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moving some definitions to pointed_pi

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Egbert Rijke 2017-07-08 15:25:17 +01:00
parent e92fb0a435
commit e6b1c49f4a
2 changed files with 43 additions and 42 deletions
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42
homotopy/spectrum.hlean
View file

@ -657,34 +657,6 @@ set_option pp.coercions true
spectrum.MK (λn, Π*a, E a n)
(λn, !ppi_loop_pequiv⁻¹ᵉ* ∘*ᵉ ppi_pequiv_right (λa, equiv_glue (E a) n))
definition ppi_assoc_compose_left {A : Type*} {B C D : A → Type*}
(f : Π (a : A), B a →* C a) (g : Π (a : A), C a →* D a)
: (ppi_compose_left g ∘* ppi_compose_left f) ~* ppi_compose_left (λ a, g a ∘* f a) :=
begin
fapply phomotopy.mk,
intro h, fapply eq_of_ppi_homotopy,
fapply ppi_homotopy.mk,
-- intro a, reflexivity,
-- esimp,
repeat exact sorry,
end /- TODO FOR SSS -/
definition psquare_of_ppi_compose_left {A : Type*} {B C D E : A → Type*}
{ftop : Π (a : A), B a →* C a} {fbot : Π (a : A), D a →* E a}
{fleft : Π (a : A), B a →* D a} {fright : Π (a : A), C a →* E a}
(psq : Π (a :A), psquare (ftop a) (fbot a) (fleft a) (fright a))
: psquare
(ppi_compose_left ftop)
(ppi_compose_left fbot)
(ppi_compose_left fleft)
(ppi_compose_left fright)
:=
begin
refine (ppi_assoc_compose_left ftop fright) ⬝* _ ⬝* (ppi_assoc_compose_left fleft fbot)⁻¹*,
refine eq_of_homotopy (λ a, eq_of_phomotopy (psq a)) ▸ phomotopy.rfl,
-- the last step is probably an unnecessary application of function extensionality.
end
definition spi_compose_left_topsq
{N : succ_str} {A : Type*} {E F : A → gen_spectrum N} (f : Π a, (E a) →ₛ (F a)) (n : N)
: psquare
@ -698,20 +670,6 @@ set_option pp.coercions true
intro a, exact glue_square (f a) n,
end
definition psquare_loop_ppi_compose_left {A : Type*} {X Y : A → Type*} (f : Π (a : A), X a →* Y a) :
psquare
(Ω→ (ppi_compose_left f))
(ppi_compose_left (λ a, Ω→ (f a)))
(ppi_loop_pequiv)
(ppi_loop_pequiv)
:=
begin
fapply psquare_of_phomotopy,
fapply phomotopy.mk, intro g,
fapply eq_of_ppi_homotopy, fapply ppi_homotopy.mk, intro a,
repeat exact sorry
end /- TODO FOR SSS -/
definition spi_compose_left_botsq
{N : succ_str} {A : Type*} {E F : A → gen_spectrum N} (f : Π a, (E a) →ₛ (F a)) (n : N)
: psquare

43
pointed_pi.hlean
View file

@ -219,6 +219,49 @@ namespace pointed
(Π*(a : A), P a) →* Π*(a : A), Q a :=
pmap.mk (pmap_compose_ppi g) (ppi_eq (pmap_compose_ppi_const_right g))
-- ppi_compose_left is associative in the following sense.
definition ppi_assoc_compose_left {A : Type*} {B C D : A → Type*}
(f : Π (a : A), B a →* C a) (g : Π (a : A), C a →* D a)
: (ppi_compose_left g ∘* ppi_compose_left f) ~* ppi_compose_left (λ a, g a ∘* f a) :=
begin
fapply phomotopy.mk,
intro h, fapply eq_of_ppi_homotopy,
fapply ppi_homotopy.mk,
-- intro a, reflexivity,
-- esimp,
repeat exact sorry,
end /- TODO FOR SSS -/
definition psquare_loop_ppi_compose_left {A : Type*} {X Y : A → Type*} (f : Π (a : A), X a →* Y a) :
psquare
(Ω→ (ppi_compose_left f))
(ppi_compose_left (λ a, Ω→ (f a)))
(ppi_loop_pequiv)
(ppi_loop_pequiv)
:=
begin
fapply psquare_of_phomotopy,
fapply phomotopy.mk, intro g,
fapply eq_of_ppi_homotopy, fapply ppi_homotopy.mk, intro a,
repeat exact sorry
end /- TODO FOR SSS -/
definition psquare_of_ppi_compose_left {A : Type*} {B C D E : A → Type*}
{ftop : Π (a : A), B a →* C a} {fbot : Π (a : A), D a →* E a}
{fleft : Π (a : A), B a →* D a} {fright : Π (a : A), C a →* E a}
(psq : Π (a :A), psquare (ftop a) (fbot a) (fleft a) (fright a))
: psquare
(ppi_compose_left ftop)
(ppi_compose_left fbot)
(ppi_compose_left fleft)
(ppi_compose_left fright)
:=
begin
refine (ppi_assoc_compose_left ftop fright) ⬝* _ ⬝* (ppi_assoc_compose_left fleft fbot)⁻¹*,
refine eq_of_homotopy (λ a, eq_of_phomotopy (psq a)) ▸ phomotopy.rfl,
-- the last step is probably an unnecessary application of function extensionality.
end
definition pmap_compose_ppi_phomotopy_left [constructor] {g g' : Π(a : A), ppmap (P a) (Q a)}
(f : Π*(a : A), P a) (p : Πa, g a ~* g' a) : pmap_compose_ppi g f ~~* pmap_compose_ppi g' f :=
ppi_homotopy.mk (λa, p a (f a))