Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Floris van Doorn
Ported from Coq HoTT
Theorems about pi-types (dependent function spaces)
-/
import types.sigma
open eq equiv is_equiv funext
namespace pi
universe variables l k
variables {A A' : Type.{l}} {B : A → Type.{k}} {B' : A' → Type.{k}} {C : Πa, B a → Type}
{D : Πa b, C a b → Type}
{a a' a'' : A} {b b₁ b₂ : B a} {b' : B a'} {b'' : B a''} {f g : Πa, B a}
/- Paths -/
/- Paths [p : f ≈ g] in a function type [Πx:X, P x] are equivalent to functions taking values in path types, [H : Πx:X, f x ≈ g x], or concisely, [H : f ∼ g].
This equivalence, however, is just the combination of [apD10] and function extensionality [funext], and as such, [path_forall], et seq. are given in axioms.funext and path: -/
/- Now we show how these things compute. -/
definition apD10_path_pi [H : funext] (h : f ∼ g) : apD10 (path_pi h) ∼ h :=
apD10 (retr apD10 h)
definition path_pi_eta [H : funext] (p : f = g) : path_pi (apD10 p) = p :=
sect apD10 p
definition path_pi_idp [H : funext] : path_pi (λx : A, refl (f x)) = refl f :=
!path_pi_eta
/- The identification of the path space of a dependent function space, up to equivalence, is of course just funext. -/
definition path_equiv_homotopy [H : funext] (f g : Πx, B x) : (f = g) ≃ (f ∼ g) :=
equiv.mk _ !funext.ap
definition is_equiv_path_pi [instance] [H : funext] (f g : Πx, B x)
: is_equiv (@path_pi _ _ _ f g) :=
inv_closed apD10
definition homotopy_equiv_path [H : funext] (f g : Πx, B x) : (f ∼ g) ≃ (f = g) :=
equiv.mk _ !is_equiv_path_pi
/- Transport -/
protected definition transport (p : a = a') (f : Π(b : B a), C a b)
: (transport (λa, Π(b : B a), C a b) p f)
∼ (λb, transport (C a') !transport_pV (transportD _ _ p _ (f (p⁻¹ ▹ b)))) :=
eq.rec_on p (λx, idp)
/- A special case of [transport_pi] where the type [B] does not depend on [A],
and so it is just a fixed type [B]. -/
definition transport_constant {C : A → A' → Type} (p : a = a') (f : Π(b : A'), C a b)
: (eq.transport (λa, Π(b : A'), C a b) p f) ∼ (λb, eq.transport (λa, C a b) p (f b)) :=
eq.rec_on p (λx, idp)
/- Maps on paths -/
/- The action of maps given by lambda. -/
definition ap_lambdaD [H : funext] {C : A' → Type} (p : a = a') (f : Πa b, C b) :
ap (λa b, f a b) p = path_pi (λb, ap (λa, f a b) p) :=
begin
apply (eq.rec_on p),
apply inverse,
apply path_pi_idp
end
/- Dependent paths -/
/- with more implicit arguments the conclusion of the following theorem is
(Π(b : B a), transportD B C p b (f b) = g (eq.transport B p b)) ≃
(eq.transport (λa, Π(b : B a), C a b) p f = g) -/
definition dpath_pi [H : funext] (p : a = a') (f : Π(b : B a), C a b) (g : Π(b' : B a'), C a' b')
: (Π(b : B a), p ▹D (f b) = g (p ▹ b)) ≃ (p ▹ f = g) :=
eq.rec_on p (λg, !homotopy_equiv_path) g
section open sigma sigma.ops
/- more implicit arguments:
(Π(b : B a), eq.transport C (sigma.path p idp) (f b) = g (p ▹ b)) ≃
(Π(b : B a), transportD B (λ(a : A) (b : B a), C ⟨a, b⟩) p b (f b) = g (eq.transport B p b)) -/
definition dpath_pi_sigma {C : (Σa, B a) → Type} (p : a = a')
(f : Π(b : B a), C ⟨a, b⟩) (g : Π(b' : B a'), C ⟨a', b'⟩) :
(Π(b : B a), (sigma.path p idp) ▹ (f b) = g (p ▹ b)) ≃ (Π(b : B a), p ▹D (f b) = g (p ▹ b)) :=
eq.rec_on p (λg, !equiv.refl) g
end
variables (f0 : A' → A) (f1 : Π(a':A'), B (f0 a') → B' a')
definition transport_V [reducible] (P : A → Type) {x y : A} (p : x = y) (u : P y) : P x :=
p⁻¹ ▹ u
protected definition functor_pi : (Π(a:A), B a) → (Π(a':A'), B' a') := (λg a', f1 a' (g (f0 a')))
/- Equivalences -/
definition isequiv_functor_pi [instance] (f0 : A' → A) (f1 : Π(a':A'), B (f0 a') → B' a')
