# CuTe Layout Algebra

CuTe provides an "algebra of `Layout`s" to support combining layouts in different ways.  This algebra includes operations such as

* `Layout` functional composition,
* a notion of `Layout` "product" to reproduce one layout according to another, and
* a notion of `Layout` "divide" to split one layout according to another.

Common utilities for building complicated layouts from simpler ones depend on the `Layout` product. Common utilities for partitioning layouts (of data, for example) across other layouts (of threads, for example) depend on the `Layout` divide. All of these utilities rely on the functional composition of `Layout`s.

In this section, we'll build up the tools of the `Layout` algebra and explain some of these core operations in detail.

## Coalesce

In the previous section, we summarized `Layout`s with
> Layouts are functions from integers to integers.

The `coalesce` operation is a "simplify" on functions from integers to integers. If we only care about input integers, then we can manipulate the shape and number of modes of the `Layout` without changing it as a function. The only thing `coalesce` can't change is the `Layout`'s `size`.

More specifically, you can find the checked post-conditions in [the `coalesce` unit test](../../../test/unit/cute/core/coalesce.cpp), which we'll reproduce here:
```cpp
// @post size(@a result) == size(@a layout)
// @post depth(@a result) <= 1
// @post for all i, 0 <= i < size(@a layout), @a result(i) == @a layout(i)
Layout coalesce(Layout const& layout)
```

For example,

```cpp
auto layout = Layout<Shape <_2,Shape <_1,_6>>,
                     Stride<_1,Stride<_6,_2>>>{};
auto result = coalesce(layout);    // _12:_1
```

where we can see the result has fewer modes and is "simpler." Indeed, this could save us a few operations in the coordinate mapping and index mapping (if those are performed dynamically).

So, how do we get there?

* We've already seen that column-major `Layout`s like `(_2,_4):(_1,_2)` act identically to `_8:_1` for 1-D coordinates.
* Modes with size static-1 will always produce a natural coordinate of static-0. They can be ignored no matter the stride.

Generalizing, consider a layout with just two integral modes, s0:d0 and s1:d1.  Denote the result of coalescing this layout as s0:d0 ++ s1:d1. Then, there are four cases:

1. `s0:d0  ++  _1:d1  =>  s0:d0`. Ignore modes with size static-1.
2. `_1:d0  ++  s1:d1  =>  s1:d1`. Ignore modes with size static-1.
3. `s0:d0  ++  s1:s0*d0  =>  s0*s1:d0`. If the second mode's stride is the product of the first mode's size and stride, then they can be combined.
4. `s0:d0  ++  s1:d1  =>  (s0,s1):(d0,d1)`. Else, nothing can be done and they must be treated separately.

That's it! We can flatten any layout and apply the above binary operation to each pair of adjacent modes in order to "coalesce" the modes of the layout.

### By-mode Coalesce

Obviously, sometimes we do care about the shape of our `Layout`, but would still like to coalesce. For example, I have a 2-D `Layout` and I would like the result to remain 2-D.

For this reason, there's an overload of `coalesce` that takes an additional parameter
```cpp
// Apply coalesce at the terminals of trg_profile
Layout coalesce(Layout const& layout, IntTuple const& trg_profile)
```

which can be used as follows

```cpp
auto a = Layout<Shape <_2,Shape <_1,_6>>,
                Stride<_1,Stride<_6,_2>>>{};
auto result = coalesce(a, Step<_1,_1>{});   // (_2,_6):(_1,_2)
// Identical to
auto same_r = make_layout(coalesce(layout<0>(a)),
                          coalesce(layout<1>(a)));
```

This function is recursing into `Step<_1,_1>{}` and applying `coalesce` to the corresponding sublayout whenever it sees an integer (the values don't matter, they're just flags) rather than a tuple.

> This theme of defining an operation that treats a `Layout` as a "1-D" function from integers to integers and then generalizing to use it for an arbitrarily shaped layout will be a common one!

## Composition

Functional composition of `Layout`s is the core of CuTe and is used in just about every higher-level operation.

Starting again from the observation that `Layout`s are just functions from integers to integers, we can define functional composition that results in another `Layout`. First, an example.

```text
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)

R( 0) = A(B( 0)) = A(B(0,0)) = A( 0) = A(0,0) =  0
R( 1) = A(B( 1)) = A(B(1,0)) = A( 3) = A(3,0) = 24
R( 2) = A(B( 2)) = A(B(2,0)) = A( 6) = A(0,1) =  2
R( 3) = A(B( 3)) = A(B(3,0)) = A( 9) = A(3,1) = 26
R( 4) = A(B( 4)) = A(B(0,1)) = A( 1) = A(1,0) =  8
R( 5) = A(B( 5)) = A(B(1,1)) = A( 4) = A(4,0) = 32
R( 6) = A(B( 6)) = A(B(2,1)) = A( 7) = A(1,1) = 10
R( 7) = A(B( 7)) = A(B(3,1)) = A(10) = A(4,1) = 34
R( 8) = A(B( 8)) = A(B(0,2)) = A( 2) = A(2,0) = 16
R( 9) = A(B( 9)) = A(B(1,2)) = A( 5) = A(5,0) = 40
R(10) = A(B(10)) = A(B(2,2)) = A( 8) = A(2,1) = 18
R(11) = A(B(11)) = A(B(3,2)) = A(11) = A(5,1) = 42
```

The absolutely amazing observation is that the function `R(c) = k` defined above can be written down as another `Layout`

```
R = ((2,2),3):((24,2),8)
```

AND

```
compatible(B, R)
```

That is, every coordinate of `B` can also be used as a coordinate of `R`. This is an expected property of functional composition because `B` defines the *domain* of `R`.

You can find many examples and checked post-conditions in [the `composition` unit test](../../../test/unit/cute/core/composition.cpp). The post-conditions are precisely as we just stated.
```cpp
// @post compatible(@a layout_b, @a result)
// @post for all i, 0 <= i < size(@a layout_b), @a result(i) == @a layout_a(@a layout_b(i)))
Layout composition(LayoutA const& layout_a, LayoutB const& layout_b)
```

### Computing Composition

First, a few observations:

* `B = (B_0, B_1, ...)`. A layout can be expressed as the concatenation of its sublayouts.

* `A o B = A o (B_0, B_1, ...) = (A o B_0, A o B_1, ...)`. When `B` is injective, composition is left-distributive with concatenation.

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.

Put into words, `A o B = A o s:d`, for integral `s` and `d` means that we want (1) every `d`th element of `A`, and then (2) keep the first `s` of those strided elements.

1. Every `d`th element of `A` can be computed by "dividing out" the first `d` elements from the shape of `A`. For an array of integers representing the shape, this is computed as
```cpp
void shape_div(int* shapeA, int N, int& strideB) {
   for (int i = 0; i < N; ++i) {
      assert(shapeA[i] %   strideB == 0 or
               strideB % shapeA[i] == 0);
      int new_shape  = ceil_div(shapeA[i], strideB);
      int new_stride = ceil_div(strideB, shapeA[i]);
      shapeA[i] = new_shape;
      strideB   = new_stride;
   }
}
```
which progressively "removes" the first `strideB` elements from `shapeA` starting from the left. For example,
* `(6,2) /  2 => (3,2)`
* `(6,2) /  3 => (2,2)`
* `(6,2) /  6 => (1,2)`
* `(6,2) / 12 => (1,1)`
* `(3,6,2,8) / 6 => (1,3,2,8)`
* `(3,6,2,8) / 9 => (1,2,2,8)`
* `(42,16,3) / 2 => (21,16,3)`
* `(42,16,3) / 6 => ( 7,16,3)`

As you may have noticed, we can only divide shapes by certain values and get a sensible result. This is called the **divisibility condition** and is enforced by the `assert` in the above code and statically checked in CuTe when possible.

2. The first `s` elements of the strided `A` layout can be computed by "modding out" the first `s` elements from the shape of `A`. For an array of integers representing the shape, this is computed as
```cpp
void shape_mod(int* shapeA, int N, int& shapeB) {
   for (int i = 0; i < N; ++i) {
      assert(shapeA[i] %    shapeB == 0 or
                shapeB % shapeA[i] == 0);
      int new_shapeA =      min(shapeA[i], shapeB);
      int new_shapeB = ceil_div(shapeB, shapeA[i]);
      shapeA[i] = new_shapeA;
      shapeB    = new_shapeB;
   }
}
```
which progressibly "keeps" the first `shapeB` elements from `shapeA` starting from the left. For example,
* `(6,2) %  2 => (2,1)`
* `(6,2) %  3 => (3,1)`
* `(6,2) %  6 => (6,1)`
* `(6,2) % 12 => (6,2)`
* `(3,6,2,8) %  6 => (3,2,1,1)`
* `(3,6,2,8) %  9 => (3,3,1,1)`
* `(1,2,2,8) %  2 => (1,2,1,1)`
* `(1,2,2,8) % 16 => (1,2,2,4)`

Again, this operation must satisfy the divisibility condition to yield a sensible result. This is enforced by the `assert` in the above code and statically checked in CuTe when possible.

Clearly, CuTe does not use arrays to store shapes or strides and the above code is for explication only. CuTe works with shapes and strides as `IntTuple`s and the implementation is expressed as algorithmic `fold`s which carefully account for static and dynamic integers.

#### Example 1 -- Reshape a layout into a matrix

`20:2  o  (5,4):(4,1)`. Composition formulation.

This describes interpreting the layout `20:2`
as a 5x4 matrix in a row-major order.

1. ` = 20:2 o (5:4,4:1)`. Layout `(5,4):(4,1)` as concatenation of sublayouts.

2. ` = (20:2 o 5:4, 20:2 o 4:1)`. Left distributivity.

    * `20:2 o 5:4  =>  5:8`. Trivial case.
    * `20:2 o 4:1  =>  4:2`. Trivial case.

3. ` = (5:8, 4:2)`. Composed Layout as concatenation of sublayouts.

4. ` = (5,4):(8,2)`. Final composed layout.

#### Example 2 -- Reshape a layout into a matrix

`(10,2):(16,4)  o  (5,4):(1,5)`

This describes interpreting the layout `(10,2):(16,4)`
as a 5x4 matrix in a column-major order.

1. ` = (10,2):(16,4) o (5:1,4:5)`. Layout `(5,4):(1,5)` as concatenation of sublayouts.

2. ` = ((10,2):(16,4) o 5:1, (10,2):(16,4) o 4:5)`. Left distributivity.

    * `(10,2):(16,4) o 5:1 => (5,1):(16,4)`. Mod out the shape `5`.
    * `(10,2):(16,4) o 4:5 => (2,2):(80,4)`. Div out the stride `5`.

3. ` = ((5,1):(16,4), (2,2):(80,4))`. Composed Layout as concatenation of sublayouts.

4. ` = (5:16, (2,2):(80,4))`. By-mode coalesce.

5. ` = (5,(2,2))):(16,(80,4))`. Final composed layout.

We get exactly this result with CuTe
if we use compile-time shapes and strides.
The following C++ code prints `(_5,(_2,_2)):(_16,(_80,_4))`.

```cpp
Layout a = make_layout(make_shape (Int<10>{}, Int<2>{}),
                       make_stride(Int<16>{}, Int<4>{}));
Layout b = make_layout(make_shape (Int< 5>{}, Int<4>{}),
                       make_stride(Int< 1>{}, Int<5>{}));
Layout c = composition(a, b);
print(c);
```

If we use dynamic integers, the following C++ code prints `((5,1),(2,2)):((16,4),(80,4))`.

```cpp
Layout a = make_layout(make_shape (10, 2),
                       make_stride(16, 4));
Layout b = make_layout(make_shape ( 5, 4),
                       make_stride( 1, 5));
Layout c = composition(a, b);
print(c);
```

The results may _look_ different but are the mathematically the same. The 1s in the shape don't affect the layout as a mathematical function from 1-D coordinates to integers or as a function from 2-D coordinates to integers. In the dynamic case, CuTe can not coalesce the dynamic size-1 modes to "simplify" the layout due to the static rank and type of the tuples containing them.

### By-mode Composition

Similar to by-mode `coalesce` and building up to a generic tiling operation, sometimes we do care about the shape of the `A` layout and would still like to apply `composition` to individual modes. For example, I have a 2-D `Layout` and would like some sublayout of the elements down the columns and another sublayout of elements across the rows.

For this reason, `composition` also works when its second parameter -- the `B` -- is a `Tiler`. In general, a tiler is a layout or a tuple-of-layouts (note the generalization on `IntTuple`), which can be used as follows
```cpp
// (12,(4,8)):(59,(13,1))
auto a = make_layout(make_shape (12,make_shape ( 4,8)),
                     make_stride(59,make_stride(13,1)));
// <3:4, 8:2>
auto tiler = make_tile(Layout<_3,_4>{},  // Apply 3:4 to mode-0
                       Layout<_8,_2>{}); // Apply 8:2 to mode-1

// (_3,(2,4)):(236,(26,1))
auto result = composition(a, tiler);
// Identical to
auto same_r = make_layout(composition(layout<0>(a), get<0>(tiler)),
                          composition(layout<1>(a), get<1>(tiler)));
```
We often use the `<LayoutA, LayoutB, ...>` notation to distinguish `Tiler`s from the concatenation-of-sublayouts notation `(LayoutA, LayoutB, ...)` that we used previously.

The `result` in the above code can be depicted as the 3x8 sublayout of the original layout highlighted in the figure below.
<p align="center">
  <img src="../../images/cute/composition1.png" alt="composition1.png" height="250"/>
</p>

For convenience, CuTe also interprets `Shape`s as a tiler as well. A `Shape` is interpreted as tuple-of-layouts-with-stride-1:
```cpp
// (12,(4,8)):(59,(13,1))
auto a = make_layout(make_shape (12,make_shape ( 4,8)),
                     make_stride(59,make_stride(13,1)));
// (8, 3)
auto tiler = make_shape(Int<3>{}, Int<8>{});
// Equivalent to <3:1, 8:1>
// auto tiler = make_tile(Layout<_3,_1>{},  // Apply 3:1 to mode-0
//                        Layout<_8,_1>{}); // Apply 8:1 to mode-1

// (_3,(4,2)):(59,(13,1))
auto result = composition(a, tiler);
```
where `result` can be depicted as the 3x8 sublayout of the original layout highlighted in the figure below.
<p align="center">
  <img src="../../images/cute/composition2.png" alt="composition2.png" height="250"/>
</p>

## Composition Tilers

In summary, a `Tiler` is one of the following objects.
1. A `Layout`.
2. A tuple of `Tiler`s.
3. A `Shape`, which will be interpreted as a tiler of `Layout`s with stride-1.

Any of the above can be used as the second argument in `composition`. With (1), we think of the `composition` as between two functions from integers to integers, no matter the ranks of the layouts. With (2) and (3), the `composition` is performed on each pair of corresponding modes of `A` and `B`, until case (1) is found.

This allows composition to be applied by-mode to retrieve arbitrary sublayouts of specified modes of a tensor ("Give me the 3x5x8 subblock of this MxNxL tensor") but also allows entire tiles of data to be reshaped and reordered as if they were 1-D vectors ("Reorder this 8x16 block of data into a 32x4 block using this weird order of elements"). We will see the by-mode cases appear often when we are tiling for threadblocks in examples that follow. We will see 1-D reshaping and reordering when we want to apply arbitrary partitioning patterns for threads and values in MMAs in examples that follow.

## Complement

Before getting to "product" and "divide," we need one more operation. We can think of `composition` as a layout `B` that is "selecting" certain coordinates from another layout `A`. But what about the coordinates that aren't "selected"? To implement generic tiling, we want to be able to select arbitrary elements -- the tile -- and to describe the layout of those tiles -- the leftovers, or the "rest."

The `complement` of a layout attempts to find another layout that represents the "rest" -- the elements that aren't touched by the layout.

You can find many examples and checked post-conditions in [the `complement` unit test](../../../test/unit/cute/core/complement.cpp). The post-conditions include
```cpp
// @post cosize(make_layout(@a layout_a, @a result))) >= size(@a cotarget)
// @post cosize(@a result) >= round_up(size(@a cotarget), cosize(@a layout_a))
// @post for all i, 1 <= i < size(@a result),
//         @a result(i-1) < @a result(i)
// @post for all i, 1 <= i < size(@a result),
//         for all j, 0 <= j < size(@a layout_a),
//           @a result(i) != @a layout_a(j)
Layout complement(LayoutA const& layout_a, Shape const& cotarget)
```
That is, the complement `R` of a layout `A` with respect to a Shape (IntTuple) `M` satisfies the following properties.
1. The size (and cosize) of `R` is *bounded* by `size(M)`.
2. `R` is *ordered*.  That is, the strides of `R` are positive and increasing.  This means that `R` is unique.
3. `A` and `R` have *disjoint* codomains. `R` attempts to "complete" the codomain of `A`.

The `cotarget` parameter above is most commonly an integer -- you can see we only use `size(cotarget)` above. However, sometimes it is useful to specify an integer that has static properties. For example, `28` is a dynamic integer and `(_4,7)` is a shape with size `28` that is statically known to be divisible by `_4`. Both will produce the same `complement` mathematically, but the extra information can used by `complement` to preserve the staticness of the result as much as possible.

### Complement Examples

`complement` is most effective on static shapes and strides, so consider all integers below to be static. Similar examples for dynamic shapes and strides as well as IntTuple `cotarget` can be found in [the unit test](../../../test/unit/cute/core/complement.cpp).

* `complement(4:1, 24)` is `6:4`. Note that `(4,6):(1,4)` has cosize `24`. The layout `4:1` is effectively repeated 6 times with `6:4`.

* `complement(6:4, 24)` is `4:1`. Note that `(6,4):(4,1)` has cosize `24`. The "hole" in `6:4` is filled with `4:1`.

* `complement((4,6):(1,4), 24)` is `1:0`. Nothing needs to be appended.

* `complement(4:2, 24)` is `(2,3):(1,8)`. Note that `(4,(2,3)):(2,(1,8))` has cosize `24`. The "hole" in `4:2` is filled with `2:1` first, then everything is repeated 3 times with `3:8`.

* `complement((2,4):(1,6), 24)` is `3:2`. Note that `((2,4),3):((1,6),2)` has cosize `24` and produces unique indices.

* `complement((2,2):(1,6), 24)` is `(3,2):(2,12)`. Note that `((2,2),(3,2)):((1,6),(2,12))` has cosize `24` and produces unique indices.

<p align="center">
  <img src="../../images/cute/complement1.png" alt="complement1.png" height="75"/>
</p>
As a visualization, the above figure depicts the codomain of the last example. The image of the original layout `(2,2):(1,6)` is colored in gray. The complement effectively "repeats" the original layout (displayed in the other colors) such that the codomain size of the result is `24`. The complement `(3,2):(2,12)` can be viewed as the "layout of the repetition."

## Division (Tiling)

Finally, we can define the division of a `Layout` by another `Layout`. Functions that divide a layout into components are useful as a basis for tiling and partitioning layouts.

In this section, we'll define `logical_divide(Layout, Layout)`, which again considers all `Layout`s as 1-D functions from integers to integers, and then use that definition to create multidimensional `Layout` divides.

Informally, `logical_divide(A, B)` splits a layout `A` into two modes -- in the first mode are all elements pointed to by `B` and in the second mode are all elements not pointed to by `B`.

Formally, this can be written as

$A \oslash B := A \circ (B,B^*)$

and implemented as
```cpp
template <class LShape, class LStride,
          class TShape, class TStride>
auto logical_divide(Layout<LShape,LStride> const& layout,
                    Layout<TShape,TStride> const& tiler)
{
  return composition(layout, make_layout(tiler, complement(tiler, size(layout))));
}
```
Note that this is defined only in terms of concatenation, composition, and complement.

So what is that?

> in the first mode are all elements pointed to by `B`

This is clearly composition, `A o B`.

> in the second mode are all elements not pointed to by `B`

The elements NOT pointed to by `B` sounds like a complement, `B*`, up to the size of `A`. As we've seen above in the `complement` section, this can be described as the "layout of the repetition of `B`." If `B` is the "tiler", then `B*` is the layout of the tiles.

### Logical Divide 1-D Example

Consider tiling the 1-D layout `A = (4,2,3):(2,1,8)` with the tiler `B = 4:2`. Informally, this means that we have a 1-D vector of 24 elements in some storage order defined by `A` and we want to extract tiles of 4 elements strided by 2.

This is computed in the three steps described in the implementation above.
* Complement of `B = 4:2` under `size(A) = 24` is `B* = (2,3):(1,8)`.
* Concantenation of `(B,B*) = (4,(2,3)):(2,(1,8))`.
* Composition of `A = (4,2,3):(2,1,8)` with `(B,B*)` is then `((2,2),(2,3)):((4,1),(2,8))`.

<p align="center">
  <img src="../../images/cute/divide1.png" alt="divide1.png" height="150"/>
</p>

The above figure depicts `A` as a 1-D layout with the elements pointed to by `B` highlighted in gray. The layout `B` describes our "tile" of data, and there are six of those tiles in `A` shown by each of the colors. After the divide, the first mode of the result is the tile of data and the second mode of the result iterates over each tile.

### Logical Divide 2-D Example

Using the `Tiler` concept defined above, this immediately generalizes to multidimensional tiling. The below example simply applies `layout_divide` by-mode to the cols and rows of a 2-D layout using a `Tiler`.

Similar to the 2-D composition example above, consider a 2-D layout `A = (9,(4,8)):(59,(13,1))` and want to apply `3:3` down the columns (mode-0) and `(2,4):(1,8)` across the rows (mode-1). This means the tiler can be written as `B = <3:3, (2,4):(1,8)>`.

<p align="center">
  <img src="../../images/cute/divide2.png" alt="divide2.png" height="450"/>
</p>

The above figure depicts `A` as a 2-D layout with the elements pointed to by `B` highlighted in gray. The layout `B` describes our "tile" of data, and there are twelve of those tiles in `A` shown by each of the colors. After the divide, the first mode of each mode of the result is the tile of data and the second mode of each mode iterates over each tile. In that sense, this operation can be viewed as a kind of `gather` operation or as simply a permutation on the rows and cols.

Note that the first mode of each mode of the result is the sublayout `(3,(2,4)):(177,(13,2))` and is precisely the result we would have received if we had applied `composition` instead of `logical_divide`.

### Zipped, Tiled, Flat Divides

It's easy to see the tiles when they are highlighted in the images above, but working with them can still be awkward. How would you slice out the `3`rd tile or the `7`th tile or the `(1,2)`th tile so you could continue working on it?

Enter the convenience flavors of `logical_divide`. Suppose we have a `Layout` and a `Tiler` of some shape, then each operation will apply `logical_divide`, but potentially rearrange the modes into more convenient forms.
```text
Layout Shape : (M, N, L, ...)
Tiler Shape  : <TileM, TileN>

logical_divide : ((TileM,RestM), (TileN,RestN), L, ...)
zipped_divide  : ((TileM,TileN), (RestM,RestN,L,...))
tiled_divide   : ((TileM,TileN), RestM, RestN, L, ...)
flat_divide    : (TileM, TileN, RestM, RestN, L, ...)
```

For example, the `zipped_divide` function applies `logical_divide`, and then gathers the "subtiles" into a single mode and the "rest" into a single mode.
```cpp
// A: shape is (9,32)
auto layout_a = make_layout(make_shape (Int< 9>{}, make_shape (Int< 4>{}, Int<8>{})),
                            make_stride(Int<59>{}, make_stride(Int<13>{}, Int<1>{})));
// B: shape is (3,8)
auto tiler = make_tile(Layout<_3,_3>{},           // Apply     3:3     to mode-0
                       Layout<Shape <_2,_4>,      // Apply (2,4):(1,8) to mode-1
                              Stride<_1,_8>>{});

// ((TileM,RestM), (TileN,RestN)) with shape ((3,3), (8,4))
auto ld = logical_divide(layout_a, tiler);
// ((TileM,TileN), (RestM,RestN)) with shape ((3,8), (3,4))
auto zd = zipped_divide(layout_a, tiler);
```
Then, the offset to the `3`rd tile is `zd(0,3)`. The offset to the `7`th tile is `zd(0,7)`. The offset to the `(1,2)`th tile is `zd(0,make_coord(1,2))`. The tile itself always has layout `layout<0>(zd)`. Indeed, it is always the case that

`layout<0>(zipped_divide(a, b)) == composition(a, b)`.

We note that `logical_divide` preserves the *semantics* of the modes while permuting the elements within those modes -- the `M`-mode of layout `A` is still the `M`-mode of the result and the `N`-mode of layout `A` is still the `N`-mode of the result.

This is not the case with `zipped_divide`. The mode-0 in the `zipped_divide` result is the `Tile` itself (of whatever rank the `Tiler` was) and mode-1 is the layout of those tiles. It doesn't always make sense to plot these as 2-D layouts, because the `M`-mode is now more aptly the "tile-mode" and the `N`-mode is more aptly the "rest-mode". Regardless, we still can plot the resulting layout as 2-D as shown below.

<p align="center">
  <img src="../../images/cute/divide3.png" alt="divide3.png" height="450"/>
</p>

We've kept each tile as its color in the previous images for clarity. Clearly, iterating across tiles is now equivalent to iterating across a row of this layout and iterating over elements within a tile is equivalent to iterating down a column of this layout. As we'll see in the `Tensor` section, this can be used to great effect in partitioning within or across tiles of data.

## Product (Tiling)

Finally, we can define the product of a Layout by another Layout. In this section, we'll define `logical_product(Layout, Layout)`, which again considers all `Layout`s as 1-D functions from integers to integers, and then use that definition to create multidimensional `Layout` products.

Informally, `logical_product(A, B)` results in a two mode layout where the first mode is the layout `A` and the second mode is the layout `B` but with each element replaced by a "unique replication" of layout `A`.

Formally, this can be written as

$A \otimes B := (A, A^* \circ B)$

and implemented in CuTe as
```cpp
template <class LShape, class LStride,
          class TShape, class TStride>
auto logical_product(Layout<LShape,LStride> const& layout,
                     Layout<TShape,TStride> const& tiler)
{
  return make_layout(layout, composition(complement(layout, size(layout)*cosize(tiler)), tiler));
}
```
Note that this is defined only in terms of concatenation, composition, and complement.

So what is that?

> where the first mode is the layout `A`

This is clearly just a copy of `A`.

> the second mode is the layout `B` but with each element replaced by a "unique replication" of layout `A`.

The "unique replication" of layout `A` sounds like complement, `A*`, up to the cosize of `B`. As we've seen in the `complement` section, this can be described as the "layout of the repetition of `A`". If `A` is the "tile", then `A*` is the layout of repetitions that are available for `B`.

### Logical Product 1-D Example

Consider reproducing the 1-D layout `A = (2,2):(4,1)` according to `B = 6:1`. Informally, this means that we have a 1-D layout of 4 elements defined by `A` and we want to reproduce it 6 times.

This is computed in the three steps described in the implementation above.
* Complement of `A = (2,2):(4,1)` under `6*4 = 24` is `A* = (2,3):(2,8)`.
* Composition of `A* = (2,3):(2,8)` with `B = 6:1` is then `(2,3):(2,8)`.
* Concatenation of `(A,A* o B) = ((2,2),(2,3)):((4,1),(2,8))`.

<p align="center">
  <img src="../../images/cute/product1.png" alt="product1.png" height="175"/>
</p>

The above figure depicts `A` and `B` as a 1-D layouts. The layout `B` describes the number and order of repetitions of `A` and they are colored for clarity. After the product, the first mode of the result is the tile of data and the second mode of the result iterates over each tile.

Note that the result is identical to the result of the 1-D Logical Divide example.

Of course, we can change the number and order of the tiles in the product by changing `B`.

<p align="center">
  <img src="../../images/cute/product2.png" alt="product2.png" height="175"/>
</p>

For example, in the above image with `B = (4,2):(2,1)`, there are 8 repeated tiles instead of 6 and the tiles are in a different order.

### Logical Product 2-D Example

We can use the by-mode `tiler` strategies previously developed to write multidimensional products as well.

<p align="center">
  <img src="../../images/cute/product2d.png" alt="product2d.png" height="250"/>
</p>

The above image demonstates the use of a `tiler` to apply `logical_product` by-mode. Despite this **not being the recommended approach**, the result is a rank-2 layout consisting of 2x5 row-major block that is tiled across a 3x4 column-major arrangement.

The reason **this is not the recommended approach** is that the `tiler B` in the above expression is highly unintuitive. In fact, it requires perfect knowledge of the shape and strides of `A` in order to construct. We would like to express "Tile Layout `A` according to Layout `B`" in a way that makes `A` and `B` independent and is much more intuitive.

#### Blocked and Raked Products

The `blocked_product(LayoutA, LayoutB)` and `raked_product(LayoutA, LayoutB)` are rank-sensitive transformations on top of 1-D `logical_product` that let us express the more intuitive `Layout` products that we most often want to express.

A key observation in the implementation of these functions are the compatibility post-conditions of `logical_product`:
```
// @post rank(result) == 2
// @post compatible(layout_a, layout<0>(result))
// @post compatible(layout_b, layout<1>(result))
```

Because `A` is always compatible with mode-0 of the result and `B` is always compatible with mode-1 of the result, if we made `A` and `B` the same rank then we could "reassociate" like-modes after the product. That is, the "column" mode in `A` could be combined with the "column" mode in `B` and the "row" mode in `A` could be combined with the "row" mode in `B`, etc.

This is exactly what `blocked_product` and `raked_product` do and it is why they are called rank-sensitive. Unlike other CuTe functions that take `Layout` arguments, these care about the top-level rank of the arguments so that each mode can be reassociated after the `logical_product`.

<p align="center">
  <img src="../../images/cute/productblocked2d.png" alt="productblocked2d.png" height="250"/>
</p>

The above image shows the same result as the `tiler` approach, but with much more intuitive arguments. A 2x5 row-major layout is arranged as a tile in a 3x4 column-major arrangement. Also note that `blocked_product` went ahead and `coalesced` mode-0 for us.

Similarly, `raked_product` combines the modes slightly differently. Instead of the resulting "column" mode being constructed from the `A` "column" mode then the `B` "column" mode, the resulting "column" mode is constructed from the `B` "column" mode then the `A` "column" mode.

<p align="center">
  <img src="../../images/cute/productraked2d.png" alt="productraked2d.png" height="250"/>
</p>

This results in the "tile" `A` now being interleaved or "raked" with the "layout-of-tiles" `B` instead of appearing as blocks. Other references call this a "cyclic distribution."

### Zipped and Tiled Products

Similar to `zipped_divide` and `tiled_divide`, the `zipped_product` and `tiled_product` simply rearrange the modes that result from a by-mode `logical_product`.

```text
Layout Shape : (M, N, L, ...)
Tiler Shape  : <TileM, TileN>

logical_product : ((M,TileM), (N,TileN), L, ...)
zipped_product  : ((M,N), (TileM,TileN,L,...))
tiled_product   : ((M,N), TileM, TileN, L, ...)
flat_product    : (M, N, TileM, TileN, L, ...)
```
