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feat(hott/trunc): clean up some theorems, prove some basic theorems

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Floris van Doorn 2014-11-06 17:26:23 -05:00 committed by Leonardo de Moura
parent 08c56188b6
commit 74779dd855
1 changed files with 44 additions and 32 deletions
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76
library/hott/trunc.lean
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@ -2,15 +2,16 @@
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Jeremy Avigad, Floris van Doorn
-- Ported from Coq HoTT
import .path data.nat data.empty data.unit
import .path data.nat.basic data.empty data.unit
open path nat
-- Truncation levels
-- -----------------
structure contr_internal [class] (A : Type₊) :=
-- TODO: make everything universe polymorphic
structure contr_internal (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,
@ -20,51 +21,54 @@ 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)
notation `-2` := trunc_index.minus_two
notation `-1` := (-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
-- Coq calls this `-2+`, but `+2+` looks more natural, since trunc_index_add 0 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
notation x <= y := trunc_index_leq x y
notation x ≤ y := trunc_index_leq x y
definition nat_to_trunc_index [coercion] (n : ℕ) : trunc_index :=
nat.rec_on n (-2.+2) (λ n k, k.+1)
definition nat_to_trunc_index [coercion] (n : nat) : trunc_index :=
nat.rec_on n (-1.+1) (λ n k, k.+1)
-- TODO: note in the Coq version, there is an internal version
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)))
structure is_trunc [class] (n : trunc_index) (A : Type) :=
mk :: (trunc_is_trunc : is_trunc_internal n A)
mk :: (to_internal : is_trunc_internal n A)
--prefix `is_contr`:max := is_trunc -2
definition is_contr := is_trunc -2
definition is_hProp := is_trunc -1
definition is_hSet := is_trunc 0
definition is_hprop := is_trunc -1
definition is_hset := is_trunc nat.zero
definition contr_to_internal {A : Type₁} [H : is_contr A] : contr_internal A :=
is_trunc.trunc_is_trunc
variable {A : Type₁}
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 :=
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 : Type₁} [H : is_contr A] : A :=
@contr_internal.center A is_trunc.trunc_is_trunc
definition center (A : Type₁) [H : is_contr A] : A :=
@contr_internal.center A is_trunc.to_internal
definition contr {A : Type₁} [H : is_contr A] (a : A) : center ≈ a :=
@contr_internal.contr A is_trunc.trunc_is_trunc a
definition contr [H : is_contr A] (a : A) : !center ≈ a :=
@contr_internal.contr A is_trunc.to_internal a
definition path_contr {A : Type₁} [H : is_contr A] (x y : A) : x ≈ y :=
definition is_trunc_succ (A : Type₁) {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 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)
definition path_contr [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 :=
@ -73,18 +77,26 @@ have K : ∀ (r : x ≈ y), path_contr x y ≈ r, from
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)
is_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 :=
definition trunc_succ (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))
)))
(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 [instance] {A : Type₁} {m n : trunc_index} (H : trunc_index_leq m n)
definition trunc_leq [instance] {A : Type₁} {m n : trunc_index} (H : m ≤ n)
[H : is_trunc m A] : is_trunc n A :=
sorry
definition is_hprop.mk (A : Type₁) (H : ∀x y : A, x ≈ y) : is_hprop A := sorry
definition is_hprop.elim [H : is_hprop A] (x y : A) : x ≈ y := sorry
definition is_trunc_is_hprop {n : trunc_index} : is_hprop (is_trunc n A) := sorry
definition is_hset.mk (A : Type₁) (H : ∀(x y : A) (p q : x ≈ y), p ≈ q) : is_hset A := sorry
definition is_hset.elim [H : is_hset A] ⦃x y : A⦄ (p q : x ≈ y) : p ≈ q := sorry
end truncation