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chore(library/hott) clean up file and add class inference

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Jakob von Raumer 2014-11-06 13:16:35 -05:00 committed by Leonardo de Moura
parent 1f5be44f51
commit 9ad75108a3
1 changed files with 43 additions and 42 deletions
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85
library/hott/equiv.lean
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@ -179,37 +179,34 @@ namespace IsEquiv
end IsEquiv
namespace IsEquiv
variables {A B C : Type} {f : A → B} {g : B → C} {f' : A → B}
variables {A : Type}
section
variables {B C : Type} (f : A → B) {f' : A → B} [Hf : IsEquiv f]
include Hf
definition cancel_R (Hf : IsEquiv f) (Hgf : IsEquiv (g ∘ f)) : (IsEquiv g) :=
definition cancel_R (g : B → C) [Hgf : IsEquiv (g ∘ f)] : (IsEquiv g) :=
homotopic (comp_closed (inv_closed Hf) Hgf) (λb, ap g (retr f b))
definition cancel_L (Hg : IsEquiv g) (Hgf : IsEquiv (g ∘ f)) : (IsEquiv f) :=
homotopic (comp_closed Hgf (inv_closed Hg)) (λa, sect g (f a))
--Transporting is an equivalence
definition transport [instance] (P : A → Type) {x y : A} (p : x ≈ y) : (IsEquiv (transport P p)) :=
IsEquiv_mk (transport P (p⁻¹)) (transport_pV P p) (transport_Vp P p) (transport_pVp P p)
definition cancel_L (g : C → A) [Hgf : IsEquiv (f ∘ g)] : (IsEquiv g) :=
homotopic (comp_closed Hgf (inv_closed Hf)) (λa, sect f (g a))
--Rewrite rules
section
definition moveR_M (Hf : IsEquiv f) {x : A} {y : B} (p : x ≈ (inv f) y) : (f x ≈ y) :=
definition moveR_M {x : A} {y : B} (p : x ≈ (inv f) y) : (f x ≈ y) :=
(ap f p) ⬝ (retr f y)
definition moveL_M (Hf : IsEquiv f) {x : A} {y : B} (p : (inv f) y ≈ x) : (y ≈ f x) :=
(moveR_M Hf (p⁻¹))⁻¹
definition moveL_M {x : A} {y : B} (p : (inv f) y ≈ x) : (y ≈ f x) :=
(moveR_M f (p⁻¹))⁻¹
definition moveR_V (Hf : IsEquiv f) {x : B} {y : A} (p : x ≈ f y) : (inv f) x ≈ y :=
definition moveR_V {x : B} {y : A} (p : x ≈ f y) : (inv f) x ≈ y :=
ap (inv f) p ⬝ sect f y
definition moveL_V (Hf : IsEquiv f) {x : B} {y : A} (p : f y ≈ x) : y ≈ (inv f) x :=
(moveR_V Hf (p⁻¹))⁻¹
definition moveL_V {x : B} {y : A} (p : f y ≈ x) : y ≈ (inv f) x :=
(moveR_V f (p⁻¹))⁻¹
definition contr (Hf : IsEquiv f) (HA: Contr A) : (Contr B) :=
Contr.Contr_mk (f (center HA)) (λb, moveR_M Hf (contr HA (inv f b)))
definition contr (HA: Contr A) : (Contr B) :=
Contr.Contr_mk (f (center HA)) (λb, moveR_M f (contr HA (inv f b)))
definition ap_closed (Hf : IsEquiv f) (x y : A) : IsEquiv (@ap A B f x y) :=
definition ap_closed (x y : A) : IsEquiv (@ap A B f x y) :=
adjointify (ap f)
(λq, (inverse (sect f x)) ⬝ ap (f⁻¹) q ⬝ sect f y)
(λq, !ap_pp
@ -233,33 +230,37 @@ namespace IsEquiv
-- once pulled back along an equivalence f : A → B, then it has a section
-- over all of B.
definition equiv_rect (Hf : IsEquiv f) (P : B -> Type) :
definition equiv_rect (P : B -> Type) :
(Πx, P (f x)) → (Πy, P y) :=
(λg y, path.transport _ (retr f y) (g (f⁻¹ y)))
definition equiv_rect_comp (Hf : IsEquiv f) (P : B → Type)
(df : Π (x : A), P (f x)) (x : A) : equiv_rect Hf P df (f x) ≈ df x :=
definition equiv_rect_comp (P : B → Type)
(df : Π (x : A), P (f x)) (x : A) : equiv_rect f P df (f x) ≈ df x :=
let eq1 := (apD df (sect f x)) in
calc equiv_rect Hf P df (f x)
≈ path.transport P (retr f (f x)) (df (f⁻¹ (f x))) : idp
... ≈ path.transport P (ap f (sect f x)) (df (f⁻¹ (f x))) : adj f
... ≈ path.transport (P ∘ f) (sect f x) (df (f⁻¹ (f x))) : transport_compose
calc equiv_rect f P df (f x)
≈ transport P (retr f (f x)) (df (f⁻¹ (f x))) : idp
... ≈ transport P (ap f (sect f x)) (df (f⁻¹ (f x))) : adj f
... ≈ transport (P ∘ f) (sect f x) (df (f⁻¹ (f x))) : transport_compose
... ≈ df x : eq1
end
--Transporting is an equivalence
definition transport [instance] (P : A → Type) {x y : A} (p : x ≈ y) : (IsEquiv (transport P p)) :=
IsEquiv_mk (transport P (p⁻¹)) (transport_pV P p) (transport_Vp P p) (transport_pVp P p)
end IsEquiv
namespace Equiv
context
parameters {A B : Type} (eqf : A ≃ B)
parameters {A B C : Type} (eqf : A ≃ B)
private definition f : A → B := equiv_fun eqf
private definition Hf [instance] : IsEquiv f := equiv_isequiv eqf
definition id : A ≃ A := Equiv_mk id IsEquiv.id_closed
protected definition id : A ≃ A := Equiv_mk id IsEquiv.id_closed
theorem compose {C : Type} (eqg: B ≃ C) : A ≃ C :=
theorem compose (eqg: B ≃ C) : A ≃ C :=
Equiv_mk ((equiv_fun eqg) ∘ f)
(IsEquiv.comp_closed Hf (equiv_isequiv eqg))
@ -269,28 +270,28 @@ namespace Equiv
theorem inv_closed : B ≃ A :=
Equiv_mk (IsEquiv.inv f) (IsEquiv.inv_closed Hf)
theorem cancel_R {C : Type} {g : B → C} (Hgf : IsEquiv (g ∘ f)) : B ≃ C :=
Equiv_mk g (IsEquiv.cancel_R Hf _)
theorem cancel_R {g : B → C} (Hgf : IsEquiv (g ∘ f)) : B ≃ C :=
Equiv_mk g (IsEquiv.cancel_R f _)
theorem cancel_L {C : Type} {g : C → A} (Hgf : IsEquiv (f ∘ g)) : C ≃ A :=
Equiv_mk g (IsEquiv.cancel_L Hf _)
theorem cancel_L {g : C → A} (Hgf : IsEquiv (f ∘ g)) : C ≃ A :=
Equiv_mk g (IsEquiv.cancel_L f _)
theorem transport (P : A → Type) {x y : A} {p : x ≈ y} : (P x) ≃ (P y) :=
Equiv_mk (transport P p) (IsEquiv.transport P p)
theorem contr_closed (HA: Contr A) : (Contr B) :=
IsEquiv.contr Hf HA
-- calc enviroment
-- TODO: find a transport lemma?
-- theorem foo (P : Type → Type) : P A → P B := sorry
-- calc_subst transport
--calc_trans Equiv.compose
calc_refl id
calc_symm inv_closed
IsEquiv.contr f HA
end
-- calc enviroment
-- TODO: transport lemma without univalence?
-- theorem foo (P : Type → Type) (A B : Type) (H : A ≃ B) : P A → P B := sorry
-- calc_subst foo
calc_trans compose
calc_refl id
calc_symm inv_closed
end Equiv
namespace Equiv