Equivalence between fintypes

This file contains some basic results on equivalences where one or both sides of the equivalence are Fintypes.

Main definitions

• Function.Embedding.toEquivRange: computably turn an embedding of a fintype into an Equiv of the domain to its range
• Equiv.Perm.viaFintypeEmbedding : Perm α → (α ↪ β) → Perm β extends the domain of a permutation, fixing everything outside the range of the embedding

Implementation details

• Function.Embedding.toEquivRange uses a computable inverse, but one that has poor computational performance, since it operates by exhaustive search over the input Fintypes.

def Function.Embedding.toEquivRange {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] (f : α ↪ β) : α ≃ ↑(Set.range ⇑f)

Computably turn an embedding f : α ↪ β into an equiv α ≃ Set.range f, if α is a Fintype. Has poor computational performance, due to exhaustive searching in constructed inverse. When a better inverse is known, use Equiv.ofLeftInverse' or Equiv.ofLeftInverse instead. This is the computable version of Equiv.ofInjective.

Equations

• f.toEquivRange = { toFun := fun (a : α) => ⟨f a, ⋯⟩, invFun := f.invOfMemRange, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem Function.Embedding.toEquivRange_apply {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] (f : α ↪ β) (a : α) : f.toEquivRange a = ⟨f a, ⋯⟩

@[simp]

theorem Function.Embedding.toEquivRange_symm_apply_self {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] (f : α ↪ β) (a : α) : f.toEquivRange.symm ⟨f a, ⋯⟩ = a

theorem Function.Embedding.toEquivRange_eq_ofInjective {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] (f : α ↪ β) : f.toEquivRange = Equiv.ofInjective ⇑f ⋯

def Equiv.Perm.viaFintypeEmbedding {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] (e : Equiv.Perm α) (f : α ↪ β) : Equiv.Perm β

Extend the domain of e : Equiv.Perm α, mapping it through f : α ↪ β. Everything outside of Set.range f is kept fixed. Has poor computational performance, due to exhaustive searching in constructed inverse due to using Function.Embedding.toEquivRange. When a better α ≃ Set.range f is known, use Equiv.Perm.viaSetRange. When [Fintype α] is not available, a noncomputable version is available as Equiv.Perm.viaEmbedding.

Equations

• e.viaFintypeEmbedding f = e.extendDomain f.toEquivRange

@[simp]

theorem Equiv.Perm.viaFintypeEmbedding_apply_image {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] (e : Equiv.Perm α) (f : α ↪ β) (a : α) : (e.viaFintypeEmbedding f) (f a) = f (e a)

theorem Equiv.Perm.viaFintypeEmbedding_apply_mem_range {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] (e : Equiv.Perm α) (f : α ↪ β) {b : β} (h : b ∈ Set.range ⇑f) : (e.viaFintypeEmbedding f) b = f (e (f.invOfMemRange ⟨b, h⟩))

theorem Equiv.Perm.viaFintypeEmbedding_apply_not_mem_range {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] (e : Equiv.Perm α) (f : α ↪ β) {b : β} (h : b ∉ Set.range ⇑f) : (e.viaFintypeEmbedding f) b = b

@[simp]

theorem Equiv.Perm.viaFintypeEmbedding_sign {α : Type u_1} {β : Type u_2} [Fintype α] [DecidableEq β] (e : Equiv.Perm α) (f : α ↪ β) [DecidableEq α] [Fintype β] : Equiv.Perm.sign (e.viaFintypeEmbedding f) = Equiv.Perm.sign e

noncomputable def Equiv.toCompl {α : Type u_1} [Finite α] {p : α → Prop} {q : α → Prop} (e : { x : α // p x } ≃ { x : α // q x }) : { x : α // ¬p x } ≃ { x : α // ¬q x }

If e is an equivalence between two subtypes of a finite type α, e.toCompl is an equivalence between the complement of those subtypes.

See also Equiv.compl, for a computable version when a term of type {e' : α ≃ α // ∀ x : {x // p x}, e' x = e x} is known.

Equations

• e.toCompl = Classical.choice ⋯

@[reducible, inline]

noncomputable abbrev Equiv.extendSubtype {α : Type u_1} [Finite α] {p : α → Prop} {q : α → Prop} [DecidablePred p] [DecidablePred q] (e : { x : α // p x } ≃ { x : α // q x }) : Equiv.Perm α

If e is an equivalence between two subtypes of a fintype α, e.extendSubtype is a permutation of α acting like e on the subtypes and doing something arbitrary outside.

Note that when p = q, Equiv.Perm.subtypeCongr e (Equiv.refl _) can be used instead.

Equations

• e.extendSubtype = e.subtypeCongr e.toCompl

theorem Equiv.extendSubtype_apply_of_mem {α : Type u_1} [Finite α] {p : α → Prop} {q : α → Prop} [DecidablePred p] [DecidablePred q] (e : { x : α // p x } ≃ { x : α // q x }) (x : α) (hx : p x) : e.extendSubtype x = ↑(e ⟨x, hx⟩)

theorem Equiv.extendSubtype_mem {α : Type u_1} [Finite α] {p : α → Prop} {q : α → Prop} [DecidablePred p] [DecidablePred q] (e : { x : α // p x } ≃ { x : α // q x }) (x : α) (hx : p x) : q (e.extendSubtype x)

theorem Equiv.extendSubtype_apply_of_not_mem {α : Type u_1} [Finite α] {p : α → Prop} {q : α → Prop} [DecidablePred p] [DecidablePred q] (e : { x : α // p x } ≃ { x : α // q x }) (x : α) (hx : ¬p x) : e.extendSubtype x = ↑(e.toCompl ⟨x, hx⟩)

theorem Equiv.extendSubtype_not_mem {α : Type u_1} [Finite α] {p : α → Prop} {q : α → Prop} [DecidablePred p] [DecidablePred q] (e : { x : α // p x } ≃ { x : α // q x }) (x : α) (hx : ¬p x) : ¬q (e.extendSubtype x)