mathlib documentation

data.fun_like

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

This typeclass is primarily for use by homomorphisms like monoid_hom and linear_map.

Basic usage of fun_like #

A typical type of morphisms should be declared as:

structure my_hom (A B : Type*) [my_class A] [my_class B] :=
(to_fun : A  B)
(map_op' :  {x y : A}, to_fun (my_class.op x y) = my_class.op (to_fun x) (to_fun y))

namespace my_hom

variables (A B : Type*) [my_class A] [my_class B]

-- This instance is optional if you follow the "Hom class" design below:
instance : fun_like (my_hom A B) A (λ _, B) :=
{ coe := my_hom.to_fun, coe_injective' := λ f g h, by cases f; cases g; congr' }

/-- Helper instance for when there's too many metavariables to apply `to_fun.to_coe_fn` directly. -/
instance : has_coe_to_fun (my_hom A B) (λ _, A  B) := to_fun.to_coe_fn

@[simp] lemma to_fun_eq_coe {f : my_hom A B} : f.to_fun = (f : A  B) := rfl

@[ext] theorem ext {f g : my_hom A B} (h :  x, f x = g x) : f = g := fun_like.ext f g h

/-- Copy of a `my_hom` with a new `to_fun` equal to the old one. Useful to fix definitional
equalities. -/
protected def copy (f : my_hom A B) (f' : A  B) (h : f' = f) : my_hom A B :=
{ to_fun := f',
  map_op' := h.symm  f.map_op' }

end my_hom

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

Hom classes extending fun_like #

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

/-- `my_hom_class F A B` states that `F` is a type of `my_class.op`-preserving morphisms.
You should extend this class when you extend `my_hom`. -/
class my_hom_class (F : Type*) (A B : out_param $ Type*) [my_class A] [my_class B]
  extends fun_like F A (λ _, B) :=
(map_op :  (f : F) (x y : A), f (my_class.op x y) = my_class.op (f x) (f y))

@[simp] lemma map_op {F A B : Type*} [my_class A] [my_class B] [my_hom_class F A B]
  (f : F) (x y : A) : f (my_class.op x y) = my_class.op (f x) (f y) :=
my_hom_class.map_op

-- You can replace `my_hom.fun_like` with the below instance, or keep both:
instance : my_hom_class (my_hom A B) A B :=
{ coe := my_hom.to_fun,
  coe_injective' := λ f g h, by cases f; cases g; congr',
  map_op := my_hom.map_op' }

-- [Insert `has_coe_to_fun`, `to_fun_eq_coe`, `ext` and `copy` here]

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

structure cooler_hom (A B : Type*) [cool_class A] [cool_class B]
  extends my_hom A B :=
(map_cool' : to_fun cool_class.cool = cool_class.cool)

class cooler_hom_class (F : Type*) (A B : out_param $ Type*) [cool_class A] [cool_class B]
  extends my_hom_class F A B :=
(map_cool :  (f : F), f cool_class.cool = cool_class.cool)

@[simp] lemma map_cool {F A B : Type*} [cool_class A] [cool_class B] [cooler_hom_class F A B]
  (f : F) : f cool_class.cool = cool_class.cool :=
my_hom_class.map_op

-- You can also replace `my_hom.fun_like` with the below instance:
instance : cool_hom_class (cool_hom A B) A B :=
{ coe := cool_hom.to_fun,
  coe_injective' := λ f g h, by cases f; cases g; congr',
  map_op := cool_hom.map_op',
  map_cool := cool_hom.map_cool' }

-- [Insert `has_coe_to_fun`, `to_fun_eq_coe`, `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 : my_hom A B) : sorry := sorry
lemma do_something {F : Type*} [my_hom_class F A B] (f : F) : sorry := sorry

This means anything set up for my_homs will automatically work for cool_hom_classes, and defining cool_hom_class only takes a constant amount of effort, instead of linearly increasing the work per my_hom-related declaration.

@[class]
structure fun_like (F : Sort u_1) (α : out_param (Sort u_2)) (β : out_param (α → Sort u_3)) :
Sort (max 1 (imax u_1 u_2 u_3))

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

This typeclass is used in the definition of the homomorphism typeclasses, such as zero_hom_class, mul_hom_class, monoid_hom_class, ....

Instances

fun_like F α β where β depends on a : α #

@[protected, nolint, instance]
def fun_like.has_coe_to_fun {F : Sort u_1} {α : Sort u_2} {β : α → Sort u_3} [i : fun_like F α β] :
has_coe_to_fun F (λ (_x : F), Π (a : α), β a)
Equations
theorem fun_like.coe_injective {F : Sort u_1} {α : Sort u_2} {β : α → Sort u_3} [i : fun_like F α β] :
@[simp, norm_cast]
theorem fun_like.coe_fn_eq {F : Sort u_1} {α : Sort u_2} {β : α → Sort u_3} [i : fun_like F α β] {f g : F} :
f = g f = g
theorem fun_like.ext' {F : Sort u_1} {α : Sort u_2} {β : α → Sort u_3} [i : fun_like F α β] {f g : F} (h : f = g) :
f = g
theorem fun_like.ext'_iff {F : Sort u_1} {α : Sort u_2} {β : α → Sort u_3} [i : fun_like F α β] {f g : F} :
f = g f = g
theorem fun_like.ext {F : Sort u_1} {α : Sort u_2} {β : α → Sort u_3} [i : fun_like F α β] (f g : F) (h : ∀ (x : α), f x = g x) :
f = g
theorem fun_like.ext_iff {F : Sort u_1} {α : Sort u_2} {β : α → Sort u_3} [i : fun_like F α β] {f g : F} :
f = g ∀ (x : α), f x = g x
@[protected]
theorem fun_like.congr_fun {F : Sort u_1} {α : Sort u_2} {β : α → Sort u_3} [i : fun_like F α β] {f g : F} (h₁ : f = g) (x : α) :
f x = g x

fun_like F α (λ _, β) where β does not depend on a : α #

@[protected]
theorem fun_like.congr {F : Sort u_1} {α : Sort u_2} {β : Sort u_3} [i : fun_like F α (λ (_x : α), β)] {f g : F} {x y : α} (h₁ : f = g) (h₂ : x = y) :
f x = g y
@[protected]
theorem fun_like.congr_arg {F : Sort u_1} {α : Sort u_2} {β : Sort u_3} [i : fun_like F α (λ (_x : α), β)] (f : F) {x y : α} (h₂ : x = y) :
f x = f y