[DOC] Add more exposition to composition example (#2536)

* Add more exposition to composition example

* Apply suggestions from code review

Co-authored-by: Cris Cecka <ccecka@users.noreply.github.com>

---------

Co-authored-by: Haicheng Wu <57973641+hwu36@users.noreply.github.com>
Co-authored-by: Cris Cecka <ccecka@users.noreply.github.com>

media/docs/cpp/cute/02_layout_algebra.md

@@ -133,11 +133,15 @@ First, a few observations:
 
 With the above, we can assume without loss of generality that `B = s:d` is a layout with integral shape and stride. We can also assume that `A` is a flattened, coalesced layout.
 
-When `A` is integral, `A = a:b`, the result is rather trivial: `R = A o B = a:b o s:d = s:(b*d)`. But when `A` is multimodal, we need to be more careful.
+When `A` is integral, `A = a:b`, the result is rather trivial: `R = A o B = a:b o s:d = s:(b*d)`. Here, the result of the composition `R` are the first `s` elements of `A` strided by `d`.
 
-Put into words, `A o B = A o s:d`, for integral `s` and `d` means that we want (1) "remove" the first `d` elements from `A`, and then (2) "keep" the first `s` of those strided elements.
+When `A` is multimodal, we need to be more careful. Put into words, `A o B = A o s:d`, for integral `s` and `d` means that we want to:
 
-1. Removing the first `d` elements of `A` can be computed by progressively "dividing out" the first `d` elements from the shape of `A` starting from the left. For example,
+1. Determine a layout that produces every `d`th element of `A`.
+
+The shape of this intermediate layout can be calculated by progressively "dividing out" the first `d` elements from the shape of `A` starting from the left.
+
+For example,
 * `(6,2) /  2 => (3,2)`
 * `(6,2) /  3 => (2,2)`
 * `(6,2) /  6 => (1,2)`
@@ -147,9 +151,14 @@ Put into words, `A o B = A o s:d`, for integral `s` and `d` means that we want (
 * `(3,6,2,8) /  9 => (1,2,2,8)`
 * `(3,6,2,8) / 72 => (1,1,1,4)`
 
+To compute the strides of the strided layout, the residues of the above operation are used to scale the strides of `A`. For instance, the last example `(3,6,2,8):(w,x,y,z) / 72` with strides `(w,x,y,z)` produces `(3*w,6*x,2*x,2*z)` as the strides of the strided layout.
+
+
 As you may have noticed, we can only divide shapes by certain values and get a sensible result. This is called the **stride divisibility condition** and is statically checked in CuTe when possible.
 
-2. Keeping the first `s` elements of the strided `A` layout can be computed by "modding out" the first `s` elements from the shape of `A` starting from the left. For example,
+2. Keep the first `s` elements of the newly strided `A` so that the result has a compatible shape with `B`. This can be computed by "modding out" the first `s` elements from the shape of `A` starting from the left.
+
+For example,
 * `(6,2) %  2 => (2,1)`
 * `(6,2) %  3 => (3,1)`
 * `(6,2) %  6 => (6,1)`
@@ -159,11 +168,63 @@ As you may have noticed, we can only divide shapes by certain values and get a s
 * `(1,2,2,8) %  2 => (1,2,1,1)`
 * `(1,2,2,8) % 16 => (1,2,2,4)`
 
+This operation causes the result to have a shape that is compatible with `B`.
+
 Again, this operation must satisfy a **shape divisibility condition** to yield a sensible result and is statically checked in CuTe when possible.
 
 From the above examples, we can construct the composition `(3,6,2,8):(w,x,y,z) o 16:9 = (1,2,2,4):(3*w,3*x,y,z)`.
 
-#### Example 1 -- Reshape a layout into a matrix
+---
+
+#### Example 1 -- Worked Example of Calculating a Composition
+
+We provide a more complex example of composition, where both operand layouts are multi-modal to illustrate the concepts introduced above.
+
+```
+Functional composition, R := A o B
+R(c) := (A o B)(c) := A(B(c))
+
+Example
+A = (6,2):(8,2)
+B = (4,3):(3,1)
+
+1. Using the left-distributive and concatenation properties for layouts we write the composition as,
+
+R = A o B
+  = (6,2):(8,2) o (4,3):(3,1)
+  = ((6,2):(8,2) o 4:3, (6,2):(8,2) o 3:1)
+
+---
+1. Compute `(6,2):(8,2) o 4:3`
+
+- First, we compute the strided layout,
+
+(6,2):(8,2) / 3 = (6/3,2):(8*3,2) = (2,2):(24,2)
+
+- Next, we keep the shape compatible,
+
+(2,2):(24,2) % 4 = (2,2):(24,2)
+
+---
+2. Compute `(6,2):(8,2) o 3:1`
+
+- First, we compute the strided layout
+
+(6,2):(8,2) / 1 = (6,2):(8,2)
+
+- Next, we keep the shape compatible,
+
+(6,2):(8,2) % 3 = (3,1):(8,2)
+
+---
+
+Putting this together and coalescing each mode, we obtain the result
+
+R = A o B
+  = ((2, 2), 3): ((24, 2), 8)
+```
+
+#### Example 2 -- Reshape a layout into a matrix
 
 `20:2  o  (5,4):(4,1)`. Composition formulation.
 
@@ -181,7 +242,7 @@ as a 5x4 matrix in a row-major order.
 
 4. ` = (5,4):(8,2)`. Final composed layout.
 
-#### Example 2 -- Reshape a layout into a matrix
+#### Example 3 -- Reshape a layout into a matrix
 
 `(10,2):(16,4)  o  (5,4):(1,5)`