{- Pi Type: - Pi; - FunExt. Copyright (c) Groupoid Infinity, 2014-2018. HoTT 1.5 Product types HoTT 2.9 Pi-types and the function extensionality axiom. -} module pi where import proto import prop import path -- Pi Formation Pi (A: U) (B: A -> U) : U = (x:A) -> B(x) -- Pi Intro lam (A: U) (B: A -> U) (b: Pi A B): Pi A B = \(x: A) -> b x -- Pi Elim app (A: U) (B: A -> U) (f: Pi A B) (a: A): B a = f a -- Pi Computation Pi_Beta (A: U) (B: A -> U) (f: Pi A B) : Path (Pi A B) (\(x:A) -> f x) f = refl (Pi A B) f -- Pi Uniqueness Pi_Eta (A: U) (B: A -> U) (f: Pi A B) : Path (Pi A B) f (\(x:A) -> f x) = refl (Pi A B) f -- FunExt Type funext_form (A B: U) (f g: A -> B): U = Path (A -> B) f g -- funext Intro funext (A B: U) (f g: A -> B) (p: (x:A) -> Path B (f x) (g x)) : funext_form A B f g = \(a: A) -> p a @ i -- funext Elim happly (A B: U) (f g: A -> B) (p: funext_form A B f g) (x: A) : Path B (f x) (g x) = cong (A -> B) B (\(h: A -> B) -> apply A B h x) f g p -- funext Computation funext_Beta (A B: U) (f g: A -> B) (p: (x:A) -> Path B (f x) (g x)) : (x:A) -> Path B (f x) (g x) = \(x:A) -> happly A B f g (funext A B f g p) x -- funext Uniqueness funext_Eta (A B: U) (f g: A -> B) (p: Path (A -> B) f g) : Path (Path (A -> B) f g) (funext A B f g (happly A B f g p)) p = refl (Path (A -> B) f g) p -- dependent funext piext (A: U) (B: A -> U) (f g: (x:A) -> B x) (p: (x:A) -> Path (B x) (f x) (g x)) : Path ((y:A) -> B y) f g = \(a: A) -> (p a) @ i -- if pi is set and two functions are equal in two ways then these ways are contractible setPi (A: U) (B: A -> U) (h: (x: A) -> isSet (B x)) (f g: Pi A B) (p q: Path (Pi A B) f g) : Path (Path (Pi A B) f g) p q = \(x: A) -> (h x (f x) (g x) ((p@i)x) ((q@i)x)) @ i @ j -- pi is set on codomain setFun' (X Y: U) (p: X -> isSet Y) : isSet (X -> Y) = setPi X (\(_: X) -> Y) p setFun (A B : U) (sB: isSet B) : isSet (A -> B) = setPi A (\(x: A) -> B) (\(x: A) -> sB) -- if pi is contractible on domain and codomain then whole space is contractible piIsContr (A: U) (B: A -> U) (u: isContr A) (q: (x: A) -> isContr (B x)) : isContr (Pi A B) = (g,r) where a: A = u.1 p: (x:A) -> Path A a x = u.2 g (x:A): B x = (q x).1 h (x:A): (y:B x) -> Path (B x) (g x) y = (q x).2 r (z:Pi A B): Path (Pi A B) g z = piext A B g z (\(x:A) -> h x (z x))