136 lines
5 KiB
Text
136 lines
5 KiB
Text
/-
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Copyright (c) 2015 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: hit.trunc
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Authors: Floris van Doorn
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n-truncation of types.
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Ported from Coq HoTT
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-/
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/- The hit n-truncation is primitive, declared in init.hit. -/
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import types.sigma
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open is_trunc eq equiv is_equiv function prod sum sigma
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namespace trunc
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protected definition elim {n : trunc_index} {A : Type} {P : Type}
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[Pt : is_trunc n P] (H : A → P) : trunc n A → P :=
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rec H
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protected definition elim_on {n : trunc_index} {A : Type} {P : Type} (aa : trunc n A)
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[Pt : is_trunc n P] (H : A → P) : P :=
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elim H aa
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exit
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variables {X Y Z : Type} {P : X → Type} (A B : Type) (n : trunc_index)
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local attribute is_trunc_eq [instance]
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definition is_equiv_tr [instance] [H : is_trunc n A] : is_equiv (@tr n A) :=
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adjointify _
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(trunc.rec id)
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(λaa, trunc.rec_on aa (λa, idp))
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(λa, idp)
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definition equiv_trunc [H : is_trunc n A] : A ≃ trunc n A :=
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equiv.mk tr _
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definition is_trunc_of_is_equiv_tr [H : is_equiv (@tr n A)] : is_trunc n A :=
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is_trunc_is_equiv_closed n tr⁻¹
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definition untrunc_of_is_trunc [reducible] [H : is_trunc n A] : trunc n A → A :=
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tr⁻¹
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/- Functoriality -/
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definition trunc_functor (f : X → Y) : trunc n X → trunc n Y :=
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λxx, trunc_rec_on xx (λx, tr (f x))
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-- by intro xx; apply (trunc_rec_on xx); intro x; exact (tr (f x))
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-- by intro xx; fapply (trunc_rec_on xx); intro x; exact (tr (f x))
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-- by intro xx; exact (trunc_rec_on xx (λx, (tr (f x))))
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definition trunc_functor_compose (f : X → Y) (g : Y → Z)
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: trunc_functor n (g ∘ f) ∼ trunc_functor n g ∘ trunc_functor n f :=
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λxx, trunc_rec_on xx (λx, idp)
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definition trunc_functor_id : trunc_functor n (@id A) ∼ id :=
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λxx, trunc_rec_on xx (λx, idp)
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definition is_equiv_trunc_functor (f : X → Y) [H : is_equiv f] : is_equiv (trunc_functor n f) :=
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adjointify _
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(trunc_functor n f⁻¹)
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(λyy, trunc_rec_on yy (λy, ap tr !retr))
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(λxx, trunc_rec_on xx (λx, ap tr !sect))
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section
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open prod.ops
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definition trunc_prod_equiv : trunc n (X × Y) ≃ trunc n X × trunc n Y :=
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sorry
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-- equiv.MK (λpp, trunc_rec_on pp (λp, (tr p.1, tr p.2)))
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-- (λp, trunc_rec_on p.1 (λx, trunc_rec_on p.2 (λy, tr (x,y))))
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-- sorry --(λp, trunc_rec_on p.1 (λx, trunc_rec_on p.2 (λy, idp)))
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-- (λpp, trunc_rec_on pp (λp, prod.rec_on p (λx y, idp)))
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-- begin
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-- fapply equiv.MK,
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-- apply sorry, --{exact (λpp, trunc_rec_on pp (λp, (tr p.1, tr p.2)))},
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-- apply sorry, /-{intro p, cases p with (xx, yy),
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-- apply (trunc_rec_on xx), intro x,
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-- apply (trunc_rec_on yy), intro y, exact (tr (x,y))},-/
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-- apply sorry, /-{intro p, cases p with (xx, yy),
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-- apply (trunc_rec_on xx), intro x,
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-- apply (trunc_rec_on yy), intro y, apply idp},-/
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-- apply sorry --{intro pp, apply (trunc_rec_on pp), intro p, cases p, apply idp},
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-- end
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end
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/- Propositional truncation -/
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-- should this live in hprop?
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definition merely [reducible] (A : Type) : Type := trunc -1 A
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notation `||`:max A `||`:0 := merely A
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notation `∥`:max A `∥`:0 := merely A
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definition Exists [reducible] (P : X → Type) : Type := ∥ sigma P ∥
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definition or [reducible] (A B : Type) : Type := ∥ A ⊎ B ∥
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notation `exists` binders `,` r:(scoped P, Exists P) := r
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notation `∃` binders `,` r:(scoped P, Exists P) := r
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notation A `\/` B := or A B
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notation A ∨ B := or A B
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definition merely.intro [reducible] (a : A) : ∥ A ∥ := tr a
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definition exists.intro [reducible] (x : X) (p : P x) : ∃x, P x := tr ⟨x, p⟩
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definition or.intro_left [reducible] (x : X) : X ∨ Y := tr (inl x)
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definition or.intro_right [reducible] (y : Y) : X ∨ Y := tr (inr y)
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definition is_contr_of_merely_hprop [H : is_hprop A] (aa : merely A) : is_contr A :=
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is_contr_of_inhabited_hprop (trunc_rec_on aa id)
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section
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open sigma.ops
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definition trunc_sigma_equiv : trunc n (Σ x, P x) ≃ trunc n (Σ x, trunc n (P x)) :=
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equiv.MK (λpp, trunc_rec_on pp (λp, tr ⟨p.1, tr p.2⟩))
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(λpp, trunc_rec_on pp (λp, trunc_rec_on p.2 (λb, tr ⟨p.1, b⟩)))
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(λpp, trunc_rec_on pp (λp, sigma.rec_on p (λa bb, trunc_rec_on bb (λb, idp))))
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(λpp, trunc_rec_on pp (λp, sigma.rec_on p (λa b, idp)))
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definition trunc_sigma_equiv_of_is_trunc [H : is_trunc n X]
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: trunc n (Σ x, P x) ≃ Σ x, trunc n (P x) :=
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calc
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trunc n (Σ x, P x) ≃ trunc n (Σ x, trunc n (P x)) : trunc_sigma_equiv
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... ≃ Σ x, trunc n (P x) : equiv.symm !equiv_trunc
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end
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end trunc
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protected definition rec_on {n : trunc_index} {A : Type} {P : trunc n A → Type} (aa : trunc n A)
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[Pt : Πaa, is_trunc n (P aa)] (H : Πa, P (tr a)) : P aa :=
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trunc.rec H aa
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