756 lines
33 KiB
Text
756 lines
33 KiB
Text
/-
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Copyright (c) 2016 Ulrik Buchholtz. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Ulrik Buchholtz, Floris van Doorn
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-/
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import homotopy.connectedness types.pointed2 .move_to_lib .pointed
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open eq pointed equiv sigma is_equiv trunc option
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/-
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In this file we define dependent pointed maps and properties of them.
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Using this, we give the truncation level
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of the type of pointed maps, giving the connectivity of
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the domain and the truncation level of the codomain.
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This is is_trunc_pmap_of_is_conn at the end.
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We also prove other properties about pointed (dependent maps), like the fact that
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(Π*a, F a) → (Π*a, X a) → (Π*a, B a)
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is a fibration sequence if (F a) → (X a) → B a) is.
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-/
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namespace pointed
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definition pointed_respect_pt [instance] [constructor] {A B : Type*} (f : A →* B) :
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pointed (f pt = pt) :=
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pointed.mk (respect_pt f)
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definition ppi_gen_of_phomotopy [constructor] {A B : Type*} {f g : A →* B} (h : f ~* g) :
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ppi_gen (λx, f x = g x) (respect_pt f ⬝ (respect_pt g)⁻¹) :=
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h
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abbreviation ppi_resp_pt [unfold 3] := @ppi.resp_pt
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definition ppi_const [constructor] {A : Type*} (P : A → Type*) : ppi P :=
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ppi.mk (λa, pt) idp
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definition pointed_ppi [instance] [constructor] {A : Type*}
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(P : A → Type*) : pointed (ppi P) :=
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pointed.mk (ppi_const P)
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definition pppi [constructor] {A : Type*} (P : A → Type*) : Type* :=
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pointed.mk' (ppi P)
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notation `Π*` binders `, ` r:(scoped P, pppi P) := r
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definition ppi_homotopy {A : Type*} {P : A → Type} {x : P pt} (f g : ppi_gen P x) : Type :=
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ppi_gen (λa, f a = g a) (ppi_gen.resp_pt f ⬝ (ppi_gen.resp_pt g)⁻¹)
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variables {A : Type*} {P Q R : A → Type*} {f g h : Π*a, P a}
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{B : A → Type} {x₀ : B pt} {k l m : ppi_gen B x₀}
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infix ` ~~* `:50 := ppi_homotopy
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definition ppi_homotopy.mk [constructor] [reducible] (h : k ~ l)
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(p : h pt ⬝ ppi_gen.resp_pt l = ppi_gen.resp_pt k) : k ~~* l :=
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ppi_gen.mk h (eq_con_inv_of_con_eq p)
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definition ppi_to_homotopy [coercion] [unfold 6] [reducible] (p : k ~~* l) : Πa, k a = l a := p
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definition ppi_to_homotopy_pt [unfold 6] [reducible] (p : k ~~* l) :
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p pt ⬝ ppi_gen.resp_pt l = ppi_gen.resp_pt k :=
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con_eq_of_eq_con_inv (ppi_gen.resp_pt p)
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variable (k)
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protected definition ppi_homotopy.refl : k ~~* k :=
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ppi_homotopy.mk homotopy.rfl !idp_con
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variable {k}
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protected definition ppi_homotopy.rfl [refl] : k ~~* k :=
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ppi_homotopy.refl k
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protected definition ppi_homotopy.symm [symm] (p : k ~~* l) : l ~~* k :=
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ppi_homotopy.mk p⁻¹ʰᵗʸ (inv_con_eq_of_eq_con (ppi_to_homotopy_pt p)⁻¹)
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protected definition ppi_homotopy.trans [trans] (p : k ~~* l) (q : l ~~* m) : k ~~* m :=
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ppi_homotopy.mk (λa, p a ⬝ q a) (!con.assoc ⬝ whisker_left (p pt) (ppi_to_homotopy_pt q) ⬝ ppi_to_homotopy_pt p)
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infix ` ⬝*' `:75 := ppi_homotopy.trans
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postfix `⁻¹*'`:(max+1) := ppi_homotopy.symm
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definition ppi_trans_refl (p : k ~~* l) : p ⬝*' ppi_homotopy.refl l = p :=
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begin
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unfold ppi_homotopy.trans,
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induction A with A a₀,
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induction k with k k₀, induction l with l l₀, induction p with p p₀, esimp at p, induction l₀, esimp at p₀, induction p₀, reflexivity,
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end
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definition ppi_equiv_pmap [constructor] (A B : Type*) : (Π*(a : A), B) ≃ (A →* B) :=
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begin
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fapply equiv.MK,
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{ intro k, induction k with k p, exact pmap.mk k p },
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{ intro k, induction k with k p, exact ppi.mk k p },
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{ intro k, induction k with k p, reflexivity },
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{ intro k, induction k with k p, reflexivity }
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end
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definition pppi_pequiv_ppmap [constructor] (A B : Type*) : (Π*(a : A), B) ≃* ppmap A B :=
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pequiv_of_equiv (ppi_equiv_pmap A B) idp
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protected definition ppi_gen.sigma_char [constructor] {A : Type*} (B : A → Type) (b₀ : B pt) :
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ppi_gen B b₀ ≃ Σ(k : Πa, B a), k pt = b₀ :=
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begin
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fapply equiv.MK: intro x,
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{ constructor, exact ppi_gen.resp_pt x },
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{ induction x, constructor, assumption },
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{ induction x, reflexivity },
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{ induction x, reflexivity }
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end
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definition ppi.sigma_char [constructor] {A : Type*} (B : A → Type*)
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: (Π*(a : A), B a) ≃ Σ(k : (Π (a : A), B a)), k pt = pt :=
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begin
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fapply equiv.MK : intros k,
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{ exact ⟨ k , ppi_resp_pt k ⟩ },
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all_goals cases k with k p,
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{ exact ppi.mk k p },
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all_goals reflexivity
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end
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variables (k l)
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definition ppi_homotopy.rec' [recursor] (B : k ~~* l → Type)
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(H : Π(h : k ~ l) (p : h pt ⬝ ppi_gen.resp_pt l = ppi_gen.resp_pt k), B (ppi_homotopy.mk h p))
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(h : k ~~* l) : B h :=
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begin
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induction h with h p,
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refine transport (λp, B (ppi_gen.mk h p)) _ (H h (con_eq_of_eq_con_inv p)),
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apply to_left_inv !eq_con_inv_equiv_con_eq p
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end
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definition ppi_homotopy.sigma_char [constructor]
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: (k ~~* l) ≃ Σ(p : k ~ l), p pt ⬝ ppi_gen.resp_pt l = ppi_gen.resp_pt k :=
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begin
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fapply equiv.MK : intros h,
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{ exact ⟨h , ppi_to_homotopy_pt h⟩ },
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{ cases h with h p, exact ppi_homotopy.mk h p },
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{ cases h with h p, exact ap (dpair h) (to_right_inv !eq_con_inv_equiv_con_eq p) },
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{ induction h using ppi_homotopy.rec' with h p,
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exact ap (ppi_homotopy.mk h) (to_right_inv !eq_con_inv_equiv_con_eq p) }
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end
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-- the same as pmap_eq_equiv
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definition ppi_eq_equiv : (k = l) ≃ (k ~~* l) :=
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calc (k = l) ≃ ppi_gen.sigma_char B x₀ k = ppi_gen.sigma_char B x₀ l
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: eq_equiv_fn_eq (ppi_gen.sigma_char B x₀) k l
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... ≃ Σ(p : k = l),
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pathover (λh, h pt = x₀) (ppi_gen.resp_pt k) p (ppi_gen.resp_pt l)
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: sigma_eq_equiv _ _
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... ≃ Σ(p : k = l),
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ppi_gen.resp_pt k = ap (λh, h pt) p ⬝ ppi_gen.resp_pt l
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: sigma_equiv_sigma_right
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(λp, eq_pathover_equiv_Fl p (ppi_gen.resp_pt k) (ppi_gen.resp_pt l))
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... ≃ Σ(p : k = l),
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ppi_gen.resp_pt k = apd10 p pt ⬝ ppi_gen.resp_pt l
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: sigma_equiv_sigma_right
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(λp, equiv_eq_closed_right _ (whisker_right _ (ap_eq_apd10 p _)))
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... ≃ Σ(p : k ~ l), ppi_gen.resp_pt k = p pt ⬝ ppi_gen.resp_pt l
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: sigma_equiv_sigma_left' eq_equiv_homotopy
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... ≃ Σ(p : k ~ l), p pt ⬝ ppi_gen.resp_pt l = ppi_gen.resp_pt k
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: sigma_equiv_sigma_right (λp, eq_equiv_eq_symm _ _)
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... ≃ (k ~~* l) : ppi_homotopy.sigma_char k l
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-- the same as pmap_eq
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variables {k l}
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definition ppi_eq (h : k ~~* l) : k = l :=
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(ppi_eq_equiv k l)⁻¹ᵉ h
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definition eq_of_ppi_homotopy (h : k ~~* l) : k = l := ppi_eq h
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definition ppi_homotopy_of_eq (p : k = l) : k ~~* l := ppi_eq_equiv k l p
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definition ppi_homotopy_of_eq_of_ppi_homotopy (h : k ~~* l) :
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ppi_homotopy_of_eq (eq_of_ppi_homotopy h) = h :=
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to_right_inv (ppi_eq_equiv k l) h
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variable (k)
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definition eq_ppi_homotopy_refl_ppi_homotopy_of_eq_refl : ppi_homotopy.refl k = ppi_homotopy_of_eq (refl k) :=
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begin
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induction k with k p,
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induction p, reflexivity
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end
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definition ppi_homotopy_of_eq_refl
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: ppi_homotopy.refl k = ppi_homotopy_of_eq (refl k)
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:=
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begin
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induction k with k k₀, induction k₀, reflexivity,
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end
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definition ppi_homotopy_of_eq_con {A : Type*} {B : A → Type*} {f g h : Π* (a : A), B a} (p : f = g) (q : g = h) :
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ppi_homotopy_of_eq (p ⬝ q) = ppi_homotopy_of_eq p ⬝*' ppi_homotopy_of_eq q :=
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begin induction q, induction p,
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fapply eq_of_ppi_homotopy,
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rewrite [!idp_con],
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refine transport (λ x, x ~~* x ⬝*' x) !ppi_homotopy_of_eq_refl _,
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fapply ppi_homotopy_of_eq,
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refine (ppi_trans_refl (ppi_homotopy.refl f))⁻¹
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end
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-- definition ppi_homotopy_of_eq_of_ppi_homotopy
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definition ppi_homotopy_mk_ppmap [constructor]
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{A : Type*} {X : A → Type*} {Y : Π (a : A), X a → Type*}
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{f g : Π* (a : A), Π*(x : (X a)), (Y a x)}
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(p : Πa, f a ~~* g a)
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(q : p pt ⬝*' ppi_homotopy_of_eq (ppi_resp_pt g) = ppi_homotopy_of_eq (ppi_resp_pt f))
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: f ~~* g :=
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begin
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apply ppi_homotopy.mk (λa, eq_of_ppi_homotopy (p a)),
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apply eq_of_fn_eq_fn (ppi_eq_equiv _ _),
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refine !ppi_homotopy_of_eq_con ⬝ _,
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repeat exact sorry
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-- refine !ppi_homotopy_of_eq_of_ppi_homotopy ◾** idp ⬝ q,
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end
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variable {k}
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definition ppi_homotopy_rec_on_eq [recursor] {k' : ppi_gen B x₀}
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{Q : (k ~~* k') → Type} (p : k ~~* k') (H : Π(q : k = k'), Q (ppi_homotopy_of_eq q)) : Q p :=
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ppi_homotopy_of_eq_of_ppi_homotopy p ▸ H (eq_of_ppi_homotopy p)
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definition ppi_homotopy_rec_on_idp [recursor]
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{Q : Π {k' : ppi_gen B x₀}, (k ~~* k') → Type}
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(q : Q (ppi_homotopy.refl k)) {k' : ppi_gen B x₀} (H : k ~~* k') : Q H :=
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begin
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induction H using ppi_homotopy_rec_on_eq with t,
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induction t, exact eq_ppi_homotopy_refl_ppi_homotopy_of_eq_refl k ▸ q,
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end
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variables (k l)
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definition ppi_loop_equiv : (k = k) ≃ Π*(a : A), Ω (pType.mk (B a) (k a)) :=
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begin
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induction k with k p, induction p,
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exact ppi_eq_equiv (ppi_gen.mk k idp) (ppi_gen.mk k idp)
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end
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variables {k l}
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-- definition eq_of_ppi_homotopy (h : k ~~* l) : k = l :=
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-- (ppi_eq_equiv k l)⁻¹ᵉ h
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definition ppi_loop_pequiv : Ω (Π*(a : A), P a) ≃* Π*(a : A), Ω (P a) :=
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pequiv_of_equiv (ppi_loop_equiv pt) idp
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definition pmap_compose_ppi [constructor] (g : Π(a : A), ppmap (P a) (Q a))
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(f : Π*(a : A), P a) : Π*(a : A), Q a :=
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proof ppi.mk (λa, g a (f a)) (ap (g pt) (ppi.resp_pt f) ⬝ respect_pt (g pt)) qed
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definition pmap_compose_ppi_const_right (g : Π(a : A), ppmap (P a) (Q a)) :
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pmap_compose_ppi g (ppi_const P) ~~* ppi_const Q :=
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proof ppi_homotopy.mk (λa, respect_pt (g a)) !idp_con⁻¹ qed
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definition pmap_compose_ppi_const_left (f : Π*(a : A), P a) :
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pmap_compose_ppi (λa, pconst (P a) (Q a)) f ~~* ppi_const Q :=
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ppi_homotopy.mk homotopy.rfl !ap_constant⁻¹
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definition ppi_compose_left [constructor] (g : Π(a : A), ppmap (P 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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-- 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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refine !idp_con ⬝ _,
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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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(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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abstract !con.assoc⁻¹ ⬝ whisker_right _ !ap_con_eq_con_ap⁻¹ ⬝ !con.assoc ⬝
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whisker_left _ (to_homotopy_pt (p pt)) end
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definition pmap_compose_ppi_pid_left [constructor]
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(f : Π*(a : A), P a) : pmap_compose_ppi (λa, pid (P a)) f ~~* f :=
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ppi_homotopy.mk homotopy.rfl idp
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definition pmap_compose_ppi_pcompose [constructor] (h : Π(a : A), ppmap (Q a) (R a))
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(g : Π(a : A), ppmap (P a) (Q a)) :
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pmap_compose_ppi (λa, h a ∘* g a) f ~~* pmap_compose_ppi h (pmap_compose_ppi g f) :=
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ppi_homotopy.mk homotopy.rfl
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abstract !idp_con ⬝ whisker_right _ (!ap_con ⬝ whisker_right _ !ap_compose'⁻¹) ⬝ !con.assoc end
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definition ppi_pequiv_right [constructor] (g : Π(a : A), P a ≃* Q a) :
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(Π*(a : A), P a) ≃* Π*(a : A), Q a :=
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begin
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apply pequiv_of_pmap (ppi_compose_left g),
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apply adjointify _ (ppi_compose_left (λa, (g a)⁻¹ᵉ*)),
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{ intro f, apply ppi_eq,
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refine !pmap_compose_ppi_pcompose⁻¹*' ⬝*' _,
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refine pmap_compose_ppi_phomotopy_left _ (λa, !pright_inv) ⬝*' _,
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apply pmap_compose_ppi_pid_left },
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{ intro f, apply ppi_eq,
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refine !pmap_compose_ppi_pcompose⁻¹*' ⬝*' _,
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refine pmap_compose_ppi_phomotopy_left _ (λa, !pleft_inv) ⬝*' _,
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apply pmap_compose_ppi_pid_left }
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end
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definition psigma_gen [constructor] {A : Type*} (P : A → Type) (x : P pt) : Type* :=
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pointed.MK (Σa, P a) ⟨pt, x⟩
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end pointed
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open fiber function
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namespace pointed
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variables {A B C : Type*}
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-- TODO: replace in types.fiber
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definition pfiber.sigma_char' (f : A →* B) :
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pfiber f ≃* psigma_gen (λa, f a = pt) (respect_pt f) :=
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pequiv_of_equiv (fiber.sigma_char f pt) idp
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/- the pointed type of unpointed (nondependent) maps -/
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definition pumap [constructor] (A : Type) (B : Type*) : Type* :=
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pointed.MK (A → B) (const A pt)
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/- the pointed type of unpointed dependent maps -/
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definition pupi [constructor] {A : Type} (B : A → Type*) : Type* :=
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pointed.MK (Πa, B a) (λa, pt)
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notation `Πᵘ*` binders `, ` r:(scoped P, pupi P) := r
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infix ` →ᵘ* `:30 := pumap
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definition ppmap.sigma_char [constructor] (A B : Type*) :
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ppmap A B ≃* @psigma_gen (A →ᵘ* B) (λf, f pt = pt) idp :=
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pequiv_of_equiv pmap.sigma_char idp
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definition pppi.sigma_char [constructor] {A : Type*} (B : A → Type*) :
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(Π*(a : A), B a) ≃* @psigma_gen (Πᵘ*a, B a) (λf, f pt = pt) idp :=
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proof pequiv_of_equiv !ppi.sigma_char idp qed
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definition psigma_gen_pequiv_psigma_gen [constructor] {A A' : Type*} {B : A → Type}
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{B' : A' → Type} {b : B pt} {b' : B' pt} (f : A ≃* A')
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(g : Πa, B a ≃ B' (f a)) (p : g pt b =[respect_pt f] b') : psigma_gen B b ≃* psigma_gen B' b' :=
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pequiv_of_equiv (sigma_equiv_sigma f g) (sigma_eq (respect_pt f) p)
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definition psigma_gen_pequiv_psigma_gen_left [constructor] {A A' : Type*} {B : A' → Type}
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{b : B pt} (f : A ≃* A') {b' : B (f pt)} (q : b' =[respect_pt f] b) :
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psigma_gen (B ∘ f) b' ≃* psigma_gen B b :=
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psigma_gen_pequiv_psigma_gen f (λa, erfl) q
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definition psigma_gen_pequiv_psigma_gen_right [constructor] {A : Type*} {B B' : A → Type}
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{b : B pt} {b' : B' pt} (f : Πa, B a ≃ B' a) (p : f pt b = b') :
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psigma_gen B b ≃* psigma_gen B' b' :=
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psigma_gen_pequiv_psigma_gen pequiv.rfl f (pathover_idp_of_eq p)
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definition psigma_gen_pequiv_psigma_gen_basepoint [constructor] {A : Type*} {B : A → Type}
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{b b' : B pt} (p : b = b') : psigma_gen B b ≃* psigma_gen B b' :=
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psigma_gen_pequiv_psigma_gen_right (λa, erfl) p
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definition ppi_gen_functor_right [constructor] {A : Type*} {B B' : A → Type}
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{b : B pt} {b' : B' pt} (f : Πa, B a → B' a) (p : f pt b = b') (g : ppi_gen B b)
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: ppi_gen B' b' :=
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ppi_gen.mk (λa, f a (g a)) (ap (f pt) (ppi_gen.resp_pt g) ⬝ p)
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definition ppi_gen_functor_right_compose [constructor] {A : Type*} {B₁ B₂ B₃ : A → Type}
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{b₁ : B₁ pt} {b₂ : B₂ pt} {b₃ : B₃ pt} (f₂ : Πa, B₂ a → B₃ a) (p₂ : f₂ pt b₂ = b₃)
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(f₁ : Πa, B₁ a → B₂ a) (p₁ : f₁ pt b₁ = b₂)
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(g : ppi_gen B₁ b₁) : ppi_gen_functor_right (λa, f₂ a ∘ f₁ a) (ap (f₂ pt) p₁ ⬝ p₂) g ~~*
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ppi_gen_functor_right f₂ p₂ (ppi_gen_functor_right f₁ p₁ g) :=
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begin
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fapply ppi_homotopy.mk,
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{ reflexivity },
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{ induction p₁, induction p₂, exact !idp_con ⬝ !ap_compose⁻¹ }
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end
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definition ppi_gen_functor_right_id [constructor] {A : Type*} {B : A → Type}
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{b : B pt} (g : ppi_gen B b) : ppi_gen_functor_right (λa, id) idp g ~~* g :=
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begin
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fapply ppi_homotopy.mk,
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{ reflexivity },
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{ reflexivity }
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end
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definition ppi_gen_functor_right_homotopy [constructor] {A : Type*} {B B' : A → Type}
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{b : B pt} {b' : B' pt} {f f' : Πa, B a → B' a} {p : f pt b = b'} {p' : f' pt b = b'}
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(h : f ~2 f') (q : h pt b ⬝ p' = p) (g : ppi_gen B b) :
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ppi_gen_functor_right f p g ~~* ppi_gen_functor_right f' p' g :=
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begin
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fapply ppi_homotopy.mk,
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{ exact λa, h a (g a) },
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{ induction g with g r, induction r, induction q,
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exact whisker_left _ !idp_con ⬝ !idp_con⁻¹ }
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end
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definition ppi_gen_equiv_ppi_gen_right [constructor] {A : Type*} {B B' : A → Type}
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{b : B pt} {b' : B' pt} (f : Πa, B a ≃ B' a) (p : f pt b = b') :
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ppi_gen B b ≃ ppi_gen B' b' :=
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equiv.MK (ppi_gen_functor_right f p) (ppi_gen_functor_right (λa, (f a)⁻¹ᵉ) (inv_eq_of_eq p⁻¹))
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abstract begin
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intro g, apply ppi_eq,
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refine !ppi_gen_functor_right_compose⁻¹*' ⬝*' _,
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refine ppi_gen_functor_right_homotopy (λa, to_right_inv (f a)) _ g ⬝*'
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!ppi_gen_functor_right_id, induction p, exact adj (f pt) b ⬝ ap02 (f pt) !idp_con⁻¹
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end end
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abstract begin
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intro g, apply ppi_eq,
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refine !ppi_gen_functor_right_compose⁻¹*' ⬝*' _,
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refine ppi_gen_functor_right_homotopy (λa, to_left_inv (f a)) _ g ⬝*'
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!ppi_gen_functor_right_id, induction p, exact (!idp_con ⬝ !idp_con)⁻¹,
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end end
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definition ppi_gen_equiv_ppi_gen_basepoint [constructor] {A : Type*} {B : A → Type} {b b' : B pt}
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(p : b = b') : ppi_gen B b ≃ ppi_gen B b' :=
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ppi_gen_equiv_ppi_gen_right (λa, erfl) p
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open sigma.ops
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definition psigma_gen_pi_pequiv_pupi_psigma_gen [constructor] {A : Type*} {B : A → Type*}
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(C : Πa, B a → Type) (c : Πa, C a pt) :
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@psigma_gen (Πᵘ*a, B a) (λf, Πa, C a (f a)) c ≃* Πᵘ*a, psigma_gen (C a) (c a) :=
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pequiv_of_equiv sigma_pi_equiv_pi_sigma idp
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definition pupi_psigma_gen_pequiv_psigma_gen_pi [constructor] {A : Type*} {B : A → Type*}
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(C : Πa, B a → Type) (c : Πa, C a pt) :
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(Πᵘ*a, psigma_gen (C a) (c a)) ≃* @psigma_gen (Πᵘ*a, B a) (λf, Πa, C a (f a)) c :=
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pequiv_of_equiv sigma_pi_equiv_pi_sigma⁻¹ᵉ idp
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definition psigma_gen_assoc [constructor] {A : Type*} {B : A → Type} (C : Πa, B a → Type)
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(b₀ : B pt) (c₀ : C pt b₀) :
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psigma_gen (λa, Σb, C a b) ⟨b₀, c₀⟩ ≃* @psigma_gen (psigma_gen B b₀) (λv, C v.1 v.2) c₀ :=
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pequiv_of_equiv !sigma_assoc_equiv idp
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definition psigma_gen_swap [constructor] {A : Type*} {B B' : A → Type}
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(C : Π⦃a⦄, B a → B' a → Type) (b₀ : B pt) (b₀' : B' pt) (c₀ : C b₀ b₀') :
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@psigma_gen (psigma_gen B b₀ ) (λv, Σb', C v.2 b') ⟨b₀', c₀⟩ ≃*
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@psigma_gen (psigma_gen B' b₀') (λv, Σb , C b v.2) ⟨b₀ , c₀⟩ :=
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|
!psigma_gen_assoc⁻¹ᵉ* ⬝e* psigma_gen_pequiv_psigma_gen_right (λa, !sigma_comm_equiv) idp ⬝e*
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!psigma_gen_assoc
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definition ppi_psigma.{u v w} {A : pType.{u}} {B : A → pType.{v}} (C : Πa, B a → Type.{w})
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(c : Πa, C a pt) : (Π*(a : A), (psigma_gen (C a) (c a))) ≃*
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psigma_gen (λ(f : Π*(a : A), B a), ppi_gen (λa, C a (f a))
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(transport (C pt) (ppi.resp_pt f)⁻¹ (c pt))) (ppi_const _) :=
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|
proof
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|
calc (Π*(a : A), psigma_gen (C a) (c a))
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≃* @psigma_gen (Πᵘ*a, psigma_gen (C a) (c a)) (λf, f pt = pt) idp : pppi.sigma_char
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... ≃* @psigma_gen (@psigma_gen (Πᵘ*a, B a) (λf, Πa, C a (f a)) c)
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(λv, Σ(p : v.1 pt = pt), v.2 pt =[p] c pt) ⟨idp, idpo⟩ :
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|
by exact psigma_gen_pequiv_psigma_gen (pupi_psigma_gen_pequiv_psigma_gen_pi C c)
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(λf, sigma_eq_equiv _ _) idpo
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... ≃* @psigma_gen (@psigma_gen (Πᵘ*a, B a) (λf, f pt = pt) idp)
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(λv, Σ(g : Πa, C a (v.1 a)), g pt =[v.2] c pt) ⟨c, idpo⟩ :
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|
by apply psigma_gen_swap
|
|
... ≃* psigma_gen (λ(f : Π*(a : A), B a), ppi_gen (λa, C a (f a))
|
|
(transport (C pt) (ppi.resp_pt f)⁻¹ (c pt)))
|
|
(ppi_const _) :
|
|
by exact (psigma_gen_pequiv_psigma_gen (pppi.sigma_char B)
|
|
(λf, !ppi_gen.sigma_char ⬝e sigma_equiv_sigma_right (λg, !pathover_equiv_eq_tr⁻¹ᵉ))
|
|
idpo)⁻¹ᵉ*
|
|
qed
|
|
|
|
definition ppmap_psigma {A B : Type*} (C : B → Type) (c : C pt) :
|
|
ppmap A (psigma_gen C c) ≃*
|
|
psigma_gen (λ(f : ppmap A B), ppi_gen (C ∘ f) (transport C (respect_pt f)⁻¹ c))
|
|
(ppi_const _) :=
|
|
!pppi_pequiv_ppmap⁻¹ᵉ* ⬝e* !ppi_psigma ⬝e*
|
|
sorry
|
|
-- psigma_gen_pequiv_psigma_gen (pppi_pequiv_ppmap A B) (λf, begin esimp, exact ppi_gen_equiv_ppi_gen_right (λa, _) _ end) _
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|
|
definition pfiber_ppcompose_left (f : B →* C) :
|
|
pfiber (@ppcompose_left A B C f) ≃* ppmap A (pfiber f) :=
|
|
calc
|
|
pfiber (@ppcompose_left A B C f) ≃*
|
|
psigma_gen (λ(g : ppmap A B), f ∘* g = pconst A C)
|
|
proof (eq_of_phomotopy (pcompose_pconst f)) qed :
|
|
by exact !pfiber.sigma_char'
|
|
... ≃* psigma_gen (λ(g : ppmap A B), f ∘* g ~* pconst A C) proof (pcompose_pconst f) qed :
|
|
by exact psigma_gen_pequiv_psigma_gen_right (λa, !pmap_eq_equiv)
|
|
!phomotopy_of_eq_of_phomotopy
|
|
... ≃* psigma_gen (λ(g : ppmap A B), ppi_gen (λa, f (g a) = pt)
|
|
(transport (λb, f b = pt) (respect_pt g)⁻¹ (respect_pt f)))
|
|
(ppi_const _) :
|
|
begin
|
|
refine psigma_gen_pequiv_psigma_gen_right
|
|
(λg, ppi_gen_equiv_ppi_gen_basepoint (_ ⬝ !eq_transport_Fl⁻¹)) _,
|
|
intro g, refine !con_idp ⬝ _, apply whisker_right,
|
|
exact ap02 f !inv_inv⁻¹ ⬝ !ap_inv,
|
|
apply ppi_eq, fapply ppi_homotopy.mk,
|
|
intro x, reflexivity,
|
|
refine !idp_con ⬝ _, symmetry, refine !ap_id ◾ !idp_con ⬝ _, apply con.right_inv
|
|
end
|
|
... ≃* ppmap A (psigma_gen (λb, f b = pt) (respect_pt f)) :
|
|
by exact (ppmap_psigma _ _)⁻¹ᵉ*
|
|
... ≃* ppmap A (pfiber f) : by exact pequiv_ppcompose_left !pfiber.sigma_char'⁻¹ᵉ*
|
|
|
|
|
|
definition pfiber_ppi_compose_left {B C : A → Type*} (f : Πa, B a →* C a) :
|
|
pfiber (ppi_compose_left f) ≃* Π*(a : A), pfiber (f a) :=
|
|
calc
|
|
pfiber (ppi_compose_left f) ≃*
|
|
psigma_gen (λ(g : Π*(a : A), B a), pmap_compose_ppi f g = ppi_const C)
|
|
proof (ppi_eq (pmap_compose_ppi_const_right f)) qed : by exact !pfiber.sigma_char'
|
|
... ≃* psigma_gen (λ(g : Π*(a : A), B a), pmap_compose_ppi f g ~~* ppi_const C)
|
|
proof (pmap_compose_ppi_const_right f) qed :
|
|
by exact psigma_gen_pequiv_psigma_gen_right (λa, !ppi_eq_equiv)
|
|
!ppi_homotopy_of_eq_of_ppi_homotopy
|
|
... ≃* psigma_gen (λ(g : Π*(a : A), B a), ppi_gen (λa, f a (g a) = pt)
|
|
(transport (λb, f pt b = pt) (ppi.resp_pt g)⁻¹ (respect_pt (f pt))))
|
|
(ppi_const _) :
|
|
begin
|
|
refine psigma_gen_pequiv_psigma_gen_right
|
|
(λg, ppi_gen_equiv_ppi_gen_basepoint (_ ⬝ !eq_transport_Fl⁻¹)) _,
|
|
intro g, refine !con_idp ⬝ _, apply whisker_right,
|
|
exact ap02 (f pt) !inv_inv⁻¹ ⬝ !ap_inv,
|
|
apply ppi_eq, fapply ppi_homotopy.mk,
|
|
intro x, reflexivity,
|
|
refine !idp_con ⬝ _, symmetry, refine !ap_id ◾ !idp_con ⬝ _, apply con.right_inv
|
|
end
|
|
... ≃* Π*(a : A), (psigma_gen (λb, f a b = pt) (respect_pt (f a))) :
|
|
by exact (ppi_psigma _ _)⁻¹ᵉ*
|
|
... ≃* Π*(a : A), pfiber (f a) : by exact ppi_pequiv_right (λa, !pfiber.sigma_char'⁻¹ᵉ*)
|
|
|
|
-- definition pppi_ppmap {A C : Type*} {B : A → Type*} :
|
|
-- ppmap (/- dependent smash of B -/) C ≃* Π*(a : A), ppmap (B a) C :=
|
|
|
|
definition ppi_add_point_over {A : Type} (B : A → Type*) :
|
|
(Π*a, add_point_over B a) ≃ Πa, B a :=
|
|
begin
|
|
fapply equiv.MK,
|
|
{ intro f a, exact f (some a) },
|
|
{ intro f, fconstructor,
|
|
intro a, cases a, exact pt, exact f a,
|
|
reflexivity },
|
|
{ intro f, reflexivity },
|
|
{ intro f, cases f with f p, apply ppi_eq, fapply ppi_homotopy.mk,
|
|
{ intro a, cases a, exact p⁻¹, reflexivity },
|
|
{ exact con.left_inv p }},
|
|
end
|
|
|
|
definition pppi_add_point_over {A : Type} (B : A → Type*) :
|
|
(Π*a, add_point_over B a) ≃* Πᵘ*a, B a :=
|
|
pequiv_of_equiv (ppi_add_point_over B) idp
|
|
|
|
definition ppmap_add_point {A : Type} (B : Type*) :
|
|
ppmap A₊ B ≃* A →ᵘ* B :=
|
|
pequiv_of_equiv (pmap_equiv_left A B) idp
|
|
|
|
-- TODO: homotopy_of_eq and apd10 should be the same
|
|
-- TODO: there is also apd10_eq_of_homotopy in both pi and eq(?)
|
|
|
|
/- stuff about the pointed type of unpointed maps (dependent and non-dependent) -/
|
|
definition sigma_pumap {A : Type} (B : A → Type) (C : Type*) :
|
|
((Σa, B a) →ᵘ* C) ≃* Πᵘ*a, B a →ᵘ* C :=
|
|
pequiv_of_equiv (equiv_sigma_rec _)⁻¹ᵉ idp
|
|
|
|
definition loop_pupi [constructor] {A : Type} (B : A → Type*) : Ω (Πᵘ*a, B a) ≃* Πᵘ*a, Ω (B a) :=
|
|
pequiv_of_equiv eq_equiv_homotopy idp
|
|
|
|
definition phomotopy_mk_pupi [constructor] {A : Type*} {B : Type} {C : B → Type*}
|
|
{f g : A →* (Πᵘ*b, C b)} (p : f ~2 g)
|
|
(q : p pt ⬝hty apd10 (respect_pt g) ~ apd10 (respect_pt f)) : f ~* g :=
|
|
begin
|
|
apply phomotopy.mk (λa, eq_of_homotopy (p a)),
|
|
apply eq_of_fn_eq_fn eq_equiv_homotopy,
|
|
apply eq_of_homotopy, intro b,
|
|
refine !apd10_con ⬝ _,
|
|
refine whisker_right _ !pi.apd10_eq_of_homotopy ⬝ q b
|
|
end
|
|
|
|
definition pupi_functor [constructor] {A A' : Type} {B : A → Type*} {B' : A' → Type*}
|
|
(f : A' → A) (g : Πa, B (f a) →* B' a) : (Πᵘ*a, B a) →* (Πᵘ*a', B' a') :=
|
|
pmap.mk (λh a, g a (h (f a))) (eq_of_homotopy (λa, respect_pt (g a)))
|
|
|
|
definition pupi_functor_compose {A A' A'' : Type}
|
|
{B : A → Type*} {B' : A' → Type*} {B'' : A'' → Type*} (f : A'' → A') (f' : A' → A)
|
|
(g' : Πa, B' (f a) →* B'' a) (g : Πa, B (f' a) →* B' a) :
|
|
pupi_functor (f' ∘ f) (λa, g' a ∘* g (f a)) ~* pupi_functor f g' ∘* pupi_functor f' g :=
|
|
begin
|
|
fapply phomotopy_mk_pupi,
|
|
{ intro h a, reflexivity },
|
|
{ intro a, refine !idp_con ⬝ _, refine !apd10_con ⬝ _ ⬝ !pi.apd10_eq_of_homotopy⁻¹, esimp,
|
|
refine (!apd10_prepostcompose ⬝ ap02 (g' a) !pi.apd10_eq_of_homotopy) ◾
|
|
!pi.apd10_eq_of_homotopy }
|
|
end
|
|
|
|
definition pupi_functor_pid (A : Type) (B : A → Type*) :
|
|
pupi_functor id (λa, pid (B a)) ~* pid (Πᵘ*a, B a) :=
|
|
begin
|
|
fapply phomotopy_mk_pupi,
|
|
{ intro h a, reflexivity },
|
|
{ intro a, refine !idp_con ⬝ !pi.apd10_eq_of_homotopy⁻¹ }
|
|
end
|
|
|
|
definition pupi_functor_phomotopy {A A' : Type} {B : A → Type*} {B' : A' → Type*}
|
|
{f f' : A' → A} {g : Πa, B (f a) →* B' a} {g' : Πa, B (f' a) →* B' a}
|
|
(p : f ~ f') (q : Πa, g a ~* g' a ∘* ptransport B (p a)) :
|
|
pupi_functor f g ~* pupi_functor f' g' :=
|
|
begin
|
|
fapply phomotopy_mk_pupi,
|
|
{ intro h a, exact q a (h (f a)) ⬝ ap (g' a) (apdt h (p a)) },
|
|
{ intro a, esimp,
|
|
exact whisker_left _ !pi.apd10_eq_of_homotopy ⬝ !con.assoc ⬝
|
|
to_homotopy_pt (q a) ⬝ !pi.apd10_eq_of_homotopy⁻¹ }
|
|
end
|
|
|
|
definition pupi_pequiv [constructor] {A A' : Type} {B : A → Type*} {B' : A' → Type*}
|
|
(e : A' ≃ A) (f : Πa, B (e a) ≃* B' a) :
|
|
(Πᵘ*a, B a) ≃* (Πᵘ*a', B' a') :=
|
|
pequiv.MK (pupi_functor e f)
|
|
(pupi_functor e⁻¹ᵉ (λa, ptransport B (right_inv e a) ∘* (f (e⁻¹ᵉ a))⁻¹ᵉ*))
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abstract begin
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refine !pupi_functor_compose⁻¹* ⬝* pupi_functor_phomotopy (to_right_inv e) _ ⬝*
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!pupi_functor_pid,
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intro a, exact !pinv_pcompose_cancel_right ⬝* !pid_pcompose⁻¹*
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end end
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abstract begin
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refine !pupi_functor_compose⁻¹* ⬝* pupi_functor_phomotopy (to_left_inv e) _ ⬝*
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!pupi_functor_pid,
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intro a, refine !passoc⁻¹* ⬝* pinv_right_phomotopy_of_phomotopy _ ⬝* !pid_pcompose⁻¹*,
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refine pwhisker_left _ _ ⬝* !ptransport_natural,
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exact ptransport_change_eq _ (adj e a) ⬝* ptransport_ap B e (to_left_inv e a)
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end end
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definition pupi_pequiv_right [constructor] {A : Type} {B B' : A → Type*} (f : Πa, B a ≃* B' a) :
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(Πᵘ*a, B a) ≃* (Πᵘ*a, B' a) :=
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pupi_pequiv erfl f
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definition loop_pumap [constructor] (A : Type) (B : Type*) : Ω (A →ᵘ* B) ≃* A →ᵘ* Ω B :=
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loop_pupi (λa, B)
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definition phomotopy_mk_pumap [constructor] {A C : Type*} {B : Type} {f g : A →* (B →ᵘ* C)}
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(p : f ~2 g) (q : p pt ⬝hty apd10 (respect_pt g) ~ apd10 (respect_pt f))
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: f ~* g :=
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phomotopy_mk_pupi p q
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definition pumap_functor [constructor] {A A' : Type} {B B' : Type*} (f : A' → A) (g : B →* B') :
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(A →ᵘ* B) →* (A' →ᵘ* B') :=
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pupi_functor f (λa, g)
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definition pumap_functor_compose {A A' A'' : Type} {B B' B'' : Type*}
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(f : A'' → A') (f' : A' → A) (g' : B' →* B'') (g : B →* B') :
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pumap_functor (f' ∘ f) (g' ∘* g) ~* pumap_functor f g' ∘* pumap_functor f' g :=
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pupi_functor_compose f f' (λa, g') (λa, g)
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definition pumap_functor_pid (A : Type) (B : Type*) :
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pumap_functor id (pid B) ~* pid (A →ᵘ* B) :=
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pupi_functor_pid A (λa, B)
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definition pumap_functor_phomotopy {A A' : Type} {B B' : Type*} {f f' : A' → A} {g g' : B →* B'}
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(p : f ~ f') (q : g ~* g') : pumap_functor f g ~* pumap_functor f' g' :=
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pupi_functor_phomotopy p (λa, q ⬝* !pcompose_pid⁻¹* ⬝* pwhisker_left _ !ptransport_constant⁻¹*)
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definition pumap_pequiv [constructor] {A A' : Type} {B B' : Type*} (e : A ≃ A') (f : B ≃* B') :
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(A →ᵘ* B) ≃* (A' →ᵘ* B') :=
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pupi_pequiv e⁻¹ᵉ (λa, f)
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definition pumap_pequiv_right [constructor] (A : Type) {B B' : Type*} (f : B ≃* B') :
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(A →ᵘ* B) ≃* (A →ᵘ* B') :=
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pumap_pequiv erfl f
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definition pumap_pequiv_left [constructor] {A A' : Type} (B : Type*) (f : A ≃ A') :
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(A →ᵘ* B) ≃* (A' →ᵘ* B) :=
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pumap_pequiv f pequiv.rfl
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end pointed open pointed
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open is_trunc is_conn
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namespace is_conn
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section
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variables (A : Type*) (n : ℕ₋₂) [H : is_conn (n.+1) A]
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include H
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definition is_contr_ppi_match (P : A → Type*) (H : Πa, is_trunc (n.+1) (P a))
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: is_contr (Π*(a : A), P a) :=
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begin
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apply is_contr.mk pt,
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intro f, induction f with f p,
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apply ppi_eq, fapply ppi_homotopy.mk,
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{ apply is_conn.elim n, exact p⁻¹ },
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{ krewrite (is_conn.elim_β n), apply con.left_inv }
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end
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-- definition is_trunc_ppi_of_is_conn (k : ℕ₋₂) (P : A → Type*)
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-- : is_trunc k.+1 (Π*(a : A), P a) :=
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definition is_trunc_ppi_of_is_conn (k l : ℕ₋₂) (H2 : l ≤ n.+1+2+k)
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(P : A → Type*) (H3 : Πa, is_trunc l (P a)) :
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is_trunc k.+1 (Π*(a : A), P a) :=
|
|
begin
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have H4 : Πa, is_trunc (n.+1+2+k) (P a), from λa, is_trunc_of_le (P a) H2,
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clear H2 H3, revert P H4,
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|
induction k with k IH: intro P H4,
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{ apply is_prop_of_imp_is_contr, intro f,
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|
apply is_contr_ppi_match A n P H4 },
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|
{ apply is_trunc_succ_of_is_trunc_loop
|
|
(trunc_index.succ_le_succ (trunc_index.minus_two_le k)),
|
|
intro f,
|
|
apply @is_trunc_equiv_closed_rev _ _ k.+1 (ppi_loop_equiv f),
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|
apply IH, intro a,
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|
apply is_trunc_loop, apply H4 }
|
|
end
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definition is_trunc_pmap_of_is_conn (k l : ℕ₋₂) (B : Type*) (H2 : l ≤ n.+1+2+k)
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|
(H3 : is_trunc l B) : is_trunc k.+1 (A →* B) :=
|
|
@is_trunc_equiv_closed _ _ k.+1 (ppi_equiv_pmap A B)
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|
(is_trunc_ppi_of_is_conn A n k l H2 (λ a, B) _)
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|
|
|
end
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-- this is probably much easier to prove directly
|
|
definition is_trunc_ppi (A : Type*) (n k : ℕ₋₂) (H : n ≤ k) (P : A → Type*)
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|
(H2 : Πa, is_trunc n (P a)) : is_trunc k (Π*(a : A), P a) :=
|
|
begin
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|
cases k with k,
|
|
{ apply is_contr_of_merely_prop,
|
|
{ exact @is_trunc_ppi_of_is_conn A -2 (is_conn_minus_one A (tr pt)) -2 _
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(trunc_index.le.step H) P H2 },
|
|
{ exact tr pt } },
|
|
{ assert K : n ≤ -1 +2+ k,
|
|
{ rewrite (trunc_index.add_plus_two_comm -1 k), exact H },
|
|
{ exact @is_trunc_ppi_of_is_conn A -2 (is_conn_minus_one A (tr pt)) k _ K P H2 } }
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|
end
|
|
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|
end is_conn
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