lean2/hott/init/trunc.hlean

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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Authors: Jeremy Avigad, Floris van Doorn
-- Ported from Coq HoTT
prelude
import .path .logic .datatypes .equiv .types.empty .types.sigma
open eq nat sigma unit
set_option pp.universes true
-- Truncation levels
-- -----------------
-- TODO: make everything universe polymorphic
-- TODO: everything definition with a hprop as codomain can be a theorem?
/- truncation indices -/
namespace truncation
inductive trunc_index : Type₁ :=
minus_two : trunc_index,
trunc_S : trunc_index → trunc_index
postfix `.+1`:(max+1) := trunc_index.trunc_S
postfix `.+2`:(max+1) := λn, (n .+1 .+1)
notation `-2` := trunc_index.minus_two
notation `-1` := (-2.+1)
namespace trunc_index
definition add (n m : trunc_index) : trunc_index :=
trunc_index.rec_on m n (λ k l, l .+1)
definition leq (n m : trunc_index) : Type₁ :=
trunc_index.rec_on n (λm, unit) (λ n p m, trunc_index.rec_on m (λ p, empty) (λ m q p, p m) p) m
end trunc_index
-- Coq calls this `-2+`, but `+2+` looks more natural, since trunc_index_add 0 0 = 2
infix `+2+`:65 := trunc_index.add
notation x <= y := trunc_index.leq x y
notation x ≤ y := trunc_index.leq x y
namespace trunc_index
definition succ_le {n m : trunc_index} (H : n ≤ m) : n.+1 ≤ m.+1 := H
definition succ_le_cancel {n m : trunc_index} (H : n.+1 ≤ m.+1) : n ≤ m := H
definition minus_two_le (n : trunc_index) : -2 ≤ n := star
definition not_succ_le_minus_two {n : trunc_index} (H : n .+1 ≤ -2) : empty := H
end trunc_index
definition nat_to_trunc_index [coercion] (n : nat) : trunc_index :=
nat.rec_on n (-1.+1) (λ n k, k.+1)
/- truncated types -/
/-
Just as in Coq HoTT we define an internal version of contractibility and is_trunc, but we only
use `is_trunc` and `is_contr`
-/
structure contr_internal (A : Type) :=
(center : A) (contr : Π(a : A), center = a)
definition is_trunc_internal (n : trunc_index) : Type → Type :=
trunc_index.rec_on n (λA, contr_internal A)
(λn trunc_n A, (Π(x y : A), trunc_n (x = y)))
structure is_trunc [class] (n : trunc_index) (A : Type) :=
(to_internal : is_trunc_internal n A)
-- should this be notation or definitions?
notation `is_contr` := is_trunc -2
notation `is_hprop` := is_trunc -1
notation `is_hset` := is_trunc (nat_to_trunc_index nat.zero)
-- definition is_contr := is_trunc -2
-- definition is_hprop := is_trunc -1
-- definition is_hset := is_trunc 0
variables {A B : Type}
-- TODO: rename to is_trunc_succ
definition is_trunc_succ (A : Type) (n : trunc_index) [H : ∀x y : A, is_trunc n (x = y)]
: is_trunc n.+1 A :=
is_trunc.mk (λ x y, !is_trunc.to_internal)
-- TODO: rename to is_trunc_path
definition succ_is_trunc (n : trunc_index) [H : is_trunc (n.+1) A] (x y : A) : is_trunc n (x = y) :=
is_trunc.mk (!is_trunc.to_internal x y)
/- contractibility -/
definition is_contr.mk (center : A) (contr : Π(a : A), center = a) : is_contr A :=
is_trunc.mk (contr_internal.mk center contr)
definition center (A : Type) [H : is_contr A] : A :=
@contr_internal.center A !is_trunc.to_internal
definition contr [H : is_contr A] (a : A) : !center = a :=
@contr_internal.contr A !is_trunc.to_internal a
definition path_contr [H : is_contr A] (x y : A) : x = y :=
contr x⁻¹ ⬝ (contr y)
definition path2_contr {A : Type} [H : is_contr A] {x y : A} (p q : x = y) : p = q :=
have K : ∀ (r : x = y), path_contr x y = r, from (λ r, eq.rec_on r !concat_Vp),
K p⁻¹ ⬝ K q
definition contr_paths_contr [instance] {A : Type} [H : is_contr A] (x y : A) : is_contr (x = y) :=
is_contr.mk !path_contr (λ p, !path2_contr)
/- truncation is upward close -/
-- n-types are also (n+1)-types
definition trunc_succ [instance] (A : Type) (n : trunc_index) [H : is_trunc n A] : is_trunc (n.+1) A :=
trunc_index.rec_on n
(λ A (H : is_contr A), !is_trunc_succ)
(λ n IH A (H : is_trunc (n.+1) A), @is_trunc_succ _ _ (λ x y, IH _ !succ_is_trunc))
A H
--in the proof the type of H is given explicitly to make it available for class inference
definition trunc_leq (A : Type) (n m : trunc_index) (Hnm : n ≤ m)
[Hn : is_trunc n A] : is_trunc m A :=
have base : ∀k A, k ≤ -2 → is_trunc k A → (is_trunc -2 A), from
λ k A, trunc_index.cases_on k
(λh1 h2, h2)
(λk h1 h2, empty.elim (is_trunc -2 A) (trunc_index.not_succ_le_minus_two h1)),
have step : Π (m : trunc_index)
(IHm : Π (n : trunc_index) (A : Type), n ≤ m → is_trunc n A → is_trunc m A)
(n : trunc_index) (A : Type)
(Hnm : n ≤ m .+1) (Hn : is_trunc n A), is_trunc m .+1 A, from
λm IHm n, trunc_index.rec_on n
(λA Hnm Hn, @trunc_succ A m (IHm -2 A star Hn))
(λn IHn A Hnm (Hn : is_trunc n.+1 A),
@is_trunc_succ A m (λx y, IHm n (x = y) (trunc_index.succ_le_cancel Hnm) !succ_is_trunc)),
trunc_index.rec_on m base step n A Hnm Hn
-- the following cannot be instances in their current form, because it is looping
definition trunc_contr (A : Type) (n : trunc_index) [H : is_contr A] : is_trunc n A :=
trunc_index.rec_on n H _
definition trunc_hprop (A : Type) (n : trunc_index) [H : is_hprop A]
: is_trunc (n.+1) A :=
trunc_leq A -1 (n.+1) star
definition trunc_hset (A : Type) (n : trunc_index) [H : is_hset A]
: is_trunc (n.+2) A :=
trunc_leq A nat.zero (n.+2) star
/- hprops -/
definition is_hprop.elim [H : is_hprop A] (x y : A) : x = y :=
@center _ !succ_is_trunc
definition contr_inhabited_hprop {A : Type} [H : is_hprop A] (x : A) : is_contr A :=
is_contr.mk x (λy, !is_hprop.elim)
--Coq has the following as instance, but doesn't look too useful
definition hprop_inhabited_contr {A : Type} (H : A → is_contr A) : is_hprop A :=
@is_trunc_succ A -2
(λx y,
have H2 [visible] : is_contr A, from H x,
!contr_paths_contr)
definition is_hprop.mk {A : Type} (H : ∀x y : A, x = y) : is_hprop A :=
hprop_inhabited_contr (λ x, is_contr.mk x (H x))
/- hsets -/
definition is_hset.mk (A : Type) (H : ∀(x y : A) (p q : x = y), p = q) : is_hset A :=
@is_trunc_succ _ _ (λ x y, is_hprop.mk (H x y))
definition is_hset.elim [H : is_hset A] ⦃x y : A⦄ (p q : x = y) : p = q :=
@is_hprop.elim _ !succ_is_trunc p q
/- instances -/
definition contr_basedpaths [instance] {A : Type} (a : A) : is_contr (Σ(x : A), a = x) :=
is_contr.mk (sigma.mk a idp) (λp, sigma.rec_on p (λ b q, eq.rec_on q idp))
definition unit_contr [instance] : is_contr unit :=
is_contr.mk star (λp, unit.rec_on p idp)
definition empty_hprop [instance] : is_hprop empty :=
is_hprop.mk (λx, !empty.elim x)
/- truncated universe -/
structure trunctype (n : trunc_index) :=
(trunctype_type : Type) (is_trunc_trunctype_type : is_trunc n trunctype_type)
coercion trunctype.trunctype_type
notation n `-Type` := trunctype n
notation `hprop` := -1-Type
notation `hset` := 0-Type
definition hprop.mk := @trunctype.mk -1
definition hset.mk := @trunctype.mk nat.zero
--what does the following line in Coq do?
--Canonical Structure default_TruncType := fun n T P => (@BuildTruncType n T P).
/- interaction with equivalences -/
section
open is_equiv equiv
--should we remove the following two theorems as they are special cases of "trunc_equiv"
definition equiv_preserves_contr (f : A → B) [Hf : is_equiv f] [HA: is_contr A] : (is_contr B) :=
is_contr.mk (f (center A)) (λp, moveR_M f !contr)
theorem contr_equiv (H : A ≃ B) [HA: is_contr A] : is_contr B :=
@equiv_preserves_contr _ _ (to_fun H) (to_is_equiv H) _
definition contr_equiv_contr [HA : is_contr A] [HB : is_contr B] : A ≃ B :=
equiv.mk
(λa, center B)
(is_equiv.adjointify (λa, center B) (λb, center A) contr contr)
definition trunc_equiv (n : trunc_index) (f : A → B) [H : is_equiv f] [HA : is_trunc n A]
: is_trunc n B :=
trunc_index.rec_on n
(λA (HA : is_contr A) B f (H : is_equiv f), !equiv_preserves_contr)
(λn IH A (HA : is_trunc n.+1 A) B f (H : is_equiv f), @is_trunc_succ _ _ (λ x y : B,
IH (f⁻¹ x = f⁻¹ y) !succ_is_trunc (x = y) ((ap (f⁻¹))⁻¹) !inv_closed))
A HA B f H
definition trunc_equiv' (n : trunc_index) (f : A ≃ B) [HA : is_trunc n A] : is_trunc n B :=
trunc_equiv n (to_fun f)
definition isequiv_iff_hprop [HA : is_hprop A] [HB : is_hprop B] (f : A → B) (g : B → A)
: is_equiv f :=
is_equiv.adjointify f g (λb, !is_hprop.elim) (λa, !is_hprop.elim)
-- definition equiv_iff_hprop_uncurried [HA : is_hprop A] [HB : is_hprop B] : (A ↔ B) → (A ≃ B) := sorry
definition equiv_iff_hprop [HA : is_hprop A] [HB : is_hprop B] (f : A → B) (g : B → A) : A ≃ B :=
equiv.mk f (isequiv_iff_hprop f g)
end
/- interaction with the Unit type -/
-- A contractible type is equivalent to [Unit]. *)
definition equiv_contr_unit [H : is_contr A] : A ≃ unit :=
equiv.mk (λ (x : A), ⋆)
(is_equiv.mk (λ (u : unit), center A)
(λ (u : unit), unit.rec_on u idp)
(λ (x : A), contr x)
(λ (x : A), (!ap_const)⁻¹))
-- TODO: port "Truncated morphisms"
end truncation