mathlib documentation

data.fun_like.equiv

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

This typeclass is primarily for use by isomorphisms like monoid_equiv and linear_equiv.

Basic usage of equiv_like #

A typical type of morphisms should be declared as:

structure my_iso (A B : Type*) [my_class A] [my_class B]
  extends equiv 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_iso

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

-- This instance is optional if you follow the "Isomorphism class" design below:
instance : equiv_like (my_iso A B) A (λ _, B) :=
{ coe := my_iso.to_equiv.to_fun,
  inv := my_iso.to_equiv.inv_fun,
  left_inv := my_iso.to_equiv.left_inv,
  right_inv := my_iso.to_equiv.right_inv,
  coe_injective' := λ f g h, by cases f; cases g; congr' }

/-- Helper instance for when there's too many metavariables to apply `equiv_like.coe` directly. -/
instance : has_coe_to_fun (my_iso A B) := to_fun.to_coe_fn

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

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

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

end my_iso

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

Isomorphism classes extending equiv_like #

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

/-- `my_iso_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_iso`. -/
class my_iso_class (F : Type*) (A B : out_param $ Type*) [my_class A] [my_class B]
  extends equiv_like F A (λ _, B), my_hom_class F A B.

-- You can replace `my_iso.equiv_like` with the below instance:
instance : my_iso_class (my_iso A B) A B :=
{ coe := my_iso.to_fun,
  inv := my_iso.inv_fun,
  left_inv := my_iso.left_inv,
  right_inv := my_iso.right_inv,
  coe_injective' := λ f g h, by cases f; cases g; congr',
  map_op := my_iso.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_iso_class for all types extending my_iso. Typically, you can just declare a new class analogous to my_iso_class:

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

class cooler_iso_class (F : Type*) (A B : out_param $ Type*) [cool_class A] [cool_class B]
  extends my_iso_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_iso_class F A B]
  (f : F) : f cool_class.cool = cool_class.cool :=
my_iso_class.map_op

-- You can also replace `my_iso.equiv_like` with the below instance:
instance : cool_iso_class (cool_iso A B) A B :=
{ coe := cool_iso.to_fun,
  coe_injective' := λ f g h, by cases f; cases g; congr',
  map_op := cool_iso.map_op',
  map_cool := cool_iso.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_iso A B) : sorry := sorry
lemma do_something {F : Type*} [my_iso_class F A B] (f : F) : sorry := sorry

This means anything set up for my_isos will automatically work for cool_iso_classes, and defining cool_iso_class only takes a constant amount of effort, instead of linearly increasing the work per my_iso-related declaration.

@[class]
structure equiv_like (E : Sort u_1) (α : out_param (Sort u_2)) (β : out_param (Sort u_3)) :
Sort (max 1 (imax u_1 u_2 u_3) (imax u_1 u_3 u_2))

The class equiv_like E α β expresses that terms of type E have an injective coercion to bijections between α and β.

This typeclass is used in the definition of the homomorphism typeclasses, such as zero_equiv_class, mul_equiv_class, monoid_equiv_class, ....

Instances of this typeclass
Instances of other typeclasses for equiv_like
  • equiv_like.has_sizeof_inst
theorem equiv_like.inv_injective {E : Sort u_1} {α : Sort u_3} {β : Sort u_4} [iE : equiv_like E α β] :
@[protected, instance]
def equiv_like.to_embedding_like {E : Sort u_1} {α : Sort u_3} {β : Sort u_4} [iE : equiv_like E α β] :
Equations
@[protected]
theorem equiv_like.injective {E : Sort u_1} {α : Sort u_3} {β : Sort u_4} [iE : equiv_like E α β] (e : E) :
@[protected]
theorem equiv_like.surjective {E : Sort u_1} {α : Sort u_3} {β : Sort u_4} [iE : equiv_like E α β] (e : E) :
@[protected]
theorem equiv_like.bijective {E : Sort u_1} {α : Sort u_3} {β : Sort u_4} [iE : equiv_like E α β] (e : E) :
theorem equiv_like.apply_eq_iff_eq {E : Sort u_1} {α : Sort u_3} {β : Sort u_4} [iE : equiv_like E α β] (f : E) {x y : α} :
f x = f y x = y
@[simp]
theorem equiv_like.injective_comp {E : Sort u_1} {α : Sort u_3} {β : Sort u_4} {γ : Sort u_5} [iE : equiv_like E α β] (e : E) (f : β → γ) :
@[simp]
theorem equiv_like.surjective_comp {E : Sort u_1} {α : Sort u_3} {β : Sort u_4} {γ : Sort u_5} [iE : equiv_like E α β] (e : E) (f : β → γ) :
@[simp]
theorem equiv_like.bijective_comp {E : Sort u_1} {α : Sort u_3} {β : Sort u_4} {γ : Sort u_5} [iE : equiv_like E α β] (e : E) (f : β → γ) :
theorem equiv_like.comp_injective {F : Sort u_2} {α : Sort u_3} {β : Sort u_4} {γ : Sort u_5} [iF : equiv_like F β γ] (f : α → β) (e : F) :
@[simp]
theorem equiv_like.comp_surjective {F : Sort u_2} {α : Sort u_3} {β : Sort u_4} {γ : Sort u_5} [iF : equiv_like F β γ] (f : α → β) (e : F) :
@[simp]
theorem equiv_like.comp_bijective {F : Sort u_2} {α : Sort u_3} {β : Sort u_4} {γ : Sort u_5} [iF : equiv_like F β γ] (f : α → β) (e : F) :