lean2/hott/init/trunc.hlean

/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.

Module: init.trunc
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

/- Truncation levels -/

-- TODO: can we replace some definitions with a hprop as codomain by theorems?

/- truncation indices -/

namespace is_trunc

  inductive trunc_index : Type₁ :=
  minus_two : trunc_index,
  succ : trunc_index → trunc_index

  /-
     notation for trunc_index is -2, -1, 0, 1, ...
     from 0 and up this comes from a coercion from num to trunc_index (via nat)
  -/
  postfix `.+1`:(max+1) := trunc_index.succ
  postfix `.+2`:(max+1) := λn, (n .+1 .+1)
  notation `-2` := trunc_index.minus_two
  notation `-1` := -2.+1
  export [coercions] nat -- does this export

  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

  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_succ {n m : trunc_index} (H : n ≤ m) : n.+1 ≤ m.+1 := H
  definition le_of_succ_le_succ {n m : trunc_index} (H : n.+1 ≤ m.+1) : n ≤ m := H
  definition minus_two_le (n : trunc_index) : -2 ≤ n := star
  definition empty_of_succ_le_minus_two {n : trunc_index} (H : n .+1 ≤ -2) : empty := H
  end trunc_index
  definition trunc_index.of_nat [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)))

end is_trunc

open is_trunc
structure is_trunc [class] (n : trunc_index) (A : Type) :=
  (to_internal : is_trunc_internal n A)
open nat num is_trunc.trunc_index
namespace is_trunc

  abbreviation is_contr := is_trunc -2
  abbreviation is_hprop := is_trunc -1
  abbreviation is_hset  := is_trunc nat.zero

  variables {A B : Type}

  definition is_trunc_succ_intro (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)

  definition is_trunc_eq (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 center_eq [H : is_contr A] (x y : A) : x = y :=
  contr x⁻¹ ⬝ (contr y)

  definition hprop_eq {A : Type} [H : is_contr A] {x y : A} (p q : x = y) : p = q :=
  have K : ∀ (r : x = y), center_eq x y = r, from (λ r, eq.rec_on r !con.right_inv),
  K p⁻¹ ⬝ K q

  definition is_contr_eq [instance] {A : Type} [H : is_contr A] (x y : A) : is_contr (x = y)
  :=
  is_contr.mk !center_eq (λ p, !hprop_eq)

  /- truncation is upward close -/

  -- n-types are also (n+1)-types
  definition is_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_intro)
    (λ n IH A (H : is_trunc (n.+1) A), @is_trunc_succ_intro _ _ (λ x y, IH _ !is_trunc_eq))
    A H
  --in the proof the type of H is given explicitly to make it available for class inference

  definition is_trunc_of_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.empty_of_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, @is_trunc_succ A m (IHm -2 A star Hn))
           (λn IHn A Hnm (Hn : is_trunc n.+1 A),
           @is_trunc_succ_intro A m (λx y, IHm n (x = y) (trunc_index.le_of_succ_le_succ Hnm) !is_trunc_eq)),
  trunc_index.rec_on m base step n A Hnm Hn

  -- the following cannot be instances in their current form, because they are looping
  definition is_trunc_of_is_contr (A : Type) (n : trunc_index) [H : is_contr A] : is_trunc n A :=
  trunc_index.rec_on n H _

  definition is_trunc_succ_of_is_hprop (A : Type) (n : trunc_index) [H : is_hprop A]
      : is_trunc (n.+1) A :=
  is_trunc_of_leq A -1 (n.+1) star

  definition is_trunc_succ_succ_of_is_hset (A : Type) (n : trunc_index) [H : is_hset A]
      : is_trunc (n.+2) A :=
  is_trunc_of_leq A nat.zero (n.+2) star

  /- hprops -/

  definition is_hprop.elim [H : is_hprop A] (x y : A) : x = y :=
  @center _ !is_trunc_eq

  definition is_contr_of_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 is_hprop_of_imp_is_contr {A : Type} (H : A → is_contr A) : is_hprop A :=
  @is_trunc_succ_intro A -2
    (λx y,
      have H2 [visible] : is_contr A, from H x,
      !is_contr_eq)

  definition is_hprop.mk {A : Type} (H : ∀x y : A, x = y) : is_hprop A :=
  is_hprop_of_imp_is_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_intro _ _ (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 _ !is_trunc_eq p q

  /- instances -/

  definition is_contr_sigma_eq [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 is_contr_unit [instance] : is_contr unit :=
  is_contr.mk star (λp, unit.rec_on p idp)

  definition is_hprop_empty [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)
  attribute trunctype.trunctype_type [coercion]
  attribute trunctype.is_trunc_trunctype_type [instance]

  notation n `-Type` := trunctype n
  abbreviation hprop := -1-Type
  abbreviation hset := (-1.+1)-Type

  protected definition hprop.mk := @trunctype.mk -1
  protected definition hset.mk := @trunctype.mk (-1.+1)

  /- interaction with equivalences -/

  section
  open is_equiv equiv

  --should we remove the following two theorems as they are special cases of
  --"is_trunc_is_equiv_closed"
  definition is_contr_is_equiv_closed (f : A → B) [Hf : is_equiv f] [HA: is_contr A] : (is_contr B) :=
    is_contr.mk (f (center A)) (λp, eq_of_eq_inv f !contr)

  theorem is_contr_equiv_closed (H : A ≃ B) [HA: is_contr A] : is_contr B :=
    @is_contr_is_equiv_closed _ _ (to_fun H) (to_is_equiv H) _

  definition equiv_of_is_contr_of_is_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 is_trunc_is_equiv_closed (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), !is_contr_is_equiv_closed)
    (λn IH A (HA : is_trunc n.+1 A) B f (H : is_equiv f), @is_trunc_succ_intro _ _ (λ x y : B,
      IH (f⁻¹ x = f⁻¹ y) !is_trunc_eq (x = y) ((ap (f⁻¹))⁻¹) !is_equiv_inv))
    A HA B f H

  definition is_trunc_equiv_closed (n : trunc_index) (f : A ≃ B) [HA : is_trunc n A]
    : is_trunc n B :=
  is_trunc_is_equiv_closed n (to_fun f)

  definition is_equiv_of_is_hprop [HA : is_hprop A] [HB : is_hprop B] (f : A → B) (g : B → A)
    : is_equiv f :=
  is_equiv.mk g (λb, !is_hprop.elim) (λa, !is_hprop.elim) (λa, !is_hset.elim)

  definition equiv_of_is_hprop [HA : is_hprop A] [HB : is_hprop B] (f : A → B) (g : B → A)
    : A ≃ B :=
  equiv.mk f (is_equiv_of_is_hprop f g)

  definition equiv_of_iff_of_is_hprop [HA : is_hprop A] [HB : is_hprop B] (H : A ↔ B) : A ≃ B :=
  equiv_of_is_hprop (iff.elim_left H) (iff.elim_right H)

  end

  /- interaction with the Unit type -/

  -- A contractible type is equivalent to [Unit]. *)
  definition equiv_unit_of_is_contr [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_constant)⁻¹))

  -- TODO: port "Truncated morphisms"

end is_trunc