98 lines
3.9 KiB
Text
98 lines
3.9 KiB
Text
/-
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Copyright (c) 2014 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: types.equiv
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Author: Floris van Doorn
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Ported from Coq HoTT
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Theorems about the types equiv and is_equiv
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-/
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import types.fiber types.arrow
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open eq is_trunc sigma sigma.ops arrow pi
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namespace is_equiv
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open equiv function
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section
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open fiber
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variables {A B : Type} (f : A → B) [H : is_equiv f]
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include H
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definition is_contr_fiber_of_is_equiv (b : B) : is_contr (fiber f b) :=
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is_contr.mk
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(fiber.mk (f⁻¹ b) (retr f b))
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(λz, fiber.rec_on z (λa p, fiber.eq_mk ((ap f⁻¹ p)⁻¹ ⬝ sect f a) (calc
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retr f b = (ap (f ∘ f⁻¹) p)⁻¹ ⬝ ((ap (f ∘ f⁻¹) p) ⬝ retr f b) : inv_con_cancel_left
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... = (ap (f ∘ f⁻¹) p)⁻¹ ⬝ (retr f (f a) ⬝ p) : by rewrite ap_con_eq_con
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... = (ap (f ∘ f⁻¹) p)⁻¹ ⬝ (ap f (sect f a) ⬝ p) : by rewrite adj
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... = (ap (f ∘ f⁻¹) p)⁻¹ ⬝ ap f (sect f a) ⬝ p : con.assoc
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... = (ap f (ap f⁻¹ p))⁻¹ ⬝ ap f (sect f a) ⬝ p : by rewrite ap_compose
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... = ap f (ap f⁻¹ p)⁻¹ ⬝ ap f (sect f a) ⬝ p : by rewrite ap_inv
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... = ap f ((ap f⁻¹ p)⁻¹ ⬝ sect f a) ⬝ p : by rewrite ap_con)))
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definition is_contr_right_inverse : is_contr (Σ(g : B → A), f ∘ g ∼ id) :=
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begin
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fapply is_trunc_equiv_closed,
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{apply sigma_equiv_sigma_id, intro g, apply eq_equiv_homotopy},
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fapply is_trunc_equiv_closed,
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{apply fiber.sigma_char},
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fapply is_contr_fiber_of_is_equiv,
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apply (to_is_equiv (arrow_equiv_arrow_right (equiv.mk f H))),
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end
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definition is_contr_right_coherence (u : Σ(g : B → A), f ∘ g ∼ id)
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: is_contr (Σ(η : u.1 ∘ f ∼ id), Π(a : A), u.2 (f a) = ap f (η a)) :=
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begin
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fapply is_trunc_equiv_closed,
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{apply equiv.symm, apply sigma_pi_equiv_pi_sigma},
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fapply is_trunc_equiv_closed,
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{apply pi_equiv_pi_id, intro a,
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apply (equiv_fiber_eq (fiber.mk (u.1 (f a)) (u.2 (f a))) (fiber.mk a idp))},
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fapply is_trunc_pi,
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intro a, fapply @is_contr_eq,
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apply is_contr_fiber_of_is_equiv
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end
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end
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variables {A B : Type} (f : A → B)
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protected definition sigma_char : (is_equiv f) ≃
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(Σ(g : B → A) (ε : f ∘ g ∼ id) (η : g ∘ f ∼ id), Π(a : A), ε (f a) = ap f (η a)) :=
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equiv.MK (λH, ⟨inv f, retr f, sect f, adj f⟩)
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(λp, is_equiv.mk p.1 p.2.1 p.2.2.1 p.2.2.2)
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(λp, begin
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cases p with (p1, p2),
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cases p2 with (p21, p22),
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cases p22 with (p221, p222),
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apply idp
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end)
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(λH, is_equiv.rec_on H (λH1 H2 H3 H4, idp))
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protected definition sigma_char' : (is_equiv f) ≃
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(Σ(u : Σ(g : B → A), f ∘ g ∼ id), Σ(η : u.1 ∘ f ∼ id), Π(a : A), u.2 (f a) = ap f (η a)) :=
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calc
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(is_equiv f) ≃
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(Σ(g : B → A) (ε : f ∘ g ∼ id) (η : g ∘ f ∼ id), Π(a : A), ε (f a) = ap f (η a))
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: is_equiv.sigma_char
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... ≃ (Σ(u : Σ(g : B → A), f ∘ g ∼ id), Σ(η : u.1 ∘ f ∼ id), Π(a : A), u.2 (f a) = ap f (η a))
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: {sigma_assoc_equiv (λu, Σ(η : u.1 ∘ f ∼ id), Π(a : A), u.2 (f a) = ap f (η a))}
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local attribute is_contr_right_inverse [instance]
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local attribute is_contr_right_coherence [instance]
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theorem is_hprop_is_equiv [instance] : is_hprop (is_equiv f) :=
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is_hprop_of_imp_is_contr (λ(H : is_equiv f), is_trunc_equiv_closed -2 (equiv.symm !sigma_char'))
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end is_equiv
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namespace equiv
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open is_equiv
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variables {A B : Type}
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protected definition eq_mk' {f f' : A → B} [H : is_equiv f] [H' : is_equiv f'] (p : f = f')
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: equiv.mk f H = equiv.mk f' H' :=
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apD011 equiv.mk p !is_hprop.elim
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protected definition eq_mk {f f' : A ≃ B} (p : to_fun f = to_fun f') : f = f' :=
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by (cases f; cases f'; apply (equiv.eq_mk' p))
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end equiv
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