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Merged
Alizter merged 4 commits into from
Nov 18, 2025
+14 −6
Merged

Edit diagram in comments in Cubical/PathSquare.v #2327

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427703b
edit diagram in comment + formulate interchange lemma
Nov 14, 2025
a7a877f
sq's are transported 1-squares
Nov 17, 2025
3aea8e2
only diagram edits
Nov 18, 2025
0d5a5eb
Update theories/Cubical/PathSquare.v
Alizter Nov 18, 2025
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21 changes: 15 additions & 6 deletions theories/Cubical/PathSquare.v
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Expand Up @@ -10,14 +10,22 @@ Local Unset Elimination Schemes.
(* Homogeneous squares *)

(*
x00 ----pi0---- x01
a00 ----p0i---- a01
| |
| > |
pi0 = pi1
| = |
| |
p0i ==> p1i
| |
| |
x01-----pi1-----x11
*)
a10-----p1i-----a11

Indexing of points in a square follows the convention for matrices,
first a row index, then a column index. Unless stated otherwise,
paths are oriented in the direction of increasing index i (or x, etc).
In PathSquare below, the order of the paths is: left, right, top, bottom.
The stylized 2-path on the antidiagonal goes from pi0 @ p1i to p0i @ pi1,
complying with the definition of equiv_sq_path below.

*)

(** Contents:

Expand Down Expand Up @@ -72,6 +80,7 @@ Definition equiv_sq_path {A} {a00 a10 a01 a11 : A}
{px0 : a00 = a10} {px1 : a01 = a11}
{p0x : a00 = a01} {p1x : a10 = a11}
: px0 @ p1x = p0x @ px1 <~> PathSquare px0 px1 p0x p1x.

Proof.
snapply Build_Equiv.
{ destruct p0x, p1x.
Expand Down
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