Typeclass for a type F with an injective map to A ↪ B

This typeclass is primarily for use by embeddings such as RelEmbedding.

Basic usage of EmbeddingLike

A typical type of embeddings should be declared as:

structure MyEmbedding (A B : Type*) [MyClass A] [MyClass B] where
  (toFun : A → B)
  (injective' : Function.Injective toFun)
  (map_op' : ∀ (x y : A), toFun (MyClass.op x y) = MyClass.op (toFun x) (toFun y))

namespace MyEmbedding

variable (A B : Type*) [MyClass A] [MyClass B]

instance : FunLike (MyEmbedding A B) A B where
  coe := MyEmbedding.toFun
  coe_injective' := fun f g h ↦ by cases f; cases g; congr

-- This instance is optional if you follow the "Embedding class" design below:
instance : EmbeddingLike (MyEmbedding A B) A B where
  injective' := MyEmbedding.injective'

@[ext] theorem ext {f g : MyEmbedding A B} (h : ∀ x, f x = g x) : f = g := DFunLike.ext f g h

/-- Copy of a `MyEmbedding` with a new `toFun` equal to the old one. Useful to fix definitional
equalities. -/
protected def copy (f : MyEmbedding A B) (f' : A → B) (h : f' = ⇑f) : MyEmbedding A B :=
  { toFun := f'
    injective' := h.symm ▸ f.injective'
    map_op' := h.symm ▸ f.map_op' }

end MyEmbedding

This file will then provide a CoeFun instance and various extensionality and simp lemmas.

Embedding classes extending EmbeddingLike

The EmbeddingLike design provides further benefits if you put in a bit more work. The first step is to extend EmbeddingLike to create a class of those types satisfying the axioms of your new type of morphisms. Continuing the example above:

/-- `MyEmbeddingClass F A B` states that `F` is a type of `MyClass.op`-preserving embeddings.
You should extend this class when you extend `MyEmbedding`. -/
class MyEmbeddingClass (F : Type*) (A B : outParam Type*) [MyClass A] [MyClass B]
    [FunLike F A B]
    extends EmbeddingLike F A B where
  map_op : ∀ (f : F) (x y : A), f (MyClass.op x y) = MyClass.op (f x) (f y)

@[simp]
lemma map_op {F A B : Type*} [MyClass A] [MyClass B] [FunLike F A B] [MyEmbeddingClass F A B]
    (f : F) (x y : A) :
    f (MyClass.op x y) = MyClass.op (f x) (f y) :=
  MyEmbeddingClass.map_op _ _ _

namespace MyEmbedding

variable {A B : Type*} [MyClass A] [MyClass B]

-- You can replace `MyEmbedding.EmbeddingLike` with the below instance:
instance : MyEmbeddingClass (MyEmbedding A B) A B where
  injective' := MyEmbedding.injective'
  map_op := MyEmbedding.map_op'

end MyEmbedding

The second step is to add instances of your new MyEmbeddingClass for all types extending MyEmbedding. Typically, you can just declare a new class analogous to MyEmbeddingClass:

structure CoolerEmbedding (A B : Type*) [CoolClass A] [CoolClass B] extends MyEmbedding A B where
  (map_cool' : toFun CoolClass.cool = CoolClass.cool)

class CoolerEmbeddingClass (F : Type*) (A B : outParam Type*) [CoolClass A] [CoolClass B]
    [FunLike F A B]
    extends MyEmbeddingClass F A B where
  (map_cool : ∀ (f : F), f CoolClass.cool = CoolClass.cool)

@[simp]
lemma map_cool {F A B : Type*} [CoolClass A] [CoolClass B]
    [FunLike F A B] [CoolerEmbeddingClass F A B] (f : F) :
    f CoolClass.cool = CoolClass.cool :=
  CoolerEmbeddingClass.map_cool _

variable {A B : Type*} [CoolClass A] [CoolClass B]

instance : FunLike (CoolerEmbedding A B) A B where
  coe f := f.toFun
  coe_injective' f g h := by cases f; cases g; congr; apply DFunLike.coe_injective; congr

instance : CoolerEmbeddingClass (CoolerEmbedding A B) A B where
  injective' f := f.injective'
  map_op f := f.map_op'
  map_cool f := f.map_cool'

-- [Insert `ext` and `copy` here]

Then any declaration taking a specific type of morphisms as parameter can instead take the class you just defined:

-- Compare with: lemma do_something (f : MyEmbedding A B) : sorry := sorry
lemma do_something {F : Type*} [FunLike F A B] [MyEmbeddingClass F A B] (f : F) : sorry := sorry

This means anything set up for MyEmbeddings will automatically work for CoolerEmbeddingClasses, and defining CoolerEmbeddingClass only takes a constant amount of effort, instead of linearly increasing the work per MyEmbedding-related declaration.

class EmbeddingLike (F : Sort u_1) (α : outParam (Sort u_2)) (β : outParam (Sort u_3)) [FunLike F α β] : Prop

The class EmbeddingLike F α β expresses that terms of type F have an injective coercion to injective functions α ↪ β.

• injective' : ∀ (f : F), Function.Injective ⇑f

  The coercion to functions must produce injective functions.

theorem EmbeddingLike.injective {F : Sort u_1} {α : Sort u_2} {β : Sort u_3} [FunLike F α β] [i : EmbeddingLike F α β] (f : F) : Function.Injective ⇑f

@[simp]

theorem EmbeddingLike.apply_eq_iff_eq {F : Sort u_1} {α : Sort u_2} {β : Sort u_3} [FunLike F α β] [i : EmbeddingLike F α β] (f : F) {x y : α} : f x = f y ↔ x = y

@[simp]

theorem EmbeddingLike.comp_injective {α : Sort u_2} {β : Sort u_3} {γ : Sort u_4} {F : Sort u_5} [FunLike F β γ] [EmbeddingLike F β γ] (f : α → β) (e : F) : Function.Injective (⇑e ∘ f) ↔ Function.Injective f