable-Prop l) → + ¬ (type-Double-Negation-Stable-Prop P ↔ + ¬ (type-Double-Negation-Stable-Prop P)) + no-fixed-points-neg-Double-Negation-Stable-Prop P = + no-fixed-points-neg (type-Double-Negation-Stable-Prop P) ``` ## See also - [The double negation modality](foundation.double-negation-modality.md) - [Irrefutable propositions](foundation.irrefutable-propositions.md) are double - negation stable. + negation connected types. diff --git a/src/foundation/double-negation.lagda.md b/src/foundation/double-negation.lagda.md index 4529283ca7..ee813f6384 100644 --- a/src/foundation/double-negation.lagda.md +++ b/src/foundation/double-negation.lagda.md @@ -7,15 +7,12 @@ module foundation.double-negation where
Imports ```agda -open import foundation.dependent-pair-types open import foundation.negation 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.empty-types -open import foundation-core.function-types open import foundation-core.propositions ``` @@ -94,45 +91,13 @@ double-negation-linearity-implication {P = P} {Q = Q} f = ( λ (p : P) → map-neg (inr {A = P → Q} {B = Q → P}) f (λ _ → p)) ``` -### Cases of double negation elimination - -```agda -double-negation-elim-neg : {l : Level} (P : UU l) → ¬¬¬ P → ¬ P -double-negation-elim-neg P f p = f (λ g → g p) - -double-negation-elim-product : - {l1 l2 : Level} {P : UU l1} {Q : UU l2} → - ¬¬ ((¬¬ P) × (¬¬ Q)) → (¬¬ P) × (¬¬ Q) -pr1 (double-negation-elim-product {P = P} {Q = Q} f) = - double-negation-elim-neg (¬ P) (map-double-negation pr1 f) -pr2 (double-negation-elim-product {P = P} {Q = Q} f) = - double-negation-elim-neg (¬ Q) (map-double-negation pr2 f) - -double-negation-elim-exp : - {l1 l2 : Level} {P : UU l1} {Q : UU l2} → - ¬¬ (P → ¬¬ Q) → (P → ¬¬ Q) -double-negation-elim-exp {P = P} {Q = Q} f p = - double-negation-elim-neg - ( ¬ Q) - ( map-double-negation (λ (g : P → ¬¬ Q) → g p) f) - -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-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) -``` - ### Maps into double negations extend along `intro-double-negation` ```agda double-negation-extend : {l1 l2 : Level} {P : UU l1} {Q : UU l2} → (P → ¬¬ Q) → (¬¬ P → ¬¬ Q) -double-negation-extend {P = P} {Q = Q} f = - double-negation-elim-neg (¬ Q) ∘ (map-double-negation f) +double-negation-extend {P = P} {Q = Q} f nnp nq = nnp (λ p → f p nq) ``` ### The double negation of a type is logically equivalent to the double negation of its propositional truncation @@ -154,6 +119,11 @@ abstract map-double-negation unit-trunc-Prop ``` +## See also + +- [Double negation elimination](logic.double-negation-elimination.md) +- [Irrefutable propositions](foundation.irrefutable-propositions.md) + ## Table of files about propositional logic The following table gives an overview of basic constructions in propositional diff --git a/src/foundation/equality-cartesian-product-types.lagda.md b/src/foundation/equality-cartesian-product-types.lagda.md index eb42462766..b500eb9d43 100644 --- a/src/foundation/equality-cartesian-product-types.lagda.md +++ b/src/foundation/equality-cartesian-product-types.lagda.md @@ -7,11 +7,13 @@ module foundation.equality-cartesian-product-types where
Imports ```agda +open import foundation.action-on-identifications-binary-functions open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.universe-levels open import foundation-core.cartesian-product-types +open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.function-types open import foundation-core.homotopies @@ -36,7 +38,7 @@ module _ where Eq-product : (s t : A × B) → UU (l1 ⊔ l2) - Eq-product s t = ((pr1 s) ＝ (pr1 t)) × ((pr2 s) ＝ (pr2 t)) + Eq-product s t = (pr1 s ＝ pr1 t) × (pr2 s ＝ pr2 t) ``` ## Properties @@ -49,23 +51,22 @@ module _ where eq-pair' : {s t : A × B} → Eq-product s t → s ＝ t - eq-pair' {pair x y} {pair .x .y} (pair refl refl) = refl + eq-pair' (p , q) = ap-binary pair p q - eq-pair : - {s t : A × B} → (pr1 s) ＝ (pr1 t) → (pr2 s) ＝ (pr2 t) → s ＝ t - eq-pair p q = eq-pair' (pair p q) + eq-pair : {s t : A × B} → pr1 s ＝ pr1 t → pr2 s ＝ pr2 t → s ＝ t + eq-pair p q = eq-pair' (p , q) pair-eq : {s t : A × B} → s ＝ t → Eq-product s t pr1 (pair-eq α) = ap pr1 α pr2 (pair-eq α) = ap pr2 α is-retraction-pair-eq : - {s t : A × B} → ((pair-eq {s} {t}) ∘ (eq-pair' {s} {t})) ~ id - is-retraction-pair-eq {pair x y} {pair .x .y} (pair refl refl) = refl + {s t : A × B} → pair-eq {s} {t} ∘ eq-pair' {s} {t} ~ id + is-retraction-pair-eq (refl , refl) = refl is-section-pair-eq : - {s t : A × B} → ((eq-pair' {s} {t}) ∘ (pair-eq {s} {t})) ~ id - is-section-pair-eq {pair x y} {pair .x .y} refl = refl + {s t : A × B} → eq-pair' {s} {t} ∘ pair-eq {s} {t} ~ id + is-section-pair-eq refl = refl abstract is-equiv-eq-pair : @@ -99,13 +100,13 @@ module _ triangle-eq-pair : {a0 a1 : A} {b0 b1 : B} (p : a0 ＝ a1) (q : b0 ＝ b1) → - eq-pair p q ＝ ((eq-pair p refl) ∙ (eq-pair refl q)) - triangle-eq-pair refl refl = refl + eq-pair p q ＝ eq-pair p refl ∙ eq-pair refl q + triangle-eq-pair refl q = refl triangle-eq-pair' : {a0 a1 : A} {b0 b1 : B} (p : a0 ＝ a1) (q : b0 ＝ b1) → - eq-pair p q ＝ ((eq-pair refl q) ∙ (eq-pair p refl)) - triangle-eq-pair' refl refl = refl + eq-pair p q ＝ eq-pair refl q ∙ eq-pair p refl + triangle-eq-pair' p refl = refl ``` ### `eq-pair` preserves concatenation @@ -114,9 +115,7 @@ module _ eq-pair-concat : {l1 l2 : Level} {A : UU l1} {B : UU l2} {x x' x'' : A} {y y' y'' : B} (p : x ＝ x') (p' : x' ＝ x'') (q : y ＝ y') (q' : y' ＝ y'') → - ( eq-pair {s = pair x y} {t = pair x'' y''} (p ∙ p') (q ∙ q')) ＝ - ( ( eq-pair {s = pair x y} {t = pair x' y'} p q) ∙ - ( eq-pair p' q')) + eq-pair (p ∙ p') (q ∙ q') ＝ eq-pair p q ∙ eq-pair p' q' eq-pair-concat refl p' refl q' = refl ``` @@ -126,13 +125,13 @@ eq-pair-concat refl p' refl q' = refl ap-pr1-eq-pair : {l1 l2 : Level} {A : UU l1} {B : UU l2} {x x' : A} (p : x ＝ x') {y y' : B} (q : y ＝ y') → - ap pr1 (eq-pair {s = pair x y} {pair x' y'} p q) ＝ p + ap pr1 (eq-pair p q) ＝ p ap-pr1-eq-pair refl refl = refl ap-pr2-eq-pair : {l1 l2 : Level} {A : UU l1} {B : UU l2} {x x' : A} (p : x ＝ x') {y y' : B} (q : y ＝ y') → - ap pr2 (eq-pair {s = pair x y} {pair x' y'} p q) ＝ q + ap pr2 (eq-pair p q) ＝ q ap-pr2-eq-pair refl refl = refl ``` @@ -155,12 +154,12 @@ module _ ```agda left-unit-law-tr-eq-pair : (C : A × B → UU l3) (q : b0 ＝ b1) (u : C (a0 , b0)) → - (tr C (eq-pair refl q) u) ＝ tr (λ x → C (a0 , x)) q u + tr C (eq-pair refl q) u ＝ tr (λ x → C (a0 , x)) q u left-unit-law-tr-eq-pair C refl u = refl right-unit-law-tr-eq-pair : (C : A × B → UU l3) (p : a0 ＝ a1) (u : C (a0 , b0)) → - (tr C (eq-pair p refl) u) ＝ tr (λ x → C (x , b0)) p u + tr C (eq-pair p refl) u ＝ tr (λ x → C (x , b0)) p u right-unit-law-tr-eq-pair C refl u = refl ``` diff --git a/src/foundation/equality-dependent-pair-types.lagda.md b/src/foundation/equality-dependent-pair-types.lagda.md index 5126671f54..6ab7a7bbb3 100644 --- a/src/foundation/equality-dependent-pair-types.lagda.md +++ b/src/foundation/equality-dependent-pair-types.lagda.md @@ -192,13 +192,13 @@ module _ coh-eq-base-Σ² : {s t : Σ A (λ x → Σ (B x) λ y → C x y)} (p : s ＝ t) → - eq-base-eq-pair-Σ p ＝ - eq-base-eq-pair-Σ (eq-base-eq-pair-Σ (ap (map-inv-associative-Σ' A B C) p)) + eq-base-eq-pair p ＝ + eq-base-eq-pair (eq-base-eq-pair (ap (map-inv-associative-Σ' A B C) p)) coh-eq-base-Σ² refl = refl dependent-eq-second-component-eq-Σ² : {s t : Σ A (λ x → Σ (B x) λ y → C x y)} (p : s ＝ t) → - dependent-identification B (eq-base-eq-pair-Σ p) (pr1 (pr2 s)) (pr1 (pr2 t)) + dependent-identification B (eq-base-eq-pair p) (pr1 (pr2 s)) (pr1 (pr2 t)) dependent-eq-second-component-eq-Σ² {s = s} {t = t} p = ( ap (λ q → tr B q (pr1 (pr2 s))) (coh-eq-base-Σ² p)) ∙ ( pr2 @@ -219,14 +219,14 @@ module _ coh-eq-base-Σ³ : { s t : Σ A (λ x → Σ (B x) (λ y → Σ (C x) (D x y)))} (p : s ＝ t) → - eq-base-eq-pair-Σ p ＝ - eq-base-eq-pair-Σ (ap (map-equiv (interchange-Σ-Σ-Σ D)) p) + eq-base-eq-pair p ＝ + eq-base-eq-pair (ap (map-equiv (interchange-Σ-Σ-Σ D)) p) coh-eq-base-Σ³ refl = refl dependent-eq-third-component-eq-Σ³ : { s t : Σ A (λ x → Σ (B x) (λ y → Σ (C x) (D x y)))} (p : s ＝ t) → dependent-identification C - ( eq-base-eq-pair-Σ p) + ( eq-base-eq-pair p) ( pr1 (pr2 (pr2 s))) ( pr1 (pr2 (pr2 t))) dependent-eq-third-component-eq-Σ³ {s = s} {t = t} p = diff --git a/src/foundation/equivalence-injective-type-families.lagda.md b/src/foundation/equivalence-injective-type-families.lagda.md index d565eb4308..5b1ea1cb5f 100644 --- a/src/foundation/equivalence-injective-type-families.lagda.md +++ b/src/foundation/equivalence-injective-type-families.lagda.md @@ -37,9 +37,7 @@ as a map. **Note.** The concept of equivalence injective type family as considered here is unrelated to the concept of "injective type" as studied by Martín Escardó in -_Injective types in univalent mathematics_ -([arXiv:1903.01211](https://arxiv.org/abs/1903.01211), -[TypeTopology](https://www.cs.bham.ac.uk/~mhe/TypeTopology/InjectiveTypes.index.html)). +_Injective types in univalent mathematics_ {{#cite Esc21b}}. ## Definition @@ -108,3 +106,7 @@ module _ pr1 is-equivalence-injective-Prop = is-equivalence-injective P pr2 is-equivalence-injective-Prop = is-prop-is-equivalence-injective ``` + +## References + +{{#bibliography}} diff --git a/src/foundation/fibers-of-maps.lagda.md b/src/foundation/fibers-of-maps.lagda.md index e0995d367b..01e4a3f77b 100644 --- a/src/foundation/fibers-of-maps.lagda.md +++ b/src/foundation/fibers-of-maps.lagda.md @@ -12,6 +12,7 @@ open import foundation-core.fibers-of-maps public open import foundation.action-on-identifications-functions open import foundation.cones-over-cospan-diagrams open import foundation.dependent-pair-types +open import foundation.postcomposition-functions open import foundation.type-arithmetic-dependent-pair-types open import foundation.type-arithmetic-unit-type open import foundation.unit-type @@ -20,6 +21,7 @@ open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.equivalences open import foundation-core.function-types +open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types @@ -109,6 +111,25 @@ module _ inv-equiv equiv-total-fiber-terminal-map ``` +### Computing the fibers of postcomposition by a projection map + +```agda +module _ + {l1 l2 : Level} {A : UU l1} (B : A → UU l2) + where + + compute-fiber-postcomp-pr1 : + {l3 : Level} {X : UU l3} (h : X → A) → + ((i : X) → B (h i)) ≃ fiber (postcomp X (pr1 {B = B})) h + compute-fiber-postcomp-pr1 {X = X} h = + compute-Π-fiber-postcomp X pr1 h ∘e + equiv-Π-equiv-family (λ i → inv-equiv-fiber-pr1 B (h i)) + + compute-fiber-id-postcomp-pr1 : + ((a : A) → B a) ≃ fiber (postcomp A (pr1 {B = B})) id + compute-fiber-id-postcomp-pr1 = compute-fiber-postcomp-pr1 id +``` + ### Transport in fibers ```agda diff --git a/src/foundation/function-types.lagda.md b/src/foundation/function-types.lagda.md index 9af9c22baf..d6fb4ed16f 100644 --- a/src/foundation/function-types.lagda.md +++ b/src/foundation/function-types.lagda.md @@ -111,6 +111,11 @@ module _ {l1 l2 l3 : Level} {A : UU l1} {x y : A} (B : A → UU l2) (C : UU l3) where + tr-function-type-fixed-codomain : + (p : x ＝ y) (f : B x → C) → + tr (λ a → B a → C) p f ＝ f ∘ tr B (inv p) + tr-function-type-fixed-codomain refl f = refl + compute-dependent-identification-function-type-fixed-codomain : (p : x ＝ y) (f : B x → C) (g : B y → C) → ((b : B x) → f b ＝ g (tr B p b)) ≃ @@ -127,6 +132,19 @@ module _ ( compute-dependent-identification-function-type-fixed-codomain p f g) ``` +### Transport in a family of function types with fixed domain + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {x y : A} (B : UU l2) (C : A → UU l3) + where + + tr-function-type-fixed-domain : + (p : x ＝ y) (f : B → C x) → + tr (λ a → B → C a) p f ＝ tr C p ∘ f + tr-function-type-fixed-domain refl f = refl +``` + ### Relation between `compute-dependent-identification-function-type` and `preserves-tr` ```agda @@ -216,7 +234,7 @@ module _ ( H s) ( k s) ( l s))) - ( λ k l s → inv-equiv (equiv-funext)))) ∙ + ( λ k l s → inv-equiv equiv-funext))) ∙ ( eq-htpy-refl-htpy (h (i s)))) ``` diff --git a/src/foundation/functoriality-dependent-function-types.lagda.md b/src/foundation/functoriality-dependent-function-types.lagda.md index bb38a2d469..5993ac6c2a 100644 --- a/src/foundation/functoriality-dependent-function-types.lagda.md +++ b/src/foundation/functoriality-dependent-function-types.lagda.md @@ -103,15 +103,8 @@ module _ ( f (map-inv-equiv e (map-equiv e a'))) ( h (map-inv-equiv e (map-equiv e a'))))) ( coherence-map-inv-equiv e a')) ∙ - ( ( tr-ap - ( map-equiv e) - ( λ _ → id) - ( is-retraction-map-inv-equiv e a') - ( map-equiv - ( f (map-inv-equiv e (map-equiv e a'))) - ( h (map-inv-equiv e (map-equiv e a'))))) ∙ - ( α ( map-inv-equiv e (map-equiv e a')) - ( is-retraction-map-inv-equiv e a'))) + ( substitution-law-tr B (map-equiv e) (is-retraction-map-inv-equiv e a')) ∙ + ( α (map-inv-equiv e (map-equiv e a')) (is-retraction-map-inv-equiv e a')) where α : (x : A') (p : x ＝ a') → diff --git a/src/foundation/hilberts-epsilon-operators.lagda.md b/src/foundation/hilberts-epsilon-operators.lagda.md index 6ad95cd052..1c23904a8b 100644 --- a/src/foundation/hilberts-epsilon-operators.lagda.md +++ b/src/foundation/hilberts-epsilon-operators.lagda.md @@ -19,8 +19,14 @@ open import foundation-core.function-types ## Idea -Hilbert's $ε$-operator at a type `A` is a map `type-trunc-Prop A → A`. Contrary -to Hilbert, we will not assume that such an operator exists for each type `A`. +{{#concept "Hilbert's $ε$-operator"}} at a type `A` is a map + +```text + ε : ║A║₋₁ → A +``` + +Some authors also refer to this as _split support_ {{#cite KECA17}}. Contrary to +Hilbert, we will not assume that such an operator exists for each type `A`. ## Definition @@ -46,3 +52,12 @@ to Hilbert, we will not assume that such an operator exists for each type `A`. ε-operator-equiv' e f = (map-inv-equiv e ∘ f) ∘ (map-trunc-Prop (map-equiv e)) ``` + +## References + +{{#bibliography}} + +## External links + +- [Epsilon calculus](https://en.wikipedia.org/wiki/Epsilon_calculus) at + Wikipedia diff --git a/src/foundation/images.lagda.md b/src/foundation/images.lagda.md index f678e73231..4ffd596c45 100644 --- a/src/foundation/images.lagda.md +++ b/src/foundation/images.lagda.md @@ -36,7 +36,9 @@ open import foundation-core.truncation-levels ## Idea -The **image** of a map is a type that satisfies the +The +{{#concept "image" Disambiguation="of a map" WD="image" WDID=Q860623 Agda=im}} +of a map is a type that satisfies the [universal property of the image](foundation.universal-property-image.md) of a map. @@ -73,7 +75,7 @@ module _ ## Properties -### We characterize the identity type of im f +### We characterize the identity type of `im f` ```agda module _ @@ -152,7 +154,7 @@ abstract ( trunc-Prop (fiber (map-unit-im f) (y , z))) ( α) where - α : fiber f y → type-Prop (trunc-Prop (fiber (map-unit-im f) (y , z))) + α : fiber f y → type-trunc-Prop (fiber (map-unit-im f) (y , z)) α (x , p) = unit-trunc-Prop (x , eq-type-subtype (trunc-Prop ∘ fiber f) p) ``` @@ -216,3 +218,8 @@ im-1-Type : (f : A → type-1-Type X) → 1-Type (l1 ⊔ l2) im-1-Type X f = im-Truncated-Type zero-𝕋 X f ``` + +## External links + +- [Image (mathematics)]() at + Wikipedia diff --git a/src/foundation/irrefutable-propositions.lagda.md b/src/foundation/irrefutable-propositions.lagda.md index 7e0dd013dc..a6a0d7fcde 100644 --- a/src/foundation/irrefutable-propositions.lagda.md +++ b/src/foundation/irrefutable-propositions.lagda.md @@ -7,18 +7,22 @@ module foundation.irrefutable-propositions where
Imports ```agda +open import foundation.cartesian-product-types +open import foundation.contractible-types open import foundation.coproduct-types open import foundation.decidable-propositions open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.double-negation +open import foundation.empty-types open import foundation.function-types open import foundation.negation +open import foundation.propositions open import foundation.subuniverses open import foundation.unit-type open import foundation.universe-levels -open import foundation-core.propositions +open import logic.double-negation-elimination ```
@@ -35,26 +39,57 @@ The [subuniverse](foundation.subuniverses.md) of ### The predicate on a proposition of being irrefutable ```agda -is-irrefutable : {l : Level} → Prop l → UU l -is-irrefutable P = ¬¬ (type-Prop P) +module _ + {l : Level} (P : Prop l) + where + + is-irrefutable : UU l + is-irrefutable = ¬¬ (type-Prop P) -is-prop-is-irrefutable : {l : Level} (P : Prop l) → is-prop (is-irrefutable P) -is-prop-is-irrefutable P = is-prop-double-negation + is-prop-is-irrefutable : is-prop is-irrefutable + is-prop-is-irrefutable = is-prop-double-negation -is-irrefutable-Prop : {l : Level} → Prop l → Prop l -is-irrefutable-Prop = double-negation-Prop + is-irrefutable-Prop : Prop l + is-irrefutable-Prop = double-negation-Prop P ``` -### The subuniverse of irrefutable propositions +### The predicate on a type of being an irrefutable proposition ```agda -subuniverse-Irrefutable-Prop : (l : Level) → subuniverse l l -subuniverse-Irrefutable-Prop l A = - product-Prop (is-prop-Prop A) (double-negation-type-Prop A) +module _ + {l : Level} (P : UU l) + where + + is-irrefutable-prop : UU l + is-irrefutable-prop = is-prop P × (¬¬ P) + + is-prop-is-irrefutable-prop : is-prop is-irrefutable-prop + is-prop-is-irrefutable-prop = + is-prop-product (is-prop-is-prop P) is-prop-double-negation + + is-irrefutable-prop-Prop : Prop l + is-irrefutable-prop-Prop = is-irrefutable-prop , is-prop-is-irrefutable-prop + +module _ + {l : Level} {P : UU l} (H : is-irrefutable-prop P) + where + + is-prop-type-is-irrefutable-prop : is-prop P + is-prop-type-is-irrefutable-prop = pr1 H + + prop-is-irrefutable-prop : Prop l + prop-is-irrefutable-prop = P , is-prop-type-is-irrefutable-prop + is-irrefutable-is-irrefutable-prop : + is-irrefutable (P , is-prop-type-is-irrefutable-prop) + is-irrefutable-is-irrefutable-prop = pr2 H +``` + +### The subuniverse of irrefutable propositions + +```agda Irrefutable-Prop : (l : Level) → UU (lsuc l) -Irrefutable-Prop l = - type-subuniverse (subuniverse-Irrefutable-Prop l) +Irrefutable-Prop l = type-subuniverse is-irrefutable-prop-Prop make-Irrefutable-Prop : {l : Level} (P : Prop l) → is-irrefutable P → Irrefutable-Prop l @@ -68,18 +103,47 @@ module _ type-Irrefutable-Prop : UU l type-Irrefutable-Prop = pr1 P + is-irrefutable-prop-type-Irrefutable-Prop : + is-irrefutable-prop type-Irrefutable-Prop + is-irrefutable-prop-type-Irrefutable-Prop = pr2 P + is-prop-type-Irrefutable-Prop : is-prop type-Irrefutable-Prop - is-prop-type-Irrefutable-Prop = pr1 (pr2 P) + is-prop-type-Irrefutable-Prop = + is-prop-type-is-irrefutable-prop is-irrefutable-prop-type-Irrefutable-Prop prop-Irrefutable-Prop : Prop l prop-Irrefutable-Prop = type-Irrefutable-Prop , is-prop-type-Irrefutable-Prop is-irrefutable-Irrefutable-Prop : is-irrefutable prop-Irrefutable-Prop - is-irrefutable-Irrefutable-Prop = pr2 (pr2 P) + is-irrefutable-Irrefutable-Prop = + is-irrefutable-is-irrefutable-prop is-irrefutable-prop-type-Irrefutable-Prop ``` ## Properties +### Provable propositions are irrefutable + +```agda +module _ + {l : Level} (P : Prop l) + where + + is-irrefutable-has-element : type-Prop P → is-irrefutable P + is-irrefutable-has-element = intro-double-negation + +is-irrefutable-unit : is-irrefutable unit-Prop +is-irrefutable-unit = is-irrefutable-has-element unit-Prop star +``` + +### Contractible types are irrefutable propositions + +```agda +is-irrefutable-prop-is-contr : + {l : Level} {P : UU l} → is-contr P → is-irrefutable-prop P +is-irrefutable-prop-is-contr H = + ( is-prop-is-contr H , intro-double-negation (center H)) +``` + ### If it is irrefutable that a proposition is irrefutable, then the proposition is irrefutable ```agda @@ -87,13 +151,13 @@ module _ {l : Level} (P : Prop l) where - is-irrefutable-is-irrefutable-is-irrefutable : + is-idempotent-is-irrefutable : is-irrefutable (is-irrefutable-Prop P) → is-irrefutable P - is-irrefutable-is-irrefutable-is-irrefutable = + is-idempotent-is-irrefutable = double-negation-elim-neg (¬ (type-Prop P)) ``` -### The decidability of a proposition is irrefutable +### Decidability is irrefutable ```agda is-irrefutable-is-decidable : {l : Level} {A : UU l} → ¬¬ (is-decidable A) @@ -111,16 +175,67 @@ module _ make-Irrefutable-Prop (is-decidable-Prop P) is-irrefutable-is-decidable-Prop ``` -### Provable propositions are irrefutable +### Double negation elimination is irrefutable + +```agda +is-irrefutable-double-negation-elim : + {l : Level} {A : UU l} → ¬¬ (has-double-negation-elim A) +is-irrefutable-double-negation-elim H = + H (λ f → ex-falso (f (λ a → H (λ _ → a)))) +``` + +### Dependent sums of irrefutable propositions ```agda module _ - {l : Level} (P : Prop l) + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} where - is-irrefutable-is-inhabited : type-Prop P → is-irrefutable P - is-irrefutable-is-inhabited = intro-double-negation + is-irrefutable-Σ : + ¬¬ A → ((x : A) → ¬¬ B x) → ¬¬ (Σ A B) + is-irrefutable-Σ nna nnb nab = nna (λ a → nnb a (λ b → nab (a , b))) + + is-irrefutable-prop-Σ : + is-irrefutable-prop A → ((x : A) → is-irrefutable-prop (B x)) → + is-irrefutable-prop (Σ A B) + is-irrefutable-prop-Σ a b = + ( is-prop-Σ + ( is-prop-type-is-irrefutable-prop a) + ( is-prop-type-is-irrefutable-prop ∘ b) , + is-irrefutable-Σ + ( is-irrefutable-is-irrefutable-prop a) + ( is-irrefutable-is-irrefutable-prop ∘ b)) +``` -is-irrefutable-unit : is-irrefutable unit-Prop -is-irrefutable-unit = is-irrefutable-is-inhabited unit-Prop star +### Products of irrefutable propositions + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-irrefutable-product : ¬¬ A → ¬¬ B → ¬¬ (A × B) + is-irrefutable-product nna nnb = is-irrefutable-Σ nna (λ _ → nnb) + + is-irrefutable-prop-product : + is-irrefutable-prop A → is-irrefutable-prop B → is-irrefutable-prop (A × B) + is-irrefutable-prop-product a b = is-irrefutable-prop-Σ a (λ _ → b) ``` + +### Coproducts of irrefutable propositions + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-irrefutable-coproduct-inl : ¬¬ A → ¬¬ (A + B) + is-irrefutable-coproduct-inl nna x = nna (x ∘ inl) + + is-irrefutable-coproduct-inr : ¬¬ B → ¬¬ (A + B) + is-irrefutable-coproduct-inr nnb x = nnb (x ∘ inr) +``` + +## See also + +- [De Morgan's law](logic.de-morgans-law.md) is irrefutable. diff --git a/src/foundation/large-locale-of-propositions.lagda.md b/src/foundation/large-locale-of-propositions.lagda.md index 816cd18c3d..d5ef3ea50a 100644 --- a/src/foundation/large-locale-of-propositions.lagda.md +++ b/src/foundation/large-locale-of-propositions.lagda.md @@ -8,6 +8,7 @@ module foundation.large-locale-of-propositions where ```agda open import foundation.conjunction +open import foundation.empty-types open import foundation.existential-quantification open import foundation.logical-equivalences open import foundation.propositional-extensionality @@ -18,6 +19,7 @@ open import foundation.universe-levels open import foundation-core.function-types open import foundation-core.propositions +open import order-theory.bottom-elements-large-posets open import order-theory.large-frames open import order-theory.large-locales open import order-theory.large-meet-semilattices @@ -87,6 +89,17 @@ is-top-element-top-has-top-element-Large-Poset star ``` +### The smallest element in the large poset of propositions + +```agda +has-bottom-element-Prop-Large-Locale : + has-bottom-element-Large-Poset Prop-Large-Poset +bottom-has-bottom-element-Large-Poset + has-bottom-element-Prop-Large-Locale = empty-Prop +is-bottom-element-bottom-has-bottom-element-Large-Poset + has-bottom-element-Prop-Large-Locale P = ex-falso +``` + ### The large poset of propositions is a large meet-semilattice ```agda @@ -98,6 +111,12 @@ has-meets-is-large-meet-semilattice-Large-Poset has-top-element-is-large-meet-semilattice-Large-Poset is-large-meet-semilattice-Prop-Large-Locale = has-top-element-Prop-Large-Locale + +Prop-Large-Meet-Semilattice : Large-Meet-Semilattice lsuc (_⊔_) +Prop-Large-Meet-Semilattice = + make-Large-Meet-Semilattice + ( Prop-Large-Poset) + ( is-large-meet-semilattice-Prop-Large-Locale) ``` ### Suprema in the large poset of propositions @@ -111,6 +130,10 @@ sup-has-least-upper-bound-family-of-elements-Large-Poset is-least-upper-bound-sup-has-least-upper-bound-family-of-elements-Large-Poset ( is-large-suplattice-Prop-Large-Locale {I = I} P) R = inv-iff (up-exists R) + +Prop-Large-Suplattice : Large-Suplattice lsuc (_⊔_) lzero +Prop-Large-Suplattice = + make-Large-Suplattice Prop-Large-Poset is-large-suplattice-Prop-Large-Locale ``` ### The large frame of propositions diff --git a/src/foundation/large-locale-of-subtypes.lagda.md b/src/foundation/large-locale-of-subtypes.lagda.md index 4bec76c43e..2661d3c83d 100644 --- a/src/foundation/large-locale-of-subtypes.lagda.md +++ b/src/foundation/large-locale-of-subtypes.lagda.md @@ -14,21 +14,25 @@ open import foundation.universe-levels open import foundation-core.identity-types open import foundation-core.sets +open import order-theory.bottom-elements-large-posets open import order-theory.greatest-lower-bounds-large-posets open import order-theory.large-locales open import order-theory.large-meet-semilattices open import order-theory.large-posets +open import order-theory.large-preorders open import order-theory.large-suplattices open import order-theory.least-upper-bounds-large-posets open import order-theory.powers-of-large-locales +open import order-theory.top-elements-large-posets ```
## Idea -The **large locale of subtypes** of a type `A` is the -[power locale](order-theory.powers-of-large-locales.md) `A → Prop-Large-Locale`. +The {{#concept "large locale of subtypes" Agda=powerset-Large-Locale}} of a type +`A` is the [power locale](order-theory.powers-of-large-locales.md) +`A → Prop-Large-Locale`. ## Definition @@ -41,6 +45,11 @@ module _ Large-Locale (λ l2 → l1 ⊔ lsuc l2) (λ l2 l3 → l1 ⊔ l2 ⊔ l3) lzero powerset-Large-Locale = power-Large-Locale A Prop-Large-Locale + large-preorder-powerset-Large-Locale : + Large-Preorder (λ l2 → l1 ⊔ lsuc l2) (λ l2 l3 → l1 ⊔ l2 ⊔ l3) + large-preorder-powerset-Large-Locale = + large-preorder-Large-Locale powerset-Large-Locale + large-poset-powerset-Large-Locale : Large-Poset (λ l2 → l1 ⊔ lsuc l2) (λ l2 l3 → l1 ⊔ l2 ⊔ l3) large-poset-powerset-Large-Locale = @@ -168,4 +177,16 @@ module _ sup-powerset-Large-Locale (λ j → meet-powerset-Large-Locale x (y j)) distributive-meet-sup-powerset-Large-Locale = distributive-meet-sup-Large-Locale (powerset-Large-Locale A) + + has-top-element-powerset-Large-Locale : + has-top-element-Large-Poset (large-poset-powerset-Large-Locale A) + has-top-element-powerset-Large-Locale = + has-top-element-Large-Locale (powerset-Large-Locale A) + + has-bottom-element-powerset-Large-Locale : + has-bottom-element-Large-Poset (large-poset-powerset-Large-Locale A) + has-bottom-element-powerset-Large-Locale = + has-bottom-element-Π-Large-Poset + ( λ _ → Prop-Large-Poset) + ( λ _ → has-bottom-element-Prop-Large-Locale) ``` diff --git a/src/foundation/law-of-excluded-middle.lagda.md b/src/foundation/law-of-excluded-middle.lagda.md index 3ee5bdb8c0..a633ea32d6 100644 --- a/src/foundation/law-of-excluded-middle.lagda.md +++ b/src/foundation/law-of-excluded-middle.lagda.md @@ -22,8 +22,9 @@ open import univalent-combinatorics.2-element-types ## Idea -The {{#concept "law of excluded middle" Agda=LEM}} asserts that any -[proposition](foundation-core.propositions.md) `P` is +The +{{#concept "law of excluded middle" WD="principle of excluded middle" WDID=Q468422 Agda=LEM}} +asserts that any [proposition](foundation-core.propositions.md) `P` is [decidable](foundation.decidable-types.md). ## Definition diff --git a/src/foundation/lesser-limited-principle-of-omniscience.lagda.md b/src/foundation/lesser-limited-principle-of-omniscience.lagda.md index a44fa6433b..242c9b27b6 100644 --- a/src/foundation/lesser-limited-principle-of-omniscience.lagda.md +++ b/src/foundation/lesser-limited-principle-of-omniscience.lagda.md @@ -10,35 +10,44 @@ module foundation.lesser-limited-principle-of-omniscience where open import elementary-number-theory.natural-numbers open import elementary-number-theory.parity-natural-numbers +open import foundation.booleans +open import foundation.dependent-pair-types open import foundation.disjunction +open import foundation.universal-quantification open import foundation.universe-levels open import foundation-core.fibers-of-maps open import foundation-core.propositions open import foundation-core.sets - -open import univalent-combinatorics.standard-finite-types ```
## Statement -The **lesser limited principle of omniscience** asserts that for any sequence -`f : ℕ → Fin 2` containing at most one `1`, either `f n ＝ 0` for all even `n` -or `f n ＝ 0` for all odd `n`. +The {{#concept "lesser limited principle of omniscience" Agda=LLPO}} (LLPO) +asserts that for any [sequence](foundation.sequences.md) of +[booleans](foundation.booleans.md) `f : ℕ → bool` such that `f n` is true for +[at most one](foundation-core.propositions.md) `n`, then either `f n` is false +for all even `n` or `f n` is false for all odd `n`. ```agda +prop-LLPO : Prop lzero +prop-LLPO = + ∀' + ( ℕ → bool) + ( λ f → + function-Prop + ( is-prop (Σ ℕ (λ n → is-true (f n)))) + ( disjunction-Prop + ( ∀' ℕ (λ n → function-Prop (is-even-ℕ n) (is-false-Prop (f n)))) + ( ∀' ℕ (λ n → function-Prop (is-odd-ℕ n) (is-false-Prop (f n)))))) + LLPO : UU lzero -LLPO = - (f : ℕ → Fin 2) → is-prop (fiber f (one-Fin 1)) → - type-disjunction-Prop - ( Π-Prop ℕ - ( λ n → - function-Prop (is-even-ℕ n) (Id-Prop (Fin-Set 2) (f n) (zero-Fin 1)))) - ( Π-Prop ℕ - ( λ n → - function-Prop (is-odd-ℕ n) (Id-Prop (Fin-Set 2) (f n) (zero-Fin 1)))) +LLPO = type-Prop prop-LLPO + +is-prop-LLPO : is-prop LLPO +is-prop-LLPO = is-prop-type-Prop prop-LLPO ``` ## See also @@ -46,3 +55,11 @@ LLPO = - [The principle of omniscience](foundation.principle-of-omniscience.md) - [The limited principle of omniscience](foundation.limited-principle-of-omniscience.md) - [The weak limited principle of omniscience](foundation.weak-limited-principle-of-omniscience.md) +- [Markov's principle](logic.markovs-principle.md) + +## External links + +- [Taboos.LLPO](https://martinescardo.github.io/TypeTopology/Taboos.LLPO.html) + at TypeTopology +- [lesser limited principle of omniscience](https://ncatlab.org/nlab/show/lesser+limited+principle+of+omniscience) + at $n$Lab diff --git a/src/foundation/limited-principle-of-omniscience.lagda.md b/src/foundation/limited-principle-of-omniscience.lagda.md index 5b6d52ed2e..e66cea9ad0 100644 --- a/src/foundation/limited-principle-of-omniscience.lagda.md +++ b/src/foundation/limited-principle-of-omniscience.lagda.md @@ -9,8 +9,12 @@ module foundation.limited-principle-of-omniscience where ```agda open import elementary-number-theory.natural-numbers +open import foundation.booleans +open import foundation.coproduct-types +open import foundation.dependent-pair-types open import foundation.disjunction open import foundation.existential-quantification +open import foundation.negation open import foundation.universal-quantification open import foundation.universe-levels @@ -27,17 +31,36 @@ open import univalent-combinatorics.standard-finite-types 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` +(LPO) asserts that for every [sequence](foundation.sequences.md) `f : ℕ → bool` there either [exists](foundation.existential-quantification.md) an `n` such that -`f n ＝ 1` or for all `n` we have `f n ＝ 0`. +`f n` is true, [or](foundation.disjunction.md) `f n` is false for all `n`. ```agda LPO : UU lzero LPO = - (f : ℕ → Fin 2) → - type-disjunction-Prop - ( ∃ 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))) + (f : ℕ → bool) → + ( exists ℕ (λ n → is-true-Prop (f n))) + + ( for-all ℕ (λ n → is-false-Prop (f n))) +``` + +## Properties + +### The limited principle of omniscience is a proposition + +```agda +is-prop-LPO : is-prop LPO +is-prop-LPO = + is-prop-Π + ( λ f → + is-prop-coproduct + ( elim-exists + ( ¬' ∀' ℕ (λ n → is-false-Prop (f n))) + ( λ n t h → not-is-false-is-true (f n) t (h n))) + ( is-prop-exists ℕ (λ n → is-true-Prop (f n))) + ( is-prop-for-all-Prop ℕ (λ n → is-false-Prop (f n)))) + +prop-LPO : Prop lzero +prop-LPO = LPO , is-prop-LPO ``` ## See also @@ -45,3 +68,11 @@ LPO = - [The principle of omniscience](foundation.principle-of-omniscience.md) - [The lesser limited principle of omniscience](foundation.lesser-limited-principle-of-omniscience.md) - [The weak limited principle of omniscience](foundation.weak-limited-principle-of-omniscience.md) +- [Markov's principle](logic.markovs-principle.md) + +## External links + +- [Taboos.LPO](https://martinescardo.github.io/TypeTopology/Taboos.LPO.html) at + TypeTopology +- [limited principle of omniscience](https://ncatlab.org/nlab/show/limited+principle+of+omniscience) + at $n$Lab diff --git a/src/foundation/locale-of-propositions.lagda.md b/src/foundation/locale-of-propositions.lagda.md new file mode 100644 index 0000000000..daef5de811 --- /dev/null +++ b/src/foundation/locale-of-propositions.lagda.md @@ -0,0 +1,132 @@ +# The locale of propositions + +```agda +module foundation.locale-of-propositions where +``` + +
Imports + +```agda +open import foundation.conjunction +open import foundation.dependent-pair-types +open import foundation.existential-quantification +open import foundation.large-locale-of-propositions +open import foundation.logical-equivalences +open import foundation.unit-type +open import foundation.universe-levels + +open import foundation-core.function-types + +open import order-theory.frames +open import order-theory.greatest-lower-bounds-posets +open import order-theory.large-posets +open import order-theory.large-preorders +open import order-theory.meet-semilattices +open import order-theory.meet-suplattices +open import order-theory.posets +open import order-theory.preorders +open import order-theory.suplattices +open import order-theory.top-elements-posets +``` + +
+ +## Idea + +The [locale](order-theory.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 +[poset](order-theory.posets.md) the structure of a +[meet-semilattice](order-theory.meet-semilattices.md), and +[existential quantification](foundation.existential-quantification.md) gives it +the structure of a [suplattice](order-theory.suplattices.md). + +## Definitions + +### The preorder of propositions + +```agda +Prop-Preorder : (l : Level) → Preorder (lsuc l) l +Prop-Preorder = preorder-Large-Preorder Prop-Large-Preorder +``` + +### The poset of propositions + +```agda +Prop-Poset : (l : Level) → Poset (lsuc l) l +Prop-Poset = poset-Large-Poset Prop-Large-Poset +``` + +### The largest element in the poset of propositions + +```agda +has-top-element-Prop-Locale : + {l : Level} → has-top-element-Poset (Prop-Poset l) +has-top-element-Prop-Locale {l} = (raise-unit-Prop l , λ _ _ → raise-star) +``` + +### Meets in the poset of propositions + +```agda +is-meet-semilattice-Prop-Locale : + {l : Level} → is-meet-semilattice-Poset (Prop-Poset l) +is-meet-semilattice-Prop-Locale P Q = + ( P ∧ Q , is-greatest-binary-lower-bound-conjunction-Prop P Q) + +Prop-Order-Theoretic-Meet-Semilattice : + (l : Level) → Order-Theoretic-Meet-Semilattice (lsuc l) l +Prop-Order-Theoretic-Meet-Semilattice l = + Prop-Poset l , is-meet-semilattice-Prop-Locale + +Prop-Meet-Semilattice : + (l : Level) → Meet-Semilattice (lsuc l) +Prop-Meet-Semilattice l = + meet-semilattice-Order-Theoretic-Meet-Semilattice + ( Prop-Order-Theoretic-Meet-Semilattice l) +``` + +### Suprema in the poset of propositions + +```agda +is-suplattice-lzero-Prop-Locale : + {l : Level} → is-suplattice-Poset lzero (Prop-Poset l) +is-suplattice-lzero-Prop-Locale I P = (∃ I P , inv-iff ∘ up-exists) + +is-suplattice-Prop-Locale : + {l : Level} → is-suplattice-Poset l (Prop-Poset l) +is-suplattice-Prop-Locale I P = (∃ I P , inv-iff ∘ up-exists) + +Prop-Suplattice : + (l : Level) → Suplattice (lsuc l) l l +Prop-Suplattice l = Prop-Poset l , is-suplattice-Prop-Locale +``` + +```text +is-meet-suplattice-Prop-Locale : + {l : Level} → is-meet-suplattice-Meet-Semilattice l (Prop-Meet-Semilattice l) +is-meet-suplattice-Prop-Locale I P = ？ + +Prop-Meet-Suplattice : + (l : Level) → Meet-Suplattice (lsuc l) l +Prop-Meet-Suplattice l = + Prop-Meet-Semilattice l , is-meet-suplattice-Prop-Locale +``` + +### The frame of propositions + +```text +Prop-Frame : (l : Level) → Frame (lsuc l) l +Prop-Frame l = Prop-Meet-Suplattice l , ？ +``` + +### The locale of propositions + +```text +Prop-Locale : Locale lsuc (_⊔_) lzero +Prop-Locale = Prop-Frame +``` + +## See also + +- [Propositional resizing](foundation.propositional-resizing.md) diff --git a/src/foundation/logical-equivalences.lagda.md b/src/foundation/logical-equivalences.lagda.md index e6208ce084..d806e1ca3d 100644 --- a/src/foundation/logical-equivalences.lagda.md +++ b/src/foundation/logical-equivalences.lagda.md @@ -220,6 +220,10 @@ iff-equiv : {l1 l2 : Level} {A : UU l1} {B : UU l2} → (A ≃ B) → (A ↔ B) pr1 (iff-equiv e) = map-equiv e pr2 (iff-equiv e) = map-inv-equiv e +iff-equiv' : {l1 l2 : Level} {A : UU l1} {B : UU l2} → (A ≃ B) → (B ↔ A) +pr1 (iff-equiv' e) = map-inv-equiv e +pr2 (iff-equiv' e) = map-equiv e + is-injective-iff-equiv : {l1 l2 : Level} {A : UU l1} {B : UU l2} → is-injective (iff-equiv {A = A} {B}) is-injective-iff-equiv p = eq-htpy-equiv (pr1 (htpy-eq-iff p)) diff --git a/src/foundation/negation.lagda.md b/src/foundation/negation.lagda.md index 3a7b5044e1..273c37d133 100644 --- a/src/foundation/negation.lagda.md +++ b/src/foundation/negation.lagda.md @@ -15,6 +15,7 @@ open import foundation.universe-levels open import foundation-core.empty-types open import foundation-core.equivalences +open import foundation-core.identity-types open import foundation-core.propositions ``` @@ -34,7 +35,7 @@ using propositions as types. Thus, the negation of a type `A` is the type ```agda is-prop-neg : {l : Level} {A : UU l} → is-prop (¬ A) -is-prop-neg {A = A} = is-prop-function-type is-prop-empty +is-prop-neg = is-prop-function-type is-prop-empty neg-type-Prop : {l1 : Level} → UU l1 → Prop l1 neg-type-Prop A = ¬ A , is-prop-neg @@ -49,6 +50,9 @@ infix 25 ¬'_ ¬'_ : {l1 : Level} → Prop l1 → Prop l1 ¬'_ = neg-Prop + +eq-neg : {l : Level} {A : UU l} {p q : ¬ A} → p ＝ q +eq-neg = eq-is-prop is-prop-neg ``` ### Reductio ad absurdum @@ -58,17 +62,30 @@ reductio-ad-absurdum : {l1 l2 : Level} {P : UU l1} {Q : UU l2} → P → ¬ P reductio-ad-absurdum p np = ex-falso (np p) ``` +### Logically equivalent types have logically equivalent negations + +```agda +module _ + {l1 l2 : Level} {X : UU l1} {Y : UU l2} + where + + iff-neg : X ↔ Y → ¬ X ↔ ¬ Y + iff-neg e = map-neg (backward-implication e) , map-neg (forward-implication e) + + equiv-iff-neg : X ↔ Y → ¬ X ≃ ¬ Y + equiv-iff-neg e = + equiv-iff' (neg-type-Prop X) (neg-type-Prop Y) (iff-neg e) +``` + ### Equivalent types have equivalent negations ```agda -equiv-neg : - {l1 l2 : Level} {X : UU l1} {Y : UU l2} → - (X ≃ Y) → (¬ X ≃ ¬ Y) -equiv-neg {l1} {l2} {X} {Y} e = - equiv-iff' - ( neg-type-Prop X) - ( neg-type-Prop Y) - ( pair (map-neg (map-inv-equiv e)) (map-neg (map-equiv e))) +module _ + {l1 l2 : Level} {X : UU l1} {Y : UU l2} + where + + equiv-neg : X ≃ Y → ¬ X ≃ ¬ Y + equiv-neg e = equiv-iff-neg (iff-equiv e) ``` ### Negation has no fixed points diff --git a/src/foundation/path-split-type-families.lagda.md b/src/foundation/path-split-type-families.lagda.md new file mode 100644 index 0000000000..690d6d1a74 --- /dev/null +++ b/src/foundation/path-split-type-families.lagda.md @@ -0,0 +1,146 @@ +# Path-split type families + +```agda +module foundation.path-split-type-families where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.dependent-pair-types +open import foundation.embeddings +open import foundation.transport-along-identifications +open import foundation.universe-levels + +open import foundation-core.contractible-types +open import foundation-core.dependent-identifications +open import foundation-core.equality-dependent-pair-types +open import foundation-core.identity-types +open import foundation-core.injective-maps +open import foundation-core.propositions +open import foundation-core.sections +open import foundation-core.subtypes +``` + +
+ +## Idea + +We say a type family `P : A → 𝒰` is +{{#concept "path-split" Disambiguation="type family" Agda=is-path-split-family}} +if, for every [identification](foundation-core.identity-types.md) `p : x ＝ y` +in `A`, and every pair of elements `u : P x` and `v : P y`, there is a +[dependent identification](foundation-core.dependent-identifications.md) of `u` +and `v` _over_ p. This condition is +[equivalent](foundation.logical-equivalences.md) to asking that `P` is a family +of [propositions](foundation-core.propositions.md). + +This condition is a direct rephrasing of stating that the +[action on identifications](foundation.action-on-identifications-functions.md) +of the first projection map `Σ A P → A` has a +[section](foundation-core.sections.md), and in this way is closely related to +the concept of [path-split maps](foundation-core.path-split-maps.md). + +## Definitions + +### Path-split type families + +```agda +is-path-split-family : + {l1 l2 : Level} {A : UU l1} → (A → UU l2) → UU (l1 ⊔ l2) +is-path-split-family {A = A} P = + {x y : A} (p : x ＝ y) (s : P x) (t : P y) → dependent-identification P p s t +``` + +## Properties + +### The first projection map of a path-split type family is an embedding + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {P : A → UU l2} + (H : is-path-split-family P) + where + + is-injective-pr1-is-path-split-family : is-injective (pr1 {B = P}) + is-injective-pr1-is-path-split-family {x} {y} p = + eq-pair-Σ p (H p (pr2 x) (pr2 y)) + + is-section-is-injective-pr1-is-path-split-family : + {x y : Σ A P} → + is-section (ap pr1) (is-injective-pr1-is-path-split-family {x} {y}) + is-section-is-injective-pr1-is-path-split-family {x , u} {.x , v} refl = + ap-pr1-eq-pair-eq-fiber (H refl u v) + + section-ap-pr1-is-path-split-family : + (x y : Σ A P) → section (ap pr1 {x} {y}) + section-ap-pr1-is-path-split-family x y = + is-injective-pr1-is-path-split-family , + is-section-is-injective-pr1-is-path-split-family + + is-emb-pr1-is-path-split-family : is-emb (pr1 {B = P}) + is-emb-pr1-is-path-split-family = + is-emb-section-ap pr1 section-ap-pr1-is-path-split-family +``` + +### The fibers of a path-split type family are propositions + +We give two proofs, one is direct and the other uses the previous result. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {P : A → UU l2} + (H : is-path-split-family P) + where + + all-elements-equal-is-path-split-family : {x : A} (u v : P x) → u ＝ v + all-elements-equal-is-path-split-family u v = H refl u v + + is-subtype-is-path-split-family : is-subtype P + is-subtype-is-path-split-family x = + is-prop-all-elements-equal all-elements-equal-is-path-split-family + + is-subtype-is-path-split-family' : is-subtype P + is-subtype-is-path-split-family' = + is-subtype-is-emb-pr1 (is-emb-pr1-is-path-split-family H) +``` + +### Path-splittings of type families are unique + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {P : A → UU l2} + where + + abstract + is-proof-irrelevant-is-path-split-family : + is-proof-irrelevant (is-path-split-family P) + is-proof-irrelevant-is-path-split-family H = + is-contr-implicit-Π + ( λ x → + is-contr-implicit-Π + ( λ y → + is-contr-Π + ( λ p → + is-contr-Π + ( λ s → + is-contr-Π + ( is-subtype-is-path-split-family H y (tr P p s)))))) + + is-prop-is-path-split-family : is-prop (is-path-split-family P) + is-prop-is-path-split-family = + is-prop-is-proof-irrelevant is-proof-irrelevant-is-path-split-family +``` + +### Subtypes are path-split + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {P : A → UU l2} + where + + is-path-split-family-is-subtype : + is-subtype P → is-path-split-family P + is-path-split-family-is-subtype H {x} refl s t = eq-is-prop (H x) +``` diff --git a/src/foundation/postcomposition-functions.lagda.md b/src/foundation/postcomposition-functions.lagda.md index 3f4e9643e9..2799d76ae4 100644 --- a/src/foundation/postcomposition-functions.lagda.md +++ b/src/foundation/postcomposition-functions.lagda.md @@ -109,6 +109,23 @@ module _ compute-fiber-postcomp h = compute-coherence-triangle-fiber-postcomp (f ∘ h) ``` +### Computing the fiber of the postcomposition function at the identity function + +```agda +module _ + {l1 l2 : Level} {X : UU l1} {Y : UU l2} + (f : X → Y) + where + + compute-fiber-id-postcomp : + ((x : Y) → fiber f x) ≃ fiber (postcomp Y f) id + compute-fiber-id-postcomp = compute-Π-fiber-postcomp Y f id + + inv-compute-fiber-id-postcomp : + fiber (postcomp Y f) id ≃ ((x : Y) → fiber f x) + inv-compute-fiber-id-postcomp = inv-compute-Π-fiber-postcomp Y f id +``` + ### Postcomposition and equivalences #### A map `f` is an equivalence if and only if postcomposing by `f` is an equivalence diff --git a/src/foundation/powersets.lagda.md b/src/foundation/powersets.lagda.md index 33631db4f1..5610d81426 100644 --- a/src/foundation/powersets.lagda.md +++ b/src/foundation/powersets.lagda.md @@ -7,15 +7,34 @@ module foundation.powersets where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.embeddings +open import foundation.empty-types +open import foundation.large-locale-of-propositions open import foundation.subtypes +open import foundation.unit-type open import foundation.universe-levels open import foundation-core.sets +open import order-theory.bottom-elements-large-posets +open import order-theory.bottom-elements-posets +open import order-theory.dependent-products-large-meet-semilattices +open import order-theory.dependent-products-large-posets +open import order-theory.dependent-products-large-preorders +open import order-theory.dependent-products-large-suplattices +open import order-theory.large-meet-semilattices open import order-theory.large-posets open import order-theory.large-preorders +open import order-theory.large-suplattices +open import order-theory.meet-semilattices +open import order-theory.order-preserving-maps-large-posets +open import order-theory.order-preserving-maps-large-preorders open import order-theory.posets open import order-theory.preorders +open import order-theory.suplattices +open import order-theory.top-elements-large-posets +open import order-theory.top-elements-posets ```
@@ -25,8 +44,7 @@ open import order-theory.preorders The {{#concept "powerset" Disambiguation="of a type" Agda=powerset WD="power set" WDID=Q205170}} of a type is the [set](foundation-core.sets.md) of all -[subtypes](foundation-core.subtypes.md) with respect to a given -[universe level](foundation.universe-levels.md). +[subtypes](foundation-core.subtypes.md). ## Definition @@ -61,10 +79,14 @@ module _ powerset-Large-Preorder : Large-Preorder (λ l → l1 ⊔ lsuc l) (λ l2 l3 → l1 ⊔ l2 ⊔ l3) - type-Large-Preorder powerset-Large-Preorder l = subtype l A - leq-prop-Large-Preorder powerset-Large-Preorder = leq-prop-subtype - refl-leq-Large-Preorder powerset-Large-Preorder = refl-leq-subtype - transitive-leq-Large-Preorder powerset-Large-Preorder = transitive-leq-subtype + powerset-Large-Preorder = Π-Large-Preorder {I = A} (λ _ → Prop-Large-Preorder) + +module _ + {l1 : Level} (l2 : Level) (A : UU l1) + where + + powerset-Preorder : Preorder (l1 ⊔ lsuc l2) (l1 ⊔ l2) + powerset-Preorder = preorder-Large-Preorder (powerset-Large-Preorder A) l2 ``` ### The powerset large poset @@ -76,33 +98,91 @@ module _ powerset-Large-Poset : Large-Poset (λ l → l1 ⊔ lsuc l) (λ l2 l3 → l1 ⊔ l2 ⊔ l3) - large-preorder-Large-Poset powerset-Large-Poset = powerset-Large-Preorder A - antisymmetric-leq-Large-Poset powerset-Large-Poset P Q = - antisymmetric-leq-subtype P Q + powerset-Large-Poset = Π-Large-Poset {I = A} (λ _ → Prop-Large-Poset) + +module _ + {l1 : Level} (l2 : Level) (A : UU l1) + where + + powerset-Poset : Poset (l1 ⊔ lsuc l2) (l1 ⊔ l2) + powerset-Poset = poset-Large-Poset (powerset-Large-Poset A) l2 ``` -### The powerset preorder at a universe level +### The powerset poset has a top element ```agda module _ {l1 : Level} (A : UU l1) where - powerset-Preorder : (l2 : Level) → Preorder (l1 ⊔ lsuc l2) (l1 ⊔ l2) - powerset-Preorder = preorder-Large-Preorder (powerset-Large-Preorder A) + has-top-element-powerset-Large-Poset : + has-top-element-Large-Poset (powerset-Large-Poset A) + has-top-element-powerset-Large-Poset = + has-top-element-Π-Large-Poset + ( λ _ → Prop-Large-Poset) + ( λ _ → has-top-element-Prop-Large-Locale) + + has-top-element-powerset-Poset : + {l2 : Level} → has-top-element-Poset (powerset-Poset l2 A) + has-top-element-powerset-Poset {l2} = + (λ _ → raise-unit-Prop l2) , (λ _ _ _ → raise-star) ``` -### The powerset poset at a universe level +### The powerset poset has a bottom element ```agda module _ {l1 : Level} (A : UU l1) where - powerset-Poset : (l2 : Level) → Poset (l1 ⊔ lsuc l2) (l1 ⊔ l2) - powerset-Poset = poset-Large-Poset (powerset-Large-Poset A) + has-bottom-element-powerset-Large-Poset : + has-bottom-element-Large-Poset (powerset-Large-Poset A) + has-bottom-element-powerset-Large-Poset = + has-bottom-element-Π-Large-Poset + ( λ _ → Prop-Large-Poset) + ( λ _ → has-bottom-element-Prop-Large-Locale) + + has-bottom-element-powerset-Poset : + {l2 : Level} → has-bottom-element-Poset (powerset-Poset l2 A) + has-bottom-element-powerset-Poset {l2} = + (λ _ → raise-empty-Prop l2) , (λ _ _ → raise-ex-falso l2) ``` +### The powerset meet semilattice + +```agda +module _ + {l1 : Level} (A : UU l1) + where + + powerset-Large-Meet-Semilattice : + Large-Meet-Semilattice (λ l2 → l1 ⊔ lsuc l2) (λ l2 l3 → l1 ⊔ l2 ⊔ l3) + powerset-Large-Meet-Semilattice = + Π-Large-Meet-Semilattice {I = A} (λ _ → Prop-Large-Meet-Semilattice) +``` + +### The powerset suplattice + +```agda +module _ + {l1 : Level} (A : UU l1) + where + + powerset-Large-Suplattice : + Large-Suplattice (λ l2 → lsuc l2 ⊔ l1) (λ l2 l3 → l2 ⊔ l3 ⊔ l1) lzero + powerset-Large-Suplattice = + Π-Large-Suplattice {I = A} (λ _ → Prop-Large-Suplattice) + + powerset-Suplattice : + (l2 l3 : Level) → Suplattice (l1 ⊔ lsuc l2 ⊔ lsuc l3) (l1 ⊔ l2 ⊔ l3) l2 + powerset-Suplattice = suplattice-Large-Suplattice powerset-Large-Suplattice +``` + +## See also + +- [the large locale of subtypes](foundation.large-locale-of-subtypes.md) +- [images of subtypes](foundation.images-subtypes.md) + ## External links - [power object](https://ncatlab.org/nlab/show/power+object) at $n$Lab diff --git a/src/foundation/preimages-of-subtypes.lagda.md b/src/foundation/preimages-of-subtypes.lagda.md deleted file mode 100644 index ac1a192b66..0000000000 --- a/src/foundation/preimages-of-subtypes.lagda.md +++ /dev/null @@ -1,29 +0,0 @@ -# Preimages of subtypes - -```agda -module foundation.preimages-of-subtypes where -``` - -
Imports - -```agda -open import foundation.universe-levels - -open import foundation-core.subtypes -``` - -
- -## Idea - -The preimage of a subtype `S ⊆ B` under a map `f : A → B` is the subtype of `A` -consisting of elements `a` such that `f a ∈ S`. - -## Definition - -```agda -preimage-subtype : - {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (f : A → B) → - subtype l3 B → subtype l3 A -preimage-subtype f S a = S (f a) -``` diff --git a/src/foundation/principle-of-omniscience.lagda.md b/src/foundation/principle-of-omniscience.lagda.md index e3a35258cb..cf03f741b2 100644 --- a/src/foundation/principle-of-omniscience.lagda.md +++ b/src/foundation/principle-of-omniscience.lagda.md @@ -19,8 +19,9 @@ open import foundation-core.propositions ## Idea -A type `X` is said to satisfy the **principle of omniscience** if every -[decidable subtype](foundation.decidable-subtypes.md) of `X` is either +A type `X` is said to satisfy the +{{#concept "principle of omniscience" Disambiguation="type" Agda=is-omniscient}} +if every [decidable subtype](foundation.decidable-subtypes.md) of `X` is either [inhabited](foundation.inhabited-types.md) or [empty](foundation-core.empty-types.md). @@ -42,3 +43,4 @@ is-omniscient X = type-Prop (is-omniscient-Prop X) - [The limited principle of omniscience](foundation.limited-principle-of-omniscience.md) - [The lesser limited principle of omniscience](foundation.lesser-limited-principle-of-omniscience.md) - [The weak limited principle of omniscience](foundation.weak-limited-principle-of-omniscience.md) +- [Markov's principle](logic.markovs-principle.md) diff --git a/src/foundation/propositional-resizing.lagda.md b/src/foundation/propositional-resizing.lagda.md index 46d48494aa..4abe732a36 100644 --- a/src/foundation/propositional-resizing.lagda.md +++ b/src/foundation/propositional-resizing.lagda.md @@ -10,7 +10,9 @@ module foundation.propositional-resizing where open import foundation.dependent-pair-types open import foundation.universe-levels +open import foundation-core.equivalences open import foundation-core.propositions +open import foundation-core.small-types open import foundation-core.subtypes ``` @@ -19,20 +21,172 @@ open import foundation-core.subtypes ## Idea 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 +{{#concept "propositional resizing" Agda=propositional-resizing-Level}} 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 +## Definitions + +### The predicate on a type of being a subtype classifier of a given universe level + +```agda +is-subtype-classifier : + {l1 l2 : Level} (l3 : Level) → Σ (UU l1) (subtype l2) → UU (l1 ⊔ l2 ⊔ lsuc l3) +is-subtype-classifier l3 Ω = + (P : Prop l3) → Σ (pr1 Ω) (λ u → type-equiv-Prop P (pr2 Ω u)) + +module _ + {l1 l2 l3 : Level} + (Ω : Σ (UU l1) (subtype l2)) (H : is-subtype-classifier l3 Ω) + (P : Prop l3) + where + + object-prop-is-subtype-classifier : pr1 Ω + object-prop-is-subtype-classifier = pr1 (H P) + + prop-is-small-prop-is-subtype-classifier : Prop l2 + prop-is-small-prop-is-subtype-classifier = + pr2 Ω object-prop-is-subtype-classifier + + type-is-small-prop-is-subtype-classifier : UU l2 + type-is-small-prop-is-subtype-classifier = + type-Prop prop-is-small-prop-is-subtype-classifier + + equiv-is-small-prop-is-subtype-classifier : + type-Prop P ≃ type-is-small-prop-is-subtype-classifier + equiv-is-small-prop-is-subtype-classifier = pr2 (H P) + + is-small-prop-is-subtype-classifier : is-small l2 (type-Prop P) + is-small-prop-is-subtype-classifier = + ( type-is-small-prop-is-subtype-classifier , + equiv-is-small-prop-is-subtype-classifier) +``` + +### Propositional resizing between specified universes + +```agda +propositional-resizing-Level' : + (l1 l2 l3 : Level) → UU (lsuc l1 ⊔ lsuc l2 ⊔ lsuc l3) +propositional-resizing-Level' l1 l2 l3 = + Σ (Σ (UU l1) (subtype l2)) (is-subtype-classifier l3) + +propositional-resizing-Level : + (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) +propositional-resizing-Level l1 l2 = + propositional-resizing-Level' l1 l1 l2 +``` + +```agda +module _ + {l1 l2 l3 : Level} (ρ : propositional-resizing-Level' l1 l2 l3) + where + + subtype-classifier-propositional-resizing-Level : Σ (UU l1) (subtype l2) + subtype-classifier-propositional-resizing-Level = pr1 ρ + + type-subtype-classifier-propositional-resizing-Level : UU l1 + type-subtype-classifier-propositional-resizing-Level = + pr1 subtype-classifier-propositional-resizing-Level + + is-subtype-classifier-subtype-classifier-propositional-resizing-Level : + is-subtype-classifier l3 subtype-classifier-propositional-resizing-Level + is-subtype-classifier-subtype-classifier-propositional-resizing-Level = pr2 ρ + +module _ + {l1 l2 l3 : Level} (ρ : propositional-resizing-Level' l1 l2 l3) (P : Prop l3) + where + + object-prop-propositional-resizing-Level : + type-subtype-classifier-propositional-resizing-Level ρ + object-prop-propositional-resizing-Level = + object-prop-is-subtype-classifier + ( subtype-classifier-propositional-resizing-Level ρ) + ( is-subtype-classifier-subtype-classifier-propositional-resizing-Level ρ) + ( P) + + prop-is-small-prop-propositional-resizing-Level : Prop l2 + prop-is-small-prop-propositional-resizing-Level = + prop-is-small-prop-is-subtype-classifier + ( subtype-classifier-propositional-resizing-Level ρ) + ( is-subtype-classifier-subtype-classifier-propositional-resizing-Level ρ) + ( P) + + type-is-small-prop-propositional-resizing-Level : UU l2 + type-is-small-prop-propositional-resizing-Level = + type-is-small-prop-is-subtype-classifier + ( subtype-classifier-propositional-resizing-Level ρ) + ( is-subtype-classifier-subtype-classifier-propositional-resizing-Level ρ) + ( P) + + equiv-is-small-prop-propositional-resizing-Level : + type-Prop P ≃ type-is-small-prop-propositional-resizing-Level + equiv-is-small-prop-propositional-resizing-Level = + equiv-is-small-prop-is-subtype-classifier + ( subtype-classifier-propositional-resizing-Level ρ) + ( is-subtype-classifier-subtype-classifier-propositional-resizing-Level ρ) + ( P) + + is-small-prop-propositional-resizing-Level : is-small l2 (type-Prop P) + is-small-prop-propositional-resizing-Level = + is-small-prop-is-subtype-classifier + ( subtype-classifier-propositional-resizing-Level ρ) + ( is-subtype-classifier-subtype-classifier-propositional-resizing-Level ρ) + ( P) +``` + +### The propositional resizing axiom + +```agda +propositional-resizing : UUω +propositional-resizing = (l1 l2 : Level) → propositional-resizing-Level l1 l2 +``` ```agda -propositional-resizing : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) -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)) +module _ + (ρ : propositional-resizing) (l1 l2 : Level) + where + + subtype-classifier-propositional-resizing : Σ (UU l1) (subtype l1) + subtype-classifier-propositional-resizing = pr1 (ρ l1 l2) + + type-subtype-classifier-propositional-resizing : UU l1 + type-subtype-classifier-propositional-resizing = + type-subtype-classifier-propositional-resizing-Level (ρ l1 l2) + + is-subtype-classifier-subtype-classifier-propositional-resizing : + is-subtype-classifier l2 subtype-classifier-propositional-resizing + is-subtype-classifier-subtype-classifier-propositional-resizing = + is-subtype-classifier-subtype-classifier-propositional-resizing-Level + ( ρ l1 l2) + +module _ + (ρ : propositional-resizing) (l1 : Level) {l2 : Level} (P : Prop l2) + where + + object-prop-propositional-resizing : + type-subtype-classifier-propositional-resizing ρ l1 l2 + object-prop-propositional-resizing = + object-prop-propositional-resizing-Level (ρ l1 l2) P + + prop-is-small-prop-propositional-resizing : Prop l1 + prop-is-small-prop-propositional-resizing = + prop-is-small-prop-propositional-resizing-Level (ρ l1 l2) P + + type-is-small-prop-propositional-resizing : UU l1 + type-is-small-prop-propositional-resizing = + type-is-small-prop-propositional-resizing-Level (ρ l1 l2) P + + equiv-is-small-prop-propositional-resizing : + type-Prop P ≃ type-is-small-prop-propositional-resizing + equiv-is-small-prop-propositional-resizing = + equiv-is-small-prop-propositional-resizing-Level (ρ l1 l2) P + + is-small-prop-propositional-resizing : is-small l1 (type-Prop P) + is-small-prop-propositional-resizing = + is-small-prop-propositional-resizing-Level (ρ l1 l2) P ``` ## See also diff --git a/src/foundation/propositions.lagda.md b/src/foundation/propositions.lagda.md index cc36431139..e4bf1eb773 100644 --- a/src/foundation/propositions.lagda.md +++ b/src/foundation/propositions.lagda.md @@ -11,12 +11,15 @@ open import foundation-core.propositions public ```agda open import foundation.contractible-types open import foundation.dependent-pair-types +open import foundation.fibers-of-maps open import foundation.logical-equivalences open import foundation.retracts-of-types +open import foundation.unit-type open import foundation.universe-levels open import foundation-core.embeddings open import foundation-core.equivalences +open import foundation-core.propositional-maps open import foundation-core.truncated-types open import foundation-core.truncation-levels ``` @@ -64,6 +67,24 @@ abstract is-prop-emb = is-trunc-emb neg-two-𝕋 ``` +### A type is a proposition if and only if it embeds into the unit type + +```agda +module _ + {l : Level} {A : UU l} + where + + abstract + is-prop-is-emb-terminal-map : is-emb (terminal-map A) → is-prop A + is-prop-is-emb-terminal-map H = + is-prop-is-emb (terminal-map A) H is-prop-unit + + abstract + is-emb-terminal-map-is-prop : is-prop A → is-emb (terminal-map A) + is-emb-terminal-map-is-prop H = + is-emb-is-prop-map (λ y → is-prop-equiv (equiv-fiber-terminal-map y) H) +``` + ### Two equivalent types are equivalently propositions ```agda diff --git a/src/foundation/retracts-of-types.lagda.md b/src/foundation/retracts-of-types.lagda.md index ce0fa62ff3..56c7476a61 100644 --- a/src/foundation/retracts-of-types.lagda.md +++ b/src/foundation/retracts-of-types.lagda.md @@ -15,12 +15,14 @@ open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.homotopy-algebra open import foundation.homotopy-induction +open import foundation.logical-equivalences open import foundation.structure-identity-principle open import foundation.univalence open import foundation.universe-levels open import foundation.whiskering-homotopies-composition open import foundation-core.contractible-types +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.torsorial-type-families @@ -166,6 +168,18 @@ module _ eq-equiv-retracts R S = map-inv-is-equiv (is-equiv-equiv-eq-retracts R S) ``` +### The underlying logical equivalence associated to a retract + +```agda +iff-retract : + {l1 l2 : Level} {A : UU l1} {B : UU l2} → A retract-of B → A ↔ B +iff-retract R = inclusion-retract R , map-retraction-retract R + +iff-retract' : + {l1 l2 : Level} {A : UU l1} {B : UU l2} → A retract-of B → B ↔ A +iff-retract' = inv-iff ∘ iff-retract +``` + ## See also - [Retracts of maps](foundation.retracts-of-maps.md) diff --git a/src/foundation/subsingleton-induction.lagda.md b/src/foundation/subsingleton-induction.lagda.md index 7007417be8..1497a0c6d6 100644 --- a/src/foundation/subsingleton-induction.lagda.md +++ b/src/foundation/subsingleton-induction.lagda.md @@ -59,12 +59,18 @@ compute-ind-is-subsingleton is-subsingleton-A a = pr2 (is-subsingleton-A a) ### Propositions satisfy subsingleton induction ```agda +abstract + ind-subsingleton' : + {l1 l2 : Level} {A : UU l1} → is-proof-irrelevant A → + {B : A → UU l2} (a : A) → B a → (x : A) → B x + ind-subsingleton' H {B} a = ind-singleton a (H a) B + abstract ind-subsingleton : - {l1 l2 : Level} {A : UU l1} (is-prop-A : is-prop A) + {l1 l2 : Level} {A : UU l1} → is-prop A → {B : A → UU l2} (a : A) → B a → (x : A) → B x - ind-subsingleton is-prop-A {B} a = - ind-singleton a (is-proof-irrelevant-is-prop is-prop-A a) B + ind-subsingleton is-prop-A = + ind-subsingleton' (is-proof-irrelevant-is-prop is-prop-A) abstract compute-ind-subsingleton : @@ -78,18 +84,30 @@ abstract ### A type satisfies subsingleton induction if and only if it is a proposition ```agda +is-subsingleton-is-proof-irrelevant : + {l1 l2 : Level} {A : UU l1} → is-proof-irrelevant A → is-subsingleton l2 A +is-subsingleton-is-proof-irrelevant H {B} a = + is-singleton-is-contr a (H a) B + is-subsingleton-is-prop : {l1 l2 : Level} {A : UU l1} → is-prop A → is-subsingleton l2 A -is-subsingleton-is-prop is-prop-A {B} a = - is-singleton-is-contr a (is-proof-irrelevant-is-prop is-prop-A a) B +is-subsingleton-is-prop is-prop-A = + is-subsingleton-is-proof-irrelevant (is-proof-irrelevant-is-prop is-prop-A) + +abstract + is-proof-irrelevant-ind-subsingleton : + {l1 : Level} (A : UU l1) → + ({l2 : Level} {B : A → UU l2} (a : A) → B a → (x : A) → B x) → + is-proof-irrelevant A + is-proof-irrelevant-ind-subsingleton A S a = + is-contr-ind-singleton A a (λ B → S {B = B} a) abstract is-prop-ind-subsingleton : {l1 : Level} (A : UU l1) → ({l2 : Level} {B : A → UU l2} (a : A) → B a → (x : A) → B x) → is-prop A is-prop-ind-subsingleton A S = - is-prop-is-proof-irrelevant - ( λ a → is-contr-ind-singleton A a (λ B → S {B = B} a)) + is-prop-is-proof-irrelevant (is-proof-irrelevant-ind-subsingleton A S) abstract is-prop-is-subsingleton : diff --git a/src/foundation/surjective-maps.lagda.md b/src/foundation/surjective-maps.lagda.md index 59d60318b5..42ab12dd78 100644 --- a/src/foundation/surjective-maps.lagda.md +++ b/src/foundation/surjective-maps.lagda.md @@ -470,6 +470,9 @@ module _ is-surjective g → is-surjective h → is-surjective (g ∘ h) is-surjective-comp {g} {h} = is-surjective-left-map-triangle (g ∘ h) g h refl-htpy + + comp-surjection : B ↠ X → A ↠ B → A ↠ X + comp-surjection (g , G) (h , H) = g ∘ h , is-surjective-comp G H ``` ### Functoriality of products preserves being surjective diff --git a/src/foundation/transport-along-homotopies.lagda.md b/src/foundation/transport-along-homotopies.lagda.md index f4dd21d96f..577c47c512 100644 --- a/src/foundation/transport-along-homotopies.lagda.md +++ b/src/foundation/transport-along-homotopies.lagda.md @@ -69,7 +69,7 @@ module _ abstract compute-tr-htpy : {f g : (x : A) → B x} (H : f ~ g) → statement-compute-tr-htpy H - compute-tr-htpy {f} {g} = + compute-tr-htpy {f} = ind-htpy f ( λ _ → statement-compute-tr-htpy) ( base-case-compute-tr-htpy) diff --git a/src/foundation/transport-along-identifications.lagda.md b/src/foundation/transport-along-identifications.lagda.md index 10987aecc0..d137762977 100644 --- a/src/foundation/transport-along-identifications.lagda.md +++ b/src/foundation/transport-along-identifications.lagda.md @@ -78,4 +78,9 @@ tr-loop : {l1 : Level} {A : UU l1} {a0 a1 : A} (p : a0 ＝ a1) (q : a0 ＝ a0) → tr (λ y → y ＝ y) p q ＝ (inv p ∙ q) ∙ p tr-loop refl q = inv right-unit + +tr-loop-self : + {l1 : Level} {A : UU l1} {a : A} (p : a ＝ a) → + tr (λ y → y ＝ y) p p ＝ p +tr-loop-self p = tr-loop p p ∙ ap (_∙ p) (left-inv p) ``` diff --git a/src/foundation/trivial-relaxed-sigma-decompositions.lagda.md b/src/foundation/trivial-relaxed-sigma-decompositions.lagda.md index 6578bdec99..8ab55c81ac 100644 --- a/src/foundation/trivial-relaxed-sigma-decompositions.lagda.md +++ b/src/foundation/trivial-relaxed-sigma-decompositions.lagda.md @@ -83,13 +83,7 @@ module _ equiv-trivial-is-trivial-Relaxed-Σ-Decomposition : equiv-Relaxed-Σ-Decomposition D (trivial-Relaxed-Σ-Decomposition l4 A) pr1 equiv-trivial-is-trivial-Relaxed-Σ-Decomposition = - ( map-equiv (compute-raise-unit l4) ∘ - terminal-map (indexing-type-Relaxed-Σ-Decomposition D) , - is-equiv-comp - ( map-equiv (compute-raise-unit l4)) - ( terminal-map (indexing-type-Relaxed-Σ-Decomposition D)) - ( is-equiv-terminal-map-is-contr is-trivial) - ( is-equiv-map-equiv ( compute-raise-unit l4))) + equiv-raise-unit-is-contr is-trivial pr1 (pr2 equiv-trivial-is-trivial-Relaxed-Σ-Decomposition) x = ( inv-equiv (matching-correspondence-Relaxed-Σ-Decomposition D)) ∘e ( inv-left-unit-law-Σ-is-contr is-trivial x) diff --git a/src/foundation/trivial-sigma-decompositions.lagda.md b/src/foundation/trivial-sigma-decompositions.lagda.md index 4e0fbf766b..e50fa85044 100644 --- a/src/foundation/trivial-sigma-decompositions.lagda.md +++ b/src/foundation/trivial-sigma-decompositions.lagda.md @@ -101,13 +101,7 @@ module _ ( A) ( is-inhabited-base-type-is-trivial-Σ-Decomposition)) pr1 equiv-trivial-is-trivial-Σ-Decomposition = - ( map-equiv (compute-raise-unit l4) ∘ - terminal-map (indexing-type-Σ-Decomposition D) , - is-equiv-comp - ( map-equiv (compute-raise-unit l4)) - ( terminal-map (indexing-type-Σ-Decomposition D)) - ( is-equiv-terminal-map-is-contr is-trivial) - ( is-equiv-map-equiv ( compute-raise-unit l4))) + equiv-raise-unit-is-contr is-trivial pr1 (pr2 equiv-trivial-is-trivial-Σ-Decomposition) = ( λ x → ( ( inv-equiv (matching-correspondence-Σ-Decomposition D)) ∘e diff --git a/src/foundation/truncation-equivalences.lagda.md b/src/foundation/truncation-equivalences.lagda.md index 8d8ecf5786..5462a0bf32 100644 --- a/src/foundation/truncation-equivalences.lagda.md +++ b/src/foundation/truncation-equivalences.lagda.md @@ -40,8 +40,10 @@ open import foundation-core.truncation-levels ## Idea -A map `f : A → B` is said to be a `k`-equivalence if the map -`map-trunc k f : trunc k A → trunc k B` is an equivalence. +A map `f : A → B` is said to be a +{{#concept "`k`-equivalence" Disambiguation="truncations of types" Agda=truncation-equivalence}} +if the map `map-trunc k f : trunc k A → trunc k B` is an +[equivalence](foundation-core.equivalences.md). ## Definition @@ -300,11 +302,11 @@ We consider the following composition of maps ```text fiber f b = Σ A (λ a → f a = b) - → Σ A (λ a → ║ f a ＝ b ║) - ≃ Σ A (λ a → | f a | = | b | - ≃ Σ A (λ a → ║ f ║ | a | = | b |) - → Σ ║ A ║ (λ t → ║ f ║ t = | b |) - = fiber ║ f ║ | b | + → Σ A (λ a → ║f a ＝ b║) + ≃ Σ A (λ a → |f a| = |b|) + ≃ Σ A (λ a → ║f║ |a| = |b|) + → Σ ║A║ (λ t → ║f║ t = |b|) + = fiber ║f║ |b| ``` where the first and last maps are `k`-equivalences. diff --git a/src/foundation/truncations.lagda.md b/src/foundation/truncations.lagda.md index 8b01d5b970..9523e78045 100644 --- a/src/foundation/truncations.lagda.md +++ b/src/foundation/truncations.lagda.md @@ -144,11 +144,8 @@ module _ unique-dependent-function-trunc B f = is-contr-equiv' ( fiber (precomp-Π-Truncated-Type unit-trunc B) f) - ( equiv-tot - ( λ h → equiv-funext)) - ( is-contr-map-is-equiv - ( dependent-universal-property-trunc B) - ( f)) + ( equiv-tot (λ h → equiv-funext)) + ( is-contr-map-is-equiv (dependent-universal-property-trunc B) f) apply-dependent-universal-property-trunc : {l2 : Level} (B : type-trunc k A → Truncated-Type l2 k) → diff --git a/src/foundation/type-arithmetic-dependent-function-types.lagda.md b/src/foundation/type-arithmetic-dependent-function-types.lagda.md index dcf9aa2c94..27533716f7 100644 --- a/src/foundation/type-arithmetic-dependent-function-types.lagda.md +++ b/src/foundation/type-arithmetic-dependent-function-types.lagda.md @@ -36,7 +36,7 @@ module _ ( left-unit-law-Π ( λ _ → B a)) ∘e ( equiv-Π ( λ _ → B a) - ( terminal-map A , is-equiv-terminal-map-is-contr C) + ( equiv-unit-is-contr C) ( λ a → equiv-eq (ap B ( eq-is-contr C)))) ``` diff --git a/src/foundation/uniformly-decidable-type-families.lagda.md b/src/foundation/uniformly-decidable-type-families.lagda.md new file mode 100644 index 0000000000..01904ea182 --- /dev/null +++ b/src/foundation/uniformly-decidable-type-families.lagda.md @@ -0,0 +1,175 @@ +# Uniformly decidable type families + +```agda +module foundation.uniformly-decidable-type-families where +``` + +
Imports + +```agda +open import foundation.contractible-types +open import foundation.coproduct-types +open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.equality-coproduct-types +open import foundation.inhabited-types +open import foundation.negation +open import foundation.propositional-truncations +open import foundation.propositions +open import foundation.truncated-types +open import foundation.truncation-levels +open import foundation.type-arithmetic-empty-type +open import foundation.universe-levels + +open import foundation-core.cartesian-product-types +open import foundation-core.contractible-maps +open import foundation-core.empty-types +open import foundation-core.equivalences +open import foundation-core.function-types +open import foundation-core.homotopies +open import foundation-core.identity-types +``` + +
+ +## Idea + +A type family `B : A → 𝒰` is +{{#concept "uniformly decidable" Agda=is-uniformly-decidable-family}} if there +either is an element of every fiber `B x`, or every fiber is +[empty](foundation-core.empty-types.md). + +## Definitions + +### The predicate of being uniformly decidable + +```agda +is-uniformly-decidable-family : + {l1 l2 : Level} {A : UU l1} → (A → UU l2) → UU (l1 ⊔ l2) +is-uniformly-decidable-family {A = A} B = + ((x : A) → B x) + ((x : A) → ¬ (B x)) +``` + +## Properties + +### The fibers of a uniformly decidable type family are decidable + +```agda +is-decidable-is-uniformly-decidable-family : + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} → + is-uniformly-decidable-family B → + (x : A) → is-decidable (B x) +is-decidable-is-uniformly-decidable-family (inl f) x = inl (f x) +is-decidable-is-uniformly-decidable-family (inr g) x = inr (g x) +``` + +### The uniform decidability predicate on a family of truncated types + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + is-contr-is-uniformly-decidable-family-is-inhabited-base'' : + is-contr ((x : A) → (B x)) → + A → + is-contr (is-uniformly-decidable-family B) + is-contr-is-uniformly-decidable-family-is-inhabited-base'' H a = + is-contr-equiv + ( (x : A) → B x) + ( right-unit-law-coproduct-is-empty + ( (x : A) → B x) + ( (x : A) → ¬ B x) + ( λ f → f a (center H a))) + ( H) + + is-contr-is-uniformly-decidable-family-is-inhabited-base' : + ((x : A) → is-contr (B x)) → + A → + is-contr (is-uniformly-decidable-family B) + is-contr-is-uniformly-decidable-family-is-inhabited-base' H = + is-contr-is-uniformly-decidable-family-is-inhabited-base'' (is-contr-Π H) + + is-contr-is-uniformly-decidable-family-is-inhabited-base : + ((x : A) → is-contr (B x)) → + is-inhabited A → + is-contr (is-uniformly-decidable-family B) + is-contr-is-uniformly-decidable-family-is-inhabited-base H = + rec-trunc-Prop + ( is-contr-Prop (is-uniformly-decidable-family B)) + ( is-contr-is-uniformly-decidable-family-is-inhabited-base' H) +``` + +### The uniform decidability predicate on a family of propositions + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + is-prop-is-uniformly-decidable-family-is-inhabited-base'' : + is-prop ((x : A) → (B x)) → + A → + is-prop (is-uniformly-decidable-family B) + is-prop-is-uniformly-decidable-family-is-inhabited-base'' H a = + is-prop-coproduct + ( λ f nf → nf a (f a)) + ( H) + ( is-prop-Π (λ x → is-prop-neg)) + + is-prop-is-uniformly-decidable-family-is-inhabited-base' : + ((x : A) → is-prop (B x)) → + A → + is-prop (is-uniformly-decidable-family B) + is-prop-is-uniformly-decidable-family-is-inhabited-base' H = + is-prop-is-uniformly-decidable-family-is-inhabited-base'' (is-prop-Π H) + + is-prop-is-uniformly-decidable-family-is-inhabited-base : + ((x : A) → is-prop (B x)) → + is-inhabited A → + is-prop (is-uniformly-decidable-family B) + is-prop-is-uniformly-decidable-family-is-inhabited-base H = + rec-trunc-Prop + ( is-prop-Prop (is-uniformly-decidable-family B)) + ( is-prop-is-uniformly-decidable-family-is-inhabited-base' H) +``` + +### The uniform decidability predicate on a family of truncated types + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + is-trunc-succ-succ-is-uniformly-decidable-family' : + (k : 𝕋) → + is-trunc (succ-𝕋 (succ-𝕋 k)) ((x : A) → (B x)) → + is-trunc (succ-𝕋 (succ-𝕋 k)) (is-uniformly-decidable-family B) + is-trunc-succ-succ-is-uniformly-decidable-family' k H = + is-trunc-coproduct k + ( H) + ( is-trunc-is-prop (succ-𝕋 k) (is-prop-Π (λ x → is-prop-neg))) + + is-trunc-succ-succ-is-uniformly-decidable-family : + (k : 𝕋) → + ((x : A) → is-trunc (succ-𝕋 (succ-𝕋 k)) (B x)) → + is-trunc (succ-𝕋 (succ-𝕋 k)) (is-uniformly-decidable-family B) + is-trunc-succ-succ-is-uniformly-decidable-family k H = + is-trunc-succ-succ-is-uniformly-decidable-family' k + ( is-trunc-Π (succ-𝕋 (succ-𝕋 k)) H) + + is-trunc-is-uniformly-decidable-family-is-inhabited-base : + (k : 𝕋) → + ((x : A) → is-trunc k (B x)) → + is-inhabited A → + is-trunc k (is-uniformly-decidable-family B) + is-trunc-is-uniformly-decidable-family-is-inhabited-base + neg-two-𝕋 = + is-contr-is-uniformly-decidable-family-is-inhabited-base + is-trunc-is-uniformly-decidable-family-is-inhabited-base + ( succ-𝕋 neg-two-𝕋) = + is-prop-is-uniformly-decidable-family-is-inhabited-base + is-trunc-is-uniformly-decidable-family-is-inhabited-base + ( succ-𝕋 (succ-𝕋 k)) H _ = + is-trunc-succ-succ-is-uniformly-decidable-family k H +``` diff --git a/src/foundation/unions-subtypes.lagda.md b/src/foundation/unions-subtypes.lagda.md index f1bd0aee75..a3ae84790a 100644 --- a/src/foundation/unions-subtypes.lagda.md +++ b/src/foundation/unions-subtypes.lagda.md @@ -17,6 +17,10 @@ open import foundation.universe-levels open import foundation-core.subtypes +open import logic.de-morgan-propositions +open import logic.de-morgan-subtypes +open import logic.double-negation-stable-subtypes + open import order-theory.least-upper-bounds-large-posets ``` @@ -49,6 +53,15 @@ module _ union-decidable-subtype P Q x = disjunction-Decidable-Prop (P x) (Q x) ``` +### Unions of De Morgan subtypes + +```agda + union-de-morgan-subtype : + de-morgan-subtype l1 X → de-morgan-subtype l2 X → + de-morgan-subtype (l1 ⊔ l2) X + union-de-morgan-subtype P Q x = disjunction-De-Morgan-Prop (P x) (Q x) +``` + ### Unions of families of subtypes ```agda diff --git a/src/foundation/unit-type.lagda.md b/src/foundation/unit-type.lagda.md index ca159a3636..cc388a234b 100644 --- a/src/foundation/unit-type.lagda.md +++ b/src/foundation/unit-type.lagda.md @@ -7,6 +7,7 @@ module foundation.unit-type where
Imports ```agda +open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.diagonal-maps-of-types open import foundation.raising-universe-levels @@ -18,6 +19,7 @@ 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.retractions open import foundation-core.sets open import foundation-core.truncated-types open import foundation-core.truncation-levels @@ -96,7 +98,12 @@ inv-compute-raise-unit l = inv-compute-raise l unit abstract is-contr-unit : is-contr unit pr1 is-contr-unit = star - pr2 is-contr-unit star = refl + pr2 is-contr-unit _ = refl + +abstract + is-contr-raise-unit : {l1 : Level} → is-contr (raise-unit l1) + is-contr-raise-unit {l1} = + is-contr-equiv' unit (compute-raise l1 unit) is-contr-unit ``` ### Any contractible type is equivalent to the unit type @@ -106,22 +113,58 @@ module _ {l : Level} {A : UU l} where - abstract - is-equiv-terminal-map-is-contr : - is-contr A → is-equiv (terminal-map A) - pr1 (pr1 (is-equiv-terminal-map-is-contr H)) = ind-unit (center H) - pr2 (pr1 (is-equiv-terminal-map-is-contr H)) = ind-unit refl - pr1 (pr2 (is-equiv-terminal-map-is-contr H)) x = center H - pr2 (pr2 (is-equiv-terminal-map-is-contr H)) = contraction H + is-equiv-terminal-map-is-contr : is-contr A → is-equiv (terminal-map A) + is-equiv-terminal-map-is-contr H = + is-equiv-is-invertible (λ _ → center H) (λ _ → refl) (contraction H) equiv-unit-is-contr : is-contr A → A ≃ unit - pr1 (equiv-unit-is-contr H) = terminal-map A - pr2 (equiv-unit-is-contr H) = is-equiv-terminal-map-is-contr H + equiv-unit-is-contr H = terminal-map A , is-equiv-terminal-map-is-contr H + + is-contr-retraction-terminal-map : retraction (terminal-map A) → is-contr A + is-contr-retraction-terminal-map (h , H) = h star , H + + is-contr-is-equiv-terminal-map : is-equiv (terminal-map A) → is-contr A + is-contr-is-equiv-terminal-map H = + is-contr-retraction-terminal-map (retraction-is-equiv H) + + is-contr-equiv-unit : A ≃ unit → is-contr A + is-contr-equiv-unit e = map-inv-equiv e star , is-retraction-map-inv-equiv e +``` + +### Any contractible type is equivalent to the raised unit type + +```agda +module _ + {l1 l2 : Level} {A : UU l1} + where - abstract - is-contr-is-equiv-terminal-map : is-equiv (terminal-map A) → is-contr A - pr1 (is-contr-is-equiv-terminal-map ((g , G) , (h , H))) = h star - pr2 (is-contr-is-equiv-terminal-map ((g , G) , (h , H))) = H + is-equiv-raise-terminal-map-is-contr : + is-contr A → is-equiv (raise-terminal-map {l2 = l2} A) + is-equiv-raise-terminal-map-is-contr H = + is-equiv-is-invertible + ( λ _ → center H) + ( λ where (map-raise x) → refl) + ( contraction H) + + equiv-raise-unit-is-contr : is-contr A → A ≃ raise-unit l2 + equiv-raise-unit-is-contr H = + raise-terminal-map A , is-equiv-raise-terminal-map-is-contr H + + is-contr-retraction-raise-terminal-map : + retraction (raise-terminal-map {l2 = l2} A) → is-contr A + is-contr-retraction-raise-terminal-map (h , H) = h raise-star , H + + is-contr-is-equiv-raise-terminal-map : + is-equiv (raise-terminal-map {l2 = l2} A) → is-contr A + is-contr-is-equiv-raise-terminal-map H = + is-contr-retraction-raise-terminal-map (retraction-is-equiv H) + + is-contr-equiv-raise-unit : A ≃ raise-unit l2 → is-contr A + is-contr-equiv-raise-unit e = + ( map-inv-equiv e raise-star) , + ( λ x → + ap (map-inv-equiv e) (eq-is-contr is-contr-raise-unit) ∙ + is-retraction-map-inv-equiv e x) ``` ### The unit type is a proposition @@ -132,8 +175,14 @@ abstract is-prop-unit = is-prop-is-contr is-contr-unit unit-Prop : Prop lzero -pr1 unit-Prop = unit -pr2 unit-Prop = is-prop-unit +unit-Prop = unit , is-prop-unit + +abstract + is-prop-raise-unit : {l1 : Level} → is-prop (raise-unit l1) + is-prop-raise-unit {l1} = is-prop-equiv' (compute-raise l1 unit) is-prop-unit + +raise-unit-Prop : (l1 : Level) → Prop l1 +raise-unit-Prop l1 = raise-unit l1 , is-prop-raise-unit ``` ### The unit type is a set @@ -144,36 +193,14 @@ abstract is-set-unit = is-trunc-succ-is-trunc neg-one-𝕋 is-prop-unit unit-Set : Set lzero -pr1 unit-Set = unit -pr2 unit-Set = is-set-unit -``` - -```agda -abstract - is-contr-raise-unit : - {l1 : Level} → is-contr (raise-unit l1) - is-contr-raise-unit {l1} = - is-contr-equiv' unit (compute-raise l1 unit) is-contr-unit - -abstract - is-prop-raise-unit : - {l1 : Level} → is-prop (raise-unit l1) - is-prop-raise-unit {l1} = - is-prop-equiv' (compute-raise l1 unit) is-prop-unit - -raise-unit-Prop : - (l1 : Level) → Prop l1 -pr1 (raise-unit-Prop l1) = raise-unit l1 -pr2 (raise-unit-Prop l1) = is-prop-raise-unit +unit-Set = unit , is-set-unit abstract - is-set-raise-unit : - {l1 : Level} → is-set (raise-unit l1) + is-set-raise-unit : {l1 : Level} → is-set (raise-unit l1) is-set-raise-unit = is-trunc-succ-is-trunc neg-one-𝕋 is-prop-raise-unit raise-unit-Set : (l1 : Level) → Set l1 -pr1 (raise-unit-Set l1) = raise-unit l1 -pr2 (raise-unit-Set l1) = is-set-raise-unit +raise-unit-Set l1 = raise-unit l1 , is-set-raise-unit ``` ### All parallel maps into `unit` are equal diff --git a/src/foundation/univalence-implies-function-extensionality.lagda.md b/src/foundation/univalence-implies-function-extensionality.lagda.md index 8a0b581a0e..8845653a49 100644 --- a/src/foundation/univalence-implies-function-extensionality.lagda.md +++ b/src/foundation/univalence-implies-function-extensionality.lagda.md @@ -21,6 +21,7 @@ open import foundation-core.fibers-of-maps open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types +open import foundation-core.retracts-of-types open import foundation-core.transport-along-identifications ``` @@ -34,14 +35,20 @@ The [univalence axiom](foundation-core.univalence.md) implies ## Theorem ```agda +retract-compute-fiber-id-postcomp-pr1 : + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} → + ((a : A) → B a) retract-of (fiber (postcomp A (pr1 {B = B})) id) +retract-compute-fiber-id-postcomp-pr1 {B = B} = + ( λ f → ((λ x → (x , f x)) , refl)) , + ( λ h x → tr B (htpy-eq (pr2 h) x) (pr2 (pr1 h x))) , + ( refl-htpy) + abstract weak-funext-univalence : {l : Level} → weak-function-extensionality-Level l l weak-funext-univalence A B is-contr-B = is-contr-retract-of ( fiber (postcomp A pr1) id) - ( ( λ f → ((λ x → (x , f x)) , refl)) , - ( λ h x → tr B (htpy-eq (pr2 h) x) (pr2 (pr1 h x))) , - ( refl-htpy)) + ( retract-compute-fiber-id-postcomp-pr1) ( is-contr-map-is-equiv ( is-equiv-postcomp-univalence A (equiv-pr1 is-contr-B)) ( id)) diff --git a/src/foundation/weak-limited-principle-of-omniscience.lagda.md b/src/foundation/weak-limited-principle-of-omniscience.lagda.md index b4a805bc24..bc9b213e06 100644 --- a/src/foundation/weak-limited-principle-of-omniscience.lagda.md +++ b/src/foundation/weak-limited-principle-of-omniscience.lagda.md @@ -9,38 +9,37 @@ module foundation.weak-limited-principle-of-omniscience where ```agda open import elementary-number-theory.natural-numbers +open import foundation.booleans open import foundation.disjunction open import foundation.negation open import foundation.universal-quantification open import foundation.universe-levels +open import foundation-core.decidable-propositions open import foundation-core.propositions open import foundation-core.sets - -open import univalent-combinatorics.standard-finite-types ```
## Statement -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 +The {{#concept "weak limited principle of omniscience"}} (WLPO) asserts that for +any [sequence](foundation.sequences.md) `f : ℕ → bool` either `f n` is true for +all `n : ℕ` or it is [not](foundation-core.negation.md). 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))))) +prop-WLPO : Prop lzero +prop-WLPO = + ∀' (ℕ → bool) (λ f → is-decidable-Prop (∀' ℕ (λ n → is-true-Prop (f n)))) WLPO : UU lzero -WLPO = type-Prop WLPO-Prop +WLPO = type-Prop prop-WLPO + +is-prop-WLPO : is-prop WLPO +is-prop-WLPO = is-prop-type-Prop prop-WLPO ``` ## See also @@ -48,3 +47,11 @@ WLPO = type-Prop WLPO-Prop - [The principle of omniscience](foundation.principle-of-omniscience.md) - [The limited principle of omniscience](foundation.limited-principle-of-omniscience.md) - [The lesser limited principle of omniscience](foundation.lesser-limited-principle-of-omniscience.md) +- [Markov's principle](logic.markovs-principle.md) + +## External links + +- [Taboos.WLPO](https://martinescardo.github.io/TypeTopology/Taboos.WLPO.html) + at TypeTopology +- [weak limited principle of omniscience](https://ncatlab.org/nlab/show/weak+limited+principle+of+omniscience) + at $n$Lab diff --git a/src/logic.lagda.md b/src/logic.lagda.md new file mode 100644 index 0000000000..4f76bcf78b --- /dev/null +++ b/src/logic.lagda.md @@ -0,0 +1,23 @@ +# Logic + +## Modules in the logic namespace + +```agda +module logic where + +open import logic.complements-de-morgan-subtypes public +open import logic.complements-decidable-subtypes public +open import logic.complements-double-negation-stable-subtypes public +open import logic.de-morgan-embeddings public +open import logic.de-morgan-maps public +open import logic.de-morgan-propositions public +open import logic.de-morgan-subtypes public +open import logic.de-morgan-types public +open import logic.de-morgans-law public +open import logic.double-negation-eliminating-maps public +open import logic.double-negation-elimination public +open import logic.double-negation-stable-embeddings public +open import logic.double-negation-stable-subtypes public +open import logic.markovian-types public +open import logic.markovs-principle public +``` diff --git a/src/logic/complements-de-morgan-subtypes.lagda.md b/src/logic/complements-de-morgan-subtypes.lagda.md new file mode 100644 index 0000000000..fbfe0b0e4a --- /dev/null +++ b/src/logic/complements-de-morgan-subtypes.lagda.md @@ -0,0 +1,101 @@ +# Complements of De Morgan subtypes + +```agda +module logic.complements-de-morgan-subtypes where +``` + +
Imports + +```agda +open import foundation.complements-subtypes +open import foundation.decidable-subtypes +open import foundation.dependent-pair-types +open import foundation.double-negation +open import foundation.full-subtypes +open import foundation.involutions +open import foundation.negation +open import foundation.postcomposition-functions +open import foundation.powersets +open import foundation.propositional-truncations +open import foundation.subtypes +open import foundation.unions-subtypes +open import foundation.universe-levels + +open import foundation-core.function-types + +open import logic.complements-decidable-subtypes +open import logic.de-morgan-propositions +open import logic.de-morgan-subtypes +``` + +
+ +## Idea + +The +{{#concept "complement" Disambiguation="of a De Morgan subtype" Agda=complement-de-morgan-subtype}} +of a [De Morgan subtype](logic.de-morgan-subtypes.md) `B ⊆ A` consists of the +elements that are not in `B`. + +## Definition + +### Complements of De Morgan subtypes + +```agda +complement-de-morgan-subtype : + {l1 l2 : Level} {A : UU l1} → de-morgan-subtype l2 A → de-morgan-subtype l2 A +complement-de-morgan-subtype P x = neg-De-Morgan-Prop (P x) +``` + +## Properties + +### Complement of De Morgan subtypes are decidable + +```agda +is-decidable-complement-de-morgan-subtype : + {l1 l2 : Level} {A : UU l1} (P : de-morgan-subtype l2 A) → + is-decidable-subtype + ( subtype-de-morgan-subtype (complement-de-morgan-subtype P)) +is-decidable-complement-de-morgan-subtype P = is-de-morgan-de-morgan-subtype P +``` + +### The union of the complement of a subtype `P` with its double complement is the full subtype if and only if `P` is De Morgan + +```agda +module _ + {l1 l2 : Level} {A : UU l1} + where + + is-full-union-complement-subtype-double-complement-subtype : + (P : subtype l2 A) → is-de-morgan-subtype P → + is-full-subtype + ( union-subtype + ( complement-subtype P) + ( complement-subtype (complement-subtype P))) + is-full-union-complement-subtype-double-complement-subtype P = + is-full-union-subtype-complement-subtype (complement-subtype P) + + is-de-morgan-subtype-is-full-union-complement-subtype-double-complement-subtype : + (P : subtype l2 A) → + is-full-subtype + ( union-subtype + ( complement-subtype P) + ( complement-subtype (complement-subtype P))) → + is-de-morgan-subtype P + is-de-morgan-subtype-is-full-union-complement-subtype-double-complement-subtype + P = + is-decidable-subtype-is-full-union-subtype-complement-subtype + ( complement-subtype P) + + is-full-union-subtype-complement-de-morgan-subtype : + (P : de-morgan-subtype l2 A) → + is-full-subtype + ( union-subtype + ( complement-subtype (subtype-de-morgan-subtype P)) + ( complement-subtype + ( complement-subtype (subtype-de-morgan-subtype P)))) + is-full-union-subtype-complement-de-morgan-subtype P = + is-full-union-complement-subtype-double-complement-subtype + ( subtype-de-morgan-subtype P) + ( is-de-morgan-de-morgan-subtype P) +``` diff --git a/src/logic/complements-decidable-subtypes.lagda.md b/src/logic/complements-decidable-subtypes.lagda.md new file mode 100644 index 0000000000..ae2a775f33 --- /dev/null +++ b/src/logic/complements-decidable-subtypes.lagda.md @@ -0,0 +1,100 @@ +# Complements of decidable subtypes + +```agda +module logic.complements-decidable-subtypes where +``` + +
Imports + +```agda +open import foundation.complements-subtypes +open import foundation.coproduct-types +open import foundation.decidable-propositions +open import foundation.decidable-subtypes +open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.double-negation-stable-propositions +open import foundation.evaluation-functions +open import foundation.full-subtypes +open import foundation.involutions +open import foundation.negation +open import foundation.postcomposition-functions +open import foundation.powersets +open import foundation.propositional-truncations +open import foundation.unions-subtypes +open import foundation.universe-levels + +open import foundation-core.function-types +open import foundation-core.subtypes + +open import logic.double-negation-stable-subtypes +``` + +
+ +## Idea + +The +{{#concept "complement" Disambiguation="of a decidable subtype" Agda=complement-decidable-subtype}} +of a [decidable subtype](foundation.decidable-subtypes.md) `B ⊆ A` consists of +the elements that are not in `B`. + +## Definition + +### Complements of decidable subtypes + +```agda +complement-decidable-subtype : + {l1 l2 : Level} {A : UU l1} → decidable-subtype l2 A → decidable-subtype l2 A +complement-decidable-subtype P x = neg-Decidable-Prop (P x) +``` + +## Properties + +### Taking complements is an involution on decidable subtypes + +```agda +is-involution-complement-decidable-subtype : + {l1 l2 : Level} {A : UU l1} → + is-involution (complement-decidable-subtype {l1} {l2} {A}) +is-involution-complement-decidable-subtype P = + eq-has-same-elements-decidable-subtype + ( complement-decidable-subtype (complement-decidable-subtype P)) + ( P) + ( λ x → + double-negation-elim-is-decidable (is-decidable-decidable-subtype P x) , + ev) +``` + +### The union of a subtype `P` with its complement is the full subtype if and only if `P` is a decidable subtype + +```agda +module _ + {l1 l2 : Level} {A : UU l1} + where + + is-full-union-subtype-complement-subtype : + (P : subtype l2 A) → is-decidable-subtype P → + is-full-subtype (union-subtype P (complement-subtype P)) + is-full-union-subtype-complement-subtype P d x = + unit-trunc-Prop (d x) + + is-decidable-subtype-is-full-union-subtype-complement-subtype : + (P : subtype l2 A) → + is-full-subtype (union-subtype P (complement-subtype P)) → + is-decidable-subtype P + is-decidable-subtype-is-full-union-subtype-complement-subtype P H x = + apply-universal-property-trunc-Prop + ( H x) + ( is-decidable-Prop (P x)) + ( id) + + is-full-union-subtype-complement-decidable-subtype : + (P : decidable-subtype l2 A) → + is-full-decidable-subtype + ( union-decidable-subtype P (complement-decidable-subtype P)) + is-full-union-subtype-complement-decidable-subtype P = + is-full-union-subtype-complement-subtype + ( subtype-decidable-subtype P) + ( is-decidable-decidable-subtype P) +``` diff --git a/src/logic/complements-double-negation-stable-subtypes.lagda.md b/src/logic/complements-double-negation-stable-subtypes.lagda.md new file mode 100644 index 0000000000..e4ba546190 --- /dev/null +++ b/src/logic/complements-double-negation-stable-subtypes.lagda.md @@ -0,0 +1,66 @@ +# Complements of double negation stable subtypes + +```agda +module logic.complements-double-negation-stable-subtypes where +``` + +
Imports + +```agda +open import foundation.dependent-pair-types +open import foundation.double-negation +open import foundation.double-negation-stable-propositions +open import foundation.full-subtypes +open import foundation.involutions +open import foundation.negation +open import foundation.postcomposition-functions +open import foundation.powersets +open import foundation.propositional-truncations +open import foundation.subtypes +open import foundation.unions-subtypes +open import foundation.universe-levels + +open import foundation-core.function-types + +open import logic.double-negation-stable-subtypes +``` + +
+ +## Idea + +The +{{#concept "complement" Disambiguation="of a double negation stable subtype" Agda=complement-double-negation-stable-subtype}} +of a [double negation stable subtype](logic.double-negation-stable-subtypes.md) +`B ⊆ A` consists of the elements that are not in `B`. + +## Definition + +### Complements of double negation stable subtypes + +```agda +complement-double-negation-stable-subtype : + {l1 l2 : Level} {A : UU l1} → + double-negation-stable-subtype l2 A → + double-negation-stable-subtype l2 A +complement-double-negation-stable-subtype P x = + neg-Double-Negation-Stable-Prop (P x) +``` + +## Properties + +### Taking complements is an involution on double negation stable subtypes + +```agda +is-involution-complement-double-negation-stable-subtype : + {l1 l2 : Level} {A : UU l1} → + is-involution (complement-double-negation-stable-subtype {l1} {l2} {A}) +is-involution-complement-double-negation-stable-subtype P = + eq-has-same-elements-double-negation-stable-subtype + ( complement-double-negation-stable-subtype + ( complement-double-negation-stable-subtype P)) + ( P) + ( λ x → + ( is-double-negation-stable-double-negation-stable-subtype P x , + intro-double-negation)) +``` diff --git a/src/logic/de-morgan-embeddings.lagda.md b/src/logic/de-morgan-embeddings.lagda.md new file mode 100644 index 0000000000..3391dea43b --- /dev/null +++ b/src/logic/de-morgan-embeddings.lagda.md @@ -0,0 +1,676 @@ +# De Morgan embeddings + +```agda +module logic.de-morgan-embeddings where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.cartesian-morphisms-arrows +open import foundation.decidable-embeddings +open import foundation.decidable-maps +open import foundation.decidable-propositions +open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.embeddings +open import foundation.fibers-of-maps +open import foundation.functoriality-cartesian-product-types +open import foundation.functoriality-coproduct-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.negation +open import foundation.propositional-maps +open import foundation.propositions +open import foundation.retracts-of-maps +open import foundation.subtype-identity-principle +open import foundation.type-arithmetic-dependent-pair-types +open import foundation.unit-type +open import foundation.universal-property-equivalences +open import foundation.universe-levels + +open import foundation-core.cartesian-product-types +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.functoriality-dependent-pair-types +open import foundation-core.homotopies +open import foundation-core.injective-maps +open import foundation-core.torsorial-type-families + +open import logic.de-morgan-maps +open import logic.de-morgan-propositions +open import logic.de-morgan-types +open import logic.double-negation-eliminating-maps +open import logic.double-negation-elimination +``` + +
+ +## Idea + +A [map](foundation-core.function-types.md) is said to be a +{{#concept "De Morgan embedding" Disambiguation="of types" Agda=is-de-morgan-emb}} +if it is an [embedding](foundation-core.embeddings.md) and the +[negations](foundation-core.negation.md) of its +[fibers](foundation-core.fibers-of-maps.md) are +[decidable](foundation.decidable-maps.md). + +Equivalently, a De Morgan embedding is a map whose fibers are +[De Morgan propositions](logic.de-morgan-propositions.md). We refer to this +condition as being a +{{#concept "De Morgan propositional map" Disambiguation="of types" Agda=is-de-morgan-prop-map}}. + +## Definitions + +### The condition on a map of being a De Morgan embedding + +```agda +is-de-morgan-emb : + {l1 l2 : Level} {X : UU l1} {Y : UU l2} → (X → Y) → UU (l1 ⊔ l2) +is-de-morgan-emb f = is-emb f × is-de-morgan-map f + +module _ + {l1 l2 : Level} {X : UU l1} {Y : UU l2} {f : X → Y} + (F : is-de-morgan-emb f) + where + + is-emb-is-de-morgan-emb : is-emb f + is-emb-is-de-morgan-emb = pr1 F + + is-de-morgan-map-is-de-morgan-emb : + is-de-morgan-map f + is-de-morgan-map-is-de-morgan-emb = pr2 F + + is-prop-map-is-de-morgan-emb : is-prop-map f + is-prop-map-is-de-morgan-emb = + is-prop-map-is-emb is-emb-is-de-morgan-emb + + is-injective-is-de-morgan-emb : is-injective f + is-injective-is-de-morgan-emb = + is-injective-is-emb is-emb-is-de-morgan-emb +``` + +### De Morgan propositional maps + +```agda +module _ + {l1 l2 : Level} {X : UU l1} {Y : UU l2} + where + + is-de-morgan-prop-map : (X → Y) → UU (l1 ⊔ l2) + is-de-morgan-prop-map f = + (y : Y) → is-de-morgan-prop (fiber f y) + + is-prop-is-de-morgan-prop-map : + (f : X → Y) → is-prop (is-de-morgan-prop-map f) + is-prop-is-de-morgan-prop-map f = + is-prop-Π (λ y → is-prop-is-de-morgan-prop (fiber f y)) + + is-de-morgan-prop-map-Prop : (X → Y) → Prop (l1 ⊔ l2) + is-de-morgan-prop-map-Prop f = + ( is-de-morgan-prop-map f , is-prop-is-de-morgan-prop-map f) + + is-prop-map-is-de-morgan-prop-map : + {f : X → Y} → is-de-morgan-prop-map f → is-prop-map f + is-prop-map-is-de-morgan-prop-map H y = + is-prop-type-is-de-morgan-prop (H y) + + is-de-morgan-map-is-de-morgan-prop-map : + {f : X → Y} → is-de-morgan-prop-map f → is-de-morgan-map f + is-de-morgan-map-is-de-morgan-prop-map H y = + is-de-morgan-type-is-de-morgan-prop (H y) +``` + +### The type of De Morgan embeddings + +```agda +infix 5 _↪ᵈᵐ_ +_↪ᵈᵐ_ : {l1 l2 : Level} (X : UU l1) (Y : UU l2) → UU (l1 ⊔ l2) +X ↪ᵈᵐ Y = Σ (X → Y) (is-de-morgan-emb) + +de-morgan-emb : {l1 l2 : Level} (X : UU l1) (Y : UU l2) → UU (l1 ⊔ l2) +de-morgan-emb = _↪ᵈᵐ_ + +module _ + {l1 l2 : Level} {X : UU l1} {Y : UU l2} (e : X ↪ᵈᵐ Y) + where + + map-de-morgan-emb : X → Y + map-de-morgan-emb = pr1 e + + is-de-morgan-emb-map-de-morgan-emb : is-de-morgan-emb map-de-morgan-emb + is-de-morgan-emb-map-de-morgan-emb = pr2 e + + is-emb-map-de-morgan-emb : is-emb map-de-morgan-emb + is-emb-map-de-morgan-emb = + is-emb-is-de-morgan-emb is-de-morgan-emb-map-de-morgan-emb + + is-de-morgan-map-map-de-morgan-emb : is-de-morgan-map map-de-morgan-emb + is-de-morgan-map-map-de-morgan-emb = + is-de-morgan-map-is-de-morgan-emb is-de-morgan-emb-map-de-morgan-emb + + emb-de-morgan-emb : X ↪ Y + emb-de-morgan-emb = map-de-morgan-emb , is-emb-map-de-morgan-emb +``` + +## Properties + +### Any map of which the fibers are De Morgan propositions is a De Morgan embedding + +```agda +module _ + {l1 l2 : Level} {X : UU l1} {Y : UU l2} {f : X → Y} + where + + abstract + is-de-morgan-emb-is-de-morgan-prop-map : + is-de-morgan-prop-map f → is-de-morgan-emb f + pr1 (is-de-morgan-emb-is-de-morgan-prop-map H) = + is-emb-is-prop-map (is-prop-map-is-de-morgan-prop-map H) + pr2 (is-de-morgan-emb-is-de-morgan-prop-map H) = + is-de-morgan-map-is-de-morgan-prop-map H + + abstract + is-de-morgan-prop-map-is-de-morgan-emb : + is-de-morgan-emb f → is-de-morgan-prop-map f + pr1 (is-de-morgan-prop-map-is-de-morgan-emb H y) = + is-prop-map-is-de-morgan-emb H y + pr2 (is-de-morgan-prop-map-is-de-morgan-emb H y) = + is-de-morgan-map-is-de-morgan-emb H y +``` + +### The first projection map of a dependent sum of De Morgan propositions is a De Morgan embedding + +```agda +module _ + {l1 l2 : Level} {A : UU l1} (Q : A → De-Morgan-Prop l2) + where + + is-de-morgan-prop-map-pr1 : + is-de-morgan-prop-map + ( pr1 {B = type-De-Morgan-Prop ∘ Q}) + is-de-morgan-prop-map-pr1 y = + is-de-morgan-prop-equiv + ( equiv-fiber-pr1 (type-De-Morgan-Prop ∘ Q) y) + ( is-de-morgan-prop-type-De-Morgan-Prop (Q y)) + + is-de-morgan-emb-pr1 : + is-de-morgan-emb + ( pr1 {B = type-De-Morgan-Prop ∘ Q}) + is-de-morgan-emb-pr1 = + is-de-morgan-emb-is-de-morgan-prop-map + ( is-de-morgan-prop-map-pr1) +``` + +### Decidable embeddings are De Morgan + +```agda +is-de-morgan-map-is-decidable-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} → + is-decidable-emb f → is-de-morgan-map f +is-de-morgan-map-is-decidable-emb H = + is-de-morgan-map-is-decidable-map + ( is-decidable-map-is-decidable-emb H) + +abstract + is-de-morgan-emb-is-decidable-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} → + is-decidable-emb f → is-de-morgan-emb f + is-de-morgan-emb-is-decidable-emb H = + ( is-emb-is-decidable-emb H , + is-de-morgan-map-is-decidable-emb H) +``` + +### Equivalences are De Morgan embeddings + +```agda +abstract + is-de-morgan-emb-is-equiv : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} → + is-equiv f → is-de-morgan-emb f + is-de-morgan-emb-is-equiv H = + ( is-emb-is-equiv H , is-de-morgan-map-is-equiv H) +``` + +### Identity maps are De Morgan embeddings + +```agda +is-de-morgan-emb-id : + {l : Level} {A : UU l} → is-de-morgan-emb (id {A = A}) +is-de-morgan-emb-id = + ( is-emb-id , is-de-morgan-map-id) + +de-morgan-emb-id : {l : Level} {A : UU l} → A ↪ᵈᵐ A +de-morgan-emb-id = (id , is-de-morgan-emb-id) + +is-de-morgan-prop-map-id : + {l : Level} {A : UU l} → is-de-morgan-prop-map (id {A = A}) +is-de-morgan-prop-map-id y = + is-de-morgan-prop-is-contr (is-torsorial-Id' y) +``` + +### Being a De Morgan embedding is a property + +```agda +abstract + is-prop-is-de-morgan-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) → + is-prop (is-de-morgan-emb f) + is-prop-is-de-morgan-emb f = + is-prop-product (is-property-is-emb f) is-prop-is-de-morgan-map +``` + +### De Morgan embeddings are closed under homotopies + +```agda +abstract + is-de-morgan-emb-htpy : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f g : A → B} → + f ~ g → is-de-morgan-emb g → is-de-morgan-emb f + is-de-morgan-emb-htpy {f = f} {g} H K = + ( is-emb-htpy H (is-emb-is-de-morgan-emb K) , + is-de-morgan-map-htpy H + ( is-de-morgan-map-is-de-morgan-emb K)) +``` + +### De Morgan embeddings are closed under composition + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} + {g : B → C} {f : A → B} + where + + is-de-morgan-map-comp-is-decidable-emb : + is-decidable-emb g → + is-de-morgan-map f → + is-de-morgan-map (g ∘ f) + is-de-morgan-map-comp-is-decidable-emb G F = + is-de-morgan-map-comp-is-decidable-map + ( is-injective-is-decidable-emb G) + ( is-decidable-map-is-decidable-emb G) + ( F) + + is-de-morgan-prop-map-comp-is-decidable-prop-map : + is-decidable-prop-map g → + is-de-morgan-prop-map f → + is-de-morgan-prop-map (g ∘ f) + is-de-morgan-prop-map-comp-is-decidable-prop-map K H z = + is-de-morgan-prop-equiv + ( compute-fiber-comp g f z) + ( is-de-morgan-prop-Σ (K z) (H ∘ pr1)) + + is-de-morgan-emb-comp-is-decidable-emb : + is-decidable-emb g → + is-de-morgan-emb f → + is-de-morgan-emb (g ∘ f) + is-de-morgan-emb-comp-is-decidable-emb K H = + is-de-morgan-emb-is-de-morgan-prop-map + ( is-de-morgan-prop-map-comp-is-decidable-prop-map + ( is-decidable-prop-map-is-decidable-emb K) + ( is-de-morgan-prop-map-is-de-morgan-emb H)) + +comp-de-morgan-emb : + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} → + B ↪ᵈ C → A ↪ᵈᵐ B → A ↪ᵈᵐ C +comp-de-morgan-emb (g , G) (f , F) = + ( g ∘ f , is-de-morgan-emb-comp-is-decidable-emb G F) +``` + +### Left cancellation for De Morgan embeddings + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {f : A → B} {g : B → C} + where + + is-de-morgan-emb-right-factor' : + is-de-morgan-emb (g ∘ f) → + is-emb g → + is-de-morgan-emb f + is-de-morgan-emb-right-factor' GF G = + ( is-emb-right-factor g f G (is-emb-is-de-morgan-emb GF) , + is-de-morgan-map-right-factor' + ( is-injective-is-emb G) + ( is-de-morgan-map-is-de-morgan-emb GF)) + + is-de-morgan-emb-right-factor : + is-de-morgan-emb (g ∘ f) → + is-de-morgan-emb g → + is-de-morgan-emb f + is-de-morgan-emb-right-factor GF G = + is-de-morgan-emb-right-factor' + ( GF) + ( is-emb-is-de-morgan-emb G) +``` + +### In a commuting triangle of maps, if the top and right maps are De Morgan embeddings so is the left map + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} + {top : A → B} {left : A → C} {right : B → C} + (H : left ~ right ∘ top) + where + + is-de-morgan-emb-left-map-triangle-is-decidable-emb-top : + is-de-morgan-emb top → + is-decidable-emb right → + is-de-morgan-emb left + is-de-morgan-emb-left-map-triangle-is-decidable-emb-top T R = + is-de-morgan-emb-htpy H (is-de-morgan-emb-comp-is-decidable-emb R T) +``` + +### In a commuting triangle of maps, if the left and right maps are De Morgan embeddings so is the top map + +In fact, the right map need only be an embedding. + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} + {top : A → B} {left : A → C} {right : B → C} + (H : left ~ right ∘ top) + where + + is-de-morgan-emb-top-map-triangle' : + is-emb right → + is-de-morgan-emb left → + is-de-morgan-emb top + is-de-morgan-emb-top-map-triangle' R' L = + is-de-morgan-emb-right-factor' + ( is-de-morgan-emb-htpy (inv-htpy H) L) + ( R') + + is-de-morgan-emb-top-map-triangle : + is-de-morgan-emb right → + is-de-morgan-emb left → + is-de-morgan-emb top + is-de-morgan-emb-top-map-triangle R L = + is-de-morgan-emb-right-factor + ( is-de-morgan-emb-htpy (inv-htpy H) L) + ( R) +``` + +### Characterizing equality in the type of De Morgan embeddings + +```agda +htpy-de-morgan-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f g : A ↪ᵈᵐ B) → UU (l1 ⊔ l2) +htpy-de-morgan-emb f g = + map-de-morgan-emb f ~ map-de-morgan-emb g + +refl-htpy-de-morgan-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A ↪ᵈᵐ B) → + htpy-de-morgan-emb f f +refl-htpy-de-morgan-emb f = refl-htpy + +htpy-eq-de-morgan-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f g : A ↪ᵈᵐ B) → + f ＝ g → htpy-de-morgan-emb f g +htpy-eq-de-morgan-emb f .f refl = + refl-htpy-de-morgan-emb f + +abstract + is-torsorial-htpy-de-morgan-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A ↪ᵈᵐ B) → + is-torsorial (htpy-de-morgan-emb f) + is-torsorial-htpy-de-morgan-emb f = + is-torsorial-Eq-subtype + ( is-torsorial-htpy (map-de-morgan-emb f)) + ( is-prop-is-de-morgan-emb) + ( map-de-morgan-emb f) + ( refl-htpy) + ( is-de-morgan-emb-map-de-morgan-emb f) + +abstract + is-equiv-htpy-eq-de-morgan-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f g : A ↪ᵈᵐ B) → + is-equiv (htpy-eq-de-morgan-emb f g) + is-equiv-htpy-eq-de-morgan-emb f = + fundamental-theorem-id + ( is-torsorial-htpy-de-morgan-emb f) + ( htpy-eq-de-morgan-emb f) + +eq-htpy-de-morgan-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f g : A ↪ᵈᵐ B) → + htpy-de-morgan-emb f g → f ＝ g +eq-htpy-de-morgan-emb f g = + map-inv-is-equiv (is-equiv-htpy-eq-de-morgan-emb f g) +``` + +### Any map out of the empty type is a De Morgan embedding + +```agda +abstract + is-de-morgan-emb-ex-falso : + {l : Level} {X : UU l} → is-de-morgan-emb (ex-falso {l} {X}) + is-de-morgan-emb-ex-falso = + ( is-emb-ex-falso , is-de-morgan-map-ex-falso) + +de-morgan-emb-ex-falso : + {l : Level} {X : UU l} → empty ↪ᵈᵐ X +de-morgan-emb-ex-falso = + ( ex-falso , is-de-morgan-emb-ex-falso) +``` + +### The map on total spaces induced by a family of De Morgan embeddings is a De Morgan embedding + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} + where + + is-de-morgan-emb-tot : + {f : (x : A) → B x → C x} → + ((x : A) → is-de-morgan-emb (f x)) → + is-de-morgan-emb (tot f) + is-de-morgan-emb-tot H = + ( is-emb-tot (is-emb-is-de-morgan-emb ∘ H) , + is-de-morgan-map-tot + ( is-de-morgan-map-is-de-morgan-emb ∘ H)) + + de-morgan-emb-tot : ((x : A) → B x ↪ᵈᵐ C x) → Σ A B ↪ᵈᵐ Σ A C + de-morgan-emb-tot f = + ( tot (map-de-morgan-emb ∘ f) , + is-de-morgan-emb-tot + ( is-de-morgan-emb-map-de-morgan-emb ∘ f)) +``` + +### The map on total spaces induced by a De Morgan embedding on the base is a De Morgan embedding + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (C : B → UU l3) + where + + is-de-morgan-emb-map-Σ-map-base : + {f : A → B} → + is-de-morgan-emb f → + is-de-morgan-emb (map-Σ-map-base f C) + is-de-morgan-emb-map-Σ-map-base H = + ( is-emb-map-Σ-map-base C (is-emb-is-de-morgan-emb H) , + is-de-morgan-map-Σ-map-base C + ( is-de-morgan-map-is-de-morgan-emb H)) + + de-morgan-emb-map-Σ-map-base : + (f : A ↪ᵈᵐ B) → Σ A (C ∘ map-de-morgan-emb f) ↪ᵈᵐ Σ B C + de-morgan-emb-map-Σ-map-base f = + ( map-Σ-map-base (map-de-morgan-emb f) C , + is-de-morgan-emb-map-Σ-map-base + ( is-de-morgan-emb-map-de-morgan-emb f)) +``` + +### The functoriality of dependent pair types on De Morgan embeddings + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : A → UU l3} (D : B → UU l4) + where + + is-de-morgan-emb-map-Σ : + {f : A → B} {g : (x : A) → C x → D (f x)} → + is-decidable-emb f → + ((x : A) → is-de-morgan-emb (g x)) → + is-de-morgan-emb (map-Σ D f g) + is-de-morgan-emb-map-Σ {f} {g} F G = + is-de-morgan-emb-left-map-triangle-is-decidable-emb-top + ( triangle-map-Σ D f g) + ( is-de-morgan-emb-tot G) + ( is-decidable-emb-map-Σ-map-base D F) + + de-morgan-emb-Σ : + (f : A ↪ᵈ B) → + ((x : A) → C x ↪ᵈᵐ D (map-decidable-emb f x)) → + Σ A C ↪ᵈᵐ Σ B D + de-morgan-emb-Σ f g = + ( ( map-Σ D (map-decidable-emb f) (map-de-morgan-emb ∘ g)) , + ( is-de-morgan-emb-map-Σ + ( is-decidable-emb-map-decidable-emb f) + ( is-de-morgan-emb-map-de-morgan-emb ∘ g))) +``` + +### Products of De Morgan embeddings are De Morgan embeddings + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {D : UU l4} + where + + is-de-morgan-emb-map-product : + {f : A → B} {g : C → D} → + is-de-morgan-emb f → + is-de-morgan-emb g → + is-de-morgan-emb (map-product f g) + is-de-morgan-emb-map-product (eF , dF) (eG , dG) = + ( is-emb-map-product eF eG , + is-de-morgan-map-product dF dG) + + de-morgan-emb-product : + A ↪ᵈᵐ B → C ↪ᵈᵐ D → A × C ↪ᵈᵐ B × D + de-morgan-emb-product (f , F) (g , G) = + ( map-product f g , is-de-morgan-emb-map-product F G) +``` + +### Coproducts of De Morgan embeddings are De Morgan embeddings + +```agda +module _ + {l1 l2 l1' l2' : Level} {A : UU l1} {B : UU l2} {A' : UU l1'} {B' : UU l2'} + where + +abstract + is-de-morgan-emb-map-coproduct : + {l1 l2 l3 l4 : Level} + {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} + {f : A → B} {g : X → Y} → + is-de-morgan-emb f → + is-de-morgan-emb g → + is-de-morgan-emb (map-coproduct f g) + is-de-morgan-emb-map-coproduct (eF , dF) (eG , dG) = + ( is-emb-map-coproduct eF eG , + is-de-morgan-map-coproduct dF dG) +``` + +### De Morgan embeddings are closed under base change + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {D : UU l4} + {f : A → B} {g : C → D} + where + + is-de-morgan-prop-map-base-change : + cartesian-hom-arrow g f → + is-de-morgan-prop-map f → + is-de-morgan-prop-map g + is-de-morgan-prop-map-base-change α F d = + is-de-morgan-prop-equiv + ( equiv-fibers-cartesian-hom-arrow g f α d) + ( F (map-codomain-cartesian-hom-arrow g f α d)) + + is-de-morgan-emb-base-change : + cartesian-hom-arrow g f → + is-de-morgan-emb f → + is-de-morgan-emb g + is-de-morgan-emb-base-change α F = + is-de-morgan-emb-is-de-morgan-prop-map + ( is-de-morgan-prop-map-base-change α + ( is-de-morgan-prop-map-is-de-morgan-emb F)) +``` + +### De Morgan embeddings are closed under retracts of maps + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} + {f : A → B} {g : X → Y} + where + + is-de-morgan-prop-map-retract-map : + f retract-of-map g → + is-de-morgan-prop-map g → + is-de-morgan-prop-map f + is-de-morgan-prop-map-retract-map R G x = + is-de-morgan-prop-retract-of + ( retract-fiber-retract-map f g R x) + ( G (map-codomain-inclusion-retract-map f g R x)) + + is-de-morgan-emb-retract-map : + f retract-of-map g → + is-de-morgan-emb g → + is-de-morgan-emb f + is-de-morgan-emb-retract-map R G = + is-de-morgan-emb-is-de-morgan-prop-map + ( is-de-morgan-prop-map-retract-map R + ( is-de-morgan-prop-map-is-de-morgan-emb G)) +``` + +### A type is a De Morgan proposition if and only if its terminal map is a De Morgan embedding + +```agda +module _ + {l : Level} {A : UU l} + where + + is-de-morgan-prop-is-de-morgan-emb-terminal-map : + is-de-morgan-emb (terminal-map A) → + is-de-morgan-prop A + is-de-morgan-prop-is-de-morgan-emb-terminal-map H = + is-de-morgan-prop-equiv' + ( equiv-fiber-terminal-map star) + ( is-de-morgan-prop-map-is-de-morgan-emb H star) + + is-de-morgan-emb-terminal-map-is-de-morgan-prop : + is-de-morgan-prop A → + is-de-morgan-emb (terminal-map A) + is-de-morgan-emb-terminal-map-is-de-morgan-prop H = + is-de-morgan-emb-is-de-morgan-prop-map + ( λ y → + is-de-morgan-prop-equiv (equiv-fiber-terminal-map y) H) +``` + +### If a dependent sum of propositions over a proposition is De Morgan, then the family is a family of De Morgan propositions + +```agda +module _ + {l1 l2 : Level} (P : Prop l1) (Q : type-Prop P → Prop l2) + where + + is-de-morgan-prop-family-is-de-morgan-Σ : + is-de-morgan (Σ (type-Prop P) (type-Prop ∘ Q)) → + (p : type-Prop P) → is-de-morgan (type-Prop (Q p)) + is-de-morgan-prop-family-is-de-morgan-Σ H p = + is-de-morgan-equiv' + ( equiv-fiber-pr1 (type-Prop ∘ Q) p) + ( is-de-morgan-map-is-de-morgan-emb + ( is-de-morgan-emb-right-factor' + ( is-de-morgan-emb-terminal-map-is-de-morgan-prop + ( is-prop-Σ (is-prop-type-Prop P) (is-prop-type-Prop ∘ Q) , H)) + ( is-emb-terminal-map-is-prop (is-prop-type-Prop P))) + ( p)) +``` diff --git a/src/logic/de-morgan-maps.lagda.md b/src/logic/de-morgan-maps.lagda.md new file mode 100644 index 0000000000..17b5a68d6e --- /dev/null +++ b/src/logic/de-morgan-maps.lagda.md @@ -0,0 +1,339 @@ +# De Morgan maps + +```agda +module logic.de-morgan-maps where +``` + +
Imports + +```agda +open import elementary-number-theory.natural-numbers + +open import foundation.action-on-identifications-functions +open import foundation.cartesian-morphisms-arrows +open import foundation.coproduct-types +open import foundation.decidable-equality +open import foundation.decidable-maps +open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.double-negation +open import foundation.embeddings +open import foundation.empty-types +open import foundation.existential-quantification +open import foundation.functoriality-cartesian-product-types +open import foundation.functoriality-coproduct-types +open import foundation.identity-types +open import foundation.injective-maps +open import foundation.negation +open import foundation.propositions +open import foundation.retractions +open import foundation.retracts-of-maps +open import foundation.retracts-of-types +open import foundation.transport-along-identifications +open import foundation.unit-type +open import foundation.universal-property-equivalences +open import foundation.universe-levels + +open import foundation-core.contractible-maps +open import foundation-core.equivalences +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 logic.de-morgan-types +open import logic.de-morgans-law +open import logic.double-negation-eliminating-maps +open import logic.double-negation-elimination +``` + +
+ +## Idea + +A [map](foundation-core.function-types.md) is said to be +{{#concept "De Morgan" Disambiguation="map of types" Agda=is-de-morgan-map}} if +the [negation](foundation-core.negation.md) of its +[fibers](foundation-core.fibers-of-maps.md) are +[decidable](foundation.decidable-types.md). I.e., the map `f : A → B` is De +Morgan if for every `y : B`, the fiber `fiber f y` is either +[empty](foundation.empty-types.md) or not empty. This is equivalent to asking +that the fibers satisfy [De Morgan's law](logic.de-morgans-law.md), but is a +[small](foundation.small-types.md) condition. + +## Definintion + +### De Morgan maps + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-de-morgan-map : (A → B) → UU (l1 ⊔ l2) + is-de-morgan-map f = (y : B) → is-de-morgan (fiber f y) + + is-prop-is-de-morgan-map : {f : A → B} → is-prop (is-de-morgan-map f) + is-prop-is-de-morgan-map {f} = + is-prop-Π (λ y → is-prop-is-de-morgan (fiber f y)) + + is-de-morgan-map-Prop : (A → B) → Prop (l1 ⊔ l2) + is-de-morgan-map-Prop f = is-de-morgan-map f , is-prop-is-de-morgan-map +``` + +### The type of De Morgan maps + +```agda +infix 5 _→ᵈᵐ_ + +_→ᵈᵐ_ : {l1 l2 : Level} (A : UU l1) (B : UU l2) → UU (l1 ⊔ l2) +A →ᵈᵐ B = Σ (A → B) (is-de-morgan-map) + +de-morgan-map : {l1 l2 : Level} (A : UU l1) (B : UU l2) → UU (l1 ⊔ l2) +de-morgan-map = _→ᵈᵐ_ + +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A →ᵈᵐ B) + where + + map-de-morgan-map : A → B + map-de-morgan-map = pr1 f + + is-de-morgan-de-morgan-map : is-de-morgan-map map-de-morgan-map + is-de-morgan-de-morgan-map = pr2 f +``` + +### Self-De-Morgan maps + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-self-de-morgan-map : (A → B) → UU (l1 ⊔ l2) + is-self-de-morgan-map f = + (y z : B) → de-morgans-law' (fiber f y) (fiber f z) +``` + +## Properties + +### De Morgan maps are closed under homotopy + +```agda +abstract + is-de-morgan-map-htpy : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f g : A → B} → + f ~ g → + is-de-morgan-map g → + is-de-morgan-map f + is-de-morgan-map-htpy H K b = + is-decidable-equiv' + ( equiv-precomp (equiv-tot (λ a → equiv-concat (inv (H a)) b)) empty) + ( K b) +``` + +### Decidable maps are De Morgan + +```agda +is-de-morgan-map-is-decidable-map : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} → + is-decidable-map f → is-de-morgan-map f +is-de-morgan-map-is-decidable-map H y = is-de-morgan-is-decidable (H y) +``` + +### Double negation eliminating De Morgan maps are decidable + +```agda +is-decidable-map-is-double-negation-eliminating-de-morgan-map : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} → + is-de-morgan-map f → is-double-negation-eliminating-map f → is-decidable-map f +is-decidable-map-is-double-negation-eliminating-de-morgan-map H K y = + is-decidable-is-decidable-neg-has-double-negation-elim (K y) (H y) +``` + +### Left cancellation for De Morgan maps + +If a composite `g ∘ f` is De Morgan and the left factor `g` is injective, then +the right factor `f` is De Morgan. + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {f : A → B} {g : B → C} + where + + is-de-morgan-map-right-factor' : + is-injective g → + is-de-morgan-map (g ∘ f) → + is-de-morgan-map f + is-de-morgan-map-right-factor' H GF y = + rec-coproduct + ( λ ngfy → inl (λ p → ngfy (pr1 p , ap g (pr2 p)))) + ( λ nngfy → inr (λ nq → nngfy (λ p → nq (pr1 p , H (pr2 p))))) + ( GF (g y)) +``` + +### Composition of De Morgan maps with decidable maps + +If `g` is a decidable injection and `f` is a De Morgan map, then `g ∘ f` is De +Morgan. + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {f : A → B} {g : B → C} + where + + is-de-morgan-map-comp-is-decidable-map : + is-injective g → + is-decidable-map g → + is-de-morgan-map f → + is-de-morgan-map (g ∘ f) + is-de-morgan-map-comp-is-decidable-map H G F y = + rec-coproduct + ( λ u → + is-de-morgan-iff + ( λ v → (pr1 v) , ap g (pr2 v) ∙ pr2 u) + ( λ w → pr1 w , H (pr2 w ∙ inv (pr2 u))) + ( F (pr1 u))) + ( λ ng → inl (λ u → ng (f (pr1 u) , pr2 u))) + ( G y) +``` + +### Any map out of the empty type is De Morgan + +```agda +abstract + is-de-morgan-map-ex-falso : + {l : Level} {X : UU l} → is-de-morgan-map (ex-falso {l} {X}) + is-de-morgan-map-ex-falso = + is-de-morgan-map-is-decidable-map is-decidable-map-ex-falso +``` + +### The identity map is De Morgan + +```agda +abstract + is-de-morgan-map-id : + {l : Level} {X : UU l} → is-de-morgan-map (id {l} {X}) + is-de-morgan-map-id = + is-de-morgan-map-is-decidable-map is-decidable-map-id +``` + +### Equivalences are De Morgan maps + +```agda +abstract + is-de-morgan-map-is-equiv : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} → + is-equiv f → is-de-morgan-map f + is-de-morgan-map-is-equiv H = + is-de-morgan-map-is-decidable-map (is-decidable-map-is-equiv H) +``` + +### The map on total spaces induced by a family of De Morgan maps is De Morgan + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} + where + + is-de-morgan-map-tot : + {f : (x : A) → B x → C x} → + ((x : A) → is-de-morgan-map (f x)) → + is-de-morgan-map (tot f) + is-de-morgan-map-tot {f} H x = + is-decidable-equiv (equiv-neg (compute-fiber-tot f x)) (H (pr1 x) (pr2 x)) +``` + +### The map on total spaces induced by a De Morgan map on the base is De Morgan + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (C : B → UU l3) + where + + is-de-morgan-map-Σ-map-base : + {f : A → B} → + is-de-morgan-map f → + is-de-morgan-map (map-Σ-map-base f C) + is-de-morgan-map-Σ-map-base {f} H x = + is-decidable-equiv' + ( equiv-neg (compute-fiber-map-Σ-map-base f C x)) + ( H (pr1 x)) +``` + +### Products of De Morgan maps are De Morgan + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {D : UU l4} + where + + is-de-morgan-map-product : + {f : A → B} {g : C → D} → + is-de-morgan-map f → + is-de-morgan-map g → + is-de-morgan-map (map-product f g) + is-de-morgan-map-product {f} {g} F G y = + is-de-morgan-equiv + ( compute-fiber-map-product f g y) + ( is-de-morgan-product (F (pr1 y)) (G (pr2 y))) +``` + +### Coproducts of De Morgan maps are De Morgan + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {D : UU l4} + where + + is-de-morgan-map-coproduct : + {f : A → B} {g : C → D} → + is-de-morgan-map f → + is-de-morgan-map g → + is-de-morgan-map (map-coproduct f g) + is-de-morgan-map-coproduct {f} {g} F G (inl x) = + is-decidable-equiv' + ( equiv-neg (compute-fiber-inl-map-coproduct f g x)) + ( F x) + is-de-morgan-map-coproduct {f} {g} F G (inr x) = + is-decidable-equiv' + ( equiv-neg (compute-fiber-inr-map-coproduct f g x)) + ( G x) +``` + +### De Morgan maps are closed under base change + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {D : UU l4} + {f : A → B} {g : C → D} + where + + is-de-morgan-map-base-change : + cartesian-hom-arrow g f → + is-de-morgan-map f → + is-de-morgan-map g + is-de-morgan-map-base-change α F d = + is-decidable-equiv + ( equiv-neg (equiv-fibers-cartesian-hom-arrow g f α d)) + ( F (map-codomain-cartesian-hom-arrow g f α d)) +``` + +### De Morgan maps are closed under retracts of maps + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} + {f : A → B} {g : X → Y} + where + + is-de-morgan-retract-map : + f retract-of-map g → + is-de-morgan-map g → + is-de-morgan-map f + is-de-morgan-retract-map R G x = + is-decidable-iff + ( map-neg (inclusion-retract (retract-fiber-retract-map f g R x))) + ( map-neg (map-retraction-retract (retract-fiber-retract-map f g R x))) + ( G (map-codomain-inclusion-retract-map f g R x)) +``` diff --git a/src/logic/de-morgan-propositions.lagda.md b/src/logic/de-morgan-propositions.lagda.md new file mode 100644 index 0000000000..5ca81ba117 --- /dev/null +++ b/src/logic/de-morgan-propositions.lagda.md @@ -0,0 +1,351 @@ +# De Morgan propositions + +```agda +module logic.de-morgan-propositions where +``` + +
Imports + +```agda +open import foundation.cartesian-product-types +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.disjunction +open import foundation.double-negation +open import foundation.embeddings +open import foundation.empty-types +open import foundation.equivalences +open import foundation.evaluation-functions +open import foundation.function-types +open import foundation.functoriality-dependent-pair-types +open import foundation.identity-types +open import foundation.irrefutable-propositions +open import foundation.logical-equivalences +open import foundation.negation +open import foundation.propositional-extensionality +open import foundation.propositional-truncations +open import foundation.propositions +open import foundation.retracts-of-types +open import foundation.sets +open import foundation.subtypes +open import foundation.transport-along-identifications +open import foundation.universe-levels + +open import foundation-core.decidable-propositions + +open import logic.de-morgan-types +open import logic.de-morgans-law +``` + +
+ +## Idea + +In classical logic, i.e., logic where we assume +[the law of excluded middle](foundation.law-of-excluded-middle.md), the _De +Morgan laws_ refers to the pair of logical equivalences + +```text + ¬ (P ∨ Q) ⇔ (¬ P) ∧ (¬ Q) + ¬ (P ∧ Q) ⇔ (¬ P) ∨ (¬ Q). +``` + +Out of these in total four logical implications, all but one are validated in +constructive mathematics. The odd one out is + +```text + ¬ (P ∧ Q) ⇒ (¬ P) ∨ (¬ Q). +``` + +Indeed, this would state that we could constructively deduce from a proof that +not both of `P` and `Q` are true, which of `P` and `Q` that is false. This +logical law is what we refer to as [De Morgan's Law](logic.de-morgans-law.md). +If a proposition `P` is such that for every other proposition `Q`, the De Morgan +implication + +```text + ¬ (P ∧ Q) ⇒ (¬ P) ∨ (¬ Q) +``` + +holds, we say `P` is {{#concept "De Morgan" Disambiguation="proposition"}}. + +Equivalently, a proposition is De Morgan if its negation is decidable. Since +this is a [small](foundation.small-types.md) condition, it is frequently more +convenient to use. + +## Definition + +### The predicate on propositions of being De Morgan + +```agda +module _ + {l : Level} (P : UU l) + where + + is-de-morgan-prop : UU l + is-de-morgan-prop = is-prop P × is-de-morgan P + + is-prop-is-de-morgan-prop : is-prop is-de-morgan-prop + is-prop-is-de-morgan-prop = + is-prop-product (is-prop-is-prop P) (is-prop-is-de-morgan P) + + is-de-morgan-prop-Prop : Prop l + is-de-morgan-prop-Prop = is-de-morgan-prop , is-prop-is-de-morgan-prop + +module _ + {l : Level} {P : UU l} (H : is-de-morgan-prop P) + where + + is-prop-type-is-de-morgan-prop : is-prop P + is-prop-type-is-de-morgan-prop = pr1 H + + is-de-morgan-type-is-de-morgan-prop : is-de-morgan P + is-de-morgan-type-is-de-morgan-prop = pr2 H +``` + +### The subuniverse of De Morgan propositions + +```agda +De-Morgan-Prop : (l : Level) → UU (lsuc l) +De-Morgan-Prop l = Σ (UU l) (is-de-morgan-prop) + +module _ + {l : Level} (A : De-Morgan-Prop l) + where + + type-De-Morgan-Prop : UU l + type-De-Morgan-Prop = pr1 A + + is-de-morgan-prop-type-De-Morgan-Prop : is-de-morgan-prop type-De-Morgan-Prop + is-de-morgan-prop-type-De-Morgan-Prop = pr2 A + + is-prop-type-De-Morgan-Prop : is-prop type-De-Morgan-Prop + is-prop-type-De-Morgan-Prop = + is-prop-type-is-de-morgan-prop is-de-morgan-prop-type-De-Morgan-Prop + + is-de-morgan-type-De-Morgan-Prop : is-de-morgan type-De-Morgan-Prop + is-de-morgan-type-De-Morgan-Prop = + is-de-morgan-type-is-de-morgan-prop is-de-morgan-prop-type-De-Morgan-Prop + + prop-De-Morgan-Prop : Prop l + prop-De-Morgan-Prop = type-De-Morgan-Prop , is-prop-type-De-Morgan-Prop + + de-morgan-type-De-Morgan-Prop : De-Morgan-Type l + de-morgan-type-De-Morgan-Prop = + type-De-Morgan-Prop , is-de-morgan-type-De-Morgan-Prop +``` + +## Properties + +### The forgetful map from De Morgan propositions to propositions is an embedding + +```agda +is-emb-prop-De-Morgan-Prop : + {l : Level} → is-emb (prop-De-Morgan-Prop {l}) +is-emb-prop-De-Morgan-Prop = + is-emb-tot + ( λ X → + is-emb-inclusion-subtype (λ _ → is-de-morgan X , is-prop-is-de-morgan X)) + +emb-prop-De-Morgan-Prop : + {l : Level} → De-Morgan-Prop l ↪ Prop l +emb-prop-De-Morgan-Prop = + ( prop-De-Morgan-Prop , is-emb-prop-De-Morgan-Prop) +``` + +### The subuniverse of De Morgan propositions is a set + +```agda +is-set-De-Morgan-Prop : {l : Level} → is-set (De-Morgan-Prop l) +is-set-De-Morgan-Prop {l} = + is-set-emb emb-prop-De-Morgan-Prop is-set-type-Prop + +set-De-Morgan-Prop : (l : Level) → Set (lsuc l) +set-De-Morgan-Prop l = (De-Morgan-Prop l , is-set-De-Morgan-Prop) +``` + +### Extensionality of De Morgan propositions + +```agda +module _ + {l : Level} (P Q : De-Morgan-Prop l) + where + + extensionality-De-Morgan-Prop : + (P ＝ Q) ≃ (type-De-Morgan-Prop P ↔ type-De-Morgan-Prop Q) + extensionality-De-Morgan-Prop = + ( propositional-extensionality + ( prop-De-Morgan-Prop P) + ( prop-De-Morgan-Prop Q)) ∘e + ( equiv-ap-emb emb-prop-De-Morgan-Prop) + + iff-eq-De-Morgan-Prop : P ＝ Q → type-De-Morgan-Prop P ↔ type-De-Morgan-Prop Q + iff-eq-De-Morgan-Prop = map-equiv extensionality-De-Morgan-Prop + + eq-iff-De-Morgan-Prop' : type-De-Morgan-Prop P ↔ type-De-Morgan-Prop Q → P ＝ Q + eq-iff-De-Morgan-Prop' = map-inv-equiv extensionality-De-Morgan-Prop + + eq-iff-De-Morgan-Prop : + (type-De-Morgan-Prop P → type-De-Morgan-Prop Q) → + (type-De-Morgan-Prop Q → type-De-Morgan-Prop P) → + P ＝ Q + eq-iff-De-Morgan-Prop f g = eq-iff-De-Morgan-Prop' (f , g) +``` + +### De Morgan propositions are closed under retracts + +```agda +is-de-morgan-prop-retract-of : + {l1 l2 : Level} {P : UU l1} {Q : UU l2} → + P retract-of Q → is-de-morgan-prop Q → is-de-morgan-prop P +is-de-morgan-prop-retract-of R H = + is-prop-retract-of R (is-prop-type-is-de-morgan-prop H) , + is-de-morgan-iff' (iff-retract' R) (is-de-morgan-type-is-de-morgan-prop H) +``` + +### De Morgan propositions are closed under equivalences + +```agda +is-de-morgan-prop-equiv : + {l1 l2 : Level} {P : UU l1} {Q : UU l2} → + P ≃ Q → is-de-morgan-prop Q → is-de-morgan-prop P +is-de-morgan-prop-equiv e = is-de-morgan-prop-retract-of (retract-equiv e) + +is-de-morgan-prop-equiv' : + {l1 l2 : Level} {P : UU l1} {Q : UU l2} → + Q ≃ P → is-de-morgan-prop Q → is-de-morgan-prop P +is-de-morgan-prop-equiv' e = is-de-morgan-prop-retract-of (retract-inv-equiv e) +``` + +### Irrefutable propositions are De Morgan + +```agda +is-de-morgan-prop-is-irrefutable-prop : + {l : Level} {P : UU l} → is-irrefutable-prop P → is-de-morgan-prop P +is-de-morgan-prop-is-irrefutable-prop = tot (λ _ → is-de-morgan-is-irrefutable) +``` + +### Contractible types are De Morgan propositions + +```agda +is-de-morgan-prop-is-contr : + {l : Level} {P : UU l} → is-contr P → is-de-morgan-prop P +is-de-morgan-prop-is-contr H = + is-de-morgan-prop-is-irrefutable-prop (is-irrefutable-prop-is-contr H) +``` + +### Empty types are De Morgan propositions + +```agda +is-de-morgan-prop-is-empty : + {l : Level} {P : UU l} → is-empty P → is-de-morgan-prop P +is-de-morgan-prop-is-empty H = is-prop-is-empty H , is-de-morgan-is-empty H +``` + +### Dependent sums of De Morgan propositions over decidable propositions + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + is-de-morgan-prop-Σ' : + is-decidable-prop A → ((x : A) → is-de-morgan (B x)) → is-de-morgan (Σ A B) + is-de-morgan-prop-Σ' (is-prop-A , inl a) b = + rec-coproduct + ( λ nb → inl λ ab → nb (tr B (eq-is-prop is-prop-A) (pr2 ab))) + ( λ x → inr (λ z → x (λ b → z (a , b)))) + ( b a) + is-de-morgan-prop-Σ' (is-prop-A , inr na) b = inl (λ ab → na (pr1 ab)) + + is-de-morgan-prop-Σ : + is-decidable-prop A → + ((x : A) → is-de-morgan-prop (B x)) → + is-de-morgan-prop (Σ A B) + is-de-morgan-prop-Σ a b = + ( is-prop-Σ + ( is-prop-type-is-decidable-prop a) + ( is-prop-type-is-de-morgan-prop ∘ b)) , + ( is-de-morgan-prop-Σ' a (is-de-morgan-type-is-de-morgan-prop ∘ b)) +``` + +### The negation operation on decidable propositions + +```agda +is-de-morgan-prop-neg : + {l1 : Level} {A : UU l1} → is-de-morgan A → is-de-morgan-prop (¬ A) +is-de-morgan-prop-neg is-de-morgan-A = + ( is-prop-neg , is-de-morgan-neg is-de-morgan-A) + +neg-type-De-Morgan-Prop : + {l1 : Level} (A : UU l1) → is-de-morgan A → De-Morgan-Prop l1 +neg-type-De-Morgan-Prop A is-de-morgan-A = + ( ¬ A , is-de-morgan-prop-neg is-de-morgan-A) + +neg-De-Morgan-Prop : + {l1 : Level} → De-Morgan-Prop l1 → De-Morgan-Prop l1 +neg-De-Morgan-Prop P = + neg-type-De-Morgan-Prop + ( type-De-Morgan-Prop P) + ( is-de-morgan-type-De-Morgan-Prop P) + +type-neg-De-Morgan-Prop : + {l1 : Level} → De-Morgan-Prop l1 → UU l1 +type-neg-De-Morgan-Prop P = type-De-Morgan-Prop (neg-De-Morgan-Prop P) +``` + +### Propositional truncations of De Morgan types are De Morgan propositions + +```agda +module _ + {l1 : Level} {A : UU l1} + where + + is-de-morgan-prop-trunc-Prop : + is-de-morgan A → is-de-morgan-prop (type-trunc-Prop A) + is-de-morgan-prop-trunc-Prop a = + ( is-prop-type-trunc-Prop , is-de-morgan-trunc a) +``` + +### Disjunctions of De Morgan types are De Morgan propositions + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-de-morgan-prop-disjunction : + is-de-morgan A → is-de-morgan B → is-de-morgan-prop (disjunction-type A B) + is-de-morgan-prop-disjunction a b = + is-de-morgan-prop-trunc-Prop (is-de-morgan-coproduct a b) + +module _ + {l1 l2 : Level} (P : De-Morgan-Prop l1) (Q : De-Morgan-Prop l2) + where + + disjunction-De-Morgan-Prop : De-Morgan-Prop (l1 ⊔ l2) + disjunction-De-Morgan-Prop = + disjunction-type (type-De-Morgan-Prop P) (type-De-Morgan-Prop Q) , + is-de-morgan-prop-disjunction + ( is-de-morgan-type-De-Morgan-Prop P) + ( is-de-morgan-type-De-Morgan-Prop Q) +``` + +### Negation has no fixed points on decidable propositions + +```agda +abstract + no-fixed-points-neg-De-Morgan-Prop : + {l : Level} (P : De-Morgan-Prop l) → + ¬ (type-De-Morgan-Prop P ↔ ¬ (type-De-Morgan-Prop P)) + no-fixed-points-neg-De-Morgan-Prop P = + no-fixed-points-neg (type-De-Morgan-Prop P) +``` + +## External links + +- [De Morgan laws, in constructive mathematics](https://ncatlab.org/nlab/show/De+Morgan+laws#in_constructive_mathematics) + at $n$Lab diff --git a/src/logic/de-morgan-subtypes.lagda.md b/src/logic/de-morgan-subtypes.lagda.md new file mode 100644 index 0000000000..8981009f85 --- /dev/null +++ b/src/logic/de-morgan-subtypes.lagda.md @@ -0,0 +1,334 @@ +# De Morgan subtypes + +```agda +module logic.de-morgan-subtypes where +``` + +
Imports + +```agda +open import foundation.1-types +open import foundation.coproduct-types +open import foundation.dependent-pair-types +open import foundation.equality-dependent-function-types +open import foundation.functoriality-cartesian-product-types +open import foundation.functoriality-dependent-pair-types +open import foundation.logical-equivalences +open import foundation.propositional-maps +open import foundation.sets +open import foundation.structured-type-duality +open import foundation.subtypes +open import foundation.type-theoretic-principle-of-choice +open import foundation.universe-levels + +open import foundation-core.embeddings +open import foundation-core.equivalences +open import foundation-core.fibers-of-maps +open import foundation-core.function-types +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 + +open import logic.de-morgan-embeddings +open import logic.de-morgan-maps +open import logic.de-morgan-propositions +open import logic.de-morgan-types +``` + +
+ +## Idea + +A +{{#concept "De Morgan subtype" Disambiguation="of a type" Agda=is-de-morgan-subtype Agda=de-morgan-subtype}} +of a type consists of a family of +[De Morgan propositions](logic.de-morgan-propositions.md) over it. + +## Definitions + +### De Morgan subtypes + +```agda +is-de-morgan-subtype-Prop : + {l1 l2 : Level} {A : UU l1} → subtype l2 A → Prop (l1 ⊔ l2) +is-de-morgan-subtype-Prop {A = A} P = + Π-Prop A (λ a → is-de-morgan-Prop (type-Prop (P a))) + +is-de-morgan-subtype : + {l1 l2 : Level} {A : UU l1} → subtype l2 A → UU (l1 ⊔ l2) +is-de-morgan-subtype P = + type-Prop (is-de-morgan-subtype-Prop P) + +is-prop-is-de-morgan-subtype : + {l1 l2 : Level} {A : UU l1} (P : subtype l2 A) → + is-prop (is-de-morgan-subtype P) +is-prop-is-de-morgan-subtype P = + is-prop-type-Prop (is-de-morgan-subtype-Prop P) + +de-morgan-subtype : + {l1 : Level} (l : Level) (X : UU l1) → UU (l1 ⊔ lsuc l) +de-morgan-subtype l X = X → De-Morgan-Prop l +``` + +### The underlying subtype of a De Morgan subtype + +```agda +module _ + {l1 l2 : Level} {A : UU l1} (P : de-morgan-subtype l2 A) + where + + subtype-de-morgan-subtype : subtype l2 A + subtype-de-morgan-subtype a = + prop-De-Morgan-Prop (P a) + + is-de-morgan-de-morgan-subtype : + is-de-morgan-subtype subtype-de-morgan-subtype + is-de-morgan-de-morgan-subtype a = + is-de-morgan-type-De-Morgan-Prop (P a) + + is-in-de-morgan-subtype : A → UU l2 + is-in-de-morgan-subtype = + is-in-subtype subtype-de-morgan-subtype + + is-prop-is-in-de-morgan-subtype : + (a : A) → is-prop (is-in-de-morgan-subtype a) + is-prop-is-in-de-morgan-subtype = + is-prop-is-in-subtype subtype-de-morgan-subtype +``` + +### The underlying type of a De Morgan subtype + +```agda +module _ + {l1 l2 : Level} {A : UU l1} (P : de-morgan-subtype l2 A) + where + + type-de-morgan-subtype : UU (l1 ⊔ l2) + type-de-morgan-subtype = + type-subtype (subtype-de-morgan-subtype P) + + inclusion-de-morgan-subtype : + type-de-morgan-subtype → A + inclusion-de-morgan-subtype = + inclusion-subtype (subtype-de-morgan-subtype P) + + is-emb-inclusion-de-morgan-subtype : + is-emb inclusion-de-morgan-subtype + is-emb-inclusion-de-morgan-subtype = + is-emb-inclusion-subtype (subtype-de-morgan-subtype P) + + is-de-morgan-map-inclusion-de-morgan-subtype : + is-de-morgan-map inclusion-de-morgan-subtype + is-de-morgan-map-inclusion-de-morgan-subtype x = + is-de-morgan-equiv + ( equiv-fiber-pr1 (type-De-Morgan-Prop ∘ P) x) + ( is-de-morgan-type-De-Morgan-Prop (P x)) + + is-injective-inclusion-de-morgan-subtype : + is-injective inclusion-de-morgan-subtype + is-injective-inclusion-de-morgan-subtype = + is-injective-inclusion-subtype (subtype-de-morgan-subtype P) + + emb-de-morgan-subtype : type-de-morgan-subtype ↪ A + emb-de-morgan-subtype = + emb-subtype (subtype-de-morgan-subtype P) + + is-de-morgan-emb-inclusion-de-morgan-subtype : + is-de-morgan-emb inclusion-de-morgan-subtype + is-de-morgan-emb-inclusion-de-morgan-subtype = + ( is-emb-inclusion-de-morgan-subtype , + is-de-morgan-map-inclusion-de-morgan-subtype) + + de-morgan-emb-de-morgan-subtype : + type-de-morgan-subtype ↪ᵈᵐ A + de-morgan-emb-de-morgan-subtype = + ( inclusion-de-morgan-subtype , + is-de-morgan-emb-inclusion-de-morgan-subtype) +``` + +### The De Morgan subtype associated to a De Morgan embedding + +```agda +module _ + {l1 l2 : Level} {X : UU l1} {Y : UU l2} (f : X ↪ᵈᵐ Y) + where + + de-morgan-subtype-de-morgan-emb : + de-morgan-subtype (l1 ⊔ l2) Y + pr1 (de-morgan-subtype-de-morgan-emb y) = + fiber (map-de-morgan-emb f) y + pr2 (de-morgan-subtype-de-morgan-emb y) = + is-de-morgan-prop-map-is-de-morgan-emb + ( is-de-morgan-emb-map-de-morgan-emb f) + ( y) + + compute-type-de-morgan-type-de-morgan-emb : + type-de-morgan-subtype + de-morgan-subtype-de-morgan-emb ≃ + X + compute-type-de-morgan-type-de-morgan-emb = + equiv-total-fiber (map-de-morgan-emb f) + + inv-compute-type-de-morgan-type-de-morgan-emb : + X ≃ + type-de-morgan-subtype + de-morgan-subtype-de-morgan-emb + inv-compute-type-de-morgan-type-de-morgan-emb = + inv-equiv-total-fiber (map-de-morgan-emb f) +``` + +## Examples + +### The De Morgan subtypes of left and right elements in a coproduct type + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-de-morgan-is-left : + (x : A + B) → is-de-morgan (is-left x) + is-de-morgan-is-left (inl x) = is-de-morgan-unit + is-de-morgan-is-left (inr x) = is-de-morgan-empty + + is-left-De-Morgan-Prop : + A + B → De-Morgan-Prop lzero + pr1 (is-left-De-Morgan-Prop x) = is-left x + pr1 (pr2 (is-left-De-Morgan-Prop x)) = is-prop-is-left x + pr2 (pr2 (is-left-De-Morgan-Prop x)) = + is-de-morgan-is-left x + + is-de-morgan-is-right : + (x : A + B) → is-de-morgan (is-right x) + is-de-morgan-is-right (inl x) = is-de-morgan-empty + is-de-morgan-is-right (inr x) = is-de-morgan-unit + + is-right-De-Morgan-Prop : + A + B → De-Morgan-Prop lzero + pr1 (is-right-De-Morgan-Prop x) = is-right x + pr1 (pr2 (is-right-De-Morgan-Prop x)) = is-prop-is-right x + pr2 (pr2 (is-right-De-Morgan-Prop x)) = + is-de-morgan-is-right x +``` + +## Properties + +### A De Morgan subtype of a `k+1`-truncated type is `k+1`-truncated + +```agda +module _ + {l1 l2 : Level} (k : 𝕋) {A : UU l1} (P : de-morgan-subtype l2 A) + where + + abstract + is-trunc-type-de-morgan-subtype : + is-trunc (succ-𝕋 k) A → is-trunc (succ-𝕋 k) + (type-de-morgan-subtype P) + is-trunc-type-de-morgan-subtype = + is-trunc-type-subtype k (subtype-de-morgan-subtype P) + +module _ + {l1 l2 : Level} {A : UU l1} (P : de-morgan-subtype l2 A) + where + + abstract + is-prop-type-de-morgan-subtype : + is-prop A → is-prop (type-de-morgan-subtype P) + is-prop-type-de-morgan-subtype = + is-prop-type-subtype (subtype-de-morgan-subtype P) + + abstract + is-set-type-de-morgan-subtype : + is-set A → is-set (type-de-morgan-subtype P) + is-set-type-de-morgan-subtype = + is-set-type-subtype (subtype-de-morgan-subtype P) + + abstract + is-1-type-type-de-morgan-subtype : + is-1-type A → is-1-type (type-de-morgan-subtype P) + is-1-type-type-de-morgan-subtype = + is-1-type-type-subtype (subtype-de-morgan-subtype P) + +prop-de-morgan-subprop : + {l1 l2 : Level} (A : Prop l1) + (P : de-morgan-subtype l2 (type-Prop A)) → + Prop (l1 ⊔ l2) +prop-de-morgan-subprop A P = + prop-subprop A (subtype-de-morgan-subtype P) + +set-de-morgan-subset : + {l1 l2 : Level} (A : Set l1) + (P : de-morgan-subtype l2 (type-Set A)) → + Set (l1 ⊔ l2) +set-de-morgan-subset A P = + set-subset A (subtype-de-morgan-subtype P) +``` + +### The type of De Morgan subtypes of a type is a set + +```agda +is-set-de-morgan-subtype : + {l1 l2 : Level} {X : UU l1} → is-set (de-morgan-subtype l2 X) +is-set-de-morgan-subtype = + is-set-function-type is-set-De-Morgan-Prop +``` + +### Extensionality of the type of De Morgan subtypes + +```agda +module _ + {l1 l2 : Level} {A : UU l1} (P : de-morgan-subtype l2 A) + where + + has-same-elements-de-morgan-subtype : + {l3 : Level} → de-morgan-subtype l3 A → UU (l1 ⊔ l2 ⊔ l3) + has-same-elements-de-morgan-subtype Q = + has-same-elements-subtype + ( subtype-de-morgan-subtype P) + ( subtype-de-morgan-subtype Q) + + extensionality-de-morgan-subtype : + (Q : de-morgan-subtype l2 A) → + (P ＝ Q) ≃ has-same-elements-de-morgan-subtype Q + extensionality-de-morgan-subtype = + extensionality-Π P + ( λ x Q → + ( type-De-Morgan-Prop (P x)) ↔ + ( type-De-Morgan-Prop Q)) + ( λ x Q → extensionality-De-Morgan-Prop (P x) Q) + + has-same-elements-eq-de-morgan-subtype : + (Q : de-morgan-subtype l2 A) → + (P ＝ Q) → has-same-elements-de-morgan-subtype Q + has-same-elements-eq-de-morgan-subtype Q = + map-equiv (extensionality-de-morgan-subtype Q) + + eq-has-same-elements-de-morgan-subtype : + (Q : de-morgan-subtype l2 A) → + has-same-elements-de-morgan-subtype Q → P ＝ Q + eq-has-same-elements-de-morgan-subtype Q = + map-inv-equiv (extensionality-de-morgan-subtype Q) + + refl-extensionality-de-morgan-subtype : + map-equiv (extensionality-de-morgan-subtype P) refl ＝ + (λ x → id , id) + refl-extensionality-de-morgan-subtype = refl +``` + +### The type of De Morgan subtypes of `A` is equivalent to the type of all De Morgan embeddings into a type `A` + +```agda +equiv-Fiber-De-Morgan-Prop : + (l : Level) {l1 : Level} (A : UU l1) → + Σ (UU (l1 ⊔ l)) (λ X → X ↪ᵈᵐ A) ≃ (de-morgan-subtype (l1 ⊔ l) A) +equiv-Fiber-De-Morgan-Prop l A = + ( equiv-Fiber-structure l is-de-morgan-prop A) ∘e + ( equiv-tot + ( λ X → + equiv-tot + ( λ f → + ( inv-distributive-Π-Σ) ∘e + ( equiv-product-left (equiv-is-prop-map-is-emb f))))) +``` diff --git a/src/logic/de-morgan-types.lagda.md b/src/logic/de-morgan-types.lagda.md new file mode 100644 index 0000000000..a87cd18593 --- /dev/null +++ b/src/logic/de-morgan-types.lagda.md @@ -0,0 +1,466 @@ +# De Morgan types + +```agda +module logic.de-morgan-types where +``` + +
Imports + +```agda +open import foundation.cartesian-product-types +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.disjunction +open import foundation.double-negation +open import foundation.empty-types +open import foundation.evaluation-functions +open import foundation.function-types +open import foundation.identity-types +open import foundation.irrefutable-propositions +open import foundation.logical-equivalences +open import foundation.negation +open import foundation.precomposition-functions +open import foundation.propositional-truncations +open import foundation.retracts-of-types +open import foundation.truncation-levels +open import foundation.truncations +open import foundation.unit-type +open import foundation.universe-levels + +open import foundation-core.decidable-propositions +open import foundation-core.equivalences +open import foundation-core.propositions + +open import logic.de-morgans-law +``` + +
+ +## Idea + +In classical logic, i.e., logic where we assume +[the law of excluded middle](foundation.law-of-excluded-middle.md), the _De +Morgan laws_ refers to the pair of logical equivalences + +```text + ¬ (P ∨ Q) ⇔ (¬ P) ∧ (¬ Q) + ¬ (P ∧ Q) ⇔ (¬ P) ∨ (¬ Q). +``` + +Out of these in total four logical implications, all but one are validated in +constructive mathematics. The odd one out is + +```text + ¬ (P ∧ Q) ⇒ (¬ P) ∨ (¬ Q). +``` + +Indeed, this would state that we could constructively deduce from a proof that +not both of `P` and `Q` are true, which of `P` and `Q` that is false. This +logical law is what we refer to as [De Morgan's Law](logic.de-morgans-law.md). +If a type `P` is such that for every other type `Q`, _the De Morgan implication_ + +```text + ¬ (P ∧ Q) ⇒ (¬ P) ∨ (¬ Q) +``` + +holds, we say `P` is +{{#concept "De Morgan" Disambiguation="type" Agda=is-de-morgan Agda=De-Morgan-Type Agda=satisfies-de-morgans-law-type}}. + +Equivalently, a type is De Morgan [iff](foundation.logical-equivalences.md) its +[negation](foundation-core.negation.md) is +[decidable](foundation.decidable-types.md). Since this is a +[small](foundation.small-types.md) condition, it is frequently more convenient +to use and is what we take as the main definition. + +## Definition + +### The small condition of being a De Morgan type + +I.e., types whose negation is decidable. + +```agda +module _ + {l : Level} (A : UU l) + where + + is-de-morgan : UU l + is-de-morgan = is-decidable (¬ A) + + is-prop-is-de-morgan : is-prop is-de-morgan + is-prop-is-de-morgan = is-prop-is-decidable is-prop-neg + + is-de-morgan-Prop : Prop l + is-de-morgan-Prop = is-decidable-Prop (neg-type-Prop A) +``` + +### The subuniverse of De Morgan types + +We use the decidability of the negation condition to define the subuniverse of +De Morgan types. + +```agda +De-Morgan-Type : (l : Level) → UU (lsuc l) +De-Morgan-Type l = Σ (UU l) (is-de-morgan) + +module _ + {l : Level} (A : De-Morgan-Type l) + where + + type-De-Morgan-Type : UU l + type-De-Morgan-Type = pr1 A + + is-de-morgan-type-De-Morgan-Type : is-de-morgan type-De-Morgan-Type + is-de-morgan-type-De-Morgan-Type = pr2 A +``` + +### Types that satisfy De Morgan's law + +```agda +satisfies-de-morgans-law-type-Level : + {l1 : Level} (l2 : Level) (A : UU l1) → UU (l1 ⊔ lsuc l2) +satisfies-de-morgans-law-type-Level l2 A = + (B : UU l2) → ¬ (A × B) → disjunction-type (¬ A) (¬ B) + +satisfies-de-morgans-law-type : {l1 : Level} (A : UU l1) → UUω +satisfies-de-morgans-law-type A = + {l2 : Level} (B : UU l2) → ¬ (A × B) → disjunction-type (¬ A) (¬ B) + +is-prop-satisfies-de-morgans-law-type-Level : + {l1 l2 : Level} {A : UU l1} → + is-prop (satisfies-de-morgans-law-type-Level l2 A) +is-prop-satisfies-de-morgans-law-type-Level {A = A} = + is-prop-Π (λ B → is-prop-Π (λ p → is-prop-disjunction-type (¬ A) (¬ B))) + +satisfies-de-morgans-law-type-Prop : + {l1 : Level} (l2 : Level) (A : UU l1) → Prop (l1 ⊔ lsuc l2) +satisfies-de-morgans-law-type-Prop l2 A = + ( satisfies-de-morgans-law-type-Level l2 A , + is-prop-satisfies-de-morgans-law-type-Level) +``` + +```agda +satisfies-de-morgans-law-type-Level' : + {l1 : Level} (l2 : Level) (A : UU l1) → UU (l1 ⊔ lsuc l2) +satisfies-de-morgans-law-type-Level' l2 A = + (B : UU l2) → ¬ (B × A) → disjunction-type (¬ B) (¬ A) + +satisfies-de-morgans-law-type' : {l1 : Level} (A : UU l1) → UUω +satisfies-de-morgans-law-type' A = + {l2 : Level} (B : UU l2) → ¬ (B × A) → disjunction-type (¬ B) (¬ A) + +is-prop-satisfies-de-morgans-law-type-Level' : + {l1 l2 : Level} {A : UU l1} → + is-prop (satisfies-de-morgans-law-type-Level' l2 A) +is-prop-satisfies-de-morgans-law-type-Level' {A = A} = + is-prop-Π (λ B → is-prop-Π (λ p → is-prop-disjunction-type (¬ B) (¬ A))) + +satisfies-de-morgans-law-type-Prop' : + {l1 : Level} (l2 : Level) (A : UU l1) → Prop (l1 ⊔ lsuc l2) +satisfies-de-morgans-law-type-Prop' l2 A = + ( satisfies-de-morgans-law-type-Level' l2 A , + is-prop-satisfies-de-morgans-law-type-Level') +``` + +## Properties + +### If a type satisfies De Morgan's law then its negation is decidable + +Indeed, one need only check that `A` and `¬ A` satisfy De Morgan's law, as then +the hypothesis of the implication + +```text + ¬ (A ∧ ¬ A) ⇒ ¬ A ∨ ¬¬ A +``` + +is true. + +```agda +module _ + {l : Level} (A : UU l) + where + + is-de-morgan-satisfies-de-morgans-law' : + ({l' : Level} (B : UU l') → ¬ (A × B) → ¬ A + ¬ B) → is-de-morgan A + is-de-morgan-satisfies-de-morgans-law' H = H (¬ A) (λ f → pr2 f (pr1 f)) + + is-merely-decidable-neg-satisfies-de-morgans-law : + satisfies-de-morgans-law-type A → is-merely-decidable (¬ A) + is-merely-decidable-neg-satisfies-de-morgans-law H = + H (¬ A) (λ f → pr2 f (pr1 f)) + + is-de-morgan-satisfies-de-morgans-law : + satisfies-de-morgans-law-type A → is-de-morgan A + is-de-morgan-satisfies-de-morgans-law H = + rec-trunc-Prop + ( is-decidable-Prop (neg-type-Prop A)) + ( id) + ( H (¬ A) (λ f → pr2 f (pr1 f))) +``` + +### If the negation of a type is decidable then it satisfies De Morgan's law + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + satisfies-de-morgans-law-is-de-morgan-left : + is-de-morgan A → ¬ (A × B) → ¬ A + ¬ B + satisfies-de-morgans-law-is-de-morgan-left (inl na) f = + inl na + satisfies-de-morgans-law-is-de-morgan-left (inr nna) f = + inr (λ y → nna (λ x → f (x , y))) + + satisfies-de-morgans-law-is-de-morgan-right : + is-de-morgan B → ¬ (A × B) → ¬ A + ¬ B + satisfies-de-morgans-law-is-de-morgan-right (inl nb) f = + inr nb + satisfies-de-morgans-law-is-de-morgan-right (inr nnb) f = + inl (λ x → nnb (λ y → f (x , y))) + +satisfies-de-morgans-law-is-de-morgan : + {l : Level} {A : UU l} → is-de-morgan A → satisfies-de-morgans-law-type A +satisfies-de-morgans-law-is-de-morgan x B q = + unit-trunc-Prop (satisfies-de-morgans-law-is-de-morgan-left x q) + +satisfies-de-morgans-law-is-de-morgan' : + {l : Level} {A : UU l} → is-de-morgan A → satisfies-de-morgans-law-type' A +satisfies-de-morgans-law-is-de-morgan' x B q = + unit-trunc-Prop (satisfies-de-morgans-law-is-de-morgan-right x q) + +module _ + {l : Level} (A : De-Morgan-Type l) + where + + satisfies-de-morgans-law-type-De-Morgan-Type : + satisfies-de-morgans-law-type (type-De-Morgan-Type A) + satisfies-de-morgans-law-type-De-Morgan-Type = + satisfies-de-morgans-law-is-de-morgan (is-de-morgan-type-De-Morgan-Type A) + + satisfies-de-morgans-law-type-De-Morgan-Type' : + satisfies-de-morgans-law-type' (type-De-Morgan-Type A) + satisfies-de-morgans-law-type-De-Morgan-Type' = + satisfies-de-morgans-law-is-de-morgan' (is-de-morgan-type-De-Morgan-Type A) +``` + +### It is irrefutable that a type is De Morgan + +```agda +is-irrefutable-is-de-morgan : {l : Level} {A : UU l} → ¬¬ (is-de-morgan A) +is-irrefutable-is-de-morgan = is-irrefutable-is-decidable +``` + +### Decidable types are De Morgan + +```agda +is-de-morgan-is-decidable : + {l : Level} {A : UU l} → is-decidable A → is-de-morgan A +is-de-morgan-is-decidable (inl x) = inr (intro-double-negation x) +is-de-morgan-is-decidable (inr x) = inl x + +satisfies-de-morgans-law-is-decidable : + {l : Level} {A : UU l} → is-decidable A → satisfies-de-morgans-law-type A +satisfies-de-morgans-law-is-decidable H = + satisfies-de-morgans-law-is-de-morgan (is-de-morgan-is-decidable H) + +satisfies-de-morgans-law-is-decidable' : + {l : Level} {A : UU l} → is-decidable A → satisfies-de-morgans-law-type' A +satisfies-de-morgans-law-is-decidable' H = + satisfies-de-morgans-law-is-de-morgan' (is-de-morgan-is-decidable H) +``` + +### Irrefutable types are De Morgan + +```agda +is-de-morgan-is-irrefutable : + {l : Level} {A : UU l} → ¬¬ A → is-de-morgan A +is-de-morgan-is-irrefutable = inr +``` + +### Contractible types are De Morgan + +```agda +is-de-morgan-is-contr : + {l : Level} {A : UU l} → is-contr A → is-de-morgan A +is-de-morgan-is-contr H = + is-de-morgan-is-irrefutable (intro-double-negation (center H)) + +is-de-morgan-unit : is-de-morgan unit +is-de-morgan-unit = is-de-morgan-is-contr is-contr-unit +``` + +### Empty types are De Morgan + +```agda +is-de-morgan-is-empty : {l : Level} {A : UU l} → is-empty A → is-de-morgan A +is-de-morgan-is-empty = inl + +is-de-morgan-empty : is-de-morgan empty +is-de-morgan-empty = is-de-morgan-is-empty id +``` + +### De Morgan types are closed under logical equivalences + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-de-morgan-iff : + (A → B) → (B → A) → is-de-morgan A → is-de-morgan B + is-de-morgan-iff f g = is-decidable-iff (map-neg g) (map-neg f) + + is-de-morgan-iff' : + (A ↔ B) → is-de-morgan A → is-de-morgan B + is-de-morgan-iff' (f , g) = is-de-morgan-iff f g + + satisfies-de-morgans-law-iff : + (A → B) → (B → A) → + satisfies-de-morgans-law-type A → satisfies-de-morgans-law-type B + satisfies-de-morgans-law-iff f g a = + satisfies-de-morgans-law-is-de-morgan + ( is-de-morgan-iff f g (is-de-morgan-satisfies-de-morgans-law A a)) + + satisfies-de-morgans-law-iff' : + (A ↔ B) → satisfies-de-morgans-law-type A → satisfies-de-morgans-law-type B + satisfies-de-morgans-law-iff' (f , g) = satisfies-de-morgans-law-iff f g +``` + +### De Morgan types are closed under retracts + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (R : B retract-of A) + where + + is-de-morgan-retract-of : is-de-morgan A → is-de-morgan B + is-de-morgan-retract-of = is-de-morgan-iff' (iff-retract' R) + + is-de-morgan-retract-of' : is-de-morgan B → is-de-morgan A + is-de-morgan-retract-of' = is-de-morgan-iff' (iff-retract R) +``` + +### De Morgan types are closed under equivalences + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (e : A ≃ B) + where + + is-de-morgan-equiv' : is-de-morgan A → is-de-morgan B + is-de-morgan-equiv' = is-de-morgan-iff' (iff-equiv e) + + is-de-morgan-equiv : is-de-morgan B → is-de-morgan A + is-de-morgan-equiv = is-de-morgan-iff' (iff-equiv' e) +``` + +### Equivalent types have equivalent De Morgan predicates + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + iff-is-de-morgan : A ↔ B → is-de-morgan A ↔ is-de-morgan B + iff-is-de-morgan e = iff-is-decidable (iff-neg e) + + equiv-iff-is-de-morgan : A ↔ B → is-de-morgan A ≃ is-de-morgan B + equiv-iff-is-de-morgan e = equiv-is-decidable (equiv-iff-neg e) + + equiv-is-de-morgan : A ≃ B → is-de-morgan A ≃ is-de-morgan B + equiv-is-de-morgan e = equiv-iff-is-de-morgan (iff-equiv e) +``` + +### The truncation of a De Morgan type is De Morgan + +```agda +module _ + {l1 : Level} {A : UU l1} + where + + is-de-morgan-trunc : {k : 𝕋} → is-de-morgan A → is-de-morgan (type-trunc k A) + is-de-morgan-trunc {neg-two-𝕋} a = + is-de-morgan-is-contr is-trunc-type-trunc + is-de-morgan-trunc {succ-𝕋 k} (inl na) = + inl (map-universal-property-trunc (empty-Truncated-Type k) na) + is-de-morgan-trunc {succ-𝕋 k} (inr nna) = + inr (λ nn|a| → nna (λ a → nn|a| (unit-trunc a))) +``` + +### If the truncation of a type is De Morgan then the type is De Morgan + +```agda +module _ + {l1 : Level} {A : UU l1} + where + + equiv-is-de-morgan-trunc : + {k : 𝕋} → is-de-morgan (type-trunc (succ-𝕋 k) A) ≃ is-de-morgan A + equiv-is-de-morgan-trunc {k} = + equiv-is-decidable + ( map-neg unit-trunc , is-truncation-trunc (empty-Truncated-Type k)) + + is-de-morgan-is-de-morgan-trunc : + {k : 𝕋} → is-de-morgan (type-trunc (succ-𝕋 k) A) → is-de-morgan A + is-de-morgan-is-de-morgan-trunc = map-equiv equiv-is-de-morgan-trunc +``` + +### Products of De Morgan types are De Morgan + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-de-morgan-product : is-de-morgan A → is-de-morgan B → is-de-morgan (A × B) + is-de-morgan-product (inl na) b = + inl (λ ab → na (pr1 ab)) + is-de-morgan-product (inr nna) (inl nb) = + inl (λ ab → nb (pr2 ab)) + is-de-morgan-product (inr nna) (inr nnb) = + inr (is-irrefutable-product nna nnb) +``` + +### Coproducts of De Morgan types are De Morgan + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-de-morgan-coproduct : + is-de-morgan A → is-de-morgan B → is-de-morgan (A + B) + is-de-morgan-coproduct (inl na) (inl nb) = + inl (rec-coproduct na nb) + is-de-morgan-coproduct (inl na) (inr nnb) = + inr (λ nab → nnb (λ nb → nab (inr nb))) + is-de-morgan-coproduct (inr nna) _ = + inr (λ nab → nna (λ na → nab (inl na))) + + is-de-morgan-disjunction : + is-de-morgan A → is-de-morgan B → is-de-morgan (disjunction-type A B) + is-de-morgan-disjunction a b = is-de-morgan-trunc (is-de-morgan-coproduct a b) +``` + +### The negation of a De Morgan type is De Morgan + +```agda +is-de-morgan-neg : {l : Level} {A : UU l} → is-de-morgan A → is-de-morgan (¬ A) +is-de-morgan-neg = is-decidable-neg +``` + +### The identity types of De Morgan types are not generally De Morgan + +Consider any type `A`, then its suspension `ΣA` is De Morgan since it is +inhabited. However, its identity type `N ＝ S` is equivalent to `A`, so cannot +be De Morgan unless `A` is. + +> This remains to be formalized. + +## External links + +- [De Morgan laws, in constructive mathematics](https://ncatlab.org/nlab/show/De+Morgan+laws#in_constructive_mathematics) + at $n$Lab diff --git a/src/logic/de-morgans-law.lagda.md b/src/logic/de-morgans-law.lagda.md new file mode 100644 index 0000000000..b680aef549 --- /dev/null +++ b/src/logic/de-morgans-law.lagda.md @@ -0,0 +1,144 @@ +# De Morgan's law + +```agda +module logic.de-morgans-law where +``` + +
Imports + +```agda +open import foundation.cartesian-product-types +open import foundation.conjunction +open import foundation.coproduct-types +open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.disjunction +open import foundation.double-negation +open import foundation.empty-types +open import foundation.evaluation-functions +open import foundation.function-types +open import foundation.logical-equivalences +open import foundation.negation +open import foundation.universe-levels + +open import foundation-core.decidable-propositions +open import foundation-core.propositions + +open import univalent-combinatorics.2-element-types +``` + +
+ +## Idea + +In classical logic, i.e., logic where we assume +[the law of excluded middle](foundation.law-of-excluded-middle.md), the _De +Morgan laws_ refers to the pair of logical equivalences + +```text + ¬ (P ∨ Q) ⇔ (¬ P) ∧ (¬ Q) + ¬ (P ∧ Q) ⇔ (¬ P) ∨ (¬ Q). +``` + +Out of these in total four logical implications, all but one are validated in +constructive mathematics. The odd one out is + +```text + ¬ (P ∧ Q) ⇒ (¬ P) ∨ (¬ Q). +``` + +Indeed, this would state that we could constructively deduce from a proof that +not both of `P` and `Q` are true, which of `P` and `Q` that is false. This +logical law is what we refer to as +{{#concept "De Morgan's Law" Agda=De-Morgans-Law}}. + +## Definition + +```agda +module _ + {l1 l2 : Level} (A : UU l1) (B : UU l2) + where + + de-morgans-law-Prop' : Prop (l1 ⊔ l2) + de-morgans-law-Prop' = + neg-type-Prop (A × B) ⇒ (neg-type-Prop A) ∨ (neg-type-Prop B) + + de-morgans-law' : UU (l1 ⊔ l2) + de-morgans-law' = ¬ (A × B) → disjunction-type (¬ A) (¬ B) + +module _ + {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) + where + + de-morgans-law-Prop : Prop (l1 ⊔ l2) + de-morgans-law-Prop = ¬' (P ∧ Q) ⇒ (¬' P) ∨ (¬' Q) + + de-morgans-law : UU (l1 ⊔ l2) + de-morgans-law = type-Prop de-morgans-law-Prop + +De-Morgans-Law-Level : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) +De-Morgans-Law-Level l1 l2 = + (P : Prop l1) (Q : Prop l2) → de-morgans-law P Q + +prop-De-Morgans-Law-Level : (l1 l2 : Level) → Prop (lsuc l1 ⊔ lsuc l2) +prop-De-Morgans-Law-Level l1 l2 = + Π-Prop + ( Prop l1) + ( λ P → Π-Prop (Prop l2) (λ Q → de-morgans-law-Prop P Q)) + +De-Morgans-Law : UUω +De-Morgans-Law = {l1 l2 : Level} → De-Morgans-Law-Level l1 l2 +``` + +## Properties + +### The constructively valid De Morgan's laws + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + forward-implication-constructive-de-morgan : ¬ (A + B) → (¬ A) × (¬ B) + forward-implication-constructive-de-morgan z = (z ∘ inl) , (z ∘ inr) + + backward-implication-constructive-de-morgan : (¬ A) × (¬ B) → ¬ (A + B) + backward-implication-constructive-de-morgan (na , nb) (inl x) = na x + backward-implication-constructive-de-morgan (na , nb) (inr y) = nb y + + constructive-de-morgan : ¬ (A + B) ↔ (¬ A) × (¬ B) + constructive-de-morgan = + ( forward-implication-constructive-de-morgan , + backward-implication-constructive-de-morgan) + + backward-implication-de-morgan : ¬ A + ¬ B → ¬ (A × B) + backward-implication-de-morgan (inl na) (x , y) = na x + backward-implication-de-morgan (inr nb) (x , y) = nb y +``` + +### Given the hypothesis of De Morgan's law, the conclusion is irrefutable + +```agda +double-negation-de-morgan : + {l1 l2 : Level} {A : UU l1} {B : UU l2} → ¬ (A × B) → ¬¬ (¬ A + ¬ B) +double-negation-de-morgan f v = + v (inl (λ x → v (inr (λ y → f (x , y))))) +``` + +### De Morgan's law is irrefutable + +```agda +is-irrefutable-de-morgans-law : + {l1 l2 : Level} {A : UU l1} {B : UU l2} → ¬¬ (¬ (A × B) → ¬ A + ¬ B) +is-irrefutable-de-morgans-law u = + u (λ _ → inl (λ x → u (λ f → inr (λ y → f (x , y))))) +``` + +## See also + +- [De Morgan types](logic.de-morgan-types.md) + +## External links + +- [De Morgan laws, in constructive mathematics](https://ncatlab.org/nlab/show/De+Morgan+laws#in_constructive_mathematics) + at $n$Lab diff --git a/src/logic/double-negation-eliminating-maps.lagda.md b/src/logic/double-negation-eliminating-maps.lagda.md new file mode 100644 index 0000000000..836e3191e1 --- /dev/null +++ b/src/logic/double-negation-eliminating-maps.lagda.md @@ -0,0 +1,335 @@ +# Double negation eliminating maps + +```agda +module logic.double-negation-eliminating-maps where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.cartesian-morphisms-arrows +open import foundation.coproduct-types +open import foundation.decidable-equality +open import foundation.decidable-maps +open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.double-negation +open import foundation.empty-types +open import foundation.functoriality-cartesian-product-types +open import foundation.functoriality-coproduct-types +open import foundation.identity-types +open import foundation.injective-maps +open import foundation.retractions +open import foundation.retracts-of-maps +open import foundation.retracts-of-types +open import foundation.transport-along-identifications +open import foundation.universe-levels + +open import foundation-core.contractible-maps +open import foundation-core.equivalences +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 logic.double-negation-elimination +``` + +
+ +## Idea + +A [map](foundation-core.function-types.md) is said to be +{{#concept "double negation eliminating" Disambiguation="map of types" Agda=is-double-negation-eliminating-map}} +if its [fibers](foundation-core.fibers-of-maps.md) satisfy +[untruncated double negation elimination](logic.double-negation-elimination.md). +I.e., for every `y : B`, if `fiber f y` is +[irrefutable](foundation.irrefutable-propositions.md), then we do in fact have +an element of the fiber `p : fiber f y`. In other words, double negation +eliminating maps come [equipped](foundation.structure.md) with a map + +```text + (y : B) → ¬¬ (fiber f y) → fiber f y. +``` + +## Definintion + +### Double negation elimination structure on a map + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-double-negation-eliminating-map : (A → B) → UU (l1 ⊔ l2) + is-double-negation-eliminating-map f = + (y : B) → has-double-negation-elim (fiber f y) +``` + +### The type of double negation eliminating maps + +```agda +infix 5 _→¬¬_ + +_→¬¬_ : {l1 l2 : Level} (A : UU l1) (B : UU l2) → UU (l1 ⊔ l2) +A →¬¬ B = Σ (A → B) (is-double-negation-eliminating-map) + +double-negation-eliminating-map : + {l1 l2 : Level} (A : UU l1) (B : UU l2) → UU (l1 ⊔ l2) +double-negation-eliminating-map = _→¬¬_ + +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A →¬¬ B) + where + + map-double-negation-eliminating-map : A → B + map-double-negation-eliminating-map = pr1 f + + is-double-negation-eliminating-double-negation-eliminating-map : + is-double-negation-eliminating-map map-double-negation-eliminating-map + is-double-negation-eliminating-double-negation-eliminating-map = pr2 f +``` + +## Properties + +### Double negation eliminating maps are closed under homotopy + +```agda +abstract + is-double-negation-eliminating-map-htpy : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f g : A → B} → + f ~ g → + is-double-negation-eliminating-map g → + is-double-negation-eliminating-map f + is-double-negation-eliminating-map-htpy H K b = + has-double-negation-elim-equiv + ( equiv-tot (λ a → equiv-concat (inv (H a)) b)) + ( K b) +``` + +### Decidable maps are double negation eliminating + +```agda +is-double-negation-eliminating-map-is-decidable-map : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} → + is-decidable-map f → is-double-negation-eliminating-map f +is-double-negation-eliminating-map-is-decidable-map H y = + double-negation-elim-is-decidable (H y) +``` + +### Composition of double negation eliminating maps + +Given a composition `g ∘ f` of double negation eliminating maps where the left +factor `g` is injective, then the composition is double negation eliminating. + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} + {g : B → C} {f : A → B} + where + + fiber-left-is-double-negation-eliminating-map-left : + is-double-negation-eliminating-map g → + (z : C) → ¬¬ (fiber (g ∘ f) z) → fiber g z + fiber-left-is-double-negation-eliminating-map-left G z nngfz = + G z (λ x → nngfz (λ w → x (f (pr1 w) , pr2 w))) + + fiber-right-is-double-negation-eliminating-map-comp : + is-injective g → + (G : is-double-negation-eliminating-map g) → + is-double-negation-eliminating-map f → + (z : C) (nngfz : ¬¬ (fiber (g ∘ f) z)) → + fiber f (pr1 (fiber-left-is-double-negation-eliminating-map-left G z nngfz)) + fiber-right-is-double-negation-eliminating-map-comp H G F z nngfz = + F ( pr1 + ( fiber-left-is-double-negation-eliminating-map-left G z nngfz)) + ( λ x → + nngfz + ( λ w → + x ( pr1 w , + H ( pr2 w ∙ + inv + ( pr2 + ( fiber-left-is-double-negation-eliminating-map-left + G z nngfz)))))) + + is-double-negation-eliminating-map-comp : + is-injective g → + is-double-negation-eliminating-map g → + is-double-negation-eliminating-map f → + is-double-negation-eliminating-map (g ∘ f) + is-double-negation-eliminating-map-comp H G F z nngfz = + map-inv-compute-fiber-comp g f z + ( ( fiber-left-is-double-negation-eliminating-map-left G z nngfz) , + ( fiber-right-is-double-negation-eliminating-map-comp H G F z nngfz)) +``` + +### Left cancellation for double negation eliminating maps + +If a composite `g ∘ f` is double negation eliminating and the left factor `g` is +injective, then the right factor `f` is double negation eliminating. + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {f : A → B} {g : B → C} + (GF : is-double-negation-eliminating-map (g ∘ f)) + where + + fiber-comp-is-double-negation-eliminating-map-right-factor' : + (y : B) → ¬¬ (fiber f y) → Σ (fiber g (g y)) (λ t → fiber f (pr1 t)) + fiber-comp-is-double-negation-eliminating-map-right-factor' y nnfy = + map-compute-fiber-comp g f (g y) + ( GF (g y) (λ ngfgy → nnfy λ x → ngfgy ((pr1 x) , ap g (pr2 x)))) + + is-double-negation-eliminating-map-right-factor' : + is-injective g → is-double-negation-eliminating-map f + is-double-negation-eliminating-map-right-factor' G y nnfy = + tr + ( fiber f) + ( G ( pr2 + ( pr1 + ( fiber-comp-is-double-negation-eliminating-map-right-factor' + ( y) + ( nnfy))))) + ( pr2 + ( fiber-comp-is-double-negation-eliminating-map-right-factor' y nnfy)) +``` + +### Any map out of the empty type is double negation eliminating + +```agda +abstract + is-double-negation-eliminating-map-ex-falso : + {l : Level} {X : UU l} → + is-double-negation-eliminating-map (ex-falso {l} {X}) + is-double-negation-eliminating-map-ex-falso x f = ex-falso (f λ ()) +``` + +### The identity map is double negation eliminating + +```agda +abstract + is-double-negation-eliminating-map-id : + {l : Level} {X : UU l} → is-double-negation-eliminating-map (id {l} {X}) + is-double-negation-eliminating-map-id x y = (x , refl) +``` + +### Equivalences are double negation eliminating maps + +```agda +abstract + is-double-negation-eliminating-map-is-equiv : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} → + is-equiv f → is-double-negation-eliminating-map f + is-double-negation-eliminating-map-is-equiv H x = + double-negation-elim-is-contr (is-contr-map-is-equiv H x) +``` + +### The map on total spaces induced by a family of double negation eliminating maps is double negation eliminating + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} + where + + is-double-negation-eliminating-map-tot : + {f : (x : A) → B x → C x} → + ((x : A) → is-double-negation-eliminating-map (f x)) → + is-double-negation-eliminating-map (tot f) + is-double-negation-eliminating-map-tot {f} H x = + has-double-negation-elim-equiv (compute-fiber-tot f x) (H (pr1 x) (pr2 x)) +``` + +### The map on total spaces induced by a double negation eliminating map on the base is double negation eliminating + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (C : B → UU l3) + where + + is-double-negation-eliminating-map-Σ-map-base : + {f : A → B} → + is-double-negation-eliminating-map f → + is-double-negation-eliminating-map (map-Σ-map-base f C) + is-double-negation-eliminating-map-Σ-map-base {f} H x = + has-double-negation-elim-equiv' + ( compute-fiber-map-Σ-map-base f C x) + ( H (pr1 x)) +``` + +### Products of double negation eliminating maps are double negation eliminating + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {D : UU l4} + where + + is-double-negation-eliminating-map-product : + {f : A → B} {g : C → D} → + is-double-negation-eliminating-map f → + is-double-negation-eliminating-map g → + is-double-negation-eliminating-map (map-product f g) + is-double-negation-eliminating-map-product {f} {g} F G x = + has-double-negation-elim-equiv + ( compute-fiber-map-product f g x) + ( double-negation-elim-product (F (pr1 x)) (G (pr2 x))) +``` + +### Coproducts of double negation eliminating maps are double negation eliminating + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {D : UU l4} + where + + is-double-negation-eliminating-map-coproduct : + {f : A → B} {g : C → D} → + is-double-negation-eliminating-map f → + is-double-negation-eliminating-map g → + is-double-negation-eliminating-map (map-coproduct f g) + is-double-negation-eliminating-map-coproduct {f} {g} F G (inl x) = + has-double-negation-elim-equiv' + ( compute-fiber-inl-map-coproduct f g x) + ( F x) + is-double-negation-eliminating-map-coproduct {f} {g} F G (inr y) = + has-double-negation-elim-equiv' + ( compute-fiber-inr-map-coproduct f g y) + ( G y) +``` + +### Double negation eliminating maps are closed under base change + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {D : UU l4} + {f : A → B} {g : C → D} + where + + is-double-negation-eliminating-map-base-change : + cartesian-hom-arrow g f → + is-double-negation-eliminating-map f → + is-double-negation-eliminating-map g + is-double-negation-eliminating-map-base-change α F d = + has-double-negation-elim-equiv + ( equiv-fibers-cartesian-hom-arrow g f α d) + ( F (map-codomain-cartesian-hom-arrow g f α d)) +``` + +### Double negation eliminating maps are closed under retracts of maps + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} + {f : A → B} {g : X → Y} + where + + is-double-negation-eliminating-retract-map : + f retract-of-map g → + is-double-negation-eliminating-map g → + is-double-negation-eliminating-map f + is-double-negation-eliminating-retract-map R G x = + has-double-negation-elim-retract + ( retract-fiber-retract-map f g R x) + ( G (map-codomain-inclusion-retract-map f g R x)) +``` diff --git a/src/logic/double-negation-elimination.lagda.md b/src/logic/double-negation-elimination.lagda.md new file mode 100644 index 0000000000..6fd70fe6e7 --- /dev/null +++ b/src/logic/double-negation-elimination.lagda.md @@ -0,0 +1,270 @@ +# Double negation elimination + +```agda +module logic.double-negation-elimination where +``` + +
Imports + +```agda +open import foundation.cartesian-product-types +open import foundation.coproduct-types +open import foundation.decidable-propositions +open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.double-negation +open import foundation.empty-types +open import foundation.evaluation-functions +open import foundation.hilberts-epsilon-operators +open import foundation.logical-equivalences +open import foundation.negation +open import foundation.retracts-of-types +open import foundation.transport-along-identifications +open import foundation.unit-type +open import foundation.universe-levels + +open import foundation-core.contractible-types +open import foundation-core.equivalences +open import foundation-core.function-types +open import foundation-core.propositions +``` + +
+ +## Idea + +We say a type `A` satisfies +{{#concept "untruncated double negation elimination" Disambiguation="on a type" Agda=has-double-negation-elim}} +if there is a map + +```text + ¬¬A → A. +``` + +## Definitions + +### Untruncated double negation elimination + +```agda +module _ + {l : Level} (A : UU l) + where + + has-double-negation-elim : UU l + has-double-negation-elim = ¬¬ A → A +``` + +## Properties + +### Having double negation elimination is a property for propositions + +```agda +is-prop-has-double-negation-elim : + {l : Level} {A : UU l} → is-prop A → is-prop (has-double-negation-elim A) +is-prop-has-double-negation-elim = is-prop-function-type +``` + +### Double negation elimination is preserved by logical equivalences + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + has-double-negation-elim-iff : + A ↔ B → has-double-negation-elim B → has-double-negation-elim A + has-double-negation-elim-iff e H = + backward-implication e ∘ H ∘ map-double-negation (forward-implication e) + + has-double-negation-elim-iff' : + B ↔ A → has-double-negation-elim B → has-double-negation-elim A + has-double-negation-elim-iff' e = has-double-negation-elim-iff (inv-iff e) +``` + +### Double negation elimination is preserved by equivalences + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + has-double-negation-elim-equiv : + A ≃ B → has-double-negation-elim B → has-double-negation-elim A + has-double-negation-elim-equiv e = + has-double-negation-elim-iff (iff-equiv e) + + has-double-negation-elim-equiv' : + B ≃ A → has-double-negation-elim B → has-double-negation-elim A + has-double-negation-elim-equiv' e = + has-double-negation-elim-iff' (iff-equiv e) +``` + +### Double negation elimination is preserved by retracts + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + has-double-negation-elim-retract : + A retract-of B → has-double-negation-elim B → has-double-negation-elim A + has-double-negation-elim-retract e = + has-double-negation-elim-iff (iff-retract e) + + has-double-negation-elim-retract' : + B retract-of A → has-double-negation-elim B → has-double-negation-elim A + has-double-negation-elim-retract' e = + has-double-negation-elim-iff' (iff-retract e) +``` + +### If the negation of a type with double negation elimination is decidable, then the type is decidable + +```agda +module _ + {l1 : Level} {A : UU l1} + where + + is-decidable-is-decidable-neg-has-double-negation-elim : + has-double-negation-elim A → is-decidable (¬ A) → is-decidable A + is-decidable-is-decidable-neg-has-double-negation-elim f (inl nx) = + inr nx + is-decidable-is-decidable-neg-has-double-negation-elim f (inr nnx) = + inl (f nnx) +``` + +**Remark.** It is an established fact that both the property of satisfying +double negation elimination, and the property of having decidable negation, are +strictly weaker conditions than being decidable. Therefore, this result +demonstrates that they are independent too. + +### Double negation elimination for empty types + +```agda +double-negation-elim-empty : has-double-negation-elim empty +double-negation-elim-empty e = e id + +double-negation-elim-is-empty : + {l : Level} {A : UU l} → is-empty A → has-double-negation-elim A +double-negation-elim-is-empty H q = ex-falso (q H) +``` + +### Double negation elimination for the unit type + +```agda +double-negation-elim-unit : has-double-negation-elim unit +double-negation-elim-unit _ = star +``` + +### Double negation elimination for contractible types + +```agda +double-negation-elim-is-contr : + {l : Level} {A : UU l} → is-contr A → has-double-negation-elim A +double-negation-elim-is-contr H _ = center H +``` + +### Double negation elimination for negations of types + +```agda +double-negation-elim-neg : + {l : Level} (A : UU l) → has-double-negation-elim (¬ A) +double-negation-elim-neg A f p = f (ev p) +``` + +### Double negation elimination for universal quantification over double negations + +```agda +module _ + {l1 l2 : Level} {P : UU l1} {Q : P → UU l2} + where + + double-negation-elim-for-all-neg-neg : + has-double-negation-elim ((p : P) → ¬¬ (Q p)) + double-negation-elim-for-all-neg-neg f p = + double-negation-elim-neg + ( ¬ (Q p)) + ( map-double-negation (λ (g : (u : P) → ¬¬ (Q u)) → g p) f) + + double-negation-elim-for-all : + ((p : P) → has-double-negation-elim (Q p)) → + has-double-negation-elim ((p : P) → Q p) + double-negation-elim-for-all H f p = H p (λ nq → f (λ g → nq (g p))) +``` + +### Double negation elimination for function types into double negations + +```agda +module _ + {l1 l2 : Level} {P : UU l1} {Q : UU l2} + where + + double-negation-elim-exp-neg-neg : + has-double-negation-elim (P → ¬¬ Q) + double-negation-elim-exp-neg-neg = + double-negation-elim-for-all-neg-neg + + double-negation-elim-exp : + has-double-negation-elim Q → + has-double-negation-elim (P → Q) + double-negation-elim-exp q = double-negation-elim-for-all (λ _ → q) +``` + +### Double negation elimination for decidable propositions + +```text +double-negation-elim-is-decidable : + {l : Level} {A : UU l} → is-decidable A → has-double-negation-elim A +double-negation-elim-is-decidable = double-negation-elim-is-decidable +``` + +### Double negation elimination for dependent sums of types with double negation elimination over a double negation stable proposition + +```agda +double-negation-elim-Σ-is-prop-base : + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} → + is-prop A → + has-double-negation-elim A → + ((a : A) → has-double-negation-elim (B a)) → + has-double-negation-elim (Σ A B) +double-negation-elim-Σ-is-prop-base {B = B} is-prop-A f g h = + ( f (λ na → h (na ∘ pr1))) , + ( g ( f (λ na → h (na ∘ pr1))) + ( λ nb → h (λ y → nb (tr B (eq-is-prop is-prop-A) (pr2 y))))) + +double-negation-elim-Σ-is-decidable-prop-base : + {l1 l2 : Level} {P : UU l1} {B : P → UU l2} → + is-decidable-prop P → + ((x : P) → has-double-negation-elim (B x)) → + has-double-negation-elim (Σ P B) +double-negation-elim-Σ-is-decidable-prop-base (H , d) = + double-negation-elim-Σ-is-prop-base H (double-negation-elim-is-decidable d) +``` + +### Double negation elimination for products of types with double negation elimination + +```agda +double-negation-elim-product : + {l1 l2 : Level} {A : UU l1} {B : UU l2} → + has-double-negation-elim A → + has-double-negation-elim B → + has-double-negation-elim (A × B) +double-negation-elim-product f g h = + f (λ na → h (na ∘ pr1)) , g (λ nb → h (nb ∘ pr2)) +``` + +### If a type satisfies untruncated double negation elimination then it has a Hilbert ε-operator + +```agda +ε-operator-Hilbert-has-double-negation-elim : + {l1 : Level} {A : UU l1} → + has-double-negation-elim A → + ε-operator-Hilbert A +ε-operator-Hilbert-has-double-negation-elim {A = A} H = + H ∘ double-negation-double-negation-type-trunc-Prop A ∘ intro-double-negation +``` + +## See also + +- [The double negation modality](foundation.double-negation-modality.md) +- [Irrefutable propositions](foundation.irrefutable-propositions.md) are double + negation connected types. diff --git a/src/logic/double-negation-stable-embeddings.lagda.md b/src/logic/double-negation-stable-embeddings.lagda.md new file mode 100644 index 0000000000..4e6a207085 --- /dev/null +++ b/src/logic/double-negation-stable-embeddings.lagda.md @@ -0,0 +1,731 @@ +# Double negation stable embeddings + +```agda +module logic.double-negation-stable-embeddings where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.cartesian-morphisms-arrows +open import foundation.decidable-embeddings +open import foundation.decidable-maps +open import foundation.decidable-propositions +open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.double-negation-stable-propositions +open import foundation.embeddings +open import foundation.fibers-of-maps +open import foundation.functoriality-cartesian-product-types +open import foundation.functoriality-coproduct-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.negation +open import foundation.propositional-maps +open import foundation.propositions +open import foundation.retracts-of-maps +open import foundation.subtype-identity-principle +open import foundation.type-arithmetic-dependent-pair-types +open import foundation.unit-type +open import foundation.universal-property-equivalences +open import foundation.universe-levels + +open import foundation-core.cartesian-product-types +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.functoriality-dependent-pair-types +open import foundation-core.homotopies +open import foundation-core.injective-maps +open import foundation-core.torsorial-type-families + +open import logic.double-negation-eliminating-maps +open import logic.double-negation-elimination +``` + +
+ +## Idea + +A [map](foundation-core.function-types.md) is said to be a +{{#concept "double negation stable embedding" Disambiguation="of types" Agda=is-double-negation-stable-emb}} +if it is an [embedding](foundation-core.embeddings.md) and its +[fibers](foundation-core.fibers-of-maps.md) satisfy +[double negation elimination](logic.double-negation-elimination.md). + +Equivalently, a double negation stable embedding is a map whose fibers are +[double negation stable propositions](foundation.double-negation-stable-propositions.md). +We refer to this condition as being a +{{#concept "double negation stable propositional map" Disambiguation="of types" Agda=is-double-negation-stable-prop-map}}. + +## Definitions + +### The condition on a map of being a double negation stable embedding + +```agda +is-double-negation-stable-emb : + {l1 l2 : Level} {X : UU l1} {Y : UU l2} → (X → Y) → UU (l1 ⊔ l2) +is-double-negation-stable-emb f = + is-emb f × is-double-negation-eliminating-map f + +module _ + {l1 l2 : Level} {X : UU l1} {Y : UU l2} {f : X → Y} + (F : is-double-negation-stable-emb f) + where + + is-emb-is-double-negation-stable-emb : is-emb f + is-emb-is-double-negation-stable-emb = pr1 F + + is-double-negation-eliminating-map-is-double-negation-stable-emb : + is-double-negation-eliminating-map f + is-double-negation-eliminating-map-is-double-negation-stable-emb = pr2 F + + is-prop-map-is-double-negation-stable-emb : is-prop-map f + is-prop-map-is-double-negation-stable-emb = + is-prop-map-is-emb is-emb-is-double-negation-stable-emb + + is-injective-is-double-negation-stable-emb : is-injective f + is-injective-is-double-negation-stable-emb = + is-injective-is-emb is-emb-is-double-negation-stable-emb +``` + +### Double negation stable propositional maps + +```agda +module _ + {l1 l2 : Level} {X : UU l1} {Y : UU l2} + where + + is-double-negation-stable-prop-map : (X → Y) → UU (l1 ⊔ l2) + is-double-negation-stable-prop-map f = + (y : Y) → is-double-negation-stable-prop (fiber f y) + + is-prop-is-double-negation-stable-prop-map : + (f : X → Y) → is-prop (is-double-negation-stable-prop-map f) + is-prop-is-double-negation-stable-prop-map f = + is-prop-Π (λ y → is-prop-is-double-negation-stable-prop (fiber f y)) + + is-double-negation-stable-prop-map-Prop : (X → Y) → Prop (l1 ⊔ l2) + is-double-negation-stable-prop-map-Prop f = + ( is-double-negation-stable-prop-map f , + is-prop-is-double-negation-stable-prop-map f) + + is-prop-map-is-double-negation-stable-prop-map : + {f : X → Y} → is-double-negation-stable-prop-map f → is-prop-map f + is-prop-map-is-double-negation-stable-prop-map H y = pr1 (H y) + + is-double-negation-eliminating-map-is-double-negation-stable-prop-map : + {f : X → Y} → + is-double-negation-stable-prop-map f → + is-double-negation-eliminating-map f + is-double-negation-eliminating-map-is-double-negation-stable-prop-map H y = + pr2 (H y) +``` + +### The type of double negation stable embeddings + +```agda +infix 5 _↪¬¬_ +_↪¬¬_ : + {l1 l2 : Level} (X : UU l1) (Y : UU l2) → UU (l1 ⊔ l2) +X ↪¬¬ Y = Σ (X → Y) is-double-negation-stable-emb + +double-negation-stable-emb : + {l1 l2 : Level} (X : UU l1) (Y : UU l2) → UU (l1 ⊔ l2) +double-negation-stable-emb = _↪¬¬_ + +module _ + {l1 l2 : Level} {X : UU l1} {Y : UU l2} (e : X ↪¬¬ Y) + where + + map-double-negation-stable-emb : X → Y + map-double-negation-stable-emb = pr1 e + + is-double-negation-stable-emb-map-double-negation-stable-emb : + is-double-negation-stable-emb map-double-negation-stable-emb + is-double-negation-stable-emb-map-double-negation-stable-emb = pr2 e + + is-emb-map-double-negation-stable-emb : + is-emb map-double-negation-stable-emb + is-emb-map-double-negation-stable-emb = + is-emb-is-double-negation-stable-emb + is-double-negation-stable-emb-map-double-negation-stable-emb + + is-double-negation-eliminating-map-map-double-negation-stable-emb : + is-double-negation-eliminating-map map-double-negation-stable-emb + is-double-negation-eliminating-map-map-double-negation-stable-emb = + is-double-negation-eliminating-map-is-double-negation-stable-emb + is-double-negation-stable-emb-map-double-negation-stable-emb + + emb-double-negation-stable-emb : X ↪ Y + emb-double-negation-stable-emb = + map-double-negation-stable-emb , is-emb-map-double-negation-stable-emb +``` + +## Properties + +### Any map of which the fibers are double negation stable propositions is a double negation stable embedding + +```agda +module _ + {l1 l2 : Level} {X : UU l1} {Y : UU l2} {f : X → Y} + where + + abstract + is-double-negation-stable-emb-is-double-negation-stable-prop-map : + is-double-negation-stable-prop-map f → is-double-negation-stable-emb f + pr1 (is-double-negation-stable-emb-is-double-negation-stable-prop-map H) = + is-emb-is-prop-map (is-prop-map-is-double-negation-stable-prop-map H) + pr2 (is-double-negation-stable-emb-is-double-negation-stable-prop-map H) = + is-double-negation-eliminating-map-is-double-negation-stable-prop-map H + + abstract + is-double-negation-stable-prop-map-is-double-negation-stable-emb : + is-double-negation-stable-emb f → is-double-negation-stable-prop-map f + pr1 (is-double-negation-stable-prop-map-is-double-negation-stable-emb H y) = + is-prop-map-is-double-negation-stable-emb H y + pr2 (is-double-negation-stable-prop-map-is-double-negation-stable-emb H y) = + is-double-negation-eliminating-map-is-double-negation-stable-emb H y +``` + +### The first projection map of a dependent sum of double negation stable propositions is a double negation stable embedding + +```agda +module _ + {l1 l2 : Level} {A : UU l1} (Q : A → Double-Negation-Stable-Prop l2) + where + + is-double-negation-stable-prop-map-pr1 : + is-double-negation-stable-prop-map + ( pr1 {B = type-Double-Negation-Stable-Prop ∘ Q}) + is-double-negation-stable-prop-map-pr1 y = + is-double-negation-stable-prop-equiv + ( equiv-fiber-pr1 (type-Double-Negation-Stable-Prop ∘ Q) y) + ( is-double-negation-stable-prop-type-Double-Negation-Stable-Prop (Q y)) + + is-double-negation-stable-emb-pr1 : + is-double-negation-stable-emb + ( pr1 {B = type-Double-Negation-Stable-Prop ∘ Q}) + is-double-negation-stable-emb-pr1 = + is-double-negation-stable-emb-is-double-negation-stable-prop-map + ( is-double-negation-stable-prop-map-pr1) +``` + +### Decidable embeddings are double negation stable + +```agda +is-double-negation-eliminating-map-is-decidable-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} → + is-decidable-emb f → is-double-negation-eliminating-map f +is-double-negation-eliminating-map-is-decidable-emb H = + is-double-negation-eliminating-map-is-decidable-map + ( is-decidable-map-is-decidable-emb H) + +abstract + is-double-negation-stable-emb-is-decidable-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} → + is-decidable-emb f → is-double-negation-stable-emb f + is-double-negation-stable-emb-is-decidable-emb H = + ( is-emb-is-decidable-emb H , + is-double-negation-eliminating-map-is-decidable-emb H) +``` + +### Equivalences are double negation stable embeddings + +```agda +abstract + is-double-negation-stable-emb-is-equiv : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} → + is-equiv f → is-double-negation-stable-emb f + is-double-negation-stable-emb-is-equiv H = + ( is-emb-is-equiv H , is-double-negation-eliminating-map-is-equiv H) +``` + +### Identity maps are double negation stable embeddings + +```agda +is-double-negation-stable-emb-id : + {l : Level} {A : UU l} → is-double-negation-stable-emb (id {A = A}) +is-double-negation-stable-emb-id = + ( is-emb-id , is-double-negation-eliminating-map-id) + +double-negation-stable-emb-id : {l : Level} {A : UU l} → A ↪¬¬ A +double-negation-stable-emb-id = (id , is-double-negation-stable-emb-id) + +is-double-negation-stable-prop-map-id : + {l : Level} {A : UU l} → is-double-negation-stable-prop-map (id {A = A}) +is-double-negation-stable-prop-map-id y = + is-double-negation-stable-prop-is-contr (is-torsorial-Id' y) +``` + +### Being a double negation stable embedding is a property + +```agda +abstract + is-prop-is-double-negation-stable-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) → + is-prop (is-double-negation-stable-emb f) + is-prop-is-double-negation-stable-emb f = + is-prop-has-element + ( λ H → + is-prop-product + ( is-property-is-emb f) + ( is-prop-Π + ( is-prop-has-double-negation-elim ∘ is-prop-map-is-emb (pr1 H)))) +``` + +### Double negation stable embeddings are closed under homotopies + +```agda +abstract + is-double-negation-stable-emb-htpy : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f g : A → B} → + f ~ g → is-double-negation-stable-emb g → is-double-negation-stable-emb f + is-double-negation-stable-emb-htpy {f = f} {g} H K = + ( is-emb-htpy H (is-emb-is-double-negation-stable-emb K) , + is-double-negation-eliminating-map-htpy H + ( is-double-negation-eliminating-map-is-double-negation-stable-emb K)) +``` + +### Double negation stable embeddings are closed under composition + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} + {g : B → C} {f : A → B} + where + + is-double-negation-eliminating-map-comp-is-double-negation-stable-emb : + is-double-negation-stable-emb g → + is-double-negation-eliminating-map f → + is-double-negation-eliminating-map (g ∘ f) + is-double-negation-eliminating-map-comp-is-double-negation-stable-emb G F = + is-double-negation-eliminating-map-comp + ( is-injective-is-double-negation-stable-emb G) + ( is-double-negation-eliminating-map-is-double-negation-stable-emb G) + ( F) + + is-double-negation-stable-prop-map-comp : + is-double-negation-stable-prop-map g → + is-double-negation-stable-prop-map f → + is-double-negation-stable-prop-map (g ∘ f) + is-double-negation-stable-prop-map-comp K H z = + is-double-negation-stable-prop-equiv + ( compute-fiber-comp g f z) + ( is-double-negation-stable-prop-Σ (K z) (H ∘ pr1)) + + is-double-negation-stable-emb-comp : + is-double-negation-stable-emb g → + is-double-negation-stable-emb f → + is-double-negation-stable-emb (g ∘ f) + is-double-negation-stable-emb-comp K H = + is-double-negation-stable-emb-is-double-negation-stable-prop-map + ( is-double-negation-stable-prop-map-comp + ( is-double-negation-stable-prop-map-is-double-negation-stable-emb K) + ( is-double-negation-stable-prop-map-is-double-negation-stable-emb H)) + +comp-double-negation-stable-emb : + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} → + B ↪¬¬ C → A ↪¬¬ B → A ↪¬¬ C +comp-double-negation-stable-emb (g , G) (f , F) = + ( g ∘ f , is-double-negation-stable-emb-comp G F) + +infixr 15 _∘¬¬_ +_∘¬¬_ : + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} → + B ↪¬¬ C → A ↪¬¬ B → A ↪¬¬ C +_∘¬¬_ = comp-double-negation-stable-emb +``` + +### Left cancellation for double negation stable embeddings + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {f : A → B} {g : B → C} + where + + is-double-negation-stable-emb-right-factor' : + is-double-negation-stable-emb (g ∘ f) → + is-emb g → + is-double-negation-stable-emb f + is-double-negation-stable-emb-right-factor' GH G = + ( is-emb-right-factor g f G (is-emb-is-double-negation-stable-emb GH) , + is-double-negation-eliminating-map-right-factor' + ( is-double-negation-eliminating-map-is-double-negation-stable-emb GH) + ( is-injective-is-emb G)) + + is-double-negation-stable-emb-right-factor : + is-double-negation-stable-emb (g ∘ f) → + is-double-negation-stable-emb g → + is-double-negation-stable-emb f + is-double-negation-stable-emb-right-factor GH G = + is-double-negation-stable-emb-right-factor' + ( GH) + ( is-emb-is-double-negation-stable-emb G) +``` + +### In a commuting triangle of maps, if the top and right maps are double negation stable embeddings so is the left map + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} + {top : A → B} {left : A → C} {right : B → C} + (H : left ~ right ∘ top) + where + + is-double-negation-stable-emb-left-map-triangle : + is-double-negation-stable-emb top → + is-double-negation-stable-emb right → + is-double-negation-stable-emb left + is-double-negation-stable-emb-left-map-triangle T R = + is-double-negation-stable-emb-htpy H + ( is-double-negation-stable-emb-comp R T) +``` + +### In a commuting triangle of maps, if the left and right maps are double negation stable embeddings so is the top map + +In fact, the right map need only be an embedding. + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} + {top : A → B} {left : A → C} {right : B → C} + (H : left ~ right ∘ top) + where + + is-double-negation-stable-emb-top-map-triangle' : + is-emb right → + is-double-negation-stable-emb left → + is-double-negation-stable-emb top + is-double-negation-stable-emb-top-map-triangle' R' L = + is-double-negation-stable-emb-right-factor' + ( is-double-negation-stable-emb-htpy (inv-htpy H) L) + ( R') + + is-double-negation-stable-emb-top-map-triangle : + is-double-negation-stable-emb right → + is-double-negation-stable-emb left → + is-double-negation-stable-emb top + is-double-negation-stable-emb-top-map-triangle R L = + is-double-negation-stable-emb-right-factor + ( is-double-negation-stable-emb-htpy (inv-htpy H) L) + ( R) +``` + +### Characterizing equality in the type of double negation stable embeddings + +```agda +htpy-double-negation-stable-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f g : A ↪¬¬ B) → UU (l1 ⊔ l2) +htpy-double-negation-stable-emb f g = + map-double-negation-stable-emb f ~ map-double-negation-stable-emb g + +refl-htpy-double-negation-stable-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A ↪¬¬ B) → + htpy-double-negation-stable-emb f f +refl-htpy-double-negation-stable-emb f = refl-htpy + +htpy-eq-double-negation-stable-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f g : A ↪¬¬ B) → + f ＝ g → htpy-double-negation-stable-emb f g +htpy-eq-double-negation-stable-emb f .f refl = + refl-htpy-double-negation-stable-emb f + +abstract + is-torsorial-htpy-double-negation-stable-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A ↪¬¬ B) → + is-torsorial (htpy-double-negation-stable-emb f) + is-torsorial-htpy-double-negation-stable-emb f = + is-torsorial-Eq-subtype + ( is-torsorial-htpy (map-double-negation-stable-emb f)) + ( is-prop-is-double-negation-stable-emb) + ( map-double-negation-stable-emb f) + ( refl-htpy) + ( is-double-negation-stable-emb-map-double-negation-stable-emb f) + +abstract + is-equiv-htpy-eq-double-negation-stable-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f g : A ↪¬¬ B) → + is-equiv (htpy-eq-double-negation-stable-emb f g) + is-equiv-htpy-eq-double-negation-stable-emb f = + fundamental-theorem-id + ( is-torsorial-htpy-double-negation-stable-emb f) + ( htpy-eq-double-negation-stable-emb f) + +eq-htpy-double-negation-stable-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f g : A ↪¬¬ B) → + htpy-double-negation-stable-emb f g → f ＝ g +eq-htpy-double-negation-stable-emb f g = + map-inv-is-equiv (is-equiv-htpy-eq-double-negation-stable-emb f g) +``` + +### Precomposing double negation stable embeddings with equivalences + +```agda +equiv-precomp-double-negation-stable-emb-equiv : + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (e : A ≃ B) → + (C : UU l3) → (B ↪¬¬ C) ≃ (A ↪¬¬ C) +equiv-precomp-double-negation-stable-emb-equiv e C = + equiv-Σ + ( is-double-negation-stable-emb) + ( equiv-precomp e C) + ( λ g → + equiv-iff-is-prop + ( is-prop-is-double-negation-stable-emb g) + ( is-prop-is-double-negation-stable-emb (g ∘ map-equiv e)) + ( λ H → + is-double-negation-stable-emb-comp H + ( is-double-negation-stable-emb-is-equiv (pr2 e))) + ( λ d → + is-double-negation-stable-emb-htpy + ( λ b → ap g (inv (is-section-map-inv-equiv e b))) + ( is-double-negation-stable-emb-comp + ( d) + ( is-double-negation-stable-emb-is-equiv + ( is-equiv-map-inv-equiv e))))) +``` + +### Any map out of the empty type is a double negation stable embedding + +```agda +abstract + is-double-negation-stable-emb-ex-falso : + {l : Level} {X : UU l} → is-double-negation-stable-emb (ex-falso {l} {X}) + is-double-negation-stable-emb-ex-falso = + ( is-emb-ex-falso , is-double-negation-eliminating-map-ex-falso) + +double-negation-stable-emb-ex-falso : + {l : Level} {X : UU l} → empty ↪¬¬ X +double-negation-stable-emb-ex-falso = + ( ex-falso , is-double-negation-stable-emb-ex-falso) + +double-negation-stable-emb-is-empty : + {l1 l2 : Level} {A : UU l1} {B : UU l2} → is-empty A → A ↪¬¬ B +double-negation-stable-emb-is-empty {A = A} f = + map-equiv + ( equiv-precomp-double-negation-stable-emb-equiv (equiv-is-empty f id) _) + ( double-negation-stable-emb-ex-falso) +``` + +### The map on total spaces induced by a family of double negation stable embeddings is a double negation stable embedding + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} + where + + is-double-negation-stable-emb-tot : + {f : (x : A) → B x → C x} → + ((x : A) → is-double-negation-stable-emb (f x)) → + is-double-negation-stable-emb (tot f) + is-double-negation-stable-emb-tot H = + ( is-emb-tot (is-emb-is-double-negation-stable-emb ∘ H) , + is-double-negation-eliminating-map-tot + ( is-double-negation-eliminating-map-is-double-negation-stable-emb ∘ H)) + + double-negation-stable-emb-tot : ((x : A) → B x ↪¬¬ C x) → Σ A B ↪¬¬ Σ A C + double-negation-stable-emb-tot f = + ( tot (map-double-negation-stable-emb ∘ f) , + is-double-negation-stable-emb-tot + ( is-double-negation-stable-emb-map-double-negation-stable-emb ∘ f)) +``` + +### The map on total spaces induced by a double negation stable embedding on the base is a double negation stable embedding + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (C : B → UU l3) + where + + is-double-negation-stable-emb-map-Σ-map-base : + {f : A → B} → + is-double-negation-stable-emb f → + is-double-negation-stable-emb (map-Σ-map-base f C) + is-double-negation-stable-emb-map-Σ-map-base H = + ( is-emb-map-Σ-map-base C (is-emb-is-double-negation-stable-emb H) , + is-double-negation-eliminating-map-Σ-map-base C + ( is-double-negation-eliminating-map-is-double-negation-stable-emb H)) + + double-negation-stable-emb-map-Σ-map-base : + (f : A ↪¬¬ B) → Σ A (C ∘ map-double-negation-stable-emb f) ↪¬¬ Σ B C + double-negation-stable-emb-map-Σ-map-base f = + ( map-Σ-map-base (map-double-negation-stable-emb f) C , + is-double-negation-stable-emb-map-Σ-map-base + ( is-double-negation-stable-emb-map-double-negation-stable-emb f)) +``` + +### The functoriality of dependent pair types preserves double negation stable embeddings + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : A → UU l3} (D : B → UU l4) + where + + is-double-negation-stable-emb-map-Σ : + {f : A → B} {g : (x : A) → C x → D (f x)} → + is-double-negation-stable-emb f → + ((x : A) → is-double-negation-stable-emb (g x)) → + is-double-negation-stable-emb (map-Σ D f g) + is-double-negation-stable-emb-map-Σ {f} {g} F G = + is-double-negation-stable-emb-left-map-triangle + ( triangle-map-Σ D f g) + ( is-double-negation-stable-emb-tot G) + ( is-double-negation-stable-emb-map-Σ-map-base D F) + + double-negation-stable-emb-Σ : + (f : A ↪¬¬ B) → + ((x : A) → C x ↪¬¬ D (map-double-negation-stable-emb f x)) → + Σ A C ↪¬¬ Σ B D + double-negation-stable-emb-Σ f g = + ( ( map-Σ D + ( map-double-negation-stable-emb f) + ( map-double-negation-stable-emb ∘ g)) , + ( is-double-negation-stable-emb-map-Σ + ( is-double-negation-stable-emb-map-double-negation-stable-emb f) + ( is-double-negation-stable-emb-map-double-negation-stable-emb ∘ g))) +``` + +### Products of double negation stable embeddings are double negation stable embeddings + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {D : UU l4} + where + + is-double-negation-stable-emb-map-product : + {f : A → B} {g : C → D} → + is-double-negation-stable-emb f → + is-double-negation-stable-emb g → + is-double-negation-stable-emb (map-product f g) + is-double-negation-stable-emb-map-product (eF , dF) (eG , dG) = + ( is-emb-map-product eF eG , + is-double-negation-eliminating-map-product dF dG) + + double-negation-stable-emb-product : + A ↪¬¬ B → C ↪¬¬ D → A × C ↪¬¬ B × D + double-negation-stable-emb-product (f , F) (g , G) = + ( map-product f g , is-double-negation-stable-emb-map-product F G) +``` + +### Coproducts of double negation stable embeddings are double negation stable embeddings + +```agda +module _ + {l1 l2 l1' l2' : Level} {A : UU l1} {B : UU l2} {A' : UU l1'} {B' : UU l2'} + where + +abstract + is-double-negation-stable-emb-map-coproduct : + {l1 l2 l3 l4 : Level} + {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} + {f : A → B} {g : X → Y} → + is-double-negation-stable-emb f → + is-double-negation-stable-emb g → + is-double-negation-stable-emb (map-coproduct f g) + is-double-negation-stable-emb-map-coproduct (eF , dF) (eG , dG) = + ( is-emb-map-coproduct eF eG , + is-double-negation-eliminating-map-coproduct dF dG) +``` + +### Double negation stable embeddings are closed under base change + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {D : UU l4} + {f : A → B} {g : C → D} + where + + is-double-negation-stable-prop-map-base-change : + cartesian-hom-arrow g f → + is-double-negation-stable-prop-map f → + is-double-negation-stable-prop-map g + is-double-negation-stable-prop-map-base-change α F d = + is-double-negation-stable-prop-equiv + ( equiv-fibers-cartesian-hom-arrow g f α d) + ( F (map-codomain-cartesian-hom-arrow g f α d)) + + is-double-negation-stable-emb-base-change : + cartesian-hom-arrow g f → + is-double-negation-stable-emb f → + is-double-negation-stable-emb g + is-double-negation-stable-emb-base-change α F = + is-double-negation-stable-emb-is-double-negation-stable-prop-map + ( is-double-negation-stable-prop-map-base-change α + ( is-double-negation-stable-prop-map-is-double-negation-stable-emb F)) +``` + +### Double negation stable embeddings are closed under retracts of maps + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} + {f : A → B} {g : X → Y} + where + + is-double-negation-stable-prop-map-retract-map : + f retract-of-map g → + is-double-negation-stable-prop-map g → + is-double-negation-stable-prop-map f + is-double-negation-stable-prop-map-retract-map R G x = + is-double-negation-stable-prop-retract + ( retract-fiber-retract-map f g R x) + ( G (map-codomain-inclusion-retract-map f g R x)) + + is-double-negation-stable-emb-retract-map : + f retract-of-map g → + is-double-negation-stable-emb g → + is-double-negation-stable-emb f + is-double-negation-stable-emb-retract-map R G = + is-double-negation-stable-emb-is-double-negation-stable-prop-map + ( is-double-negation-stable-prop-map-retract-map R + ( is-double-negation-stable-prop-map-is-double-negation-stable-emb G)) +``` + +### A type is a double negation stable proposition if and only if its terminal map is a double negation stable embedding + +```agda +module _ + {l : Level} {A : UU l} + where + + is-double-negation-stable-prop-is-double-negation-stable-emb-terminal-map : + is-double-negation-stable-emb (terminal-map A) → + is-double-negation-stable-prop A + is-double-negation-stable-prop-is-double-negation-stable-emb-terminal-map H = + is-double-negation-stable-prop-equiv' + ( equiv-fiber-terminal-map star) + ( is-double-negation-stable-prop-map-is-double-negation-stable-emb H star) + + is-double-negation-stable-emb-terminal-map-is-double-negation-stable-prop : + is-double-negation-stable-prop A → + is-double-negation-stable-emb (terminal-map A) + is-double-negation-stable-emb-terminal-map-is-double-negation-stable-prop H = + is-double-negation-stable-emb-is-double-negation-stable-prop-map + ( λ y → + is-double-negation-stable-prop-equiv (equiv-fiber-terminal-map y) H) +``` + +### If a dependent sum of propositions over a proposition is double negation stable, then the family is a family of double negation stable propositions + +```agda +module _ + {l1 l2 : Level} (P : Prop l1) (Q : type-Prop P → Prop l2) + where + + is-double-negation-stable-prop-family-has-double-negation-elim-Σ : + has-double-negation-elim (Σ (type-Prop P) (type-Prop ∘ Q)) → + (p : type-Prop P) → has-double-negation-elim (type-Prop (Q p)) + is-double-negation-stable-prop-family-has-double-negation-elim-Σ H p = + has-double-negation-elim-equiv' + ( equiv-fiber-pr1 (type-Prop ∘ Q) p) + ( is-double-negation-eliminating-map-is-double-negation-stable-emb + ( is-double-negation-stable-emb-right-factor' + ( is-double-negation-stable-emb-terminal-map-is-double-negation-stable-prop + ( is-prop-Σ (is-prop-type-Prop P) (is-prop-type-Prop ∘ Q) , H)) + ( is-emb-terminal-map-is-prop (is-prop-type-Prop P))) + ( p)) +``` diff --git a/src/logic/double-negation-stable-subtypes.lagda.md b/src/logic/double-negation-stable-subtypes.lagda.md new file mode 100644 index 0000000000..6614510a1b --- /dev/null +++ b/src/logic/double-negation-stable-subtypes.lagda.md @@ -0,0 +1,338 @@ +# Double negation stable subtypes + +```agda +module logic.double-negation-stable-subtypes where +``` + +
Imports + +```agda +open import foundation.1-types +open import foundation.coproduct-types +open import foundation.dependent-pair-types +open import foundation.double-negation-stable-propositions +open import foundation.equality-dependent-function-types +open import foundation.functoriality-cartesian-product-types +open import foundation.functoriality-dependent-function-types +open import foundation.functoriality-dependent-pair-types +open import foundation.logical-equivalences +open import foundation.propositional-maps +open import foundation.sets +open import foundation.structured-type-duality +open import foundation.subtypes +open import foundation.type-theoretic-principle-of-choice +open import foundation.universe-levels + +open import foundation-core.embeddings +open import foundation-core.equivalences +open import foundation-core.fibers-of-maps +open import foundation-core.function-types +open import foundation-core.identity-types +open import foundation-core.injective-maps +open import foundation-core.propositions +open import foundation-core.transport-along-identifications +open import foundation-core.truncated-types +open import foundation-core.truncation-levels + +open import logic.double-negation-eliminating-maps +open import logic.double-negation-elimination +open import logic.double-negation-stable-embeddings +``` + +
+ +## Idea + +A +{{#concept "double negation stable subtype" Disambiguation="of a type" Agda=is-double-negation-stable-subtype Agda=double-negation-stable-subtype}} +of a type consists of a family of +[double negation stable propositions](foundation.double-negation-stable-propositions.md) +over it. + +## Definitions + +### Decidable subtypes + +```agda +is-double-negation-stable-subtype-Prop : + {l1 l2 : Level} {A : UU l1} → subtype l2 A → Prop (l1 ⊔ l2) +is-double-negation-stable-subtype-Prop {A = A} P = + Π-Prop A (λ a → is-double-negation-stable-Prop (P a)) + +is-double-negation-stable-subtype : + {l1 l2 : Level} {A : UU l1} → subtype l2 A → UU (l1 ⊔ l2) +is-double-negation-stable-subtype P = + type-Prop (is-double-negation-stable-subtype-Prop P) + +is-prop-is-double-negation-stable-subtype : + {l1 l2 : Level} {A : UU l1} (P : subtype l2 A) → + is-prop (is-double-negation-stable-subtype P) +is-prop-is-double-negation-stable-subtype P = + is-prop-type-Prop (is-double-negation-stable-subtype-Prop P) + +double-negation-stable-subtype : + {l1 : Level} (l : Level) (X : UU l1) → UU (l1 ⊔ lsuc l) +double-negation-stable-subtype l X = X → Double-Negation-Stable-Prop l +``` + +### The underlying subtype of a double negation stable subtype + +```agda +module _ + {l1 l2 : Level} {A : UU l1} (P : double-negation-stable-subtype l2 A) + where + + subtype-double-negation-stable-subtype : subtype l2 A + subtype-double-negation-stable-subtype a = + prop-Double-Negation-Stable-Prop (P a) + + is-double-negation-stable-double-negation-stable-subtype : + is-double-negation-stable-subtype subtype-double-negation-stable-subtype + is-double-negation-stable-double-negation-stable-subtype a = + has-double-negation-elim-type-Double-Negation-Stable-Prop (P a) + + is-in-double-negation-stable-subtype : A → UU l2 + is-in-double-negation-stable-subtype = + is-in-subtype subtype-double-negation-stable-subtype + + is-prop-is-in-double-negation-stable-subtype : + (a : A) → is-prop (is-in-double-negation-stable-subtype a) + is-prop-is-in-double-negation-stable-subtype = + is-prop-is-in-subtype subtype-double-negation-stable-subtype +``` + +### The underlying type of a double negation stable subtype + +```agda +module _ + {l1 l2 : Level} {A : UU l1} (P : double-negation-stable-subtype l2 A) + where + + type-double-negation-stable-subtype : UU (l1 ⊔ l2) + type-double-negation-stable-subtype = + type-subtype (subtype-double-negation-stable-subtype P) + + inclusion-double-negation-stable-subtype : + type-double-negation-stable-subtype → A + inclusion-double-negation-stable-subtype = + inclusion-subtype (subtype-double-negation-stable-subtype P) + + is-emb-inclusion-double-negation-stable-subtype : + is-emb inclusion-double-negation-stable-subtype + is-emb-inclusion-double-negation-stable-subtype = + is-emb-inclusion-subtype (subtype-double-negation-stable-subtype P) + + is-double-negation-eliminating-map-inclusion-double-negation-stable-subtype : + is-double-negation-eliminating-map inclusion-double-negation-stable-subtype + is-double-negation-eliminating-map-inclusion-double-negation-stable-subtype + x = + has-double-negation-elim-equiv + ( equiv-fiber-pr1 (type-Double-Negation-Stable-Prop ∘ P) x) + ( has-double-negation-elim-type-Double-Negation-Stable-Prop (P x)) + + is-injective-inclusion-double-negation-stable-subtype : + is-injective inclusion-double-negation-stable-subtype + is-injective-inclusion-double-negation-stable-subtype = + is-injective-inclusion-subtype (subtype-double-negation-stable-subtype P) + + emb-double-negation-stable-subtype : type-double-negation-stable-subtype ↪ A + emb-double-negation-stable-subtype = + emb-subtype (subtype-double-negation-stable-subtype P) + + is-double-negation-stable-emb-inclusion-double-negation-stable-subtype : + is-double-negation-stable-emb inclusion-double-negation-stable-subtype + is-double-negation-stable-emb-inclusion-double-negation-stable-subtype = + ( is-emb-inclusion-double-negation-stable-subtype , + is-double-negation-eliminating-map-inclusion-double-negation-stable-subtype) + + double-negation-stable-emb-double-negation-stable-subtype : + type-double-negation-stable-subtype ↪¬¬ A + double-negation-stable-emb-double-negation-stable-subtype = + ( inclusion-double-negation-stable-subtype , + is-double-negation-stable-emb-inclusion-double-negation-stable-subtype) +``` + +### The double negation stable subtype associated to a double negation stable embedding + +```agda +module _ + {l1 l2 : Level} {X : UU l1} {Y : UU l2} (f : X ↪¬¬ Y) + where + + double-negation-stable-subtype-double-negation-stable-emb : + double-negation-stable-subtype (l1 ⊔ l2) Y + pr1 (double-negation-stable-subtype-double-negation-stable-emb y) = + fiber (map-double-negation-stable-emb f) y + pr2 (double-negation-stable-subtype-double-negation-stable-emb y) = + is-double-negation-stable-prop-map-is-double-negation-stable-emb + ( is-double-negation-stable-emb-map-double-negation-stable-emb f) + ( y) + + compute-type-double-negation-stable-type-double-negation-stable-emb : + type-double-negation-stable-subtype + double-negation-stable-subtype-double-negation-stable-emb ≃ + X + compute-type-double-negation-stable-type-double-negation-stable-emb = + equiv-total-fiber (map-double-negation-stable-emb f) + + inv-compute-type-double-negation-stable-type-double-negation-stable-emb : + X ≃ + type-double-negation-stable-subtype + double-negation-stable-subtype-double-negation-stable-emb + inv-compute-type-double-negation-stable-type-double-negation-stable-emb = + inv-equiv-total-fiber (map-double-negation-stable-emb f) +``` + +## Examples + +### The double negation stable subtypes of left and right elements in a coproduct type + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-double-negation-stable-is-left : + (x : A + B) → has-double-negation-elim (is-left x) + is-double-negation-stable-is-left (inl x) = double-negation-elim-unit + is-double-negation-stable-is-left (inr x) = double-negation-elim-empty + + is-left-Double-Negation-Stable-Prop : + A + B → Double-Negation-Stable-Prop lzero + pr1 (is-left-Double-Negation-Stable-Prop x) = is-left x + pr1 (pr2 (is-left-Double-Negation-Stable-Prop x)) = is-prop-is-left x + pr2 (pr2 (is-left-Double-Negation-Stable-Prop x)) = + is-double-negation-stable-is-left x + + is-double-negation-stable-is-right : + (x : A + B) → has-double-negation-elim (is-right x) + is-double-negation-stable-is-right (inl x) = double-negation-elim-empty + is-double-negation-stable-is-right (inr x) = double-negation-elim-unit + + is-right-Double-Negation-Stable-Prop : + A + B → Double-Negation-Stable-Prop lzero + pr1 (is-right-Double-Negation-Stable-Prop x) = is-right x + pr1 (pr2 (is-right-Double-Negation-Stable-Prop x)) = is-prop-is-right x + pr2 (pr2 (is-right-Double-Negation-Stable-Prop x)) = + is-double-negation-stable-is-right x +``` + +## Properties + +### A double negation stable subtype of a `k+1`-truncated type is `k+1`-truncated + +```agda +module _ + {l1 l2 : Level} (k : 𝕋) {A : UU l1} (P : double-negation-stable-subtype l2 A) + where + + abstract + is-trunc-type-double-negation-stable-subtype : + is-trunc (succ-𝕋 k) A → is-trunc (succ-𝕋 k) + (type-double-negation-stable-subtype P) + is-trunc-type-double-negation-stable-subtype = + is-trunc-type-subtype k (subtype-double-negation-stable-subtype P) + +module _ + {l1 l2 : Level} {A : UU l1} (P : double-negation-stable-subtype l2 A) + where + + abstract + is-prop-type-double-negation-stable-subtype : + is-prop A → is-prop (type-double-negation-stable-subtype P) + is-prop-type-double-negation-stable-subtype = + is-prop-type-subtype (subtype-double-negation-stable-subtype P) + + abstract + is-set-type-double-negation-stable-subtype : + is-set A → is-set (type-double-negation-stable-subtype P) + is-set-type-double-negation-stable-subtype = + is-set-type-subtype (subtype-double-negation-stable-subtype P) + + abstract + is-1-type-type-double-negation-stable-subtype : + is-1-type A → is-1-type (type-double-negation-stable-subtype P) + is-1-type-type-double-negation-stable-subtype = + is-1-type-type-subtype (subtype-double-negation-stable-subtype P) + +prop-double-negation-stable-subprop : + {l1 l2 : Level} (A : Prop l1) + (P : double-negation-stable-subtype l2 (type-Prop A)) → + Prop (l1 ⊔ l2) +prop-double-negation-stable-subprop A P = + prop-subprop A (subtype-double-negation-stable-subtype P) + +set-double-negation-stable-subset : + {l1 l2 : Level} (A : Set l1) + (P : double-negation-stable-subtype l2 (type-Set A)) → + Set (l1 ⊔ l2) +set-double-negation-stable-subset A P = + set-subset A (subtype-double-negation-stable-subtype P) +``` + +### The type of double negation stable subtypes of a type is a set + +```agda +is-set-double-negation-stable-subtype : + {l1 l2 : Level} {X : UU l1} → is-set (double-negation-stable-subtype l2 X) +is-set-double-negation-stable-subtype = + is-set-function-type is-set-Double-Negation-Stable-Prop +``` + +### Extensionality of the type of double negation stable subtypes + +```agda +module _ + {l1 l2 : Level} {A : UU l1} (P : double-negation-stable-subtype l2 A) + where + + has-same-elements-double-negation-stable-subtype : + {l3 : Level} → double-negation-stable-subtype l3 A → UU (l1 ⊔ l2 ⊔ l3) + has-same-elements-double-negation-stable-subtype Q = + has-same-elements-subtype + ( subtype-double-negation-stable-subtype P) + ( subtype-double-negation-stable-subtype Q) + + extensionality-double-negation-stable-subtype : + (Q : double-negation-stable-subtype l2 A) → + (P ＝ Q) ≃ has-same-elements-double-negation-stable-subtype Q + extensionality-double-negation-stable-subtype = + extensionality-Π P + ( λ x Q → + ( type-Double-Negation-Stable-Prop (P x)) ↔ + ( type-Double-Negation-Stable-Prop Q)) + ( λ x Q → extensionality-Double-Negation-Stable-Prop (P x) Q) + + has-same-elements-eq-double-negation-stable-subtype : + (Q : double-negation-stable-subtype l2 A) → + (P ＝ Q) → has-same-elements-double-negation-stable-subtype Q + has-same-elements-eq-double-negation-stable-subtype Q = + map-equiv (extensionality-double-negation-stable-subtype Q) + + eq-has-same-elements-double-negation-stable-subtype : + (Q : double-negation-stable-subtype l2 A) → + has-same-elements-double-negation-stable-subtype Q → P ＝ Q + eq-has-same-elements-double-negation-stable-subtype Q = + map-inv-equiv (extensionality-double-negation-stable-subtype Q) + + refl-extensionality-double-negation-stable-subtype : + map-equiv (extensionality-double-negation-stable-subtype P) refl ＝ + (λ x → id , id) + refl-extensionality-double-negation-stable-subtype = refl +``` + +### The type of double negation stable subtypes of `A` is equivalent to the type of all double negation stable embeddings into a type `A` + +```agda +equiv-Fiber-Double-Negation-Stable-Prop : + (l : Level) {l1 : Level} (A : UU l1) → + Σ (UU (l1 ⊔ l)) (λ X → X ↪¬¬ A) ≃ (double-negation-stable-subtype (l1 ⊔ l) A) +equiv-Fiber-Double-Negation-Stable-Prop l A = + ( equiv-Fiber-structure l is-double-negation-stable-prop A) ∘e + ( equiv-tot + ( λ X → + equiv-tot + ( λ f → + ( inv-distributive-Π-Σ) ∘e + ( equiv-product-left (equiv-is-prop-map-is-emb f))))) +``` diff --git a/src/logic/markovian-types.lagda.md b/src/logic/markovian-types.lagda.md new file mode 100644 index 0000000000..54b779cddd --- /dev/null +++ b/src/logic/markovian-types.lagda.md @@ -0,0 +1,101 @@ +# Markovian types + +```agda +module logic.markovian-types where +``` + +
Imports + +```agda +open import elementary-number-theory.natural-numbers + +open import foundation.booleans +open import foundation.decidable-subtypes +open import foundation.dependent-pair-types +open import foundation.disjunction +open import foundation.existential-quantification +open import foundation.function-types +open import foundation.inhabited-types +open import foundation.negation +open import foundation.universal-quantification +open import foundation.universe-levels + +open import foundation-core.identity-types +open import foundation-core.propositions +open import foundation-core.sets + +open import univalent-combinatorics.standard-finite-types +``` + +
+ +## Idea + +A type `A` is {{#concept "Markovian" Disambiguation="type" Agda=is-markovian}} +if, for every [decidable subtype](foundation.decidable-subtypes.md) `𝒫` of `A`, +if `𝒫` is [not](foundation-core.negation.md) [full](foundation.full-subtypes.md) +then there is an element of `A` that is +[not in](foundation.complements-subtypes.md) `𝒫`. + +## Definitions + +### The predicate of being Markovian + +We phrase the condition using the [type of booleans](foundation.booleans.md) so +that the predicate is small. + +```agda +is-markovian : {l : Level} → UU l → UU l +is-markovian A = + (𝒫 : A → bool) → + ¬ ((x : A) → is-true (𝒫 x)) → + exists A (λ x → is-false-Prop (𝒫 x)) + +is-prop-is-markovian : {l : Level} (A : UU l) → is-prop (is-markovian A) +is-prop-is-markovian A = + is-prop-Π (λ 𝒫 → is-prop-function-type (is-prop-exists A (is-false-Prop ∘ 𝒫))) +``` + +### The predicate of being Markovian at a universe level + +```agda +module _ + {l1 : Level} (l2 : Level) (A : UU l1) + where + + is-markovian-prop-Level : Prop (l1 ⊔ lsuc l2) + is-markovian-prop-Level = + Π-Prop + ( decidable-subtype l2 A) + ( λ P → + ¬' (∀' A (subtype-decidable-subtype P)) ⇒ + ∃ A (¬'_ ∘ subtype-decidable-subtype P)) + + is-markovian-Level : UU (l1 ⊔ lsuc l2) + is-markovian-Level = + (P : decidable-subtype l2 A) → + ¬ ((x : A) → is-in-decidable-subtype P x) → + exists A (¬'_ ∘ subtype-decidable-subtype P) + + is-prop-is-markovian-Level : is-prop is-markovian-Level + is-prop-is-markovian-Level = is-prop-type-Prop is-markovian-prop-Level +``` + +## Properties + +### A type is Markovian if and only if it is Markovian at any universe level + +> This remains to be formalized. + +### A type is Markovian if and only if it is Markovian at all universe levels + +> This remains to be formalized. + +## See also + +- [Markov's principle](logic.markovs-principle.md) + +## External links + +- [limited principle of omniscience](https://ncatlab.org/nlab/show/limited+principle+of+omniscience) + at $n$Lab diff --git a/src/logic/markovs-principle.lagda.md b/src/logic/markovs-principle.lagda.md new file mode 100644 index 0000000000..370789da8e --- /dev/null +++ b/src/logic/markovs-principle.lagda.md @@ -0,0 +1,86 @@ +# Markov's principle + +```agda +module logic.markovs-principle where +``` + +
Imports + +```agda +open import elementary-number-theory.natural-numbers + +open import foundation.booleans +open import foundation.decidable-subtypes +open import foundation.dependent-pair-types +open import foundation.disjunction +open import foundation.existential-quantification +open import foundation.function-types +open import foundation.inhabited-types +open import foundation.negation +open import foundation.universal-quantification +open import foundation.universe-levels + +open import foundation-core.identity-types +open import foundation-core.propositions +open import foundation-core.sets + +open import logic.markovian-types + +open import univalent-combinatorics.standard-finite-types +``` + +
+ +## Idea + +{{#concept "Markov's principle" WDID=Q3922074 WD="Markov's principle" Agda=Markov's-Principle}} +asserts that if a [decidable subtype](foundation.decidable-subtypes.md) `𝒫` of +the [natural numbers](elementary-number-theory.natural-numbers.md) `ℕ` is not +[full](foundation.full-subtypes.md), then +[there is](foundation.existential-quantification.md) a natural number `n` that +is not in `𝒫`. + +Markov's principle is an example of a _constructive taboo_. It is a consequence +of the [law of excluded middle](foundation.law-of-excluded-middle.md) that is +not provable generally in constructive mathematics. + +## Definitions + +### Markov's principle + +```agda +Markov's-Principle : UU lzero +Markov's-Principle = is-markovian ℕ +``` + +## Properties + +### Markov's principle is constructively valid for ascending chains of decidable propositions + +**Proof.** Assume given an ascending chain of decidable propositions `Pᵢ ⇒ Pᵢ₊₁` +indexed by the natural numbers `ℕ`. This gives a decidable subtype `𝒫` of `ℕ` +given by `i ∈ 𝒫` iff `Pᵢ` is true. Observe that if `i ∈ 𝒫` then every `j ≥ i` is +also in `𝒫`, and there must exist a least such `i ∈ 𝒫`. Therefore, +`𝒫 = Σ (m ∈ ℕ) (m ≥ k)` for some `k`. So, if `¬ (∀ᵢ Pᵢ)` it is necessarily the +case that `¬ P₀`. + +```agda +markovs-principle-ascending-chains : + {l : Level} (P : ℕ → UU l) + (H : (n : ℕ) → P n → P (succ-ℕ n)) → ¬ ((n : ℕ) → P n) → Σ ℕ (¬_ ∘ P) +markovs-principle-ascending-chains P H q = (0 , λ x → q (ind-ℕ x H)) +``` + +## See also + +- [The principle of omniscience](foundation.principle-of-omniscience.md) +- [The limited principle of omniscience](foundation.limited-principle-of-omniscience.md) +- [The lesser limited principle of omniscience](foundation.lesser-limited-principle-of-omniscience.md) +- [The weak limited principle of omniscience](foundation.weak-limited-principle-of-omniscience.md) + +## External links + +- [Taboos.MarkovsPrinciple](https://martinescardo.github.io/TypeTopology/Taboos.MarkovsPrinciple.html) + at TypeTopology +- [limited principle of omniscience](https://ncatlab.org/nlab/show/limited+principle+of+omniscience) + at $n$Lab diff --git a/src/modal-type-theory/transport-along-crisp-identifications.lagda.md b/src/modal-type-theory/transport-along-crisp-identifications.lagda.md index 5ca52cc131..3405430500 100644 --- a/src/modal-type-theory/transport-along-crisp-identifications.lagda.md +++ b/src/modal-type-theory/transport-along-crisp-identifications.lagda.md @@ -10,6 +10,7 @@ module modal-type-theory.transport-along-crisp-identifications where ```agda open import foundation.action-on-identifications-functions +open import foundation.function-types open import foundation.identity-types open import foundation.universe-levels @@ -119,6 +120,17 @@ crisp-tr-ap {A = A} {B} {C} {D} f g {x} p z = ( p) ``` +### Substitution law for crisp transport + +```agda +substitution-law-crisp-tr : + {@♭ l1 l2 l3 : Level} {@♭ X : UU l1} {@♭ A : UU l2} (@♭ B : @♭ A → UU l3) + (@♭ f : X → A) + {@♭ x y : X} (@♭ p : x ＝ y) {@♭ x' : B (f x)} → + crisp-tr B (ap f p) x' ＝ crisp-tr (λ (@♭ x : X) → B (f x)) p x' +substitution-law-crisp-tr B f p {x'} = crisp-tr-ap f (λ _ → id) p x' +``` + ### Computing maps out of crisp identity types as crisp transports ```agda @@ -152,18 +164,3 @@ crisp-tr-Id-right : crisp-tr-Id-right {a = a} q p = crisp-based-ind-Id (λ y q → crisp-tr (a ＝_) q p ＝ (p ∙ q)) (inv right-unit) q ``` - -### Substitution law for crisp transport - -```agda -substitution-law-crisp-tr : - {@♭ l1 l2 l3 : Level} {@♭ X : UU l1} {@♭ A : UU l2} (@♭ B : @♭ A → UU l3) - (@♭ f : X → A) - {@♭ x y : X} (@♭ p : x ＝ y) {@♭ x' : B (f x)} → - crisp-tr B (ap f p) x' ＝ crisp-tr (λ (@♭ x : X) → B (f x)) p x' -substitution-law-crisp-tr {X = X} B f p {x'} = - crisp-based-ind-Id - ( λ y p → crisp-tr B (ap f p) x' ＝ crisp-tr (λ (@♭ x : X) → B (f x)) p x') - ( refl) - ( p) -``` diff --git a/src/species/small-cauchy-composition-species-of-types-in-subuniverses.lagda.md b/src/species/small-cauchy-composition-species-of-types-in-subuniverses.lagda.md index b535b78e80..78eac4cbc1 100644 --- a/src/species/small-cauchy-composition-species-of-types-in-subuniverses.lagda.md +++ b/src/species/small-cauchy-composition-species-of-types-in-subuniverses.lagda.md @@ -193,12 +193,9 @@ module _ pr2 (equiv-Σ-extension-small-cauchy-composition-unit-subuniverse X) = is-equiv-is-invertible ( λ S → - ( tr + ( inv-tr ( is-in-subuniverse P) - ( eq-equiv - ( ( inv-equiv - ( terminal-map X , is-equiv-terminal-map-is-contr S)) ∘e - ( inv-equiv (compute-raise-unit l1)))) + ( eq-equiv (equiv-raise-unit-is-contr S)) ( C4) , map-equiv-is-small (C5 X) S)) ( λ x → eq-is-prop is-property-is-contr) diff --git a/src/synthetic-homotopy-theory.lagda.md b/src/synthetic-homotopy-theory.lagda.md index 454cf9d9d3..ab877c2eea 100644 --- a/src/synthetic-homotopy-theory.lagda.md +++ b/src/synthetic-homotopy-theory.lagda.md @@ -120,6 +120,7 @@ open import synthetic-homotopy-theory.spheres public open import synthetic-homotopy-theory.suspension-prespectra public open import synthetic-homotopy-theory.suspension-structures public open import synthetic-homotopy-theory.suspensions-of-pointed-types public +open import synthetic-homotopy-theory.suspensions-of-propositions public open import synthetic-homotopy-theory.suspensions-of-types public open import synthetic-homotopy-theory.tangent-spheres public open import synthetic-homotopy-theory.total-cocones-families-sequential-diagrams public diff --git a/src/synthetic-homotopy-theory/codiagonals-of-maps.lagda.md b/src/synthetic-homotopy-theory/codiagonals-of-maps.lagda.md index 20311a86ad..b483d3e9a8 100644 --- a/src/synthetic-homotopy-theory/codiagonals-of-maps.lagda.md +++ b/src/synthetic-homotopy-theory/codiagonals-of-maps.lagda.md @@ -103,12 +103,8 @@ module _ ( fiber f b) ( unit) ( unit) - ( inv-equiv - ( terminal-map (fiber id b) , - ( is-equiv-terminal-map-is-contr (is-torsorial-Id' b)))) - ( inv-equiv - ( terminal-map (fiber id b) , - ( is-equiv-terminal-map-is-contr (is-torsorial-Id' b)))) + ( inv-equiv (equiv-unit-is-contr (is-torsorial-Id' b))) + ( inv-equiv (equiv-unit-is-contr (is-torsorial-Id' b))) ( id-equiv) ( terminal-map (fiber f b)) ( terminal-map (fiber f b)) diff --git a/src/synthetic-homotopy-theory/suspension-structures.lagda.md b/src/synthetic-homotopy-theory/suspension-structures.lagda.md index 761190971f..7d579d350f 100644 --- a/src/synthetic-homotopy-theory/suspension-structures.lagda.md +++ b/src/synthetic-homotopy-theory/suspension-structures.lagda.md @@ -86,7 +86,7 @@ module _ where suspension-structure : UU (l1 ⊔ l2) - suspension-structure = Σ Y (λ N → Σ Y (λ S → (x : X) → N ＝ S)) + suspension-structure = Σ Y (λ N → Σ Y (λ S → X → N ＝ S)) module _ {l1 l2 : Level} {X : UU l1} {Y : UU l2} diff --git a/src/synthetic-homotopy-theory/suspensions-of-propositions.lagda.md b/src/synthetic-homotopy-theory/suspensions-of-propositions.lagda.md new file mode 100644 index 0000000000..0ad717c882 --- /dev/null +++ b/src/synthetic-homotopy-theory/suspensions-of-propositions.lagda.md @@ -0,0 +1,503 @@ +# Suspensions of propositions + +```agda +module synthetic-homotopy-theory.suspensions-of-propositions where +``` + +
Imports + +```agda +open import foundation.booleans +open import foundation.contractible-types +open import foundation.dependent-pair-types +open import foundation.equivalences +open import foundation.existential-quantification +open import foundation.function-extensionality +open import foundation.function-types +open import foundation.fundamental-theorem-of-identity-types +open import foundation.homotopies +open import foundation.identity-types +open import foundation.propositions +open import foundation.sets +open import foundation.subsingleton-induction +open import foundation.surjective-maps +open import foundation.torsorial-type-families +open import foundation.transport-along-identifications +open import foundation.unit-type +open import foundation.univalence +open import foundation.universe-levels + +open import synthetic-homotopy-theory.dependent-suspension-structures +open import synthetic-homotopy-theory.suspension-structures +open import synthetic-homotopy-theory.suspensions-of-types + +open import univalent-combinatorics.kuratowski-finite-sets +``` + +
+ +## Idea + +The +{{#concept "suspension" Disambiguation="of a proposition" Agda=suspension-Prop}} +of a [proposition](foundation-core.propositions.md) `P` is the +[set](foundation.sets.md) satisfying the +[universal property of the pushout](synthetic-homotopy-theory.universal-property-pushouts.md) +of the [span](foundation.spans.md) + +```text + 1 <--- P ---> 1. +``` + +## Definitions + +### The suspension of a proposition + +```agda +module _ + {l : Level} (P : Prop l) + where + + suspension-Prop : UU l + suspension-Prop = suspension (type-Prop P) +``` + +### Suspensions of propositions are sets + +We characterize equality in the suspension of a proposition `P` by the following +four equations + +```text + (N ＝ N) ≃ 1 + (N ＝ S) ≃ P + (S ＝ N) ≃ P + (S ＝ S) ≃ 1. +``` + +Since these are all propositions, `ΣP` forms a set. This is Lemma 10.1.13 in +{{#cite UF13}}. + +First we define the observational equality family on `ΣP`. + +```agda +module _ + {l : Level} (P : Prop l) + where + + suspension-structure-Eq-north-suspension-Prop : + suspension-structure (type-Prop P) (UU l) + suspension-structure-Eq-north-suspension-Prop = + ( raise-unit l) , + ( type-Prop P) , + ( λ x → + inv + ( eq-equiv + ( equiv-raise-unit-is-contr (is-proof-irrelevant-type-Prop P x)))) + + Eq-north-suspension-Prop : + suspension (type-Prop P) → UU l + Eq-north-suspension-Prop = + cogap-suspension suspension-structure-Eq-north-suspension-Prop + + suspension-structure-Eq-south-suspension-Prop : + suspension-structure (type-Prop P) (UU l) + suspension-structure-Eq-south-suspension-Prop = + ( type-Prop P) , + ( raise-unit l) , + ( λ x → + eq-equiv (equiv-raise-unit-is-contr (is-proof-irrelevant-type-Prop P x))) + + Eq-south-suspension-Prop : + suspension (type-Prop P) → UU l + Eq-south-suspension-Prop = + cogap-suspension suspension-structure-Eq-south-suspension-Prop + + abstract + coh-Eq-suspension-Prop : + type-Prop P → Eq-north-suspension-Prop ~ Eq-south-suspension-Prop + coh-Eq-suspension-Prop x = + dependent-cogap-suspension + ( eq-value + ( Eq-north-suspension-Prop) + ( Eq-south-suspension-Prop)) + ( ( ( compute-north-cogap-suspension + ( suspension-structure-Eq-north-suspension-Prop)) ∙ + ( inv + ( eq-equiv + ( equiv-raise-unit-is-contr + ( is-proof-irrelevant-type-Prop P x)))) ∙ + ( inv + ( compute-north-cogap-suspension + ( suspension-structure-Eq-south-suspension-Prop)))) , + ( ( compute-south-cogap-suspension + ( suspension-structure-Eq-north-suspension-Prop)) ∙ + ( eq-equiv + ( equiv-raise-unit-is-contr + ( is-proof-irrelevant-type-Prop P x))) ∙ + ( inv + ( compute-south-cogap-suspension + ( suspension-structure-Eq-south-suspension-Prop)))) , + ind-subsingleton + ( is-prop-type-Prop P) + ( x) + ( eq-is-contr + ( is-contr-equiv + ( type-Prop P ≃ raise-unit l) + ( equiv-univalence ∘e + equiv-binary-concat + ( inv + ( compute-south-cogap-suspension + ( suspension-structure-Eq-north-suspension-Prop))) + ( compute-south-cogap-suspension + ( suspension-structure-Eq-south-suspension-Prop))) + ( is-contr-equiv-is-contr + ( is-proof-irrelevant-type-Prop P x) + ( is-contr-raise-unit))))) + + suspension-structure-Eq-suspension-Prop : + suspension-structure (type-Prop P) (suspension-Prop P → UU l) + suspension-structure-Eq-suspension-Prop = + ( Eq-north-suspension-Prop , + Eq-south-suspension-Prop , + eq-htpy ∘ coh-Eq-suspension-Prop) + + Eq-suspension-Prop : + suspension-Prop P → suspension-Prop P → UU l + Eq-suspension-Prop = + cogap-suspension suspension-structure-Eq-suspension-Prop +``` + +We compute the observational equality of the suspension of a proposition at the +poles. + +```agda + eq-compute-Eq-north-north-suspension-Prop : + Eq-suspension-Prop north-suspension north-suspension ＝ raise-unit l + eq-compute-Eq-north-north-suspension-Prop = + htpy-eq + ( compute-north-cogap-suspension suspension-structure-Eq-suspension-Prop) + ( north-suspension) ∙ + compute-north-cogap-suspension suspension-structure-Eq-north-suspension-Prop + + compute-Eq-north-north-suspension-Prop : + Eq-suspension-Prop north-suspension north-suspension ≃ unit + compute-Eq-north-north-suspension-Prop = + inv-compute-raise-unit l ∘e + equiv-eq eq-compute-Eq-north-north-suspension-Prop + + is-contr-Eq-north-north-suspension-Prop : + is-contr (Eq-suspension-Prop north-suspension north-suspension) + is-contr-Eq-north-north-suspension-Prop = + is-contr-equiv-raise-unit + ( equiv-eq eq-compute-Eq-north-north-suspension-Prop) + + eq-compute-Eq-south-south-suspension-Prop : + Eq-suspension-Prop south-suspension south-suspension ＝ raise-unit l + eq-compute-Eq-south-south-suspension-Prop = + htpy-eq + ( compute-south-cogap-suspension suspension-structure-Eq-suspension-Prop) + ( south-suspension) ∙ + compute-south-cogap-suspension suspension-structure-Eq-south-suspension-Prop + + compute-Eq-south-south-suspension-Prop : + Eq-suspension-Prop south-suspension south-suspension ≃ unit + compute-Eq-south-south-suspension-Prop = + inv-compute-raise-unit l ∘e + equiv-eq eq-compute-Eq-south-south-suspension-Prop + + is-contr-Eq-south-south-suspension-Prop : + is-contr (Eq-suspension-Prop south-suspension south-suspension) + is-contr-Eq-south-south-suspension-Prop = + is-contr-equiv-raise-unit + ( equiv-eq eq-compute-Eq-south-south-suspension-Prop) + + eq-compute-Eq-north-south-suspension-Prop : + Eq-suspension-Prop north-suspension south-suspension ＝ type-Prop P + eq-compute-Eq-north-south-suspension-Prop = + htpy-eq + ( compute-north-cogap-suspension suspension-structure-Eq-suspension-Prop) + ( south-suspension) ∙ + compute-south-cogap-suspension suspension-structure-Eq-north-suspension-Prop + + eq-compute-Eq-south-north-suspension-Prop : + Eq-suspension-Prop south-suspension north-suspension ＝ type-Prop P + eq-compute-Eq-south-north-suspension-Prop = + htpy-eq + ( compute-south-cogap-suspension suspension-structure-Eq-suspension-Prop) + ( north-suspension) ∙ + compute-north-cogap-suspension suspension-structure-Eq-south-suspension-Prop +``` + +The observational equality on the suspension of a proposition is propositional. + +```agda + abstract + is-prop-Eq-north-suspension-Prop : + (y : suspension-Prop P) → is-prop (Eq-north-suspension-Prop y) + is-prop-Eq-north-suspension-Prop = + dependent-cogap-suspension + ( λ x → is-prop (Eq-north-suspension-Prop x)) + ( ( inv-tr + ( is-prop) + ( compute-north-cogap-suspension + ( suspension-structure-Eq-north-suspension-Prop)) + ( is-prop-raise-unit)) , + ( inv-tr + ( is-prop) + ( compute-south-cogap-suspension + ( suspension-structure-Eq-north-suspension-Prop)) + ( is-prop-type-Prop P)) , + ( λ _ → + eq-is-prop + ( is-prop-is-prop (Eq-north-suspension-Prop south-suspension)))) + + abstract + is-prop-Eq-south-suspension-Prop : + (y : suspension-Prop P) → is-prop (Eq-south-suspension-Prop y) + is-prop-Eq-south-suspension-Prop = + dependent-cogap-suspension + ( λ x → is-prop (Eq-south-suspension-Prop x)) + ( ( inv-tr + ( is-prop) + ( compute-north-cogap-suspension + ( suspension-structure-Eq-south-suspension-Prop)) + ( is-prop-type-Prop P)) , + ( inv-tr + ( is-prop) + ( compute-south-cogap-suspension + ( suspension-structure-Eq-south-suspension-Prop)) + ( is-prop-raise-unit)) , + ( λ _ → + eq-is-prop + ( is-prop-is-prop (Eq-south-suspension-Prop south-suspension)))) + + abstract + is-prop-Eq-suspension-Prop : + (x y : suspension-Prop P) → is-prop (Eq-suspension-Prop x y) + is-prop-Eq-suspension-Prop x y = + dependent-cogap-suspension + ( λ x → is-prop (Eq-suspension-Prop x y)) + ( ( inv-tr + ( is-prop) + ( htpy-eq + ( compute-north-cogap-suspension + ( suspension-structure-Eq-suspension-Prop)) + ( y)) + ( is-prop-Eq-north-suspension-Prop y)) , + ( inv-tr + ( is-prop) + ( htpy-eq + ( compute-south-cogap-suspension + ( suspension-structure-Eq-suspension-Prop)) + ( y)) + ( is-prop-Eq-south-suspension-Prop y)) , + ( λ _ → + eq-is-prop + ( is-prop-is-prop (Eq-suspension-Prop south-suspension y)))) + ( x) +``` + +The observational equality characterizes equality in `ΣP`. + +```agda + refl-Eq-suspension-Prop : (x : suspension-Prop P) → Eq-suspension-Prop x x + refl-Eq-suspension-Prop = + dependent-cogap-suspension + ( λ x → Eq-suspension-Prop x x) + ( ( map-inv-eq + ( ( htpy-eq + ( compute-north-cogap-suspension + ( suspension-structure-Eq-suspension-Prop)) + ( north-suspension)) ∙ + ( compute-north-cogap-suspension + ( suspension-structure-Eq-north-suspension-Prop))) + ( raise-star)) , + ( map-inv-eq + ( ( htpy-eq + ( compute-south-cogap-suspension + ( suspension-structure-Eq-suspension-Prop)) + ( south-suspension)) ∙ + ( compute-south-cogap-suspension + ( suspension-structure-Eq-south-suspension-Prop))) + ( raise-star)) , + ( λ _ → + eq-is-contr + ( inv-tr + ( is-contr) + ( eq-compute-Eq-south-south-suspension-Prop) + ( is-contr-raise-unit)))) + + Eq-eq-suspension-Prop : + (x y : suspension-Prop P) → x ＝ y → Eq-suspension-Prop x y + Eq-eq-suspension-Prop x .x refl = refl-Eq-suspension-Prop x + + dependent-suspension-structure-eq-north-Eq-north-suspension-Prop : + dependent-suspension-structure + ( λ y → Eq-suspension-Prop north-suspension y → north-suspension ＝ y) + ( suspension-structure-suspension (type-Prop P)) + dependent-suspension-structure-eq-north-Eq-north-suspension-Prop = + ( ( λ _ → refl) , + ( λ u → + meridian-suspension + ( map-eq eq-compute-Eq-north-south-suspension-Prop u)) , + ( λ p → + eq-is-contr + ( is-contr-function-type + ( is-prop-is-contr + ( is-contr-suspension-is-contr + ( is-proof-irrelevant-type-Prop P p)) + ( north-suspension) + ( south-suspension))))) + + eq-north-Eq-north-suspension-Prop : + (y : suspension-Prop P) → + Eq-suspension-Prop north-suspension y → north-suspension ＝ y + eq-north-Eq-north-suspension-Prop = + dependent-cogap-suspension + ( λ y → Eq-suspension-Prop north-suspension y → north-suspension ＝ y) + ( dependent-suspension-structure-eq-north-Eq-north-suspension-Prop) + + dependent-suspension-structure-eq-south-Eq-south-suspension-Prop : + dependent-suspension-structure + ( λ y → Eq-suspension-Prop south-suspension y → south-suspension ＝ y) + ( suspension-structure-suspension (type-Prop P)) + dependent-suspension-structure-eq-south-Eq-south-suspension-Prop = + ( ( λ u → + inv + ( meridian-suspension + ( map-eq eq-compute-Eq-south-north-suspension-Prop u))) , + ( λ _ → refl) , + ( λ p → + eq-is-contr + ( is-contr-function-type + ( is-prop-is-contr + ( is-contr-suspension-is-contr + ( is-proof-irrelevant-type-Prop P p)) + ( south-suspension) + ( south-suspension))))) + + eq-south-Eq-south-suspension-Prop : + (y : suspension-Prop P) → + Eq-suspension-Prop south-suspension y → south-suspension ＝ y + eq-south-Eq-south-suspension-Prop = + dependent-cogap-suspension + ( λ y → Eq-suspension-Prop south-suspension y → south-suspension ＝ y) + ( dependent-suspension-structure-eq-south-Eq-south-suspension-Prop) + + eq-Eq-suspension-Prop : + (x y : suspension-Prop P) → Eq-suspension-Prop x y → x ＝ y + eq-Eq-suspension-Prop = + dependent-cogap-suspension + ( λ x → (y : suspension-Prop P) → Eq-suspension-Prop x y → x ＝ y) + ( ( eq-north-Eq-north-suspension-Prop) , + ( eq-south-Eq-south-suspension-Prop) , + ( λ p → + eq-is-contr + ( is-contr-Π + ( λ y → + is-contr-function-type + ( is-prop-is-contr + ( is-contr-suspension-is-contr + ( is-proof-irrelevant-type-Prop P p)) + ( south-suspension) + ( y)))))) + + abstract + is-equiv-Eq-eq-suspension-Prop : + (x y : suspension-Prop P) → is-equiv (Eq-eq-suspension-Prop x y) + is-equiv-Eq-eq-suspension-Prop x = + fundamental-theorem-id-section + ( x) + ( Eq-eq-suspension-Prop x) + ( λ y → + ( eq-Eq-suspension-Prop x y) , + ( λ _ → eq-is-prop (is-prop-Eq-suspension-Prop x y))) + + abstract + is-equiv-eq-Eq-suspension-Prop : + (x y : suspension-Prop P) → is-equiv (eq-Eq-suspension-Prop x y) + is-equiv-eq-Eq-suspension-Prop x = + fundamental-theorem-id-retraction + ( x) + ( eq-Eq-suspension-Prop x) + ( λ y → + ( Eq-eq-suspension-Prop x y) , + ( λ _ → eq-is-prop (is-prop-Eq-suspension-Prop x y))) + + equiv-Eq-eq-suspension-Prop : + (x y : suspension-Prop P) → (x ＝ y) ≃ Eq-suspension-Prop x y + equiv-Eq-eq-suspension-Prop x y = + ( Eq-eq-suspension-Prop x y , is-equiv-Eq-eq-suspension-Prop x y) + + equiv-eq-Eq-suspension-Prop : + (x y : suspension-Prop P) → Eq-suspension-Prop x y ≃ (x ＝ y) + equiv-eq-Eq-suspension-Prop x y = + ( eq-Eq-suspension-Prop x y , is-equiv-eq-Eq-suspension-Prop x y) + + abstract + is-torsorial-Eq-suspension-Prop : + (x : suspension-Prop P) → is-torsorial (Eq-suspension-Prop x) + is-torsorial-Eq-suspension-Prop x = + fundamental-theorem-id' + ( Eq-eq-suspension-Prop x) + ( is-equiv-Eq-eq-suspension-Prop x) +``` + +Piecing it all together, we conclude that suspensions of propositions are sets. + +```agda + abstract + is-set-suspension-Prop : is-set (suspension-Prop P) + is-set-suspension-Prop x y = + is-prop-equiv + ( equiv-Eq-eq-suspension-Prop x y) + ( is-prop-Eq-suspension-Prop x y) + + suspension-set-Prop : Set l + suspension-set-Prop = suspension-Prop P , is-set-suspension-Prop +``` + +```agda + compute-eq-north-south-suspension-Prop : + (north-suspension {X = type-Prop P} ＝ south-suspension) ≃ type-Prop P + compute-eq-north-south-suspension-Prop = + equiv-eq eq-compute-Eq-north-south-suspension-Prop ∘e + equiv-Eq-eq-suspension-Prop north-suspension south-suspension + + compute-eq-south-north-suspension-Prop : + (south-suspension {X = type-Prop P} ＝ north-suspension) ≃ type-Prop P + compute-eq-south-north-suspension-Prop = + equiv-eq eq-compute-Eq-south-north-suspension-Prop ∘e + equiv-Eq-eq-suspension-Prop south-suspension north-suspension +``` + +### Suspensions of propositions are Kuratowski finite sets + +```agda +module _ + {l : Level} (P : Prop l) + where + + is-kuratowski-finite-set-suspension-Prop : + is-kuratowski-finite-set (suspension-set-Prop P) + is-kuratowski-finite-set-suspension-Prop = + intro-exists 2 + ( comp-surjection + ( surjection-bool-suspension) + ( surjection-equiv equiv-bool-Fin-two-ℕ)) +``` + +## See also + +- Suspensions of propositions are used in the proof of + [Diaconescu's theorem](foundation.diaconescus-theorem.md), which states that + the axiom of choice implies the law of excluded middle. + +## References + +{{#bibliography}} + +## External links + +- [Truncatedness, properties of suspensions](https://1lab.dev/Homotopy.Space.Suspension.Properties.html#truncatedness) + at 1lab diff --git a/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md b/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md index b66b91c00a..b0d6084cbc 100644 --- a/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md +++ b/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md @@ -9,6 +9,7 @@ module synthetic-homotopy-theory.suspensions-of-types where ```agda open import foundation.action-on-identifications-dependent-functions open import foundation.action-on-identifications-functions +open import foundation.booleans open import foundation.commuting-squares-of-identifications open import foundation.connected-types open import foundation.constant-maps @@ -18,6 +19,7 @@ open import foundation.dependent-pair-types open import foundation.diagonal-maps-of-types open import foundation.equivalence-extensionality open import foundation.equivalences +open import foundation.fibers-of-maps open import foundation.function-extensionality open import foundation.function-types open import foundation.functoriality-dependent-function-types @@ -25,8 +27,11 @@ open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types open import foundation.path-algebra +open import foundation.propositional-truncations +open import foundation.propositions open import foundation.retractions open import foundation.sections +open import foundation.surjective-maps open import foundation.torsorial-type-families open import foundation.transport-along-identifications open import foundation.truncated-types @@ -50,12 +55,12 @@ open import synthetic-homotopy-theory.universal-property-suspensions ## Idea -The **suspension** of a type `X` is the -[pushout](synthetic-homotopy-theory.pushouts.md) of the +The {{#concept "suspension" WD="suspension" WDID=Q1307987 Agda=suspension}} of a +type `X` is the [pushout](synthetic-homotopy-theory.pushouts.md) of the [span](foundation.spans.md) ```text -unit <-- X --> unit + 1 <--- X ---> 1 ``` Suspensions play an important role in synthetic homotopy theory. For example, @@ -488,9 +493,9 @@ module _ ( c))) is-section-htpy-htpy-function-out-of-suspension : - ( ( htpy-function-out-of-suspension-htpy) ∘ - ( htpy-htpy-function-out-of-suspension)) ~ - ( id) + is-section + htpy-function-out-of-suspension-htpy + htpy-htpy-function-out-of-suspension is-section-htpy-htpy-function-out-of-suspension c = ( ap ( htpy-function-out-of-suspension-htpy) @@ -516,22 +521,54 @@ module _ ( is-section-htpy-htpy-function-out-of-suspension c)) ``` +### The booleans surject onto suspensions + +```agda +module _ + {l : Level} {X : UU l} + where + + map-surjection-bool-suspension : bool → suspension X + map-surjection-bool-suspension true = north-suspension + map-surjection-bool-suspension false = south-suspension + + is-surjective-map-surjection-bool-suspension : + is-surjective map-surjection-bool-suspension + is-surjective-map-surjection-bool-suspension = + dependent-cogap-suspension + ( λ x → ║ fiber map-surjection-bool-suspension x ║₋₁) + ( unit-trunc-Prop (true , refl) , + unit-trunc-Prop (false , refl) , + ( λ _ → + eq-type-Prop + ( trunc-Prop + ( fiber + ( map-surjection-bool-suspension) + ( south-suspension))))) + + surjection-bool-suspension : bool ↠ suspension X + surjection-bool-suspension = + map-surjection-bool-suspension , + is-surjective-map-surjection-bool-suspension +``` + ### The suspension of a contractible type is contractible ```agda -is-contr-suspension-is-contr : - {l : Level} {X : UU l} → is-contr X → is-contr (suspension X) -is-contr-suspension-is-contr {l} {X} is-contr-X = - is-contr-is-equiv' - ( unit) - ( pr1 (pr2 (cocone-suspension X))) - ( is-equiv-universal-property-pushout - ( terminal-map X) - ( terminal-map X) - ( cocone-suspension X) - ( is-equiv-is-contr (terminal-map X) is-contr-X is-contr-unit) - ( up-suspension' X)) - ( is-contr-unit) +abstract + is-contr-suspension-is-contr : + {l : Level} {X : UU l} → is-contr X → is-contr (suspension X) + is-contr-suspension-is-contr {X = X} is-contr-X = + is-contr-is-equiv' + ( unit) + ( pr1 (pr2 (cocone-suspension X))) + ( is-equiv-universal-property-pushout + ( terminal-map X) + ( terminal-map X) + ( cocone-suspension X) + ( is-equiv-is-contr (terminal-map X) is-contr-X is-contr-unit) + ( up-suspension' X)) + ( is-contr-unit) ``` ### Suspensions increase connectedness diff --git a/src/univalent-combinatorics.lagda.md b/src/univalent-combinatorics.lagda.md index d0099b7719..afd6a44b8c 100644 --- a/src/univalent-combinatorics.lagda.md +++ b/src/univalent-combinatorics.lagda.md @@ -43,6 +43,7 @@ open import univalent-combinatorics.cubes public open import univalent-combinatorics.cycle-partitions public open import univalent-combinatorics.cycle-prime-decomposition-natural-numbers public open import univalent-combinatorics.cyclic-finite-types public +open import univalent-combinatorics.de-morgans-law public open import univalent-combinatorics.decidable-dependent-function-types public open import univalent-combinatorics.decidable-dependent-pair-types public open import univalent-combinatorics.decidable-equivalence-relations public diff --git a/src/univalent-combinatorics/de-morgans-law.lagda.md b/src/univalent-combinatorics/de-morgans-law.lagda.md new file mode 100644 index 0000000000..bca666f108 --- /dev/null +++ b/src/univalent-combinatorics/de-morgans-law.lagda.md @@ -0,0 +1,98 @@ +# De Morgan's law for finite families of propositions + +```agda +module univalent-combinatorics.de-morgans-law where +``` + +
Imports + +```agda +open import elementary-number-theory.natural-numbers + +open import foundation.coproduct-types +open import foundation.decidable-dependent-pair-types +open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.empty-types +open import foundation.equivalences +open import foundation.existential-quantification +open import foundation.function-types +open import foundation.functoriality-dependent-pair-types +open import foundation.negation +open import foundation.unit-type +open import foundation.universe-levels + +open import logic.de-morgan-propositions +open import logic.de-morgan-types + +open import univalent-combinatorics.counting +open import univalent-combinatorics.standard-finite-types +``` + +
+ +## Idea + +Given a [finite](univalent-combinatorics.finite-types.md) family of +[De Morgan types](logic.de-morgan-types.md) `P : Fin k → De-Morgan-Type`, then +the _finitary De Morgan's law_ + +```text + ¬ (∀ i, P i) ⇒ ∃ i, ¬ (P i) +``` + +holds. + +## Result + +```agda +cases-decide-de-morgans-law-family-Fin-is-de-morgan-family : + {l : Level} (k : ℕ) {P : Fin (succ-ℕ k) → UU l} → + ((x : Fin (succ-ℕ k)) → is-de-morgan (P x)) → + ¬ ((x : Fin (succ-ℕ k)) → P x) → + ¬ P (inr star) + ¬ ((x : Fin k) → P (inl x)) +cases-decide-de-morgans-law-family-Fin-is-de-morgan-family k {P} H q = + rec-coproduct + ( inl) + ( λ z → inr (λ y → z (λ y' → q (ind-coproduct P y (λ _ → y'))))) + ( H (inr star)) + +decide-de-morgans-law-family-Fin-is-de-morgan-family : + {l : Level} (k : ℕ) {P : Fin k → UU l} → + ((x : Fin k) → is-de-morgan (P x)) → + ¬ ((x : Fin k) → P x) → + Σ (Fin k) (¬_ ∘ P) +decide-de-morgans-law-family-Fin-is-de-morgan-family zero-ℕ {P} H q = + q (λ ()) , (λ x → q (λ ())) +decide-de-morgans-law-family-Fin-is-de-morgan-family (succ-ℕ k) {P} H q = + rec-coproduct + ( inr star ,_) + ( λ q' → + map-Σ-map-base inl + ( ¬_ ∘ P) + ( decide-de-morgans-law-family-Fin-is-de-morgan-family k (H ∘ inl) q')) + ( cases-decide-de-morgans-law-family-Fin-is-de-morgan-family k H q) +``` + +```agda +module _ + {l : Level} {k : ℕ} (P : Fin k → De-Morgan-Type l) + where + + decide-de-morgans-law-family-family-Fin : + ¬ ((x : Fin k) → type-De-Morgan-Type (P x)) → + Σ (Fin k) (¬_ ∘ type-De-Morgan-Type ∘ P) + decide-de-morgans-law-family-family-Fin = + decide-de-morgans-law-family-Fin-is-de-morgan-family k + ( is-de-morgan-type-De-Morgan-Type ∘ P) +``` + +## Comment + +It is likely that this finitary De Morgan's law can be generalized to families +of De Morgan types indexed by _searchable types_ in the sense of Escardó +{{#cite Esc08}}. + +## References + +{{#bibliography}} diff --git a/src/univalent-combinatorics/kuratowski-finite-sets.lagda.md b/src/univalent-combinatorics/kuratowski-finite-sets.lagda.md index 8255b3ad10..1917de6c09 100644 --- a/src/univalent-combinatorics/kuratowski-finite-sets.lagda.md +++ b/src/univalent-combinatorics/kuratowski-finite-sets.lagda.md @@ -89,6 +89,10 @@ has-decidable-equality-is-finite-type-𝔽-Kuratowski X = has-decidable-equality-is-finite ``` +### Kuratowski finite sets are Markovian + +> This remains to be formalized. ([Markovian types](logic.markovian-types.md)) + ## See also - [Finite types](univalent-combinatorics.finite-types.md)