86 lines
2.4 KiB
Text
86 lines
2.4 KiB
Text
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/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: hit.circle
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Authors: Floris van Doorn
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Declaration of the n-spheres
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-/
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import .suspension
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open eq nat suspension bool is_trunc unit
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/- We can define spheres with the following possible indices:
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- trunc_index
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- nat
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- nat, but counting wrong (S^0 = empty, S^1 = bool, ...)
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- some new type "integers >= 1"
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-/
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/- Sphere levels -/
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inductive sphere_index : Type₀ :=
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| minus_one : sphere_index
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| succ : sphere_index → sphere_index
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namespace sphere_index
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/-
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notation for sphere_index is -1, 0, 1, ...
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from 0 and up this comes from a coercion from num to sphere_index (via nat)
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-/
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postfix `.+1`:(max+1) := sphere_index.succ
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postfix `.+2`:(max+1) := λ(n : sphere_index), (n .+1 .+1)
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notation `-1` := minus_one
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export [coercions] nat
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definition add (n m : sphere_index) : sphere_index :=
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sphere_index.rec_on m n (λ k l, l .+1)
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definition leq (n m : sphere_index) : Type₁ :=
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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
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infix `+1+`:65 := sphere_index.add
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notation x <= y := sphere_index.leq x y
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notation x ≤ y := sphere_index.leq x y
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definition succ_le_succ {n m : sphere_index} (H : n ≤ m) : n.+1 ≤ m.+1 := H
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definition le_of_succ_le_succ {n m : sphere_index} (H : n.+1 ≤ m.+1) : n ≤ m := H
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definition minus_two_le (n : sphere_index) : -1 ≤ n := star
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definition empty_of_succ_le_minus_two {n : sphere_index} (H : n .+1 ≤ -1) : empty := H
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definition of_nat [coercion] [reducible] (n : nat) : sphere_index :=
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nat.rec_on n (-1.+1) (λ n k, k.+1)
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definition trunc_index_of_sphere_index [coercion] [reducible] (n : sphere_index) : trunc_index :=
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sphere_index.rec_on n -1 (λ n k, k.+1)
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end sphere_index
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open sphere_index equiv
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definition sphere : sphere_index → Type₀
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| -1 := empty
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| n.+1 := suspension (sphere n)
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namespace sphere
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namespace ops
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abbreviation S := sphere
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end ops
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definition bool_of_sphere [reducible] : sphere 0 → bool :=
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suspension.rec tt ff (λx, empty.elim _ x)
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definition sphere_of_bool [reducible] : bool → sphere 0
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| tt := !north
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| ff := !south
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definition sphere_equiv_bool : sphere 0 ≃ bool :=
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equiv.MK bool_of_sphere
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sphere_of_bool
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sorry --idp
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sorry --idp
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end sphere
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