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-- The circle as a HIT. | |
module circle where | |
import bool | |
import int | |
data S1 = base | |
| loop <i> [ (i=0) -> base | |
, (i=1) -> base] | |
loopS1 : U = Path S1 base base | |
loop1 : loopS1 = <i> loop{S1} @ i | |
invLoop : loopS1 = inv S1 base base loop1 | |
moebius : S1 -> U = split | |
base -> bool | |
loop @ i -> negBoolEq @ i | |
helix : S1 -> U = split | |
base -> Z | |
loop @ i -> sucPathZ @ i | |
winding (p : loopS1) : Z = trans Z Z rem zeroZ | |
where | |
rem : Path U Z Z = <i> helix (p @ i) | |
compS1 : loopS1 -> loopS1 -> loopS1 = compPath S1 base base base | |
-- All of these should be equal to "posZ (suc zero)": | |
loop2 : loopS1 = compS1 loop1 loop1 | |
loop2' : loopS1 = compPath' S1 base base base loop1 loop1 | |
loop2'' : loopS1 = compPath'' S1 base base loop1 base loop1 | |
-- More examples: | |
loopZ1 : Z = winding loop1 | |
loopZ2 : Z = winding (compS1 loop1 loop1) | |
loopZ3 : Z = winding (compS1 loop1 (compS1 loop1 loop1)) | |
loopZN1 : Z = winding invLoop | |
loopZ0 : Z = winding (compS1 loop1 invLoop) | |
loopZ5 : Z = winding (compS1 loop1 (compS1 loop1 (compS1 loop1 (compS1 loop1 loop1)))) | |
mLoop : (x : S1) -> Path S1 x x = split | |
base -> loop1 | |
loop @ i -> constSquare S1 base loop1 @ i | |
mult (x : S1) : S1 -> S1 = split | |
base -> x | |
loop @ i -> mLoop x @ i | |
square (x : S1) : S1 = mult x x | |
doubleLoop (l : loopS1) : loopS1 = <i> square (l @ i) | |
tripleLoop (l : loopS1) : loopS1 = <i> mult (l @ i) (square (l @ i)) | |
loopZ4 : Z = winding (doubleLoop (compS1 loop1 loop1)) | |
loopZ8 : Z = winding (doubleLoop (doubleLoop (compS1 loop1 loop1))) | |
triv : loopS1 = <i> base | |
-- A nice example of a homotopy on the circle. The path going halfway | |
-- around the circle and then back is contractible: | |
hmtpy : Path loopS1 (<i> base) (<i> loop{S1} @ (i /\ -i)) = | |
<j i> loop{S1} @ j /\ i /\ -i | |
circleelim (X : U) (x : X) (p : Path X x x) : S1 -> X = split | |
base -> x | |
loop @ i -> p @ i | |
apcircleelim (A B : U) (x : A) (p : Path A x x) (f : A -> B) : | |
(z : S1) -> Path B (f (circleelim A x p z)) | |
(circleelim B (f x) (<i> f (p @ i)) z) = split | |
base -> <_> f x | |
loop @ i -> <_> f (p @ i) | |
-- a special case, Lemmas 6.2.5-6.2.9 in the book | |
aLoop (A:U) : U = (a:A) * Path A a a | |
phi (A:U) (al : aLoop A) : S1 -> A = split | |
base -> al.1 | |
loop @ i -> (al.2)@ i | |
psi (A:U) (f:S1 -> A) : aLoop A = (f base,<i>f (loop1@i)) | |
rem (A:U) (f : S1 -> A) : (u : S1) -> Path A (phi A (psi A f) u) (f u) = split | |
base -> refl A (f base) | |
loop @ i -> <j>f (loop1@i) | |
lem (A:U) (f : S1 -> A) : Path (S1 -> A) (phi A (psi A f)) f = | |
<i> \ (x:S1) -> (rem A f x) @ i | |
thm (A:U) : Path U (aLoop A) (S1 -> A) = isoPath T0 T1 f g t s | |
where T0 : U = aLoop A | |
T1 : U = S1 -> A | |
f : T0 -> T1 = phi A | |
g : T1 -> T0 = psi A | |
s (x:T0) : Path T0 (g (f x)) x = refl T0 x | |
t : (y:T1) -> Path T1 (f (g y)) y = lem A | |