7cfac38eda
This also involves: - adding definitions about logic and natural numbers existing in the standard library to init - porting the current algebraic hierarchy
88 lines
2.5 KiB
Text
88 lines
2.5 KiB
Text
/-
|
|
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
|