Add "Exact mechanism" section to derivation.md

Author: Sporarum

Diff for: docs/_docs/reference/contextual/derivation.md

@@ -5,8 +5,8 @@ nightlyOf: https://docs.scala-lang.org/scala3/reference/contextual/derivation.ht
 ---
 
 Type class derivation is a way to automatically generate given instances for type classes which satisfy some simple
-conditions. A type class in this sense is any trait or class with a type parameter determining the type being operated
-on. Common examples are `Eq`, `Ordering`, or `Show`. For example, given the following `Tree` algebraic data type
+conditions. A type class in this sense is any trait or class with a single type parameter determining the type being operated
+on, and the special case `CanEqual`. Common examples are `Eq`, `Ordering`, or `Show`. For example, given the following `Tree` algebraic data type
 (ADT):
 
 ```scala
@@ -26,17 +26,130 @@ given [T: Show]     : Show[Tree[T]]     = Show.derived
 
 We say that `Tree` is the _deriving type_ and that the `Eq`, `Ordering` and `Show` instances are _derived instances_.
 
-**Note:** The access to `derived` above is a normal access, therefore if there are multiple definitions of `derived` in the type class, overloading resolution applies.
-
-**Note:** `derived` can be used manually, this is useful when you do not have control over the definition. For example we can implement an `Ordering` for `Option`s like so:
+**Note:** `derived` can be used manually, this is useful when you do not have control over the definition. For example we can implement `Ordering` for `Option`s like so:
 
 ```scala
 given [T: Ordering]: Ordering[Option[T]] = Ordering.derived
 ```
 
 It is discouraged to directly refer to the `derived` member if you can use a `derives` clause instead.
 
-All data types can have a `derives` clause. This document focuses primarily on data types which also have a given instance
+## Exact mechanism
+In the following, when type arguments are enumerated and the first index evaluates to a larger value than the last, then there are actually no arguments, for example: `A[T_2, ..., T_1]` means `A`.
+
+For a class/trait/object/enum `DerivingType[T_1, ..., T_N] derives TC`, a derived instance is created in `DerivingType`'s companion object (or `DerivingType` itself if it is an object).
+
+The general "shape" of the derived instance is as follows:
+```scala
+given [...](using ...): TC[ ... DerivingType[...] ... ] = TC.derived
+```
+`TC.derived` should be an expression that conforms to the expected type on the left, potentially elaborated using term and/or type inference.
+
+**Note:** `TC.derived` is a normal access, therefore if there are multiple definitions of `TC.derived`, overloading resolution applies.
+
+What the derived instance precisely looks like depends on the specifics of `DerivingType` and `TC`, we first examine `TC`:
+
+### `TC` takes 1 parameter `F`
+
+Therefore `TC` is defined as `TC[F[A_1, ..., A_K]]` (`TC[F]` if `K == 0`) for some `F`.
+There are two further cases depending on the kinds of arguments:
+
+#### `F` and all arguments of `DerivingType` have kind `*`
+**Note:** `K == 0` in this case.
+
+The generated instance is then:
+```scala
+given [T_1: TC, ..., T_N: TC]: TC[DerivingType[T_1, ..., T_N]] = TC.derived
+```
+
+This is the most common case, and is the one that was highlighted in the introduction.
+
+**Note:** The `[T_i: TC, ...]` introduces a `(using TC[T_i], ...)`, more information in [Context Bounds](./context-bounds.md).
+This allows the `derived` member to access these evidences.
+
+**Note:** If `N == 0` the above means:
+```scala
+given TC[DerivingType] = TC.derived
+```
+For example, the class
+```scala
+case class Point(x: Int, y: Int) derives Ordering
+```
+generates the instance
+```scala
+object Point:
+  ...
+  given Ordering[Point] = Ordering.derived
+```
+
+
+#### `F` and `DerivingType` have parameters of matching kind on the right
+This section covers cases where you can pair arguments of `F` and `DerivingType` starting from the right such that they have the same kinds pairwise, and all arguments of `F` or `DerivingType` (or both) are used up.
+`F` must also have at least one parameter.
+
+The general shape will then be:
+```scala
+given [...]: TC[ [...] =>> DerivingType[...] ] = TC.derived
+```
+Where of course `TC` and `DerivingType` are applied to types of the correct kind.
+
+To make this work, we split it into 3 cases:
+
+If `F` and `DerivingType` take the same number of arguments (`N == K`):
+```scala
+given TC[DerivingType] = TC.derived
+// simplified form of:
+given TC[ [A_1, ..., A_K] => DerivingType[A_1, ..., A_K] ] = TC.derived
+```
+If `DerivingType` takes less arguments than `F` (`N < K`), we use only the rightmost parameters from the type lambda:
+```scala
+given TC[ [A_1, ..., A_K] =>> DerivingType[A_(K-N+1), ..., A_K] ] = TC.derived
+
+// if DerivingType takes no arguments (N == 0), the above simplifies to:
+given TC[ [A_1, ..., A_K] =>> DerivingType ] = TC.derived
+```
+
+If `F` takes less arguments than `DerivingType` (`K < N`), we fill in the remaining leftmost slots with type parameters of the given:
+```scala
+given [T_1, ... T_(N-K)]: TC[[A_1, ..., A_K] =>> DerivingType[T_1, ... T_(N-K), A_1, ..., A_K]] = TC.derived
+```
+
+### `TC` is the `CanEqual` type class
+
+We have therefore: `DerivingType[T_1, ..., T_N] derives CanEqual`.
+
+Let `U_1`, ..., `U_M` be the parameters of `DerivingType` of kind `*`.
+(These are a subset of the `T_i`s)
+
+The generated instance is then:
+```scala
+given [T_1L, T_1R, ..., T_NL, T_NR]                            // every parameter of DerivingType twice
+      (using CanEqual[U_1L, U_1R], ..., CanEqual[U_ML, U_MR]): // only parameters of DerivingType with kind *
+        CanEqual[DerivingType[T_1L, ..., T_NL], DerivingType[T_1R, ..., T_NR]] = // again, every parameter
+          CanEqual.derived
+```
+
+The bounds of `T_i`s are handled correctly, for example: `T_2 <: T_1` becomes `T_2L <: T_1L`.
+
+For example, the class
+```scala
+class MyClass[A, G[_]](a: A, b: G[B]) derives CanEqual
+```
+generates the following given instance:
+```scala
+object MyClass:
+  ...
+  given [A_L, A_R, G_L[_], G_R[_]](using CanEqual[A_L, A_R]): CanEqual[MyClass[A_L, G_L], MyClass[A_R, G_R]] = CanEqual.derived
+```
+
+### `TC` is not valid for automatic derivation
+
+Throw an error.
+
+The exact error depends on which of the above conditions failed.
+As an example, if `TC` takes more than 1 parameter and is not `CanEqual`, the error is `DerivingType cannot be unified with the type argument of TC`.
+
+All data types can have a `derives` clause. The rest of this document focuses primarily on data types which also have a given instance
 of the `Mirror` type class available.
 
 ## `Mirror`