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redtt/library/paths/s1.red
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import prelude | |
import data.s1 | |
import data.int | |
import basics.isotoequiv | |
import basics.retract | |
import paths.pi | |
import paths.int | |
import paths.hlevel | |
def s1-univ-cover : s1 → type = | |
elim [ | |
| base → int | |
| loop i → ua _ _ isuc/equiv i | |
] | |
def Ω1s1 : type = path s1 base base | |
def loopn : int → Ω1s1 = | |
elim [ | |
| pos n → | |
elim n [ | |
| zero → refl | |
| suc (n → loopn) → | |
-- this is trans, but let's expand the definition | |
λ i → comp 0 1 (loopn i) [ i=0 → refl | i=1 j → loop j ] | |
] | |
| negsuc n → | |
elim n [ | |
| zero → | |
λ i → comp 1 0 base [ i=0 → refl | i=1 j → loop j ] | |
| suc (n → loopn) → | |
λ i → comp 1 0 (loopn i) [ i=0 → refl | i=1 j → loop j ] | |
] | |
] | |
def encode (x : s1) (p : path s1 base x) : s1-univ-cover x = | |
coe 0 1 (pos zero) in λ i → s1-univ-cover (p i) | |
def winding (l : path s1 base base) : int = encode base l | |
def winding-loopn : (n : int) → path int (winding (loopn n)) n = | |
elim [ | |
| pos n → | |
elim n [ | |
| zero → refl | |
| suc (n → loopn) → λ i → isuc (loopn i) | |
] | |
| negsuc n → | |
elim n [ | |
| zero → refl | |
| suc (n → loopn) → λ i → pred (loopn i) | |
] | |
] | |
def decode-square | |
: (n : int) | |
→ [i j] s1 [ | |
| i=0 → loopn (pred n) j | |
| i=1 → loopn n j | |
| j=0 → base | |
| j=1 → loop i | |
] | |
= | |
elim [ | |
| pos n → | |
elim n [ | |
| zero → λ i j → comp 1 i base [ j=0 → refl | j=1 i → loop i ] | |
| suc n → λ i j → comp 0 i (loopn (pos n) j) [ j=0 → refl | j=1 i → loop i ] | |
] | |
| negsuc n → λ i j → comp 1 i (loopn (negsuc n) j) [ j=0 → refl | j=1 i → loop i ] | |
] | |
def decode : (x : s1) → s1-univ-cover x → path s1 base x = | |
elim [ | |
| base → loopn | |
| loop i → λ y j → | |
let n : int = y .vproj in | |
comp 0 1 (decode-square n i j) [ | |
| ∂[j] | i=1 → refl | |
| i=0 k → loopn (pred-isuc y k) j | |
] | |
] | |
def loopn-winding (l : Ω1s1) : path _ (loopn (winding l)) l = | |
J _ l (λ p → path (path s1 base (p 1)) (decode (p 1) (encode (p 1) p)) p) refl | |
def winding/equiv : equiv Ω1s1 int = | |
iso→equiv _ _ (winding, (loopn, (winding-loopn, loopn-winding))) | |
def winding/path : path^1 _ Ω1s1 int = | |
ua Ω1s1 int winding/equiv | |
opaque | |
def Ω1s1/set : has-hlevel set Ω1s1 = | |
retract/hlevel set Ω1s1 int (winding, loopn, loopn-winding) int/set | |
opaque | |
def s1/groupoid : is-groupoid s1 = | |
let from-base : (s : s1) → is-set (path s1 base s) = | |
elim [ | |
| base → Ω1s1/set | |
| loop i → | |
prop→prop-over (λ j → is-set (path s1 base (loop j))) | |
(has-hlevel/prop set Ω1s1) | |
Ω1s1/set Ω1s1/set i | |
] | |
in | |
elim [ | |
| base → from-base | |
| loop i → | |
prop→prop-over (λ j → (s : s1) → is-set (path s1 (loop j) s)) | |
(pi/hlevel prop s1 (λ s → is-set (path s1 base s)) | |
(λ s → has-hlevel/prop set (path s1 base s))) | |
from-base from-base i | |
] |