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