style(hott/types): some style fixes in prod and sigma
This commit is contained in:
Floris van Doorn
Leonardo de Moura
parent
ff5e3d4403
commit
2913035a61
2 changed files with 47 additions and 29 deletions
|
|
@ -28,6 +28,6 @@ namespace prod
|
|||
(λu, destruct u (λa b, idp))
|
||||
|
||||
definition equiv_prod_symm (A B : Type) : A × B ≃ B × A :=
|
||||
equiv.mk flip _
|
||||
equiv.mk flip _
|
||||
|
||||
end prod
|
||||
|
|
|
|||
|
|
@ -11,7 +11,6 @@ import ..trunc .prod
|
|||
open path sigma sigma.ops equiv is_equiv
|
||||
|
||||
namespace sigma
|
||||
-- remove the ₁'s (globally)
|
||||
variables {A A' : Type} {B : A → Type} {B' : A' → Type} {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''} {u v w : Σa, B a}
|
||||
|
|
@ -51,9 +50,9 @@ namespace sigma
|
|||
definition dpair_path_sigma (p : u.1 ≈ v.1) (q : p ▹ u.2 ≈ v.2)
|
||||
: dpair (path_sigma p q)..1 (path_sigma p q)..2 ≈ ⟨p, q⟩ :=
|
||||
begin
|
||||
generalize q, generalize p,
|
||||
reverts (p, q),
|
||||
apply (destruct u), intros (u1, u2),
|
||||
apply (destruct v), intros (v1, v2, p), generalize v2,
|
||||
apply (destruct v), intros (v1, v2, p), generalize v2, --change to revert
|
||||
apply (path.rec_on p), intros (v2, q),
|
||||
apply (path.rec_on q), apply idp
|
||||
end
|
||||
|
|
@ -75,7 +74,7 @@ namespace sigma
|
|||
definition transport_pr1_path_sigma {B' : A → Type} (p : u.1 ≈ v.1) (q : p ▹ u.2 ≈ v.2)
|
||||
: transport (λx, B' x.1) (path_sigma p q) ≈ transport B' p :=
|
||||
begin
|
||||
generalize q, generalize p,
|
||||
reverts (p, q),
|
||||
apply (destruct u), intros (u1, u2),
|
||||
apply (destruct v), intros (v1, v2, p), generalize v2,
|
||||
apply (path.rec_on p), intros (v2, q),
|
||||
|
|
@ -121,7 +120,7 @@ namespace sigma
|
|||
path_sigma_dpair (p1 ⬝ p2) (transport_pp B p1 p2 b ⬝ ap (transport B p2) q1 ⬝ q2)
|
||||
≈ path_sigma_dpair p1 q1 ⬝ path_sigma_dpair p2 q2 :=
|
||||
begin
|
||||
generalize q2, generalize q1, generalize b'', generalize p2, generalize b',
|
||||
reverts (b', p2, b'', q1, q2),
|
||||
apply (path.rec_on p1), intros (b', p2),
|
||||
apply (path.rec_on p2), intros (b'', q1),
|
||||
apply (path.rec_on q1), intro q2,
|
||||
|
|
@ -133,7 +132,7 @@ namespace sigma
|
|||
path_sigma (p1 ⬝ p2) (transport_pp B p1 p2 u.2 ⬝ ap (transport B p2) q1 ⬝ q2)
|
||||
≈ path_sigma p1 q1 ⬝ path_sigma p2 q2 :=
|
||||
begin
|
||||
generalize q2, generalize p2, generalize q1, generalize p1,
|
||||
reverts (p1, q1, p2, q2),
|
||||
apply (destruct u), intros (u1, u2),
|
||||
apply (destruct v), intros (v1, v2),
|
||||
apply (destruct w), intros,
|
||||
|
|
@ -143,7 +142,7 @@ namespace sigma
|
|||
definition path_sigma_dpair_p1_1p (p : a ≈ a') (q : p ▹ b ≈ b') :
|
||||
path_sigma_dpair p q ≈ path_sigma_dpair p idp ⬝ path_sigma_dpair idp q :=
|
||||
begin
|
||||
generalize q, generalize b',
|
||||
reverts (b', q),
|
||||
apply (path.rec_on p), intros (b', q),
|
||||
apply (path.rec_on q), apply idp
|
||||
end
|
||||
|
|
@ -172,7 +171,7 @@ namespace sigma
|
|||
q2
|
||||
s
|
||||
-- begin
|
||||
-- generalize s, generalize q2,
|
||||
-- reverts (q2, s),
|
||||
-- apply (path.rec_on r), intros (q2, s),
|
||||
-- apply (path.rec_on s), apply idp
|
||||
-- end
|
||||
|
|
@ -181,7 +180,7 @@ namespace sigma
|
|||
/- A path between paths in a total space is commonly shown component wise. -/
|
||||
definition path_path_sigma {p q : u ≈ v} (r : p..1 ≈ q..1) (s : r ▹ p..2 ≈ q..2) : p ≈ q :=
|
||||
begin
|
||||
generalize s, generalize r, generalize q,
|
||||
reverts (q, r, s),
|
||||
apply (path.rec_on p),
|
||||
apply (destruct u), intros (u1, u2, q, r, s),
|
||||
apply concat, rotate 1,
|
||||
|
|
@ -189,7 +188,6 @@ namespace sigma
|
|||
apply (path_path_sigma_path_sigma r s)
|
||||
end
|
||||
|
||||
|
||||
/- In Coq they often have to give u and v explicitly when using the following definition -/
|
||||
definition path_path_sigma_uncurried {p q : u ≈ v}
|
||||
(rs : Σ(r : p..1 ≈ q..1), transport (λx, transport B x u.2 ≈ v.2) r p..2 ≈ q..2) : p ≈ q :=
|
||||
|
|
@ -223,7 +221,7 @@ namespace sigma
|
|||
definition transport_sigma_' {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
|
||||
generalize bcd,
|
||||
revert bcd,
|
||||
apply (path.rec_on p), intro bcd,
|
||||
apply (destruct bcd), intros (b, cd),
|
||||
apply (destruct cd), intros (c, d),
|
||||
|
|
@ -236,15 +234,35 @@ namespace sigma
|
|||
definition functor_sigma (u : Σa, B a) : Σa', B' a' :=
|
||||
⟨f u.1, g u.1 u.2⟩
|
||||
|
||||
/- Equivalences -/
|
||||
-- variables {A A' : Type} {B : A → Type} {B' : A' → Type} (f : A → A') (g : Πa, B a → B' (f a))
|
||||
-- (H1 : is_equiv f) (H2 : Π (a : A), is_equiv (g a)) (u' : Σ (a' : A'), B' a')
|
||||
-- (a' : A') (b' : B' a')
|
||||
-- check retr f a' ▹ (g (f⁻¹ a') (g (f⁻¹ a')⁻¹ ((retr f a')⁻¹ ▹ b'))) ≈ b'
|
||||
-- check retr f a' ▹ (g (f⁻¹ a') (g (f⁻¹ a')⁻¹ ((retr f a')⁻¹ ▹ b')))
|
||||
-- check (g (f⁻¹ a') (g (f⁻¹ a')⁻¹ ((retr f a')⁻¹ ▹ b')))
|
||||
-- check retr f a'
|
||||
|
||||
--remove explicit arguments of IsEquiv
|
||||
/- Equivalences -/
|
||||
irreducible inv --function.compose
|
||||
--TODO: remove explicit arguments of IsEquiv
|
||||
definition isequiv_functor_sigma [H1 : is_equiv f] [H2 : Π a, @is_equiv (B a) (B' (f a)) (g a)]
|
||||
: is_equiv (functor_sigma f g) :=
|
||||
/-adjointify (functor_sigma f g)
|
||||
(functor_sigma (f⁻¹) (λ(x : A') (y : B' x), ((g (f⁻¹ x))⁻¹ ((retr f x)⁻¹ ▹ y))))
|
||||
sorry-/
|
||||
sorry
|
||||
adjointify (functor_sigma f g)
|
||||
(functor_sigma (f⁻¹) (λ(x : A') (y : B' x),
|
||||
((g (f⁻¹ x))⁻¹ (transport B' (retr f x)⁻¹ y))))
|
||||
-- begin
|
||||
-- intro u',
|
||||
-- apply (destruct u'), intros (a', b'),
|
||||
-- apply (path_sigma (retr f a')),
|
||||
-- -- show retr f a' ▹ (g (f⁻¹ a') (g (f⁻¹ a')⁻¹ ((retr f a')⁻¹ ▹ b'))) ≈ b',
|
||||
-- -- from sorry
|
||||
-- -- exact (calc
|
||||
-- -- retr f a' ▹ g (f⁻¹ a') (g (f⁻¹ a')⁻¹ ((retr f a')⁻¹ ▹ b'))
|
||||
-- -- ≈ retr f a' ▹ ((retr f a')⁻¹ ▹ b') : {retr (g (f⁻¹ a')) _}
|
||||
-- -- ... ≈ b' : transport_pV)
|
||||
-- end
|
||||
proof (λu', sorry) qed
|
||||
proof (λu, sorry) qed
|
||||
|
||||
definition equiv_functor_sigma [H1 : is_equiv f] [H2 : Π a, is_equiv (g a)] : (Σa, B a) ≃ (Σa', B' a') :=
|
||||
equiv.mk (functor_sigma f g) !isequiv_functor_sigma
|
||||
|
|
@ -333,25 +351,25 @@ namespace sigma
|
|||
... ≃ Σ(b : B), A : equiv_sigma0_prod
|
||||
|
||||
/- truncatedness -/
|
||||
definition sigma_trunc (n : trunc_index) [HA : is_trunc n A] [HB : Πa, is_trunc n (B a)]
|
||||
: is_trunc n (Σa, B a) :=
|
||||
definition trunc_sigma [instance] (B : A → Type) (n : trunc_index)
|
||||
[HA : is_trunc n A] [HB : Πa, is_trunc n (B a)] : is_trunc n (Σa, B a) :=
|
||||
begin
|
||||
generalize HB, generalize HA, generalize B, generalize A,
|
||||
reverts (A, B, HA, HB),
|
||||
apply (truncation.trunc_index.rec_on n),
|
||||
intros (A, B, HA, HB),
|
||||
apply trunc_equiv',
|
||||
fapply trunc_equiv',
|
||||
apply equiv.symm,
|
||||
apply equiv_contr_sigma, apply HA,
|
||||
apply HB,
|
||||
apply equiv_contr_sigma, apply HA, --should be solved by term synthesis
|
||||
apply HB,
|
||||
intros (n, IH, A, B, HA, HB),
|
||||
apply is_trunc_succ, intros (u, v),
|
||||
apply trunc_equiv',
|
||||
fapply is_trunc_succ, intros (u, v),
|
||||
fapply trunc_equiv',
|
||||
apply equiv_path_sigma,
|
||||
apply IH,
|
||||
apply succ_is_trunc,
|
||||
intro aa,
|
||||
show is_trunc n (aa ▹ u .2 ≈ v .2), from
|
||||
succ_is_trunc (aa ▹ u.2) (v.2),
|
||||
intro p,
|
||||
show is_trunc n (p ▹ u .2 ≈ v .2), from
|
||||
succ_is_trunc (p ▹ u.2) (v.2),
|
||||
end
|
||||
|
||||
end sigma
|
||||
|
|
|
|||
Loading…
Reference in a new issue