{- Definitions for functions -} {-# OPTIONS --safe #-} module Cubical.Foundations.Function where open import Cubical.Foundations.Prelude private variable ℓ ℓ' ℓ'' : Level A : Type ℓ B : A → Type ℓ C : (a : A) → B a → Type ℓ D : (a : A) (b : B a) → C a b → Type ℓ -- The identity function idfun : (A : Type ℓ) → A → A idfun _ x = x infixr 9 _∘_ _∘_ : (g : {a : A} → (b : B a) → C a b) → (f : (a : A) → B a) → (a : A) → C a (f a) g ∘ f = λ x → g (f x) {-# INLINE _∘_ #-} ∘-assoc : (h : {a : A} {b : B a} → (c : C a b) → D a b c) (g : {a : A} → (b : B a) → C a b) (f : (a : A) → B a) → (h ∘ g) ∘ f ≡ h ∘ (g ∘ f) ∘-assoc h g f i x = h (g (f x)) ∘-idˡ : (f : (a : A) → B a) → f ∘ idfun A ≡ f ∘-idˡ f i x = f x ∘-idʳ : (f : (a : A) → B a) → (λ {a} → idfun (B a)) ∘ f ≡ f ∘-idʳ f i x = f x flip : {B : Type ℓ'} {C : A → B → Type ℓ''} → ((x : A) (y : B) → C x y) → ((y : B) (x : A) → C x y) flip f y x = f x y {-# INLINE flip #-} const : {B : Type ℓ} → A → B → A const x = λ _ → x {-# INLINE const #-} case_of_ : ∀ {ℓ ℓ'} {A : Type ℓ} {B : A → Type ℓ'} → (x : A) → (∀ x → B x) → B x case x of f = f x {-# INLINE case_of_ #-} case_return_of_ : ∀ {ℓ ℓ'} {A : Type ℓ} (x : A) (B : A → Type ℓ') → (∀ x → B x) → B x case x return P of f = f x {-# INLINE case_return_of_ #-} uncurry : ((x : A) → (y : B x) → C x y) → (p : Σ A B) → C (fst p) (snd p) uncurry f (x , y) = f x y curry : ((p : Σ A B) → C (fst p) (snd p)) → (x : A) → (y : B x) → C x y curry f x y = f (x , y) module _ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} where -- Notions of 'coherently constant' functions for low dimensions. -- These are the properties of functions necessary to e.g. eliminate -- from the propositional truncation. -- 2-Constant functions are coherently constant if B is a set. 2-Constant : (A → B) → Type _ 2-Constant f = ∀ x y → f x ≡ f y 2-Constant-isProp : isSet B → (f : A → B) → isProp (2-Constant f) 2-Constant-isProp Bset f link1 link2 i x y j = Bset (f x) (f y) (link1 x y) (link2 x y) i j -- 3-Constant functions are coherently constant if B is a groupoid. record 3-Constant (f : A → B) : Type (ℓ-max ℓ ℓ') where field link : 2-Constant f coh₁ : ∀ x y z → Square (link x y) (link x z) refl (link y z) coh₂ : ∀ x y z → Square (link x z) (link y z) (link x y) refl coh₂ x y z i j = hcomp (λ k → λ { (j = i0) → link x y i ; (i = i0) → link x z (j ∧ k) ; (j = i1) → link x z (i ∨ k) ; (i = i1) → link y z j }) (coh₁ x y z j i) link≡refl : ∀ x → refl ≡ link x x link≡refl x i j = hcomp (λ k → λ { (i = i0) → link x x (j ∧ ~ k) ; (i = i1) → link x x j ; (j = i0) → f x ; (j = i1) → link x x (~ i ∧ ~ k) }) (coh₁ x x x (~ i) j) downleft : ∀ x y → Square (link x y) refl refl (link y x) downleft x y i j = hcomp (λ k → λ { (i = i0) → link x y j ; (i = i1) → link≡refl x (~ k) j ; (j = i0) → f x ; (j = i1) → link y x i }) (coh₁ x y x i j) link≡symlink : ∀ x y → link x y ≡ sym (link y x) link≡symlink x y i j = hcomp (λ k → λ { (i = i0) → link x y j ; (i = i1) → link y x (~ j ∨ ~ k) ; (j = i0) → f x ; (j = i1) → link y x (i ∧ ~ k) }) (downleft x y i j) homotopyNatural : {B : Type ℓ'} {f g : A → B} (H : ∀ a → f a ≡ g a) {x y : A} (p : x ≡ y) → H x ∙ cong g p ≡ cong f p ∙ H y homotopyNatural {f = f} {g = g} H {x} {y} p i j = hcomp (λ k → λ { (i = i0) → compPath-filler (H x) (cong g p) k j ; (i = i1) → compPath-filler' (cong f p) (H y) k j ; (j = i0) → cong f p (i ∧ ~ k) ; (j = i1) → cong g p (i ∨ k) }) (H (p i) j) homotopySymInv : {f : A → A} (H : ∀ a → f a ≡ a) (a : A) → Path (f a ≡ f a) (λ i → H (H a (~ i)) i) refl homotopySymInv {f = f} H a j i = hcomp (λ k → λ { (i = i0) → f a ; (i = i1) → H a (j ∧ ~ k) ; (j = i0) → H (H a (~ i)) i ; (j = i1) → H a (i ∧ ~ k)}) (H (H a (~ i ∨ j)) i)