1. How to instantiate a typeclass?

   Take CoeOut as an example :

   /--
       `CoeOut α β` is for coercions that are applied from left-to-right.
   -/
   class CoeOut (α : Sort u) (β : semiOutParam (Sort v)) where
     /-- Coerces a value of type `α` to type `β`. Accessible by the notation `↑x`,
     or by double type ascription `((x : α) : β)`. -/
     coe : α → β

   there is only one field to be filled, which is coe

   instance : CoeOut MyType MyOtherType := {
       coe := myCoerceFunction
   }

   or

   instance : CoeOut MyType MyOtherType :=
     ⟨myCoerceFunction⟩

2. How to define and instantiate a subtype?

   Take MySubtype as an example:

   def MySubtype := { x : MyType // x.property }

   Here, MySubtype is a subtype of MyType consisting of elements x that satisfy x.property.

   To instantiate a value of MySubtype, you need to provide a value of MyType along with a proof that it satisfies the property:

   def mySubtypeInstance : MySubtype :=
     ⟨myValueOfMyType, myProofThatItSatisfiesProperty⟩

3. How to define a linear map?

   For example :

   def RSEvalLinear [Semiring F] (α : ι → F) : F[X] →ₗ[F] ι → F where
       toFun := RSEval α
       -- `simp` uses registered theorem `Eval_add`
       map_add' := by
         intro p q
         simp
       -- `simp` uses registered theorem `Eval_smul`
       map_smul' := by
         intro c p
         simp

   linear map is a homomorphism between vector spaces, preserving both vector addition and scalar multiplication. So besides defining the function itself toFun, you also need to provide proofs that it respects addition and scalar multiplication, which are map_add' and map_smul' in the LinearMap structure.