Equivalences on embeddings

This file shows some advanced equivalences on embeddings, useful for constructing larger embeddings from smaller ones.

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def Equiv.sumEmbeddingEquivProdEmbeddingDisjoint {α : Type u_1} {β : Type u_2} {γ : Type u_3} : (α ⊕ β ↪ γ) ≃ { f // Disjoint (Set.range ↑f.fst) (Set.range ↑f.snd) }

Embeddings from a sum type are equivalent to two separate embeddings with disjoint ranges.

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def Equiv.codRestrict (α : Type u_1) {β : Type u_2} (bs : Set β) : { f // ∀ (a : α), ↑f a ∈ bs } ≃ (α ↪ ↑bs)

Embeddings whose range lies within a set are equivalent to embeddings to that set. This is Function.Embedding.codRestrict as an equiv.

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def Equiv.prodEmbeddingDisjointEquivSigmaEmbeddingRestricted {α : Type u_1} {β : Type u_2} {γ : Type u_3} : { f // Disjoint (Set.range ↑f.fst) (Set.range ↑f.snd) } ≃ (f : α ↪ γ) × (β ↪ ↑(Set.range ↑fᶜ))

Pairs of embeddings with disjoint ranges are equivalent to a dependent sum of embeddings, in which the second embedding cannot take values in the range of the first.

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def Equiv.sumEmbeddingEquivSigmaEmbeddingRestricted {α : Type u_1} {β : Type u_2} {γ : Type u_3} : (α ⊕ β ↪ γ) ≃ (f : α ↪ γ) × (β ↪ ↑(Set.range ↑fᶜ))

A combination of the above results, allowing us to turn one embedding over a sum type into two dependent embeddings, the second of which avoids any members of the range of the first. This is helpful for constructing larger embeddings out of smaller ones.

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• Equiv.sumEmbeddingEquivSigmaEmbeddingRestricted = Equiv.trans Equiv.sumEmbeddingEquivProdEmbeddingDisjoint Equiv.prodEmbeddingDisjointEquivSigmaEmbeddingRestricted

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def Equiv.uniqueEmbeddingEquivResult {α : Type u_1} {β : Type u_2} [inst : Unique α] : (α ↪ β) ≃ β

Embeddings from a single-member type are equivalent to members of the target type.

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