330 lines
12 KiB
Text
330 lines
12 KiB
Text
/-
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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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Definition of is_trunc (n-truncatedness)
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Ported from Coq HoTT.
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-/
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--TODO: can we replace some definitions with a hprop as codomain by theorems?
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prelude
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import .logic .equiv .types .pathover
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open eq nat sigma unit
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namespace is_trunc
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/- Truncation levels -/
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inductive trunc_index : Type₀ :=
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| minus_two : trunc_index
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| succ : trunc_index → trunc_index
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/-
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notation for trunc_index is -2, -1, 0, 1, ...
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from 0 and up this comes from a coercion from num to trunc_index (via nat)
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-/
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postfix ` .+1`:(max+1) := trunc_index.succ
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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 -- ISSUE: -1 gets printed as -2.+1
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export [coercions] nat
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notation `ℕ₋₂` := trunc_index
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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) : Type₀ :=
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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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infix <= := trunc_index.leq
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infix ≤ := trunc_index.leq
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end trunc_index
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infix `+2+`:65 := trunc_index.add
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namespace trunc_index
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definition succ_le_succ {n m : trunc_index} (H : n ≤ m) : n.+1 ≤ m.+1 := H
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definition le_of_succ_le_succ {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 le.refl (n : trunc_index) : n ≤ n := by induction n with n IH; exact star; exact IH
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definition empty_of_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 trunc_index.of_nat [coercion] [reducible] (n : nat) : trunc_index :=
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(nat.rec_on n -2 (λ n k, k.+1)).+2
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definition sub_two [reducible] (n : nat) : trunc_index :=
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nat.rec_on n -2 (λ n k, k.+1)
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postfix ` .-2`:(max+1) := sub_two
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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)
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(center_eq : 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
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(λA, contr_internal A)
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(λn trunc_n A, (Π(x y : A), trunc_n (x = y)))
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end is_trunc open is_trunc
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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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open nat num is_trunc.trunc_index
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namespace is_trunc
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abbreviation is_contr := is_trunc -2
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abbreviation is_hprop := is_trunc -1
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abbreviation is_hset := is_trunc 0
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variables {A B : Type}
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definition is_trunc_succ_intro (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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definition is_trunc_eq [instance] [priority 1200]
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(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 (n.+1) A x y)
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/- contractibility -/
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definition is_contr.mk (center : A) (center_eq : Π(a : A), center = a) : is_contr A :=
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is_trunc.mk (contr_internal.mk center center_eq)
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definition center (A : Type) [H : is_contr A] : A :=
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contr_internal.center (is_trunc.to_internal -2 A)
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definition center_eq [H : is_contr A] (a : A) : !center = a :=
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contr_internal.center_eq (is_trunc.to_internal -2 A) a
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definition eq_of_is_contr [H : is_contr A] (x y : A) : x = y :=
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(center_eq x)⁻¹ ⬝ (center_eq y)
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definition hprop_eq_of_is_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), eq_of_is_contr x y = r, from (λ r, eq.rec_on r !con.left_inv),
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(K p)⁻¹ ⬝ K q
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theorem is_contr_eq {A : Type} [H : is_contr A] (x y : A) : is_contr (x = y) :=
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is_contr.mk !eq_of_is_contr (λ p, !hprop_eq_of_is_contr)
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local attribute is_contr_eq [instance]
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/- truncation is upward close -/
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-- n-types are also (n+1)-types
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theorem is_trunc_succ [instance] [priority 900] (A : Type) (n : trunc_index)
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[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_intro)
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(λ n IH A (H : is_trunc (n.+1) A), @is_trunc_succ_intro _ _ (λ x y, IH _ _))
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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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theorem is_trunc_of_leq.{l} (A : Type.{l}) {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 (trunc_index.empty_of_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, @is_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_intro A m (λx y, IHm n (x = y) (trunc_index.le_of_succ_le_succ Hnm) _)),
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trunc_index.rec_on m base step n A Hnm Hn
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definition is_trunc_of_imp_is_trunc {n : trunc_index} (H : A → is_trunc (n.+1) A)
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: is_trunc (n.+1) A :=
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@is_trunc_succ_intro _ _ (λx y, @is_trunc_eq _ _ (H x) x y)
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definition is_trunc_of_imp_is_trunc_of_leq {n : trunc_index} (Hn : -1 ≤ n) (H : A → is_trunc n A)
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: is_trunc n A :=
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trunc_index.rec_on n (λHn H, empty.rec _ Hn)
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(λn IH Hn, is_trunc_of_imp_is_trunc)
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Hn H
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-- these must be definitions, because we need them to compute sometimes
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definition is_trunc_of_is_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 is_trunc_succ_of_is_hprop (A : Type) (n : trunc_index) [H : is_hprop A]
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: is_trunc (n.+1) A :=
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is_trunc_of_leq A (show -1 ≤ n.+1, from star)
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definition is_trunc_succ_succ_of_is_hset (A : Type) (n : trunc_index) [H : is_hset A]
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: is_trunc (n.+2) A :=
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is_trunc_of_leq A (show 0 ≤ n.+2, from 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
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definition is_contr_of_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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theorem is_hprop_of_imp_is_contr {A : Type} (H : A → is_contr A) : is_hprop A :=
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@is_trunc_succ_intro 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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!is_contr_eq)
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theorem is_hprop.mk {A : Type} (H : ∀x y : A, x = y) : is_hprop A :=
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is_hprop_of_imp_is_contr (λ x, is_contr.mk x (H x))
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theorem is_hprop_elim_self {A : Type} {H : is_hprop A} (x : A) : is_hprop.elim x x = idp :=
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!is_hprop.elim
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/- hsets -/
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theorem 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_intro _ _ (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
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/- instances -/
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definition is_contr_sigma_eq [instance] [priority 800] {A : Type} (a : A)
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: 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 is_contr_sigma_eq' [instance] [priority 800] {A : Type} (a : A)
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: is_contr (Σ(x : A), x = a) :=
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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 is_contr_unit : is_contr unit :=
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is_contr.mk star (λp, unit.rec_on p idp)
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definition is_hprop_empty : is_hprop empty :=
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is_hprop.mk (λx, !empty.elim x)
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local attribute is_contr_unit is_hprop_empty [instance]
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definition is_trunc_unit [instance] (n : trunc_index) : is_trunc n unit :=
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!is_trunc_of_is_contr
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definition is_trunc_empty [instance] (n : trunc_index) : is_trunc (n.+1) empty :=
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!is_trunc_succ_of_is_hprop
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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
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--"is_trunc_is_equiv_closed"
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definition is_contr_is_equiv_closed (f : A → B) [Hf : is_equiv f] [HA: is_contr A]
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: (is_contr B) :=
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is_contr.mk (f (center A)) (λp, eq_of_eq_inv !center_eq)
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definition is_contr_equiv_closed (H : A ≃ B) [HA: is_contr A] : is_contr B :=
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is_contr_is_equiv_closed (to_fun H)
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definition equiv_of_is_contr_of_is_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) center_eq center_eq)
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theorem is_trunc_is_equiv_closed (n : trunc_index) (f : A → B) [H : is_equiv f]
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[HA : is_trunc n A] : 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), is_contr_is_equiv_closed f)
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(λn IH A (HA : is_trunc n.+1 A) B f (H : is_equiv f), @is_trunc_succ_intro _ _ (λ x y : B,
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IH (f⁻¹ x = f⁻¹ y) _ (x = y) (ap f⁻¹)⁻¹ !is_equiv_inv))
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A HA B f H
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definition is_trunc_is_equiv_closed_rev (n : trunc_index) (f : A → B) [H : is_equiv f]
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[HA : is_trunc n B] : is_trunc n A :=
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is_trunc_is_equiv_closed n f⁻¹
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definition is_trunc_equiv_closed (n : trunc_index) (f : A ≃ B) [HA : is_trunc n A]
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: is_trunc n B :=
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is_trunc_is_equiv_closed n (to_fun f)
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definition is_trunc_equiv_closed_rev (n : trunc_index) (f : A ≃ B) [HA : is_trunc n B]
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: is_trunc n A :=
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is_trunc_is_equiv_closed n (to_inv f)
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definition is_equiv_of_is_hprop [constructor] [HA : is_hprop A] [HB : is_hprop B]
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(f : A → B) (g : B → A) : is_equiv f :=
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is_equiv.mk f g (λb, !is_hprop.elim) (λa, !is_hprop.elim) (λa, !is_hset.elim)
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definition equiv_of_is_hprop [constructor] [HA : is_hprop A] [HB : is_hprop B]
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(f : A → B) (g : B → A) : A ≃ B :=
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equiv.mk f (is_equiv_of_is_hprop f g)
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definition equiv_of_iff_of_is_hprop [unfold 5] [HA : is_hprop A] [HB : is_hprop B] (H : A ↔ B) : A ≃ B :=
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equiv_of_is_hprop (iff.elim_left H) (iff.elim_right H)
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/- truncatedness of lift -/
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definition is_trunc_lift [instance] [priority 1450] (A : Type) (n : trunc_index)
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[H : is_trunc n A] : is_trunc n (lift A) :=
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is_trunc_equiv_closed _ !equiv_lift
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end
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/- interaction with the Unit type -/
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open equiv
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-- A contractible type is equivalent to [Unit]. *)
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definition equiv_unit_of_is_contr [H : is_contr A] : A ≃ unit :=
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equiv.MK (λ (x : A), ⋆)
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(λ (u : unit), center A)
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(λ (u : unit), unit.rec_on u idp)
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(λ (x : A), center_eq x)
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/- interaction with pathovers -/
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variables {C : A → Type}
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{a a₂ : A} (p : a = a₂)
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(c : C a) (c₂ : C a₂)
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definition is_hprop.elimo [H : is_hprop (C a)] : c =[p] c₂ :=
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pathover_of_eq_tr !is_hprop.elim
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definition is_trunc_pathover [instance]
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(n : trunc_index) [H : is_trunc (n.+1) (C a)] : is_trunc n (c =[p] c₂) :=
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is_trunc_equiv_closed_rev n !pathover_equiv_eq_tr
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variables {p c c₂}
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theorem is_hset.elimo (q q' : c =[p] c₂) [H : is_hset (C a)] : q = q' :=
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!is_hprop.elim
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-- TODO: port "Truncated morphisms"
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/- truncated universe -/
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-- TODO: move to root namespace?
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structure trunctype (n : trunc_index) :=
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(carrier : Type) (struct : is_trunc n carrier)
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attribute trunctype.carrier [coercion]
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attribute trunctype.struct [instance] [priority 1400]
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notation n `-Type` := trunctype n
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abbreviation hprop := -1-Type
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abbreviation hset := 0-Type
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protected abbreviation hprop.mk := @trunctype.mk -1
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protected abbreviation hset.mk := @trunctype.mk (-1.+1)
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protected abbreviation trunctype.mk' [parsing_only] (n : trunc_index) (A : Type)
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[H : is_trunc n A] : n-Type :=
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trunctype.mk A H
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definition tlift.{u v} [constructor] {n : trunc_index} (A : trunctype.{u} n)
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: trunctype.{max u v} n :=
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trunctype.mk (lift A) (is_trunc_lift _ _)
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end is_trunc
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