feat(library/hott/trunc): prove that n-types are (n+1)-types.

library/hott/trunc.lean

@@ -1,35 +1,90 @@
 -- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
 -- Released under Apache 2.0 license as described in the file LICENSE.
--- Author: Jeremy Avigad
+-- Author: Jeremy Avigad, Floris van Doorn
 -- Ported from Coq HoTT
-import .path
-open path
+import .path data.nat data.empty data.unit
+open path nat
 
 -- Truncation levels
 -- -----------------
 
-inductive Contr (A : Type) : Type :=
-Contr_mk : Π
-  (center : A)
-  (contr : Πy : A, center ≈ y),
-Contr A
-
-definition center {A : Type} (C : Contr A) : A := Contr.rec (λcenter contr, center) C
-
-definition contr {A : Type} (C : Contr A) : Πy : A, center C ≈ y :=
-Contr.rec (λcenter contr, contr) C
+structure contr_internal [class] (A : Type₊) :=
+mk :: (center : A) (contr : Π(a : A), center ≈ a)
+-- TODO: center and contr should live in different namespaces
 
 inductive trunc_index : Type :=
 minus_two : trunc_index,
 trunc_S : trunc_index → trunc_index
 
--- TODO: add coercions to / from nat
+namespace truncation
+
+postfix `.+1`:max := trunc_index.trunc_S
+postfix `.+2`:max := λn, (n .+1 .+1)
+notation `-2`:max := trunc_index.minus_two
+notation `-1`:max := (-2.+1)
+
+definition trunc_index_add (n m : trunc_index) : trunc_index :=
+trunc_index.rec_on m n (λ k l, l .+1)
+
+-- Coq calls this `-2+`, but this looks more natural, since 0 +2+ 0 = 2
+infix `+2+`:65 := trunc_index_add
+
+definition trunc_index_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
+
+notation `<=` := trunc_index_leq
+notation `≤` := trunc_index_leq
+
+definition nat_to_trunc_index [coercion] (n : ℕ) : trunc_index :=
+nat.rec_on n (-2.+2) (λ n k, k.+1)
 
 -- TODO: note in the Coq version, there is an internal version
-definition IsTrunc (n : trunc_index) : Type.{1} → Type.{1} :=
-trunc_index.rec (λA, Contr A) (λn trunc_n A, (Π(x y : A), trunc_n (x ≈ y))) n
+definition is_trunc_internal (n : trunc_index) : Type₁ → Type₁ :=
+trunc_index.rec_on n (λA, contr_internal A) (λn trunc_n A, (Π(x y : A), trunc_n (x ≈ y)))
 
--- TODO: in the Coq version, this is notation
-definition minus_one := trunc_index.trunc_S trunc_index.minus_two
-definition IsHProp := IsTrunc minus_one
-definition IsHSet := IsTrunc (trunc_index.trunc_S minus_one)
+structure is_trunc [class] (n : trunc_index) (A : Type) :=
+mk :: (trunc_is_trunc : is_trunc_internal n A)
+
+definition is_contr := is_trunc -2
+definition is_hProp := is_trunc -1
+definition is_hSet := is_trunc 0
+
+definition contr_to_internal {A : Type₁} [H : is_contr A] : contr_internal A :=
+is_trunc.trunc_is_trunc
+
+definition internal_to_contr {A : Type₁} [H : contr_internal A] : is_contr A :=
+is_trunc.mk H
+
+definition contr_mk {A : Type₁} (center : A) (contr : Π(a : A), center ≈ a) : is_contr A :=
+is_trunc.mk (contr_internal.mk center contr)
+
+definition center {A : Type₁} [H : is_contr A] : A :=
+@contr_internal.center A is_trunc.trunc_is_trunc
+
+definition contr {A : Type₁} [H : is_contr A] (a : A) : center ≈ a :=
+@contr_internal.contr A is_trunc.trunc_is_trunc a
+
+definition path_contr {A : Type₁} [H : is_contr A] (x y : A) : x ≈ y :=
+(contr x)⁻¹ ⬝ (contr y)
+
+definition path2_contr {A : Type₁} [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, path.rec_on r !concat_Vp),
+K p⁻¹ ⬝ K q
+
+definition contr_paths_contr [instance] {A : Type₁} [H : is_contr A] (x y : A) : is_contr (x ≈ y) :=
+contr_mk !path_contr (λ p, !path2_contr)
+
+definition trunc_succ [instance] {A : Type₁} (n : trunc_index) [H : is_trunc n A] : is_trunc (n.+1) A :=
+trunc_index.rec_on n
+  (λ A H, @is_trunc.mk -1 _ (λ x y, @contr_to_internal _ (@contr_paths_contr _ H _ _)))
+  (λ n IH A H, is_trunc.mk (λ x y, @is_trunc.trunc_is_trunc (n.+1) (x≈y) (IH _
+    (@is_trunc.mk n (x≈y) (@is_trunc.trunc_is_trunc (n.+1) _ H x y))
+    )))
+  A H
+
+definition trunc_leq [instance] {A : Type₁} {m n : trunc_index} (H : trunc_index_leq m n)
+  [H : is_trunc m A] : is_trunc n A :=
+sorry
+
+end truncation