moving some definitions to pointed_pi
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Egbert Rijke
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e92fb0a435
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2 changed files with 43 additions and 42 deletions
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@ -657,34 +657,6 @@ set_option pp.coercions true
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spectrum.MK (λn, Π*a, E a n)
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spectrum.MK (λn, Π*a, E a n)
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(λn, !ppi_loop_pequiv⁻¹ᵉ* ∘*ᵉ ppi_pequiv_right (λa, equiv_glue (E a) n))
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(λn, !ppi_loop_pequiv⁻¹ᵉ* ∘*ᵉ ppi_pequiv_right (λa, equiv_glue (E a) n))
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definition ppi_assoc_compose_left {A : Type*} {B C D : A → Type*}
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(f : Π (a : A), B a →* C a) (g : Π (a : A), C a →* D a)
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: (ppi_compose_left g ∘* ppi_compose_left f) ~* ppi_compose_left (λ a, g a ∘* f a) :=
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begin
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fapply phomotopy.mk,
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intro h, fapply eq_of_ppi_homotopy,
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fapply ppi_homotopy.mk,
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-- intro a, reflexivity,
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-- esimp,
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repeat exact sorry,
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end /- TODO FOR SSS -/
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definition psquare_of_ppi_compose_left {A : Type*} {B C D E : A → Type*}
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{ftop : Π (a : A), B a →* C a} {fbot : Π (a : A), D a →* E a}
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{fleft : Π (a : A), B a →* D a} {fright : Π (a : A), C a →* E a}
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(psq : Π (a :A), psquare (ftop a) (fbot a) (fleft a) (fright a))
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: psquare
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(ppi_compose_left ftop)
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(ppi_compose_left fbot)
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(ppi_compose_left fleft)
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(ppi_compose_left fright)
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:=
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begin
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refine (ppi_assoc_compose_left ftop fright) ⬝* _ ⬝* (ppi_assoc_compose_left fleft fbot)⁻¹*,
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refine eq_of_homotopy (λ a, eq_of_phomotopy (psq a)) ▸ phomotopy.rfl,
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-- the last step is probably an unnecessary application of function extensionality.
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end
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definition spi_compose_left_topsq
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definition spi_compose_left_topsq
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{N : succ_str} {A : Type*} {E F : A → gen_spectrum N} (f : Π a, (E a) →ₛ (F a)) (n : N)
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{N : succ_str} {A : Type*} {E F : A → gen_spectrum N} (f : Π a, (E a) →ₛ (F a)) (n : N)
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: psquare
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: psquare
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@ -698,20 +670,6 @@ set_option pp.coercions true
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intro a, exact glue_square (f a) n,
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intro a, exact glue_square (f a) n,
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end
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end
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definition psquare_loop_ppi_compose_left {A : Type*} {X Y : A → Type*} (f : Π (a : A), X a →* Y a) :
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psquare
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(Ω→ (ppi_compose_left f))
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(ppi_compose_left (λ a, Ω→ (f a)))
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(ppi_loop_pequiv)
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(ppi_loop_pequiv)
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:=
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begin
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fapply psquare_of_phomotopy,
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fapply phomotopy.mk, intro g,
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fapply eq_of_ppi_homotopy, fapply ppi_homotopy.mk, intro a,
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repeat exact sorry
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end /- TODO FOR SSS -/
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definition spi_compose_left_botsq
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definition spi_compose_left_botsq
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{N : succ_str} {A : Type*} {E F : A → gen_spectrum N} (f : Π a, (E a) →ₛ (F a)) (n : N)
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{N : succ_str} {A : Type*} {E F : A → gen_spectrum N} (f : Π a, (E a) →ₛ (F a)) (n : N)
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: psquare
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: psquare
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@ -219,6 +219,49 @@ namespace pointed
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(Π*(a : A), P a) →* Π*(a : A), Q a :=
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(Π*(a : A), P a) →* Π*(a : A), Q a :=
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pmap.mk (pmap_compose_ppi g) (ppi_eq (pmap_compose_ppi_const_right g))
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pmap.mk (pmap_compose_ppi g) (ppi_eq (pmap_compose_ppi_const_right g))
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-- ppi_compose_left is associative in the following sense.
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definition ppi_assoc_compose_left {A : Type*} {B C D : A → Type*}
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(f : Π (a : A), B a →* C a) (g : Π (a : A), C a →* D a)
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: (ppi_compose_left g ∘* ppi_compose_left f) ~* ppi_compose_left (λ a, g a ∘* f a) :=
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begin
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fapply phomotopy.mk,
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intro h, fapply eq_of_ppi_homotopy,
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fapply ppi_homotopy.mk,
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-- intro a, reflexivity,
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-- esimp,
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repeat exact sorry,
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end /- TODO FOR SSS -/
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definition psquare_loop_ppi_compose_left {A : Type*} {X Y : A → Type*} (f : Π (a : A), X a →* Y a) :
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psquare
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(Ω→ (ppi_compose_left f))
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(ppi_compose_left (λ a, Ω→ (f a)))
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(ppi_loop_pequiv)
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(ppi_loop_pequiv)
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:=
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begin
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fapply psquare_of_phomotopy,
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fapply phomotopy.mk, intro g,
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fapply eq_of_ppi_homotopy, fapply ppi_homotopy.mk, intro a,
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repeat exact sorry
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end /- TODO FOR SSS -/
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definition psquare_of_ppi_compose_left {A : Type*} {B C D E : A → Type*}
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{ftop : Π (a : A), B a →* C a} {fbot : Π (a : A), D a →* E a}
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{fleft : Π (a : A), B a →* D a} {fright : Π (a : A), C a →* E a}
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(psq : Π (a :A), psquare (ftop a) (fbot a) (fleft a) (fright a))
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: psquare
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(ppi_compose_left ftop)
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(ppi_compose_left fbot)
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(ppi_compose_left fleft)
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(ppi_compose_left fright)
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:=
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begin
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refine (ppi_assoc_compose_left ftop fright) ⬝* _ ⬝* (ppi_assoc_compose_left fleft fbot)⁻¹*,
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refine eq_of_homotopy (λ a, eq_of_phomotopy (psq a)) ▸ phomotopy.rfl,
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-- the last step is probably an unnecessary application of function extensionality.
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end
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definition pmap_compose_ppi_phomotopy_left [constructor] {g g' : Π(a : A), ppmap (P a) (Q a)}
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definition pmap_compose_ppi_phomotopy_left [constructor] {g g' : Π(a : A), ppmap (P a) (Q a)}
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(f : Π*(a : A), P a) (p : Πa, g a ~* g' a) : pmap_compose_ppi g f ~~* pmap_compose_ppi g' f :=
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(f : Π*(a : A), P a) (p : Πa, g a ~* g' a) : pmap_compose_ppi g f ~~* pmap_compose_ppi g' f :=
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ppi_homotopy.mk (λa, p a (f a))
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ppi_homotopy.mk (λa, p a (f a))
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