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rename nondep to constant

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This commit is contained in:
Floris van Doorn 2018-09-20 15:53:05 +02:00
parent 98fb55e428
commit 3468ab8a9f
3 changed files with 8 additions and 14 deletions
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10
hott/types/fiber.hlean
View file

@ -165,9 +165,7 @@ namespace fiber
definition fiber_pr1 (B : A → Type) (a : A) : fiber (pr1 : (Σa, B a) → A) a ≃ B a :=
calc
fiber pr1 a ≃ Σu, u.1 = a : fiber.sigma_char
... ≃ Σa' (b : B a'), a' = a : sigma_assoc_equiv
... ≃ Σa' (p : a' = a), B a' : sigma_equiv_sigma_right (λa', !comm_equiv_nondep)
... ≃ Σu, B u.1 : sigma_assoc_equiv
... ≃ Σu, B u.1 : sigma_assoc_comm_equiv
... ≃ B a : sigma_equiv_of_is_contr_left _ _
definition sigma_fiber_equiv (f : A → B) : (Σb, fiber f b) ≃ A :=
@ -266,12 +264,8 @@ namespace fiber
apply sigma_equiv_sigma_right, intro x,
apply sigma_comm_equiv
end
... ≃ Σ(w : Σx, x = a), Σ(p : P w.1), f w.1 p =[w.2] q
: sigma_assoc_equiv
... ≃ Σ(p : P (center (Σx, x=a)).1), f (center (Σx, x=a)).1 p =[(center (Σx, x=a)).2] q
: sigma_equiv_of_is_contr_left
... ≃ Σ(p : P a), f a p =[idpath a] q
: equiv_of_eq idp
: sigma_sigma_eq_left
... ≃ Σ(p : P a), f a p = q
:
begin

10
hott/types/sigma.hlean
View file

@ -218,13 +218,13 @@ namespace sigma
by induction p; induction bc; reflexivity
/- The special case when the second variable doesn't depend on the first is simpler. -/
definition sigma_transport_nondep {B : Type} {C : A → B → Type} (p : a = a')
definition sigma_transport_constant {B : Type} {C : A → B → Type} (p : a = a')
(bc : Σ(b : B), C a b) : p ▸ bc = ⟨bc.1, p ▸ bc.2⟩ :=
by induction p; induction bc; reflexivity
/- Or if the second variable contains a first component that doesn't depend on the first. -/
definition sigma_transport2_nondep {C : A → Type} {D : Π a:A, B a → C a → Type} (p : a = a')
definition sigma_transport2_constant {C : A → Type} {D : Π a:A, B a → C a → Type} (p : a = a')
(bcd : Σ(b : B a) (c : C a), D a b c) : p ▸ bcd = ⟨p ▸ bcd.1, p ▸ bcd.2.1, p ▸D2 bcd.2.2⟩ :=
begin
induction p, induction bcd with b cd, induction cd, reflexivity
@ -250,7 +250,7 @@ namespace sigma
induction s using idp_rec_on, apply idpo
end
definition sigma_pathover_nondep {B : Type} {C : A → B → Type} (p : a = a')
definition sigma_pathover_constant {B : Type} {C : A → B → Type} (p : a = a')
(u : Σ(b : B), C a b) (v : Σ(b : B), C a' b)
(r : u.1 = v.1) (s : pathover (λx, C (prod.pr1 x) (prod.pr2 x)) u.2 (prod.prod_eq p r) v.2) : u =[p] v :=
begin
@ -462,7 +462,7 @@ namespace sigma
proof (λp, prod.destruct p (λa b, idp)) qed
proof (λs, destruct s (λa b, idp)) qed)
definition comm_equiv_nondep (A B : Type) : (Σ(a : A), B) ≃ Σ(b : B), A :=
definition comm_equiv_constant (A B : Type) : (Σ(a : A), B) ≃ Σ(b : B), A :=
calc
(Σ(a : A), B) ≃ A × B : equiv_prod
... ≃ B × A : prod_comm_equiv
@ -472,7 +472,7 @@ namespace sigma
: (Σ(v : Σa, B a), C v.1) ≃ (Σ(u : Σa, C a), B u.1) :=
calc (Σ(v : Σa, B a), C v.1)
≃ (Σa (b : B a), C a) : sigma_assoc_equiv
... ≃ (Σa (c : C a), B a) : sigma_equiv_sigma_right (λa, !comm_equiv_nondep)
... ≃ (Σa (c : C a), B a) : sigma_equiv_sigma_right (λa, !comm_equiv_constant)
... ≃ (Σ(u : Σa, C a), B u.1) : sigma_assoc_equiv
/- Interaction with other type constructors -/

2
hott/types/univ.hlean
View file

@ -115,7 +115,7 @@ namespace univ
(λb, !sigma_assoc_equiv)
... ≃ Σb X (x : X), X = fiber f b : sigma_equiv_sigma_right
(λb, sigma_equiv_sigma_right
(λX, !comm_equiv_nondep))
(λX, !comm_equiv_constant))
... ≃ Σb (v : ΣX, X), v.1 = fiber f b : sigma_equiv_sigma_right
(λb, !sigma_assoc_equiv⁻¹ᵉ)
... ≃ Σb (Y : Type*), Y = fiber f b : sigma_equiv_sigma_right