Partial equivalences

This files defines equivalences between subsets of given types. An element e of PartialEquiv α β is made of two maps e.toFun and e.invFun respectively from α to β and from β to α (just like equivs), which are inverse to each other on the subsets e.source and e.target of respectively α and β.

They are designed in particular to define charts on manifolds.

The main functionality is e.trans f, which composes the two partial equivalences by restricting the source and target to the maximal set where the composition makes sense.

As for equivs, we register a coercion to functions and use it in our simp normal form: we write e x and e.symm y instead of e.toFun x and e.invFun y.

Main definitions

• Equiv.toPartialEquiv: associating a partial equiv to an equiv, with source = target = univ
• PartialEquiv.symm: the inverse of a partial equivalence
• PartialEquiv.trans: the composition of two partial equivalences
• PartialEquiv.refl: the identity partial equivalence
• PartialEquiv.ofSet: the identity on a set s
• EqOnSource: equivalence relation describing the "right" notion of equality for partial equivalences (see below in implementation notes)

Implementation notes

There are at least three possible implementations of partial equivalences:

• equivs on subtypes
• pairs of functions taking values in Option α and Option β, equal to none where the partial equivalence is not defined
• pairs of functions defined everywhere, keeping the source and target as additional data

Each of these implementations has pros and cons.

• When dealing with subtypes, one still need to define additional API for composition and restriction of domains. Checking that one always belongs to the right subtype makes things very tedious, and leads quickly to DTT hell (as the subtype u ∩ v is not the "same" as v ∩ u, for instance).
• With option-valued functions, the composition is very neat (it is just the usual composition, and the domain is restricted automatically). These are implemented in PEquiv.lean. For manifolds, where one wants to discuss thoroughly the smoothness of the maps, this creates however a lot of overhead as one would need to extend all classes of smoothness to option-valued maps.
• The PartialEquiv version as explained above is easier to use for manifolds. The drawback is that there is extra useless data (the values of toFun and invFun outside of source and target). In particular, the equality notion between partial equivs is not "the right one", i.e., coinciding source and target and equality there. Moreover, there are no partial equivs in this sense between an empty type and a nonempty type. Since empty types are not that useful, and since one almost never needs to talk about equal partial equivs, this is not an issue in practice. Still, we introduce an equivalence relation EqOnSource that captures this right notion of equality, and show that many properties are invariant under this equivalence relation.

Local coding conventions

If a lemma deals with the intersection of a set with either source or target of a PartialEquiv, then it should use e.source ∩ s or e.target ∩ t, not s ∩ e.source or t ∩ e.target.

Implementation of the mfld_set_tac tactic for working with the domains of partially-defined functions (PartialEquiv, PartialHomeomorph, etc).

This is in a separate file from Mathlib.Logic.Equiv.MfldSimpsAttr because attributes need a new file to become functional.

def mfld_cfg : Simps.Config

Common @[simps] configuration options used for manifold-related declarations.

Equations

• One or more equations did not get rendered due to their size.

def Tactic.MfldSetTac.mfldSetTac : Lean.ParserDescr

A very basic tactic to show that sets showing up in manifolds coincide or are included in one another.

Equations

• Tactic.MfldSetTac.mfldSetTac = Lean.ParserDescr.node `Tactic.MfldSetTac.mfldSetTac 1024 (Lean.ParserDescr.nonReservedSymbol "mfld_set_tac" false)

structure PartialEquiv (α : Type u_5) (β : Type u_6) : Type (max u_5 u_6)

Local equivalence between subsets source and target of α and β respectively. The (global) maps toFun : α → β and invFun : β → α map source to target and conversely, and are inverse to each other there. The values of toFun outside of source and of invFun outside of target are irrelevant.

• toFun : α → β

  The global function which has a partial inverse. Its value outside of the source subset is irrelevant.

• invFun : β → α

  The partial inverse to toFun. Its value outside of the target subset is irrelevant.

• source : Set α

  The domain of the partial equivalence.

• target : Set β

  The codomain of the partial equivalence.

• map_source' : ∀ ⦃x : α⦄, x ∈ self.source → ↑self x ∈ self.target

  The proposition that elements of source are mapped to elements of target.

• map_target' : ∀ ⦃x : β⦄, x ∈ self.target → self.invFun x ∈ self.source

  The proposition that elements of target are mapped to elements of source.

• left_inv' : ∀ ⦃x : α⦄, x ∈ self.source → self.invFun (↑self x) = x

  The proposition that invFun is a left-inverse of toFun on source.

• right_inv' : ∀ ⦃x : β⦄, x ∈ self.target → ↑self (self.invFun x) = x

  The proposition that invFun is a right-inverse of toFun on target.

instance PartialEquiv.instInhabitedPartialEquiv {α : Type u_1} {β : Type u_2} [Inhabited α] [Inhabited β] : Inhabited (PartialEquiv α β)

Equations

• One or more equations did not get rendered due to their size.

def PartialEquiv.symm {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : PartialEquiv β α

The inverse of a partial equivalence

Equations

• PartialEquiv.symm e = { toFun := e.invFun, invFun := ↑e, source := e.target, target := e.source, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯ }

instance PartialEquiv.instCoeFunPartialEquivForAll {α : Type u_1} {β : Type u_2} : CoeFun (PartialEquiv α β) fun (x : PartialEquiv α β) => α → β

Equations

• PartialEquiv.instCoeFunPartialEquivForAll = { coe := PartialEquiv.toFun }

def PartialEquiv.Simps.symm_apply {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : β → α

See Note [custom simps projection]

Equations

• PartialEquiv.Simps.symm_apply e = ↑(PartialEquiv.symm e)

@[simp]

theorem PartialEquiv.coe_symm_mk {α : Type u_1} {β : Type u_2} (f : α → β) (g : β → α) (s : Set α) (t : Set β) (ml : ∀ ⦃x : α⦄, x ∈ s → f x ∈ t) (mr : ∀ ⦃x : β⦄, x ∈ t → g x ∈ s) (il : ∀ ⦃x : α⦄, x ∈ s → g (f x) = x) (ir : ∀ ⦃x : β⦄, x ∈ t → f (g x) = x) : ↑(PartialEquiv.symm { toFun := f, invFun := g, source := s, target := t, map_source' := ml, map_target' := mr, left_inv' := il, right_inv' := ir }) = g

@[simp]

theorem PartialEquiv.invFun_as_coe {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : e.invFun = ↑(PartialEquiv.symm e)

@[simp]

theorem PartialEquiv.map_source {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) {x : α} (h : x ∈ e.source) : ↑e x ∈ e.target

theorem PartialEquiv.map_source'' {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : ↑e '' e.source ⊆ e.target

Variant of e.map_source and map_source', stated for images of subsets of source.

@[simp]

theorem PartialEquiv.map_target {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) {x : β} (h : x ∈ e.target) : ↑(PartialEquiv.symm e) x ∈ e.source

@[simp]

theorem PartialEquiv.left_inv {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) {x : α} (h : x ∈ e.source) : ↑(PartialEquiv.symm e) (↑e x) = x

@[simp]

theorem PartialEquiv.right_inv {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) {x : β} (h : x ∈ e.target) : ↑e (↑(PartialEquiv.symm e) x) = x

theorem PartialEquiv.eq_symm_apply {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) {x : α} {y : β} (hx : x ∈ e.source) (hy : y ∈ e.target) : x = ↑(PartialEquiv.symm e) y ↔ ↑e x = y

theorem PartialEquiv.mapsTo {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : Set.MapsTo (↑e) e.source e.target

theorem PartialEquiv.symm_mapsTo {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : Set.MapsTo (↑(PartialEquiv.symm e)) e.target e.source

theorem PartialEquiv.leftInvOn {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : Set.LeftInvOn (↑(PartialEquiv.symm e)) (↑e) e.source

theorem PartialEquiv.rightInvOn {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : Set.RightInvOn (↑(PartialEquiv.symm e)) (↑e) e.target

theorem PartialEquiv.invOn {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : Set.InvOn (↑(PartialEquiv.symm e)) (↑e) e.source e.target

theorem PartialEquiv.injOn {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : Set.InjOn (↑e) e.source

theorem PartialEquiv.bijOn {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : Set.BijOn (↑e) e.source e.target

theorem PartialEquiv.surjOn {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : Set.SurjOn (↑e) e.source e.target

@[simp]

theorem Equiv.toPartialEquivOfImageEq_source {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) (t : Set β) (h : ⇑e '' s = t) : (Equiv.toPartialEquivOfImageEq e s t h).source = s

@[simp]

theorem Equiv.toPartialEquivOfImageEq_apply {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) (t : Set β) (h : ⇑e '' s = t) : ↑(Equiv.toPartialEquivOfImageEq e s t h) = ⇑e

@[simp]

theorem Equiv.toPartialEquivOfImageEq_symm_apply {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) (t : Set β) (h : ⇑e '' s = t) : ↑(PartialEquiv.symm (Equiv.toPartialEquivOfImageEq e s t h)) = ⇑e.symm

@[simp]

theorem Equiv.toPartialEquivOfImageEq_target {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) (t : Set β) (h : ⇑e '' s = t) : (Equiv.toPartialEquivOfImageEq e s t h).target = t

def Equiv.toPartialEquivOfImageEq {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) (t : Set β) (h : ⇑e '' s = t) : PartialEquiv α β

Interpret an Equiv as a PartialEquiv by restricting it to s in the domain and to t in the codomain.

Equations

• Equiv.toPartialEquivOfImageEq e s t h = { toFun := ⇑e, invFun := ⇑e.symm, source := s, target := t, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯ }

@[simp]

theorem Equiv.toPartialEquiv_target {α : Type u_1} {β : Type u_2} (e : α ≃ β) : (Equiv.toPartialEquiv e).target = Set.univ

@[simp]

theorem Equiv.toPartialEquiv_apply {α : Type u_1} {β : Type u_2} (e : α ≃ β) : ↑(Equiv.toPartialEquiv e) = ⇑e

@[simp]

theorem Equiv.toPartialEquiv_source {α : Type u_1} {β : Type u_2} (e : α ≃ β) : (Equiv.toPartialEquiv e).source = Set.univ

@[simp]

theorem Equiv.toPartialEquiv_symm_apply {α : Type u_1} {β : Type u_2} (e : α ≃ β) : ↑(PartialEquiv.symm (Equiv.toPartialEquiv e)) = ⇑e.symm

def Equiv.toPartialEquiv {α : Type u_1} {β : Type u_2} (e : α ≃ β) : PartialEquiv α β

Associate a PartialEquiv to an Equiv.

Equations

• Equiv.toPartialEquiv e = Equiv.toPartialEquivOfImageEq e Set.univ Set.univ ⋯

instance PartialEquiv.inhabitedOfEmpty {α : Type u_1} {β : Type u_2} [IsEmpty α] [IsEmpty β] : Inhabited (PartialEquiv α β)

Equations

• PartialEquiv.inhabitedOfEmpty = { default := Equiv.toPartialEquiv ((Equiv.equivEmpty α).trans (Equiv.equivEmpty β).symm) }

@[simp]

theorem PartialEquiv.copy_symm_apply {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (f : α → β) (hf : ↑e = f) (g : β → α) (hg : ↑(PartialEquiv.symm e) = g) (s : Set α) (hs : e.source = s) (t : Set β) (ht : e.target = t) : ↑(PartialEquiv.symm (PartialEquiv.copy e f hf g hg s hs t ht)) = g

@[simp]

theorem PartialEquiv.copy_apply {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (f : α → β) (hf : ↑e = f) (g : β → α) (hg : ↑(PartialEquiv.symm e) = g) (s : Set α) (hs : e.source = s) (t : Set β) (ht : e.target = t) : ↑(PartialEquiv.copy e f hf g hg s hs t ht) = f

@[simp]

theorem PartialEquiv.copy_source {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (f : α → β) (hf : ↑e = f) (g : β → α) (hg : ↑(PartialEquiv.symm e) = g) (s : Set α) (hs : e.source = s) (t : Set β) (ht : e.target = t) : (PartialEquiv.copy e f hf g hg s hs t ht).source = s

@[simp]

theorem PartialEquiv.copy_target {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (f : α → β) (hf : ↑e = f) (g : β → α) (hg : ↑(PartialEquiv.symm e) = g) (s : Set α) (hs : e.source = s) (t : Set β) (ht : e.target = t) : (PartialEquiv.copy e f hf g hg s hs t ht).target = t

def PartialEquiv.copy {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (f : α → β) (hf : ↑e = f) (g : β → α) (hg : ↑(PartialEquiv.symm e) = g) (s : Set α) (hs : e.source = s) (t : Set β) (ht : e.target = t) : PartialEquiv α β

Create a copy of a PartialEquiv providing better definitional equalities.

Equations

• PartialEquiv.copy e f hf g hg s hs t ht = { toFun := f, invFun := g, source := s, target := t, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯ }

theorem PartialEquiv.copy_eq {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (f : α → β) (hf : ↑e = f) (g : β → α) (hg : ↑(PartialEquiv.symm e) = g) (s : Set α) (hs : e.source = s) (t : Set β) (ht : e.target = t) : PartialEquiv.copy e f hf g hg s hs t ht = e

def PartialEquiv.toEquiv {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : ↑e.source ≃ ↑e.target

Associate to a PartialEquiv an Equiv between the source and the target.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem PartialEquiv.symm_source {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : (PartialEquiv.symm e).source = e.target

@[simp]

theorem PartialEquiv.symm_target {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : (PartialEquiv.symm e).target = e.source

@[simp]

theorem PartialEquiv.symm_symm {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : PartialEquiv.symm (PartialEquiv.symm e) = e

theorem PartialEquiv.symm_bijective {α : Type u_1} {β : Type u_2} : Function.Bijective PartialEquiv.symm

theorem PartialEquiv.image_source_eq_target {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : ↑e '' e.source = e.target

theorem PartialEquiv.forall_mem_target {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) {p : β → Prop} : (∀ y ∈ e.target, p y) ↔ ∀ x ∈ e.source, p (↑e x)

theorem PartialEquiv.exists_mem_target {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) {p : β → Prop} : (∃ y ∈ e.target, p y) ↔ ∃ x ∈ e.source, p (↑e x)

def PartialEquiv.IsImage {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (s : Set α) (t : Set β) : Prop

We say that t : Set β is an image of s : Set α under a partial equivalence if any of the following equivalent conditions hold:

• e '' (e.source ∩ s) = e.target ∩ t;
• e.source ∩ e ⁻¹ t = e.source ∩ s;
• ∀ x ∈ e.source, e x ∈ t ↔ x ∈ s (this one is used in the definition).

Equations

• PartialEquiv.IsImage e s t = ∀ ⦃x : α⦄, x ∈ e.source → (↑e x ∈ t ↔ x ∈ s)

theorem PartialEquiv.IsImage.apply_mem_iff {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} {x : α} (h : PartialEquiv.IsImage e s t) (hx : x ∈ e.source) : ↑e x ∈ t ↔ x ∈ s

theorem PartialEquiv.IsImage.symm_apply_mem_iff {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} (h : PartialEquiv.IsImage e s t) ⦃y : β⦄ : y ∈ e.target → (↑(PartialEquiv.symm e) y ∈ s ↔ y ∈ t)

theorem PartialEquiv.IsImage.symm {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} (h : PartialEquiv.IsImage e s t) : PartialEquiv.IsImage (PartialEquiv.symm e) t s

@[simp]

theorem PartialEquiv.IsImage.symm_iff {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} : PartialEquiv.IsImage (PartialEquiv.symm e) t s ↔ PartialEquiv.IsImage e s t

theorem PartialEquiv.IsImage.mapsTo {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} (h : PartialEquiv.IsImage e s t) : Set.MapsTo (↑e) (e.source ∩ s) (e.target ∩ t)

theorem PartialEquiv.IsImage.symm_mapsTo {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} (h : PartialEquiv.IsImage e s t) : Set.MapsTo (↑(PartialEquiv.symm e)) (e.target ∩ t) (e.source ∩ s)

@[simp]

theorem PartialEquiv.IsImage.restr_apply {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} (h : PartialEquiv.IsImage e s t) : ↑(PartialEquiv.IsImage.restr h) = ↑e

@[simp]

theorem PartialEquiv.IsImage.restr_target {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} (h : PartialEquiv.IsImage e s t) : (PartialEquiv.IsImage.restr h).target = e.target ∩ t

@[simp]

theorem PartialEquiv.IsImage.restr_symm_apply {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} (h : PartialEquiv.IsImage e s t) : ↑(PartialEquiv.symm (PartialEquiv.IsImage.restr h)) = ↑(PartialEquiv.symm e)

@[simp]

theorem PartialEquiv.IsImage.restr_source {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} (h : PartialEquiv.IsImage e s t) : (PartialEquiv.IsImage.restr h).source = e.source ∩ s

def PartialEquiv.IsImage.restr {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} (h : PartialEquiv.IsImage e s t) : PartialEquiv α β

Restrict a PartialEquiv to a pair of corresponding sets.

Equations

• One or more equations did not get rendered due to their size.

theorem PartialEquiv.IsImage.image_eq {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} (h : PartialEquiv.IsImage e s t) : ↑e '' (e.source ∩ s) = e.target ∩ t

theorem PartialEquiv.IsImage.symm_image_eq {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} (h : PartialEquiv.IsImage e s t) : ↑(PartialEquiv.symm e) '' (e.target ∩ t) = e.source ∩ s

theorem PartialEquiv.IsImage.iff_preimage_eq {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} : PartialEquiv.IsImage e s t ↔ e.source ∩ ↑e ⁻¹' t = e.source ∩ s

theorem PartialEquiv.IsImage.preimage_eq {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} : PartialEquiv.IsImage e s t → e.source ∩ ↑e ⁻¹' t = e.source ∩ s

Alias of the forward direction of PartialEquiv.IsImage.iff_preimage_eq.

theorem PartialEquiv.IsImage.of_preimage_eq {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} : e.source ∩ ↑e ⁻¹' t = e.source ∩ s → PartialEquiv.IsImage e s t

Alias of the reverse direction of PartialEquiv.IsImage.iff_preimage_eq.

theorem PartialEquiv.IsImage.iff_symm_preimage_eq {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} : PartialEquiv.IsImage e s t ↔ e.target ∩ ↑(PartialEquiv.symm e) ⁻¹' s = e.target ∩ t

theorem PartialEquiv.IsImage.symm_preimage_eq {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} : PartialEquiv.IsImage e s t → e.target ∩ ↑(PartialEquiv.symm e) ⁻¹' s = e.target ∩ t

Alias of the forward direction of PartialEquiv.IsImage.iff_symm_preimage_eq.

theorem PartialEquiv.IsImage.of_symm_preimage_eq {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} : e.target ∩ ↑(PartialEquiv.symm e) ⁻¹' s = e.target ∩ t → PartialEquiv.IsImage e s t

Alias of the reverse direction of PartialEquiv.IsImage.iff_symm_preimage_eq.

theorem PartialEquiv.IsImage.of_image_eq {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} (h : ↑e '' (e.source ∩ s) = e.target ∩ t) : PartialEquiv.IsImage e s t

theorem PartialEquiv.IsImage.of_symm_image_eq {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} (h : ↑(PartialEquiv.symm e) '' (e.target ∩ t) = e.source ∩ s) : PartialEquiv.IsImage e s t

theorem PartialEquiv.IsImage.compl {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} (h : PartialEquiv.IsImage e s t) : PartialEquiv.IsImage e sᶜ tᶜ

theorem PartialEquiv.IsImage.inter {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} {s' : Set α} {t' : Set β} (h : PartialEquiv.IsImage e s t) (h' : PartialEquiv.IsImage e s' t') : PartialEquiv.IsImage e (s ∩ s') (t ∩ t')

theorem PartialEquiv.IsImage.union {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} {s' : Set α} {t' : Set β} (h : PartialEquiv.IsImage e s t) (h' : PartialEquiv.IsImage e s' t') : PartialEquiv.IsImage e (s ∪ s') (t ∪ t')

theorem PartialEquiv.IsImage.diff {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} {s' : Set α} {t' : Set β} (h : PartialEquiv.IsImage e s t) (h' : PartialEquiv.IsImage e s' t') : PartialEquiv.IsImage e (s \ s') (t \ t')

theorem PartialEquiv.IsImage.leftInvOn_piecewise {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} {e' : PartialEquiv α β} [(i : α) → Decidable (i ∈ s)] [(i : β) → Decidable (i ∈ t)] (h : PartialEquiv.IsImage e s t) (h' : PartialEquiv.IsImage e' s t) : Set.LeftInvOn (Set.piecewise t ↑(PartialEquiv.symm e) ↑(PartialEquiv.symm e')) (Set.piecewise s ↑e ↑e') (Set.ite s e.source e'.source)

theorem PartialEquiv.IsImage.inter_eq_of_inter_eq_of_eqOn {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} {e' : PartialEquiv α β} (h : PartialEquiv.IsImage e s t) (h' : PartialEquiv.IsImage e' s t) (hs : e.source ∩ s = e'.source ∩ s) (heq : Set.EqOn (↑e) (↑e') (e.source ∩ s)) : e.target ∩ t = e'.target ∩ t

theorem PartialEquiv.IsImage.symm_eq_on_of_inter_eq_of_eqOn {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} {e' : PartialEquiv α β} (h : PartialEquiv.IsImage e s t) (hs : e.source ∩ s = e'.source ∩ s) (heq : Set.EqOn (↑e) (↑e') (e.source ∩ s)) : Set.EqOn (↑(PartialEquiv.symm e)) (↑(PartialEquiv.symm e')) (e.target ∩ t)

theorem PartialEquiv.isImage_source_target {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : PartialEquiv.IsImage e e.source e.target

theorem PartialEquiv.isImage_source_target_of_disjoint {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (e' : PartialEquiv α β) (hs : Disjoint e.source e'.source) (ht : Disjoint e.target e'.target) : PartialEquiv.IsImage e e'.source e'.target

theorem PartialEquiv.image_source_inter_eq' {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (s : Set α) : ↑e '' (e.source ∩ s) = e.target ∩ ↑(PartialEquiv.symm e) ⁻¹' s

theorem PartialEquiv.image_source_inter_eq {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (s : Set α) : ↑e '' (e.source ∩ s) = e.target ∩ ↑(PartialEquiv.symm e) ⁻¹' (e.source ∩ s)

theorem PartialEquiv.image_eq_target_inter_inv_preimage {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) {s : Set α} (h : s ⊆ e.source) : ↑e '' s = e.target ∩ ↑(PartialEquiv.symm e) ⁻¹' s

theorem PartialEquiv.symm_image_eq_source_inter_preimage {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) {s : Set β} (h : s ⊆ e.target) : ↑(PartialEquiv.symm e) '' s = e.source ∩ ↑e ⁻¹' s

theorem PartialEquiv.symm_image_target_inter_eq {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (s : Set β) : ↑(PartialEquiv.symm e) '' (e.target ∩ s) = e.source ∩ ↑e ⁻¹' (e.target ∩ s)

theorem PartialEquiv.symm_image_target_inter_eq' {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (s : Set β) : ↑(PartialEquiv.symm e) '' (e.target ∩ s) = e.source ∩ ↑e ⁻¹' s

theorem PartialEquiv.source_inter_preimage_inv_preimage {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (s : Set α) : e.source ∩ ↑e ⁻¹' (↑(PartialEquiv.symm e) ⁻¹' s) = e.source ∩ s

theorem PartialEquiv.source_inter_preimage_target_inter {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (s : Set β) : e.source ∩ ↑e ⁻¹' (e.target ∩ s) = e.source ∩ ↑e ⁻¹' s

theorem PartialEquiv.target_inter_inv_preimage_preimage {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (s : Set β) : e.target ∩ ↑(PartialEquiv.symm e) ⁻¹' (↑e ⁻¹' s) = e.target ∩ s

theorem PartialEquiv.symm_image_image_of_subset_source {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) {s : Set α} (h : s ⊆ e.source) : ↑(PartialEquiv.symm e) '' (↑e '' s) = s

theorem PartialEquiv.image_symm_image_of_subset_target {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) {s : Set β} (h : s ⊆ e.target) : ↑e '' (↑(PartialEquiv.symm e) '' s) = s

theorem PartialEquiv.source_subset_preimage_target {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : e.source ⊆ ↑e ⁻¹' e.target

theorem PartialEquiv.symm_image_target_eq_source {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : ↑(PartialEquiv.symm e) '' e.target = e.source

theorem PartialEquiv.target_subset_preimage_source {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : e.target ⊆ ↑(PartialEquiv.symm e) ⁻¹' e.source

theorem PartialEquiv.ext {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {e' : PartialEquiv α β} (h : ∀ (x : α), ↑e x = ↑e' x) (hsymm : ∀ (x : β), ↑(PartialEquiv.symm e) x = ↑(PartialEquiv.symm e') x) (hs : e.source = e'.source) : e = e'

Two partial equivs that have the same source, same toFun and same invFun, coincide.

def PartialEquiv.restr {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (s : Set α) : PartialEquiv α β

Restricting a partial equivalence to e.source ∩ s

Equations

• PartialEquiv.restr e s = PartialEquiv.IsImage.restr ⋯

@[simp]

theorem PartialEquiv.restr_coe {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (s : Set α) : ↑(PartialEquiv.restr e s) = ↑e

@[simp]

theorem PartialEquiv.restr_coe_symm {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (s : Set α) : ↑(PartialEquiv.symm (PartialEquiv.restr e s)) = ↑(PartialEquiv.symm e)

@[simp]

theorem PartialEquiv.restr_source {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (s : Set α) : (PartialEquiv.restr e s).source = e.source ∩ s

@[simp]

theorem PartialEquiv.restr_target {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (s : Set α) : (PartialEquiv.restr e s).target = e.target ∩ ↑(PartialEquiv.symm e) ⁻¹' s

theorem PartialEquiv.restr_eq_of_source_subset {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} (h : e.source ⊆ s) : PartialEquiv.restr e s = e

@[simp]

theorem PartialEquiv.restr_univ {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} : PartialEquiv.restr e Set.univ = e

def PartialEquiv.refl (α : Type u_5) : PartialEquiv α α

The identity partial equiv

Equations

• PartialEquiv.refl α = Equiv.toPartialEquiv (Equiv.refl α)

@[simp]

theorem PartialEquiv.refl_source {α : Type u_1} : (PartialEquiv.refl α).source = Set.univ

@[simp]

theorem PartialEquiv.refl_target {α : Type u_1} : (PartialEquiv.refl α).target = Set.univ

@[simp]

theorem PartialEquiv.refl_coe {α : Type u_1} : ↑(PartialEquiv.refl α) = id

@[simp]

theorem PartialEquiv.refl_symm {α : Type u_1} : PartialEquiv.symm (PartialEquiv.refl α) = PartialEquiv.refl α

theorem PartialEquiv.refl_restr_source {α : Type u_1} (s : Set α) : (PartialEquiv.restr (PartialEquiv.refl α) s).source = s

theorem PartialEquiv.refl_restr_target {α : Type u_1} (s : Set α) : (PartialEquiv.restr (PartialEquiv.refl α) s).target = s

def PartialEquiv.ofSet {α : Type u_1} (s : Set α) : PartialEquiv α α

The identity partial equivalence on a set s

Equations

• PartialEquiv.ofSet s = { toFun := id, invFun := id, source := s, target := s, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯ }

@[simp]

theorem PartialEquiv.ofSet_source {α : Type u_1} (s : Set α) : (PartialEquiv.ofSet s).source = s

@[simp]

theorem PartialEquiv.ofSet_target {α : Type u_1} (s : Set α) : (PartialEquiv.ofSet s).target = s

@[simp]

theorem PartialEquiv.ofSet_coe {α : Type u_1} (s : Set α) : ↑(PartialEquiv.ofSet s) = id

@[simp]

theorem PartialEquiv.ofSet_symm {α : Type u_1} (s : Set α) : PartialEquiv.symm (PartialEquiv.ofSet s) = PartialEquiv.ofSet s

@[simp]

theorem PartialEquiv.trans'_source {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) (h : e.target = e'.source) : (PartialEquiv.trans' e e' h).source = e.source

@[simp]

theorem PartialEquiv.trans'_symm_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) (h : e.target = e'.source) : ∀ (a : γ), ↑(PartialEquiv.symm (PartialEquiv.trans' e e' h)) a = (↑(PartialEquiv.symm e) ∘ ↑(PartialEquiv.symm e')) a

@[simp]

theorem PartialEquiv.trans'_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) (h : e.target = e'.source) : ∀ (a : α), ↑(PartialEquiv.trans' e e' h) a = (↑e' ∘ ↑e) a

@[simp]

theorem PartialEquiv.trans'_target {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) (h : e.target = e'.source) : (PartialEquiv.trans' e e' h).target = e'.target

def PartialEquiv.trans' {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) (h : e.target = e'.source) : PartialEquiv α γ

Composing two partial equivs if the target of the first coincides with the source of the second.

Equations

• One or more equations did not get rendered due to their size.

def PartialEquiv.trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) : PartialEquiv α γ

Composing two partial equivs, by restricting to the maximal domain where their composition is well defined.

Equations

• PartialEquiv.trans e e' = PartialEquiv.trans' (PartialEquiv.symm (PartialEquiv.restr (PartialEquiv.symm e) e'.source)) (PartialEquiv.restr e' e.target) ⋯

@[simp]

theorem PartialEquiv.coe_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) : ↑(PartialEquiv.trans e e') = ↑e' ∘ ↑e

@[simp]

theorem PartialEquiv.coe_trans_symm {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) : ↑(PartialEquiv.symm (PartialEquiv.trans e e')) = ↑(PartialEquiv.symm e) ∘ ↑(PartialEquiv.symm e')

theorem PartialEquiv.trans_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) {x : α} : ↑(PartialEquiv.trans e e') x = ↑e' (↑e x)

theorem PartialEquiv.trans_symm_eq_symm_trans_symm {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) : PartialEquiv.symm (PartialEquiv.trans e e') = PartialEquiv.trans (PartialEquiv.symm e') (PartialEquiv.symm e)

@[simp]

theorem PartialEquiv.trans_source {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) : (PartialEquiv.trans e e').source = e.source ∩ ↑e ⁻¹' e'.source

theorem PartialEquiv.trans_source' {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) : (PartialEquiv.trans e e').source = e.source ∩ ↑e ⁻¹' (e.target ∩ e'.source)

theorem PartialEquiv.trans_source'' {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) : (PartialEquiv.trans e e').source = ↑(PartialEquiv.symm e) '' (e.target ∩ e'.source)

theorem PartialEquiv.image_trans_source {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) : ↑e '' (PartialEquiv.trans e e').source = e.target ∩ e'.source

@[simp]

theorem PartialEquiv.trans_target {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) : (PartialEquiv.trans e e').target = e'.target ∩ ↑(PartialEquiv.symm e') ⁻¹' e.target

theorem PartialEquiv.trans_target' {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) : (PartialEquiv.trans e e').target = e'.target ∩ ↑(PartialEquiv.symm e') ⁻¹' (e'.source ∩ e.target)

theorem PartialEquiv.trans_target'' {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) : (PartialEquiv.trans e e').target = ↑e' '' (e'.source ∩ e.target)

theorem PartialEquiv.inv_image_trans_target {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) : ↑(PartialEquiv.symm e') '' (PartialEquiv.trans e e').target = e'.source ∩ e.target

theorem PartialEquiv.trans_assoc {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (e : PartialEquiv α β) (e' : PartialEquiv β γ) (e'' : PartialEquiv γ δ) : PartialEquiv.trans (PartialEquiv.trans e e') e'' = PartialEquiv.trans e (PartialEquiv.trans e' e'')

@[simp]

theorem PartialEquiv.trans_refl {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : PartialEquiv.trans e (PartialEquiv.refl β) = e

@[simp]

theorem PartialEquiv.refl_trans {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : PartialEquiv.trans (PartialEquiv.refl α) e = e

theorem PartialEquiv.trans_ofSet {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (s : Set β) : PartialEquiv.trans e (PartialEquiv.ofSet s) = PartialEquiv.restr e (↑e ⁻¹' s)

theorem PartialEquiv.trans_refl_restr {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (s : Set β) : PartialEquiv.trans e (PartialEquiv.restr (PartialEquiv.refl β) s) = PartialEquiv.restr e (↑e ⁻¹' s)

theorem PartialEquiv.trans_refl_restr' {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (s : Set β) : PartialEquiv.trans e (PartialEquiv.restr (PartialEquiv.refl β) s) = PartialEquiv.restr e (e.source ∩ ↑e ⁻¹' s)

theorem PartialEquiv.restr_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : PartialEquiv β γ) (s : Set α) : PartialEquiv.trans (PartialEquiv.restr e s) e' = PartialEquiv.restr (PartialEquiv.trans e e') s

theorem PartialEquiv.mem_symm_trans_source {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) {e' : PartialEquiv α γ} {x : α} (he : x ∈ e.source) (he' : x ∈ e'.source) : ↑e x ∈ (PartialEquiv.trans (PartialEquiv.symm e) e').source

A lemma commonly useful when e and e' are charts of a manifold.

def PartialEquiv.EqOnSource {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (e' : PartialEquiv α β) : Prop

EqOnSource e e' means that e and e' have the same source, and coincide there. Then e and e' should really be considered the same partial equiv.

Equations

• PartialEquiv.EqOnSource e e' = (e.source = e'.source ∧ Set.EqOn (↑e) (↑e') e.source)

instance PartialEquiv.eqOnSourceSetoid {α : Type u_1} {β : Type u_2} : Setoid (PartialEquiv α β)

EqOnSource is an equivalence relation. This instance provides the ≈ notation between two PartialEquivs.

Equations

• PartialEquiv.eqOnSourceSetoid = { r := PartialEquiv.EqOnSource, iseqv := ⋯ }

theorem PartialEquiv.eqOnSource_refl {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : e ≈ e

theorem PartialEquiv.EqOnSource.source_eq {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {e' : PartialEquiv α β} (h : e ≈ e') : e.source = e'.source

Two equivalent partial equivs have the same source.

theorem PartialEquiv.EqOnSource.eqOn {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {e' : PartialEquiv α β} (h : e ≈ e') : Set.EqOn (↑e) (↑e') e.source

Two equivalent partial equivs coincide on the source.

theorem PartialEquiv.EqOnSource.target_eq {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {e' : PartialEquiv α β} (h : e ≈ e') : e.target = e'.target

Two equivalent partial equivs have the same target.

theorem PartialEquiv.EqOnSource.symm' {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {e' : PartialEquiv α β} (h : e ≈ e') : PartialEquiv.symm e ≈ PartialEquiv.symm e'

If two partial equivs are equivalent, so are their inverses.

theorem PartialEquiv.EqOnSource.symm_eqOn {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {e' : PartialEquiv α β} (h : e ≈ e') : Set.EqOn (↑(PartialEquiv.symm e)) (↑(PartialEquiv.symm e')) e.target

Two equivalent partial equivs have coinciding inverses on the target.

theorem PartialEquiv.EqOnSource.trans' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {e : PartialEquiv α β} {e' : PartialEquiv α β} {f : PartialEquiv β γ} {f' : PartialEquiv β γ} (he : e ≈ e') (hf : f ≈ f') : PartialEquiv.trans e f ≈ PartialEquiv.trans e' f'

Composition of partial equivs respects equivalence.

theorem PartialEquiv.EqOnSource.restr {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {e' : PartialEquiv α β} (he : e ≈ e') (s : Set α) : PartialEquiv.restr e s ≈ PartialEquiv.restr e' s

Restriction of partial equivs respects equivalence.

theorem PartialEquiv.EqOnSource.source_inter_preimage_eq {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {e' : PartialEquiv α β} (he : e ≈ e') (s : Set β) : e.source ∩ ↑e ⁻¹' s = e'.source ∩ ↑e' ⁻¹' s

Preimages are respected by equivalence.

theorem PartialEquiv.self_trans_symm {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : PartialEquiv.trans e (PartialEquiv.symm e) ≈ PartialEquiv.ofSet e.source

Composition of a partial equivlance and its inverse is equivalent to the restriction of the identity to the source.

theorem PartialEquiv.symm_trans_self {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) : PartialEquiv.trans (PartialEquiv.symm e) e ≈ PartialEquiv.ofSet e.target

Composition of the inverse of a partial equivalence and this partial equivalence is equivalent to the restriction of the identity to the target.

theorem PartialEquiv.eq_of_eqOnSource_univ {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (e' : PartialEquiv α β) (h : e ≈ e') (s : e.source = Set.univ) (t : e.target = Set.univ) : e = e'

Two equivalent partial equivs are equal when the source and target are univ.

def PartialEquiv.prod {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (e : PartialEquiv α β) (e' : PartialEquiv γ δ) : PartialEquiv (α × γ) (β × δ)

The product of two partial equivalences, as a partial equivalence on the product.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem PartialEquiv.prod_source {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (e : PartialEquiv α β) (e' : PartialEquiv γ δ) : (PartialEquiv.prod e e').source = e.source ×ˢ e'.source

@[simp]

theorem PartialEquiv.prod_target {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (e : PartialEquiv α β) (e' : PartialEquiv γ δ) : (PartialEquiv.prod e e').target = e.target ×ˢ e'.target

@[simp]

theorem PartialEquiv.prod_coe {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (e : PartialEquiv α β) (e' : PartialEquiv γ δ) : ↑(PartialEquiv.prod e e') = fun (p : α × γ) => (↑e p.1, ↑e' p.2)

theorem PartialEquiv.prod_coe_symm {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (e : PartialEquiv α β) (e' : PartialEquiv γ δ) : ↑(PartialEquiv.symm (PartialEquiv.prod e e')) = fun (p : β × δ) => (↑(PartialEquiv.symm e) p.1, ↑(PartialEquiv.symm e') p.2)

@[simp]

theorem PartialEquiv.prod_symm {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (e : PartialEquiv α β) (e' : PartialEquiv γ δ) : PartialEquiv.symm (PartialEquiv.prod e e') = PartialEquiv.prod (PartialEquiv.symm e) (PartialEquiv.symm e')

@[simp]

theorem PartialEquiv.refl_prod_refl {α : Type u_1} {β : Type u_2} : PartialEquiv.prod (PartialEquiv.refl α) (PartialEquiv.refl β) = PartialEquiv.refl (α × β)

@[simp]

theorem PartialEquiv.prod_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {η : Type u_5} {ε : Type u_6} (e : PartialEquiv α β) (f : PartialEquiv β γ) (e' : PartialEquiv δ η) (f' : PartialEquiv η ε) : PartialEquiv.trans (PartialEquiv.prod e e') (PartialEquiv.prod f f') = PartialEquiv.prod (PartialEquiv.trans e f) (PartialEquiv.trans e' f')

@[simp]

theorem PartialEquiv.piecewise_symm_apply {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (e' : PartialEquiv α β) (s : Set α) (t : Set β) [(x : α) → Decidable (x ∈ s)] [(y : β) → Decidable (y ∈ t)] (H : PartialEquiv.IsImage e s t) (H' : PartialEquiv.IsImage e' s t) : ↑(PartialEquiv.symm (PartialEquiv.piecewise e e' s t H H')) = Set.piecewise t ↑(PartialEquiv.symm e) ↑(PartialEquiv.symm e')

@[simp]

theorem PartialEquiv.piecewise_target {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (e' : PartialEquiv α β) (s : Set α) (t : Set β) [(x : α) → Decidable (x ∈ s)] [(y : β) → Decidable (y ∈ t)] (H : PartialEquiv.IsImage e s t) (H' : PartialEquiv.IsImage e' s t) : (PartialEquiv.piecewise e e' s t H H').target = Set.ite t e.target e'.target

@[simp]

theorem PartialEquiv.piecewise_source {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (e' : PartialEquiv α β) (s : Set α) (t : Set β) [(x : α) → Decidable (x ∈ s)] [(y : β) → Decidable (y ∈ t)] (H : PartialEquiv.IsImage e s t) (H' : PartialEquiv.IsImage e' s t) : (PartialEquiv.piecewise e e' s t H H').source = Set.ite s e.source e'.source

@[simp]

theorem PartialEquiv.piecewise_apply {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (e' : PartialEquiv α β) (s : Set α) (t : Set β) [(x : α) → Decidable (x ∈ s)] [(y : β) → Decidable (y ∈ t)] (H : PartialEquiv.IsImage e s t) (H' : PartialEquiv.IsImage e' s t) : ↑(PartialEquiv.piecewise e e' s t H H') = Set.piecewise s ↑e ↑e'

def PartialEquiv.piecewise {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (e' : PartialEquiv α β) (s : Set α) (t : Set β) [(x : α) → Decidable (x ∈ s)] [(y : β) → Decidable (y ∈ t)] (H : PartialEquiv.IsImage e s t) (H' : PartialEquiv.IsImage e' s t) : PartialEquiv α β

Combine two PartialEquivs using Set.piecewise. The source of the new PartialEquiv is s.ite e.source e'.source = e.source ∩ s ∪ e'.source \ s, and similarly for target. The function sends e.source ∩ s to e.target ∩ t using e and e'.source \ s to e'.target \ t using e', and similarly for the inverse function. The definition assumes e.isImage s t and e'.isImage s t.

Equations

• One or more equations did not get rendered due to their size.

theorem PartialEquiv.symm_piecewise {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (e' : PartialEquiv α β) {s : Set α} {t : Set β} [(x : α) → Decidable (x ∈ s)] [(y : β) → Decidable (y ∈ t)] (H : PartialEquiv.IsImage e s t) (H' : PartialEquiv.IsImage e' s t) : PartialEquiv.symm (PartialEquiv.piecewise e e' s t H H') = PartialEquiv.piecewise (PartialEquiv.symm e) (PartialEquiv.symm e') t s ⋯ ⋯

@[simp]

theorem PartialEquiv.disjointUnion_apply {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (e' : PartialEquiv α β) (hs : Disjoint e.source e'.source) (ht : Disjoint e.target e'.target) [(x : α) → Decidable (x ∈ e.source)] [(y : β) → Decidable (y ∈ e.target)] : ↑(PartialEquiv.disjointUnion e e' hs ht) = Set.piecewise e.source ↑e ↑e'

@[simp]

theorem PartialEquiv.disjointUnion_target {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (e' : PartialEquiv α β) (hs : Disjoint e.source e'.source) (ht : Disjoint e.target e'.target) [(x : α) → Decidable (x ∈ e.source)] [(y : β) → Decidable (y ∈ e.target)] : (PartialEquiv.disjointUnion e e' hs ht).target = e.target ∪ e'.target

@[simp]

theorem PartialEquiv.disjointUnion_symm_apply {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (e' : PartialEquiv α β) (hs : Disjoint e.source e'.source) (ht : Disjoint e.target e'.target) [(x : α) → Decidable (x ∈ e.source)] [(y : β) → Decidable (y ∈ e.target)] : ↑(PartialEquiv.symm (PartialEquiv.disjointUnion e e' hs ht)) = Set.piecewise e.target ↑(PartialEquiv.symm e) ↑(PartialEquiv.symm e')

@[simp]

theorem PartialEquiv.disjointUnion_source {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (e' : PartialEquiv α β) (hs : Disjoint e.source e'.source) (ht : Disjoint e.target e'.target) [(x : α) → Decidable (x ∈ e.source)] [(y : β) → Decidable (y ∈ e.target)] : (PartialEquiv.disjointUnion e e' hs ht).source = e.source ∪ e'.source

def PartialEquiv.disjointUnion {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (e' : PartialEquiv α β) (hs : Disjoint e.source e'.source) (ht : Disjoint e.target e'.target) [(x : α) → Decidable (x ∈ e.source)] [(y : β) → Decidable (y ∈ e.target)] : PartialEquiv α β

Combine two PartialEquivs with disjoint sources and disjoint targets. We reuse PartialEquiv.piecewise, then override source and target to ensure better definitional equalities.

Equations

• One or more equations did not get rendered due to their size.

theorem PartialEquiv.disjointUnion_eq_piecewise {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) (e' : PartialEquiv α β) (hs : Disjoint e.source e'.source) (ht : Disjoint e.target e'.target) [(x : α) → Decidable (x ∈ e.source)] [(y : β) → Decidable (y ∈ e.target)] : PartialEquiv.disjointUnion e e' hs ht = PartialEquiv.piecewise e e' e.source e.target ⋯ ⋯

@[simp]

theorem PartialEquiv.pi_apply {ι : Type u_5} {αi : ι → Type u_6} {βi : ι → Type u_7} (ei : (i : ι) → PartialEquiv (αi i) (βi i)) : ↑(PartialEquiv.pi ei) = fun (f : (i : ι) → αi i) (i : ι) => ↑(ei i) (f i)

@[simp]

theorem PartialEquiv.pi_target {ι : Type u_5} {αi : ι → Type u_6} {βi : ι → Type u_7} (ei : (i : ι) → PartialEquiv (αi i) (βi i)) : (PartialEquiv.pi ei).target = Set.pi Set.univ fun (i : ι) => (ei i).target

@[simp]

theorem PartialEquiv.pi_source {ι : Type u_5} {αi : ι → Type u_6} {βi : ι → Type u_7} (ei : (i : ι) → PartialEquiv (αi i) (βi i)) : (PartialEquiv.pi ei).source = Set.pi Set.univ fun (i : ι) => (ei i).source

def PartialEquiv.pi {ι : Type u_5} {αi : ι → Type u_6} {βi : ι → Type u_7} (ei : (i : ι) → PartialEquiv (αi i) (βi i)) : PartialEquiv ((i : ι) → αi i) ((i : ι) → βi i)

The product of a family of partial equivalences, as a partial equivalence on the pi type.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem PartialEquiv.pi_symm {ι : Type u_5} {αi : ι → Type u_6} {βi : ι → Type u_7} (ei : (i : ι) → PartialEquiv (αi i) (βi i)) : PartialEquiv.symm (PartialEquiv.pi ei) = PartialEquiv.pi fun (i : ι) => PartialEquiv.symm (ei i)

theorem PartialEquiv.pi_symm_apply {ι : Type u_5} {αi : ι → Type u_6} {βi : ι → Type u_7} (ei : (i : ι) → PartialEquiv (αi i) (βi i)) : ↑(PartialEquiv.symm (PartialEquiv.pi ei)) = fun (f : (i : ι) → βi i) (i : ι) => ↑(PartialEquiv.symm (ei i)) (f i)

@[simp]

theorem PartialEquiv.pi_refl {ι : Type u_5} {αi : ι → Type u_6} : (PartialEquiv.pi fun (i : ι) => PartialEquiv.refl (αi i)) = PartialEquiv.refl ((i : ι) → αi i)

@[simp]

theorem PartialEquiv.pi_trans {ι : Type u_5} {αi : ι → Type u_6} {βi : ι → Type u_7} {γi : ι → Type u_8} (ei : (i : ι) → PartialEquiv (αi i) (βi i)) (ei' : (i : ι) → PartialEquiv (βi i) (γi i)) : PartialEquiv.trans (PartialEquiv.pi ei) (PartialEquiv.pi ei') = PartialEquiv.pi fun (i : ι) => PartialEquiv.trans (ei i) (ei' i)

@[simp]

theorem Set.BijOn.toPartialEquiv_source {α : Type u_1} {β : Type u_2} [Nonempty α] (f : α → β) (s : Set α) (t : Set β) (hf : Set.BijOn f s t) : (Set.BijOn.toPartialEquiv f s t hf).source = s

@[simp]

theorem Set.BijOn.toPartialEquiv_symm_apply {α : Type u_1} {β : Type u_2} [Nonempty α] (f : α → β) (s : Set α) (t : Set β) (hf : Set.BijOn f s t) : ↑(PartialEquiv.symm (Set.BijOn.toPartialEquiv f s t hf)) = Function.invFunOn f s

@[simp]

theorem Set.BijOn.toPartialEquiv_apply {α : Type u_1} {β : Type u_2} [Nonempty α] (f : α → β) (s : Set α) (t : Set β) (hf : Set.BijOn f s t) : ↑(Set.BijOn.toPartialEquiv f s t hf) = f

@[simp]

theorem Set.BijOn.toPartialEquiv_target {α : Type u_1} {β : Type u_2} [Nonempty α] (f : α → β) (s : Set α) (t : Set β) (hf : Set.BijOn f s t) : (Set.BijOn.toPartialEquiv f s t hf).target = t

noncomputable def Set.BijOn.toPartialEquiv {α : Type u_1} {β : Type u_2} [Nonempty α] (f : α → β) (s : Set α) (t : Set β) (hf : Set.BijOn f s t) : PartialEquiv α β

A bijection between two sets s : Set α and t : Set β provides a partial equivalence between α and β.

Equations

• Set.BijOn.toPartialEquiv f s t hf = { toFun := f, invFun := Function.invFunOn f s, source := s, target := t, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯ }

noncomputable def Set.InjOn.toPartialEquiv {α : Type u_1} {β : Type u_2} [Nonempty α] (f : α → β) (s : Set α) (hf : Set.InjOn f s) : PartialEquiv α β

A map injective on a subset of its domain provides a partial equivalence.

Equations

• Set.InjOn.toPartialEquiv f s hf = Set.BijOn.toPartialEquiv f s (f '' s) ⋯

@[simp]

theorem Equiv.refl_toPartialEquiv {α : Type u_1} : Equiv.toPartialEquiv (Equiv.refl α) = PartialEquiv.refl α

@[simp]

theorem Equiv.symm_toPartialEquiv {α : Type u_1} {β : Type u_2} (e : α ≃ β) : Equiv.toPartialEquiv e.symm = PartialEquiv.symm (Equiv.toPartialEquiv e)

@[simp]

theorem Equiv.trans_toPartialEquiv {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : α ≃ β) (e' : β ≃ γ) : Equiv.toPartialEquiv (e.trans e') = PartialEquiv.trans (Equiv.toPartialEquiv e) (Equiv.toPartialEquiv e')

@[simp]

theorem Equiv.transPartialEquiv_source {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : α ≃ β) (f' : PartialEquiv β γ) : (Equiv.transPartialEquiv e f').source = ⇑e ⁻¹' f'.source

@[simp]

theorem Equiv.transPartialEquiv_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : α ≃ β) (f' : PartialEquiv β γ) : ∀ (a : α), ↑(Equiv.transPartialEquiv e f') a = ↑f' (e a)

@[simp]

theorem Equiv.transPartialEquiv_symm_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : α ≃ β) (f' : PartialEquiv β γ) : ∀ (a : γ), ↑(PartialEquiv.symm (Equiv.transPartialEquiv e f')) a = e.symm (↑(PartialEquiv.symm f') a)

@[simp]

theorem Equiv.transPartialEquiv_target {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : α ≃ β) (f' : PartialEquiv β γ) : (Equiv.transPartialEquiv e f').target = f'.target

def Equiv.transPartialEquiv {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : α ≃ β) (f' : PartialEquiv β γ) : PartialEquiv α γ

Precompose a partial equivalence with an equivalence. We modify the source and target to have better definitional behavior.

Equations

• One or more equations did not get rendered due to their size.

theorem Equiv.transPartialEquiv_eq_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : α ≃ β) (f' : PartialEquiv β γ) : Equiv.transPartialEquiv e f' = PartialEquiv.trans (Equiv.toPartialEquiv e) f'

@[simp]

theorem Equiv.transPartialEquiv_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (e : α ≃ β) (f' : PartialEquiv β γ) (f'' : PartialEquiv γ δ) : PartialEquiv.trans (Equiv.transPartialEquiv e f') f'' = Equiv.transPartialEquiv e (PartialEquiv.trans f' f'')

@[simp]

theorem Equiv.trans_transPartialEquiv {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (e : α ≃ β) (e' : β ≃ γ) (f'' : PartialEquiv γ δ) : Equiv.transPartialEquiv (e.trans e') f'' = Equiv.transPartialEquiv e (Equiv.transPartialEquiv e' f'')

@[simp]

theorem PartialEquiv.transEquiv_symm_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (f' : β ≃ γ) : ∀ (a : γ), ↑(PartialEquiv.symm (PartialEquiv.transEquiv e f')) a = ↑(PartialEquiv.symm e) (f'.symm a)

@[simp]

theorem PartialEquiv.transEquiv_source {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (f' : β ≃ γ) : (PartialEquiv.transEquiv e f').source = e.source

@[simp]

theorem PartialEquiv.transEquiv_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (f' : β ≃ γ) : ∀ (a : α), ↑(PartialEquiv.transEquiv e f') a = f' (↑e a)

@[simp]

theorem PartialEquiv.transEquiv_target {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (f' : β ≃ γ) : (PartialEquiv.transEquiv e f').target = ⇑f'.symm ⁻¹' e.target

def PartialEquiv.transEquiv {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (f' : β ≃ γ) : PartialEquiv α γ

Postcompose a partial equivalence with an equivalence. We modify the source and target to have better definitional behavior.

Equations

• One or more equations did not get rendered due to their size.

theorem PartialEquiv.transEquiv_eq_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : PartialEquiv α β) (e' : β ≃ γ) : PartialEquiv.transEquiv e e' = PartialEquiv.trans e (Equiv.toPartialEquiv e')

@[simp]

theorem PartialEquiv.transEquiv_transEquiv {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (e : PartialEquiv α β) (f' : β ≃ γ) (f'' : γ ≃ δ) : PartialEquiv.transEquiv (PartialEquiv.transEquiv e f') f'' = PartialEquiv.transEquiv e (f'.trans f'')

@[simp]

theorem PartialEquiv.trans_transEquiv {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (e : PartialEquiv α β) (e' : PartialEquiv β γ) (f'' : γ ≃ δ) : PartialEquiv.transEquiv (PartialEquiv.trans e e') f'' = PartialEquiv.trans e (PartialEquiv.transEquiv e' f'')