Equivalence relations: partitions

This file comprises properties of equivalence relations viewed as partitions. There are two implementations of partitions here:

• A collection c : Set (Set α) of sets is a partition of α if ∅ ∉ c and each element a : α belongs to a unique set b ∈ c. This is expressed as IsPartition c
• An indexed partition is a map s : ι → α whose image is a partition. This is expressed as IndexedPartition s.

Of course both implementations are related to Quotient and Setoid.

Setoid.isPartition.partition and Finpartition.isPartition_parts furnish a link between Setoid.IsPartition and Finpartition.

TODO

Could the design of Finpartition inform the one of Setoid.IsPartition? Maybe bundling it and changing it from Set (Set α) to Set α where [Lattice α] [OrderBot α] would make it more usable.

Tags

setoid, equivalence, iseqv, relation, equivalence relation, partition, equivalence class

theorem Setoid.eq_of_mem_eqv_class {α : Type u_1} {c : Set (Set α)} (H : ∀ (a : α), ∃! b : Set α, b ∈ c ∧ a ∈ b) {x : α} {b b' : Set α} (hc : b ∈ c) (hb : x ∈ b) (hc' : b' ∈ c) (hb' : x ∈ b') : b = b'

If x ∈ α is in 2 elements of a set of sets partitioning α, those 2 sets are equal.

def Setoid.mkClasses {α : Type u_1} (c : Set (Set α)) (H : ∀ (a : α), ∃! b : Set α, b ∈ c ∧ a ∈ b) : Setoid α

Makes an equivalence relation from a set of sets partitioning α.

Equations

• Setoid.mkClasses c H = { r := fun (x y : α) => ∀ s ∈ c, x ∈ s → y ∈ s, iseqv := ⋯ }

def Setoid.classes {α : Type u_1} (r : Setoid α) : Set (Set α)

Makes the equivalence classes of an equivalence relation.

Equations

• r.classes = {s : Set α | ∃ (y : α), s = {x : α | r x y}}

theorem Setoid.mem_classes {α : Type u_1} (r : Setoid α) (y : α) : {x : α | r x y} ∈ r.classes

theorem Setoid.classes_ker_subset_fiber_set {α : Type u_1} {β : Type u_2} (f : α → β) : (ker f).classes ⊆ Set.range fun (y : β) => {x : α | f x = y}

theorem Setoid.finite_classes_ker {α : Type u_2} {β : Type u_3} [Finite β] (f : α → β) : (ker f).classes.Finite

theorem Setoid.card_classes_ker_le {α : Type u_2} {β : Type u_3} [Fintype β] (f : α → β) [Fintype ↑(ker f).classes] : Fintype.card ↑(ker f).classes ≤ Fintype.card β

theorem Setoid.eq_iff_classes_eq {α : Type u_1} {r₁ r₂ : Setoid α} : r₁ = r₂ ↔ ∀ (x : α), {y : α | r₁ x y} = {y : α | r₂ x y}

Two equivalence relations are equal iff all their equivalence classes are equal.

theorem Setoid.rel_iff_exists_classes {α : Type u_1} (r : Setoid α) {x y : α} : r x y ↔ ∃ c ∈ r.classes, x ∈ c ∧ y ∈ c

theorem Setoid.classes_inj {α : Type u_1} {r₁ r₂ : Setoid α} : r₁ = r₂ ↔ r₁.classes = r₂.classes

Two equivalence relations are equal iff their equivalence classes are equal.

theorem Setoid.empty_not_mem_classes {α : Type u_1} {r : Setoid α} : ∅ ∉ r.classes

The empty set is not an equivalence class.

theorem Setoid.classes_eqv_classes {α : Type u_1} {r : Setoid α} (a : α) : ∃! b : Set α, b ∈ r.classes ∧ a ∈ b

Equivalence classes partition the type.

theorem Setoid.eq_of_mem_classes {α : Type u_1} {r : Setoid α} {x : α} {b : Set α} (hc : b ∈ r.classes) (hb : x ∈ b) {b' : Set α} (hc' : b' ∈ r.classes) (hb' : x ∈ b') : b = b'

If x ∈ α is in 2 equivalence classes, the equivalence classes are equal.

theorem Setoid.eq_eqv_class_of_mem {α : Type u_1} {c : Set (Set α)} (H : ∀ (a : α), ∃! b : Set α, b ∈ c ∧ a ∈ b) {s : Set α} {y : α} (hs : s ∈ c) (hy : y ∈ s) : s = {x : α | (mkClasses c H) x y}

The elements of a set of sets partitioning α are the equivalence classes of the equivalence relation defined by the set of sets.

theorem Setoid.eqv_class_mem {α : Type u_1} {c : Set (Set α)} (H : ∀ (a : α), ∃! b : Set α, b ∈ c ∧ a ∈ b) {y : α} : {x : α | (mkClasses c H) x y} ∈ c

The equivalence classes of the equivalence relation defined by a set of sets partitioning α are elements of the set of sets.

theorem Setoid.eqv_class_mem' {α : Type u_1} {c : Set (Set α)} (H : ∀ (a : α), ∃! b : Set α, b ∈ c ∧ a ∈ b) {x : α} : {y : α | (mkClasses c H) x y} ∈ c

theorem Setoid.eqv_classes_disjoint {α : Type u_1} {c : Set (Set α)} (H : ∀ (a : α), ∃! b : Set α, b ∈ c ∧ a ∈ b) : c.PairwiseDisjoint id

Distinct elements of a set of sets partitioning α are disjoint.

theorem Setoid.eqv_classes_of_disjoint_union {α : Type u_1} {c : Set (Set α)} (hu : ⋃₀ c = Set.univ) (H : c.PairwiseDisjoint id) (a : α) : ∃! b : Set α, b ∈ c ∧ a ∈ b

A set of disjoint sets covering α partition α (classical).

def Setoid.setoidOfDisjointUnion {α : Type u_1} {c : Set (Set α)} (hu : ⋃₀ c = Set.univ) (H : c.PairwiseDisjoint id) : Setoid α

Makes an equivalence relation from a set of disjoints sets covering α.

Equations

• Setoid.setoidOfDisjointUnion hu H = Setoid.mkClasses c ⋯

theorem Setoid.mkClasses_classes {α : Type u_1} (r : Setoid α) : mkClasses r.classes ⋯ = r

The equivalence relation made from the equivalence classes of an equivalence relation r equals r.

@[simp]

theorem Setoid.sUnion_classes {α : Type u_1} (r : Setoid α) : ⋃₀ r.classes = Set.univ

noncomputable def Setoid.quotientEquivClasses {α : Type u_1} (r : Setoid α) : Quotient r ≃ ↑r.classes

The equivalence between the quotient by an equivalence relation and its type of equivalence classes.

Equations

• r.quotientEquivClasses = Equiv.ofBijective (Quot.lift (fun (a : α) => ⟨{x : α | r x a}, ⋯⟩) ⋯) ⋯

@[simp]

theorem Setoid.quotientEquivClasses_mk_eq {α : Type u_1} (r : Setoid α) (a : α) : ↑(r.quotientEquivClasses ⟦a⟧) = {x : α | r x a}

def Setoid.IsPartition {α : Type u_1} (c : Set (Set α)) : Prop

A collection c : Set (Set α) of sets is a partition of α into pairwise disjoint sets if ∅ ∉ c and each element a : α belongs to a unique set b ∈ c.

Equations

• Setoid.IsPartition c = (∅ ∉ c ∧ ∀ (a : α), ∃! b : Set α, b ∈ c ∧ a ∈ b)

theorem Setoid.nonempty_of_mem_partition {α : Type u_1} {c : Set (Set α)} (hc : IsPartition c) {s : Set α} (h : s ∈ c) : s.Nonempty

A partition of α does not contain the empty set.

theorem Setoid.isPartition_classes {α : Type u_1} (r : Setoid α) : IsPartition r.classes

theorem Setoid.IsPartition.pairwiseDisjoint {α : Type u_1} {c : Set (Set α)} (hc : IsPartition c) : c.PairwiseDisjoint id

theorem Set.PairwiseDisjoint.isPartition_of_exists_of_ne_empty {α : Type u_2} {s : Set (Set α)} (h₁ : s.PairwiseDisjoint id) (h₂ : ∀ (a : α), ∃ x ∈ s, a ∈ x) (h₃ : ∅ ∉ s) : Setoid.IsPartition s

theorem Setoid.IsPartition.sUnion_eq_univ {α : Type u_1} {c : Set (Set α)} (hc : IsPartition c) : ⋃₀ c = Set.univ

theorem Setoid.exists_of_mem_partition {α : Type u_1} {c : Set (Set α)} (hc : IsPartition c) {s : Set α} (hs : s ∈ c) : ∃ (y : α), s = {x : α | (mkClasses c ⋯) x y}

All elements of a partition of α are the equivalence class of some y ∈ α.

theorem Setoid.classes_mkClasses {α : Type u_1} (c : Set (Set α)) (hc : IsPartition c) : (mkClasses c ⋯).classes = c

The equivalence classes of the equivalence relation defined by a partition of α equal the original partition.

instance Setoid.Partition.le {α : Type u_1} : LE (Subtype IsPartition)

Defining ≤ on partitions as the ≤ defined on their induced equivalence relations.

Equations

• Setoid.Partition.le = { le := fun (x y : Subtype Setoid.IsPartition) => Setoid.mkClasses ↑x ⋯ ≤ Setoid.mkClasses ↑y ⋯ }

instance Setoid.Partition.partialOrder {α : Type u_1} : PartialOrder (Subtype IsPartition)

Defining a partial order on partitions as the partial order on their induced equivalence relations.

Equations

• Setoid.Partition.partialOrder = PartialOrder.mk ⋯

def Setoid.Partition.orderIso (α : Type u_1) : Setoid α ≃o { C : Set (Set α) // IsPartition C }

The order-preserving bijection between equivalence relations on a type α, and partitions of α into subsets.

Equations

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

instance Setoid.Partition.completeLattice {α : Type u_1} : CompleteLattice (Subtype IsPartition)

A complete lattice instance for partitions; there is more infrastructure for the equivalent complete lattice on equivalence relations.

Equations

• Setoid.Partition.completeLattice = (Setoid.Partition.orderIso α).toGaloisInsertion.liftCompleteLattice

def Setoid.IsPartition.finpartition {α : Type u_1} {c : Finset (Set α)} (hc : IsPartition ↑c) : Finpartition Set.univ

A finite setoid partition furnishes a finpartition

Equations

• hc.finpartition = { parts := c, supIndep := ⋯, sup_parts := ⋯, not_bot_mem := ⋯ }

@[simp]

theorem Setoid.IsPartition.finpartition_parts {α : Type u_1} {c : Finset (Set α)} (hc : IsPartition ↑c) : hc.finpartition.parts = c

theorem Finpartition.isPartition_parts {α : Type u_1} (f : Finpartition Set.univ) : Setoid.IsPartition ↑f.parts

A finpartition gives rise to a setoid partition

structure IndexedPartition {ι : Type u_1} {α : Type u_2} (s : ι → Set α) : Type (max u_1 u_2)

Constructive information associated with a partition of a type α indexed by another type ι, s : ι → Set α.

IndexedPartition.index sends an element to its index, while IndexedPartition.some sends an index to an element of the corresponding set.

This type is primarily useful for definitional control of s - if this is not needed, then Setoid.ker index by itself may be sufficient.

• eq_of_mem {x : α} {i j : ι} : x ∈ s i → x ∈ s j → i = j

  two indexes are equal if they are equal in membership

• some : ι → α

  sends an index to an element of the corresponding set

• some_mem (i : ι) : self.some i ∈ s i

  membership invariance for some

• index : α → ι

  index for type α

• mem_index (x : α) : x ∈ s (self.index x)

  membership invariance for index

noncomputable def IndexedPartition.mk' {ι : Type u_1} {α : Type u_2} (s : ι → Set α) (dis : Pairwise (Function.onFun Disjoint s)) (nonempty : ∀ (i : ι), (s i).Nonempty) (ex : ∀ (x : α), ∃ (i : ι), x ∈ s i) : IndexedPartition s

The non-constructive constructor for IndexedPartition.

Equations

• IndexedPartition.mk' s dis nonempty ex = { eq_of_mem := ⋯, some := fun (i : ι) => ⋯.some, some_mem := ⋯, index := fun (x : α) => ⋯.choose, mem_index := ⋯ }

instance IndexedPartition.instInhabitedUnivOfUnique {ι : Type u_1} {α : Type u_2} [Unique ι] [Inhabited α] : Inhabited (IndexedPartition fun (_i : ι) => Set.univ)

On a unique index set there is the obvious trivial partition

Equations

• IndexedPartition.instInhabitedUnivOfUnique = { default := { eq_of_mem := ⋯, some := default, some_mem := ⋯, index := default, mem_index := ⋯ } }

theorem IndexedPartition.exists_mem {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) (x : α) : ∃ (i : ι), x ∈ s i

theorem IndexedPartition.iUnion {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) : ⋃ (i : ι), s i = Set.univ

theorem IndexedPartition.disjoint {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) : Pairwise (Function.onFun Disjoint s)

theorem IndexedPartition.mem_iff_index_eq {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) {x : α} {i : ι} : x ∈ s i ↔ hs.index x = i

theorem IndexedPartition.eq {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) (i : ι) : s i = {x : α | hs.index x = i}

@[reducible, inline]

abbrev IndexedPartition.setoid {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) : Setoid α

The equivalence relation associated to an indexed partition. Two elements are equivalent if they belong to the same set of the partition.

Equations

• hs.setoid = Setoid.ker hs.index

@[simp]

theorem IndexedPartition.index_some {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) (i : ι) : hs.index (hs.some i) = i

theorem IndexedPartition.some_index {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) (x : α) : hs.setoid (hs.some (hs.index x)) x

def IndexedPartition.Quotient {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) : Type u_2

The quotient associated to an indexed partition.

Equations

• hs.Quotient = Quotient hs.setoid

def IndexedPartition.proj {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) : α → hs.Quotient

The projection onto the quotient associated to an indexed partition.

Equations

• hs.proj = Quotient.mk''

instance IndexedPartition.instInhabitedQuotient {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) [Inhabited α] : Inhabited hs.Quotient

Equations

• hs.instInhabitedQuotient = { default := hs.proj default }

theorem IndexedPartition.proj_eq_iff {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) {x y : α} : hs.proj x = hs.proj y ↔ hs.index x = hs.index y

@[simp]

theorem IndexedPartition.proj_some_index {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) (x : α) : hs.proj (hs.some (hs.index x)) = hs.proj x

def IndexedPartition.equivQuotient {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) : ι ≃ hs.Quotient

The obvious equivalence between the quotient associated to an indexed partition and the indexing type.

Equations

• hs.equivQuotient = (Setoid.quotientKerEquivOfRightInverse hs.index hs.some ⋯).symm

@[simp]

theorem IndexedPartition.equivQuotient_index_apply {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) (x : α) : hs.equivQuotient (hs.index x) = hs.proj x

@[simp]

theorem IndexedPartition.equivQuotient_symm_proj_apply {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) (x : α) : hs.equivQuotient.symm (hs.proj x) = hs.index x

theorem IndexedPartition.equivQuotient_index {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) : ⇑hs.equivQuotient ∘ hs.index = hs.proj

def IndexedPartition.out {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) : hs.Quotient ↪ α

A map choosing a representative for each element of the quotient associated to an indexed partition. This is a computable version of Quotient.out using IndexedPartition.some.

Equations

• hs.out = hs.equivQuotient.symm.toEmbedding.trans { toFun := hs.some, inj' := ⋯ }

@[simp]

theorem IndexedPartition.out_proj {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) (x : α) : hs.out (hs.proj x) = hs.some (hs.index x)

This lemma is analogous to Quotient.mk_out'.

theorem IndexedPartition.index_out {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) (x : hs.Quotient) : hs.index (Quotient.out x) = hs.index (hs.out x)

The indices of Quotient.out and IndexedPartition.out are equal.

@[deprecated IndexedPartition.index_out (since := "2024-10-19")]

theorem IndexedPartition.index_out' {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) (x : hs.Quotient) : hs.index (Quotient.out x) = hs.index (hs.out x)

Alias of IndexedPartition.index_out.

---

The indices of Quotient.out and IndexedPartition.out are equal.

@[simp]

theorem IndexedPartition.proj_out {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) (x : hs.Quotient) : hs.proj (hs.out x) = x

This lemma is analogous to Quotient.out_eq'.

theorem IndexedPartition.class_of {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) {x : α} : setOf (hs.setoid x) = s (hs.index x)

theorem IndexedPartition.proj_fiber {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) (x : hs.Quotient) : hs.proj ⁻¹' {x} = s (hs.equivQuotient.symm x)

def IndexedPartition.piecewise {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) {β : Type u_3} (f : ι → α → β) : α → β

Combine functions with disjoint domains into a new function. You can use the regular expression def.*piecewise to search for other ways to define piecewise functions in mathlib4.

Equations

• hs.piecewise f x = f (hs.index x) x

theorem IndexedPartition.piecewise_apply {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) {β : Type u_3} {f : ι → α → β} (x : α) : hs.piecewise f x = f (hs.index x) x

theorem IndexedPartition.piecewise_inj {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) {β : Type u_3} {f : ι → α → β} (h_injOn : ∀ (i : ι), Set.InjOn (f i) (s i)) (h_disjoint : Set.univ.PairwiseDisjoint fun (i : ι) => f i '' s i) : Function.Injective (hs.piecewise f)

A family of injective functions with pairwise disjoint domains and pairwise disjoint ranges can be glued together to form an injective function.

theorem IndexedPartition.piecewise_bij {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) {β : Type u_3} {f : ι → α → β} {t : ι → Set β} (ht : IndexedPartition t) (hf : ∀ (i : ι), Set.BijOn (f i) (s i) (t i)) : Function.Bijective (hs.piecewise f)

A family of bijective functions with pairwise disjoint domains and pairwise disjoint ranges can be glued together to form a bijective function.