lean2/hott/hit/sphere.hlean
2015-05-07 16:39:03 -07:00

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/-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module: hit.circle
Authors: Floris van Doorn
Declaration of the n-spheres
-/
import .suspension
open eq nat suspension bool is_trunc unit
/-
We can define spheres with the following possible indices:
- trunc_index (defining S^-2 = S^-1 = empty)
- nat (forgetting that S^1 = empty)
- nat, but counting wrong (S^0 = empty, S^1 = bool, ...)
- some new type "integers >= -1"
We choose the last option here.
-/
/- Sphere levels -/
inductive sphere_index : Type₀ :=
| minus_one : sphere_index
| succ : sphere_index → sphere_index
namespace sphere_index
/-
notation for sphere_index is -1, 0, 1, ...
from 0 and up this comes from a coercion from num to sphere_index (via nat)
-/
postfix `.+1`:(max+1) := sphere_index.succ
postfix `.+2`:(max+1) := λ(n : sphere_index), (n .+1 .+1)
notation `-1` := minus_one
export [coercions] nat
definition add (n m : sphere_index) : sphere_index :=
sphere_index.rec_on m n (λ k l, l .+1)
definition leq (n m : sphere_index) : Type₀ :=
sphere_index.rec_on n (λm, unit) (λ n p m, sphere_index.rec_on m (λ p, empty) (λ m q p, p m) p) m
infix `+1+`:65 := sphere_index.add
notation x <= y := sphere_index.leq x y
notation x ≤ y := sphere_index.leq x y
definition succ_le_succ {n m : sphere_index} (H : n ≤ m) : n.+1 ≤ m.+1 := H
definition le_of_succ_le_succ {n m : sphere_index} (H : n.+1 ≤ m.+1) : n ≤ m := H
definition minus_two_le (n : sphere_index) : -1 ≤ n := star
definition empty_of_succ_le_minus_two {n : sphere_index} (H : n .+1 ≤ -1) : empty := H
definition of_nat [coercion] [reducible] (n : nat) : sphere_index :=
nat.rec_on n (-1.+1) (λ n k, k.+1)
definition trunc_index_of_sphere_index [coercion] [reducible] (n : sphere_index) : trunc_index :=
sphere_index.rec_on n -1 (λ n k, k.+1)
end sphere_index
open sphere_index equiv
definition sphere : sphere_index → Type₀
| -1 := empty
| n.+1 := suspension (sphere n)
namespace sphere
namespace ops
abbreviation S := sphere
end ops
definition bool_of_sphere [reducible] : sphere 0 → bool :=
suspension.rec tt ff (λx, empty.elim _ x)
definition sphere_of_bool [reducible] : bool → sphere 0
| tt := !north
| ff := !south
definition sphere_equiv_bool : sphere 0 ≃ bool :=
equiv.MK bool_of_sphere
sphere_of_bool
(λb, match b with | tt := idp | ff := idp end)
(λx, suspension.rec_on x idp idp (empty.rec _))
end sphere