104 lines
3.1 KiB
Text
104 lines
3.1 KiB
Text
/-
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Copyright (c) 2014 Jakob von Raumer. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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Ported from Coq HoTT
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-/
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import .equiv
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open eq is_equiv equiv equiv.ops pointed is_trunc
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-- structure pequiv (A B : Type*) :=
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-- (to_pmap : A →* B)
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-- (is_equiv_to_pmap : is_equiv to_pmap)
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structure pequiv (A B : Type*) extends equiv A B, pmap A B
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section
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universe variable u
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structure ptrunctype (n : trunc_index) extends trunctype.{u} n, Pointed.{u}
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end
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namespace pointed
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variables {A B C : Type*}
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/- pointed equivalences -/
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infix ` ≃* `:25 := pequiv
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attribute pequiv.to_pmap [coercion]
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attribute pequiv.to_is_equiv [instance]
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definition pequiv_of_pmap [constructor] (f : A →* B) (H : is_equiv f) : A ≃* B :=
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pequiv.mk f _ (respect_pt f)
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definition pequiv_of_equiv [constructor] (f : A ≃ B) (H : f pt = pt) : A ≃* B :=
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pequiv.mk f _ H
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definition equiv_of_pequiv [constructor] (f : A ≃* B) : A ≃ B :=
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equiv.mk f _
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definition to_pinv [constructor] (f : A ≃* B) : B →* A :=
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pmap.mk f⁻¹ (ap f⁻¹ (respect_pt f)⁻¹ ⬝ !left_inv)
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definition pua {A B : Type*} (f : A ≃* B) : A = B :=
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Pointed_eq (equiv_of_pequiv f) !respect_pt
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protected definition pequiv.refl [refl] [constructor] (A : Type*) : A ≃* A :=
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pequiv_of_pmap !pid !is_equiv_id
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protected definition pequiv.rfl [constructor] : A ≃* A :=
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pequiv.refl A
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protected definition pequiv.symm [symm] (f : A ≃* B) : B ≃* A :=
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pequiv_of_pmap (to_pinv f) !is_equiv_inv
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protected definition pequiv.trans [trans] (f : A ≃* B) (g : B ≃* C) : A ≃* C :=
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pequiv_of_pmap (pcompose g f) !is_equiv_compose
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postfix `⁻¹ᵉ*`:(max + 1) := pequiv.symm
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infix ` ⬝e* `:75 := pequiv.trans
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definition pequiv_rect' (f : A ≃* B) (P : A → B → Type)
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(g : Πb, P (f⁻¹ b) b) (a : A) : P a (f a) :=
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left_inv f a ▸ g (f a)
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definition pequiv_of_eq [constructor] {A B : Type*} (p : A = B) : A ≃* B :=
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pequiv_of_pmap (pcast p) !is_equiv_tr
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definition peconcat_eq {A B C : Type*} (p : A ≃* B) (q : B = C) : A ≃* C :=
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p ⬝e* pequiv_of_eq q
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definition eq_peconcat {A B C : Type*} (p : A = B) (q : B ≃* C) : A ≃* C :=
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pequiv_of_eq p ⬝e* q
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definition eq_of_pequiv {A B : Type*} (p : A ≃* B) : A = B :=
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Pointed_eq (equiv_of_pequiv p) !respect_pt
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definition peap {A B : Type*} (F : Type* → Type*) (p : A ≃* B) : F A ≃* F B :=
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pequiv_of_pmap (pcast (ap F (eq_of_pequiv p))) begin cases eq_of_pequiv p, apply is_equiv_id end
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definition loop_space_pequiv [constructor] (p : A ≃* B) : Ω A ≃* Ω B :=
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pequiv_of_pmap (ap1 p) (is_equiv_ap1 p)
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definition pequiv_eq {p q : A ≃* B} (H : p = q :> (A →* B)) : p = q :=
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begin
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cases p with f Hf, cases q with g Hg, esimp at *,
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exact apd011 pequiv_of_pmap H !is_hprop.elim
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end
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definition loop_space_pequiv_rfl
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: loop_space_pequiv (@pequiv.refl A) = @pequiv.refl (Ω A) :=
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begin
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apply pequiv_eq, fapply pmap_eq: esimp,
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{ intro p, exact !idp_con ⬝ !ap_id},
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{ reflexivity}
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end
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infix ` ⬝e*p `:75 := peconcat_eq
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infix ` ⬝pe* `:75 := eq_peconcat
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end pointed
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